Finite Group Databasebeta

← #29736929

Subgroups — #29736929

Order 15616.

1028 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840488C211yesmaximal, characteristic, complemented
2C2×#16840117C211yesmaximal, characteristic
3C2×#16839984C211yesmaximal, characteristic
4#21451302C211yesmaximal, characteristic
5#21446969C211yesmaximal, characteristic, complemented
6#21446964C211yesmaximal, characteristic
7#21446959C211yesmaximal, characteristic, complemented
8#21446954C211yesmaximal, characteristic
9#21446776C211yesmaximal, characteristic, complemented
10#21446755C211yesmaximal, characteristic, complemented
11#21446172C211yesmaximal, characteristic
12#21446172C211yesmaximal, characteristic
13#21446158C211yesmaximal, characteristic, complemented
14#21446158C211yesmaximal, characteristic, complemented
15C4×#103×C61C211yesmaximal, characteristic, complemented
16C22×#11758202C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18#16840488—24no—
19#16840488—24no—
20C2×#14519776—21no—
21C2×#14519776C2211yescharacteristic
22C2×#14519769—21no—
23C2×#14519769C2211yescharacteristic
24C2×#14519768—21no—
25C2×#14519768C2211yescharacteristic
26C2×#14519765C2211yescharacteristic
27C2×Q8×Dic61C2211yescharacteristic
28C2×Q8×Dic61C2211yescharacteristic
29C2×C4×Dic122—21no—
30C2×C4×Dic122C2211yescharacteristic
31#16840117—24no—
32#16839984—24no—
33C2×C4⋊#92418C2211yescharacteristic
34C2×#14519702C2211yescharacteristic
35C2×#14519702C2211yescharacteristic
36C2×#14519700C2211yescharacteristic
37C2×#14519699C2211yescharacteristic
38C2×#14519699C2211yescharacteristic
39C2×#14519697C2211yescharacteristic
40#16839689—21no—
41#16839675—21no—
42#16839667C2211yescharacteristic
43#16839667C2211yescharacteristic
44#16839663C2211yescharacteristic
45#16839663C2211yescharacteristic
46#16839653—21no—
47#16839653—21no—
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×C4×Q8×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×C4×C8×C61C2211yescharacteristic
60C4×C4⋊C4×C61C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#340C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122D411yescharacteristic
65C22×Dic122C2311yescharacteristic
66C22×D244C2311yescharacteristic
67C2×#11758202—22no—
68C2×#11758202—22no—
69C2×#11758202—22no—
70C2×#11758201—22no—
71C2×#11758201—22no—
72C2×#11758201—22no—
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75#14519776—28no—
76#14519776—22no—
77#14519776—22no—
78#14519769—28no—
79#14519769—22no—
80#14519769D414yes—
81#14519768—28no—
82#14519768—22no—
83#14519768—22no—
84#14519765—22no—
85#14519765—22no—
86Q8×Dic61—22no—
87Q8×Dic61—22no—
88Q8×Dic61—22no—
89Q8×Dic61—22no—
90C4×Dic122—28no—
91C4×Dic122—22no—
92C4×Dic122—22no—
93C2×C4⋊Dic61—21no—
94C2×C4⋊Dic61—21no—
95C2×C4⋊Dic61—21no—
96C2×C4⋊Dic61D411yescharacteristic
97C2×C4⋊Dic61C2311yescharacteristic
98C2×C4⋊Dic61C2311yescharacteristic
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196D411yescharacteristic
104C2×#11758196D411yescharacteristic
105C2×#11758198—21no—
106C2×#11758198—21no—
107C2×#11758198—21no—
108C2×#11758198—21no—
109C2×#11758198—21no—
110C2×#11758198—21no—
111C2×C4×Dic61—21no—
112C2×C4×Dic61—21no—
113C2×C4×Dic61C2311yescharacteristic
114C2×C4×Dic61C2311yescharacteristic
115C4⋊#92418—22no—
116#14519702—22no—
