Finite Group Databasebeta

← #29737531

Subgroups — #29737531

Order 15616.

920 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839769C211yesmaximal, characteristic
2#21449528C211yesmaximal, characteristic
3Q8×#11758198C211yesmaximal, characteristic
4#21447354C211yesmaximal, characteristic, complemented
5#21447199C211yesmaximal, characteristic, complemented
6#21447195C211yesmaximal, characteristic, complemented
7#21446967C211yesmaximal, characteristic, complemented
8#21446089C211yesmaximal, characteristic, complemented
9#21446073C211yesmaximal, characteristic, complemented
10#21446017C211yesmaximal, characteristic
11#21446000C211yesmaximal, characteristic
12#21445853C211yesmaximal, characteristic
13#21445822C211yesmaximal, characteristic, complemented
14#21445752C211yesmaximal, characteristic
15C2×C61×#479C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519710C2211yescharacteristic
21C2×#14519709C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23#16839769—24no—
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519689C2211yescharacteristic
31#16839697—21no—
32#16839684—21no—
33#16839668—21no—
34#16839668C2211yescharacteristic
35#16839667—21no—
36#16839667C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839658—21no—
40#16839643—21no—
41#16839631C2211yescharacteristic
42#16839619—21no—
43#16839619C2211yescharacteristic
44#16839619C2211yescharacteristic
45C4×#11758198—21no—
46C4×#11758198C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#125×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C61×#479—24no—
56C2×#107×C61C2211yescharacteristic
57C2×#106×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#103×C61C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C2×Q8×D61—22no—
64C2×Q8×D61—22no—
65C2×Q8×D61—22no—
66C2×Q8×D61—41no—
67C2×Q8×D61—41no—
68C2×Q8×D61Q814yes—
69C22×C4×D61—21no—
70C22×C4×D61C2311yescharacteristic
71C2×#11758189—22no—
72C2×#11758189—22no—
73C2×#11758189—41no—
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76Q8×Dic61—22no—
77Q8×Dic61—41no—
78#14519735—22no—
79#14519735—41no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198D411yescharacteristic
92C2×#11758198D411yescharacteristic
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61C2311yescharacteristic
96#14519710—41no—
97#14519709—41no—
98#14519706—41no—
99#14519702—22no—
100#14519702—41no—
101#14519701—22no—
102#14519701—22no—
103#14519700—22no—
104#14519700—41no—
105#14519689—22no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684D411yescharacteristic
115#14519684D411yescharacteristic
116C22×Q8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119#125×C61—22no—
120#125×C61—41no—
121C4×Q8×C61—22no—
122C4×Q8×C61—41no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61—21no—
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C42×C61—21no—
130C2×C42×C61C2311yescharacteristic
131#107×C61—22no—
132#106×C61—22no—
133C4⋊C8×C61—22no—
134#103×C61—22no—
135#103×C61D414yes—
136C23×D61C2×D411yescharacteristic
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61C2×D412yes—
141C2×Q8×C61C2×Q812yes—
142C2×Q8×C61#4112yes—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2411yescharacteristic
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157Q8×D61—28no—
158Q8×D61—44no—
159Q8×D61—44no—
160Q8×D61—44no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—24no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×C4×D61C2×Q812yes—
169C2×C4×D61#4112yes—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—24no—
176C2×Dic122—42no—
177C2×Dic122—42no—
178C2×Dic122C2×Q812yes—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61C2×D412yes—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—42no—
191C4⋊C4×C61—42no—
192C4⋊C4×C61—42no—
193C4⋊C4×C61C2×D412yes—
194C42×C61—21no—
195C42×C61—21no—
196C42×C61—42no—
197#11758198—22no—
198#11758198—41no—
199#11758198—42no—
200#11758198—42no—
