Finite Group Databasebeta

← #30670909

Subgroups — #30670909

Order 16768.

676 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 171 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×#16862245C211yesmaximal, characteristic, complemented
2C2×#16862224C212yesmaximal, complemented
3C2×#16862223C212yesmaximal, complemented
4C2×(C22⋊D131)⋊C4C211yesmaximal, characteristic
5#22553249C211yesmaximal, characteristic, complemented
6#22553248C211yesmaximal, characteristic, complemented
7#22553247C211yesmaximal, characteristic, complemented
8#22553212C212yesmaximal
9#22553181C212yesmaximal, complemented
10#22553163C211yesmaximal, characteristic, complemented
11#399×C131C211yesmaximal, characteristic, complemented
12C22×Dic262C2212yes—
13C22×C22⋊D131C2211yescharacteristic
14C23×Dic131C2211yescharacteristic
15C22×C4×D131C2211yescharacteristic
16#16862245—24no—
17#16862224—28no—
18#16862223—28no—
19(C22⋊D131)⋊C4—24no—
20C2×C4⋊Dic131C2212yes—
21C2×C4⋊Dic131C2211yescharacteristic
22C2×Dic131⋊C4C2211yescharacteristic
23C2×Dic131⋊C4C2212yes—
24C2×Dic131⋊C4C2212yes—
25C2×Dic131⋊C4C2212yes—
26C2×Dic131⋊C4C2212yes—
27C2×C22⋊Dic131C2212yes—
28C2×(C2×D131)⋊C4C2212yes—
29C2×(C2×D131)⋊C4C2212yes—
30C2×(C2×D131)⋊C4C2211yescharacteristic
31C2×C4×Dic131C2212yes—
32C2×C4×Dic131C2211yescharacteristic
33C131⋊#95C2211yescharacteristic
34C2×C4⋊C4×C131C2212yes—
35C2×C4⋊C4×C131C2211yescharacteristic
36C2×C22⋊C4×C131C2212yes—
37C2×C22⋊C4×C131C2211yescharacteristic
38#95×C131C2211yescharacteristic
39C23×D131C2311yescharacteristic
40C2×Dic262—24no—
41C2×Dic262—24no—
42C2×Dic262—24no—
43C2×C22⋊D131—22no—
44C2×C22⋊D131—22no—
45C2×C22⋊D131Q814yescomplemented
46C22×Dic131—24no—
47C22×Dic131—22no—
48C22×Dic131C2311yescharacteristic
49C22×Dic131C2312yes—
50C22×Dic131C2312yes—
51C22×Dic131C2311yescharacteristic
52C22×Dic131C2311yescharacteristic
53C2×C4×D131—24no—
54C2×C4×D131—22no—
55C4⋊Dic131—24no—
56C4⋊Dic131—22no—
57Dic131⋊C4—24no—
58Dic131⋊C4—24no—
59Dic131⋊C4—24no—
60Dic131⋊C4—22no—
61Dic131⋊C4D418yescomplemented
62C22⋊Dic131—24no—
63(C2×D131)⋊C4—24no—
64(C2×D131)⋊C4—24no—
65(C2×D131)⋊C4—22no—
66C4×Dic131—24no—
67C4×Dic131D414yes—
68C24×C131C2311yescharacteristic
69C22×C4×C131C2311yescharacteristic
70C22×C4×C131C2312yes—
71C22×C4×C131C2311yescharacteristic
72C22×C4×C131C2312yes—
73C4⋊C4×C131—24no—
74C4⋊C4×C131—22no—
75C22⋊C4×C131—24no—
76C22⋊C4×C131—22no—
77C22×D131—22no—
78C22×D131—21no—
79C22×D131—21no—
80C22×D131—22no—
81C22×D131C2×Q812yes—
82Dic262—48no—
83C22⋊D131—28no—
84C2×Dic131—22no—
85C2×Dic131—22no—
86C2×Dic131—21no—
87C2×Dic131—22no—
88C2×Dic131—22no—
89C2×Dic131—22no—
90C2×Dic131—22no—
91C2×Dic131—44no—
92C2×Dic131C2×D414yes—
93C2×Dic131C2×D412yes—
94C2×Dic131C2×D412yes—
95C2×Dic131C2×D414yes—
96C2×Dic131C2×Q812yes—
97C2×Dic131#4112yes—
98C4×D131—44no—
99C23×C131—22no—
100C23×C131—21no—
101C23×C131—21no—
102C23×C131—22no—
103C23×C131C2×Q812yes—
104C23×C131C2411yescharacteristic
105C2×C4×C131—22no—
106C2×C4×C131—22no—
107C2×C4×C131—22no—
108C2×C4×C131—21no—
109C2×C4×C131—22no—
110C2×C4×C131—22no—
111C2×C4×C131—22no—
112C2×C4×C131—22no—
113C2×C4×C131C2×D412yes—
114C2×C4×C131C2×D414yes—
115C22×C131—22no—
116C22×C131—24no—
117C22×C131—41no—
118C22×C131—41no—
119C22×C131—42no—
120C22×C131C22×D411yescharacteristic
121C22×C131C22×D412yes—
122C22×C131C22×Q811yescharacteristic
123C22×C131C2×#4111yescharacteristic
124C22×C1312−1+412yes—
125C2×D131—22no—
126C2×D131—24no—
127C2×D131—41no—
128C2×D131—41no—
129C2×D131—42no—
130C4×C131—22no—
131C4×C131—24no—
132C4×C131—41no—
133C4×C131—42no—
134Dic131—24no—
135Dic131—41no—
136Dic131—42no—
137Dic131C22⋊Q814yescomplemented
138Dic131#12714yes—
