Finite Group Databasebeta

← #49130519

Subgroups — #49130519

Order 36224.

754 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 153 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
1C22×#17985392C211yesmaximal, characteristic
2C2×#22583641C212yesmaximal, complemented
3#31826625C214yesmaximal, complemented
4#31826620C212yesmaximal, complemented
5#31826615C212yesmaximal, complemented
6#31826464C212yesmaximal, complemented
7#31826429C211yesmaximal, characteristic, complemented
8C2×C22⋊Q8×C283C211yesmaximal, characteristic, complemented
9C2×#17985392—28no—
10C2×#17985392—21no—
11C2×#17985392—21no—
12C2×#17985392C2212yes—
13C2×#17985392C2212yes—
14C22×Dic566C2211yescharacteristic
15C22×D1132C2211yescharacteristic
16C22×C22⋊D283—21no—
17C22×C4×D283—21no—
18#22583641—28no—
19C2×#17985382C2214yescomplemented
20C2×#17985379C2214yescomplemented
21C2×#17985376C2212yescomplemented
22#22583533C2214yescomplemented
23#22583530C2214yescomplemented
24C22⋊(C283⋊C8)C2212yes—
25C22⋊(C283⋊C8)C2212yescomplemented
26C22×Q8×C283C2211yescharacteristic, complemented
27C23×C4×C283C2211yescharacteristic
28C22⋊Q8×C283—21no—
29C22⋊Q8×C283—21no—
30C22⋊Q8×C283C2212yes—
31C22⋊Q8×C283C2212yescomplemented
32C2×C4⋊C4×C283—21no—
33C2×C4⋊C4×C283C2211yescharacteristic, complemented
34C2×C22⋊C4×C283—21no—
35C23×D283—21no—
36#17985392—28no—
37#17985392—28no—
38#17985392—24no—
39#17985392—24no—
40#17985392—24no—
41#17985392D418yes—
42C2×Dic566—24no—
43C2×Dic566—21no—
44C2×Dic566C2312yes—
45C2×D1132—24no—
46C2×D1132—21no—
47C2×D1132C2312yes—
48C2×C22⋊D283—28no—
49C2×C22⋊D283—22no—
50C2×C22⋊D283—22no—
51C22×Dic283—21no—
52C2×C4×D283—28no—
53C2×C4×D283—22no—
54C2×C4×D283—22no—
55#17985382—28no—
56#17985379—28no—
57#17985376—24no—
58C2×C283⋊C8C2314yes—
59C24×C283D411yescharacteristic
60C2×Q8×C283—21no—
61C2×Q8×C283—24no—
62C2×Q8×C283C2312yes—
63C22×C4×C283—24no—
64C22×C4×C283—21no—
65C22×C4×C283—21no—
66C22×C4×C283D411yescharacteristic
67C22×C4×C283D411yescharacteristic
68C22×C4×C283D411yescharacteristic
69C22×C4×C283C2311yescharacteristic
70C22×C4×C283C2311yescharacteristic
71C22×C4×C283C2311yescharacteristic
72C4⋊C4×C283—22no—
73C4⋊C4×C283—22no—
74C4⋊C4×C283—21no—
75C4⋊C4×C283C2312yes—
76C22⋊C4×C283—22no—
77C22⋊C4×C283—22no—
78C22×D283—24no—
79C22×D283—22no—
80C22×D283—42no—
81C22×D283—42no—
82Dic566—24no—
83Dic566—22no—
84Dic566C2×D414yes—
85D1132—24no—
86D1132—22no—
87D1132C2×D414yes—
88C22⋊D283—28no—
89C22⋊D283—28no—
90C2×Dic283—24no—
91C2×Dic283—22no—
92C4×D283—28no—
93C4×D283—28no—
94C283⋊C8—24no—
95C23×C283—22no—
96C23×C283—22no—
97C23×C283—41no—
98C23×C283C2×D411yescharacteristic
99C23×C283C2×D411yescharacteristic
100C23×C283C2×D411yescharacteristic
101Q8×C283—24no—
102Q8×C283—42no—
103C2×C4×C283—22no—
104C2×C4×C283—22no—
105C2×C4×C283—22no—
106C2×C4×C283—22no—
107C2×C4×C283—22no—
108C2×C4×C283—22no—
109C2×C4×C283—22no—
110C2×C4×C283—42no—
111C2×C4×C283—42no—
112C2×C4×C283C2×D414yes—
113C2×C4×C283C2×D411yescharacteristic
114C2×C4×C283C2×D411yescharacteristic
115C2×C4×C283C2×D411yescharacteristic
116C2×C4×C283C2411yescharacteristic
117C22×C283—21no—
118C22×C283—21no—
119C22×C283—21no—
120C22×C283—21no—
121C22×C283—21no—
122C22×C283—21no—
123C22×C283—21no—
124C22×C283—44no—
125C22×C283#12012yes—
126C22×C283#13712yescomplemented
127C22×C283C22×D411yescharacteristic
128C2×D283—24no—
129C2×D283—48no—
130C2×D283—42no—
131C2×D283—42no—
132C4×C283—21no—
133C4×C283—21no—
134C4×C283—42no—
135C4×C283—42no—
136C4×C283#12012yes—
137C4×C283C22×D412yes—
138Dic283—24no—
139C2×C283—22no—
140C2×C283—22no—
141C2×C283—41no—
142C2×C283C2×#12011yescharacteristic
143C2×C283C2×#13711yescharacteristic
144C2×C283#57511yescharacteristic
145D283—44no—
146C283#300711yescharacteristic, complemented, Sylow
147#3007—2831nomaximal, Sylow
