Finite Group Databasebeta

← #30671121

Subgroups — #30671121

Order 16768.

532 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 532 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16862236C211yesmaximal, characteristic, complemented
2C2×D4×Dic131C211yesmaximal, characteristic
3#22553252C211yesmaximal, characteristic, complemented
4#22553251C211yesmaximal, characteristic, complemented
5#22553226C211yesmaximal, characteristic
6#22553205C212yesmaximal, complemented
7#22553200C212yesmaximal, complemented
8#22553170C212yesmaximal, complemented
9#22553170C212yesmaximal, complemented
10#22553163C211yesmaximal, characteristic, complemented
11#394×C131C211yesmaximal, characteristic
12C23×Dic131C2212yes—
13#16862236—24no—
14D4×Dic131—24no—
15C2×C4⋊Dic131C2211yescharacteristic
16C2×C4⋊Dic131C2211yescharacteristic
17C2×Dic131⋊C4C2212yes—
18C2×Dic131⋊C4C2211yescharacteristic
19C2×Dic131⋊C4C2211yescharacteristic
20C2×Dic131⋊C4C2211yescharacteristic
21C2×Dic131⋊C4C2211yescharacteristic
22C2×C22⋊Dic131C2212yes—
23C2×C22⋊Dic131C2212yes—
24C2×C22⋊Dic131C2212yes—
25C2×C4×Dic131C2212yes—
26C2×C4×Dic131C2211yescharacteristic
27C131⋊#95C2211yescharacteristic
28C131⋊#95C2212yes—
29C131⋊#95C2212yes—
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C22×D4×C131C2211yescharacteristic
33C2×C4⋊C4×C131C2211yescharacteristic
34C2×C22⋊C4×C131C2212yes—
35C2×C22⋊C4×C131C2212yes—
36#95×C131C2211yescharacteristic
37C22×Dic131—24no—
38C22×Dic131—24no—
39C22×Dic131—24no—
40C22×Dic131C2312yes—
41C22×Dic131C2312yes—
42C22×Dic131C2311yescharacteristic
43C22×Dic131C2311yescharacteristic
44C22×Dic131C2311yescharacteristic
45C22×Dic131C2311yescharacteristic
46C4⋊Dic131—22no—
47C4⋊Dic131—22no—
48Dic131⋊C4—22no—
49Dic131⋊C4—22no—
50Dic131⋊C4—22no—
51Dic131⋊C4—24no—
52Dic131⋊C4—22no—
53C22⋊Dic131—24no—
54C22⋊Dic131—24no—
55C22⋊Dic131—42no—
56C4×Dic131—42no—
57C4×Dic131D414yes—
58C24×C131C2312yes—
59C2×D4×C131—24no—
60C2×D4×C131—22no—
61C22×C4×C131C2312yes—
62C22×C4×C131C2311yescharacteristic
63C22×C4×C131C2311yescharacteristic
64C22×C4×C131C2311yescharacteristic
65C4⋊C4×C131—22no—
66C22⋊C4×C131—24no—
67C22⋊C4×C131—42no—
68C2×Dic131—21no—
69C2×Dic131—21no—
70C2×Dic131—22no—
71C2×Dic131—21no—
72C2×Dic131—21no—
73C2×Dic131—22no—
74C2×Dic131—22no—
75C2×Dic131—22no—
76C2×Dic131—22no—
77C2×Dic131—22no—
78C2×Dic131—22no—
79C2×Dic131—21no—
80C2×Dic131—21no—
81C2×Dic131—48no—
82C2×Dic131C2×D412yes—
83C2×Dic131C2×D412yes—
84C2×Dic131#4114yes—
85C23×C131—22no—
86C23×C131—22no—
87C23×C131—22no—
88C23×C131—22no—
89C23×C131—22no—
90C23×C131—22no—
91C23×C131#4114yes—
92C23×C131C2411yescharacteristic
93D4×C131—28no—
94C2×C4×C131—22no—
95C2×C4×C131—21no—
96C2×C4×C131—21no—
97C2×C4×C131—22no—
98C2×C4×C131—22no—
99C2×C4×C131—22no—
100C2×C4×C131—21no—
101C2×C4×C131—22no—
102C2×C4×C131—21no—
103C2×C4×C131C2×D412yes—
104C22×C131—24no—
105C22×C131—24no—
106C22×C131—24no—
107C22×C131—42no—
108C22×C131—42no—
109C22×C131—42no—
110C22×C131—42no—
111C22×C131C22×D411yescharacteristic
112C22×C131C2×#4112yes—
113C22×C131C2×#4112yes—
114C22×C131H(5,𝔽2)11yescharacteristic
115C22×C1312−1+411yescharacteristic
116C4×C131—22no—
117C4×C131—41no—
118C4×C131—41no—
119C4×C131—42no—
120Dic131—22no—
121Dic131—41no—
122Dic131—42no—
123Dic131—41no—