201#11758198—81no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—42no—
209C4⋊Dic61—42no—
210#11758196—42no—
211#11758196—81no—
212#11758196—81no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—42no—
216C4×Dic61—42no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223#11758189—44no—
224C23×C61C22×D411yescharacteristic
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61C22⋊Q812yes—
232Q8×C61—22no—
233Q8×C61—22no—
234Q8×C61—22no—
235Q8×C61C22⋊Q814yes—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
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×C61C22×D411yescharacteristic
255C2×C4×C61C22×Q811yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61H(5,𝔽2)11yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—22no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61C22⋊Q812yes—
280C2×Dic61#13712yes—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—22no—
284C4×D61—22no—
285C4×D61—48no—
286Dic122—22no—
287Dic122—22no—
288Dic122—48no—
289Dic122—41no—
290Dic122—41no—
291C8×C61—22no—
292C8×C61—41no—
293#92418—41no—
294#92418—41no—
295#25981—611nomaximal, Sylow
296C22×C61C2×C22⋊Q811yescharacteristic
297C22×C61D4211yescharacteristic
298C22×C61#54611yescharacteristic
299C22×C61C2×#13711yescharacteristic
300C22×C61#57511yescharacteristic
301C22×C61#57711yescharacteristic
302C22×C61#57811yescharacteristic
303C2×D61—22no—
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—41no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61—41no—
310C4×C61—24no—
311C4×C61—41no—
312C4×C61—41no—
313C4×C61—42no—
314C4×C61—81no—
315C4×C61—81no—
316C4×C61C2×C22⋊Q811yescharacteristic
317C4×C61#54311yescharacteristic
318C4×C61#54611yescharacteristic
319C4×C61#55111yescharacteristic
320Dic61—22no—
321Dic61—22no—
322Dic61—42no—
323Dic61—41no—
324Dic61—81no—
325Dic61—81no—
326Dic61—81no—
327C2×#479D6111yescharacteristic, complemented
328C2×#419—611no—
329#2441—611no—
330Q8×C22⋊C4—611no—
331#2095—611no—
332#2066—611no—
333#2066—611no—
334#2026—611no—
335#1865—611no—
336#1863—611no—
337#1854—611no—
338#1851—611no—
339#1809—611no—
340#1803—611no—
341#1787—611no—
342C2×C61#243711yescharacteristic
343C2×C61#307811yescharacteristic
344C2×C61#308211yescharacteristic
345C2×C61#327111yescharacteristic
346C2×C61#327611yescharacteristic
347C2×C61#332011yescharacteristic
348C2×C61#332711yescharacteristic
349D61—42no—
350C23×Q8—611no—
351C22×M16—611no—
352C2×#125C2×D6111yescharacteristic
353C2×C4×Q8—611no—
354C2×C4×Q8C2×D6111yescharacteristic
355C2×#117—611no—
356#479—24no—
357C2×#107—611no—
358C2×#107C2×D6111yescharacteristic
359C2×#106—611no—
360C2×#106C2×D6111yescharacteristic
361C2×C4⋊C8—611no—
362C2×C4⋊C8C2×D6111yescharacteristic
363#419—1224no—
364C2×#103—611no—
365C2×#103—611no—
366C2×#103—611no—
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103C2×D6111yescharacteristic
371C2×#103C2×D6111yescharacteristic
372C2×C22⋊C8—611no—
373#399—1221no—
374#396—1221no—
375#391—611no—
376#391—611no—
377#391—1221no—
378#391—1221no—
379#389—611no—
380#389—611no—
381#389—1221no—
382#387—1221no—
383#384—611no—
384#380—611no—
385#380—611no—
386#380—1221no—
387C4×C22⋊C4—611no—
388C4×C22⋊C4—1221no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#336—611no—
394#336—611no—
395C61#2598111yescharacteristic, complemented, Sylow
396C22×Q8—614no—
397C22×Q8—611no—
398C22×Q8—1221no—
399C22×Q8—1222no—
400C22×Q8—1222no—
401C22×Q8—1222no—
402C22×Q8—2441no—
403C22×Q8—2441no—
404C22×Q8C22×D6111yescharacteristic
405C23×C4—611no—
406C23×C4—1221no—
407C2×M16—1222no—
408C2×M16—1222no—
409C2×M16—2441no—
410C22×C8—611no—
411C22×C8—611no—
412C22×C8C22×D6111yescharacteristic
413C22×C8C22×D6111yescharacteristic
414#125—22no—
415#125—41no—
416C4×Q8—22no—
417C4×Q8—41no—
418C4×Q8—1222no—
419C4×Q8—2441no—
420#117—1222no—
421#117—2441no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C22⋊C4—611no—
437C2×C22⋊C4—611no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C42—21no—
