Finite Group Databasebeta

← #29689081

Subgroups — #29689081

Order 15616.

1486 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840273C211yesmaximal, characteristic
2#21448202C211yesmaximal, characteristic, complemented
3#21446257C212yesmaximal, complemented
4#21446252C212yesmaximal, complemented
5#21446139C212yesmaximal, complemented
6#21446129C212yesmaximal, complemented
7#21445915C212yesmaximal, complemented
8#21445760C212yesmaximal, complemented
9C2×C61×#407C211yesmaximal, characteristic, complemented
10C22×#11758222C2211yescharacteristic
11#16840273—24no—
12#16840273C418yescomplemented
13C2×#14519760—21no—
14C2×#14519756C2211yescharacteristic
15C2×#14519756C2211yescharacteristic
16C2×#14519751C2211yescharacteristic
17C2×#14519751C2211yescharacteristic
18C2×#14519746—21no—
19C2×C4×C22⋊D61—21no—
20C2×C4×C22⋊D61—21no—
21C2×#14519736C2211yescharacteristic
22C2×#14519736C2212yes—
23C2×#14519702C2214yescomplemented
24C2×#14519699C2214yescomplemented
25#16839625—21no—
26#16839622C2212yes—
27#16839622C2212yes—
28#16839618—21no—
29C4×C22⋊Dic61—21no—
30C4×C22⋊Dic61—21no—
31#16839525C2214yescomplemented
32#16839525C2214yescomplemented
33C24×C4×C61C2211yescharacteristic
34C22×M16×C61C2212yes—
35C61×#407—24no—
36C2×C22⋊C8×C61C2212yes—
37C2×C22⋊C8×C61C2212yes—
38C2×#11758222—22no—
39C2×#11758222—44no—
40C2×#11758222C2×C414yes—
41C22×Dic122C2311yescharacteristic
42C22×D244C2311yescharacteristic
43C22×C22⋊D61—21no—
44C22×C22⋊D61—21no—
45C22×C22⋊D61—21no—
46C22×C4×D61—21no—
47#14519760—28no—
48#14519756—22no—
49#14519756—22no—
50#14519756C2×C414yes—
51#14519756C2×C414yes—
52#14519751—22no—
53#14519751—22no—
54#14519751C2×C414yes—
55#14519751C2×C414yes—
56#14519746—28no—
57C4×C22⋊D61—28no—
58C4×C22⋊D61—28no—
59#14519736—22no—
60#14519736—44no—
61#14519736C2×C414yes—
62C2×C4⋊Dic61C2314yes—
63C2×C4⋊Dic61C2311yescharacteristic
64C2×C4⋊Dic61C2311yescharacteristic
65C2×#11758196—21no—
66C2×#11758196—21no—
67C2×C22⋊Dic61—21no—
68C2×C22⋊Dic61—21no—
69C2×C22⋊Dic61—21no—
70C2×C22⋊Dic61—22no—
71C2×C22⋊Dic61—22no—
72C2×C22⋊Dic61—22no—
73C2×#11758198—21no—
74C2×#11758198—21no—
75C2×C4×Dic61—21no—
76C2×C4×Dic61—22no—
77#14519702—44no—
78#14519699—44no—
79#14519684—22no—
80#14519684—22no—
81C25×C61D411yescharacteristic
82C23×C4×C61—44no—
83C23×C4×C61C2×C414yes—
84C23×C4×C61D411yescharacteristic
85C23×C4×C61D411yescharacteristic
86C23×C4×C61D411yescharacteristic
87C23×C4×C61D414yes—
88C23×C4×C61C2311yescharacteristic
89C23×C4×C61C2311yescharacteristic
90C23×C4×C61C2311yescharacteristic
91C2×M16×C61—24no—
92C2×M16×C61—44no—
93C22×C8×C61C2314yes—
94C22⋊C8×C61—24no—
95C22⋊C8×C61—24no—
96C24×C61—42no—
97C24×C61—42no—
98C24×C61—42no—
99C24×C61C22⋊C414yes—
100C24×C61C2×D411yescharacteristic
101C24×C61C2×D411yescharacteristic
102C24×C61C2×D411yescharacteristic
103C23×D61—21no—
104C22×C4×C61—24no—
