Finite Group Databasebeta

← #29662456

Subgroups — #29662456

Order 15616.

618 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456403C211yesmaximal, characteristic, complemented
2#21455602C212yesmaximal
3C8×#11758226C211yesmaximal, characteristic
4#21445275C212yesmaximal, complemented
5#21445269C212yesmaximal, complemented
6#21445223C211yesmaximal, characteristic, complemented
7#21445219C211yesmaximal, characteristic, complemented
8#21445181C211yesmaximal, characteristic, complemented
9#335×D61C211yesmaximal, characteristic, complemented
10#21444475C212yesmaximal, complemented
11C61×#1670C211yesmaximal, characteristic, complemented
12#16840317C2212yes—
13C4×#11758226C2211yescharacteristic
14#16840167C2211yescharacteristic
15#16840112C2212yes—
16#16840067—22no—
17#16840061—22no—
18#16840057C2212yes—
19#16840053C2212yes—
20#16840039C2212yes—
21#16840000—21no—
22#16840000—21no—
23Q8×#92418C2211yescharacteristic
24Q16×Dic61—21no—
25Q16×Dic61—21no—
26C8×D244—21no—
27C8×D244C2211yescharacteristic
28#16839827—21no—
29#16839827C2211yescharacteristic
30#16839814—21no—
31#16839814C2211yescharacteristic
32#16839797—22no—
33C4⋊C8×D61C2212yes—
34#16839757—22no—
35C4×C8×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839515C2211yescharacteristic
38#16839514C2212yes—
39#16839510C2212yes—
40#16839510C2212yes—
41#16839508C2212yes—
42C61×#494C2211yescharacteristic
43C61×#475C2212yes—
44C8×Q8×C61C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#332C2212yes—
47C2×#11758226D411yescharacteristic
48C2×C8×D61—22no—
49C2×C8×D61—21no—
50C2×C8×D61—21no—
51C2×C8×D61D411yescharacteristic
52C2×C8×D61D411yescharacteristic
53#14519777C2312yes—
54#14519768—22no—
55#14519757—22no—
56#14519753—22no—
57Q8×Dic61—22no—
58Q8×Dic61D411yescharacteristic
59#14519742—22no—
60C4×D244—21no—
61C4×D244C2311yescharacteristic
62#14519739—21no—
63#14519739D411yescharacteristic
64#14519735—21no—
65#14519735—22no—
66#14519735D411yescharacteristic
67C42×D61—21no—
68C42×D61C2311yescharacteristic
69#14519711—21no—
70#14519711—21no—
71#14519710—22no—
72#14519710—22no—
73#14519709—22no—
74C4⋊#92418—21no—
75C4⋊#92418C2311yescharacteristic
76C4⋊#92418C2312yes—
77#14519706—22no—
78#14519701—22no—
79#14519701—22no—
80#14519701—22no—
81#14519699—22no—
82#14519697—22no—
83#14519689—21no—
84#14519689—21no—
85#14519689—21no—
86C4×#92418—21no—
87C4×#92418C2311yescharacteristic
88C8×Dic61—21no—
89C8×Dic61D411yescharacteristic
90C8×Dic61D411yescharacteristic
91C2×Q16×C61—21no—
92C2×Q16×C61—21no—
93C4⋊Q8×C61C2312yes—
94C4×Q8×C61C2311yescharacteristic
95#106×C61—22no—
96C4⋊C8×C61—21no—
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2312yes—
99#103×C61—22no—
100C4×C8×C61—21no—
101C4×C8×C61C2311yescharacteristic
102C2×Q8×C61—22no—
103C2×Q8×C61C2×D411yescharacteristic
104#11758226—22no—
105#11758226Q1614yescomplemented
106C2×D244—21no—
107C2×D244C2×D411yescharacteristic
108C2×C4×D61—21no—
109C2×C4×D61—21no—
110C2×C4×D61—21no—
111C2×C4×D61C2×D411yescharacteristic
112C2×C4×D61C2×D411yescharacteristic
113C2×C4×D61#4111yescharacteristic
114C2×Dic122—22no—
115Q16×C61—42no—
116Q16×C61—42no—
117C2×C8×C61—21no—
118C2×C8×C61—22no—
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D411yescharacteristic
121C4⋊C4×C61—21no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61C2×D411yescharacteristic
125C42×C61—21no—
