Finite Group Databasebeta

← #30666243

Subgroups — #30666243

Order 16768.

374 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 83 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
1#22553785C211yesmaximal, characteristic
2C2×#16862160C211yesmaximal, characteristic
3#22553189C211yesmaximal, characteristic
4#22553187C211yesmaximal, characteristic, complemented
5#22553184C211yesmaximal, characteristic, complemented
6#22553078C211yesmaximal, characteristic, complemented
7#342×C131C211yesmaximal, characteristic, complemented
8C22×C22⋊D131C2211yescharacteristic
9C23×Dic131C2211yescharacteristic
10C4×C22⋊D131—21no—
11C4×C22⋊D131—21no—
12(C22⋊D131)⋊C4—21no—
13(C22⋊D131)⋊C4—21no—
14#16862202—21no—
15C2×C4⋊Dic131—21no—
16C2×Dic131⋊C4—21no—
17C2×C22⋊Dic131C2211yescharacteristic
18C2×C22⋊Dic131C2211yescharacteristic
19C2×(C2×D131)⋊C4—21no—
20C22⋊C4×D131—21no—
21#16862160C414yescomplemented
22C131⋊#95—21no—
23C131⋊#95—21no—
24C131⋊#95—21no—
25C2×C22⋊C4×C131C2211yescharacteristic
26C2×C22⋊C4×C131C2211yescharacteristic
27C2×C22⋊C4×C131C2211yescharacteristic
28C23×D131D411yescharacteristic
29C2×C22⋊D131—21no—
30C2×C22⋊D131—21no—
31C2×C22⋊D131—21no—
32C2×C22⋊D131—21no—
33C2×C22⋊D131C2×C412yes—
34C2×C22⋊D131Q812yes—
35C22×Dic131—21no—
36C22×Dic131—21no—
37C22×Dic131—21no—
38C22×Dic131—21no—
39C22×Dic131—42no—
40C22×Dic131D411yescharacteristic
41C22×Dic131D411yescharacteristic
42C22×Dic131D411yescharacteristic
43C2×C4×D131—21no—
44C2×C4×D131—21no—
45C4⋊Dic131—42no—
46Dic131⋊C4—21no—
47Dic131⋊C4—21no—
48Dic131⋊C4—42no—
49C22⋊Dic131—21no—
50C22⋊Dic131—21no—
51C22⋊Dic131—21no—
52C22⋊Dic131C2×C412yescomplemented
53(C2×D131)⋊C4—21no—
54(C2×D131)⋊C4—21no—
55(C2×D131)⋊C4—42no—
56C4×Dic131—21no—
57C4×Dic131—21no—
58C24×C131C2311yescharacteristic
59C22×C4×C131—21no—
60C22×C4×C131—21no—
61C22×C4×C131D411yescharacteristic
62C22×C4×C131D411yescharacteristic
63C22⋊C4×C131—21no—
64C22⋊C4×C131—21no—
65C22⋊C4×C131—21no—
66C22⋊C4×C131—21no—
67C22⋊C4×C131—21no—
68C22⋊C4×C131C2×C412yescomplemented
69C22×D131—21no—
70C22×D131—21no—
71C22×D131—41no—
72C22×D131—41no—
73C22×D131C4⋊C412yescomplemented
74C22⋊D131—22no—
75C22⋊D131—22no—
76C22⋊D131—22no—
77C22⋊D131—22no—
78C2×Dic131—21no—
79C2×Dic131—21no—
80C2×Dic131—22no—
81C2×Dic131—22no—
82C2×Dic131—22no—
83C2×Dic131—22no—
84C2×Dic131—21no—
85C2×Dic131—21no—
86C2×Dic131—41no—
87C2×Dic131—41no—
88C2×Dic131—41no—
89C2×Dic131—41no—
90C2×Dic131—41no—
91C2×Dic131—41no—
92C2×Dic131—41no—
93C2×Dic131—41no—
94C2×Dic131C4⋊C412yescomplemented
95C4×D131—81no—
96C4×D131—81no—
97C23×C131—41no—
98C23×C131—41no—
99C23×C131C22×C411yescharacteristic
100C23×C131C2×D411yescharacteristic
101C23×C131C2×D411yescharacteristic
102C23×C131C2×D411yescharacteristic
103C23×C131C2×Q811yescharacteristic
104C23×C131#4111yescharacteristic
105C23×C131#4111yescharacteristic
106C2×C4×C131—22no—
107C2×C4×C131—21no—
108C2×C4×C131—21no—
109C2×C4×C131—41no—
110C2×C4×C131—41no—
111C2×C4×C131—41no—
112C2×C4×C131—41no—
113C2×C4×C131—41no—
114C2×C4×C131—41no—
115C2×C4×C131—41no—
116C22×C131—21no—
117C22×C131—21no—
118C22×C131—21no—
119C22×C131—21no—
120C22×C131—21no—
121C22×C131—21no—
122C22×C131—41no—
123C22×C131—41no—
124C22×C131—41no—
125C22×C131—41no—
126C22×C131C2×C4⋊C411yescharacteristic
127C22×C131C4×D411yescharacteristic
128C22×C131C4×D411yescharacteristic
129C22×C131#12011yescharacteristic
130C22×C131C22⋊Q811yescharacteristic
131C22×C131C22⋊Q811yescharacteristic
132C22×C131#12311yescharacteristic
133C2×D131—22no—
134C2×D131—41no—
135C2×D131—41no—
136C2×D131—42no—
137C2×D131—81no—
138C4×C131—42no—
139C4×C131—41no—
140C4×C131—41no—
141C4×C131—81no—
142Dic131—22no—
143Dic131—42no—
144Dic131—41no—
145Dic131—42no—
146Dic131—41no—
147C2×C131—21no—
148C2×C131—21no—
149C2×C131—21no—
150C2×C131—21no—
