Finite Group Databasebeta

← #29737540

Subgroups — #29737540

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×#16840070C211yesmaximal, characteristic, complemented
2C2×#16839772C211yesmaximal, characteristic
3#21449532C211yesmaximal, characteristic
4C22⋊C4×Dic122C211yesmaximal, characteristic
5#21447356C211yesmaximal, characteristic, complemented
6#21447199C211yesmaximal, characteristic, complemented
7#21447199C211yesmaximal, characteristic, complemented
8#21446971C211yesmaximal, characteristic, complemented
9#21446090C211yesmaximal, characteristic, complemented
10#21446076C211yesmaximal, characteristic, complemented
11#21446020C211yesmaximal, characteristic
12#21445852C211yesmaximal, characteristic
13#21445825C211yesmaximal, characteristic, complemented
14#21445755C211yesmaximal, characteristic
15C61×#1851C211yesmaximal, characteristic
16C23×Dic122C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16840070—24no—
22C2×#14519711C2211yescharacteristic
23C2×#14519711C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26#16839772—24no—
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519702C2211yescharacteristic
35#16839698—21no—
36#16839685—21no—
37#16839668C2211yescharacteristic
38#16839668C2211yescharacteristic
39#16839666—21no—
40#16839666—21no—
41#16839660C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839659—21no—
44#16839645—21no—
45#16839632C2211yescharacteristic
46#16839622C2211yescharacteristic
47#16839622C2211yescharacteristic
48#16839620—21no—
49C22⋊C4×Dic61—21no—
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54C22×M16×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×C22⋊C8×C61C2211yescharacteristic
58C4×C22⋊C4×C61C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—22no—
62C22×Dic122—22no—
63C22×Dic122—21no—
64C22×Dic122—22no—
65C22×Dic122—41no—
66C22×Dic122—41no—
67C22×Dic122Q814yes—
68C22×Dic122C2311yescharacteristic
69C22×Dic122C2311yescharacteristic
70C23×Dic61—21no—
71#14519769—22no—
72#14519769—41no—
73C4×Dic122—22no—
74C4×Dic122—22no—
75C4×Dic122—41no—
76C4×Dic122—41no—
77#14519736—22no—
78#14519736—41no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×C22⋊Dic61—21no—
94C2×C22⋊Dic61—21no—
95C2×C22⋊Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99#14519711—22no—
100#14519711—41no—
101#14519709—22no—
102#14519709—41no—
103#14519702—22no—
104#14519702—22no—
105#14519702—22no—
106#14519702—22no—
107#14519702—22no—
108#14519702—41no—
109#14519702—41no—
110#14519702D414yes—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C23×C4×C61C2311yescharacteristic
120C2×M16×C61—22no—
121C2×M16×C61—22no—
122C2×M16×C61—41no—
123C22×C8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C2×C22⋊C4×C61D411yescharacteristic
128C2×C22⋊C4×C61D411yescharacteristic
129C2×C42×C61C2311yescharacteristic
130C2×C42×C61C2311yescharacteristic
131C4⋊C8×C61—22no—
132C4⋊C8×C61—41no—
133C22⋊C8×C61—22no—
134#95×C61D411yescharacteristic
135#95×C61D411yescharacteristic
136C24×C61C2×D411yescharacteristic
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×Q812yes—
147C22×C4×C61#4112yes—
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—21no—
150C22×Dic61—24no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—42no—
159C22×Dic61—42no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—28no—
168C2×Dic122—24no—
169C2×Dic122—21no—
170C2×Dic122—44no—
171C2×Dic122—44no—
172C2×Dic122—44no—
173C2×Dic122—42no—
174C2×Dic122—42no—
175C2×Dic122C2×D412yes—
176C2×Dic122C2×Q812yes—
177C2×Dic122C2×Q812yes—
178C2×Dic122#4112yes—
179M16×C61—44no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61C2×D412yes—
192C22⋊C4×C61—22no—
193C22⋊C4×C61—41no—
194C42×C61—21no—
195C42×C61—21no—
196C42×C61—21no—
197C42×C61—21no—
198C22⋊Dic61—42no—
199C22⋊Dic61—42no—
