Finite Group Databasebeta

← #30666371

Subgroups — #30666371

Order 16768.

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

normal subgroups only

This group has 386 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
1C2×#16862229C211yesmaximal, characteristic
2#22553785C211yesmaximal, characteristic
3C2×#16862160C211yesmaximal, characteristic
4#22553175C211yesmaximal, characteristic, complemented
5#22553079C211yesmaximal, characteristic, complemented
6#22553078C211yesmaximal, characteristic, complemented
7#381×C131C211yesmaximal, characteristic, complemented
8C22×C22⋊D131C2211yescharacteristic
9#16862229—22no—
10#16862229C414yescomplemented
11C4×C22⋊D131—22no—
12(C22⋊D131)⋊C4—22no—
13#16862202—21no—
14C2×Dic131⋊C4—21no—
15C2×C22⋊Dic131C2211yescharacteristic
16C2×C22⋊Dic131C2211yescharacteristic
17C2×C22⋊Dic131C2211yescharacteristic
18C2×(C2×D131)⋊C4—21no—
19C22⋊C4×D131—21no—
20#16862160—22no—
21C131⋊#95—22no—
22C23×C4×C131C2211yescharacteristic
23C2×C22⋊C4×C131C2211yescharacteristic
24C2×C22⋊C4×C131C2211yescharacteristic
25#95×C131—22no—
26C23×D131D411yescharacteristic
27C2×C22⋊D131—21no—
28C2×C22⋊D131—21no—
29C2×C22⋊D131—21no—
30C2×C22⋊D131—21no—
31C2×C22⋊D131C2×C412yes—
32C2×C22⋊D131D412yes—
33C22×Dic131—21no—
34C22×Dic131—21no—
35C22×Dic131D411yescharacteristic
36C22×Dic131D412yes—
37C2×C4×D131—22no—
38Dic131⋊C4—22no—
39Dic131⋊C4—24no—
40C22⋊Dic131—21no—
41C22⋊Dic131—21no—
42C22⋊Dic131—21no—
43C22⋊Dic131—21no—
44C22⋊Dic131—21no—
45C22⋊Dic131C2×C412yescomplemented
46(C2×D131)⋊C4—22no—
47(C2×D131)⋊C4—24no—
48C4×Dic131—22no—
49C24×C131C2311yescharacteristic
50C22×C4×C131—21no—
51C22×C4×C131—21no—
52C22×C4×C131—21no—
53C22×C4×C131—21no—
54C22×C4×C131—21no—
55C22×C4×C131—21no—
56C22×C4×C131—21no—
57C22×C4×C131C2×C412yescomplemented
58C22×C4×C131D412yes—
59C22⋊C4×C131—21no—
60C22⋊C4×C131—21no—
61C22⋊C4×C131—21no—
62C22⋊C4×C131—21no—
63C22×D131—21no—
64C22×D131—21no—
65C22×D131—42no—
66C22×D131C22⋊C412yes—
67C22⋊D131—22no—
68C22⋊D131—22no—
69C22⋊D131—22no—
70C22⋊D131—22no—
71C2×Dic131—21no—
72C2×Dic131—21no—
73C2×Dic131—22no—
74C2×Dic131—22no—
75C2×Dic131—42no—
76C2×Dic131—42no—
77C2×Dic131—42no—
78C2×Dic131—41no—
79C2×Dic131C22⋊C412yes—
80C4×D131—82no—
81C23×C131—42no—
82C23×C131C22×C411yescharacteristic
83C23×C131C2×D411yescharacteristic
84C23×C131C2×D411yescharacteristic
85C23×C131C2×D411yescharacteristic
86C23×C131C2×D411yescharacteristic
87C23×C131#4111yescharacteristic
88C23×C131#4111yescharacteristic
89C2×C4×C131—22no—
90C2×C4×C131—22no—
91C2×C4×C131—22no—
92C2×C4×C131—22no—
93C2×C4×C131—41no—
94C2×C4×C131—41no—
95C2×C4×C131—41no—
96C2×C4×C131—41no—
97C2×C4×C131—42no—
98C2×C4×C131—42no—
99C2×C4×C131—41no—
100C22×C131—22no—
101C22×C131—22no—
102C22×C131—22no—
103C22×C131—41no—
104C22×C131—41no—
105C22×C131—41no—
106C22×C131—41no—
107C22×C131C2×C22⋊C411yescharacteristic
108C22×C131C4×D411yescharacteristic
109C22×C131C4×D411yescharacteristic
110C22×C131#12011yescharacteristic
111C22×C131#12111yescharacteristic
112C22×C131#12111yescharacteristic
113C22×C131#12311yescharacteristic
114C2×D131—22no—
115C2×D131—42no—
116C2×D131—42no—
117C2×D131—81no—
118C4×C131—42no—
119C4×C131—42no—
120C4×C131—81no—
121Dic131—22no—
122Dic131—42no—
123Dic131—42no—
124Dic131—81no—
125C2×C131—21no—
126C2×C131—21no—
127C2×C131—21no—
128C2×C131—21no—
