Finite Group Databasebeta

← #30669271

Subgroups — #30669271

Order 16768.

1564 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 213 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊D131C211yesmaximal, characteristic, complemented
2C2×#16862215C212yesmaximal, complemented
3C2×#16862214C212yesmaximal, complemented
4#22553220C214yesmaximal, complemented
5#22553192C212yesmaximal, complemented
6#22553191C212yesmaximal, complemented
7#22553168C211yesmaximal, characteristic, complemented
8C23×D4×C131C211yesmaximal, characteristic, complemented
9C24×D131C2211yescharacteristic, complemented
10C22×C22⋊D131—28no—
11C22×C22⋊D131—24no—
12C22×C22⋊D131C2212yes—
13C22×C22⋊D131C2212yescomplemented
14C22×C22⋊D131C2214yescomplemented
15C23×Dic131C2211yescharacteristic, complemented
16#16862215—28no—
17#16862214—28no—
18C2×C22⋊Dic131C2212yescomplemented
19C2×C22⋊Dic131C2212yescomplemented
20C2×C22⋊Dic131C2214yescomplemented
21C2×C22⋊Dic131C2212yescomplemented
22C2×(C2×D131)⋊C4C2214yescomplemented
23C131⋊#95C2214yescomplemented
24C25×C131C2211yescharacteristic
25C25×C131C2211yescharacteristic, complemented
26C22×D4×C131—24no—
27C22×D4×C131—28no—
28C22×D4×C131C2212yes—
29C22×D4×C131C2212yescomplemented
30C23×C4×C131C2211yescharacteristic, complemented
31C23×D131—24no—
32C23×D131—22no—
33C23×D131—24no—
34C23×D131—42no—
35C23×D131C2312yes—
36C2×C22⋊D131—232no—
37C2×C22⋊D131—24no—
38C2×C22⋊D131—28no—
39C2×C22⋊D131—24no—
40C2×C22⋊D131—24no—
41C2×C22⋊D131—24no—
42C2×C22⋊D131—24no—
43C2×C22⋊D131—28no—
44C2×C22⋊D131—44no—
45C2×C22⋊D131D418yescomplemented
46C22×Dic131—24no—
47C22×Dic131—22no—
48C22×Dic131C2314yes—
49C22×Dic131C2312yes—
50C22⋊Dic131—28no—
51C22⋊Dic131—24no—
52C22⋊Dic131—24no—
53C22⋊Dic131—42no—
54(C2×D131)⋊C4—28no—
55C24×C131—24no—
56C24×C131—24no—
57C24×C131—22no—
58C24×C131—24no—
59C24×C131—24no—
60C24×C131—42no—
61C24×C131—42no—
62C24×C131D414yescomplemented
63C24×C131C2311yescharacteristic
64C24×C131C2312yes—
65C24×C131C2311yescharacteristic
66C24×C131C2311yescharacteristic
67C2×D4×C131—232no—
68C2×D4×C131—24no—
69C2×D4×C131—24no—
70C2×D4×C131—24no—
71C2×D4×C131—24no—
72C2×D4×C131—24no—
73C2×D4×C131D418yescomplemented
74C22×C4×C131—22no—
75C22×C4×C131—24no—
76C22×C4×C131C2312yes—
77C22×D131—28no—
78C22×D131—24no—
79C22×D131—22no—
80C22×D131—22no—
81C22×D131—24no—
82C22×D131—416no—
83C22×D131—44no—
84C22×D131—44no—
85C22×D131C2×D414yescomplemented
86C22⋊D131—216no—
87C22⋊D131—216no—
88C22⋊D131—416no—
89C2×Dic131—28no—
90C2×Dic131—22no—
91C2×Dic131—22no—
92C2×Dic131—24no—
93C2×Dic131—24no—
94C2×Dic131—24no—
95C2×Dic131C2×D414yescomplemented
96C23×C131—22no—
97C23×C131—22no—
98C23×C131—22no—
99C23×C131—22no—
100C23×C131—28no—
101C23×C131—28no—
102C23×C131—28no—
103C23×C131—22no—
104C23×C131—22no—
105C23×C131—22no—
106C23×C131—24no—
107C23×C131—22no—
108C23×C131—22no—
109C23×C131—24no—
110C23×C131—44no—
111C23×C131—416no—
112C23×C131—416no—
113C23×C131—44no—
