Finite Group Databasebeta

← #29689668

Subgroups — #29689668

Order 15616.

1200 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459115C211yesmaximal, characteristic
2C4×#14519736C211yesmaximal, characteristic, complemented
3#21446795C212yesmaximal, complemented
4#21446646C212yesmaximal, complemented
5#21446121C214yesmaximal, complemented
6#21445836C214yesmaximal, complemented
7C2×C61×#423C211yesmaximal, characteristic, complemented
8#16840319C411yescharacteristic, complemented
9#16840319C411yescharacteristic, complemented
10#16840319C2211yescharacteristic
11#16840312C411yescharacteristic, complemented
12#16840312C411yescharacteristic, complemented
13#16840312C2211yescharacteristic
14#16840296—21no—
15#16840296—21no—
16#16840296—21no—
17#16840296—21no—
18#16840296C412yescomplemented
19#16840296C412yescomplemented
20#16840273—21no—
21#16840273—21no—
22#16840273C2212yes—
23C4×#11758222—21no—
24C4×#11758222—21no—
25C4×#11758222C2212yes—
26C2×#14519736—21no—
27C2×#14519703C2218yes—
28C4×C4⋊Dic61—21no—
29C4×C4⋊Dic61C2212yes—
30C4×C22⋊Dic61—21no—
31C4×C22⋊Dic61C2212yes—
32#16839608—21no—
33C42×Dic61C2212yes—
34#16839523C2218yes—
35C22×M16×C61C2212yes—
36C22×C42×C61C2211yescharacteristic
37C61×#423—21no—
38C61×#423—21no—
39C61×#423C2212yes—
40C61×#423C2212yes—
41C2×C4⋊C8×C61—21no—
42C2×C4⋊C8×C61—21no—
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222C2312yes—
46#14519777C2×C411yescharacteristic
47#14519777C2×C411yescharacteristic
48#14519777C2×C411yescharacteristic
49#14519774C2×C411yescharacteristic
50#14519771—22no—
51#14519771—22no—
52#14519769—21no—
53#14519769—21no—
54#14519769C2×C412yes—
55#14519765—21no—
56#14519765—21no—
57#14519765C2×C412yes—
58#14519760—22no—
59#14519760—22no—
60#14519760—22no—
61#14519756—21no—
62#14519756—21no—
63#14519756—21no—
64#14519756—21no—
65#14519756C2×C412yes—
66#14519756C2×C412yes—
67#14519751—21no—
68#14519751—21no—
69#14519751—21no—
70#14519751—21no—
71#14519751C2×C412yes—
72#14519751C2×C412yes—
73#14519746—22no—
74C4×Dic122—21no—
75C4×Dic122—21no—
76C4×Dic122—21no—
77C4×Dic122—21no—
78C4×Dic122C2×C412yes—
79C4×Dic122C2×C412yes—
80C4×D244—21no—
81C4×D244—21no—
82C4×D244—21no—
83C4×D244—21no—
84C4×D244C2×C412yes—
85C4×D244C2×C412yes—
86C4×C22⋊D61—22no—
87C4×C22⋊D61—22no—
88C4×C22⋊D61—22no—
89C4×C22⋊D61—22no—
90C4×C22⋊D61—22no—
91C4×C22⋊D61—22no—
92#14519736—28no—
93#14519736—21no—
94#14519736—21no—
95#14519736C2312yes—
96#14519734—22no—
97#14519734—22no—
98#14519734—22no—
99C2×C4⋊Dic61—22no—
100C2×C22⋊Dic61—22no—
101C2×C4×Dic61—22no—
102C2×C4×Dic61—22no—
103C2×C4×Dic61C2314yes—
104C42×D61—22no—
105#14519703—48no—
106#14519684—22no—
107#14519684—22no—
108C23×C4×C61D411yescharacteristic
109C23×C4×C61D411yescharacteristic
110C23×C4×C61C2311yescharacteristic
111C2×M16×C61—22no—
112C2×M16×C61—44no—
113C2×M16×C61C2314yes—
114C22×C8×C61—22no—
115C2×C42×C61C2×C411yescharacteristic
116C2×C42×C61C2×C411yescharacteristic
117C2×C42×C61C2×C411yescharacteristic
118C2×C42×C61C2×C411yescharacteristic
119C2×C42×C61D411yescharacteristic
120C2×C42×C61D411yescharacteristic
121C2×C42×C61D411yescharacteristic
122C2×C42×C61D411yescharacteristic
123C2×C42×C61D411yescharacteristic
124C2×C42×C61D411yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127C4⋊C8×C61—22no—
128C4⋊C8×C61—22no—
129C4⋊C8×C61—22no—
130C4⋊C8×C61—22no—
131C24×C61C2×D411yescharacteristic
132C22×C4×C61—24no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—42no—