442C2×C42—611no—
443C2×C42—1221no—
444C2×C42—1221no—
445C2×C42C22×D6111yescharacteristic
446#107—22no—
447#107—2441no—
448#106—22no—
449#106—2441no—
450C4⋊C8—22no—
451C4⋊C8—2441no—
452#103—22no—
453#103—1222no—
454#103—1222no—
455#103—1222no—
456#103—1222no—
457#103—2441no—
458#103—2441no—
459#103C22⋊D6114yes—
460C22⋊C8—1222no—
461#95—611no—
462#95—611no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471C24—611no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—21no—
475C2×Q8—612no—
476C2×Q8—1224no—
477C2×Q8—1221no—
478C2×Q8—1228no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8—2442no—
484C2×Q8—2442no—
485C2×Q8—2444no—
486C2×Q8—2444no—
487C2×Q8—2444no—
488C2×Q8Q8×D6112yes—
489C2×Q8#1175822612yes—
490C2×Q8C2×C22⋊D6112yes—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—612no—
498C22×C4—612no—
499C22×C4—611no—
500C22×C4—1224no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—2442no—
510C22×C4—2442no—
511C22×C4D4×D6111yescharacteristic
512C22×C4D4×D6111yescharacteristic
513C22×C4C23×D6111yescharacteristic
514M16—2444no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8C2×C22⋊D6112yes—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—4881no—
542C4⋊C4—4881no—
543C4⋊C4—2442no—
544C4⋊C4—2442no—
545C4⋊C4—2442no—
546C4⋊C4C2×C22⋊D6112yes—
547C22⋊C4—1222no—
548C22⋊C4—4881no—
549C22⋊C4—2441no—
550C22⋊C4—2442no—
551C22⋊C4—2442no—
552C42—21no—
553C42—21no—
554C42—42no—
555C42—1221no—
556C42—1221no—
557C42—2442no—
558C42—2442no—
559C23—612no—
560C23—1221no—
561C23—1221no—
562C23—1221no—
563C23—1221no—
564C23—1221no—
565C23—1221no—
566C23C2×D4×D6111yescharacteristic
567Q8—22no—
568Q8—22no—
569Q8—22no—
570Q8—1222no—
571Q8—1222no—
572Q8—2448no—
573Q8—2441no—
574Q8—2441no—
575Q8#1451975314yes—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—612no—
595C2×C4—612no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—2448no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4C22×C22⋊D6111yescharacteristic
621C2×C4C2×Q8×D6111yescharacteristic
622C2×C4#41×D6111yescharacteristic
623C2×C4C2×#1175822611yescharacteristic
624C2×C4#1451982211yescharacteristic
625C2×C4#1451982311yescharacteristic
626C8—22no—
627C8—41no—
628C8—2441no—
629C8—2441no—
630C22—1222no—
631C22—1222no—
632C22—1222no—
633C22—2441no—
634C22—2441no—
635C22—2441no—
636C22—2441no—
637C22C22⋊Q8×D6111yescharacteristic
638C22D4×C22⋊D6111yescharacteristic
639C22#1684040311yescharacteristic
640C22#137×D6111yescharacteristic
641C22#1684055911yescharacteristic
642C22#1684056911yescharacteristic
643C22#1684057811yescharacteristic
644C4—24no—
645C4—41no—
646C4—42no—
647C4—41no—
648C4—81no—
649C4—81no—
650C4—1222no—
651C4—1222no—
652C4—4881no—
653C4—4881no—
654C4—4881no—
655C4—2442no—
656C4—2441no—
657C4C2×#1451975311yescharacteristic
658C4#1684038711yescharacteristic
659C4#1684040011yescharacteristic
660C4#1684043911yescharacteristic
661C2—2442no—
662C2#2144948211yescharacteristic
663C2#2145562211yescharacteristic
664C2#2145565811yescharacteristic
665C2#2145740411yescharacteristic
666C2#2145745911yescharacteristic
667C2#2145798711yescharacteristic
668C2#2145808411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#479C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#479D61
hypocenterC61#25981
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#479

Complements

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

NH ≅ G/Ntype
4. #21447354C2semidirect
5. #21447199C2semidirect
6. #21447195C2semidirect
7. #21446967C2semidirect
8. #21446089C2semidirect
9. #21446073C2semidirect
13. #21445822C2semidirect
15. C2×C61×#479C2semidirect
327. C2×#479D61semidirect
395. C61#25981semidirect

Maximal subgroups

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

subgroupindex
C2×#168397692
#214495282
Q8×#117581982
#214473542
#214471992
#214471952
#214469672
#214460892
#214460732
#214460172
#214460002
#214458532
#214458222
#214457522
C2×C61×#4792
#2598161