105C22×C4×C61—24no—
106C22×C4×C61—24no—
107C22×C4×C61—48no—
108C22×C4×C61—48no—
109C22×C4×C61—42no—
110C22×C4×C61—42no—
111C22×C4×C61—42no—
112C22×C4×C61C22⋊C414yes—
113C22×C4×C61C22⋊C414yes—
114C22×C4×C61C22⋊C414yes—
115C22×C4×C61C22×C412yes—
116C22×C4×C61C22×C412yes—
117C22×C4×C61C22×C412yes—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2×D412yes—
123C22×C4×C61C2×D412yes—
124C22×C4×C61C2411yescharacteristic
125C2×C22⋊D61—24no—
126C2×C22⋊D61—24no—
127C2×C22⋊D61—24no—
128C2×C22⋊D61—44no—
129C2×C22⋊D61—44no—
130C2×C22⋊D61—44no—
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—24no—
135#11758222—48no—
136#11758222—48no—
137C2×D244—24no—
138C2×D244—21no—
139C2×D244C22×C412yes—
140C2×C4×D61—24no—
141C2×C4×D61—44no—
142C2×Dic122—24no—
143C2×Dic122—21no—
144C2×Dic122C22×C412yes—
145M16×C61—48no—
146C2×C8×C61—24no—
147C2×C8×C61—24no—
148C2×C8×C61—24no—
149C22⋊Dic61—24no—
150C22⋊Dic61—24no—
151C22⋊Dic61—24no—
152C22⋊Dic61—44no—
153C22⋊Dic61—44no—
154C22⋊Dic61—44no—
155#11758198—24no—
156#11758198—24no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—24no—
160C4⋊Dic61—24no—
161C4⋊Dic61C22×C412yes—
162C4⋊Dic61C22×C412yes—
163#11758196—24no—
164#11758196—24no—
165C4×Dic61—24no—
166C4×Dic61—44no—
167C23×C61—22no—
168C23×C61—22no—
169C23×C61—22no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61—44no—
177C23×C61—44no—
178C23×C61—44no—
179C23×C61—416no—
180C23×C61C2×C22⋊C412yes—
181C23×C61C2×C22⋊C412yes—
182C23×C61C2×C22⋊C412yes—
183C23×C61C22×D411yescharacteristic
184C22×D61—22no—
185C22×D61—42no—
186C22×D61—42no—
187C22×D61—42no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—24no—
192C2×C4×C61—24no—
193C2×C4×C61—24no—
194C2×C4×C61—24no—
195C2×C4×C61—24no—
196C2×C4×C61—24no—
197C2×C4×C61—44no—
198C2×C4×C61—44no—
199C2×C4×C61—44no—
200C2×C4×C61C2×C22⋊C412yes—
201C2×C4×C61C2×C22⋊C412yes—
202C2×C4×C61C2×C22⋊C412yes—
203C2×C4×C61C23×C411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61H(5,𝔽2)14yes—
206C22⋊D61—48no—
207C22⋊D61—48no—
208C22⋊D61—48no—
209C2×Dic61—22no—
210C2×Dic61—22no—
211C2×Dic61—22no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—44no—
216C2×Dic61—44no—
217C2×Dic61—44no—
218D244—24no—
219D244—42no—
220C4×D61—48no—
221Dic122—24no—
222Dic122—42no—
223C8×C61—44no—
224#21766—611nomaximal, Sylow
225C22×C61—22no—
226C22×C61—22no—
227C22×C61—22no—
228C22×C61—22no—
229C22×C61—22no—
230C22×C61—22no—
231C22×C61—22no—
232C22×C61—22no—
233C22×C61—22no—
234C22×C61—416no—
235C22×C61—44no—
236C22×C61—44no—
237C22×C61—44no—
238C22×C61C22×C22⋊C411yescharacteristic
239C22×C61#53412yes—
240C22×C61#57512yes—
241C22×C61#57512yes—
242C2×D61—42no—
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—84no—
246C4×C61—22no—
247C4×C61—22no—
248C4×C61—22no—
249C4×C61#51812yes—
250C4×C61#53412yes—
251Dic61—42no—
252Dic61—42no—
253Dic61—42no—
254Dic61—84no—
