Finite Group Databasebeta

← #46824018

Subgroups — #46824018

Order 34432.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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#135×D269C212yesmaximal, complemented
2#31771047C212yesmaximal, complemented
3#31771044C212yesmaximal, complemented
4#31771042C212yesmaximal, complemented
5#31771039C212yesmaximal, complemented
6#31771037C211yesmaximal, characteristic, complemented
7#31771035C212yesmaximal, complemented
8C2×#22575392C211yesmaximal, characteristic, complemented
9#508×C269C211yesmaximal, characteristic, complemented
10#41×D269—22no—
11#22575476C2212yes—
12#22575475C2212yescomplemented
13#22575474—22no—
14#22575473—22no—
15#22575472C2212yes—
16Q16×D269C2212yescomplemented
17SD16×D269—22no—
18D8×D269C2212yescomplemented
19C2×C8×D269C2211yescharacteristic
20Dic2152C2211yescharacteristic, complemented
21#22575399C2212yescomplemented
22#22575398C2212yescomplemented
23#22575397C2212yescomplemented
24#22575396C2212yescomplemented
25D4304C2211yescharacteristic, complemented
26#22575394C2212yescomplemented
27#22575392C2212yescomplemented
28#22575392C2212yescomplemented
29C2×#16864377C2211yescharacteristic, complemented
30#135×C269C2212yescomplemented
31Q32×C269C2211yescharacteristic, complemented
32SD32×C269C2212yescomplemented
33D16×C269C2211yescharacteristic, complemented
34C2×C16×C269C2211yescharacteristic, complemented
35#16864403—22no—
36#16864402—22no—
37#16864402—42no—
38#16864401—22no—
39#16864401—42no—
40Q8×D269—22no—
41D4×D269—22no—
42D4×D269—42no—
43C2×C4×D269—42no—
44C2×C4×D269D411yescharacteristic
45Dic1076C2312yes—
46#16864392—22no—
47#16864391—22no—
48#16864390—22no—
49#16864389—22no—
50D2152C2312yes—
51#16864387—22no—
52C2×#14524130D411yescharacteristic
53C8×D269D412yes—
54C8×D269C2312yes—
55#16864377C2312yes—
56#41×C269—22no—
57Q16×C269C2312yes—
58SD16×C269—22no—
59D8×C269C2312yes—
60C2×C8×C269C2311yescharacteristic
61C16×C269C2311yescharacteristic
62C16×C269C2311yescharacteristic
63C22×D269—42no—
64C22×D269D811yescharacteristic
65Dic538—22no—
66Dic538—42no—
67D1076—22no—
68D1076—42no—
69C22⋊D269—42no—
70C22⋊D269—42no—
71C22⋊D269—42no—
72C2×Dic269—42no—
73C2×Dic269D811yescharacteristic
74C4×D269—42no—
75C4×D269—42no—
76C4×D269—42no—
77C4×D269D812yes—
78C4×D269C2×D412yes—
79#14524130C2×D412yes—
80Q8×C269—22no—
81D4×C269—22no—
82D4×C269—42no—
83C2×C4×C269—42no—
84C2×C4×C269C2×D411yescharacteristic
85C8×C269C2×D411yescharacteristic
86C8×C269C2411yescharacteristic
87C22×C269—42no—
88C22×C269C2×D811yescharacteristic
89C2×D269—41no—
90C2×D269—42no—
91C2×D269—84no—
92C2×D269C2×D812yes—
93C4×C269—42no—
94C4×C269C2×D811yescharacteristic
95C4×C269C22×D411yescharacteristic
96Dic269—42no—
97Dic269C2×D812yes—
98C2×C269—21no—
99C2×C269—82no—
100C2×C269C22×D811yescharacteristic
101D269—22no—
102D269—82no—
103C269#341411yescharacteristic, complemented, Sylow
104#3414—2691nomaximal, Sylow
105C2×#135—2692no—
106#510—2692no—
107#510—2692no—
108#509—2692no—
109#509—2692no—
110#508—2692no—
111#508—2691no—
112#508D26911yescharacteristic, complemented
113C2×M32—2691no—
