Finite Group Databasebeta

← #30670268

Subgroups — #30670268

Order 16768.

624 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 157 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×#16862230C211yesmaximal, characteristic
2C2×#16862229C211yesmaximal, characteristic
3C2×(C22⋊D131)⋊C4C211yesmaximal, characteristic
4C22×Dic131⋊C4C211yesmaximal, characteristic, complemented
5#22553246C212yesmaximal, complemented
6#22553241C212yesmaximal, complemented
7#22553209C212yesmaximal, complemented
8#22553173C211yesmaximal, characteristic, complemented
9#22553171C212yesmaximal, complemented
10#22553170C211yesmaximal, characteristic, complemented
11#381×C131C211yesmaximal, characteristic, complemented
12C22×C22⋊D131C2211yescharacteristic
13C23×Dic131C2211yescharacteristic
14C23×Dic131C2211yescharacteristic
15C22×C4×D131C2211yescharacteristic
16#16862230—24no—
17#16862229—24no—
18(C22⋊D131)⋊C4—24no—
19C2×C4⋊Dic131C2212yes—
20C2×Dic131⋊C4—24no—
21C2×Dic131⋊C4C2212yes—
22C2×Dic131⋊C4C2212yes—
23C2×Dic131⋊C4C2212yes—
24C2×Dic131⋊C4C2212yes—
25C2×Dic131⋊C4C2211yescharacteristic
26C2×C22⋊Dic131C2211yescharacteristic
27C2×(C2×D131)⋊C4C2212yes—
28C2×(C2×D131)⋊C4C2212yes—
29C2×(C2×D131)⋊C4C2211yescharacteristic
30C2×C4×Dic131C2211yescharacteristic
31C131⋊#95C2212yes—
32C131⋊#95C2212yes—
33C131⋊#95C2212yes—
34C23×C4×C131C2211yescharacteristic
35C2×C22⋊C4×C131C2211yescharacteristic
36C2×C22⋊C4×C131C2211yescharacteristic
37#95×C131C2212yes—
38#95×C131C2212yes—
39C23×D131C2311yescharacteristic
40C2×C22⋊D131—22no—
41C2×C22⋊D131—22no—
42C2×C22⋊D131D414yes—
43C22×Dic131—24no—
44C22×Dic131—22no—
45C22×Dic131—22no—
46C22×Dic131—22no—
47C22×Dic131D414yes—
48C22×Dic131C2312yes—
49C22×Dic131C2312yes—
50C22×Dic131C2311yescharacteristic
51C22×Dic131C2311yescharacteristic
52C22×Dic131C2311yescharacteristic
53C2×C4×D131—22no—
54C2×C4×D131—24no—
55C4⋊Dic131—24no—
56Dic131⋊C4—24no—
57Dic131⋊C4—24no—
58Dic131⋊C4—24no—
59Dic131⋊C4—22no—
60Dic131⋊C4—24no—
61C22⋊Dic131—22no—
62(C2×D131)⋊C4—24no—
63(C2×D131)⋊C4—24no—
64(C2×D131)⋊C4—22no—
65C4×Dic131—22no—
66C24×C131C2311yescharacteristic
67C22×C4×C131—22no—
68C22×C4×C131—22no—
69C22×C4×C131—22no—
70C22×C4×C131C2311yescharacteristic
71C22×C4×C131C2311yescharacteristic
72C22×C4×C131C2312yes—
73C22×C4×C131C2312yes—
74C22⋊C4×C131—22no—
75C22⋊C4×C131—22no—
76C22×D131—21no—
77C22×D131—22no—
78C22×D131—21no—
79C22×D131—22no—
80C22×D131C2×D412yes—
81C22⋊D131—28no—
82C2×Dic131—21no—
83C2×Dic131—22no—
84C2×Dic131—21no—
85C2×Dic131—28no—
86C2×Dic131—22no—
87C2×Dic131—22no—
88C2×Dic131—22no—
89C2×Dic131—22no—
90C2×Dic131—22no—
91C2×Dic131—22no—
92C2×Dic131—44no—
93C2×Dic131C2×D412yes—
94C2×Dic131C2×D412yes—
95C2×Dic131C2×D412yes—
96C2×Dic131#4114yes—
97C4×D131—44no—
98C23×C131—22no—
99C23×C131—22no—
100C23×C131C2×D412yes—
101C23×C131C2×D412yes—
102C23×C131#4112yes—
103C23×C131C2411yescharacteristic
104C2×C4×C131—21no—
105C2×C4×C131—22no—
106C2×C4×C131—21no—
107C2×C4×C131—22no—
108C2×C4×C131—22no—
109C2×C4×C131—22no—
110C2×C4×C131—22no—
111C2×C4×C131—22no—
112C2×C4×C131—22no—
113C2×C4×C131—22no—
114C2×C4×C131—44no—
115C22×C131—22no—
116C22×C131—22no—
117C22×C131—24no—
118C22×C131—24no—
119C22×C131#12114yes—
120C22×C131C22×D411yescharacteristic
121C22×C131C22×D411yescharacteristic
122C22×C131C2×#4112yes—
123C22×C131C2×#4111yescharacteristic
124C22×C1312−1+412yes—
125C2×D131—22no—
126C2×D131—24no—
127C2×D131—41no—
128C2×D131—41no—
129C2×D131—42no—
130C4×C131—42no—
131C4×C131—41no—
132C4×C131—41no—
133C4×C131—42no—
134Dic131—22no—
135Dic131—22no—
136Dic131—42no—
137Dic131—42no—