200C22⋊Dic61—81no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—42no—
213C4⋊Dic61—81no—
214C4⋊Dic61C2×D412yes—
215#11758196—42no—
216#11758196—42no—
217#11758196—42no—
218#11758196—42no—
219#11758196—42no—
220#11758196—81no—
221C4×Dic61—42no—
222C4×Dic61—42no—
223C4×Dic61—42no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61C22⋊Q812yes—
231C23×C61C22×D411yescharacteristic
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
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—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61C22⋊Q812yes—
250C2×C4×C61#13712yes—
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×Q811yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×C4×C61H(5,𝔽2)11yescharacteristic
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—48no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
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×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282Dic122—22no—
283Dic122—22no—
284Dic122—22no—
285Dic122—22no—
286Dic122—22no—
287Dic122—48no—
288Dic122—41no—
289Dic122—41no—
290Dic122C22⋊Q814yes—
291C8×C61—22no—
292C8×C61—41no—
293C8×C61—41no—
294C8×C61—41no—
295#25981—611nomaximal, Sylow
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61C2×C22⋊Q811yescharacteristic
304C22×C61D4211yescharacteristic
305C22×C61#54611yescharacteristic
306C22×C61C2×#13711yescharacteristic
307C22×C61#57511yescharacteristic
308C22×C61#57711yescharacteristic
309C22×C61#57811yescharacteristic
310C4×C61—22no—
311C4×C61—22no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61—41no—
315C4×C61C2×C22⋊Q811yescharacteristic
316C4×C61#54311yescharacteristic
317C4×C61#54611yescharacteristic
318C4×C61#55111yescharacteristic
319Dic61—24no—
320Dic61—42no—
321Dic61—42no—
322Dic61—81no—
323Dic61—81no—
324Dic61—81no—
325Dic61—81no—
326Dic61—81no—
327C2×#479—611no—
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#1851D6111yescharacteristic
339#1809—611no—
340#1803—611no—
341#1787—611no—
342C2×C61—42no—
343C2×C61#243711yescharacteristic
344C2×C61#307811yescharacteristic
345C2×C61#308211yescharacteristic
346C2×C61#327111yescharacteristic
347C2×C61#327611yescharacteristic
348C2×C61#332011yescharacteristic
349C2×C61#332711yescharacteristic
350C23×Q8—611no—
351C22×M16C2×D6111yescharacteristic
352C2×#125—611no—
353C2×C4×Q8—611no—
354C2×C4×Q8—611no—
355C2×#117—611no—
356#479—1224no—
357C2×#107—611no—
358C2×#107—611no—
359C2×#106—611no—
360C2×#106—611no—
361C2×C4⋊C8C2×D6111yescharacteristic
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×#103—611no—
371C2×#103—611no—
372C2×C22⋊C8C2×D6111yescharacteristic
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—1221no—
388C4×C22⋊C4C2×D6111yescharacteristic
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#336C2×D6111yescharacteristic
394#336C2×D6111yescharacteristic
395C61#2598111yescharacteristic, complemented, Sylow
396C22×Q8—614no—
397C22×Q8—611no—
398C22×Q8—611no—
399C22×Q8—1222no—
400C22×Q8—1222no—
401C22×Q8—1221no—
402C22×Q8—1222no—
403C22×Q8—2441no—
404C22×Q8—2441no—
405C23×C4—1221no—
406C23×C4C22×D6111yescharacteristic
407C2×M16—22no—
408C2×M16—22no—
409C2×M16—41no—
410C22×C8C22×D6111yescharacteristic
411C22×C8C22×D6111yescharacteristic
412C22×C8C22×D6111yescharacteristic
413C22×C8C22×D6111yescharacteristic
414#125—1222no—
415#125—2441no—
416C4×Q8—1222no—
417C4×Q8—1222no—
418C4×Q8—2441no—
419C4×Q8—2441no—
420#117—1222no—
421#117—2441no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
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⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4D24411yescharacteristic
440C2×C22⋊C4D24411yescharacteristic
441C2×C42—1221no—
442C2×C42—1221no—
443C2×C42—1221no—
444C2×C42C22×D6111yescharacteristic
445C2×C42C22×D6111yescharacteristic
446#107—1222no—
