Finite Group Databasebeta

← #30677793

Subgroups — #30677793

Order 16768.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22554000C211yesmaximal, characteristic, complemented
2#22553957C211yesmaximal, characteristic
3#22553669C211yesmaximal, characteristic, complemented
4#22553644C211yesmaximal, characteristic, complemented
5#22553639C211yesmaximal, characteristic, complemented
6#22553554C211yesmaximal, characteristic, complemented
7#22553542C211yesmaximal, characteristic, complemented
8#22553535C211yesmaximal, characteristic, complemented
9#22553475C211yesmaximal, characteristic, complemented
10#22553464C211yesmaximal, characteristic, complemented
11C4×#14523501C211yesmaximal, characteristic, complemented
12#22553377C211yesmaximal, characteristic, complemented
13#22553305C211yesmaximal, characteristic, complemented
14#102×D131C211yesmaximal, characteristic, complemented
15#478×C131C211yesmaximal, characteristic, complemented
16C2×#14523512C2211yescharacteristic
17C2×D4×D131C2211yescharacteristic
18C2×Dic524C2211yescharacteristic, complemented
19C2×#14523502C2211yescharacteristic
20C2×#14523501C2211yescharacteristic
21C2×C8×D131C2211yescharacteristic
22C2×C8×D131C2211yescharacteristic, complemented
23#16862241—21no—
24#16862237—21no—
25#16862234C2211yescharacteristic
26#16862231—21no—
27#16862230—21no—
28#16862226C2211yescharacteristic
29#16862226C2211yescharacteristic
30#16862226C2211yescharacteristic, complemented
31#16862221C2211yescharacteristic
32#16862220—21no—
33#16862216—21no—
34C4×Dic262C2211yescharacteristic, complemented
35C4×D524C2211yescharacteristic
36#16862204—21no—
37#16862204C2211yescharacteristic, complemented
38C4⋊C4×D131C2211yescharacteristic
39C2×(C2×D131)⋊C4—21no—
40#16862177C2211yescharacteristic, complemented
41#16862171C2211yescharacteristic, complemented
42#16862169C2211yescharacteristic, complemented
43#16862169C2211yescharacteristic, complemented
44#16862169C2211yescharacteristic, complemented
45#16862168C2211yescharacteristic
46#16862167C2211yescharacteristic
47#16862166C2211yescharacteristic
48#16862166C2211yescharacteristic
49(C2×D131)⋊C8C2211yescharacteristic
50(C2×D131)⋊C8C2211yescharacteristic, complemented
51C4×C131⋊C8C2211yescharacteristic, complemented
52#125×C131C2211yescharacteristic, complemented
53C4×D4×C131C2211yescharacteristic
54#107×C131C2211yescharacteristic, complemented
55#106×C131C2211yescharacteristic, complemented
56C4⋊C8×C131C2211yescharacteristic, complemented
57#102×C131C2211yescharacteristic
58#102×C131C2211yescharacteristic
59C23×D131—21no—
60#14523512—22no—
61#14523512—41no—
62C2×Dic262C2311yescharacteristic
63C2×Dic262C2311yescharacteristic
64C2×D524—21no—
65C2×D524D411yescharacteristic
66C2×D524C2311yescharacteristic
67C2×C22⋊D131—21no—
68C2×C22⋊D131—21no—
69D4×D131—22no—
70D4×D131—41no—
71C22×Dic131—21no—
72C2×C4×D131—21no—
73C2×C4×D131C2311yescharacteristic
74C2×C4×D131C2311yescharacteristic
75Dic524—41no—
76#14523502—41no—
77#14523501—22no—
78C2×C131⋊C8C2311yescharacteristic
79C2×C131⋊C8C2311yescharacteristic
80C8×D131—41no—
81C8×D131—41no—
82C4⋊Dic131—21no—
83C4⋊Dic131—21no—
84C4⋊Dic131—21no—
85C4⋊Dic131D411yescharacteristic
86C4⋊Dic131C2311yescharacteristic
87Dic131⋊C4—21no—
88Dic131⋊C4—21no—
89(C2×D131)⋊C4—21no—
90(C2×D131)⋊C4—21no—
91(C2×D131)⋊C4—21no—
92(C2×D131)⋊C4—21no—
93(C2×D131)⋊C4—21no—
94(C2×D131)⋊C4—21no—
95(C2×D131)⋊C4—21no—
96(C2×D131)⋊C4D411yescharacteristic
