Finite Group Databasebeta

← #30677795

Subgroups — #30677795

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#22553955C211yesmaximal, characteristic
3#22553641C211yesmaximal, characteristic, complemented
4#22553635C211yesmaximal, characteristic, complemented
5#22553609C211yesmaximal, characteristic, complemented
6#22553552C211yesmaximal, characteristic, complemented
7#22553540C211yesmaximal, characteristic, complemented
8#22553531C211yesmaximal, characteristic, complemented
9#22553477C211yesmaximal, characteristic, complemented
10#22553464C211yesmaximal, characteristic, complemented
11C4×#14523502C211yesmaximal, characteristic, complemented
12C8×D524C211yesmaximal, characteristic, complemented
13#22553307C211yesmaximal, characteristic, complemented
14#102×D131C211yesmaximal, characteristic, complemented
15#488×C131C211yesmaximal, characteristic, complemented
16C2×#14523514C2211yescharacteristic
17C2×D4×D131C2211yescharacteristic
18C2×#14523503C2211yescharacteristic, complemented
19C2×#14523502C2211yescharacteristic
20C2×#14523501C2211yescharacteristic
21C2×C8×D131C2211yescharacteristic
22C2×C8×D131C2211yescharacteristic, complemented
23#16862241—21no—
24#16862237—21no—
25#16862231—21no—
26#16862228—21no—
27#16862226C2211yescharacteristic
28#16862226C2211yescharacteristic, complemented
29#16862222C2211yescharacteristic
30#16862219—21no—
31#16862217C2211yescharacteristic
32#16862216—21no—
33C4×Dic262C2211yescharacteristic, complemented
34C4×D524C2211yescharacteristic
35C4×D524C2211yescharacteristic
36#16862204—21no—
37#16862204C2211yescharacteristic, complemented
38C4⋊C4×D131C2211yescharacteristic
39C22⋊C4×D131—21no—
40#16862179C2211yescharacteristic, complemented
41#16862178C2211yescharacteristic, complemented
42#16862177C2211yescharacteristic, complemented
43C4⋊(C131⋊C8)C2211yescharacteristic, complemented
44#16862171C2211yescharacteristic, complemented
45#16862170C2211yescharacteristic, complemented
46#16862168C2211yescharacteristic
47#16862167C2211yescharacteristic
48#16862166C2211yescharacteristic
49#16862166C2211yescharacteristic
50(C2×D131)⋊C8C2211yescharacteristic
51(C2×D131)⋊C8C2211yescharacteristic, complemented
52#125×C131C2211yescharacteristic, complemented
53#124×C131C2211yescharacteristic
54#103×C131C2211yescharacteristic, complemented
55#103×C131C2211yescharacteristic, complemented
56#102×C131C2211yescharacteristic
57#102×C131C2211yescharacteristic
58C4×C8×C131C2211yescharacteristic, complemented
59C23×D131—21no—
60#14523514—22no—
61#14523514—41no—
62C2×Dic262C2311yescharacteristic
63C2×D524—21no—
64C2×D524D411yescharacteristic
65C2×D524C2311yescharacteristic
66C2×C22⋊D131—21no—
67C2×C22⋊D131—21no—
68D4×D131—22no—
69D4×D131—41no—
70C2×C4×D131—21no—
71C2×C4×D131—21no—
72C2×C4×D131—21no—
73C2×C4×D131C2311yescharacteristic
74C2×C4×D131C2311yescharacteristic
75#14523503—41no—
76#14523502—22no—
77#14523501—41no—
78C2×C131⋊C8C2311yescharacteristic
79C2×C131⋊C8C2311yescharacteristic
80C8×D131—41no—
81C8×D131—41no—
82C4⋊Dic131—21no—
83C4⋊Dic131—21no—
84C4⋊Dic131D411yescharacteristic
85C4⋊Dic131C2311yescharacteristic
86C4⋊Dic131C2311yescharacteristic
87Dic131⋊C4—21no—
88Dic131⋊C4—21no—
89Dic131⋊C4—21no—
90C22⋊Dic131—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×Q8×C131C2311yescharacteristic
101C2×D4×C131C2311yescharacteristic
102C2×C8×C131C2311yescharacteristic
103C2×C8×C131C2311yescharacteristic
104C4⋊C4×C131—21no—
105C4⋊C4×C131—21no—
106C4⋊C4×C131C2311yescharacteristic
107C4⋊C4×C131C2311yescharacteristic
108C22⋊C4×C131—21no—
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—
119D524—22no—
120D524—21no—
121D524—42no—
122D524#4112yes—
123C22⋊D131—42no—
124C22⋊D131—81no—
125C2×Dic131—21no—
126C2×Dic131—21no—
127C2×Dic131—21no—
128C2×Dic131C2×D411yescharacteristic
129C2×Dic131C2×D411yescharacteristic
130C4×D131—21no—
131C4×D131—21no—
132C4×D131—21no—
133C4×D131—21no—
134C4×D131—42no—
135C4×D131—42no—
136C4×D131—81no—
137C131⋊C8—21no—
138C131⋊C8—21no—
139C23×C131—21no—
140Q8×C131—21no—
141Q8×C131—21no—
142D4×C131—21no—
143D4×C131—21no—
