Finite Group Databasebeta

← #46824297

Subgroups — #46824297

Order 34432.

1160 conjugacy classes of subgroups; only the 471 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 471 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#31771204C216yesmaximal, complemented
2#31771195C216yesmaximal, complemented
3#31771167C213yesmaximal, complemented
4#31771141C216yesmaximal, complemented
5#31771118C216yesmaximal, complemented
6C2×#22575413C213yesmaximal, complemented
7#543×C269C211yesmaximal, characteristic, complemented
8C2×#16864403C2212yescomplemented
9C22×C22⋊D269C2213yes—
10C23×Dic269C2211yescharacteristic
11C22×C4×D269C2213yescomplemented
12#22575448C22112yescomplemented
13#22575444C2216yescomplemented
14#22575442C2216yescomplemented
15#22575438C22112yescomplemented
16#22575437C22112yescomplemented
17#22575429C2216yescomplemented
18#22575428C2216yescomplemented
19#22575427C2216yes—
20#22575427C2216yes—
21#22575423C22112yescomplemented
22Q8×Dic269C2212yescomplemented
23#22575415C2216yescomplemented
24#22575413C22112yescomplemented
25#22575413C22112yescomplemented
26#22575411C2216yescomplemented
27C2×#16864381C2213yescomplemented
28C2×#16864379C2213yescomplemented
29C2×C4×Dic269C2213yescomplemented
30C22⋊Q8×C269C2216yescomplemented
31C22⋊Q8×C269C2216yescomplemented
32C2×C22⋊C4×C269C2213yes—
33C23×D269C2313yes—
34#16864403—44no—
35C2×D1076C2316yes—
36C2×D1076C2316yes—
37C2×C22⋊D269—212no—
38C2×C22⋊D269C23112yes—
39C22×Dic269—42no—
40C22×Dic269C2313yes—
41C22×Dic269C2313yes—
42C22×Dic269C2313yes—
43C2×C4×D269—46no—
44C2×C4×D269C2316yes—
45C2×C4×D269C2316yes—
46C4⋊Dic269C2316yes—
47C4⋊Dic269C2316yes—
48#16864381C23112yes—
49#16864379C23112yes—
50#16864379C23112yes—
51#16864379C23112yes—
52C4×Dic269C2316yes—
53C4×Dic269C2316yes—
54C24×C269C2311yescharacteristic
55C2×Q8×C269C2312yes—
56C2×Q8×C269C2312yes—
57C22×C4×C269C2313yes—
58C22×C4×C269C2313yes—
59C4⋊C4×C269C2316yes—
60C4⋊C4×C269C2316yes—
61C22⋊C4×C269C23112yes—
62C22×D269—26no—
63C22×D269—43no—
64C22×D269—43no—
65C22×D269C2416yes—
66D1076—46no—
67D1076—46no—
68C22⋊D269—224no—
69C2×Dic269—26no—
70C2×Dic269—26no—
71C2×Dic269—43no—
72C2×Dic269C2413yes—
73C2×Dic269C2416yes—
74C2×Dic269C2411yescharacteristic
75C4×D269—212no—
76C4×D269—46no—
77C23×C269—41no—
78C23×C269—41no—
79C23×C269#4114yescomplemented
80C23×C269C2413yes—
81Q8×C269—42no—
82Q8×C269—42no—
83C2×C4×C269—43no—
84C2×C4×C269—43no—
85C2×C4×C269C2416yes—
86C2×C4×C269C2416yes—
87C22×C269—23no—
88C22×C269—23no—
89C22×C269—44no—
90C22×C269C2×#4116yes—
91C22×C269C2511yescharacteristic
92C2×D269—26no—
93C2×D269—26no—
94C2×D269—26no—
95C2×D269—412no—
96C4×C269—26no—
97C4×C269—26no—
98Dic269—26no—
99Dic269—23no—
100Dic269H(5,𝔽2)12yes—
101C2×C269—26no—
102C2×C269C22×#4111yescharacteristic
103C2×C269C2×H(5,𝔽2)11yescharacteristic
104C2×C269#58511yescharacteristic
105D269—46no—
106C269#345911yescharacteristic, complemented, Sylow
107#3459—2691nomaximal, Sylow
108#543D26911yescharacteristic, complemented
109#539—2696no—
110#538—2696no—
111#534—2693no—
112#529—2696no—
113#525—2696no—
114C2×C4×D4—2693no—
115C2×#41—2692no—
