Finite Group Databasebeta

← #29686199

Subgroups — #29686199

Order 15616.

1104 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839966C211yesmaximal, characteristic
2C2×#16839849C211yesmaximal, characteristic
3C2×#16839847C211yesmaximal, characteristic
4#21448388C211yesmaximal, characteristic, complemented
5C4×#14519739C211yesmaximal, characteristic
6#21446740C211yesmaximal, characteristic, complemented
7#21446579C211yesmaximal, characteristic, complemented
8#21446453C211yesmaximal, characteristic, complemented
9#21446156C211yesmaximal, characteristic
10#21446152C211yesmaximal, characteristic
11#21445818C211yesmaximal, characteristic
12#21445667C211yesmaximal, characteristic, complemented
13#103×Dic61C211yesmaximal, characteristic, complemented
14#21445638C211yesmaximal, characteristic, complemented
15C61×#1927C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519775C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23C2×#14519739C2211yescharacteristic
24C2×C42×D61—21no—
25#16839966C418yescomplemented
26#16839849—24no—
27#16839847—24no—
28C2×#14519708C2211yescharacteristic
29C2×#14519706C2211yescharacteristic
30C2×#14519706C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519701C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519700C2211yescharacteristic
35C2×#14519699C2211yescharacteristic
36C2×#14519697C2211yescharacteristic
37C2×C8×Dic61C2211yescharacteristic
38C2×C8×Dic61C2211yescharacteristic
39#16839668C2211yescharacteristic
40#16839659—21no—
41#16839659—21no—
42#16839631C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45#16839613—21no—
46#16839613—21no—
47C4×#11758198—21no—
48#16839609—21no—
49#16839609—21no—
50C42×Dic61C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839517C2211yescharacteristic
55C2×#106×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C2×#11758190—22no—
68C2×#11758190—22no—
69C2×#11758190C2×C414yes—
70C2×#11758201—22no—
71C2×#11758201—22no—
72C2×#11758201C2×C414yes—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75#14519775—22no—
76#14519775C2×C414yescomplemented
77Q8×Dic61—22no—
78Q8×Dic61—22no—
79#14519742—22no—
80#14519742—22no—
81C4×D244—22no—
82C4×D244—22no—
83#14519739—22no—
84#14519739—22no—
85#14519739C2×C414yes—
86#14519739D414yes—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61D411yescharacteristic
103C2×C4×Dic61D411yescharacteristic
104C2×C4×Dic61D411yescharacteristic
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61C2311yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108C42×D61—44no—
109#14519708—22no—
110#14519706—22no—
111#14519706C2×C414yescomplemented
112#14519702—22no—
113#14519701—22no—
114#14519700—22no—
115#14519700C2×C414yescomplemented
116#14519699—22no—
117#14519697—22no—
118C8×Dic61—22no—
119C8×Dic61—22no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123#14519684—21no—
124C22×Q8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C42×C61—21no—
131C2×C42×C61C2311yescharacteristic
132#106×C61—22no—
133#103×C61—22no—
134#103×C61C2×C414yescomplemented
135#95×C61—21no—
136#95×C61—21no—
137#95×C61—21no—
138C23×D61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61C22×C412yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244C22×C412yes—
163C2×D244C2×D412yes—
164C2×D244#4112yes—
165C2×C4×D61—24no—
166C2×C4×D61—24no—
167C2×C4×D61—24no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61—42no—
172C2×C4×D61—42no—
173C2×C4×D61—42no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122C22×C412yes—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61C22×C412yes—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61C22×C412yes—
191C4⋊C4×C61C2×D412yes—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—42no—
195#11758201—28no—
196#11758198—24no—
197#11758198—24no—
198#11758198—24no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—42no—
203C4⋊Dic61#4112yes—
204#11758196—24no—
205#11758196—24no—
206#11758196—42no—
207#11758196—42no—
208#11758196—42no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—41no—
212C4×Dic61—42no—
213C4×Dic61—42no—
214C4×Dic61—42no—
215C4×Dic61—42no—
216C4×Dic61SD1614yescomplemented
217C4×Dic61C22×C412yes—
218C4×Dic61C2×D412yes—
219C4×Dic61#4112yes—