139C2×C131—42no—
140C2×C131C2×C22⋊Q811yescharacteristic
141C2×C131C2×#12711yescharacteristic
142C2×C131#53611yescharacteristic
143C2×C131#54712yes—
144C2×C131D4×Q812yes—
145D131—42no—
146C131#254211yescharacteristic, complemented, Sylow
147#2542—1311nomaximal, Sylow
148C2×C4⋊Q8—1311no—
149C2×C22⋊Q8—1312no—
150C2×C22⋊Q8—1312no—
151C2×C4×D4—1311no—
152#399—1311no—
153#399—1311no—
154#399—1311no—
155#399D13111yescharacteristic, complemented
156#390—1312no—
157#384—1312no—
158C4×C4⋊C4—1311no—
159C22×Q8—1312no—
160C22×D4—1311no—
161C23×C4—1311no—
162C23×C4—1311no—
163C4⋊Q8—2624no—
164C22⋊Q8—2628no—
165C22⋊Q8—2628no—
166C4×D4—2624no—
167C2×C4⋊C4—1311no—
168C2×C4⋊C4—1312no—
169C2×C4⋊C4—1312no—
170C2×C4⋊C4—1312no—
171C2×C4⋊C4—1312no—
172C2×C4⋊C4—1311no—
173C2×C4⋊C4—1312no—
174C2×C4⋊C4C2×D13112yes—
175C2×C4⋊C4C2×D13111yescharacteristic
176C2×C22⋊C4—1312no—
177C2×C22⋊C4—1311no—
178C2×C22⋊C4—1312no—
179C2×C22⋊C4—1312no—
180C2×C22⋊C4C2×D13111yescharacteristic
181C2×C22⋊C4C2×D13112yes—
182C2×C42—1312no—
183C2×C42—1311no—
184#95—1311no—
185#95C2×D13111yescharacteristic
186C24—1311no—
187C24C22×D13111yescharacteristic
188C2×Q8—2624no—
189C2×Q8—2624no—
190C2×Q8—2624no—
191C2×D4—2622no—
192C2×D4—2622no—
193C2×D4—1314no—
194C22×C4—2622no—
195C22×C4—2624no—
196C22×C4—2622no—
197C22×C4—2624no—
198C22×C4—1312no—
199C22×C4—1311no—
200C22×C4—1312no—
201C22×C4—1311no—
202C22×C4—1311no—
203C22×C4C22×D13112yes—
204C22×C4C22×D13111yescharacteristic
205C22×C4C22×D13111yescharacteristic
206C22×C4C22×D13112yes—
207C4⋊C4—24no—
208C4⋊C4—22no—
209C4⋊C4—2622no—
210C4⋊C4—2624no—
211C4⋊C4—2624no—
212C4⋊C4—2624no—
213C4⋊C4—2624no—
214C4⋊C4—2622no—
215C4⋊C4—1318no—
216C22⋊C4—22no—
217C22⋊C4—24no—
218C22⋊C4—2624no—
219C22⋊C4—2624no—
220C22⋊C4—2622no—
221C22⋊C4—2624no—
222C42—2624no—
223C42—1314no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—2622no—
229C23—2621no—
230C23—2621no—
231C23—2622no—
232C23—1312no—
233C23Q8×D13112yes—
234C23C23×D13111yescharacteristic
235Q8—5248no—
236D4—2628no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—2622no—
246C2×C4—2622no—
247C2×C4—2621no—
248C2×C4—2622no—
249C2×C4—2622no—
250C2×C4—2622no—
251C2×C4—2622no—
252C2×C4—5244no—
253C2×C4—5244no—
254C2×C4—1314no—
255C2×C4—1312no—
256C2×C4—1312no—
257C2×C4—1314no—
258C2×C4—1312no—
259C2×C4—1312no—
260C2×C4D4×D13112yes—
261C2×C4D4×D13114yes—
262C22—24no—
263C22—22no—
264C22—41no—
265C22—42no—
266C22—41no—
267C22—2624no—
268C22—2622no—
269C22—5241no—
270C22—5242no—
271C22—5241no—
272C22C2×D4×D13111yescharacteristic
273C22C2×D4×D13112yes—
274C22C2×Q8×D13111yescharacteristic
275C22#41×D13111yescharacteristic
276C22#1686229312yes—
277C4—22no—
278C4—24no—
279C4—41no—
280C4—42no—
281C4—2624no—
282C4—5241no—
283C4—5242no—
284C4—1314no—
285C4—1314no—
286C2—42no—
287C2—5242no—
288C2C22⋊Q8×D13111yescharacteristic
289C2#127×D13111yescharacteristic
290C2#2255388411yescharacteristic
291C2#2255396012yes—
292C2#2255397512yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#399×C131C2
prime radicalC22×C22⋊D131C22
hypercenter#399D131
hypocenterC131#2542
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#16862245C2semidirect
2. C2×#16862224C2semidirect
3. C2×#16862223C2semidirect
5. #22553249C2semidirect
6. #22553248C2semidirect
7. #22553247C2semidirect
9. #22553181C2semidirect
10. #22553163C2semidirect
11. #399×C131C2semidirect
45. C2×C22⋊D131Q8semidirect
61. Dic131⋊C4D4semidirect
137. Dic131C22⋊Q8semidirect
146. C131#2542semidirect
155. #399D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622452
C2×#168622242
C2×#168622232
C2×(C22⋊D131)⋊C42
#225532492
#225532482
#225532472
#225532122
#225531812
#225531632
#399×C1312
#2542131