116C22×D4—2693no—
117C23×C4—2691no—
118C23×C4—2693no—
119#126—26912no—
120#125—2696no—
121#124—2696no—
122#123—26912no—
123#123—26912no—
124C22⋊Q8—2696no—
125C22⋊Q8C2×D26916yescomplemented
126C22⋊Q8C2×D26916yescomplemented
127#121—2696no—
128#121—2696no—
129#121—2696no—
130#120—26912no—
131C4×Q8—2692no—
132C4×D4—2696no—
133C4×D4—26912no—
134C4×D4—26912no—
135#117—2696no—
136C2×C4⋊C4—2693no—
137C2×C22⋊C4—2693no—
138C2×C22⋊C4C2×D26913yes—
139C2×C42—2693no—
140C24—2693no—
141C24C22×D26911yescharacteristic
142#41—10764no—
143C2×Q8C22×D26912yes—
144C2×Q8C22×D26912yes—
145C2×D4—26912no—
146C2×D4—2696no—
147C2×D4—2696no—
148C2×D4—53812no—
149C22×C4—2693no—
150C22×C4—2693no—
151C22×C4—2693no—
152C22×C4—2696no—
153C22×C4—2696no—
154C22×C4—10766no—
155C22×C4—10762no—
156C22×C4C22×D26913yes—
157C22×C4C22×D26913yes—
158C4⋊C4—26912no—
159C4⋊C4—2696no—
160C4⋊C4—2696no—
161C4⋊C4C22×D26916yes—
162C4⋊C4C22×D26916yes—
163C22⋊C4—26912no—
164C22⋊C4—26912no—
165C22⋊C4—26912no—
166C22⋊C4C22×D269112yes—
167C42—2696no—
168C42—2696no—
169C23—41no—
170C23—41no—
171C23—2696no—
172C23—5386no—
173C23—10763no—
174C23—10763no—
175C23#1686440314yescomplemented
176C23C23×D26913yes—
177Q8—42no—
178Q8—42no—
179D4—53824no—
180D4—10766no—
181D4—10766no—
182C2×C4—43no—
183C2×C4—43no—
184C2×C4—2691no—
185C2×C4—2696no—
186C2×C4—2693no—
187C2×C4—53812no—
188C2×C4—5386no—
189C2×C4—5386no—
190C2×C4—10766no—
191C2×C4—10763no—
192C2×C4C23×D26916yes—
193C2×C4C23×D26916yes—
194C22—23no—
195C22—23no—
196C22—44no—
197C22—5386no—
198C22—5386no—
199C22—5386no—
200C22—107612no—
201C22C2×#1686440316yes—
202C22C24×D26911yescharacteristic
203C4—26no—
204C4—26no—
205C4—2692no—
206C4—5386no—
207C4—5383no—
208C2—26no—
209C2—10766no—
210C2C22×#1686440311yescharacteristic
211C2H(5,𝔽2)×D26911yescharacteristic
212C2#3177144011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D269
commutator, socle, cosocleC22×C269C25
Fitting#543×C269C2
hypercenter#543D269
hypocenterC269#3459
LCSG ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #31771204C2semidirect
2. #31771195C2semidirect
3. #31771167C2semidirect
4. #31771141C2semidirect
5. #31771118C2semidirect
6. C2×#22575413C2semidirect
7. #543×C269C2semidirect
8. C2×#16864403C22semidirect
11. C22×C4×D269C22semidirect
12. #22575448C22semidirect
13. #22575444C22semidirect
14. #22575442C22semidirect
15. #22575438C22semidirect
16. #22575437C22semidirect
17. #22575429C22semidirect
18. #22575428C22semidirect
21. #22575423C22semidirect
22. Q8×Dic269C22semidirect
23. #22575415C22semidirect
24. #22575413C22semidirect
25. #22575413C22semidirect
26. #22575411C22semidirect
27. C2×#16864381C22semidirect
28. C2×#16864379C22semidirect
29. C2×C4×Dic269C22semidirect
30. C22⋊Q8×C269C22semidirect
31. C22⋊Q8×C269C22semidirect
79. C23×C269#41semidirect
106. C269#3459semidirect
108. #543D269semidirect
125. C22⋊Q8C2×D269semidirect
126. C22⋊Q8C2×D269semidirect
175. C23#16864403semidirect

Maximal subgroups

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

subgroupindex
#317712042
#317711952
#317711672
#317711412
#317711182
C2×#225754132
#543×C2692
#3459269