447#107—2441no—
448#106—1222no—
449#106—2441no—
450C4⋊C8—22no—
451C4⋊C8—41no—
452#103—614no—
453#103—1222no—
454#103—1222no—
455#103—1222no—
456#103—1222no—
457#103—1222no—
458#103—2441no—
459#103—2441no—
460C22⋊C8—22no—
461#95—1221no—
462#95—1221no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95D24411yescharacteristic
470#95D24411yescharacteristic
471C24C2×D24411yescharacteristic
472C2×Q8—612no—
473C2×Q8—612no—
474C2×Q8—612no—
475C2×Q8—612no—
476C2×Q8—1228no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1224no—
482C2×Q8—1221no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—2442no—
487C2×Q8—2442no—
488C2×Q8—2444no—
489C2×Q8—2444no—
490C2×Q8—2444no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1224no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—2442no—
505C22×C4—2442no—
506C22×C4C2×D24411yescharacteristic
507C22×C4C2×D24411yescharacteristic
508C22×C4C2×D24411yescharacteristic
509C22×C4C2×D24411yescharacteristic
510C22×C4C2×D24411yescharacteristic
511C22×C4#1175822412yes—
512C22×C4Q8×D6112yes—
513C22×C4C23×D6111yescharacteristic
514M16—44no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8D4×D6112yes—
527C4⋊C4—612no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—4881no—
540C4⋊C4—4881no—
541C4⋊C4—2442no—
542C4⋊C4—2442no—
543C4⋊C4—2442no—
544C4⋊C4—2442no—
545C4⋊C4—2442no—
546C4⋊C4—2442no—
547C22⋊C4—22no—
548C22⋊C4—41no—
549C22⋊C4—4881no—
550C22⋊C4—2442no—
551C22⋊C4—2442no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—2442no—
557C42—2442no—
558C42—2442no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23#1451975712yes—
566C23C22×D24411yescharacteristic
567Q8—614no—
568Q8—1222no—
569Q8—1222no—
570Q8—1222no—
571Q8—1222no—
572Q8—1222no—
573Q8—2448no—
574Q8—2441no—
575Q8—2441no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—2448no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
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#1451975712yes—
619C2×C4#1451981012yes—
620C2×C4C2×D4×D6111yescharacteristic
621C2×C4C2×Q8×D6111yescharacteristic
622C2×C4#41×D6111yescharacteristic
623C2×C4C2×#1175822411yescharacteristic
624C2×C4#1451982211yescharacteristic
625C2×C4#1451982311yescharacteristic
626C8—22no—
627C8—41no—
628C8—41no—
629C8—41no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22C2×#1451975711yescharacteristic
638C22D4×D24411yescharacteristic
639C22#1684040711yescharacteristic
640C22C2×#1451981011yescharacteristic
641C22#1684056211yescharacteristic
642C22#1684057411yescharacteristic
643C22#1684057911yescharacteristic
644C4—22no—
645C4—22no—
646C4—41no—
647C4—41no—
648C4—41no—
649C4—1224no—
650C4—4881no—
651C4—4881no—
652C4—4881no—
653C4—4881no—
654C4—4881no—
655C4—2442no—
656C4—2442no—
657C4C22⋊Q8×D6111yescharacteristic
658C4#1684038711yescharacteristic
659C4#1684039911yescharacteristic
660C4#1684043511yescharacteristic
661C2—42no—
662C2#2144948311yescharacteristic
663C2#2145562611yescharacteristic
664C2#2145566211yescharacteristic
665C2#2145740911yescharacteristic
666C2#2145746711yescharacteristic
667C2#2145799211yescharacteristic
668C2#2145808911yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1851C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1851D61
hypocenterC61#25981
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
1. C2×#16840070C2semidirect
5. #21447356C2semidirect
6. #21447199C2semidirect
7. #21447199C2semidirect
8. #21446971C2semidirect
9. #21446090C2semidirect
10. #21446076C2semidirect
13. #21445825C2semidirect
395. C61#25981semidirect

Maximal subgroups

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

subgroupindex
C2×#168400702
C2×#168397722
#214495322
C22⋊C4×Dic1222
#214473562
#214471992
#214469712
#214460902
#214460762
#214460202
#214458522
#214458252
#214457552
C61×#18512
#2598161