117#14519702—22no—
118#14519700—22no—
119#14519699—22no—
120#14519699—22no—
121#14519697—22no—
122#14519684—21no—
123#14519684—21no—
124#14519684—21no—
125#14519684—21no—
126C22×Q8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C22×C8×C61C2311yescharacteristic
129C4×Q8×C61—22no—
130C4×Q8×C61D414yes—
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C42×C61—21no—
136C2×C42×C61—21no—
137C2×C42×C61C2311yescharacteristic
138#103×C61—22no—
139#103×C61—22no—
140C4×C8×C61—22no—
141#95×C61—21no—
142C23×D61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61C2×D412yes—
147C2×Q8×C61#4112yes—
148C2×Q8×C61#4112yes—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244—21no—
169C2×Dic122—24no—
170C2×Dic122—22no—
171C2×Dic122—22no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—41no—
179C2×Dic122—42no—
180C2×Dic122—42no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C4⋊C4×C61—24no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61C2×D412yes—
194C42×C61—24no—
195C42×C61—81no—
196C42×C61C2×D412yes—
197C42×C61C2×D412yes—
198#11758202—44no—
199#11758201—44no—
200#11758198—42no—
201#11758198—42no—
202#11758198—42no—
203#11758198—42no—
204#11758198—81no—
205#11758198—81no—
206C4⋊Dic61—24no—
207C4⋊Dic61—24no—
208C4⋊Dic61—24no—
209C4⋊Dic61—22no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61C2×D412yes—
213C4⋊Dic61C2×D412yes—
214#11758196—24no—
215#11758196—24no—
216#11758196—24no—
217#11758196—24no—
218#11758196—22no—
219#11758196—22no—
220C4×Dic61—24no—
221C4×Dic61—24no—
222C4×Dic61—21no—
223C4×Dic61—21no—
224C4×Dic61—21no—
225C4×Dic61—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232C23×C61C22×D411yescharacteristic
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61—41no—
240Q8×C61—22no—
241Q8×C61—22no—
242Q8×C61—22no—
243Q8×C61#12414yes—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61#12012yes—
261C2×C4×C61#12012yes—
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C612−1+411yescharacteristic
267C2×C4×C612−1+411yescharacteristic
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—22no—
273C2×Dic61—21no—
274C2×Dic61—22no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61#13512yes—
289D244—41no—
290D244—41no—
291D244—41no—
292D244—41no—
293Dic122—44no—
294Dic122—48no—
295Dic122—41no—
296Dic122—41no—
297Dic122—41no—
298Dic122—41no—
299C8×C61—41no—
300C8×C61—41no—
301#92418—41no—
302#92418—41no—
303#25937—611nomaximal, Sylow
304C22×C61C2×#12011yescharacteristic
305C22×C61#54811yescharacteristic
306C22×C61#54811yescharacteristic
307C22×C61C2×#13511yescharacteristic
308C22×C61#57511yescharacteristic
309C22×C61#57711yescharacteristic
310C22×C61#57811yescharacteristic
311C2×D61—81no—
312C2×D61—81no—
313C2×D61—81no—
314C2×D61—81no—
315C2×D61—81no—
316C2×D61—81no—
317C2×D61—81no—
318C4×C61—24no—
319C4×C61—22no—
320C4×C61—22no—
321C4×C61—42no—
322C4×C61—81no—
323C4×C61—81no—
324C4×C61C2×#12411yescharacteristic
325C4×C61#53611yescharacteristic
326C4×C61#54811yescharacteristic
327C4×C61#54811yescharacteristic
328Dic61—22no—
329Dic61—41no—
330Dic61—42no—
331Dic61—41no—
332Dic61—41no—
333Dic61—42no—
334Dic61—81no—
335C2×#557—611no—
336C2×#488—611no—
337C2×#464—611no—
338#2592—611no—
339#2026—611no—
340#2025—611no—
341#2024—611no—
342#2023—611no—
