Finite Group Databasebeta

← #29696273

Subgroups — #29696273

Order 15616.

1206 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840477C211yesmaximal, characteristic
2C2×D4×#92418C211yesmaximal, characteristic, complemented
3#21446779C211yesmaximal, characteristic, complemented
4#21446767C211yesmaximal, characteristic, complemented
5#21446245C212yesmaximal, complemented
6#21446141C212yesmaximal, complemented
7#21445911C212yesmaximal, complemented
8#21445816C211yesmaximal, characteristic, complemented
9C4×#14519697C211yesmaximal, characteristic, complemented
10#21445535C212yesmaximal, complemented
11C61×#2354C211yesmaximal, characteristic, complemented
12C23×#92418C2212yes—
13#16840477—24no—
14#16840477C418yescomplemented
15C2×#14519777C2211yescharacteristic
16C2×#14519763—22no—
17C2×#14519751C2212yes—
18C2×#14519748—21no—
19C2×C4×D244C2211yescharacteristic
20C2×#14519736C2212yes—
21C2×#14519735—21no—
22D4×#92418—24no—
23C2×C4⋊#92418C2211yescharacteristic
24C2×#14519697C2211yescharacteristic
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2212yes—
27C2×C22⋊#92418C2212yes—
28C2×C4×#92418C2211yescharacteristic
29#16839524C2211yescharacteristic
30#16839524C2211yescharacteristic
31#16839524C2212yes—
32#16839524C2212yes—
33#16839524C2212yes—
34#16839524C2212yes—
35#16839524C2212yes—
36C2×C4×D4×C61C2211yescharacteristic
37C2×#117×C61C2212yes—
38C61×#390—21no—
39C61×#388—22no—
40C61×#384C2211yescharacteristic
41C61×#380C2212yes—
42C4×C4⋊C4×C61—21no—
43C4×C4⋊C4×C61C2211yescharacteristic
44C22×Dic122—21no—
45C22×D244C2311yescharacteristic
46C22×C22⋊D61—22no—
47C22×C4×D61—21no—
48C22×#92418—24no—
49C22×#92418—24no—
50C22×#92418—42no—
51C22×#92418C2311yescharacteristic
52C22×#92418C2312yes—
53C22×#92418C2311yescharacteristic
54#14519777—22no—
55#14519777C2×C414yes—
56#14519763—216no—
57#14519751—24no—
58#14519751C2×C418yes—
59#14519748—28no—
60C4×D244—22no—
61C4×D244C2×C414yes—
62#14519736—24no—
63#14519736C2×C418yes—
64#14519735—28no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61C2311yescharacteristic
67C2×C4⋊Dic61C2312yes—
68C2×C22⋊Dic61—22no—
69C2×#11758198—22no—
70C2×#11758198—21no—
71C2×C4×Dic61—22no—
72C4⋊#92418—22no—
73#14519697—22no—
74#14519697—22no—
75#14519697—42no—
76C22⋊#92418—24no—
77C4×#92418—22no—
78C22×D4×C61D411yescharacteristic
79C23×C4×C61C2312yes—
80C4×D4×C61C2×C414yes—
81C4×D4×C61D414yes—
82#117×C61—24no—
83#117×C61—42no—
84C2×C4⋊C4×C61—21no—
85C2×C4⋊C4×C61—21no—
86C2×C4⋊C4×C61D411yescharacteristic
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2312yes—
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C22⋊C4×C61—22no—
91C2×C22⋊C4×C61—22no—
92C2×C22⋊C4×C61D412yes—
93C2×C42×C61—21no—
94C2×C42×C61—22no—
95C2×C42×C61C2311yescharacteristic
96#95×C61—22no—
97#95×C61—21no—
98C24×C61C2×D412yes—
99C23×D61—21no—
100C2×D4×C61—22no—
101C2×D4×C61—22no—
102C2×D4×C61C22⋊C414yes—
103C22×C4×C61—22no—
104C22×C4×C61—22no—
105C22×C4×C61—22no—
106C22×C4×C61—22no—
107C22×C4×C61—21no—
108C22×C4×C61—21no—
109C22×C4×C61—22no—