97(C2×D131)⋊C4D411yescharacteristic
98C4×Dic131—21no—
99C4×Dic131—21no—
100C2×D4×C131C2311yescharacteristic
101C22×C4×C131—21no—
102C2×C8×C131C2311yescharacteristic
103C2×C8×C131C2311yescharacteristic
104C4⋊C4×C131—21no—
105C4⋊C4×C131—21no—
106C4⋊C4×C131C2311yescharacteristic
107C4⋊C4×C131C2311yescharacteristic
108C4⋊C4×C131C2311yescharacteristic
109C22⋊C4×C131—21no—
110C42×C131C2311yescharacteristic
111C22×D131—21no—
112C22×D131—42no—
113C22×D131—42no—
114C22×D131—81no—
115C22×D131C2×D411yescharacteristic
116C22×D131C2×D411yescharacteristic
117Dic262—21no—
118Dic262—21no—
119Dic262—21no—
120Dic262—21no—
121D524—22no—
122D524—21no—
123D524—21no—
124D524—81no—
125C22⋊D131—42no—
126C22⋊D131—42no—
127C2×Dic131—21no—
128C2×Dic131—21no—
129C2×Dic131—21no—
130C2×Dic131—42no—
131C2×Dic131C2×D411yescharacteristic
132C2×Dic131C2×D411yescharacteristic
133C4×D131—21no—
134C4×D131—21no—
135C4×D131—21no—
136C4×D131—21no—
137C4×D131—42no—
138C131⋊C8—21no—
139C131⋊C8—21no—
140C23×C131—21no—
141D4×C131—21no—
142D4×C131#4112yes—
143C2×C4×C131—21no—
144C2×C4×C131—21no—
145C2×C4×C131—21no—
146C2×C4×C131—81no—
147C2×C4×C131C2×D411yescharacteristic
148C2×C4×C131C2×D411yescharacteristic
149C2×C4×C131C2411yescharacteristic
150C8×C131—21no—
151C8×C131—21no—
152C22×C131—22no—
153C22×C131—41no—
154C22×C131—41no—
155C22×C131C22×D411yescharacteristic
156C2×D131—21no—
157C2×D131—21no—
158C2×D131—21no—
159C2×D131—21no—
160C2×D131—21no—
161C2×D131—41no—
162C2×D131—41no—
163C2×D131—41no—
164C2×D131—84no—
165C2×D131#13512yes—
166C4×C131—21no—
167C4×C131—21no—
168C4×C131—41no—
169C4×C131—41no—
170C4×C131—41no—
171C4×C131C2×#4111yescharacteristic
172C4×C131H(5,𝔽2)11yescharacteristic
173Dic131—21no—
174Dic131—21no—
175Dic131—41no—
176Dic131—41no—
177Dic131—41no—
178C2×C131—42no—
179C2×C131#54611yescharacteristic
180C2×C131C2×#13511yescharacteristic
181C2×C131#57711yescharacteristic
182D131—22no—
183D131—41no—
184D131—81no—
185C131#329611yescharacteristic, complemented, Sylow
186#3296—1311nomaximal, Sylow
187#552—1311no—
188#546—1311no—
189#488—1311no—
190#483—1311no—
191#482—1311no—
192#478D13111yescharacteristic, complemented
193#467—1311no—
194#465—1311no—
195#464—1311no—
196#452—1311no—
197#450—1311no—
198C4×SD16—1311no—
199C8×D4—1311no—
200#416—1311no—
201C2×#102—1311no—
202C2×#41—1311no—
203C22×D4—1311no—
204C2×Q16—1311no—
205C2×SD16—1311no—
206C2×SD16—1311no—
207C22×C8—1311no—
208C22×C8—1311no—
209#126—2621no—
210#125—2621no—
211#125C2×D13111yescharacteristic, complemented
212#124—1311no—
213#123—2621no—
214#123—2621no—
215C22⋊Q8—1311no—
216C22⋊Q8—1311no—
217C22⋊Q8—1311no—
218#121—2621no—
219#121—1311no—
220#120—2621no—
221C4×Q8—1311no—
222C4×D4—1311no—
223C4×D4C2×D13111yescharacteristic
224#117—2621no—
225#117—1311no—
226C2×C4⋊C4—1311no—
227C2×C22⋊C4—2621no—
228#107C2×D13111yescharacteristic, complemented
229#106—1311no—
230#106C2×D13111yescharacteristic, complemented
231C4⋊C8C2×D13111yescharacteristic, complemented
232#103—1311no—
233#103—1311no—
234#103—1311no—
235#103—1311no—
236#102—1311no—
237#102—1311no—
238#102—1311no—
239#102—1311no—
240#102C2×D13111yescharacteristic
241#102C2×D13111yescharacteristic
242C22⋊C8—1311no—
243C22⋊C8—1311no—
244C4×C8—1311no—
245C24—2621no—
246#41—2622no—
247#41—5241no—
248C2×Q8—1311no—
249C2×Q8—1311no—
250C2×D4—2621no—
251C2×D4—2622no—
252C2×D4—2621no—
253C2×D4—2621no—
254C2×D4—5241no—
255C2×D4—1311no—
256C2×D4—1311no—
