Finite Group Databasebeta

← #49124498

Subgroups — #49124498

Order 36224.

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

normal subgroups only

This group has 696 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×#22583572C211yesmaximal, characteristic, complemented
2C2×(C22⋊D283)⋊C4C212yesmaximal, complemented
3C2×#22583566C211yesmaximal, characteristic, complemented
4#31826396C211yesmaximal, characteristic, complemented
5#31826389C212yesmaximal, complemented
6#31826383C211yesmaximal, characteristic, complemented
7#31826380C212yesmaximal, complemented
8#31826374C211yesmaximal, characteristic
9#31826316C211yesmaximal, characteristic, complemented
10#31826312C212yesmaximal, complemented
11#397×C283C211yesmaximal, characteristic, complemented
12C22×D1132C2211yescharacteristic
13C22×C22⋊D283C2212yes—
14C23×Dic283C2211yescharacteristic
15C22×C4×D283C2212yes—
16#22583572—24no—
17(C22⋊D283)⋊C4—28no—
18#22583566—24no—
19C2×Dic283⋊C4C2212yes—
20C2×Dic283⋊C4C2211yescharacteristic
21C2×Dic283⋊C4C2211yescharacteristic
22C2×Dic283⋊C4C2212yes—
23C2×C22⋊Dic283C2211yescharacteristic
24C2×(C2×D283)⋊C4C2212yes—
25C2×(C2×D283)⋊C4C2212yes—
26C2×(C2×D283)⋊C4C2212yes—
27C2×(C2×D283)⋊C4C2212yes—
28C2×C4×Dic283C2211yescharacteristic
29C2×C4×Dic283C2212yes—
30C2×C4×Dic283C2211yescharacteristic
31C283⋊#95C2212yes—
32C283⋊#95C2211yescharacteristic
33C2×C4⋊C4×C283C2211yescharacteristic
34C2×C22⋊C4×C283C2211yescharacteristic
35C2×C22⋊C4×C283C2212yes—
36#95×C283C2212yes—
37#95×C283C2211yescharacteristic
38C23×D283C2312yes—
39C2×D1132—24no—
40C2×D1132—22no—
41C2×C22⋊D283—24no—
42C2×C22⋊D283—24no—
43C2×C22⋊D283—24no—
44C22×Dic283—22no—
45C22×Dic283—22no—
46C22×Dic283D414yescomplemented
47C22×Dic283C2311yescharacteristic
48C22×Dic283C2311yescharacteristic
49C22×Dic283C2311yescharacteristic
50C22×Dic283C2312yes—
51C22×Dic283C2311yescharacteristic
52C2×C4×D283—24no—
53C2×C4×D283—24no—
54C2×C4×D283—24no—
55Dic283⋊C4—22no—
56Dic283⋊C4—24no—
57Dic283⋊C4—24no—
58Dic283⋊C4—22no—
59C22⋊Dic283—22no—
60(C2×D283)⋊C4—24no—
61(C2×D283)⋊C4—24no—
62(C2×D283)⋊C4—24no—
63(C2×D283)⋊C4—24no—
64C4×Dic283—22no—
65C4×Dic283—24no—
66C4×Dic283D414yescomplemented
67C24×C283C2311yescharacteristic
68C22×C4×C283C2312yes—
69C22×C4×C283C2311yescharacteristic
70C22×C4×C283C2312yes—
71C22×C4×C283C2311yescharacteristic
72C4⋊C4×C283—22no—
73C22⋊C4×C283—24no—
74C22⋊C4×C283—22no—
75C22×D283—22no—
76C22×D283—22no—
77C22×D283—22no—
78C22×D283—22no—
79C22×D283—22no—
80C22×D283—22no—
81C22×D283#4114yes—
82D1132—28no—
83C22⋊D283—216no—
84C2×Dic283—22no—
85C2×Dic283—21no—
86C2×Dic283—22no—
87C2×Dic283—21no—
88C2×Dic283—22no—
89C2×Dic283—28no—
90C2×Dic283—21no—
91C2×Dic283—22no—
92C2×Dic283C2×D412yes—
93C2×Dic283C2×D412yes—
94C2×Dic283C2×D412yes—
95C2×Dic283C2×D412yes—
96C2×Dic283#4114yes—
97C2×Dic283#4112yes—
98C4×D283—48no—
99C23×C283—21no—
100C23×C283—21no—
101C23×C283—22no—
102C23×C283—22no—
103C23×C283C2×D412yes—
104C23×C283C2411yescharacteristic
105C2×C4×C283—22no—
106C2×C4×C283—22no—
107C2×C4×C283—21no—
108C2×C4×C283—22no—
109C2×C4×C283—22no—
110C2×C4×C283—22no—
111C2×C4×C283—22no—
112C2×C4×C283—22no—
113C2×C4×C283—22no—
114C2×C4×C283C2×D412yes—
115C22×C283—22no—
116C22×C283—24no—
117C22×C283—41no—
118C22×C283—42no—
119C22×C283—41no—
120C22×C283C22×D411yescharacteristic
121C22×C283C22×D411yescharacteristic
122C22×C283C2×#4111yescharacteristic
123C22×C283C2×#4112yes—
124C22×C283C2×#4112yes—
125C2×D283—24no—
126C2×D283—24no—
127C2×D283—24no—
128C2×D283—42no—
129C2×D283—42no—
130C2×D283—42no—
131C2×D283—42no—
132C4×C283—22no—
133C4×C283—42no—
134C4×C283—42no—
135C4×C283—41no—
136Dic283—22no—
137Dic283—24no—
138Dic283—22no—
139Dic283—41no—
140Dic283#12114yes—
141C2×C283—42no—