110C22×C4×C61C22×C414yes—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D412yes—
113C22×C4×C61C2×D414yes—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2411yescharacteristic
116C2×C22⋊D61—28no—
117C2×C22⋊D61—82no—
118C2×C22⋊D61—82no—
119C22×Dic61—21no—
120C22×Dic61—22no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244—21no—
125C2×D244—21no—
126C2×D244C22×C412yes—
127C2×C4×D61—24no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×Dic122—24no—
131C2×Dic122—81no—
132C2×Dic122—81no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—42no—
139C4⋊C4×C61—81no—
140C4⋊C4×C61C22⋊C414yes—
141C4⋊C4×C61#4112yes—
142C22⋊C4×C61—44no—
143C22⋊C4×C61—82no—
144C22⋊C4×C61C22⋊C418yes—
145C42×C61—44no—
146C42×C61—42no—
147C42×C61C22×C412yes—
148C42×C61C2×D412yes—
149C22⋊Dic61—28no—
150#11758198—28no—
151#11758198—24no—
152C4⋊Dic61—24no—
153C4⋊Dic61—21no—
154C4⋊Dic61—22no—
155C4⋊Dic61C22×C412yes—
156C4⋊Dic61C22×C414yes—
157C4×Dic61—28no—
158C2×#92418—21no—
159C2×#92418—21no—
160C2×#92418—21no—
161C2×#92418—22no—
162C2×#92418—21no—
163C2×#92418—22no—
164C2×#92418—22no—
165C2×#92418—21no—
166C2×#92418—48no—
167C2×#92418#4112yes—
168C23×C61—22no—
169C23×C61—22no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61C2×C22⋊C414yes—
175C23×C61C22×D411yescharacteristic
176C22×D61—22no—
177C22×D61—41no—
178C22×D61—41no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—41no—
182C22×D61—41no—
183D4×C61—28no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—24no—
189C2×C4×C61—24no—
190C2×C4×C61—24no—
191C2×C4×C61—24no—
192C2×C4×C61—22no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—42no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—42no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61C2×C22⋊C414yes—
202C2×C4×C61C2×C22⋊C412yes—
203C2×C4×C61C2×C22⋊C412yes—
204C2×C4×C61#13512yes—
205C2×C4×C61#13512yes—
206C2×C4×C61C23×C411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C61H(5,𝔽2)11yescharacteristic
211C2×C4×C612−1+411yescharacteristic
212C22⋊D61—88no—
213C2×Dic61—22no—
214C2×Dic61—24no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219D244—22no—
220D244—22no—
221D244—41no—
222D244—41no—
223C4×D61—48no—
224Dic122—84no—
225#92418—22no—
226#92418—42no—
227#92418—41no—
228#22489—611nomaximal, Sylow
229C22×C61—24no—
230C22×C61—24no—
231C22×C61—24no—
232C22×C61—42no—
233C22×C61—42no—
234C22×C61—42no—
235C22×C61—42no—
236C22×C61C22×C22⋊C411yescharacteristic
237C22×C61#54611yescharacteristic
238C22×C61#54711yescharacteristic
239C22×C61C2×#13511yescharacteristic
240C22×C61C2×#13511yescharacteristic
241C22×C61#57611yescharacteristic
242C22×C61#57711yescharacteristic
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—81no—
247C2×D61—81no—
248C2×D61—81no—
249C2×D61—81no—
250C4×C61—24no—
251C4×C61—22no—
252C4×C61—42no—
253C4×C61—81no—
254C4×C61—82no—
255C4×C61#41614yes—
256C4×C61C4×#4111yescharacteristic
257C4×C61#52011yescharacteristic
258C4×C61#54611yescharacteristic
259C4×C61#54711yescharacteristic
260Dic61—42no—
261Dic61—44no—
262C2×#555—611no—