257C2×D4C22×D13111yescharacteristic
258C22×C4—21no—
259C22×C4—2621no—
260C22×C4—2621no—
261C22×C4—1311no—
262C22×C4—1311no—
263Q16—5241no—
264SD16—2622no—
265SD16—5241no—
266C2×C8—5241no—
267C2×C8—5241no—
268C2×C8—1311no—
269C2×C8—1311no—
270C2×C8C22×D13111yescharacteristic
271C2×C8C22×D13111yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—2621no—
275C4⋊C4—2621no—
276C4⋊C4—2621no—
277C4⋊C4—2621no—
278C4⋊C4—2621no—
279C4⋊C4—1311no—
280C4⋊C4—1311no—
281C4⋊C4C22×D13111yescharacteristic
282C4⋊C4C22×D13111yescharacteristic
283C4⋊C4C22×D13111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—2621no—
286C22⋊C4—2621no—
287C22⋊C4—2621no—
288C22⋊C4—2621no—
289C22⋊C4—2621no—
290C22⋊C4—2621no—
291C22⋊C4—2621no—
292C22⋊C4—1311no—
293C22⋊C4—1311no—
294C42—2621no—
295C42—2621no—
296C42C22×D13111yescharacteristic
297C23—21no—
298C23—2621no—
299C23—5242no—
300C23—5242no—
301C23—10481no—
302C23—1311no—
303C23—1311no—
304Q8—2621no—
305Q8—2621no—
306Q8—2621no—
307Q8—2621no—
308D4—21no—
309D4—2622no—
310D4—2621no—
311D4—2621no—
312D4—5242no—
313D4—5242no—
314D4—10481no—
315D4#1452351412yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—2621no—
320C2×C4—2621no—
321C2×C4—2621no—
322C2×C4—2621no—
323C2×C4—2621no—
324C2×C4—2621no—
325C2×C4—2621no—
326C2×C4—81no—
327C2×C4—5242no—
328C2×C4—5242no—
329C2×C4—1311no—
330C2×C4—1311no—
331C2×C4D4×D13111yescharacteristic
332C2×C4D4×D13111yescharacteristic
333C2×C4C23×D13111yescharacteristic
334C8—21no—
335C8—21no—
336C8—2621no—
337C8—2621no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—2621no—
342C22—2621no—
343C22—2621no—
344C22—2621no—
345C22—2621no—
346C22—5241no—
347C22—5241no—
348C22—5241no—
349C22—10484no—
350C22—1312no—
351C22C2×D4×D13111yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—2621no—
358C4—2621no—
359C4—5241no—
360C4—5241no—
361C4—5241no—
362C4C2×#1452351411yescharacteristic
363C4#1686229211yescharacteristic
364C2—42no—
365C2—2622no—
366C2—5241no—
367C2—10481no—
368C2#2255395311yescharacteristic
369C2#135×D13111yescharacteristic
370C2#2255412211yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#478×C131C2
socleC22×C131C22×D4
prime radical#22553957C2
hypercenter#478D131
hypocenterC131#3296
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #22554000C2semidirect
3. #22553669C2semidirect
4. #22553644C2semidirect
5. #22553639C2semidirect
6. #22553554C2semidirect
7. #22553542C2semidirect
8. #22553535C2semidirect
9. #22553475C2semidirect
10. #22553464C2semidirect
11. C4×#14523501C2semidirect
12. #22553377C2semidirect
13. #22553305C2semidirect
14. #102×D131C2semidirect
15. #478×C131C2semidirect
18. C2×Dic524C22semidirect
22. C2×C8×D131C22semidirect
30. #16862226C22semidirect
34. C4×Dic262C22semidirect
37. #16862204C22semidirect
40. #16862177C22semidirect
41. #16862171C22semidirect
42. #16862169C22semidirect
43. #16862169C22semidirect
44. #16862169C22semidirect
50. (C2×D131)⋊C8C22semidirect
51. C4×C131⋊C8C22semidirect
52. #125×C131C22semidirect
54. #107×C131C22semidirect
55. #106×C131C22semidirect
56. C4⋊C8×C131C22semidirect
185. C131#3296semidirect
192. #478D131semidirect
211. #125C2×D131semidirect
228. #107C2×D131semidirect
230. #106C2×D131semidirect
231. C4⋊C8C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225540002
#225539572
#225536692
#225536442
#225536392
#225535542
#225535422
#225535352
#225534752
#225534642
C4×#145235012
#225533772
#225533052
#102×D1312
#478×C1312
#3296131