255C2×#525—611no—
256C2×#407D6111yescharacteristic, complemented
257#2280—611no—
258#1895—612no—
259#1894—612no—
260#1875—612no—
261#1873—612no—
262#1825—612no—
263#1788—612no—
264C2×C61—42no—
265C2×C61—42no—
266C2×C61—42no—
267C2×C61#230811yescharacteristic
268C2×C61#289611yescharacteristic
269C2×C61#289611yescharacteristic
270C2×C61#319714yes—
271D61—82no—
272C22×#41—611no—
273C24×C4C2×D6111yescharacteristic
274C22×M16C2×D6112yes—
275#525—618no—
276#525—1224no—
277C2×#123—1221no—
278C2×C22⋊Q8—611no—
279C2×C22⋊Q8—611no—
280C2×#121—611no—
281C2×#121—611no—
282C2×#120—1221no—
283C2×C4×D4—1221no—
284C2×C4×D4—1221no—
285C2×#117—611no—
286C2×#117—612no—
287C2×#103—614no—
288C2×#102—614no—
289#407—24no—
290C2×C22⋊C8C2×D6112yes—
291C2×C22⋊C8C2×D6112yes—
292#381—1221no—
293#380—612no—
294#380—612no—
295#379—1221no—
296C4×C22⋊C4—1221no—
297C4×C22⋊C4—1221no—
298#340—614no—
299#340—614no—
300C61#2176611yescharacteristic, complemented, Sylow
301C25D24411yescharacteristic
302C2×#41—614no—
303C2×#41—1222no—
304C2×#41—2444no—
305C22×Q8—611no—
306C22×D4—611no—
307C22×D4—1221no—
308C22×D4—1221no—
309C22×D4—1221no—
310C23×C4—44no—
311C23×C4—1221no—
312C23×C4C4×D6114yes—
313C23×C4D24411yescharacteristic
314C23×C4D24411yescharacteristic
315C23×C4D24411yescharacteristic
316C23×C4C22⋊D6114yes—
317C23×C4C22×D6111yescharacteristic
318C23×C4C22×D6111yescharacteristic
319C23×C4C22×D6111yescharacteristic
320C2×M16—24no—
321C2×M16—44no—
322C22×C8C22×D6114yes—
323#123—1228no—
324C22⋊Q8—614no—
325C22⋊Q8—614no—
326C22⋊Q8—1222no—
327C22⋊Q8—1222no—
328#121—614no—
329#121—614no—
330#121—1222no—
331#121—1222no—
332#120—1228no—
333C4×D4—1228no—
334C4×D4—1228no—
335#117—614no—
336#117—1222no—
337#117—2444no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—611no—
340C2×C4⋊C4—614no—
341C2×C4⋊C4—1221no—
342C2×C4⋊C4—1221no—
343C2×C22⋊C4—1221no—
344C2×C22⋊C4—1221no—
345C2×C22⋊C4—1221no—
346C2×C22⋊C4—1221no—
347C2×C22⋊C4—1221no—
348C2×C22⋊C4—1222no—
349C2×C22⋊C4—1222no—
350C2×C22⋊C4—1222no—
351C2×C42—1221no—
352C2×C42—1222no—
353#103—2444no—
354#102—2444no—
355C22⋊C8—24no—
356C22⋊C8—24no—
357#95—1222no—
358#95—1222no—
359C24—42no—
360C24—42no—
361C24—42no—
362C24—1221no—
363C24#1175819814yes—
364C24C2×D24411yescharacteristic
365C24C2×D24411yescharacteristic
366C24C2×D24411yescharacteristic
367#41—2448no—
368#41—2448no—
369C2×Q8—612no—
370C2×Q8—1221no—
371C2×Q8—1224no—
372C2×D4—612no—
373C2×D4—1224no—
374C2×D4—1224no—
375C2×D4—1224no—
376C2×D4—1221no—
377C2×D4—1224no—
378C2×D4—2444no—
379C2×D4—2444no—
380C2×D4—2444no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—1221no—
390C22×C4—1221no—
391C22×C4—1221no—
392C22×C4—1224no—
393C22×C4—1224no—
394C22×C4—2444no—
395C22×C4#1175819814yes—
396C22×C4#1175819814yes—
397C22×C4#1175819814yes—
398C22×C4C2×C4×D6112yes—
399C22×C4C2×C4×D6112yes—
400C22×C4C2×C4×D6112yes—