138Dic131#12314yescomplemented
139C2×C131—24no—
140C2×C131C2×#12111yescharacteristic
141C2×C131C2×#12311yescharacteristic
142C2×C131#53611yescharacteristic
143C2×C131#54712yes—
144C2×C131#55212yes—
145D131—42no—
146C131#248611yescharacteristic, complemented, Sylow
147#2486—1311nomaximal, Sylow
148C2×#123—1311no—
149C2×#123—1311no—
150C2×C4×D4—1311no—
151C22×C4⋊C4—1311no—
152#398—1312no—
153#397—1312no—
154#389—1312no—
155#381—1311no—
156#381D13111yescharacteristic, complemented
157#380—1311no—
158#380—1312no—
159C22×D4—1311no—
160C23×C4—1311no—
161C23×C4—1311no—
162C23×C4—1311no—
163C23×C4C2×D13111yescharacteristic
164#123—2624no—
165#123—2624no—
166C4×D4—2624no—
167C2×C4⋊C4—2624no—
168C2×C4⋊C4—1312no—
169C2×C4⋊C4—1312no—
170C2×C4⋊C4—1311no—
171C2×C4⋊C4—1312no—
172C2×C4⋊C4—1312no—
173C2×C4⋊C4—1312no—
174C2×C22⋊C4—1311no—
175C2×C22⋊C4—1311no—
176C2×C22⋊C4—1312no—
177C2×C22⋊C4—1312no—
178C2×C22⋊C4C2×D13111yescharacteristic
179C2×C22⋊C4C2×D13111yescharacteristic
180C2×C42—1311no—
181#95—1312no—
182#95—1312no—
183#95—1312no—
184#95C2×D13112yes—
185#95C2×D13112yes—
186C24—1311no—
187C24C22×D13111yescharacteristic
188C2×D4—2622no—
189C2×D4—2622no—
190C2×D4—1314no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—2622no—
195C22×C4—2624no—
196C22×C4—2624no—
197C22×C4—2622no—
198C22×C4—2622no—
199C22×C4—2622no—
200C22×C4—1312no—
201C22×C4—1312no—
202C22×C4—1314no—
203C22×C4—1311no—
204C22×C4—1311no—
205C22×C4—1311no—
206C22×C4C22×D13111yescharacteristic
207C22×C4C22×D13111yescharacteristic
208C22×C4C22×D13112yes—
209C22×C4C22×D13112yes—
210C4⋊C4—2624no—
211C4⋊C4—2624no—
212C4⋊C4—2624no—
213C4⋊C4—2622no—
214C4⋊C4—2624no—
215C4⋊C4—2624no—
216C22⋊C4—22no—
217C22⋊C4—22no—
218C22⋊C4—2624no—
219C22⋊C4—2624no—
220C22⋊C4—2622no—
221C22⋊C4—2622no—
222C42—2622no—
223C23—22no—
224C23—22no—
225C23—2621no—
226C23—2622no—
227C23—2621no—
228C23—2622no—
229C23—1312no—
230C23D4×D13112yes—
231C23D4×D13112yes—
232C23#1452351312yes—
233C23C23×D13111yescharacteristic
234D4—2628no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—2621no—
247C2×C4—2622no—
248C2×C4—2621no—
249C2×C4—2628no—
250C2×C4—2622no—
251C2×C4—2622no—
252C2×C4—2622no—
253C2×C4—2622no—
254C2×C4—2622no—
255C2×C4—2622no—
256C2×C4—5244no—
257C2×C4—5244no—
258C2×C4—1312no—
259C2×C4—1312no—
260C2×C4—1312no—
261C2×C4—1314no—
262C22—22no—
263C22—24no—
264C22—24no—
265C22—22no—
266C22—2624no—
267C22—2622no—
268C22—5241no—
269C22—5242no—
270C22—5241no—
271C22#1686221914yes—
272C22C2×D4×D13111yescharacteristic
273C22C2×D4×D13111yescharacteristic
274C22#41×D13112yes—
275C22C2×#1452351311yescharacteristic
276C22#1686229312yes—
277C4—41no—
278C4—42no—
279C4—42no—
280C4—41no—
281C4—2622no—
282C4—2622no—
283C4—5242no—
284C4—5242no—
285C4—1314no—
286C2—24no—
287C2—5242no—
288C2C2×#1686221911yescharacteristic
289C2#123×D13111yescharacteristic
290C2#2255388411yescharacteristic
291C2#2255396012yes—
292C2#2255399512yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#381×C131C2
prime radicalC22×C22⋊D131C22
hypercenter#381D131
hypocenterC131#2486
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
4. C22×Dic131⋊C4C2semidirect
5. #22553246C2semidirect
6. #22553241C2semidirect
7. #22553209C2semidirect
8. #22553173C2semidirect
9. #22553171C2semidirect
10. #22553170C2semidirect
11. #381×C131C2semidirect
138. Dic131#123semidirect
146. C131#2486semidirect
156. #381D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622302
C2×#168622292
C2×(C22⋊D131)⋊C42
C22×Dic131⋊C42
#225532462
#225532412
#225532092
#225531732
#225531712
#225531702
#381×C1312
#2486131