144C2×C4×C131—21no—
145C2×C4×C131—21no—
146C2×C4×C131—21no—
147C2×C4×C131C2×D411yescharacteristic
148C2×C4×C131C2×D411yescharacteristic
149C2×C4×C131C2411yescharacteristic
150C8×C131—21no—
151C8×C131—21no—
152C22×C131—41no—
153C22×C131—41no—
154C22×C131—41no—
155C22×C131C22×D411yescharacteristic
156C2×D131—21no—
157C2×D131—21no—
158C2×D131—21no—
159C2×D131—22no—
160C2×D131—21no—
161C2×D131—21no—
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—81no—
179C2×C131#54611yescharacteristic
180C2×C131C2×#13511yescharacteristic
181C2×C131#57711yescharacteristic
182D131—22no—
183D131—41no—
184D131—42no—
185C131#329611yescharacteristic, complemented, Sylow
186#3296—1311nomaximal, Sylow
187#552—1311no—
188#546—1311no—
189#488D13111yescharacteristic, complemented
190#483—1311no—
191#482—1311no—
192#478—1311no—
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#124C2×D13111yescharacteristic
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×D4—1311no—
224#117—2621no—
225#117—1311no—
226C2×C4⋊C4—1311no—
227C2×C22⋊C4—2621no—
228#107—1311no—
229#106—1311no—
230#106—1311no—
231C4⋊C8—1311no—
232#103—1311no—
233#103—1311no—
234#103C2×D13111yescharacteristic, complemented
235#103C2×D13111yescharacteristic, complemented
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×C8C2×D13111yescharacteristic, complemented
245C24—2621no—
246#41—2622no—
247#41—5241no—
248C2×Q8—1311no—
249C2×Q8C22×D13111yescharacteristic
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—2621no—
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⋊C4—1311no—
282C4⋊C4C22×D13111yescharacteristic
283C4⋊C4C22×D13111yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
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—21no—
305Q8—21no—
306Q8—2621no—
307Q8—2621no—
308D4—21no—
309D4—21no—
310D4—2622no—
311D4—2621no—
312D4—5242no—
313D4—5242no—
314D4—10481no—
315D4—1312no—
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—5242no—
327C2×C4—5242no—
328C2×C4—10481no—
329C2×C4—1311no—
330C2×C4—1311no—
331C2×C4D4×D13111yescharacteristic
332C2×C4D4×D13111yescharacteristic
333C2×C4C23×D13111yescharacteristic
334C8—21no—
335C8—21no—
336C8—2621no—
337C8—2621no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—2621no—
342C22—2621no—
343C22—2621no—
344C22—2621no—
345C22—2622no—
346C22—2621no—
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—
362C4#41×D13111yescharacteristic
363C4#1686229211yescharacteristic
364C2—2622no—
365C2—81no—
366C2—5241no—
367C2—5242no—
368C2#2255395511yescharacteristic
369C2#135×D13111yescharacteristic
370C2#2255411911yescharacteristic
subgroupG/H
centerC22C2×D4×D131
commutator, cosocleC2×C4×C131C24
FrattiniC2×C4C23×D131
Fitting#488×C131C2
socleC22×C131C22×D4
prime radical#22553955C2
hypercenter#488D131
hypocenterC131#3296
LCSG ▷ C2×C4×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #22554000C2semidirect
3. #22553641C2semidirect
4. #22553635C2semidirect
5. #22553609C2semidirect
6. #22553552C2semidirect
7. #22553540C2semidirect
8. #22553531C2semidirect
9. #22553477C2semidirect
10. #22553464C2semidirect
11. C4×#14523502C2semidirect
12. C8×D524C2semidirect
13. #22553307C2semidirect
14. #102×D131C2semidirect
15. #488×C131C2semidirect
18. C2×#14523503C22semidirect
22. C2×C8×D131C22semidirect
28. #16862226C22semidirect
33. C4×Dic262C22semidirect
37. #16862204C22semidirect
40. #16862179C22semidirect
41. #16862178C22semidirect
42. #16862177C22semidirect
43. C4⋊(C131⋊C8)C22semidirect
44. #16862171C22semidirect
45. #16862170C22semidirect
51. (C2×D131)⋊C8C22semidirect
52. #125×C131C22semidirect
54. #103×C131C22semidirect
55. #103×C131C22semidirect
58. C4×C8×C131C22semidirect
185. C131#3296semidirect
189. #488D131semidirect
211. #125C2×D131semidirect
234. #103C2×D131semidirect
235. #103C2×D131semidirect
244. C4×C8C2×D131semidirect

Maximal subgroups

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

subgroupindex
#225540002
#225539552
#225536412
#225536352
#225536092
#225535522
#225535402
#225535312
#225534772
#225534642
C4×#145235022
C8×D5242
#225533072
#102×D1312
#488×C1312
#3296131