114C2×#41—5382no—
115#135—2692no—
116#135—2692no—
117#135—2692no—
118#135—5382no—
119#135—5382no—
120#135C2×D26912yescomplemented
121C2×Q16—2692no—
122C2×SD16—5382no—
123C2×D8—2692no—
124C22×C8—2691no—
125Q32—2692no—
126Q32—2691no—
127Q32C2×D26911yescharacteristic, complemented
128SD32—2692no—
129SD32—2692no—
130SD32—2692no—
131SD32C2×D26912yescomplemented
132D16—2692no—
133D16—2691no—
134D16C2×D26911yescharacteristic, complemented
135M32—2692no—
136M32—2692no—
137C2×C16—2691no—
138C2×C16C2×D26911yescharacteristic, complemented
139#41—22no—
140#41—5382no—
141#41—5382no—
142#41—5382no—
143#41—10762no—
144#41—10762no—
145C2×Q8—5382no—
146C2×D4—5382no—
147C2×D4—10762no—
148C22×C4—2691no—
149C22×C4—10762no—
150Q16—2692no—
151Q16—5382no—
152Q16C22×D26912yes—
153SD16—22no—
154SD16—5382no—
155SD16—5382no—
156SD16—5382no—
157D8—2692no—
158D8—5382no—
159D8C22×D26912yes—
160C2×C8—2691no—
161C2×C8—2692no—
162C2×C8—2692no—
163C2×C8C22×D26911yescharacteristic
164C16—2692no—
165C16C22×D26911yescharacteristic
166C16C22×D26911yescharacteristic
167C23—2691no—
168C23—10762no—
169Q8—22no—
170Q8—5382no—
171Q8—10762no—
172D4—22no—
173D4—42no—
174D4—5382no—
175D4—10762no—
176D4—10762no—
177D4—10762no—
178D4—10762no—
179C2×C4—42no—
180C2×C4—2691no—
181C2×C4—2692no—
182C2×C4—2692no—
183C2×C4—10762no—
184C2×C4—10762no—
185C2×C4—10762no—
186C2×C4—10762no—
187C2×C4D4×D26911yescharacteristic
188C8—2692no—
189C8D4×D26911yescharacteristic
190C8C23×D26911yescharacteristic
191C22—42no—
192C22—2692no—
193C22—10761no—
194C22—10762no—
195C22—21524no—
196C22D8×D26911yescharacteristic
197C4—42no—
198C4—2692no—
199C4—10762no—
200C4D8×D26911yescharacteristic
201C4C2×D4×D26911yescharacteristic
202C2—21no—
203C2—82no—
204C2—5382no—
205C2—21522no—
206C2C2×D8×D26911yescharacteristic
subgroupG/H
centerC4D8×D269
commutator, cosocleC8×C269C24
FrattiniC8C23×D269
Fitting#508×C269C2
socleC2×C269C22×D8
hypercenter#508D269
hypocenterC269#3414
LCSG ▷ C8×C269 ▷ C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D269C2semidirect
2. #31771047C2semidirect
3. #31771044C2semidirect
4. #31771042C2semidirect
5. #31771039C2semidirect
6. #31771037C2semidirect
7. #31771035C2semidirect
8. C2×#22575392C2semidirect
9. #508×C269C2semidirect
12. #22575475C22semidirect
16. Q16×D269C22semidirect
18. D8×D269C22semidirect
20. Dic2152C22semidirect
21. #22575399C22semidirect
22. #22575398C22semidirect
23. #22575397C22semidirect
24. #22575396C22semidirect
25. D4304C22semidirect
26. #22575394C22semidirect
27. #22575392C22semidirect
28. #22575392C22semidirect
29. C2×#16864377C22semidirect
30. #135×C269C22semidirect
31. Q32×C269C22semidirect
32. SD32×C269C22semidirect
33. D16×C269C22semidirect
34. C2×C16×C269C22semidirect
103. C269#3414semidirect
112. #508D269semidirect
120. #135C2×D269semidirect
127. Q32C2×D269semidirect
131. SD32C2×D269semidirect
134. D16C2×D269semidirect
138. C2×C16C2×D269semidirect

Maximal subgroups

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

subgroupindex
#135×D2692
#317710472
#317710442
#317710422
#317710392
#317710372
#317710352
C2×#225753922
#508×C2692
#3414269