129C2×C131—21no—
130C2×C131—21no—
131C2×C131#38611yescharacteristic
132C2×C131#45812yes—
133D131—42no—
134C131#189011yescharacteristic, complemented, Sylow
135#1890—1311nomaximal, Sylow
136C2×#123—1311no—
137#518—1311no—
138C2×#99—1311no—
139#381—1311no—
140#381D13111yescharacteristic, complemented
141#342—1311no—
142#342—1311no—
143C22×D4—1311no—
144C23×C4C2×D13111yescharacteristic
145#123—2622no—
146#123—1314no—
147C4×D4—2622no—
148C4×D4—2622no—
149#117—2621no—
150C2×C4⋊C4—2621no—
151C2×C22⋊C4—2621no—
152C2×C22⋊C4—2621no—
153C2×C22⋊C4—1311no—
154C2×C22⋊C4—1311no—
155C2×C22⋊C4—1311no—
156C2×C22⋊C4C2×D13111yescharacteristic
157C2×C22⋊C4C2×D13111yescharacteristic
158#99—2622no—
159#95—22no—
160#95—2622no—
161C24—1311no—
162C24C22×D13111yescharacteristic
163C2×D4—2621no—
164C2×D4—2621no—
165C2×D4—2621no—
166C2×D4—2621no—
167C2×D4—1312no—
168C2×D4—1312no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—2622no—
177C22×C4—2621no—
178C22×C4—2621no—
179C22×C4—1311no—
180C22×C4—1312no—
181C22×C4C4×D13112yescomplemented
182C22×C4C22⋊D13112yes—
183C4⋊C4—2622no—
184C4⋊C4—2624no—
185C22⋊C4—21no—
186C22⋊C4—21no—
187C22⋊C4—21no—
188C22⋊C4—21no—
189C22⋊C4—2622no—
190C22⋊C4—2624no—
191C22⋊C4—2621no—
192C22⋊C4—2621no—
193C22⋊C4—2621no—
194C22⋊C4—2621no—
195C22⋊C4—2621no—
196C22⋊C4—1312no—
197C42—2622no—
198C23—42no—
199C23—2621no—
200C23—2621no—
201C23—5242no—
202C23—1312no—
203C23C2×C4×D13111yescharacteristic
204C23D4×D13111yescharacteristic
205C23D4×D13111yescharacteristic
206C23D4×D13111yescharacteristic
207C23C2×C22⋊D13111yescharacteristic
208C23#1452351211yescharacteristic
209C23#1452351311yescharacteristic
210D4—2622no—
211D4—2622no—
212D4—2622no—
213D4—2622no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—2621no—
226C2×C4—2621no—
227C2×C4—2622no—
228C2×C4—2622no—
229C2×C4—5242no—
230C2×C4—5242no—
231C2×C4—5242no—
232C2×C4—5241no—
233C2×C4—10482no—
234C2×C4—1312no—
235C22—22no—
236C22—22no—
237C22—22no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—41no—
242C22—2622no—
243C22—5242no—
244C22—5242no—
245C22—10481no—
246C22C22⋊C4×D13111yescharacteristic
247C22(C22⋊D131)⋊C411yescharacteristic
248C22C4×C22⋊D13111yescharacteristic
249C22#1686221411yescharacteristic
250C22#1686221711yescharacteristic
251C22#1686221911yescharacteristic
252C22#1686222811yescharacteristic
253C4—42no—
254C4—42no—
255C4—2622no—
256C4—81no—
257C4—5242no—
258C4—5242no—
259C4—10481no—
260C2—21no—
261C2—21no—
262C2—21no—
263C2—21no—
264C2—21no—
265C2—21no—
266C2—5242no—
267C2#2255318811yescharacteristic
268C2#2255350112yes—
subgroupG/H
centerC22#16862214
commutatorC23×C131C22×C4
FrattiniC24C22×D131
Fitting#381×C131C2
socleC22×C131#120
prime radicalC22×C22⋊D131C22
cosocleC24×C131C23
hypercenter#381D131
hypocenterC131#1890
LCSG ▷ C23×C131 ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C24 ◁ #381

Complements

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

NH ≅ G/Ntype
4. #22553175C2semidirect
5. #22553079C2semidirect
6. #22553078C2semidirect
7. #381×C131C2semidirect
10. #16862229C4semidirect
45. C22⋊Dic131C2×C4semidirect
57. C22×C4×C131C2×C4semidirect
134. C131#1890semidirect
140. #381D131semidirect
181. C22×C4C4×D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622292
#225537852
C2×#168621602
#225531752
#225530792
#225530782
#381×C1312
#1890131