Finite Group Databasebeta

← #46783250

Subgroups — #46783250

Order 33664.

332 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 93 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
1#41×Dic263C211yesmaximal, characteristic, complemented
2#30681955C211yesmaximal, characteristic, complemented
3C2×#22554397C211yesmaximal, characteristic, complemented
4#30681912C211yesmaximal, characteristic, complemented
5#30681710C211yesmaximal, characteristic, complemented
6#30681699C211yesmaximal, characteristic, complemented
7#575×C263C211yesmaximal, characteristic
8C2×#16862360C2211yescharacteristic
9Q8×Dic263—21no—
10D4×Dic263—22no—
11D4×Dic263—21no—
12#22554429—21no—
13#22554429C2211yescharacteristic
14#22554429C2211yescharacteristic
15C2×C4×Dic263—21no—
16C2×C4×Dic263C2211yescharacteristic
17#22554397—21no—
18#22554397—21no—
19#22554397—22no—
20#22554385—22no—
21C2×#41×C263C2211yescharacteristic
22C2×#41×C263C2211yescharacteristic
23#137×C263—21no—
24#137×C263C412yescomplemented
25AGL(1,ℤ/8)×C263—21no—
26AGL(1,ℤ/8)×C263C412yescomplemented
27#135×C263—22no—
28C2×M16×C263C2211yescharacteristic
29C22×Dic263—22no—
30C22×Dic263—21no—
31#16862360—21no—
32#16862360—21no—
33C2×C263⋊C8—21no—
34C4⋊Dic263—22no—
35C4⋊Dic263—21no—
36C4⋊Dic263—21no—
37C22⋊Dic263—22no—
38C22⋊Dic263—21no—
39C22⋊Dic263D412yes—
40C4×Dic263—22no—
41C4×Dic263—21no—
42C4×Dic263—21no—
43C4×Dic263—21no—
44C4×Dic263D411yescharacteristic
45C4×Dic263D411yescharacteristic
46#41×C263—21no—
47#41×C263—21no—
48#41×C263—21no—
49#41×C263—21no—
50#41×C263—21no—
51#41×C263—21no—
52#41×C263C2×C412yes—
53#41×C263D412yes—
54C2×Q8×C263C2×C411yescharacteristic, complemented
55C2×Q8×C263D411yescharacteristic
56C2×D4×C263—21no—
57C2×D4×C263—21no—
58C2×D4×C263C2×C411yescharacteristic, complemented
59C2×D4×C263D411yescharacteristic
60C22×C4×C263—21no—
61C22×C4×C263—21no—
62C22×C4×C263C2311yescharacteristic
63Q16×C263—22no—
64SD16×C263—22no—
65SD16×C263—22no—
66D8×C263—22no—
67M16×C263—21no—
68M16×C263C2×C412yescomplemented
69C2×C8×C263—21no—
70C2×Dic263—21no—
71C2×Dic263—22no—
72C2×Dic263—21no—
73C2×Dic263—42no—
74C2×Dic263—44no—
75C2×Dic263—41no—
76C263⋊C8—41no—
77C23×C263—21no—
78C23×C263—21no—
79C23×C263C2×D411yescharacteristic
80Q8×C263—21no—
81Q8×C263—21no—
82Q8×C263—21no—
83Q8×C263C22⋊C412yes—
84D4×C263—22no—
85D4×C263—22no—
86D4×C263—21no—
87D4×C263—21no—
88D4×C263—21no—
89D4×C263—41no—
90D4×C263—41no—
91D4×C263C22⋊C412yes—
92C2×C4×C263—21no—
93C2×C4×C263—22no—
94C2×C4×C263—22no—
95C2×C4×C263—21no—
96C2×C4×C263—41no—
97C2×C4×C263—41no—
98C2×C4×C263C22×C411yescharacteristic
99C2×C4×C263C2×D411yescharacteristic
100C2×C4×C263C2×D411yescharacteristic
101C2×C4×C263C2×D411yescharacteristic
102C2×C4×C263#4111yescharacteristic
103C2×C4×C263#4111yescharacteristic
104C8×C263—22no—
105C22×C263—22no—
106C22×C263—41no—
107C22×C263—42no—
108C22×C263—41no—
109C22×C263—81no—
110C22×C263C4×D411yescharacteristic
111C22×C263#12011yescharacteristic
112C22×C263#12111yescharacteristic
113C4×C263—22no—
114C4×C263—41no—
115C4×C263C2×C22⋊C411yescharacteristic
116C4×C263C4×D411yescharacteristic
117C4×C263#12111yescharacteristic
118C4×C263#12311yescharacteristic
119Dic263—22no—
120Dic263—42no—
121Dic263—41no—
122Dic263—81no—
123C2×C263—21no—
124C2×C263—21no—
125C2×C263—21no—
126C2×C263—42no—
127C2×C263—81no—
128C2×C263#38611yescharacteristic
129C263#190611yescharacteristic, complemented, Sylow
130#1906—2631nomaximal, Sylow
