Finite Group Databasebeta

← #29691326

Subgroups — #29691326

Order 15616.

920 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839920C211yesmaximal, characteristic
2#21448501C211yesmaximal, characteristic
3#21448403C211yesmaximal, characteristic
4#21446690C211yesmaximal, characteristic, complemented
5#21446680C211yesmaximal, characteristic, complemented
6#21446509C211yesmaximal, characteristic, complemented
7#21446495C211yesmaximal, characteristic, complemented
8#21446461C211yesmaximal, characteristic, complemented
9#21446454C211yesmaximal, characteristic, complemented
10#21446159C211yesmaximal, characteristic, complemented
11#21446159C211yesmaximal, characteristic, complemented
12#21446129C211yesmaximal, characteristic
13#21446020C211yesmaximal, characteristic
14#21445755C211yesmaximal, characteristic
15C61×#1750C211yesmaximal, characteristic
16C23×Dic122C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18C22×Dic244C2211yescharacteristic
19C2×#14519756C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×#14519736C2211yescharacteristic
22#16839920C418yescomplemented
23C2×#14519711C2211yescharacteristic
24C2×#14519709C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33#16839668C2211yescharacteristic
34#16839668C2211yescharacteristic
35#16839668C2211yescharacteristic
36#16839668C2211yescharacteristic
37#16839668C2211yescharacteristic
38#16839667—21no—
39#16839667—21no—
40#16839660C2211yescharacteristic
41#16839657—21no—
42#16839656—21no—
43#16839650—21no—
44#16839643—21no—
45#16839623—21no—
46#16839623—21no—
47#16839622C2211yescharacteristic
48C4×C4⋊Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53C2×#117×C61C2211yescharacteristic
54C2×C22⋊C8×C61C2211yescharacteristic
55C2×C22⋊C8×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×Dic122—21no—
61C22×Dic122—22no—
62C22×Dic122—41no—
63C22×Dic122—41no—
64C22×Dic122—41no—
65C22×Dic122—41no—
66C22×Dic122C2×C414yes—
67C22×Dic122C2311yescharacteristic
68C22×Dic122C2311yescharacteristic
69C22×Dic122C2311yescharacteristic
70C23×Dic61—21no—
71C2×Dic244—41no—
72C2×Dic244—41no—
73C2×Dic244—41no—
74C2×Dic244—41no—
75C2×Dic244C2×C414yes—
76C2×Dic244C2×C414yes—
77#14519756—22no—
78#14519756C2×C414yes—
79C4×Dic122—41no—
80C4×Dic122—41no—
81#14519736—41no—
82#14519736—41no—
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—21no—
93C2×C22⋊Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97#14519711—41no—
98#14519709—41no—
99#14519702—41no—
100#14519702—41no—
101#14519702—41no—
102#14519702—41no—
103#14519702—41no—
104#14519702—41no—
105#14519702C2×C414yes—
106#14519702C2×C414yes—
107#14519684—21no—
108#14519684—21no—
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#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120C23×C4×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125#117×C61—41no—
126#117×C61—41no—
127C2×C4⋊C4×C61D411yescharacteristic
128C2×C4⋊C4×C61D411yescharacteristic
129C2×C22⋊C4×C61D411yescharacteristic
130C2×C22⋊C4×C61D411yescharacteristic
131C2×C42×C61C2311yescharacteristic
132C2×C42×C61C2311yescharacteristic
133C22⋊C8×C61—22no—
134C22⋊C8×C61C2×C414yes—
135#97×C61—41no—
136C4×C8×C61—41no—
137C24×C61C2×D411yescharacteristic
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C22×C412yes—
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61#4112yes—
149C22×C4×C61C2411yescharacteristic
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—21no—
159C22×Dic61—42no—
160C22×Dic61—42no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—24no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—44no—
178C2×Dic122—44no—
179C2×Dic122—42no—
180C2×Dic122—42no—
181C2×Dic122—44no—
182C2×Dic122—44no—
183C2×Dic122C22×C412yes—
184C2×Dic122C22×C412yes—
185C2×Dic122C22×C412yes—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61—21no—
194C2×C8×C61—21no—
195C2×C8×C61—21no—
196C2×C8×C61C22×C412yes—
197C2×C8×C61C22×C412yes—
198C4⋊C4×C61—41no—
199C4⋊C4×C61—41no—
200C22⋊C4×C61—41no—
201C22⋊C4×C61—41no—
202C42×C61—21no—
203C42×C61—21no—
204C42×C61—21no—
205C42×C61—21no—
206C22⋊Dic61—24no—
