Finite Group Databasebeta

← #46821436

Subgroups — #46821436

Order 34432.

348 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 137 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#31771160C211yesmaximal, characteristic
2#125×D269C211yesmaximal, characteristic
3#31771015C211yesmaximal, characteristic, complemented
4#31771014C211yesmaximal, characteristic, complemented
5#31771011C211yesmaximal, characteristic, complemented
6#31771006C212yesmaximal, complemented
7#31771005C212yesmaximal, complemented
8#31770646C212yesmaximal
9#31770644C212yesmaximal
10#31770557C211yesmaximal, characteristic
11#501×C269C211yesmaximal, characteristic, complemented
12C2×#16864385C2212yes—
13#22575453C2211yescharacteristic
14#22575445C2211yescharacteristic
15#22575444C2212yes—
16#22575444C2211yescharacteristic
17#22575444C2211yescharacteristic
18#22575443—21no—
19#22575443—21no—
20#22575433—21no—
21#22575429—21no—
22Q8×Dic269—21no—
23#22575418—21no—
24#22575411C2212yes—
25C4⋊C4×D269—21no—
26C4⋊C4×D269—21no—
27C4⋊C4×D269C2211yescharacteristic
28C4⋊C4×D269C2211yescharacteristic
29C42×D269C2211yescharacteristic
30#22575387C2212yes—
31#22575386C2212yes—
32#22575386C2212yes—
33#22575385C2212yes—
34#22575384C2212yes—
35#22575384C2212yes—
36#22575364C2211yescharacteristic
37C8×Dic269C2212yes—
38C4⋊Q8×C269C2211yescharacteristic
39#125×C269C2211yescharacteristic
40#107×C269C2212yes—
41#106×C269C2212yes—
42#97×C269C2211yescharacteristic
43C2×Dic538—21no—
44C2×C4×D269—21no—
45C2×C4×D269—21no—
46C2×C4×D269D411yescharacteristic
47C2×C4×D269D411yescharacteristic
48C2×C4×D269C2311yescharacteristic
49#16864385Q818yescomplemented
50C2×#14524130C2312yes—
51C4⋊Dic269—21no—
52C4⋊Dic269—21no—
53C4⋊Dic269—22no—
54C4⋊Dic269—21no—
55C4⋊Dic269C2311yescharacteristic
56C4⋊Dic269C2312yes—
57C4⋊Dic269C2311yescharacteristic
58#16864381—21no—
59#16864381—21no—
60#16864381—21no—
61#16864381—21no—
62#16864381—22no—
63#16864381—21no—
64#16864381—21no—
65#16864379—22no—
66C4×Dic269—22no—
67C4×Dic269D411yescharacteristic
68C4×Dic269D411yescharacteristic
69C4×Dic269C2311yescharacteristic
70C2×Q8×C269—21no—
71C2×C8×C269C2312yes—
72C4⋊C4×C269—21no—
73C4⋊C4×C269—21no—
74C4⋊C4×C269—21no—
75C4⋊C4×C269C2311yescharacteristic
76C4⋊C4×C269C2312yes—
77C4⋊C4×C269C2311yescharacteristic
78C42×C269C2311yescharacteristic
79C22×D269C2×D411yescharacteristic
80Dic538—42no—
81C2×Dic269—22no—
82C2×Dic269—21no—
83C2×Dic269—21no—
84C2×Dic269C2×D411yescharacteristic
85C2×Dic269C2×D411yescharacteristic
86C2×Dic269C2×D411yescharacteristic
87C4×D269—41no—
88C4×D269—41no—
89C4×D269—42no—
90C4×D269—42no—
91C4×D269C2×Q814yes—
92#14524130C2×Q814yes—
93Q8×C269—42no—
94C2×C4×C269—21no—
95C2×C4×C269—21no—
96C2×C4×C269—22no—
97C2×C4×C269C2×D411yescharacteristic
98C2×C4×C269C2×D411yescharacteristic
99C2×C4×C269C2411yescharacteristic
100C8×C269C2×Q814yes—
101C22×C269C22×D411yescharacteristic
102C2×D269—21no—
103C2×D269—21no—
104C2×D269C4⋊Q812yes—
105C4×C269—21no—
106C4×C269—21no—
107C4×C269—42no—
108C4×C269—41no—
109C4×C269—41no—
110C4×C269C22×Q812yes—
111Dic269—21no—
112Dic269—21no—
113Dic269—41no—
114Dic269—41no—
115Dic269—42no—
