Finite Group Databasebeta

← #29686418

Subgroups — #29686418

Order 15616.

1008 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839966C211yesmaximal, characteristic
2C2×#16839860C211yesmaximal, characteristic
3C2×C4×#11758190C211yesmaximal, characteristic
4#21448535C211yesmaximal, characteristic, complemented
5#21448179C211yesmaximal, characteristic
6#21446740C211yesmaximal, characteristic, complemented
7#21446586C211yesmaximal, characteristic, complemented
8#21446458C211yesmaximal, characteristic, complemented
9#21446455C211yesmaximal, characteristic, complemented
10#21446154C211yesmaximal, characteristic
11#21446150C211yesmaximal, characteristic
12#21445828C211yesmaximal, characteristic
13#21445677C211yesmaximal, characteristic, complemented
14#21445652C211yesmaximal, characteristic, complemented
15C4×#103×C61C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519775C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×#14519739C2211yescharacteristic
22C2×#14519739C2211yescharacteristic
23#16839966C418yescomplemented
24#16839860—24no—
25C4×#11758190—24no—
26C2×#14519710C2211yescharacteristic
27C2×#14519709C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519706C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519699C2211yescharacteristic
35C2×#14519697C2211yescharacteristic
36C2×#14519688C2211yescharacteristic
37#16839669C2211yescharacteristic
38#16839667C2211yescharacteristic
39#16839646—21no—
40#16839646—21no—
41#16839643—21no—
42#16839643—21no—
43#16839627C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46C4×#11758198—21no—
47C4×#11758198—21no—
48#16839609—21no—
49#16839608C2211yescharacteristic
50#95×D61—21no—
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839516C2211yescharacteristic
54#16839516C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×C4×C8×C61C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
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
77C4×Dic122—22no—
78C4×Dic122—22no—
79C4×D244—22no—
80C4×D244D414yes—
81#14519739—22no—
82#14519739—22no—
83#14519739—22no—
84#14519739C2×C414yes—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61C2311yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98#14519710—22no—
99#14519709—22no—
100#14519706—22no—
101#14519706C2×C414yescomplemented
102#14519702—22no—
103#14519701—22no—
104#14519700—22no—
105#14519700C2×C414yescomplemented
106#14519699—22no—
107#14519697—22no—
108#14519688—22no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684D411yescharacteristic
118#14519684D411yescharacteristic
119#14519684D411yescharacteristic
120#14519684D411yescharacteristic
121C22×Q8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C4×Q8×C61—22no—
125C4×Q8×C61—22no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C42×C61—21no—
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133#103×C61—22no—
134#103×C61C2×C414yescomplemented
135C4×C8×C61—22no—
136#95×C61—21no—
137C23×D61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244C22×C412yes—
162C2×D244C2×D412yes—
163C2×D244#4112yes—
164C2×C4×D61—24no—
165C2×C4×D61—24no—
166C2×C4×D61—24no—
167C2×C4×D61—81no—
168C2×C4×D61—81no—
169C2×C4×D61—81no—
170C2×C4×D61—81no—
171C2×C4×D61—81no—
172C2×C4×D61—81no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122C22×C412yes—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61C22×C412yes—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—42no—
190C4⋊C4×C61C22×C412yes—
191C4⋊C4×C61#4112yes—
192C42×C61—42no—
193C42×C61—81no—
194C42×C61C2×D412yes—
195C42×C61#4112yes—
196#11758201—28no—
197#11758198—24no—
198#11758198—24no—
199#11758198—24no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61C2×D412yes—
204#11758196—24no—
205#11758196—24no—
206#11758196—42no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—42no—
210C4×Dic61—81no—
211C4×Dic61—81no—
212C4×Dic61C22×C412yes—
