Finite Group Databasebeta

← #30669243

Subgroups — #30669243

Order 16768.

568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 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×Q8×Dic131C211yesmaximal, characteristic, complemented
2C2×(C22⋊D131)⋊C4C211yesmaximal, characteristic
3#22553254C212yesmaximal, complemented
4#22553239C211yesmaximal, characteristic, complemented
5#22553237C211yesmaximal, characteristic, complemented
6#22553228C212yesmaximal
7#22553223C211yesmaximal, characteristic, complemented
8C22⋊C4×Dic131C212yesmaximal, complemented
9#22553159C212yesmaximal, complemented
10#22553157C211yesmaximal, characteristic, complemented
11#393×C131C211yesmaximal, characteristic, complemented
12C22×C22⋊D131C2211yescharacteristic
13C23×Dic131C2211yescharacteristic
14C22×C4×D131C2211yescharacteristic
15Q8×Dic131—24no—
16(C22⋊D131)⋊C4—24no—
17C2×C4⋊Dic131C2212yes—
18C2×C4⋊Dic131C2211yescharacteristic
19C2×Dic131⋊C4C2211yescharacteristic
20C2×C22⋊Dic131C2212yes—
21C2×(C2×D131)⋊C4C2212yes—
22C2×(C2×D131)⋊C4C2212yes—
23C2×(C2×D131)⋊C4C2211yescharacteristic
24C2×C4×Dic131C2212yes—
25C2×C4×Dic131C2212yes—
26C2×C4×Dic131C2211yescharacteristic
27C131⋊#95C2211yescharacteristic
28C131⋊#95C2212yes—
29C131⋊#95C2212yes—
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C22×Q8×C131C2211yescharacteristic
33C2×C22⋊C4×C131C2212yes—
34C2×C22⋊C4×C131C2211yescharacteristic
35#95×C131C2211yescharacteristic
36#95×C131C2212yes—
37C23×D131C2311yescharacteristic
38C2×C22⋊D131—22no—
39C2×C22⋊D131—22no—
40C2×C22⋊D131—22no—
41C22×Dic131—22no—
42C22×Dic131—24no—
43C22×Dic131C2311yescharacteristic
44C22×Dic131C2312yes—
45C22×Dic131C2312yes—
46C22×Dic131C2311yescharacteristic
47C22×Dic131C2311yescharacteristic
48C2×C4×D131—22no—
49C2×C4×D131—24no—
50C4⋊Dic131—24no—
51C4⋊Dic131—22no—
52Dic131⋊C4—22no—
53C22⋊Dic131—24no—
54(C2×D131)⋊C4—24no—
55(C2×D131)⋊C4—22no—
56(C2×D131)⋊C4—24no—
57C4×Dic131—24no—
58C4×Dic131—24no—
59C4×Dic131D414yes—
60C24×C131C2311yescharacteristic
61C2×Q8×C131—22no—
62C2×Q8×C131—24no—
63C22×C4×C131C2311yescharacteristic
64C22×C4×C131C2312yes—
65C22×C4×C131C2311yescharacteristic
66C22×C4×C131C2312yes—
67C22⋊C4×C131—22no—
68C22⋊C4×C131—24no—
69C22×D131—21no—
70C22×D131—22no—
71C22×D131—22no—
72C22×D131—21no—
73C22×D131#4112yes—
74C22⋊D131—28no—
75C2×Dic131—21no—
76C2×Dic131—21no—
77C2×Dic131—22no—
78C2×Dic131—21no—
79C2×Dic131—22no—
80C2×Dic131—22no—
81C2×Dic131—22no—
82C2×Dic131—22no—
83C2×Dic131—22no—
84C2×Dic131—22no—
85C2×Dic131—44no—
86C2×Dic131C2×D412yes—
87C2×Dic131C2×D412yes—
88C2×Dic131#4114yes—
89C4×D131—44no—
90C23×C131—21no—
91C23×C131—22no—
92C23×C131—22no—
93C23×C131—21no—
94C23×C131#4112yes—
95C23×C131C2411yescharacteristic
96Q8×C131—28no—
97C2×C4×C131—22no—
98C2×C4×C131—22no—
99C2×C4×C131—22no—
100C2×C4×C131—21no—
101C2×C4×C131—22no—
102C2×C4×C131—22no—
103C2×C4×C131—22no—
104C2×C4×C131—22no—
105C2×C4×C131C2×D412yes—
106C2×C4×C131#4114yes—
107C22×C131—22no—
108C22×C131—24no—
109C22×C131—41no—
110C22×C131—41no—
111C22×C131—42no—
112C22×C131C22×D411yescharacteristic
113C22×C131C2×#4111yescharacteristic
114C22×C131C2×#4112yes—
115C22×C131C2×#4112yes—
116C22×C131C2×#4111yescharacteristic
117C2×D131—22no—
118C2×D131—24no—
119C2×D131—41no—
120C2×D131—41no—
121C2×D131—42no—
122C4×C131—22no—
123C4×C131—24no—
124C4×C131—41no—
125C4×C131—42no—
126Dic131—22no—
127Dic131—42no—
128Dic131—41no—
129Dic131—42no—