114C23×C131—44no—
115C23×C131C2×D414yes—
116C23×C131C2×D412yes—
117C23×C131C2×D412yes—
118C23×C131C2×D412yes—
119C23×C131C2×D414yes—
120C23×C131C2×D414yescomplemented
121C23×C131C2411yescharacteristic
122D4×C131—216no—
123D4×C131—216no—
124C2×C4×C131—28no—
125C2×C4×C131—22no—
126C2×C4×C131—22no—
127C2×C4×C131C2×D414yescomplemented
128C22×C131—22no—
129C22×C131—22no—
130C22×C131—22no—
131C22×C131—22no—
132C22×C131—28no—
133C22×C131—28no—
134C22×C131—22no—
135C22×C131—24no—
136C22×C131—22no—
137C22×C131—22no—
138C22×C131—28no—
139C22×C131—416no—
140C22×C131—416no—
141C22×C131—42no—
142C22×C131—42no—
143C22×C131—44no—
144C22×C131—44no—
145C22×C131#12014yescomplemented
146C22×C131#12014yescomplemented
147C22×C131C22×D412yes—
148C22×C131C22×D411yescharacteristic
149C22×C131C22×D412yes—
150C22×C131H(5,𝔽2)12yes—
151C2×D131—24no—
152C2×D131—28no—
153C2×D131—416no—
154C2×D131—42no—
155C2×D131—42no—
156C2×D131—44no—
157C4×C131—24no—
158Dic131—24no—
159Dic131—44no—
160C2×C131—24no—
161C2×C131—24no—
162C2×C131—42no—
163C2×C131—44no—
164C2×C131C2×#12011yescharacteristic
165C2×C131C2×#12011yescharacteristic
166C2×C131#53411yescharacteristic
167C2×C131D4214yes—
168D131—44no—
169C131#240311yescharacteristic, complemented, Sylow
170#2403—1311nomaximal, Sylow
171C23×D4—1311no—
172C23×D4D13111yescharacteristic, complemented
173C2×#120—1312no—
174C2×#120—1312no—
175#392—1314no—
176#386—1312no—
177#386—1312no—
178#379—1311no—
179C25—1311no—
180C25C2×D13111yescharacteristic
181C25C2×D13111yescharacteristic, complemented
182C22×D4—24no—
183C22×D4—28no—
184C22×D4—2624no—
185C22×D4—2628no—
186C22×D4—1312no—
187C22×D4—1314no—
188C22×D4—1312no—
189C22×D4C2×D13112yes—
190C22×D4C2×D13112yescomplemented
191C23×C4—1311no—
192C23×C4C2×D13111yescharacteristic, complemented
193#120—2628no—
194#120—2628no—
195C2×C22⋊C4—1312no—
196C2×C22⋊C4—1312no—
197C2×C22⋊C4—1314no—
198C2×C22⋊C4—1314no—
199C2×C22⋊C4—1312no—
200#95—1314no—
201C24—24no—
202C24—24no—
203C24—22no—
204C24—24no—
205C24—24no—
206C24—42no—
207C24—42no—
208C24—2624no—
209C24—2622no—
210C24—2624no—
211C24—5242no—
212C24—1312no—
213C24C22⋊D13114yescomplemented
214C24C22×D13111yescharacteristic
215C24C22×D13112yes—
216C24C22×D13111yescharacteristic
217C24C22×D13111yescharacteristic
218C2×D4—232no—
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—26232no—
225C2×D4—2624no—
226C2×D4—2624no—
227C2×D4—2624no—
228C2×D4—2624no—
229C2×D4—2628no—
230C2×D4—2628no—
231C2×D4—2624no—
232C2×D4—5244no—
233C2×D4—1318no—
234C2×D4C22⋊D13118yescomplemented
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—2622no—
238C22×C4—2624no—
239C22×C4—1314no—
240C22×C4—1312no—
241C22×C4C22×D13112yes—
242C22⋊C4—2624no—
243C22⋊C4—2624no—
244C22⋊C4—2628no—
245C22⋊C4—2628no—
246C22⋊C4—5242no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—28no—
252C23—28no—
253C23—28no—
254C23—24no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—24no—