142C2×C283C2×#12111yescharacteristic
143C2×C283#52512yes—
144C2×C283#52511yescharacteristic
145C2×C283#52912yes—
146C2×C283#53211yescharacteristic
147D283—44no—
148C283#239211yescharacteristic, complemented, Sylow
149#2392—2831nomaximal, Sylow
150C2×C4×D4—2831no—
151C2×C4×D4—2832no—
152C2×#117—2831no—
153#398—2831no—
154#397D28311yescharacteristic, complemented
155#396—2832no—
156#394—2832no—
157#394—2831no—
158#392—2831no—
159C4×C4⋊C4—2831no—
160C4×C22⋊C4—2832no—
161C22×D4—2832no—
162C22×D4—2831no—
163C23×C4—2831no—
164C23×C4—2832no—
165C4×D4—5664no—
166C4×D4—5668no—
167#117—5664no—
168C2×C4⋊C4—2831no—
169C2×C4⋊C4—2831no—
170C2×C4⋊C4—2832no—
171C2×C4⋊C4—2832no—
172C2×C4⋊C4C2×D28311yescharacteristic
173C2×C22⋊C4—2831no—
174C2×C22⋊C4—2832no—
175C2×C22⋊C4—2832no—
176C2×C22⋊C4—2832no—
177C2×C22⋊C4—2832no—
178C2×C22⋊C4C2×D28311yescharacteristic
179C2×C22⋊C4C2×D28312yes—
180C2×C42—2831no—
181C2×C42—2831no—
182C2×C42—2832no—
183#95—2832no—
184#95—2831no—
185#95C2×D28312yes—
186#95C2×D28311yescharacteristic
187C24—2832no—
188C24C22×D28311yescharacteristic
189C2×D4—5664no—
190C2×D4—5664no—
191C2×D4—5664no—
192C2×D4—5664no—
193C2×D4—5662no—
194C22×C4—2831no—
195C22×C4—2834no—
196C22×C4—2832no—
197C22×C4—2831no—
198C22×C4—2831no—
199C22×C4—2831no—
200C22×C4—5664no—
201C22×C4—5664no—
202C22×C4—5664no—
203C22×C4—5662no—
204C22×C4—5662no—
205C22×C4C22×D28312yes—
206C22×C4C22×D28312yes—
207C22×C4C22×D28311yescharacteristic
208C22×C4C22×D28311yescharacteristic
209C4⋊C4—22no—
210C4⋊C4—5664no—
211C4⋊C4—5662no—
212C4⋊C4—5664no—
213C4⋊C4—5662no—
214C22⋊C4—24no—
215C22⋊C4—22no—
216C22⋊C4—5664no—
217C22⋊C4—5664no—
218C22⋊C4—5664no—
219C22⋊C4—5664no—
220C22⋊C4—5662no—
221C42—2834no—
222C42—5662no—
223C42—5664no—
224C23—21no—
225C23—21no—
226C23—22no—
227C23—22no—
228C23—2834no—
229C23—5662no—
230C23—5662no—
231C23—5662no—
232C23—5662no—
233C23—5662no—
234C23—5662no—
235C23D4×D28312yes—
236C23C23×D28311yescharacteristic
237D4—5668no—
238D4—56616no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—2832no—
249C2×C4—2832no—
250C2×C4—2832no—
251C2×C4—2832no—
252C2×C4—2832no—
253C2×C4—2834no—
254C2×C4—5668no—
255C2×C4—5662no—
256C2×C4—5661no—
257C2×C4—5662no—
258C2×C4—5661no—
259C2×C4—5662no—
260C2×C4—5661no—
261C2×C4—5662no—
262C2×C4—11328no—
263C2×C4D4×D28312yes—
264C22—22no—
265C22—24no—
266C22—41no—
267C22—42no—
268C22—41no—
269C22—5664no—
270C22—5664no—
271C22—5664no—
272C22—11322no—
273C22—11322no—
274C22—11322no—
275C22—11322no—
276C22C2×D4×D28311yescharacteristic
277C22C2×D4×D28311yescharacteristic
278C22#41×D28311yescharacteristic
279C22#41×D28312yes—
280C22#41×D28312yes—
281C4—22no—
282C4—42no—
283C4—42no—
284C4—41no—
285C4—2834no—
286C4—5662no—
287C4—5664no—
288C4—5662no—
289C4—11321no—
290C2—42no—
291C2—11324no—
292C2#121×D28311yescharacteristic
293C2#3182697311yescharacteristic
294C2#3182697312yes—
295C2#3182699812yes—
296C2#3182701411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D283
commutator, socle, cosocleC23×C283C24
Fitting#397×C283C2
prime radical#31826374C2
hypercenter#397D283
hypocenterC283#2392
LCSG ▷ C23×C283 ▷ C283
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#22583572C2semidirect
2. C2×(C22⋊D283)⋊C4C2semidirect
3. C2×#22583566C2semidirect
4. #31826396C2semidirect
5. #31826389C2semidirect
6. #31826383C2semidirect
7. #31826380C2semidirect
9. #31826316C2semidirect
10. #31826312C2semidirect
11. #397×C283C2semidirect
46. C22×Dic283D4semidirect
66. C4×Dic283D4semidirect
148. C283#2392semidirect
154. #397D283semidirect

Maximal subgroups

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

subgroupindex
C2×#225835722
C2×(C22⋊D283)⋊C42
C2×#225835662
#318263962
#318263892
#318263832
#318263802
#318263742
#318263162
#318263122
#397×C2832
#2392283