263C2×C8×D4—611no—
264#2354D6111yescharacteristic, complemented
265#1990—611no—
266#1987—611no—
267#1893—612no—
268#1875—612no—
269#1825—612no—
270#1802—611no—
271C4×#102—611no—
272#1735—612no—
273C2×C61—44no—
274C2×C61#233911yescharacteristic
275C2×C61C2×#41611yescharacteristic
276C2×C61#289711yescharacteristic
277C2×C61#330311yescharacteristic
278C2×C61#330511yescharacteristic
279C2×C61#334011yescharacteristic
280C2×C61#334111yescharacteristic
281D61—82no—
282C23×C8—612no—
283#555—618no—
284#555—1224no—
285C2×C4⋊Q8—611no—
286C2×#124—1222no—
287C2×#121—612no—
288C2×#121—1221no—
289C2×C4×D4—611no—
290C2×C4×D4C2×D6111yescharacteristic
291C2×#117—612no—
292C2×#117—1221no—
293C2×#117C2×D6112yes—
294C8×D4—1224no—
295C2×C4⋊C8—611no—
296C2×#102—611no—
297C2×#102—611no—
298C2×#102—612no—
299C2×C22⋊C8—612no—
300C2×C4×C8—611no—
301#390—21no—
302#388—22no—
303#384C2×D6111yescharacteristic
304#380C2×D6112yes—
305C4×C4⋊C4—21no—
306C4×C4⋊C4C2×D6111yescharacteristic
307#340—612no—
308#340—611no—
309#340—612no—
310#340—612no—
311#340—612no—
312#340—612no—
313#340—611no—
314C61#2248911yescharacteristic, complemented, Sylow
315C22×Q8—1221no—
316C22×D4—611no—
317C22×D4—1222no—
318C22×D4C22⋊D6111yescharacteristic
319C23×C4—1221no—
320C23×C4C22×D6112yes—
321C22×C8—611no—
322C22×C8—612no—
323C22×C8—611no—
324C22×C8—1224no—
325C22×C8—1224no—
326C22×C8—2442no—
327C4⋊Q8—614no—
328C4⋊Q8—1222no—
329#124—12216no—
330#121—618no—
331#121—1228no—
332#121—1224no—
333C4×D4—614no—
334C4×D4—1222no—
335C4×D4C4×D6114yes—
336C4×D4D24414yes—
337#117—24no—
338#117—42no—
339#117—618no—
340#117—1224no—
341#117—1228no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—611no—
345C2×C4⋊C4—612no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4C22⋊D6111yescharacteristic
348C2×C4⋊C4C22×D6112yes—
349C2×C4⋊C4C22×D6111yescharacteristic
350C2×C4⋊C4C22×D6111yescharacteristic
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—1222no—
354C2×C22⋊C4—1222no—
355C2×C22⋊C4—1221no—
356C2×C22⋊C4C22⋊D6112yes—
357C2×C42—22no—
358C2×C42—21no—
359C2×C42—1222no—
360C2×C42C22×D6111yescharacteristic
361C4⋊C8—1222no—
362#102—1222no—
363#102—1222no—
364#102—2442no—
365C22⋊C8—1224no—
366C4×C8—1222no—
367#95—22no—
368#95—21no—
369C24—1221no—
370C24C2×C22⋊D6112yes—
371C2×Q8—1224no—
372C2×Q8—4881no—
373C2×Q8—4881no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—612no—
377C2×D4—1228no—
378C2×D4—1221no—
379C2×D4—1221no—
380C2×D4—1221no—
381C2×D4—1221no—
382C2×D4—1221no—
383C2×D4—4882no—
384C2×D4—4882no—
385C2×D4#1175819814yes—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—1221no—
394C22×C4—1222no—
395C22×C4—1224no—
396C22×C4—2442no—
397C22×C4—2442no—
398C22×C4C2×C4×D6114yes—
399C22×C4C2×D24414yes—
400C22×C4C2×C22⋊D6112yes—
401C22×C4C2×C22⋊D6111yescharacteristic
402C22×C4C2×C22⋊D6111yescharacteristic
403C22×C4C23×D6111yescharacteristic
404C2×C8—612no—
405C2×C8—1221no—
406C2×C8—1221no—
407C2×C8—1221no—
408C2×C8—1222no—
409C2×C8—1222no—
410C2×C8—1221no—
411C2×C8—1222no—
412C2×C8—1221no—