130Dic131#12414yes—
131C2×C131—42no—
132C2×C131#52511yescharacteristic
133C2×C131#52511yescharacteristic
134C2×C131C2×#12411yescharacteristic
135C2×C131#52912yes—
136C2×C131#52912yes—
137D131—42no—
138C131#239811yescharacteristic, complemented, Sylow
139#2398—1311nomaximal, Sylow
140C2×C4×Q8—1311no—
141C2×C4×D4—1311no—
142#400—1312no—
143#397—1311no—
144#397—1311no—
145#394—1312no—
146#393—1311no—
147#393D13111yescharacteristic, complemented
148C4×C22⋊C4—1312no—
149C4×C22⋊C4—1312no—
150#376—1311no—
151C22×Q8C2×D13111yescharacteristic
152C22×D4—1311no—
153C23×C4—1311no—
154C23×C4—1311no—
155C4×Q8—2624no—
156C4×D4—2624no—
157C2×C4⋊C4—1311no—
158C2×C4⋊C4—1312no—
159C2×C4⋊C4—1311no—
160C2×C22⋊C4—1312no—
161C2×C22⋊C4—1311no—
162C2×C22⋊C4—1312no—
163C2×C22⋊C4—1312no—
164C2×C22⋊C4C2×D13111yescharacteristic
165C2×C22⋊C4C2×D13112yes—
166C2×C42—1312no—
167C2×C42—1312no—
168C2×C42—1311no—
169#95—1311no—
170#95—1312no—
171#95—1312no—
172#95—1312no—
173#95—1312no—
174#95C2×D13111yescharacteristic
175#95C2×D13112yes—
176C24—1311no—
177C24C22×D13111yescharacteristic
178C2×Q8—22no—
179C2×Q8—24no—
180C2×D4—2622no—
181C2×D4—2622no—
182C2×D4—2622no—
183C22×C4—2622no—
184C22×C4—2624no—
185C22×C4—2622no—
186C22×C4—2624no—
187C22×C4—1312no—
188C22×C4—1311no—
189C22×C4—1312no—
190C22×C4—1311no—
191C22×C4—1311no—
192C22×C4C22×D13112yes—
193C22×C4C22×D13111yescharacteristic
194C22×C4C22×D13111yescharacteristic
195C22×C4C22×D13112yes—
196C4⋊C4—2622no—
197C4⋊C4—2624no—
198C4⋊C4—2622no—
199C22⋊C4—22no—
200C22⋊C4—24no—
201C22⋊C4—2624no—
202C22⋊C4—2624no—
203C22⋊C4—2622no—
204C22⋊C4—2624no—
205C42—2624no—
206C42—2624no—
207C42—1314no—
208C23—21no—
209C23—22no—
210C23—21no—
211C23—22no—
212C23—2621no—
213C23—2622no—
214C23—2622no—
215C23—2621no—
216C23—1312no—
217C23#1452351412yes—
218C23C23×D13111yescharacteristic
219Q8—28no—
220D4—2628no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—2621no—
230C2×C4—2621no—
231C2×C4—2622no—
232C2×C4—2621no—
233C2×C4—2622no—
234C2×C4—2622no—
235C2×C4—2622no—
236C2×C4—2622no—
237C2×C4—2622no—
238C2×C4—2622no—
239C2×C4—5244no—
240C2×C4—5244no—
241C2×C4—1312no—
242C2×C4—1312no—
243C2×C4—1314no—
244C2×C4D4×D13112yes—
245C2×C4#1452351214yes—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—2622no—
252C22—2624no—
253C22—5241no—
254C22—5241no—
255C22—5242no—
256C22C2×D4×D13111yescharacteristic
257C22#41×D13111yescharacteristic
258C22#41×D13112yes—
259C22C2×#1452351212yes—
260C22C2×#1452351411yescharacteristic
261C4—22no—
262C4—24no—
263C4—41no—
264C4—42no—
265C4—2622no—
266C4—5242no—
267C4—5241no—
268C4—5242no—
269C4—1314no—
270C2—42no—
271C2—5242no—
272C2#2255381811yescharacteristic
273C2#2255382511yescharacteristic
274C2#124×D13111yescharacteristic
275C2#2255384112yes—
276C2#2255384812yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#393×C131C2
prime radicalC22×C22⋊D131C22
hypercenter#393D131
hypocenterC131#2398
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic131C2semidirect
3. #22553254C2semidirect
4. #22553239C2semidirect
5. #22553237C2semidirect
7. #22553223C2semidirect
8. C22⋊C4×Dic131C2semidirect
9. #22553159C2semidirect
10. #22553157C2semidirect
11. #393×C131C2semidirect
138. C131#2398semidirect
147. #393D131semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1312
C2×(C22⋊D131)⋊C42
#225532542
#225532392
#225532372
#225532282
#225532232
C22⋊C4×Dic1312
#225531592
#225531572
#393×C1312
#2398131