Finite Group Databasebeta

← #30669359

Subgroups — #30669359

Order 16768.

814 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 814 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C2×#16862229C211yesmaximal, characteristic, complemented
2C2×#16862216C211yesmaximal, characteristic
3C22×(C2×D131)⋊C4C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D131C211yesmaximal, characteristic, complemented
5#22553241C212yesmaximal, complemented
6#22553230C212yesmaximal, complemented
7#22553190C212yesmaximal, complemented
8#22553189C211yesmaximal, characteristic, complemented
9#22553184C211yesmaximal, characteristic, complemented
10#22553171C212yesmaximal, complemented
11#381×C131C211yesmaximal, characteristic, complemented
12C24×D131C2211yescharacteristic
13C22×D524C2211yescharacteristic
14C22×D524C2211yescharacteristic
15C22×C22⋊D131C2211yescharacteristic
16C23×Dic131C2211yescharacteristic
17C22×C4×D131C2212yes—
18#16862229—24no—
19#16862216—24no—
20C2×Dic131⋊C4C2212yescomplemented
21C2×Dic131⋊C4C2212yescomplemented
22C2×C22⋊Dic131C2211yescharacteristic
23C2×(C2×D131)⋊C4—24no—
24C2×(C2×D131)⋊C4C2211yescharacteristic
25C2×(C2×D131)⋊C4C2212yes—
26C2×(C2×D131)⋊C4C2212yes—
27C2×(C2×D131)⋊C4C2212yes—
28C2×(C2×D131)⋊C4C2211yescharacteristic
29C2×(C2×D131)⋊C4C2212yescomplemented
30C2×(C2×D131)⋊C4C2212yescomplemented
31C22⋊C4×D131—24no—
32C131⋊#95C2212yescomplemented
33C131⋊#95C2212yescomplemented
34C23×C4×C131C2211yescharacteristic
35C2×C22⋊C4×C131C2211yescharacteristic
36C2×C22⋊C4×C131C2211yescharacteristic
37#95×C131C2212yescomplemented
38#95×C131C2212yescomplemented
39C23×D131—24no—
40C23×D131—22no—
41C23×D131—22no—
42C23×D131—42no—
43C23×D131D414yes—
44C23×D131C2311yescharacteristic
45C23×D131C2311yescharacteristic
46C23×D131C2311yescharacteristic
47C2×D524—22no—
48C2×D524—22no—
49C2×D524—24no—
50C2×D524—42no—
51C2×C22⋊D131—22no—
52C2×C22⋊D131—22no—
53C2×C22⋊D131—22no—
54C22×Dic131—22no—
55C22×Dic131—42no—
56C22×Dic131C2312yes—
57C22×Dic131C2311yescharacteristic
58C22×Dic131C2312yes—
59C2×C4×D131—24no—
60C2×C4×D131—24no—
61C2×C4×D131—42no—
62Dic131⋊C4—24no—
63Dic131⋊C4—42no—
64C22⋊Dic131D414yescomplemented
65(C2×D131)⋊C4—22no—
66(C2×D131)⋊C4—22no—
67(C2×D131)⋊C4—24no—
68(C2×D131)⋊C4—24no—
69(C2×D131)⋊C4—24no—
70(C2×D131)⋊C4—24no—
71(C2×D131)⋊C4—42no—
72C24×C131C2311yescharacteristic
73C22×C4×C131—22no—
74C22×C4×C131—22no—
75C22×C4×C131—22no—
76C22×C4×C131C2312yes—
77C22×C4×C131C2311yescharacteristic
78C22×C4×C131C2311yescharacteristic
79C22×C4×C131C2312yes—
80C22⋊C4×C131—22no—
81C22⋊C4×C131—22no—
82C22×D131—21no—
83C22×D131—21no—
84C22×D131—22no—
85C22×D131—28no—
86C22×D131—21no—
87C22×D131—22no—
88C22×D131—21no—
89C22×D131—22no—
90C22×D131—28no—
91C22×D131—21no—
92C22×D131—21no—
93C22×D131—21no—
94C22×D131—22no—
95C22×D131—22no—
96C22×D131—44no—
97C22×D131—48no—
98C22×D131—48no—
99C22×D131C2×D412yes—