131#575D26311yescharacteristic
132C4×#41—2631no—
133#421—2631no—
134C2×#104—2631no—
135#410—2631no—
136#344—2631no—
137#339—2631no—
138C2×#41C2×D26311yescharacteristic
139C2×#41C2×D26311yescharacteristic
140#137—21no—
141#137Dic26312yescomplemented
142AGL(1,ℤ/8)—21no—
143AGL(1,ℤ/8)Dic26312yescomplemented
144#135—22no—
145C2×M16—2631no—
146C2×M16C2×D26311yescharacteristic
147C4×Q8—5261no—
148C4×D4—5262no—
149C4×D4—5261no—
150#117—2631no—
151#117—2631no—
152#117—5261no—
153C2×C42—2631no—
154C2×C42—5261no—
155#104—5261no—
156#104—5261no—
157#104—5262no—
158#99—5262no—
159#41—21no—
160#41—21no—
161#41—21no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41C2×Dic26312yes—
166#41C22⋊D26312yes—
167C2×Q8C2×Dic26311yescharacteristic, complemented
168C2×Q8C22⋊D26311yescharacteristic
169C2×D4—21no—
170C2×D4—21no—
171C2×D4C2×Dic26311yescharacteristic, complemented
172C2×D4C22⋊D26311yescharacteristic
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—5262no—
176C22×C4—5261no—
177C22×C4C22×D26311yescharacteristic
178Q16—22no—
179SD16—22no—
180SD16—22no—
181D8—22no—
182M16—21no—
183M16—5261no—
184M16—5261no—
185M16C2×Dic26312yescomplemented
186C2×C8—21no—
187C2×C8—5261no—
188C4⋊C4—5262no—
189C4⋊C4—5261no—
190C4⋊C4—5261no—
191C22⋊C4—2632no—
192C22⋊C4—5262no—
193C22⋊C4—5261no—
194C42—2631no—
195C42—2631no—
196C42—5262no—
197C42—5261no—
198C42—5261no—
199C42—5261no—
200C23—21no—
201C23—21no—
202C23D4×D26311yescharacteristic
203Q8—21no—
204Q8—21no—
205Q8—21no—
206Q8C22⋊Dic26312yes—
207D4—22no—
208D4—22no—
209D4—21no—
210D4—21no—
211D4—21no—
212D4—41no—
213D4—41no—
214D4C22⋊Dic26312yes—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—5261no—
222C2×C4—5261no—
223C2×C4—5262no—
224C2×C4—10522no—
225C2×C4—10524no—
226C2×C4—10521no—
227C2×C4C22×Dic26311yescharacteristic
228C2×C4D4×D26311yescharacteristic
229C2×C4C2×C22⋊D26311yescharacteristic
230C2×C4C2×C22⋊D26311yescharacteristic
231C2×C4#1686237511yescharacteristic
232C2×C4#1686237511yescharacteristic
233C8—22no—
234C8—10521no—
235C22—22no—
236C22—41no—
237C22—42no—
238C22—41no—
239C22—81no—
240C22D4×Dic26311yescharacteristic
241C22#2255443811yescharacteristic
242C22#2255444211yescharacteristic
243C4—22no—
244C4—41no—
245C4—5262no—
246C4—10521no—
247C4—10522no—
248C4—21041no—
249C4C2×C22⋊Dic26311yescharacteristic
250C4D4×Dic26311yescharacteristic
251C4#2255444211yescharacteristic
252C4#2255445111yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—42no—
257C2—81no—
258C2#3068181811yescharacteristic
subgroupG/H
centerC4#22554442
commutatorC2×C4×C263C22×C4
FrattiniC22×C4C22×D263
Fitting, prime radical#575×C263C2
socleC2×C263#386
cosocleC22×C4×C263C23
hypercenter#575D263
hypocenterC263#1906
LCSG ▷ C2×C4×C263 ▷ C2×C263 ▷ C263
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #41×Dic263C2semidirect
2. #30681955C2semidirect
3. C2×#22554397C2semidirect
4. #30681912C2semidirect
5. #30681710C2semidirect
6. #30681699C2semidirect
24. #137×C263C4semidirect
26. AGL(1,ℤ/8)×C263C4semidirect
54. C2×Q8×C263C2×C4semidirect
58. C2×D4×C263C2×C4semidirect
68. M16×C263C2×C4semidirect
129. C263#1906semidirect
141. #137Dic263semidirect
143. AGL(1,ℤ/8)Dic263semidirect
167. C2×Q8C2×Dic263semidirect
171. C2×D4C2×Dic263semidirect
185. M16C2×Dic263semidirect

Maximal subgroups

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

subgroupindex
#41×Dic2632
#306819552
C2×#225543972
#306819122
#306817102
#306816992
#575×C2632
#1906263