401C22×C4C2×D24411yescharacteristic
402C22×C4C2×D24411yescharacteristic
403C22×C4C2×D24411yescharacteristic
404C22×C4C2×C22⋊D6112yes—
405C22×C4C2×C22⋊D6112yes—
406C22×C4C2×C22⋊D6112yes—
407C22×C4C23×D6111yescharacteristic
408M16—48no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C4⋊C4—612no—
413C4⋊C4—612no—
414C4⋊C4—1221no—
415C4⋊C4—1221no—
416C4⋊C4—1224no—
417C4⋊C4—1224no—
418C4⋊C4—1224no—
419C4⋊C4—1224no—
420C22⋊C4—1224no—
421C22⋊C4—1224no—
422C22⋊C4—1224no—
423C22⋊C4—1224no—
424C22⋊C4—1224no—
425C22⋊C4—2444no—
426C22⋊C4—2444no—
427C22⋊C4—2444no—
428C42—1224no—
429C42—2444no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—416no—
443C23—1222no—
444C23—2442no—
445C23—2442no—
446C23—2442no—
447C23C2×#1175819812yes—
448C23C2×#1175819812yes—
449C23C2×#1175819812yes—
450C23C22×D24411yescharacteristic
451Q8—1224no—
452Q8—2442no—
453D4—1224no—
454D4—2448no—
455D4—2448no—
456D4—2448no—
457D4—2442no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—1222no—
471C2×C4—1222no—
472C2×C4—1222no—
473C2×C4—2448no—
474C2×C4—2444no—
475C2×C4—2444no—
476C2×C4—2444no—
477C2×C4—2442no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4C2×#1175819812yes—
481C2×C4C2×#1175819812yes—
482C2×C4C2×#1175819812yes—
483C2×C4C22×C4×D6111yescharacteristic
484C2×C4C22×C22⋊D6111yescharacteristic
485C2×C4#1451982214yes—
486C8—44no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—416no—
497C22—44no—
498C22—44no—
499C22—44no—
500C22—4884no—
501C22—2442no—
502C22—2442no—
503C22—2442no—
504C22C22×#1175819811yescharacteristic
505C22#1684032412yes—
506C22#1684056212yes—
507C22#1684056212yes—
508C4—22no—
509C4—22no—
510C4—22no—
511C4—4884no—
512C4—2442no—
513C4—2442no—
514C4—2442no—
515C4#1684023512yes—
516C4#1684032112yes—
517C2—42no—
518C2—42no—
519C2—42no—
520C2—4882no—
521C2#2144839411yescharacteristic
522C2#2145433911yescharacteristic
523C2#2145433911yescharacteristic
524C2#2145666114yes—
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#407C2
socleC23×C61C22×D4
prime radicalC2×#16840273C2
cosocleC22×C4×C61C24
hypercenterC2×#407D61
hypocenterC61#21766
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#407

Complements

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

NH ≅ G/Ntype
2. #21448202C2semidirect
3. #21446257C2semidirect
4. #21446252C2semidirect
5. #21446139C2semidirect
6. #21446129C2semidirect
7. #21445915C2semidirect
8. #21445760C2semidirect
9. C2×C61×#407C2semidirect
12. #16840273C4semidirect
23. C2×#14519702C22semidirect
24. C2×#14519699C22semidirect
31. #16839525C22semidirect
32. #16839525C22semidirect
256. C2×#407D61semidirect
300. C61#21766semidirect

Maximal subgroups

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

subgroupindex
C2×#168402732
#214482022
#214462572
#214462522
#214461392
#214461292
#214459152
#214457602
C2×C61×#4072
#2176661