136C22×C4×C61C22⋊C411yescharacteristic
137C22×C4×C61C22⋊C411yescharacteristic
138C22×C4×C61C22⋊C411yescharacteristic
139C22×C4×C61C22⋊C411yescharacteristic
140C22×C4×C61C22⋊C411yescharacteristic
141C22×C4×C61C22⋊C411yescharacteristic
142C22×C4×C61C22⋊C411yescharacteristic
143C22×C4×C61C22⋊C411yescharacteristic
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C22×C411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C2×C22⋊D61—22no—
157C22×Dic61—24no—
158#11758222—24no—
159#11758222—24no—
160#11758222—44no—
161#11758222—44no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244C22×C412yes—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61—22no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122C22×C412yes—
171M16×C61—28no—
172M16×C61—44no—
173C2×C8×C61—24no—
174C2×C8×C61—44no—
175C42×C61C22⋊C411yescharacteristic
176C42×C61C22⋊C411yescharacteristic
177C42×C61C22⋊C411yescharacteristic
178C42×C61C22⋊C411yescharacteristic
179C42×C61C22⋊C411yescharacteristic
180C42×C61C22⋊C411yescharacteristic
181C42×C61C22⋊C411yescharacteristic
182C42×C61C22⋊C411yescharacteristic
183C42×C61C22×C411yescharacteristic
184C42×C61C22×C411yescharacteristic
185C42×C61C22×C411yescharacteristic
186C42×C61C22×C411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C22⋊Dic61—28no—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201C4⋊Dic61—28no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61C22×C412yes—
207C4⋊Dic61C22×C412yes—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196—22no—
214C4×Dic61—28no—
215C4×Dic61—22no—
216C4×Dic61—22no—
217C4×Dic61—22no—
218C4×Dic61—24no—
219C4×Dic61—24no—
220C4×Dic61—44no—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61—41no—
224C23×C61C2×C22⋊C411yescharacteristic
225C23×C61C2×C22⋊C411yescharacteristic
226C23×C61C22×D411yescharacteristic
227C22×D61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
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—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61#10414yescomplemented
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C23×C411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61H(5,𝔽2)12yes—
260C2×C4×C612−1+412yes—
261C22⋊D61—44no—
262C22⋊D61—44no—
263C22⋊D61—44no—
264C2×Dic61—28no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—44no—
269C2×Dic61—44no—
270D244—24no—
271D244—42no—
272D244—42no—
273C4×D61—44no—
274C4×D61—42no—
275C4×D61—42no—
276C4×D61—44no—
277Dic122—24no—
278Dic122—42no—
279Dic122—42no—
280C8×C61—44no—
281#21856—611nomaximal, Sylow
282C22×C61—21no—
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—44no—
290C22×C61C2×#10412yes—
291C22×C61C22×C22⋊C411yescharacteristic
292C22×C61#53712yes—
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—42no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61C2×#10412yes—
304C4×C61C2×#10412yes—
305C4×C61#42112yes—
306C4×C61#52012yes—
307C4×C61#53511yescharacteristic
308C4×C61#53611yescharacteristic
309Dic61—48no—
310Dic61—42no—
311Dic61—42no—
312Dic61—42no—
313#3444—611no—
314C2×#423D6111yescharacteristic, complemented
315C4×#117—611no—
316#1994—612no—
317#1962—612no—
318#1871—614no—
319#1806—614no—
320C2×C61—22no—
321C2×C61—22no—
322C2×C61—41no—
323C2×C61#231111yescharacteristic
324C2×C61C22×#10411yescharacteristic
325C2×C61C2×#42111yescharacteristic
326D61—82no—
327C22×M16C2×D6112yes—
328#533—611no—
329#533—611no—
330#533—611no—
331#532—611no—
332#532—611no—
