Finite Group Databasebeta

← #29701807

Subgroups — #29701807

Order 15616.

926 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840040C211yesmaximal, characteristic
2#21448872C211yesmaximal, characteristic, complemented
3D4×#11758196C211yesmaximal, characteristic
4#21446778C211yesmaximal, characteristic, complemented
5#21446574C211yesmaximal, characteristic, complemented
6#21446550C211yesmaximal, characteristic, complemented
7#21446549C211yesmaximal, characteristic, complemented
8#21446391C211yesmaximal, characteristic, complemented
9#21446388C211yesmaximal, characteristic, complemented
10#21446064C211yesmaximal, characteristic
11#21445680C211yesmaximal, characteristic, complemented
12#102×Dic61C211yesmaximal, characteristic
13#102×Dic61C211yesmaximal, characteristic
14C4×#14519698C211yesmaximal, characteristic
15C61×#1862C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519742—21no—
18C2×D4×Dic61C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20C22×#11758196—21no—
21#16840040C418yescomplemented
22C2×#14519711C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519698C2211yescharacteristic
28C2×C4×#92418C2211yescharacteristic
29C2×C8×Dic61C2211yescharacteristic
30C2×C8×Dic61C2211yescharacteristic
31#16839666—21no—
32#16839660C2211yescharacteristic
33#16839658C2211yescharacteristic
34#16839658C2211yescharacteristic
35#16839644—21no—
36#16839643—21no—
37#16839635—21no—
38#16839634—21no—
39#16839631C2211yescharacteristic
40#16839620—21no—
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43C4×#11758196C2211yescharacteristic
44C4×C4⋊Dic61C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×C4×D4×C61C2211yescharacteristic
54C2×#107×C61C2211yescharacteristic
55C2×#102×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×Dic122—21no—
61C23×Dic61—21no—
62C23×Dic61—21no—
63C22×#92418C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65#14519776—22no—
66#14519776C2×C414yescomplemented
67#14519742—44no—
68D4×Dic61—22no—
69D4×Dic61—22no—
70D4×Dic61—41no—
71D4×Dic61C2×C414yes—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—44no—
83C2×#11758196D411yescharacteristic
84C2×#11758196D411yescharacteristic
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61C2311yescharacteristic
91C2×C4×Dic61C2311yescharacteristic
92#14519711C2×C414yescomplemented
93#14519710—41no—
94#14519710C2×C414yescomplemented
95#14519706C2×C414yescomplemented
96#14519698C2×C414yes—
97#14519698Q814yescomplemented
98C4×#92418—41no—
99C8×Dic61—22no—
100C8×Dic61—41no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684D411yescharacteristic
108#14519684D411yescharacteristic
109C22×D4×C61C2311yescharacteristic
110C23×C4×C61—21no—
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C4×D4×C61—22no—
114C4×D4×C61—41no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C22⋊C4×C61—21no—
120C2×C42×C61C2311yescharacteristic
121#107×C61—41no—
122#102×C61—22no—
123#102×C61C2×C414yes—
124#95×C61—21no—
125C24×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61—21no—
128C2×D4×C61—21no—
129C2×D4×C61C22×C412yes—
130C2×D4×C61C2×Q812yes—
131C2×D4×C61#4112yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—42no—
136C22×C4×C61—42no—
137C22×C4×C61—81no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—24no—
145C22×Dic61—21no—
146C22×Dic61—42no—
147C22×Dic61—42no—
148C22×Dic61—42no—
149C22×Dic61—42no—
150C22×Dic61—81no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C2×Dic122—24no—
156C2×Dic122—42no—
157C2×Dic122—42no—
158C2×C8×C61—21no—
159C2×C8×C61—21no—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61C22×C412yes—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—42no—
170C4⋊C4×C61C22×C412yes—
171C22⋊C4×C61—42no—
172C42×C61—21no—
173C42×C61—21no—
174C22⋊Dic61—24no—
175C22⋊Dic61—42no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61C22×C412yes—
182C4⋊Dic61C22×C412yes—
183C4⋊Dic61C2×Q812yes—
184#11758196—24no—
185#11758196—22no—
186#11758196—42no—
187#11758196—42no—
188#11758196—42no—
189#11758196—42no—
190#11758196—81no—
191#11758196D814yes—
192C4×Dic61—21no—