213C4×Dic61#4112yes—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418C22×C412yes—
220#11758190—28no—
221C23×C61C22×D411yescharacteristic
222C22×D61—22no—
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229Q8×C61—22no—
230Q8×C61—22no—
231Q8×C61—41no—
232Q8×C61—41no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61#12112yes—
250C2×C4×C61#12112yes—
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61C4×D412yes—
277C2×Dic61C4×D412yes—
278D244—22no—
279D244—22no—
280D244—22no—
281D244C4×D414yes—
282C4×D61—84no—
283C4×D61—84no—
284C4×D61—84no—
285Dic122—22no—
286Dic122—22no—
287Dic122—41no—
288Dic122—41no—
289C8×C61—22no—
290C8×C61—41no—
291#92418—22no—
292#92418—41no—
293#21514—611nomaximal, Sylow
294C22×C61C2×C4×D411yescharacteristic
295C22×C61C2×#12111yescharacteristic
296C22×C61#52511yescharacteristic
297C22×C61#57711yescharacteristic
298C22×C61#57711yescharacteristic
299C22×C61#57711yescharacteristic
300C22×C61#57811yescharacteristic
301C2×D61—24no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61—42no—
308C4×C61—22no—
309C4×C61—22no—
310C4×C61—42no—
311C4×C61—42no—
312C4×C61—42no—
313C4×C61—81no—
314C4×C61C2×C4×D411yescharacteristic
315C4×C61C4×#4111yescharacteristic
316C4×C61#52911yescharacteristic
317C4×C61#53211yescharacteristic
318Dic61—22no—
319Dic61—22no—
320Dic61—41no—
321Dic61—41no—
322Dic61—42no—
323Dic61—42no—
324Dic61—81no—
325C2×#461—611no—
326C2×#440—611no—
327C2×C4×SD16—611no—
328#2327—611no—
329#2275—611no—
330#1981—611no—
331#1949—611no—
332#1928—611no—
333#1927—611no—
334#1877—611no—
335#1877—611no—
336#1804—611no—
337#1768—611no—
338#1763—611no—
339C4×#103D6111yescharacteristic, complemented
340C2×C61C4×#12111yescharacteristic
341C2×C61#294711yescharacteristic
342C2×C61#294911yescharacteristic
343C2×C61#304211yescharacteristic
344C2×C61#304611yescharacteristic
345C2×C61#312011yescharacteristic
346C2×C61#312311yescharacteristic
347D61—44no—
348C22×SD16—611no—
349C22×SD16—611no—
350C2×C4⋊Q8—611no—
351C2×C4×Q8—611no—
352C2×C4×Q8C2×D6111yescharacteristic
353C2×C4×D4—611no—
354C2×C4×D4—611no—
355C2×C4×D4—611no—
356#461—618no—
357#440—1224no—
358C4×SD16—1224no—
359C2×#107—611no—
360C2×#106—611no—
361C2×C4⋊C8—611no—
362C2×C4⋊C8—611no—
363C2×#103—611no—
364C2×#103—611no—
365C2×#103—611no—
366C2×#103—611no—
367C2×#103C2×D6111yescharacteristic
368C2×#103C2×D6111yescharacteristic
369C2×#102—611no—
370C2×#102—611no—
371C2×#97—611no—
372C2×C4×C8C2×D6111yescharacteristic
373#391—611no—
374#391—611no—
375#387—1221no—
376#387—1221no—
377#387—1221no—
378#387—1221no—
379#382—611no—
380C4×C4⋊C4—611no—
381C4×C4⋊C4—611no—
382C4×C4⋊C4C2×D6111yescharacteristic
383C4×C22⋊C4—1221no—
384C4×C22⋊C4—1221no—
385C4×C22⋊C4—1221no—
386#376—611no—
387C2×#95—1221no—
388#340—611no—
389#340—611no—
390#340C2×D6111yescharacteristic
391#340C2×D6111yescharacteristic
392#336—611no—
393#336—611no—
394C61#2151411yescharacteristic, complemented, Sylow
395C22×Q8—611no—
396C22×Q8C22×D6111yescharacteristic
397C22×D4—611no—
398C23×C4—1221no—
399C23×C4—1221no—
400C23×C4—1221no—
401C2×SD16—614no—
402C2×SD16—614no—
403C2×SD16—1222no—
404C2×SD16—1222no—
405C2×SD16—1222no—
406C2×SD16—1222no—
407C22×C8—611no—
408C22×C8—611no—
409C22×C8C22×D6111yescharacteristic
410C22×C8C22×D6111yescharacteristic
411C4⋊Q8—614no—
412C4⋊Q8—1222no—
413C4×Q8—22no—
414C4×Q8—22no—
415C4×Q8—1222no—
416C4×Q8—1222no—
417C4×D4—614no—
418C4×D4—614no—
419C4×D4—1222no—
420C4×D4—1222no—
421C4×D4—1222no—
422C4×D4—1222no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C42—21no—
436C2×C42—21no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—1221no—
440C2×C42—1221no—
441C2×C42—1221no—
442C2×C42C22×D6111yescharacteristic
443#107—1222no—
444#106—1222no—
445C4⋊C8—614no—
446C4⋊C8—1222no—
447#103—22no—
448#103—614no—
449#103—1222no—
450#103—1222no—
451#103—1222no—
452#103C4×D6114yescomplemented
453#102—1222no—
454#102—1222no—
455#97—1222no—
456C4×C8—22no—
457#95—21no—
458#95—611no—
459#95—611no—