333#532—611no—
334#529—612no—
335#529—612no—
336#529—1221no—
337#529—1221no—
338#529—1221no—
339#529—1221no—
340#525—612no—
341#525—1221no—
342#525—1221no—
343C4×#41—612no—
344C4×#41—1221no—
345C4×#41—1221no—
346C2×#117—1221no—
347C22×C42C2×D6111yescharacteristic
348#423—21no—
349#423—21no—
350#423C2×D6112yes—
351#423C2×D6112yes—
352C2×C4⋊C8—21no—
353C2×C4⋊C8—21no—
354C2×#104—618no—
355C4×C4⋊C4—612no—
356C4×C4⋊C4—1221no—
357C4×C22⋊C4—612no—
358C4×C22⋊C4—1221no—
359#376—1221no—
360C43—612no—
361#339—618no—
362C61#2185611yescharacteristic, complemented, Sylow
363C2×#41—612no—
364C2×#41—1221no—
365C2×#41—1221no—
366C23×C4D24411yescharacteristic
367C23×C4D24411yescharacteristic
368C23×C4C22×D6111yescharacteristic
369C2×M16—22no—
370C2×M16—44no—
371C2×M16C22×D6114yes—
372C22×C8—22no—
373C4⋊Q8—611no—
374C4⋊Q8—611no—
375C4⋊Q8—611no—
376#127—611no—
377#126—1222no—
378#126—1222no—
379#125—612no—
380#125—1221no—
381#125—1221no—
382#124—612no—
383#124—1221no—
384#124—1221no—
385#123—1222no—
386#123—1222no—
387#123—1222no—
388C22⋊Q8—612no—
389C22⋊Q8—612no—
390C22⋊Q8—1221no—
391C22⋊Q8—1221no—
392C22⋊Q8—1221no—
393C22⋊Q8—1221no—
394#121—612no—
395#121—612no—
396#121—1221no—
397#121—1221no—
398#121—1221no—
399#121—1221no—
400#120—1222no—
401C4×Q8—612no—
402C4×Q8—612no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8—1221no—
406C4×Q8—1221no—
407C4×D4—612no—
408C4×D4—612no—
409C4×D4—1222no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—1222no—
414C4×D4—1222no—
415C4×D4—1221no—
416C4×D4—1221no—
417C4×D4—1221no—
418C4×D4—1221no—
419#117—612no—
420#117—1228no—
421#117—1221no—
422#117—1221no—
423#117—1222no—
424#117—1222no—
425#117—1222no—
426C2×C4⋊C4—1222no—
427C2×C22⋊C4—1222no—
428C2×C42—614no—
429C2×C42—1222no—
430C2×C42—1222no—
431C2×C42—1222no—
432C2×C42C4×D6111yescharacteristic
433C2×C42C4×D6111yescharacteristic
434C2×C42C4×D6111yescharacteristic
435C2×C42C4×D6111yescharacteristic
436C2×C42D24411yescharacteristic
437C2×C42D24411yescharacteristic
438C2×C42C22⋊D6111yescharacteristic
439C2×C42C22⋊D6111yescharacteristic
440C2×C42C22⋊D6111yescharacteristic
441C2×C42C22⋊D6111yescharacteristic
442C2×C42C22×D6111yescharacteristic
443C2×C42C22×D6111yescharacteristic
444C4⋊C8—22no—
445C4⋊C8—22no—
446C4⋊C8—22no—
447C4⋊C8—22no—
448#104—2448no—
449#95—1222no—
450#95—1222no—
451C24C2×D24411yescharacteristic
452#41—1224no—
453#41—1224no—
454#41—2444no—
455#41—2444no—
456C2×Q8—612no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×D4—612no—
460C2×D4—1222no—
461C2×D4—1222no—
462C2×D4—1222no—
463C2×D4—1221no—
464C2×D4—1221no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—42no—
469C22×C4—1224no—
470C22×C4—1222no—
471C22×C4—1222no—
472C22×C4—1222no—
473C22×C4#1175819811yescharacteristic
474C22×C4#1175819811yescharacteristic
475C22×C4#1175819811yescharacteristic
476C22×C4#1175819811yescharacteristic
477C22×C4#1175819811yescharacteristic
478C22×C4#1175819811yescharacteristic
479C22×C4#1175819811yescharacteristic
480C22×C4#1175819811yescharacteristic
481C22×C4C2×C4×D6111yescharacteristic
482C22×C4C2×C4×D6111yescharacteristic
483C22×C4C2×D24411yescharacteristic
484C22×C4C2×D24411yescharacteristic
485C22×C4C2×D24411yescharacteristic
486C22×C4C2×D24411yescharacteristic
487C22×C4C2×D24411yescharacteristic
488C22×C4C2×C22⋊D6111yescharacteristic
489C22×C4C2×C22⋊D6111yescharacteristic
490C22×C4C23×D6111yescharacteristic
491M16—28no—
492M16—44no—
493C2×C8—24no—
494C2×C8—44no—