151C2×C131—21no—
152C2×C131—21no—
153C2×C131#38511yescharacteristic
154C2×C131B(4,𝔽2)11yescharacteristic
155C2×C131#45811yescharacteristic
156D131—42no—
157C131#186711yescharacteristic, complemented, Sylow
158#1867—1311nomaximal, Sylow
159#518—1311no—
160C2×#99—1311no—
161#386—1311no—
162#385—1311no—
163#385—1311no—
164#342—1311no—
165#342D13111yescharacteristic, complemented
166C22×D4—1311no—
167C23×C4—1311no—
168C4×D4—2621no—
169C4×D4—2621no—
170C4×D4—2621no—
171C4×D4—2621no—
172#117—2621no—
173C2×C4⋊C4—2621no—
174C2×C4⋊C4—2621no—
175C2×C22⋊C4—2621no—
176C2×C22⋊C4—2621no—
177C2×C22⋊C4—1311no—
178C2×C22⋊C4—1311no—
179C2×C22⋊C4C2×D13111yescharacteristic
180C2×C22⋊C4C2×D13111yescharacteristic
181C2×C22⋊C4C2×D13111yescharacteristic
182#99—1314no—
183#95—2621no—
184#95—2621no—
185#95—2621no—
186C24—1311no—
187C24C22×D13111yescharacteristic
188C2×D4—2621no—
189C2×D4—2621no—
190C2×D4—2621no—
191C2×D4—2621no—
192C2×D4—1312no—
193C2×D4—1312no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—2621no—
197C22×C4—2621no—
198C22×C4—2621no—
199C22×C4—2621no—
200C22×C4—2621no—
201C22×C4—2621no—
202C22×C4—5242no—
203C22×C4—1311no—
204C22×C4—1311no—
205C22×C4—1311no—
206C22×C4D52411yescharacteristic
207C22×C4D52411yescharacteristic
208C4⋊C4—2621no—
209C4⋊C4—2621no—
210C4⋊C4—5242no—
211C4⋊C4—5242no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—2621no—
218C22⋊C4—2621no—
219C22⋊C4—2621no—
220C22⋊C4—2621no—
221C22⋊C4—2621no—
222C22⋊C4—5242no—
223C22⋊C4—1312no—
224C22⋊C4C4×D13112yescomplemented
225C42—2621no—
226C42—2621no—
227C23—41no—
228C23—41no—
229C23—2621no—
230C23—2621no—
231C23—5241no—
232C23—5241no—
233C23—1312no—
234C23C2×C4×D13111yescharacteristic
235C23D4×D13111yescharacteristic
236C23D4×D13111yescharacteristic
237C23C2×D52411yescharacteristic
238C23Q8×D13111yescharacteristic
239C23#1452351211yescharacteristic
240C23#1452351311yescharacteristic
241D4—2622no—
242D4—2622no—
243D4—2622no—
244D4—2622no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—2621no—
256C2×C4—2621no—
257C2×C4—2622no—
258C2×C4—2622no—
259C2×C4—2622no—
260C2×C4—2622no—
261C2×C4—2621no—
262C2×C4—2621no—
263C2×C4—5241no—
264C2×C4—5241no—
265C2×C4—5241no—
266C2×C4—5241no—
267C2×C4—5241no—
268C2×C4—5241no—
269C2×C4—5241no—
270C2×C4—5241no—
271C2×C4—10481no—
272C2×C4—10481no—
273C2×C4—1312no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—2622no—
285C22—5242no—
286C22—5241no—
287C22—5241no—
288C22—10481no—
289C22C4⋊C4×D13111yescharacteristic
290C22(C22⋊D131)⋊C411yescharacteristic
291C22C4×D52411yescharacteristic
292C22#1686221611yescharacteristic
293C22#1686222411yescharacteristic
294C22#1686222611yescharacteristic
295C22#1686222811yescharacteristic
296C4—42no—
297C4—41no—
298C4—41no—
299C4—2622no—
300C4—81no—
301C4—5242no—
302C4—5242no—
303C4—5241no—
304C4—5241no—
305C2—21no—
306C2—21no—
307C2—21no—
308C2—21no—
309C2—21no—
310C2—21no—
311C2—5242no—
312C2#2255318611yescharacteristic
313C2#2255350011yescharacteristic
314C2#2255350311yescharacteristic
subgroupG/H
centerC22#16862216
commutatorC23×C131C22×C4
FrattiniC24C22×D131
Fitting#342×C131C2
socleC22×C131#120
prime radicalC22×C22⋊D131C22
cosocleC24×C131C23
hypercenter#342D131
hypocenterC131#1867
LCSG ▷ C23×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
4. #22553187C2semidirect
5. #22553184C2semidirect
6. #22553078C2semidirect
7. #342×C131C2semidirect
21. #16862160C4semidirect
52. C22⋊Dic131C2×C4semidirect
68. C22⋊C4×C131C2×C4semidirect
73. C22×D131C4⋊C4semidirect
94. C2×Dic131C4⋊C4semidirect
157. C131#1867semidirect
165. #342D131semidirect
224. C22⋊C4C4×D131semidirect

Maximal subgroups

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

subgroupindex
#225537852
C2×#168621602
#225531892
#225531872
#225531842
#225530782
#342×C1312
#1867131