100C22×D131C2×D412yes—
101C22×D131#4114yes—
102D524—44no—
103D524—44no—
104C22⋊D131—44no—
105C2×Dic131—22no—
106C2×Dic131—22no—
107C2×Dic131—22no—
108C2×Dic131—21no—
109C2×Dic131—21no—
110C2×Dic131—21no—
111C2×Dic131—22no—
112C2×Dic131—22no—
113C2×Dic131—44no—
114C2×Dic131C2×D414yes—
115C4×D131—48no—
116C23×C131—22no—
117C23×C131—22no—
118C23×C131C2×D412yes—
119C23×C131C2×D412yes—
120C23×C131#4112yes—
121C23×C131C2411yescharacteristic
122C2×C4×C131—22no—
123C2×C4×C131—21no—
124C2×C4×C131—21no—
125C2×C4×C131—22no—
126C2×C4×C131—22no—
127C2×C4×C131—22no—
128C2×C4×C131—22no—
129C2×C4×C131—22no—
130C2×C4×C131—22no—
131C2×C4×C131—22no—
132C2×C4×C131—44no—
133C22×C131—22no—
134C22×C131—22no—
135C22×C131—24no—
136C22×C131—24no—
137C22×C131#12114yes—
138C22×C131C22×D411yescharacteristic
139C22×C131C22×D411yescharacteristic
140C22×C131C2×#4112yes—
141C22×C131C2×#4111yescharacteristic
142C22×C131H(5,𝔽2)12yes—
143C2×D131—22no—
144C2×D131—22no—
145C2×D131—24no—
146C2×D131—24no—
147C2×D131—22no—
148C2×D131—22no—
149C2×D131—22no—
150C2×D131—416no—
151C2×D131—42no—
152C2×D131—42no—
153C2×D131—41no—
154C2×D131—41no—
155C2×D131—41no—
156C2×D131—42no—
157C2×D131—42no—
158C2×D131#12314yescomplemented
159C4×C131—42no—
160C4×C131—41no—
161C4×C131—41no—
162C4×C131—42no—
163Dic131—24no—
164Dic131—41no—
165Dic131—42no—
166C2×C131—24no—
167C2×C131C2×#12111yescharacteristic
168C2×C131C2×#12311yescharacteristic
169C2×C131#53411yescharacteristic
170C2×C131#54612yes—
171C2×C131#55112yes—
172D131—24no—
173D131—42no—
174D131—81no—
175C131#241211yescharacteristic, complemented, Sylow
176#2412—1311nomaximal, Sylow
177C2×#123—1311no—
178C2×#120—1311no—
179C22×C22⋊C4—1311no—
180C22×C22⋊C4—1311no—
181#397—1312no—
182#394—1312no—
183#386—1311no—
184#386—1312no—
185#385—1311no—
186#381D13111yescharacteristic, complemented
187#380—1312no—
188C25—1311no—
189C22×D4—1311no—
190C22×D4—1311no—
191C22×D4—1311no—
192C23×C4—1312no—
193C23×C4—1311no—
194C23×C4C2×D13111yescharacteristic
195#123—2624no—
196#120—2624no—
197C2×C4⋊C4—1312no—
198C2×C4⋊C4—1312no—
199C2×C22⋊C4—2624no—
200C2×C22⋊C4—2624no—
201C2×C22⋊C4—1312no—
202C2×C22⋊C4—1312no—
203C2×C22⋊C4—1312no—
204C2×C22⋊C4—1311no—
205C2×C22⋊C4—1311no—
206C2×C22⋊C4—1312no—
207C2×C22⋊C4—1312no—
208C2×C22⋊C4—1311no—
209C2×C22⋊C4C2×D13111yescharacteristic
210C2×C22⋊C4C2×D13111yescharacteristic
211#95—1312no—
212#95—1312no—
213#95C2×D13112yescomplemented
214#95C2×D13112yescomplemented
215C24—2624no—
216C24—2622no—
217C24—2622no—
218C24—5242no—
219C24—1311no—
220C24—1311no—
221C24—1314no—
222C24—1311no—
223C24C22×D13111yescharacteristic
224C2×D4—2622no—
225C2×D4—2622no—
226C2×D4—2624no—
227C2×D4—2622no—
228C2×D4—2622no—
229C2×D4—2622no—
230C2×D4—5242no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—2624no—
235C22×C4—2624no—
236C22×C4—2622no—
237C22×C4—5242no—
238C22×C4—5242no—
239C22×C4—1312no—