343#1989—611no—
344#1984—611no—
345#1880—611no—
346#1880—611no—
347#1878—611no—
348#1878—611no—
349C4×#103D6111yescharacteristic, complemented
350C2×C61#243611yescharacteristic
351C2×C61#300211yescharacteristic
352C2×C61#301311yescharacteristic
353C2×C61#336411yescharacteristic
354C2×C61#336511yescharacteristic
355C2×C61#337111yescharacteristic
356C2×C61#337711yescharacteristic
357D61—161no—
358C22×Q16—611no—
359C22×SD16—611no—
360#557—1224no—
361#557—1224no—
362C2×C4⋊Q8—611no—
363C2×C4⋊Q8—1221no—
364C2×#125—611no—
365C2×#125—611no—
366C2×#125—1221no—
367C2×#125—1221no—
368C2×#124—611no—
369C2×C4×Q8—611no—
370C2×C4×Q8—611no—
371C2×C4×Q8—611no—
372C2×C4×Q8—1221no—
373C2×C4×Q8C2×D6111yescharacteristic
374#488—1224no—
375#464—1224no—
376C2×C4⋊C8—611no—
377C2×#103—611no—
378C2×#103—611no—
379C2×#103—611no—
380C2×#103C2×D6111yescharacteristic
381C2×#103C2×D6111yescharacteristic
382C2×#102—611no—
383C2×#102—611no—
384C2×#102—611no—
385C2×C4×C8C2×D6111yescharacteristic
386#397—1221no—
387#393—1221no—
388#391—611no—
389#391—611no—
390#390—611no—
391#390—611no—
392#388—1221no—
393#388—1221no—
394C4×C4⋊C4C2×D6111yescharacteristic
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340C2×D6111yescharacteristic
400#340C2×D6111yescharacteristic
401#336—611no—
402#336—611no—
403#336—611no—
404#336—611no—
405C61#2593711yescharacteristic, complemented, Sylow
406C22×Q8—611no—
407C22×Q8—611no—
408C22×Q8—1221no—
409C22×Q8C22×D6111yescharacteristic
410C22×D4—611no—
411C2×Q16—1222no—
412C2×Q16—1222no—
413C2×Q16—1222no—
414C2×SD16—1222no—
415C2×SD16—1222no—
416C2×SD16—1222no—
417C22×C8—611no—
418C22×C8—611no—
419C22×C8C22×D6111yescharacteristic
420C22×C8C22×D6111yescharacteristic
421C4⋊Q8—1222no—
422C4⋊Q8—1228no—
423C4⋊Q8—1222no—
424#125—614no—
425#125—1228no—
426#125—1222no—
427#125—1228no—
428#125—1222no—
429#125—1222no—
430#124—1222no—
431#124—1222no—
432C4×Q8—22no—
433C4×Q8—1228no—
434C4×Q8—1222no—
435C4×Q8—1222no—
436C4×Q8—1222no—
437C4×Q8—1222no—
438C4×Q8—1222no—
439C4×Q8—1222no—
440C4×Q8D24414yes—
441C2×C4⋊C4—21no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—611no—
445C2×C4⋊C4—611no—
446C2×C4⋊C4—611no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C4⋊C4C22×D6111yescharacteristic
456C2×C4⋊C4C22×D6111yescharacteristic
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1221no—
459C2×C22⋊C4—1221no—
460C2×C22⋊C4—1221no—
461C2×C22⋊C4—1221no—
462C2×C22⋊C4—1221no—
463C2×C42—21no—
464C2×C42—21no—
465C2×C42—611no—
466C2×C42—611no—
467C2×C42—1221no—
468C2×C42—1221no—
469C2×C42C22×D6111yescharacteristic
470C4⋊C8—1222no—
471#103—22no—
472#103—22no—
473#103—1222no—
474#103—1222no—
475#103—1222no—
476#102—1222no—
477#102—1222no—
478#102—1222no—
479C4×C8—22no—
480#95—21no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485C24—1221no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—1222no—
490C2×Q8—1224no—
491C2×Q8—1222no—
492C2×Q8—1221no—
493C2×Q8—1221no—
494C2×Q8—1221no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—2441no—
499C2×Q8—2442no—
500C2×Q8—2442no—
501C2×Q8C2×D24412yes—
502C2×Q8#1175822212yes—
503C2×Q8#1175822212yes—
504C2×D4—1221no—
505C2×D4—1221no—
506C2×D4—1221no—
507C2×D4—1221no—
508C2×D4—1221no—
509C2×D4—1221no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4D4×D6111yescharacteristic