193C4×Dic61—21no—
194C4×Dic61—42no—
195C4×Dic61—42no—
196C4×Dic61—42no—
197C4×Dic61C22×C412yes—
198C4×Dic61#4112yes—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418C22×C412yes—
204C2×#92418C2×Q812yes—
205C23×C61—22no—
206C23×C61—22no—
207C23×C61—22no—
208C23×C61—41no—
209C23×C61—41no—
210C23×C61—41no—
211C23×C61—41no—
212C23×C61C22×D411yescharacteristic
213D4×C61—22no—
214D4×C61—22no—
215D4×C61—22no—
216D4×C61C4×Q814yescomplemented
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—84no—
232C2×C4×C61C2×D812yes—
233C2×C4×C61C23×C411yescharacteristic
234C2×C4×C61C22×Q811yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61H(5,𝔽2)11yescharacteristic
238C2×C4×C612−1+411yescharacteristic
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—21no—
246C2×Dic61—22no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—48no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—84no—
260C2×Dic61C4×D412yes—
261C2×Dic61C4×D412yes—
262C2×Dic61C2×D812yes—
263C2×Dic61C2×D812yes—
264C2×Dic61#13512yes—
265Dic122—48no—
266C8×C61—22no—
267C8×C61—41no—
268#92418—22no—
269#92418—22no—
270#22962—611nomaximal, Sylow
271C22×C61—24no—
272C22×C61—42no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—42no—
276C22×C61—42no—
277C22×C61—42no—
278C22×C61C2×C4×D411yescharacteristic
279C22×C61#54611yescharacteristic
280C22×C61D4×Q811yescharacteristic
281C22×C61C22×D811yescharacteristic
282C22×C61C2×#13511yescharacteristic
283C22×C61#57711yescharacteristic
284C22×C61#57811yescharacteristic
285C4×C61—22no—
286C4×C61—42no—
287C4×C61—41no—
288C4×C61—81no—
289C4×C61—81no—
290C4×C61C2×C4×Q811yescharacteristic
291C4×C61#52011yescharacteristic
292C4×C61#55211yescharacteristic
293C4×C61#55411yescharacteristic
294Dic61—22no—
295Dic61—22no—
296Dic61—42no—
297Dic61—42no—
298Dic61—42no—
299Dic61—41no—
300Dic61—81no—
301Dic61C4×D814yes—
302C2×#474—611no—
303#2365—611no—
304D4×C4⋊C4—611no—
305#1990—611no—
306#1947—611no—
307#1942—611no—
308#1942—611no—
309#1918—611no—
310#1918—611no—
311#1862—611no—
312#1862D6111yescharacteristic
313C8×C4⋊C4—611no—
314C4×#102—611no—
315C4×#102—611no—
316C4×#102—611no—
317C2×C61—44no—
318C2×C61#236011yescharacteristic
319C2×C61C2×C4×D811yescharacteristic
320C2×C61#294711yescharacteristic
321C2×C61#329411yescharacteristic
322C2×C61#330211yescharacteristic
323C2×C61Q8×D811yescharacteristic
324C2×C61#338111yescharacteristic
325C2×C4⋊Q8—611no—
326C2×C4×Q8—1221no—
327C2×C4×D4—611no—
328C2×C4×D4—611no—
329C2×C4×D4C2×D6111yescharacteristic
330C22×C4⋊C4—1221no—
331#474—618no—
332C2×#107—611no—
333C2×#107—611no—
334C2×#107—611no—
335C2×#107C2×D6111yescharacteristic
336C2×C4⋊C8—611no—
337C2×#102—611no—
338C2×#102—611no—
339C2×#102C2×D6111yescharacteristic
340C2×#102C2×D6111yescharacteristic
341C2×C4×C8—611no—
342C2×C4×C8—611no—
343C2×C4×C8—611no—
344#391—1221no—
345#389—611no—
346#389—611no—
347#389—611no—
348#389C2×D6111yescharacteristic
349#387—1221no—
350#387—1221no—
351#385—1221no—
352#385—1221no—
353#384—611no—
354#380—1221no—
355C4×C4⋊C4—611no—
356C4×C4⋊C4—611no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340C2×D6111yescharacteristic
367#336—611no—
368#336C2×D6111yescharacteristic
369C61#2296211yescharacteristic, complemented, Sylow
370C22×Q8—1221no—
371C22×D4C22×D6111yescharacteristic
372C23×C4—21no—
373C23×C4—1221no—
374C23×C4—1221no—
375C22×C8—611no—
376C22×C8—611no—
377C22×C8C22×D6111yescharacteristic
378C22×C8C22×D6111yescharacteristic
379C4⋊Q8—614no—
380C4⋊Q8—1222no—
381C4×Q8—2444no—
382C4×D4—22no—
383C4×D4—41no—
384C4×D4—614no—
385C4×D4—1222no—
386C4×D4—1222no—
387C4×D4—2441no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—611no—
392C2×C4⋊C4—611no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—611no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—2444no—
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C4⋊C4C22×D6111yescharacteristic
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—1221no—
411C2×C42—1221no—
412C2×C42—1221no—
413C2×C42C22×D6111yescharacteristic
414#107—41no—
415#107—614no—
416#107—614no—
417#107—2441no—