126C42×C61C2411yescharacteristic
127#11758198—21no—
128#11758198—21no—
129#11758198—21no—
130#11758198—42no—
131C4⋊Dic61—21no—
132C4⋊Dic61—22no—
133C4⋊Dic61—22no—
134C4⋊Dic61—42no—
135C4⋊Dic61C2×D411yescharacteristic
136#11758196—42no—
137C4×Dic61—21no—
138C4×Dic61—21no—
139C4×Dic61—42no—
140C4×Dic61C2×D411yescharacteristic
141C4×Dic61C2×D411yescharacteristic
142C4×Dic61#4111yescharacteristic
143C2×#92418—21no—
144C2×#92418—22no—
145C2×#92418C2×D411yescharacteristic
146C2×#92418C2×D411yescharacteristic
147C8×D61—22no—
148C8×D61—41no—
149C8×D61—42no—
150C8×D61—44no—
151C8×D61—42no—
152C22×D61—21no—
153C22×D61#12111yescharacteristic
154Q8×C61—21no—
155Q8×C61—42no—
156Q8×C61—42no—
157Q8×C61C2×Q1612yes—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—42no—
161C2×C4×C61C22×D411yescharacteristic
162C2×C4×C61C22×D411yescharacteristic
163C2×C4×C61C2×#4111yescharacteristic
164C2×Dic61—21no—
165C2×Dic61—42no—
166C2×Dic61#12111yescharacteristic
167C2×Dic61#12111yescharacteristic
168C2×Dic61#12111yescharacteristic
169D244—24no—
170D244—21no—
171D244C2×Q1612yes—
172C4×D61—21no—
173C4×D61—21no—
174C4×D61—24no—
175C4×D61—41no—
176C4×D61—42no—
177C4×D61—41no—
178C4×D61—41no—
179C4×D61C2×Q1612yes—
180C4×D61#13512yes—
181Dic122—82no—
182C8×C61—21no—
183C8×C61—42no—
184C8×C61—41no—
185C8×C61#13512yes—
186#92418—21no—
187#92418—21no—
188#92418—42no—
189#92418—41no—
190#19126—611nomaximal, Sylow
191C22×C61C2×#12111yescharacteristic
192C2×D61—21no—
193C2×D61—21no—
194C2×D61—22no—
195C2×D61—41no—
196C2×D61—41no—
197C2×D61#46712yes—
198C4×C61—22no—
199C4×C61—21no—
200C4×C61—41no—
201C4×C61—82no—
202C4×C61C22×Q1611yescharacteristic
203C4×C61C2×#13511yescharacteristic
204C4×C61C2×#13511yescharacteristic
205C4×C61#57511yescharacteristic
206Dic61—21no—
207Dic61—21no—
208Dic61—41no—
209Dic61—82no—
210Dic61#46712yes—
211#3170—611no—
212#3076—612no—
213C8×#41—611no—
214#1684—612no—
215#1683—612no—
216#1676—611no—
217#1675—611no—
218#1670D6111yescharacteristic, complemented
219#1669—611no—
220C2×#335—611no—
221#1542—612no—
222C2×C61#303511yescharacteristic
223C2×C61C2×#46711yescharacteristic
224C2×C61#306711yescharacteristic
225D61—22no—
226D61—42no—
227#533—612no—
228C4×#41—611no—
229#500—611no—
230#494C2×D6111yescharacteristic
231#487—612no—
232#479—1222no—
233#478—1222no—
234#477—612no—
235#476—612no—
236#475C2×D6112yes—
237#474—612no—
238#467—1221no—
239#467—1221no—
240C8×Q8—611no—
241C8×Q8C2×D6111yescharacteristic
242C4×Q16—1221no—
243C4×Q16—1221no—
244C8×D4—611no—
245C8×D4—611no—
246C8×D4—1221no—
247C8×D4—1221no—
248#431—611no—
249#431—1221no—
250#427—1222no—
251C2×C4⋊C8—612no—
252#416—1222no—
253C2×C4×C8—611no—
254C2×C4×C8—1221no—
255#335—612no—
256#335—611no—
257#335C2×D6111yescharacteristic
258#333—612no—
259#333—612no—
260#332—612no—
261#332C2×D6112yes—
262C61#1912611yescharacteristic, complemented, Sylow
263C2×#41—611no—
264C2×Q16—21no—
265C2×Q16—21no—
266C22×C8—611no—
267C22×C8—611no—
268C22×C8—1221no—
269C22×C8—1222no—
270C22×C8—1221no—
271C4⋊Q8—612no—
272C4⋊Q8C22×D6112yes—
273#125—1222no—
274C22⋊Q8—1222no—
275C22⋊Q8—1222no—
276C4×Q8—611no—
277C4×Q8—1222no—
278C4×Q8—1222no—
279C4×Q8C22×D6111yescharacteristic
280C4×D4—611no—
281C4×D4—611no—
282C4×D4—1221no—
283C4×D4—1221no—
284#117—611no—
285#117—1221no—
286#117—1222no—
287C2×C42—611no—
288C2×C42—1221no—
289#107—1222no—