220C4×Dic61#4112yes—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418C22×C412yes—
227#11758190—28no—
228C23×C61C22×D411yescharacteristic
229C22×D61—22no—
230C22×D61—22no—
231C22×D61—22no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—41no—
239Q8×C61—41no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61C2×SD1612yes—
258C2×C4×C61C23×C411yescharacteristic
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61C4×D412yes—
278C2×Dic61C4×D412yes—
279C2×Dic61#12112yes—
280C2×Dic61#12112yes—
281C2×Dic61C2×SD1612yes—
282C2×Dic61C2×SD1612yes—
283C2×Dic61#13512yes—
284C2×Dic61#13712yes—
285D244—22no—
286D244—22no—
287D244—22no—
288D244C4×D414yes—
289C4×D61—48no—
290C4×D61—48no—
291C4×D61—48no—
292Dic122—22no—
293Dic122—22no—
294Dic122—41no—
295Dic122—41no—
296C8×C61—22no—
297C8×C61—41no—
298#92418—22no—
299#92418—41no—
300C4×#461—611nomaximal, Sylow
301C22×C61C2×C4×D411yescharacteristic
302C22×C61C2×#12111yescharacteristic
303C22×C61#52511yescharacteristic
304C22×C61C22×SD1611yescharacteristic
305C22×C61C2×#13511yescharacteristic
306C22×C61C2×#13711yescharacteristic
307C22×C61#57511yescharacteristic
308C2×D61—24no—
309C2×D61—42no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—42no—
314C2×D61—42no—
315C4×C61—22no—
316C4×C61—41no—
317C4×C61—42no—
318C4×C61—42no—
319C4×C61—42no—
320C4×C61—81no—
321C4×C61C2×C4×D411yescharacteristic
322C4×C61C4×#4111yescharacteristic
323C4×C61#52911yescharacteristic
324C4×C61#53211yescharacteristic
325Dic61—22no—
326Dic61—22no—
327Dic61—42no—
328Dic61—42no—
329Dic61—81no—
330Dic61C4×SD1614yescomplemented
331Dic61#46114yescomplemented
332C2×#461—611no—
333C2×C4×SD16—611no—
334C2×C4×SD16—611no—
335C4×C4⋊Q8—611no—
336C42×D4—611no—
337#1981—611no—
338#1948—611no—
339#1927—611no—
340#1927D6111yescharacteristic, complemented
341#1877—611no—
342#1877—611no—
343#1802—611no—
344C4×C4⋊C8—611no—
345C4×#103—611no—
346C4×#103—611no—
347C2×C61C4×#12111yescharacteristic
348C2×C61C2×C4×SD1611yescharacteristic
349C2×C61C4×#13711yescharacteristic
350C2×C61C2×#46111yescharacteristic
351C2×C61#303511yescharacteristic
352C2×C61#310111yescharacteristic
353C2×C61#310711yescharacteristic
354D61—44no—
355C22×SD16—611no—
356C22×SD16—611no—
357C2×C4⋊Q8—611no—
358C2×C4×Q8—611no—
359C2×C4×Q8—611no—
360C2×C4×D4—611no—
361C2×C4×D4—611no—
362C2×C4×D4—611no—
363C22×C42—1221no—
364#461—618no—
365C4×SD16—1224no—
366C4×SD16—1224no—
367C2×#106—611no—
368C2×#106C2×D6111yescharacteristic
369C2×C4⋊C8—611no—
370C2×C4⋊C8—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103C2×D6111yescharacteristic
376C2×#103C2×D6111yescharacteristic
377C2×#102—611no—
378C2×#102—611no—
379C2×C4×C8—611no—
380C2×C4×C8—611no—
381#391—611no—
382#391C2×D6111yescharacteristic
383#389—1221no—
384#389—1221no—
385#384—611no—
386C4×C4⋊C4—611no—
387C4×C4⋊C4—611no—
388C4×C4⋊C4—1221no—
389C4×C4⋊C4—1221no—
390C4×C4⋊C4C2×D6111yescharacteristic
391C4×C22⋊C4—1221no—
392C4×C22⋊C4—1221no—
393C4×C22⋊C4—1221no—
394C43—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340C2×D6111yescharacteristic
399#336—611no—
400#336C2×D6111yescharacteristic
401C61C4×#46111yescharacteristic, complemented, Sylow
402C22×Q8—611no—
403C22×Q8C22×D6111yescharacteristic
404C22×D4—611no—
405C23×C4—1221no—
406C23×C4—1221no—
407C23×C4—1221no—
408C2×SD16—614no—
409C2×SD16—614no—
410C2×SD16—1222no—
411C2×SD16—1222no—
412C2×SD16—1222no—
413C2×SD16—1222no—
414C22×C8—611no—
415C22×C8—611no—
416C22×C8C22×D6111yescharacteristic
417C22×C8C22×D6111yescharacteristic
418C4⋊Q8—614no—
419C4⋊Q8—1222no—
420C4×Q8—1222no—
421C4×Q8—1222no—
422C4×Q8—1222no—
423C4×Q8—1222no—
424C4×D4—614no—
425C4×D4—614no—
426C4×D4—1222no—
427C4×D4—1222no—
428C4×D4—1222no—
429C4×D4—1222no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C4⋊C4C22×D6111yescharacteristic
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C42—21no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—611no—
451C2×C42—1221no—
452C2×C42—1221no—
453C2×C42—1221no—
454C2×C42—1221no—
455C2×C42—2444no—
456C2×C42C22×D6111yescharacteristic
457#106—22no—
458#106—1222no—
459C4⋊C8—614no—
460C4⋊C8—1222no—
461#103—22no—
462#103—614no—
463#103—1222no—
464#103—1222no—
465#103—1222no—
466#103C4×D6114yescomplemented