207C22⋊Dic61—42no—
208C22⋊Dic61—81no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61C22×C412yes—
219#11758196—24no—
220#11758196—42no—
221#11758196—81no—
222C4×Dic61—42no—
223C4×Dic61—42no—
224C4×Dic61—81no—
225Dic244—44no—
226Dic244—44no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—21no—
233C23×C61C4×D412yes—
234C23×C61C22×D411yescharacteristic
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
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×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61C4×D412yes—
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61H(5,𝔽2)11yescharacteristic
257C2×C4×C61H(5,𝔽2)11yescharacteristic
258C2×C4×C61H(5,𝔽2)11yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—48no—
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—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285Dic122—22no—
286Dic122—22no—
287Dic122—22no—
288Dic122—22no—
289Dic122—22no—
290Dic122—22no—
291Dic122—48no—
292Dic122—41no—
293Dic122—41no—
294Dic122—41no—
295Dic122—41no—
296Dic122—41no—
297Dic122—41no—
298C8×C61—22no—
299C8×C61—22no—
300C8×C61—41no—
301C8×C61—41no—
302#22054—611nomaximal, Sylow
303C22×C61—22no—
304C22×C61—22no—
305C22×C61—22no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61C2×C4×D411yescharacteristic
311C22×C61#53411yescharacteristic
312C22×C61#53511yescharacteristic
313C22×C61#57711yescharacteristic
314C22×C61#57811yescharacteristic
315C22×C61#57811yescharacteristic
316C22×C61#57811yescharacteristic
317C4×C61—22no—
318C4×C61—41no—
319C4×C61—41no—
320C4×C61—41no—
321C4×C61—41no—
322C4×C61#51811yescharacteristic
323C4×C61#51811yescharacteristic
324C4×C61#53811yescharacteristic
325C4×C61#53811yescharacteristic
326Dic61—42no—
327Dic61—42no—
328Dic61—42no—
329Dic61—42no—
330Dic61—81no—
331Dic61—81no—
332Dic61—81no—
333Dic61—81no—
334C2×#451—611no—
335#2323—611no—
336#2310—611no—
337#1971—611no—
338#1969—611no—
339#1935—611no—
340#1933—611no—
341#1928—611no—
342#1927—611no—
343#1878—611no—
344#1878—611no—
345#1873—611no—
346#1854—611no—
347#1787—611no—
348#1750D6111yescharacteristic
349C2×C61—42no—
350C2×C61#232111yescharacteristic
351C2×C61#294711yescharacteristic
352C2×C61#295011yescharacteristic
353C2×C61#320111yescharacteristic
354C2×C61#320211yescharacteristic
355C2×C61#322311yescharacteristic
356C2×C61#322511yescharacteristic
357C23×Q8—611no—
358C22×Q16—611no—
359C22×Q16—611no—
360C2×C22⋊Q8—611no—
361C2×C4×Q8—611no—
362C2×#117—611no—
363C2×#117C2×D6111yescharacteristic
364#451—618no—
365C2×#107—611no—
366C2×#106—611no—
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×C22⋊C8C2×D6111yescharacteristic
376C2×C22⋊C8C2×D6111yescharacteristic
377C2×#97C2×D6111yescharacteristic
378C2×C4×C8C2×D6111yescharacteristic
379#391—611no—
380#391—611no—
381#391—611no—
382#391—611no—
383#391—611no—
384#391—1221no—
385#391—1221no—
386#389—611no—
387#389—1221no—
388#388—1221no—
389#388—1221no—
390#387—1221no—
391#381—1221no—
392#381—1221no—
393#380—611no—
394C4×C4⋊C4—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#336C2×D6111yescharacteristic
400#336C2×D6111yescharacteristic
401C61#2205411yescharacteristic, complemented, Sylow
402C22×Q8—614no—
403C22×Q8—611no—
404C22×Q8—611no—
405C22×Q8—611no—
406C22×Q8—1221no—
407C22×Q8—1222no—
408C22×Q8—2441no—
409C22×Q8—2441no—
410C22×Q8—2441no—
411C22×Q8—2441no—
412C23×C4—1221no—
413C23×C4C22×D6111yescharacteristic
414C2×Q16—614no—
415C2×Q16—614no—
416C2×Q16—2441no—
417C2×Q16—2441no—
418C2×Q16—2441no—
419C2×Q16—2441no—
420C22×C8C22×D6111yescharacteristic
421C22×C8C22×D6111yescharacteristic
422C22×C8C22×D6111yescharacteristic
423C22×C8C22×D6111yescharacteristic
424C22⋊Q8—614no—
425C22⋊Q8—1222no—
426C4×Q8—2441no—
427C4×Q8—2441no—
428#117—41no—
429#117—41no—
430#117—2441no—
431#117—2441no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4D24411yescharacteristic
441C2×C4⋊C4D24411yescharacteristic
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4D24411yescharacteristic
446C2×C22⋊C4D24411yescharacteristic
447C2×C42—1221no—
448C2×C42—1221no—
449C2×C42—1221no—
450C2×C42C22×D6111yescharacteristic
