Finite Group Databasebeta

← #30670952

Subgroups — #30670952

Order 16768.

604 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 151 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×#16862232C212yesmaximal
3C2×(C22⋊D131)⋊C4C211yesmaximal, characteristic
4#22553239C211yesmaximal, characteristic, complemented
5#22553238C211yesmaximal, characteristic, complemented
6#22553237C211yesmaximal, characteristic, complemented
7#22553216C212yesmaximal, complemented
8#22553171C212yesmaximal, complemented
9#22553170C212yesmaximal, complemented
10#22553163C211yesmaximal, characteristic, complemented
11#397×C131C211yesmaximal, characteristic, complemented
12C22×Dic262C2212yes—
13C22×C22⋊D131C2211yescharacteristic
14C23×Dic131C2211yescharacteristic
15C22×C4×D131C2211yescharacteristic
16#16862245—24no—
17#16862232—28no—
18(C22⋊D131)⋊C4—24no—
19C2×C4⋊Dic131C2211yescharacteristic
20C2×Dic131⋊C4C2212yes—
21C2×Dic131⋊C4C2212yes—
22C2×Dic131⋊C4C2211yescharacteristic
23C2×C22⋊Dic131C2212yes—
24C2×(C2×D131)⋊C4C2212yes—
25C2×(C2×D131)⋊C4C2212yes—
26C2×(C2×D131)⋊C4C2211yescharacteristic
27C2×C4×Dic131C2212yes—
28C2×C4×Dic131C2211yescharacteristic
29C131⋊#95C2211yescharacteristic
30C131⋊#95C2212yes—
31C131⋊#95C2212yes—
32C131⋊#95C2212yes—
33C2×C4⋊C4×C131C2211yescharacteristic
34C2×C22⋊C4×C131C2212yes—
35C2×C22⋊C4×C131C2211yescharacteristic
36#95×C131C2211yescharacteristic
37#95×C131C2212yes—
38C23×D131C2311yescharacteristic
39C2×Dic262—24no—
40C2×Dic262—24no—
41C2×Dic262—24no—
42C2×C22⋊D131—22no—
43C2×C22⋊D131—22no—
44C2×C22⋊D131D414yes—
45C22×Dic131—22no—
46C22×Dic131—24no—
47C22×Dic131C2311yescharacteristic
48C22×Dic131C2312yes—
49C22×Dic131C2312yes—
50C22×Dic131C2311yescharacteristic
51C22×Dic131C2311yescharacteristic
52C2×C4×D131—22no—
53C2×C4×D131—24no—
54C4⋊Dic131—22no—
55Dic131⋊C4—24no—
56Dic131⋊C4—22no—
57Dic131⋊C4—24no—
58C22⋊Dic131—24no—
59(C2×D131)⋊C4—24no—
60(C2×D131)⋊C4—24no—
61(C2×D131)⋊C4—22no—
62C4×Dic131—24no—
63C4×Dic131D414yes—
64C24×C131C2311yescharacteristic
65C22×C4×C131C2311yescharacteristic
66C22×C4×C131C2312yes—
67C22×C4×C131C2312yes—
68C22×C4×C131C2311yescharacteristic
69C4⋊C4×C131—22no—
70C22⋊C4×C131—24no—
71C22⋊C4×C131—22no—
72C22×D131—21no—
73C22×D131—22no—
74C22×D131—21no—
75C22×D131—22no—
76C22×D131C2×D412yes—
77Dic262—48no—
78C22⋊D131—28no—
79C2×Dic131—22no—
80C2×Dic131—22no—
81C2×Dic131—21no—
82C2×Dic131—22no—
83C2×Dic131—22no—
84C2×Dic131—22no—
85C2×Dic131—22no—
86C2×Dic131—22no—
87C2×Dic131—44no—
88C2×Dic131C2×D412yes—
89C2×Dic131C2×D412yes—
90C2×Dic131C2×D412yes—
91C2×Dic131#4112yes—
92C2×Dic131#4114yes—
93C4×D131—44no—
94C23×C131—21no—
95C23×C131—22no—
96C23×C131—21no—
97C23×C131—22no—
98C23×C131C2×D412yes—
99C23×C131C2411yescharacteristic
100C2×C4×C131—22no—
101C2×C4×C131—22no—
102C2×C4×C131—22no—
103C2×C4×C131—21no—
104C2×C4×C131—22no—
105C2×C4×C131—22no—
106C2×C4×C131—22no—
107C2×C4×C131—22no—
108C2×C4×C131—22no—
109C2×C4×C131C2×D412yes—
110C22×C131—22no—
111C22×C131—24no—
112C22×C131—41no—
113C22×C131—41no—
114C22×C131—42no—
115C22×C131C22×D411yescharacteristic
116C22×C131C22×D411yescharacteristic
117C22×C131C2×#4111yescharacteristic
118C22×C131C2×#4112yes—
119C22×C1312−1+412yes—
120C2×D131—22no—
121C2×D131—24no—
122C2×D131—41no—
123C2×D131—41no—
124C2×D131—42no—
125C4×C131—22no—
126C4×C131—42no—
127C4×C131—41no—
128C4×C131—42no—
129Dic131—42no—
130Dic131—41no—
131Dic131—42no—
132Dic131#12114yes—
133Dic131#12414yescomplemented
134C2×C131—42no—
135C2×C131C2×#12111yescharacteristic