261C23—44no—
262C23—416no—
263C23—416no—
264C23—44no—
265C23—44no—
266C23—2628no—
267C23—2622no—
268C23—2624no—
269C23—2622no—
270C23—2624no—
271C23—52416no—
272C23—5244no—
273C23—5244no—
274C23—1314no—
275C23D4×D13114yes—
276C23C2×C22⋊D13112yes—
277C23C2×C22⋊D13112yes—
278C23C2×C22⋊D13114yes—
279C23C2×C22⋊D13112yes—
280C23C2×C22⋊D13114yescomplemented
281C23C23×D13111yescharacteristic
282D4—216no—
283D4—216no—
284D4—26216no—
285D4—26216no—
286D4—52416no—
287C2×C4—28no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—2628no—
291C2×C4—2622no—
292C2×C4—2622no—
293C2×C4—2624no—
294C2×C4—2624no—
295C2×C4—2624no—
296C2×C4—1314no—
297C2×C4C2×C22⋊D13114yescomplemented
298C22—22no—
299C22—22no—
300C22—28no—
301C22—28no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—28no—
306C22—22no—
307C22—22no—
308C22—24no—
309C22—416no—
310C22—416no—
311C22—42no—
312C22—44no—
313C22—44no—
314C22—42no—
315C22—2628no—
316C22—2624no—
317C22—52416no—
318C22—5242no—
319C22—5244no—
320C22—5242no—
321C22#1686221414yescomplemented
322C22#1686221514yescomplemented
323C22C2×D4×D13112yes—
324C22C22×C22⋊D13111yescharacteristic
325C22C22×C22⋊D13112yes—
326C22#1686229112yes—
327C4—24no—
328C4—2624no—
329C4—5244no—
330C2—24no—
331C2—24no—
332C2—42no—
333C2—44no—
334C2—5244no—
335C2C2×#1686221411yescharacteristic
336C2C2×#1686221511yescharacteristic
337C2#2255387111yescharacteristic
338C2D4×C22⋊D13114yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
FittingC23×D4×C131C2
hypercenterC23×D4D131
hypocenterC131#2403
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D131C2semidirect
2. C2×#16862215C2semidirect
3. C2×#16862214C2semidirect
4. #22553220C2semidirect
5. #22553192C2semidirect
6. #22553191C2semidirect
7. #22553168C2semidirect
8. C23×D4×C131C2semidirect
9. C24×D131C22semidirect
13. C22×C22⋊D131C22semidirect
14. C22×C22⋊D131C22semidirect
15. C23×Dic131C22semidirect
18. C2×C22⋊Dic131C22semidirect
19. C2×C22⋊Dic131C22semidirect
20. C2×C22⋊Dic131C22semidirect
21. C2×C22⋊Dic131C22semidirect
22. C2×(C2×D131)⋊C4C22semidirect
23. C131⋊#95C22semidirect
25. C25×C131C22semidirect
29. C22×D4×C131C22semidirect
30. C23×C4×C131C22semidirect
45. C2×C22⋊D131D4semidirect
62. C24×C131D4semidirect
73. C2×D4×C131D4semidirect
85. C22×D131C2×D4semidirect
95. C2×Dic131C2×D4semidirect
120. C23×C131C2×D4semidirect
127. C2×C4×C131C2×D4semidirect
145. C22×C131#120semidirect
146. C22×C131#120semidirect
169. C131#2403semidirect
172. C23×D4D131semidirect
181. C25C2×D131semidirect
190. C22×D4C2×D131semidirect
192. C23×C4C2×D131semidirect
213. C24C22⋊D131semidirect
234. C2×D4C22⋊D131semidirect
280. C23C2×C22⋊D131semidirect
297. C2×C4C2×C22⋊D131semidirect
321. C22#16862214semidirect
322. C22#16862215semidirect

Maximal subgroups

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

subgroupindex
C23×C22⋊D1312
C2×#168622152
C2×#168622142
#225532202
#225531922
#225531912
#225531682
C23×D4×C1312
#2403131