451C2×C42C22×D6111yescharacteristic
452#107—2441no—
453#106—2441no—
454#103—614no—
455#103—614no—
456#103—2441no—
457#103—2441no—
458#103—2441no—
459#103—2441no—
460#103—2441no—
461#103—2441no—
462C22⋊C8—22no—
463C22⋊C8C4×D6114yes—
464#97—41no—
465C4×C8—41no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479C24C2×D24411yescharacteristic
480C2×Q8—612no—
481C2×Q8—612no—
482C2×Q8—612no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1224no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8—1221no—
491C2×Q8—1221no—
492C2×Q8—1221no—
493C2×Q8—1221no—
494C2×Q8—1221no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—2444no—
500C2×Q8—2444no—
501C2×Q8—2442no—
502C2×Q8—2442no—
503C2×Q8—2444no—
504C2×Q8—2444no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1221no—
513C22×C4—1224no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—2442no—
519C22×C4—2442no—
520C22×C4C2×C4×D6112yes—
521C22×C4C2×D24411yescharacteristic
522C22×C4C2×D24411yescharacteristic
523C22×C4C2×D24411yescharacteristic
524C22×C4C2×D24411yescharacteristic
525C22×C4C2×D24411yescharacteristic
526C22×C4#1175822212yes—
527C22×C4C23×D6111yescharacteristic
528Q16—2444no—
529Q16—2444no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8C2×C4×D6112yes—
541C2×C8C2×C4×D6112yes—
542C4⋊C4—41no—
543C4⋊C4—41no—
544C4⋊C4—612no—
545C4⋊C4—1221no—
546C4⋊C4—1224no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—4881no—
556C4⋊C4—2442no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—1224no—
560C22⋊C4—4881no—
561C22⋊C4—2442no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—4881no—
567C42—2442no—
568C42—2442no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23C4×D24412yes—
576C23C22×D24411yescharacteristic
577Q8—1222no—
578Q8—1222no—
579Q8—1222no—
580Q8—1222no—
581Q8—1222no—
582Q8—1222no—
583Q8—2448no—
584Q8—2441no—
585Q8—2441no—
586Q8—2441no—
587Q8—2441no—
588Q8—2441no—
589Q8—2441no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—1222no—
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×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2448no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4C4×D24412yes—
634C2×C4C22×C4×D6111yescharacteristic
635C2×C4C2×#1175822211yescharacteristic
636C2×C4#1451982211yescharacteristic
637C2×C4#1451982211yescharacteristic
638C2×C4#1451982311yescharacteristic
639C2×C4#1451982311yescharacteristic
640C8—22no—
641C8—22no—
642C8—41no—
643C8—41no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22C2×C4×D24411yescharacteristic
652C22#1684032411yescharacteristic
653C22#1684033111yescharacteristic
654C22#1684057411yescharacteristic
655C22#1684057911yescharacteristic
656C22#1684057911yescharacteristic
657C22#1684057911yescharacteristic
658C4—22no—
659C4—41no—
660C4—41no—
661C4—41no—
662C4—41no—
663C4—4881no—
664C4—4881no—
665C4—4881no—
666C4—4881no—
667C4—2442no—
668C4—2442no—
669C4—2442no—
670C4—2442no—
671C4#1684023411yescharacteristic
672C4#1684023411yescharacteristic
673C4#1684034511yescharacteristic
674C4#1684034511yescharacteristic
675C2—42no—
676C2#2144847911yescharacteristic
677C2#2145466411yescharacteristic
678C2#2145468811yescharacteristic
679C2#2145670311yescharacteristic
680C2#2145671411yescharacteristic
681C2#2145690411yescharacteristic
682C2#2145693011yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1750C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1750D61
hypocenterC61#22054
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1750

Complements

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

NH ≅ G/Ntype
4. #21446690C2semidirect
5. #21446680C2semidirect
6. #21446509C2semidirect
7. #21446495C2semidirect
8. #21446461C2semidirect
9. #21446454C2semidirect
10. #21446159C2semidirect
11. #21446159C2semidirect
22. #16839920C4semidirect
401. C61#22054semidirect

Maximal subgroups

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

subgroupindex
C2×#168399202
#214485012
#214484032
#214466902
#214466802
#214465092
#214464952
#214464612
#214464542
#214461592
#214461292
#214460202
#214457552
C61×#17502
#2205461