495C4⋊C4—612no—
496C4⋊C4—612no—
497C4⋊C4—1228no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C22⋊C4—1228no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C42—1228no—
519C42—1222no—
520C42—1222no—
521C42—1222no—
522C42—1224no—
523C42—1224no—
524C42—2444no—
525C42#1175819811yescharacteristic
526C42#1175819811yescharacteristic
527C42#1175819811yescharacteristic
528C42#1175819811yescharacteristic
529C42#1175819811yescharacteristic
530C42#1175819811yescharacteristic
531C42#1175819811yescharacteristic
532C42#1175819811yescharacteristic
533C42C2×C4×D6111yescharacteristic
534C42C2×C4×D6111yescharacteristic
535C42C2×C4×D6111yescharacteristic
536C42C2×C4×D6111yescharacteristic
537C42C2×C22⋊D6111yescharacteristic
538C42C2×C22⋊D6111yescharacteristic
539C42C2×C22⋊D6111yescharacteristic
540C42C2×C22⋊D6111yescharacteristic
541C23—22no—
542C23—22no—
543C23—41no—
544C23—1222no—
545C23C2×#1175819811yescharacteristic
546C23C2×#1175819811yescharacteristic
547C23C22×D24411yescharacteristic
548Q8—1224no—
549Q8—2442no—
550Q8—2442no—
551D4—1224no—
552D4—2444no—
553D4—2444no—
554D4—2444no—
555D4—2442no—
556D4—2442no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—1228no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—2444no—
580C2×C4—2444no—
581C2×C4—2444no—
582C2×C4—2444no—
583C2×C4—2442no—
584C2×C4—2442no—
585C2×C4#1451970314yescomplemented
586C2×C4C2×#1175819811yescharacteristic
587C2×C4C2×#1175819811yescharacteristic
588C2×C4C2×#1175819811yescharacteristic
589C2×C4C2×#1175819811yescharacteristic
590C2×C4C2×#1175819811yescharacteristic
591C2×C4C2×#1175819811yescharacteristic
592C2×C4C2×#1175819811yescharacteristic
593C2×C4C2×#1175819811yescharacteristic
594C2×C4C2×#1175819811yescharacteristic
595C2×C4C2×#1175819811yescharacteristic
596C2×C4C22×C4×D6111yescharacteristic
597C2×C4C22×C22⋊D6111yescharacteristic
598C2×C4#1451982212yes—
599C2×C4#1451982512yes—
600C8—44no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—44no—
609C22—2442no—
610C22—2442no—
611C22—2442no—
612C22C2×#1451970312yes—
613C22C22×#1175819811yescharacteristic
614C22#1684034312yes—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—2448no—
623C4—2442no—
624C4—2442no—
625C4—2442no—
626C4C2×#1451970312yes—
627C4C2×#1451970312yes—
628C4#1683977812yes—
629C4#1684024312yes—
630C4#1684032911yescharacteristic
631C4#1684033611yescharacteristic
632C2—22no—
633C2—22no—
634C2—41no—
635C2—4882no—
636C2#2144841211yescharacteristic
637C2C22×#1451970311yescharacteristic
638C2C2×#1683977811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#423C2
socleC22×C61C22×C22⋊C4
prime radical#21459115C2
cosocleC22×C4×C61C24
hypercenterC2×#423D61
hypocenterC61#21856
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ C2×#423

Complements

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

NH ≅ G/Ntype
2. C4×#14519736C2semidirect
3. #21446795C2semidirect
4. #21446646C2semidirect
5. #21446121C2semidirect
6. #21445836C2semidirect
7. C2×C61×#423C2semidirect
8. #16840319C4semidirect
9. #16840319C4semidirect
11. #16840312C4semidirect
12. #16840312C4semidirect
18. #16840296C4semidirect
19. #16840296C4semidirect
246. C2×C4×C61#104semidirect
314. C2×#423D61semidirect
362. C61#21856semidirect
585. C2×C4#14519703semidirect

Maximal subgroups

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

subgroupindex
#214591152
C4×#145197362
#214467952
#214466462
#214461212
#214458362
C2×C61×#4232
#2185661