413C2×C8—2448no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—21no—
419C4⋊C4—42no—
420C4⋊C4—81no—
421C4⋊C4—612no—
422C4⋊C4—614no—
423C4⋊C4—1221no—
424C4⋊C4—1222no—
425C4⋊C4—1224no—
426C4⋊C4#1175819814yes—
427C4⋊C4#1175822412yes—
428C22⋊C4—44no—
429C22⋊C4—82no—
430C22⋊C4—1228no—
431C22⋊C4—1228no—
432C22⋊C4—1224no—
433C22⋊C4#1175819818yes—
434C42—44no—
435C42—42no—
436C42—1228no—
437C42C2×C4×D6112yes—
438C42C2×D24412yes—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—1222no—
446C23—2441no—
447C23—2441no—
448C23—2441no—
449C23—2441no—
450C23—2441no—
451C23—2441no—
452C23C2×#1175819814yes—
453C23C22×C22⋊D6111yescharacteristic
454Q8—4884no—
455D4—28no—
456D4—1222no—
457D4—1222no—
458D4—4888no—
459D4—2441no—
460D4—2441no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—1222no—
479C2×C4—1224no—
480C2×C4—2441no—
481C2×C4—2441no—
482C2×C4—2442no—
483C2×C4—2442no—
484C2×C4—2448no—
485C2×C4C2×#1175819814yes—
486C2×C4C2×#1175819812yes—
487C2×C4C2×#1175819812yes—
488C2×C4#1451979712yes—
489C2×C4#1451979712yes—
490C2×C4C22×C4×D6111yescharacteristic
491C2×C4C22×D24411yescharacteristic
492C2×C4#41×D6111yescharacteristic
493C2×C4C2×#1175822411yescharacteristic
494C2×C4#1451982211yescharacteristic
495C2×C4#1451982411yescharacteristic
496C8—1222no—
497C8—2442no—
498C8—2441no—
499C22—24no—
500C22—24no—
501C22—24no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—4881no—
507C22—4881no—
508C22—4881no—
509C22—4881no—
510C22—2442no—
511C22—2442no—
512C22—2442no—
513C22C22×#1175819811yescharacteristic
514C22#1684040611yescharacteristic
515C22#1684041111yescharacteristic
516C22C2×#1451979711yescharacteristic
517C22C2×#1451979711yescharacteristic
518C22#1684056511yescharacteristic
519C22#1684057011yescharacteristic
520C4—22no—
521C4—24no—
522C4—42no—
523C4—81no—
524C4—82no—
525C4—2444no—
526C4—2442no—
527C4#1683975614yes—
528C4C4×#1175822411yescharacteristic
529C4#1684024111yescharacteristic
530C4#1684040711yescharacteristic
531C4#1684041611yescharacteristic
532C2—44no—
533C2—4882no—
534C2#2144864411yescharacteristic
535C2C2×#1683975611yescharacteristic
536C2#2145434211yescharacteristic
537C2#2145774711yescharacteristic
538C2#2145776911yescharacteristic
539C2#2145822611yescharacteristic
540C2#2145823711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2354C2
socleC23×C61C22×D4
prime radicalC2×#16840477C2
cosocleC22×C4×C61C24
hypercenter#2354D61
hypocenterC61#22489
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. C2×D4×#92418C2semidirect
3. #21446779C2semidirect
4. #21446767C2semidirect
5. #21446245C2semidirect
6. #21446141C2semidirect
7. #21445911C2semidirect
8. #21445816C2semidirect
9. C4×#14519697C2semidirect
10. #21445535C2semidirect
11. C61×#2354C2semidirect
14. #16840477C4semidirect
264. #2354D61semidirect
314. C61#22489semidirect

Maximal subgroups

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

subgroupindex
C2×#168404772
C2×D4×#924182
#214467792
#214467672
#214462452
#214461412
#214459112
#214458162
C4×#145196972
#214455352
C61×#23542
#2248961