148C22×#41—2831no—
149C2×#137—2832no—
150C2×C22⋊Q8D28311yescharacteristic, complemented
151#452—2834no—
152#451—2832no—
153#450—2832no—
154#417—2832no—
155#407—2831no—
156C2×#41—2832no—
157C2×#41—2832no—
158C2×#41—5668no—
159C2×#41—5661no—
160C2×#41—5661no—
161C22×Q8—2831no—
162C22×Q8C2×D28311yescharacteristic, complemented
163C22×D4—2831no—
164C22×D4—5661no—
165C23×C4—5661no—
166C23×C4C2×D28311yescharacteristic
167#137—5668no—
168C2×Q16—2834no—
169C2×SD16—2834no—
170C2×M16—2832no—
171C22⋊Q8—21no—
172C22⋊Q8—21no—
173C22⋊Q8C2×D28312yes—
174C22⋊Q8C2×D28312yescomplemented
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×D28311yescharacteristic, complemented
177C2×C22⋊C4—21no—
178#103—2834no—
179#102—2834no—
180C22⋊C8—2832no—
181C22⋊C8—2832no—
182C24—5661no—
183C24C22⋊D28311yescharacteristic
184#41—2838no—
185#41—5664no—
186#41—5664no—
187#41—5668no—
188#41—5668no—
189#41—5664no—
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—2832no—
193C2×Q8—5664no—
194C2×Q8—5661no—
195C2×Q8C22×D28312yes—
196C2×D4—2832no—
197C2×D4—5664no—
198C2×D4—5668no—
199C2×D4—5662no—
200C2×D4—5662no—
201C2×D4—5661no—
202C22×C4—24no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—5668no—
206C22×C4—5661no—
207C22×C4—5662no—
208C22×C4—5662no—
209C22×C4C22⋊D28311yescharacteristic
210C22×C4C22⋊D28311yescharacteristic
211C22×C4C22⋊D28311yescharacteristic
212C22×C4C22×D28311yescharacteristic
213C22×C4C22×D28311yescharacteristic
214C22×C4C22×D28311yescharacteristic
215Q16—5668no—
216SD16—5668no—
217M16—5664no—
218C2×C8—2834no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4C22×D28312yes—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—5664no—
229C23—5662no—
230C23—11322no—
231C23—11322no—
232C23C2×C22⋊D28311yescharacteristic
233C23C2×C22⋊D28311yescharacteristic
234C23C2×C22⋊D28311yescharacteristic
235Q8—24no—
236Q8—42no—
237Q8—2834no—
238Q8—5664no—
239Q8—5662no—
240D4—2834no—
241D4—5664no—
242D4—5668no—
243D4—5668no—
244D4—5662no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—5668no—
255C2×C4—5664no—
256C2×C4—5662no—
257C2×C4—5668no—
258C2×C4D4×D28314yes—
259C2×C4C2×C22⋊D28311yescharacteristic
260C2×C4C2×C22⋊D28311yescharacteristic
261C2×C4C2×C22⋊D28311yescharacteristic
262C2×C4C23×D28311yescharacteristic
263C8—5664no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22—5664no—
273C22—11328no—
274C22—11322no—
275C22—11322no—
276C22#2258357812yes—
277C22#2258364112yescomplemented
278C22C22×C22⋊D28311yescharacteristic
279C4—21no—
280C4—21no—
281C4—42no—
282C4—42no—
283C4—5664no—
284C4#2258357812yes—
285C4C2×D4×D28312yes—
286C2—22no—
287C2—22no—
288C2—41no—
289C2—11324no—
290C2C2×#2258357811yescharacteristic
291C2C2×#2258364111yescharacteristic
292C2#3182726311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
FittingC2×C22⋊Q8×C283C2
socleC22×C283C22×D4
prime radicalC22×#17985392C2
hypercenterC2×C22⋊Q8D283
hypocenterC283#3007
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#22583641C2semidirect
3. #31826625C2semidirect
4. #31826620C2semidirect
5. #31826615C2semidirect
6. #31826464C2semidirect
7. #31826429C2semidirect
8. C2×C22⋊Q8×C283C2semidirect
19. C2×#17985382C22semidirect
20. C2×#17985379C22semidirect
21. C2×#17985376C22semidirect
22. #22583533C22semidirect
23. #22583530C22semidirect
25. C22⋊(C283⋊C8)C22semidirect
26. C22×Q8×C283C22semidirect
31. C22⋊Q8×C283C22semidirect
33. C2×C4⋊C4×C283C22semidirect
126. C22×C283#137semidirect
146. C283#3007semidirect
150. C2×C22⋊Q8D283semidirect
162. C22×Q8C2×D283semidirect
174. C22⋊Q8C2×D283semidirect
176. C2×C4⋊C4C2×D283semidirect
277. C22#22583641semidirect

Maximal subgroups

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

subgroupindex
C22×#179853922
C2×#225836412
#318266252
#318266202
#318266152
#318264642
#318264292
C2×C22⋊Q8×C2832
#3007283