136C2×C131C2×#12411yescharacteristic
137C2×C131#53611yescharacteristic
138C2×C131#54712yes—
139C2×C131#55612yes—
140D131—42no—
141C131#254711yescharacteristic, complemented, Sylow
142#2547—1311nomaximal, Sylow
143C2×C4⋊Q8—1311no—
144C2×#124—1312no—
145C2×C4×D4—1311no—
146#397—1311no—
147#397—1311no—
148#397—1311no—
149#397D13111yescharacteristic, complemented
150#391—1312no—
151#380—1312no—
152#380—1312no—
153C4×C4⋊C4—1311no—
154C22×Q8—1312no—
155C22×D4—1311no—
156C23×C4—1311no—
157C23×C4—1311no—
158C4⋊Q8—2624no—
159#124—2628no—
160C4×D4—2624no—
161C2×C4⋊C4—1311no—
162C2×C4⋊C4—1312no—
163C2×C4⋊C4—1311no—
164C2×C4⋊C4—1312no—
165C2×C4⋊C4C2×D13111yescharacteristic
166C2×C22⋊C4—1312no—
167C2×C22⋊C4—1311no—
168C2×C22⋊C4—1312no—
169C2×C22⋊C4—1312no—
170C2×C22⋊C4C2×D13111yescharacteristic
171C2×C22⋊C4C2×D13112yes—
172C2×C42—1312no—
173C2×C42—1311no—
174#95—1311no—
175#95—1312no—
176#95—1312no—
177#95—1312no—
178#95C2×D13111yescharacteristic
179#95C2×D13112yes—
180C24—1311no—
181C24C22×D13111yescharacteristic
182C2×Q8—2624no—
183C2×Q8—2624no—
184C2×Q8—2624no—
185C2×D4—2622no—
186C2×D4—2622no—
187C2×D4—1314no—
188C22×C4—2622no—
189C22×C4—2624no—
190C22×C4—2622no—
191C22×C4—2624no—
192C22×C4—1311no—
193C22×C4—1312no—
194C22×C4—1312no—
195C22×C4—1311no—
196C22×C4—1311no—
197C22×C4C22×D13111yescharacteristic
198C22×C4C22×D13112yes—
199C22×C4C22×D13112yes—
200C22×C4C22×D13111yescharacteristic
201C4⋊C4—22no—
202C4⋊C4—2624no—
203C4⋊C4—2622no—
204C4⋊C4—2624no—
205C4⋊C4—2622no—
206C22⋊C4—24no—
207C22⋊C4—22no—
208C22⋊C4—2624no—
209C22⋊C4—2624no—
210C22⋊C4—2624no—
211C22⋊C4—2622no—
212C42—2624no—
213C42—1314no—
214C23—21no—
215C23—22no—
216C23—21no—
217C23—22no—
218C23—2621no—
219C23—2622no—
220C23—2621no—
221C23—2622no—
222C23—1312no—
223C23D4×D13112yes—
224C23C23×D13111yescharacteristic
225Q8—5248no—
226D4—2628no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—2622no—
237C2×C4—2622no—
238C2×C4—2621no—
239C2×C4—2622no—
240C2×C4—2622no—
241C2×C4—2622no—
242C2×C4—2622no—
243C2×C4—2622no—
244C2×C4—5244no—
245C2×C4—5244no—
246C2×C4—1312no—
247C2×C4—1312no—
248C2×C4—1312no—
249C2×C4—1314no—
250C2×C4—1312no—
251C2×C4D4×D13112yes—
252C22—24no—
253C22—22no—
254C22—41no—
255C22—42no—
256C22—41no—
257C22—2624no—
258C22—2622no—
259C22—5241no—
260C22—5242no—
261C22—5241no—
262C22C2×D4×D13111yescharacteristic
263C22C2×D4×D13111yescharacteristic
264C22#41×D13112yes—
265C22#41×D13111yescharacteristic
266C22#1686229312yes—
267C4—22no—
268C4—42no—
269C4—42no—
270C4—41no—
271C4—5242no—
272C4—5242no—
273C4—5241no—
274C4—1314no—
275C4—1314no—
276C2—42no—
277C2—5242no—
278C2#121×D13111yescharacteristic
279C2#124×D13111yescharacteristic
280C2#2255388411yescharacteristic
281C2#2255396012yes—
282C2#2255402912yes—
subgroupG/H
center, FrattiniC23C23×D131
commutator, socle, cosocleC23×C131C24
Fitting#397×C131C2
prime radicalC22×C22⋊D131C22
hypercenter#397D131
hypocenterC131#2547
LCSG ▷ C23×C131 ▷ C131
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#16862245C2semidirect
4. #22553239C2semidirect
5. #22553238C2semidirect
6. #22553237C2semidirect
7. #22553216C2semidirect
8. #22553171C2semidirect
9. #22553170C2semidirect
10. #22553163C2semidirect
11. #397×C131C2semidirect
133. Dic131#124semidirect
141. C131#2547semidirect
149. #397D131semidirect

Maximal subgroups

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

subgroupindex
C2×#168622452
C2×#168622322
C2×(C22⋊D131)⋊C42
#225532392
#225532382
#225532372
#225532162
#225531712
#225531702
#225531632
#397×C1312
#2547131