522C22×C4D4×D6111yescharacteristic
523C22×C4C23×D6111yescharacteristic
524Q16—2444no—
525SD16—2444no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C4⋊C4—21no—
539C4⋊C4—24no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—612no—
545C4⋊C4—612no—
546C4⋊C4—1224no—
547C4⋊C4—1224no—
548C4⋊C4—1224no—
549C4⋊C4—1224no—
550C4⋊C4—1224no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1222no—
554C4⋊C4—1222no—
555C4⋊C4—1222no—
556C4⋊C4—1224no—
557C4⋊C4—1224no—
558C4⋊C4C2×D24412yes—
559C22⋊C4—4881no—
560C22⋊C4—4881no—
561C22⋊C4—2442no—
562C22⋊C4—2442no—
563C22⋊C4—2442no—
564C22⋊C4—2442no—
565C42—24no—
566C42—81no—
567C42—1224no—
568C42—1224no—
569C42—1221no—
570C42—1221no—
571C42—1221no—
572C42—1221no—
573C42C2×D24412yes—
574C42D4×D6112yes—
575C23—2441no—
576C23—2441no—
577C23—2441no—
578C23—2441no—
579C23—2441no—
580C23—2441no—
581C23—2441no—
582C23C2×D4×D6111yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—22no—
586Q8—2444no—
587Q8—2448no—
588Q8—2441no—
589Q8—2441no—
590Q8—2441no—
591Q8—2441no—
592Q8#1451976514yes—
593D4—2441no—
594D4—2441no—
595D4—2441no—
596D4—2441no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—612no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—1221no—
625C2×C4—1221no—
626C2×C4—1221no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4#1451974712yes—
635C2×C4#1451974712yes—
636C2×C4C2×D4×D6111yescharacteristic
637C2×C4C22×D24411yescharacteristic
638C2×C4C2×#1175822211yescharacteristic
639C2×C4C2×#1175822211yescharacteristic
640C2×C4#1451982411yescharacteristic
641C2×C4#1451982411yescharacteristic
642C8—41no—
643C8—41no—
644C8—2441no—
645C8—2441no—
646C22—4881no—
647C22—4881no—
648C22—4881no—
649C22—4881no—
650C22—4881no—
651C22—4881no—
652C22—4881no—
653C22C2×#1451974711yescharacteristic
654C22#1684042111yescharacteristic
655C22#1684042111yescharacteristic
656C22#135×D6111yescharacteristic
657C22#1684055911yescharacteristic
658C22#1684056811yescharacteristic
659C22#1684057511yescharacteristic
660C4—24no—
661C4—22no—
662C4—22no—
663C4—42no—
664C4—81no—
665C4—81no—
666C4—1222no—
667C4—4881no—
668C4—2441no—
669C4—2441no—
670C4—2442no—
671C4—2441no—
672C4—2442no—
673C4C2×#1451976511yescharacteristic
674C4#1684033811yescharacteristic
675C4#1684042111yescharacteristic
676C4#1684042111yescharacteristic
677C2—9761no—
678C2#2144947711yescharacteristic
679C2#2145503911yescharacteristic
680C2#2145513911yescharacteristic
681C2#2145850511yescharacteristic
682C2#2145851611yescharacteristic
683C2#2145858311yescharacteristic
684C2#2145865711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC4×#103×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC4×#103D61
hypocenterC61#25937
LCSG ▷ C22×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
1. C2×#16840488C2semidirect
5. #21446969C2semidirect
7. #21446959C2semidirect
9. #21446776C2semidirect
10. #21446755C2semidirect
13. #21446158C2semidirect
14. #21446158C2semidirect
15. C4×#103×C61C2semidirect
349. C4×#103D61semidirect
405. C61#25937semidirect

Maximal subgroups

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

subgroupindex
C2×#168404882
C2×#168401172
C2×#168399842
#214513022
#214469692
#214469642
#214469592
#214469542
#214467762
#214467552
#214461722
#214461582
C4×#103×C612
#2593761