290#107—1221no—
291#107—1221no—
292#107—1222no—
293#106—22no—
294#106—1222no—
295C4⋊C8—21no—
296C4⋊C8—611no—
297C4⋊C8—612no—
298C4⋊C8—1221no—
299C4⋊C8—1222no—
300C4⋊C8C22×D6111yescharacteristic
301C4⋊C8C22×D6112yes—
302#103—22no—
303#103—1222no—
304#103—1222no—
305#103—1222no—
306#102—1222no—
307#102—1222no—
308C22⋊C8—1221no—
309C22⋊C8—1221no—
310C22⋊C8—1221no—
311C4×C8—21no—
312C4×C8—611no—
313C4×C8—611no—
314C4×C8—611no—
315C4×C8—1221no—
316C4×C8—1221no—
317C4×C8C22×D6111yescharacteristic
318#41—614no—
319#41—1222no—
320C2×Q8—22no—
321C2×Q8—1222no—
322C2×Q8D4×D6111yescharacteristic
323C2×D4—611no—
324C2×D4—1221no—
325C22×C4—611no—
326C22×C4—611no—
327C22×C4—611no—
328C22×C4—1221no—
329C22×C4—1221no—
330C22×C4—1221no—
331Q16—42no—
332Q16—42no—
333C2×C8—21no—
334C2×C8—22no—
335C2×C8—611no—
336C2×C8—611no—
337C2×C8—1222no—
338C2×C8—1221no—
339C2×C8—1222no—
340C2×C8—2441no—
341C2×C8—2442no—
342C2×C8—2444no—
343C2×C8—2442no—
344C2×C8D4×D6111yescharacteristic
345C2×C8D4×D6111yescharacteristic
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—611no—
350C4⋊C4—1221no—
351C4⋊C4—1222no—
352C4⋊C4—1222no—
353C4⋊C4—2442no—
354C4⋊C4—2442no—
355C4⋊C4D4×D6111yescharacteristic
356C22⋊C4—1221no—
357C22⋊C4—1221no—
358C22⋊C4—1221no—
359C22⋊C4—2442no—
360C42—21no—
361C42—611no—
362C42—611no—
363C42—611no—
364C42—1221no—
365C42—1221no—
366C42—2442no—
367C42C23×D6111yescharacteristic
368C23—611no—
369C23—1221no—
370Q8—21no—
371Q8—42no—
372Q8—42no—
373Q8—4882no—
374Q8Q16×D6112yes—
375D4—612no—
376D4—1224no—
377D4—1221no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—611no—
382C2×C4—612no—
383C2×C4—612no—
384C2×C4—611no—
385C2×C4—611no—
386C2×C4—1224no—
387C2×C4—1221no—
388C2×C4—1221no—
389C2×C4—1221no—
390C2×C4—2441no—
391C2×C4—2442no—
392C2×C4—2442no—
393C2×C4—2441no—
394C2×C4—2441no—
395C2×C4C2×D4×D6111yescharacteristic
396C2×C4C2×D4×D6111yescharacteristic
397C2×C4#41×D6111yescharacteristic
398C8—21no—
399C8—42no—
400C8—41no—
401C8—1221no—
402C8—1221no—
403C8—2442no—
404C8—2441no—
405C8#1451979812yes—
406C22—612no—
407C22—1221no—
408C22—1221no—
409C22—1222no—
410C22—2441no—
411C22—2441no—
412C22#121×D6111yescharacteristic
413C4—22no—
414C4—21no—
415C4—41no—
416C4—82no—
417C4—612no—
418C4—1221no—
419C4—1221no—
420C4—4882no—
421C4—2441no—
422C4C2×Q16×D6111yescharacteristic
423C4#135×D6111yescharacteristic
424C4C2×#1451979811yescharacteristic
425C4#1684055811yescharacteristic
426C2—1222no—
427C2—2442no—
428C2#2145526911yescharacteristic
429C2#467×D6111yescharacteristic
430C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1670C2
socleC22×C61C2×#121
prime radicalC2×#11758226D4
hypercenter#1670D61
hypocenterC61#19126
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1670

Complements

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

NH ≅ G/Ntype
1. #21456403C2semidirect
4. #21445275C2semidirect
5. #21445269C2semidirect
6. #21445223C2semidirect
7. #21445219C2semidirect
8. #21445181C2semidirect
9. #335×D61C2semidirect
10. #21444475C2semidirect
11. C61×#1670C2semidirect
105. #11758226Q16semidirect
218. #1670D61semidirect
262. C61#19126semidirect

Maximal subgroups

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

subgroupindex
#214564032
#214556022
C8×#117582262
#214452752
#214452692
#214452232
#214452192
#214451812
#335×D612
#214444752
C61×#16702
#1912661