460#95—611no—
461#95—611no—
462#95—1221no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470C24—1221no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—612no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8C2×C4×D6112yes—
483C2×D4—612no—
484C2×D4—612no—
485C2×D4—612no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4—1221no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—1224no—
498C22×C4—1224no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1224no—
502C22×C4—1221no—
503C22×C4—4881no—
504C22×C4—4881no—
505C22×C4—4881no—
506C22×C4—4881no—
507C22×C4—4881no—
508C22×C4—4881no—
509C22×C4D4×D6111yescharacteristic
510C22×C4D4×D6111yescharacteristic
511C22×C4C23×D6111yescharacteristic
512SD16—1228no—
513SD16—1228no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—612no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8C2×C4×D6112yes—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—42no—
531C4⋊C4—612no—
532C4⋊C4—1224no—
533C4⋊C4—1224no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—2442no—
538C4⋊C4C2×C4×D6112yes—
539C4⋊C4#1175822212yes—
540C22⋊C4—1224no—
541C22⋊C4—1224no—
542C22⋊C4—1224no—
543C42—42no—
544C42—81no—
545C42—612no—
546C42—612no—
547C42—1221no—
548C42—1221no—
549C42—4881no—
550C42—4881no—
551C42—2442no—
552C42#1175822212yes—
553C42D4×D6112yes—
554C23—1222no—
555C23—1222no—
556C23—1222no—
557C23—2441no—
558C23—2441no—
559C23—2441no—
560C23—2441no—
561C23C2×D4×D6111yescharacteristic
562Q8—22no—
563Q8—22no—
564Q8—41no—
565Q8—41no—
566Q8—1222no—
567Q8—1222no—
568Q8—2441no—
569Q8—2441no—
570D4—614no—
571D4—1222no—
572D4—1222no—
573D4—1222no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—1221no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—4884no—
605C2×C4—4884no—
606C2×C4—4884no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4#1451975012yes—
615C2×C4#1451975012yes—
616C2×C4C22×C4×D6111yescharacteristic
617C2×C4C2×D4×D6111yescharacteristic
618C2×C4#41×D6111yescharacteristic
619C2×C4#41×D6111yescharacteristic
620C2×C4C2×#1175822211yescharacteristic
621C2×C4C2×#1175822211yescharacteristic
622C8—22no—
623C8—41no—
624C8—1222no—
625C8—2441no—
626C22—1224no—
627C22—2442no—
628C22—2442no—
629C22—2442no—
630C22—2442no—
631C22—2442no—
632C22—2442no—
633C22C4×D4×D6111yescharacteristic
634C22C2×#1451975011yescharacteristic
635C22#1684027111yescharacteristic
636C22#1684056811yescharacteristic
637C22#1684056911yescharacteristic
638C22#1684056911yescharacteristic
639C22#1684057511yescharacteristic
640C4—22no—
641C4—22no—
642C4—42no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—1222no—
647C4—1222no—
648C4—4881no—
649C4—2442no—
650C4—2441no—
651C4—2441no—
652C4—2442no—
653C4C4×D4×D6111yescharacteristic
654C4C4×#1175822211yescharacteristic
655C4#1684029011yescharacteristic
656C4#1684030911yescharacteristic
657C2—2444no—
658C2C4×#1451975011yescharacteristic
659C2#2145465711yescharacteristic
660C2#2145467311yescharacteristic
661C2#2145533411yescharacteristic
662C2#2145537411yescharacteristic
663C2#2145603911yescharacteristic
664C2#2145606811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#103×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenterC4×#103D61
hypocenterC61#21514
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
4. #21448535C2semidirect
6. #21446740C2semidirect
7. #21446586C2semidirect
8. #21446458C2semidirect
9. #21446455C2semidirect
13. #21445677C2semidirect
14. #21445652C2semidirect
15. C4×#103×C61C2semidirect
23. #16839966C4semidirect
76. #14519775C2×C4semidirect
101. #14519706C2×C4semidirect
105. #14519700C2×C4semidirect
134. #103×C61C2×C4semidirect
339. C4×#103D61semidirect
394. C61#21514semidirect
452. #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×#168398602
C2×C4×#117581902
#214485352
#214481792
#214467402
#214465862
#214464582
#214464552
#214461542
#214461502
#214458282
#214456772
#214456522
C4×#103×C612
#2151461