418C4⋊C8—614no—
419#102—22no—
420#102—614no—
421#102—614no—
422#102C4×D6114yes—
423C4×C8—1222no—
424C4×C8—2441no—
425C4×C8—2441no—
426#95—21no—
427#95—611no—
428#95—611no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95—1221no—
435C24—21no—
436C2×Q8—1224no—
437C2×Q8—2442no—
438C2×Q8—2442no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4C2×C4×D6112yes—
443C2×D4#1175822412yes—
444C2×D4Q8×D6112yes—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—81no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—611no—
454C22×C4—611no—
455C22×C4—1224no—
456C22×C4—1221no—
457C22×C4—1221no—
458C22×C4—1221no—
459C22×C4—1221no—
460C22×C4—4881no—
461C22×C4—2442no—
462C22×C4—2442no—
463C22×C4—2442no—
464C22×C4—2442no—
465C22×C4D4×D6111yescharacteristic
466C22×C4D4×D6111yescharacteristic
467C22×C4C23×D6111yescharacteristic
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—612no—
474C2×C8—612no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C2×C8C2×C4×D6112yes—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—42no—
486C4⋊C4—614no—
487C4⋊C4—612no—
488C4⋊C4—612no—
489C4⋊C4—612no—
490C4⋊C4—1222no—
491C4⋊C4—1224no—
492C4⋊C4—1221no—
493C4⋊C4—1221no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—4881no—
498C4⋊C4—2442no—
499C4⋊C4—2442no—
500C4⋊C4—2442no—
501C4⋊C4—2442no—
502C4⋊C4C2×C4×D6112yes—
503C22⋊C4—42no—
504C22⋊C4—1224no—
505C22⋊C4—2442no—
506C42—21no—
507C42—21no—
508C42—612no—
509C42—612no—
510C42—1221no—
511C42—1221no—
512C42—2442no—
513C42—2442no—
514C42—2442no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23C2×D4×D6111yescharacteristic
523Q8—2448no—
524D4—22no—
525D4—22no—
526D4—22no—
527D4#1451974214yescomplemented
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—84no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—612no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—1221no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1222no—
558C2×C4—4884no—
559C2×C4—2448no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4D8×D6112yes—
570C2×C4C22×C4×D6111yescharacteristic
571C2×C4C2×Q8×D6111yescharacteristic
572C2×C4#41×D6111yescharacteristic
573C2×C4C2×#1175822411yescharacteristic
574C2×C4#1451982211yescharacteristic
575C2×C4#1451982411yescharacteristic
576C8—22no—
577C8—41no—
578C8—1222no—
579C8—1222no—
580C22—24no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22C4×D4×D6111yescharacteristic
588C22#1684039911yescharacteristic
589C22#1684042511yescharacteristic
590C22C2×D8×D6111yescharacteristic
591C22#135×D6111yescharacteristic
592C22#1684057211yescharacteristic
593C22#1684057811yescharacteristic
594C4—22no—
595C4—41no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—614no—
600C4—1222no—
601C4—1222no—
602C4—4881no—
603C4—2441no—
604C4—2442no—
605C4—2442no—
606C4—2442no—
607C4C2×#1451974211yescharacteristic
608C4#1684024111yescharacteristic
609C4#1684044911yescharacteristic
610C4#1684046611yescharacteristic
611C2—44no—
612C2#2144881911yescharacteristic
613C2C4×D8×D6111yescharacteristic
614C2#2145465911yescharacteristic
615C2#2145762511yescharacteristic
616C2#2145772911yescharacteristic
617C2#2145866911yescharacteristic
618C2#2145868911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1862C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1862D61
hypocenterC61#22962
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
2. #21448872C2semidirect
4. #21446778C2semidirect
5. #21446574C2semidirect
6. #21446550C2semidirect
7. #21446549C2semidirect
8. #21446391C2semidirect
9. #21446388C2semidirect
11. #21445680C2semidirect
21. #16840040C4semidirect
66. #14519776C2×C4semidirect
92. #14519711C2×C4semidirect
94. #14519710C2×C4semidirect
95. #14519706C2×C4semidirect
97. #14519698Q8semidirect
216. D4×C61C4×Q8semidirect
369. C61#22962semidirect
527. D4#14519742semidirect

Maximal subgroups

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

subgroupindex
C2×#168400402
#214488722
D4×#117581962
#214467782
#214465742
#214465502
#214465492
#214463912
#214463882
#214460642
#214456802
#102×Dic612
C4×#145196982
C61×#18622
#2296261