240C22×C4—1311no—
241C22×C4—1312no—
242C22×C4C22×D13111yescharacteristic
243C22×C4C22×D13112yes—
244C22×C4C22×D13112yes—
245C22×C4C22×D13111yescharacteristic
246C4⋊C4—2624no—
247C4⋊C4—5242no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—2624no—
251C22⋊C4—2624no—
252C22⋊C4—2622no—
253C22⋊C4—2622no—
254C22⋊C4—2624no—
255C22⋊C4—2624no—
256C22⋊C4—5242no—
257C22⋊C4—1314no—
258C23—22no—
259C23—22no—
260C23—2628no—
261C23—2621no—
262C23—2621no—
263C23—2622no—
264C23—2628no—
265C23—2621no—
266C23—2622no—
267C23—2621no—
268C23—2622no—
269C23—2621no—
270C23—2621no—
271C23—2622no—
272C23—2621no—
273C23—2622no—
274C23—5244no—
275C23—5248no—
276C23—5248no—
277C23—1312no—
278C23—1312no—
279C23—1314no—
280C23D4×D13112yes—
281C23D4×D13112yes—
282C23#1452351312yes—
283C23C23×D13111yescharacteristic
284D4—5244no—
285D4—5244no—
286D4—5244no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—44no—
298C2×C4—2622no—
299C2×C4—2622no—
300C2×C4—2622no—
301C2×C4—2621no—
302C2×C4—2621no—
303C2×C4—2621no—
304C2×C4—2622no—
305C2×C4—2622no—
306C2×C4—5248no—
307C2×C4—5244no—
308C2×C4—1314no—
309C22—22no—
310C22—24no—
311C22—24no—
312C22—22no—
313C22—2622no—
314C22—2624no—
315C22—2624no—
316C22—2622no—
317C22—2622no—
318C22—2622no—
319C22—2622no—
320C22—52416no—
321C22—5242no—
322C22—5242no—
323C22—5241no—
324C22—5242no—
325C22—5242no—
326C22—5241no—
327C22—5241no—
328C22—1314no—
329C22#1686221914yes—
330C22C2×D4×D13111yescharacteristic
331C22C2×D4×D13111yescharacteristic
332C22#41×D13112yes—
333C22C2×#1452351311yescharacteristic
334C22#1686229212yes—
335C4—41no—
336C4—42no—
337C4—42no—
338C4—41no—
339C4—2624no—
340C4—5242no—
341C4—5241no—
342C2—24no—
343C2—2624no—
344C2—5242no—
345C2—10481no—
346C2C2×#1686221911yescharacteristic
347C2#123×D13111yescharacteristic
348C2#2255387311yescharacteristic
349C2#2255395412yes—
350C2#2255398612yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#381×C131C2
prime radicalC2×#16862216C2
hypercenter#381D131
hypocenterC131#2412
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
1. C2×#16862229C2semidirect
3. C22×(C2×D131)⋊C4C2semidirect
4. C2×C22⋊C4×D131C2semidirect
5. #22553241C2semidirect
6. #22553230C2semidirect
7. #22553190C2semidirect
8. #22553189C2semidirect
9. #22553184C2semidirect
10. #22553171C2semidirect
11. #381×C131C2semidirect
20. C2×Dic131⋊C4C22semidirect
21. C2×Dic131⋊C4C22semidirect
29. C2×(C2×D131)⋊C4C22semidirect
30. C2×(C2×D131)⋊C4C22semidirect
32. C131⋊#95C22semidirect
33. C131⋊#95C22semidirect
37. #95×C131C22semidirect
38. #95×C131C22semidirect
64. C22⋊Dic131D4semidirect
158. C2×D131#123semidirect
175. C131#2412semidirect
186. #381D131semidirect
213. #95C2×D131semidirect
214. #95C2×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622292
C2×#168622162
C22×(C2×D131)⋊C42
C2×C22⋊C4×D1312
#225532412
#225532302
#225531902
#225531892
#225531842
#225531712
#381×C1312
#2412131