124Dic131—42no—
125Dic131#12414yes—
126C2×C131—44no—
127C2×C131#52511yescharacteristic
128C2×C131C2×#12411yescharacteristic
129C2×C131#53711yescharacteristic
130C2×C131#55212yes—
131C2×C131#55312yes—
132C131#256211yescharacteristic, complemented, Sylow
133#2562—1311nomaximal, Sylow
134C2×#125—1311no—
135C2×C4×D4—1311no—
136#400—1311no—
137#400—1311no—
138#394—1311no—
139#394D13111yescharacteristic
140#388—1312no—
141#388—1312no—
142#380—1312no—
143#380—1312no—
144C4×C4⋊C4—1311no—
145C22×D4C2×D13111yescharacteristic
146C23×C4—1312no—
147#125—2624no—
148C4×D4—2624no—
149C2×C4⋊C4—1311no—
150C2×C4⋊C4—1311no—
151C2×C4⋊C4—1311no—
152C2×C4⋊C4—1311no—
153C2×C4⋊C4—1311no—
154C2×C4⋊C4—1311no—
155C2×C4⋊C4—1312no—
156C2×C4⋊C4C2×D13111yescharacteristic
157C2×C22⋊C4—1312no—
158C2×C22⋊C4—1312no—
159C2×C22⋊C4—1312no—
160C2×C22⋊C4C2×D13112yes—
161C2×C22⋊C4C2×D13112yes—
162C2×C42—1312no—
163C2×C42—1311no—
164#95—1311no—
165#95—1312no—
166#95—1312no—
167#95—1312no—
168#95—1312no—
169#95C2×D13111yescharacteristic
170C24C22×D13112yes—
171C2×D4—24no—
172C2×D4—22no—
173C22×C4—2624no—
174C22×C4—2624no—
175C22×C4—2624no—
176C22×C4—1312no—
177C22×C4—1312no—
178C22×C4—1311no—
179C22×C4—1311no—
180C22×C4—1311no—
181C22×C4—1311no—
182C22×C4C22×D13112yes—
183C22×C4C22×D13111yescharacteristic
184C22×C4C22×D13111yescharacteristic
185C22×C4C22×D13111yescharacteristic
186C4⋊C4—22no—
187C4⋊C4—2624no—
188C4⋊C4—2622no—
189C4⋊C4—2622no—
190C4⋊C4—2622no—
191C4⋊C4—2622no—
192C4⋊C4—2622no—
193C4⋊C4—2622no—
194C22⋊C4—24no—
195C22⋊C4—42no—
196C22⋊C4—2624no—
197C22⋊C4—2624no—
198C22⋊C4—5242no—
199C42—5242no—
200C42—1314no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23#1452351214yes—
208C23C23×D13111yescharacteristic
209D4—28no—
210C2×C4—22no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—2621no—
220C2×C4—2621no—
221C2×C4—2622no—
222C2×C4—2621no—
223C2×C4—2621no—
224C2×C4—2622no—
225C2×C4—2622no—
226C2×C4—2622no—
227C2×C4—2622no—
228C2×C4—2622no—
229C2×C4—2622no—
230C2×C4—2621no—
231C2×C4—2621no—
232C2×C4—5248no—
233C2×C4—1312no—
234C2×C4—1312no—
235C2×C4—1314no—
236C2×C4D4×D13112yes—
237C22—24no—
238C22—24no—
239C22—24no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22C2×D4×D13111yescharacteristic
245C22#41×D13112yes—
246C22C2×#1452351212yes—
247C22#1686229111yescharacteristic
248C22#1686229311yescharacteristic
249C4—22no—
250C4—41no—
251C4—42no—
252C4—41no—
253C4—2622no—
254C4—5241no—
255C4—5242no—
256C4—5241no—
257C4—5242no—
258C4—1314no—
259C2—44no—
260C2#2255381811yescharacteristic
261C2#124×D13111yescharacteristic
262C2#2255389111yescharacteristic
263C2#2255399912yes—
264C2#2255400512yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#394×C131C2
prime radicalC22×D4×C131C22
hypercenter#394D131
hypocenterC131#2562
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#16862236C2semidirect
3. #22553252C2semidirect
4. #22553251C2semidirect
6. #22553205C2semidirect
7. #22553200C2semidirect
8. #22553170C2semidirect
9. #22553170C2semidirect
10. #22553163C2semidirect
132. C131#2562semidirect

Maximal subgroups

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

subgroupindex
C2×#168622362
C2×D4×Dic1312
#225532522
#225532512
#225532262
#225532052
#225532002
#225531702
#225531632
#394×C1312
#2562131