116Dic269C4⋊Q812yes—
117C2×C269C2×C4⋊Q811yescharacteristic
118C2×C269#57511yescharacteristic
119C2×C269#57511yescharacteristic
120D269—22no—
121C269#316611yescharacteristic, complemented, Sylow
122#3166—2691nomaximal, Sylow
123#533—2691no—
124C2×#125—2691no—
125#501—2691no—
126#501—2691no—
127#501—2691no—
128#501D26911yescharacteristic, complemented
129#499—2692no—
130#499—2692no—
131#428—2692no—
132#428—2692no—
133C4×M16—2691no—
134C2×M16—2692no—
135C4⋊Q8—2691no—
136C4⋊Q8C2×D26911yescharacteristic
137#125—2691no—
138#125—2691no—
139#125—2692no—
140#125—2691no—
141#125—5381no—
142#125—5381no—
143#125C2×D26911yescharacteristic
144C22⋊Q8—5381no—
145C22⋊Q8—5381no—
146C4×Q8—5381no—
147C4×Q8—5381no—
148#117—2692no—
149C2×C4⋊C4—2691no—
150C2×C4⋊C4—2691no—
151C2×C4⋊C4—5381no—
152C2×C4⋊C4—5381no—
153C2×C42—2691no—
154#107—2692no—
155#107—2692no—
156#107—2692no—
157#107C2×D26912yes—
158#106—2692no—
159#106—2692no—
160#106—2692no—
161#106C2×D26912yes—
162#97—2691no—
163#97C2×D26911yescharacteristic
164C4×C8—2692no—
165C2×Q8—21no—
166C2×Q8—5381no—
167C22×C4—2691no—
168C22×C4—2691no—
169C22×C4—2691no—
170C22×C4—5381no—
171C22×C4—5381no—
172M16—2698no—
173C2×C8—2692no—
174C2×C8C22×D26912yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—2691no—
179C4⋊C4—2691no—
180C4⋊C4—2692no—
181C4⋊C4—5381no—
182C4⋊C4—5381no—
183C4⋊C4—5381no—
184C4⋊C4—5381no—
185C4⋊C4—5381no—
186C4⋊C4—5381no—
187C4⋊C4—5382no—
188C4⋊C4—5382no—
189C4⋊C4—5381no—
190C4⋊C4—5381no—
191C4⋊C4—5381no—
192C4⋊C4C22×D26911yescharacteristic
193C4⋊C4C22×D26912yes—
194C4⋊C4C22×D26911yescharacteristic
195C22⋊C4—5382no—
196C42—2691no—
197C42—2691no—
198C42—2691no—
199C42—5382no—
200C42C22×D26911yescharacteristic
201C23—2691no—
202Q8—42no—
203Q8—10762no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—2691no—
208C2×C4—2694no—
209C2×C4—2691no—
210C2×C4—2691no—
211C2×C4—5382no—
212C2×C4—5381no—
213C2×C4—5381no—
214C2×C4—10761no—
215C2×C4—10761no—
216C2×C4—10762no—
217C2×C4—10762no—
218C2×C4D4×D26911yescharacteristic
219C2×C4D4×D26911yescharacteristic
220C2×C4C23×D26911yescharacteristic
221C8—2694no—
222C8Q8×D26914yes—
223C22—2692no—
224C22—5381no—
225C22—5381no—
226C22C2×D4×D26911yescharacteristic
227C4—21no—
228C4—21no—
229C4—41no—
230C4—42no—
231C4—41no—
232C4—2692no—
233C4—5381no—
234C4—5381no—
235C4—10761no—
236C4—10761no—
237C4—10762no—
238C4C2×Q8×D26912yes—
239C2—5382no—
240C2C4⋊Q8×D26911yescharacteristic
241C2#3177140111yescharacteristic
242C2#3177140411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#501×C269C2
socleC22×C269C22×D4
prime radicalC22×D269C2×D4
hypercenter#501D269
hypocenterC269#3166
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. #31771015C2semidirect
4. #31771014C2semidirect
5. #31771011C2semidirect
6. #31771006C2semidirect
7. #31771005C2semidirect
11. #501×C269C2semidirect
49. #16864385Q8semidirect
121. C269#3166semidirect
128. #501D269semidirect

Maximal subgroups

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

subgroupindex
#317711602
#125×D2692
#317710152
#317710142
#317710112
#317710062
#317710052
#317706462
#317706442
#317705572
#501×C2692
#3166269