467#102—1222no—
468#102—1222no—
469C4×C8—1222no—
470C4×C8—1222no—
471#95—21no—
472#95—21no—
473#95—21no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478C24—1221no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—612no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8C2×C4×D6112yes—
491C2×D4—612no—
492C2×D4—612no—
493C2×D4—612no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—1224no—
506C22×C4—1224no—
507C22×C4—1224no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—2442no—
512C22×C4—2442no—
513C22×C4—2442no—
514C22×C4—2442no—
515C22×C4—2442no—
516C22×C4—2442no—
517C22×C4D4×D6111yescharacteristic
518C22×C4D4×D6111yescharacteristic
519C22×C4C23×D6111yescharacteristic
520SD16—1228no—
521SD16—1228no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—612no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C2×C8C2×C4×D6112yes—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—612no—
539C4⋊C4—1224no—
540C4⋊C4—1224no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—2442no—
545C4⋊C4—2442no—
546C4⋊C4—2442no—
547C4⋊C4—2442no—
548C4⋊C4C2×C4×D6112yes—
549C4⋊C4D4×D6112yes—
550C22⋊C4—1224no—
551C22⋊C4—1224no—
552C22⋊C4—1224no—
553C42—21no—
554C42—21no—
555C42—42no—
556C42—612no—
557C42—612no—
558C42—612no—
559C42—612no—
560C42—614no—
561C42—1222no—
562C42—1222no—
563C42—2441no—
564C42—2442no—
565C42—2442no—
566C42—2442no—
567C42—2442no—
568C23—1222no—
569C23—1222no—
570C23—1222no—
571C23—2441no—
572C23—2441no—
573C23—2441no—
574C23—2441no—
575C23C2×D4×D6111yescharacteristic
576Q8—22no—
577Q8—22no—
578Q8—41no—
579Q8—41no—
580Q8—1222no—
581Q8—1222no—
582Q8—2441no—
583Q8—2441no—
584D4—614no—
585D4—1222no—
586D4—1222no—
587D4—1222no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—612no—
606C2×C4—612no—
607C2×C4—612no—
608C2×C4—612no—
609C2×C4—612no—
610C2×C4—612no—
611C2×C4—612no—
612C2×C4—612no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—2448no—
620C2×C4—2448no—
621C2×C4—2448no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4SD16×D6112yes—
630C2×C4C22×C4×D6111yescharacteristic
631C2×C4C2×D4×D6111yescharacteristic
632C2×C4#41×D6111yescharacteristic
633C2×C4#41×D6111yescharacteristic
634C2×C4#41×D6111yescharacteristic
635C2×C4#41×D6111yescharacteristic
636C8—22no—
637C8—41no—
638C8—1222no—
639C8—2441no—
640C22—1224no—
641C22—2442no—
642C22—2442no—
643C22—2442no—
644C22—2442no—
645C22—2442no—
646C22—2442no—
647C22C4×D4×D6111yescharacteristic
648C22#121×D6111yescharacteristic
649C22#1684027011yescharacteristic
650C22C2×SD16×D6111yescharacteristic
651C22#135×D6111yescharacteristic
652C22#137×D6111yescharacteristic
653C22#1684056111yescharacteristic
654C4—22no—
655C4—41no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—81no—
660C4—614no—
661C4—614no—
662C4—1222no—
663C4—1222no—
664C4—4881no—
665C4—2442no—
666C4—2442no—
667C4C4×D4×D6111yescharacteristic
668C4#1684023011yescharacteristic
669C4#1684029511yescharacteristic
670C4#1684031111yescharacteristic
671C2—2444no—
672C2#2144833111yescharacteristic
673C2C4×SD16×D6111yescharacteristic
674C2#2145463511yescharacteristic
675C2#461×D6111yescharacteristic
676C2#2145527311yescharacteristic
677C2#2145582411yescharacteristic
678C2#2145588511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1927C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1927D61
hypocenterC61C4×#461
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1927

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #21448388C2semidirect
6. #21446740C2semidirect
7. #21446579C2semidirect
8. #21446453C2semidirect
12. #21445667C2semidirect
13. #103×Dic61C2semidirect
14. #21445638C2semidirect
15. C61×#1927C2semidirect
25. #16839966C4semidirect
76. #14519775C2×C4semidirect
111. #14519706C2×C4semidirect
115. #14519700C2×C4semidirect
134. #103×C61C2×C4semidirect
216. C4×Dic61SD16semidirect
330. Dic61C4×SD16semidirect
331. Dic61#461semidirect
340. #1927D61semidirect
401. C61C4×#461semidirect
466. #103C4×D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168399662
C2×#168398492
C2×#168398472
#214483882
C4×#145197392
#214467402
#214465792
#214464532
#214461562
#214461522
#214458182
#214456672
#103×Dic612
#214456382
C61×#19272
C4×#46161