Finite Group Databasebeta

← #46821268

Subgroups — #46821268

Order 34432.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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#31771161C211yesmaximal, characteristic, complemented
2#124×D269C211yesmaximal, characteristic
3#31770966C212yesmaximal, complemented
4#31770963C212yesmaximal, complemented
5#31770962C211yesmaximal, characteristic, complemented
6#31770960C211yesmaximal, characteristic, complemented
7#31770959C211yesmaximal, characteristic, complemented
8#31770612C212yesmaximal, complemented
9#31770608C212yesmaximal
10C4×#16864385C211yesmaximal, characteristic, complemented
11#488×C269C211yesmaximal, characteristic, complemented
12C2×Q8×D269C2211yescharacteristic
13C2×D4×D269C2211yescharacteristic
14C2×#16864385C2212yes—
15#22575452C2212yescomplemented
16#22575445C2211yescharacteristic, complemented
17#22575442C2211yescharacteristic
18#22575441C2211yescharacteristic
19#22575440C2211yescharacteristic
20#22575439—22no—
21#22575431—22no—
22#22575418—22no—
23#22575411C2212yes—
24C22⋊C4×D269—22no—
25C42×D269C2211yescharacteristic
26#22575378C2212yescomplemented
27#22575377C2212yescomplemented
28#22575376C2212yescomplemented
29#22575375C2212yes—
30#22575374C2212yes—
31#22575373C2212yes—
32#22575363C2212yescomplemented
33C4×#14524130C2211yescharacteristic, complemented
34#125×C269C2211yescharacteristic, complemented
35#124×C269C2211yescharacteristic
36#103×C269C2212yescomplemented
37#102×C269C2212yes—
38C4×C8×C269C2211yescharacteristic, complemented
39C23×D269—21no—
40C2×Dic538—22no—
41C2×Dic538—21no—
42C2×Dic538C2311yescharacteristic
43Q8×D269—24no—
44C2×D1076C2311yescharacteristic
45C2×C22⋊D269—21no—
46D4×D269—24no—
47C2×C4×D269—21no—
48C2×C4×D269D412yes—
49C2×C4×D269C2311yescharacteristic
50#16864385—24no—
51C2×#14524130C2312yes—
52C4⋊Dic269C2312yes—
53#16864381—22no—
54#16864381—22no—
55C22⋊Dic269—22no—
56#16864379—22no—
57#16864379—22no—
58#16864379—22no—
59C4×Dic269—22no—
60C4×Dic269D412yes—
61C4×Dic269C2311yescharacteristic
62C2×Q8×C269C2311yescharacteristic
63C2×D4×C269C2311yescharacteristic
64C2×C8×C269C2312yes—
65C4⋊C4×C269—22no—
66C4⋊C4×C269C2312yes—
67C22⋊C4×C269—22no—
68C42×C269C2311yescharacteristic
69C22×D269—21no—
70C22×D269—44no—
71C22×D269—42no—
72C22×D269C2×D411yescharacteristic
73Dic538—22no—
74Dic538—44no—
75Dic538—42no—
76D1076—22no—
77C22⋊D269—42no—
78C2×Dic269—22no—
79C2×Dic269—21no—
80C2×Dic269C2×D412yes—
81C2×Dic269C2×D411yescharacteristic
82C4×D269—42no—
83C4×D269—42no—
84C4×D269#4114yes—
85#14524130—22no—
86C23×C269—21no—
87Q8×C269—22no—
88D4×C269—22no—
89C2×C4×C269—21no—
90C2×C4×C269—22no—
91C2×C4×C269C2×D412yes—
92C2×C4×C269C2411yescharacteristic
93C8×C269—22no—
94C22×C269—41no—
95C22×C269—42no—
96C22×C269C22×D411yescharacteristic
97C2×D269—22no—
98C2×D269—41no—
99C2×D269—42no—
100C2×D269—84no—
101C2×D269#12412yes—
102C4×C269—22no—
103C4×C269—41no—
104C4×C269—42no—
105C4×C269C2×#4112yes—
106Dic269—22no—
107Dic269—41no—
108Dic269—42no—
109Dic269#12412yes—
110C2×C269—81no—
111C2×C269C2×#12411yescharacteristic
112C2×C269#57512yes—
113D269—22no—
114D269—81no—
115C269#314011yescharacteristic, complemented, Sylow
116#3140—2691nomaximal, Sylow
117#533—2691no—
118C2×#124—2691no—
119#489—2692no—
120#489—2692no—
121#488—2691no—
122#488—2691no—
123#488—2691no—
124#488D26911yescharacteristic, complemented
125#419—2692no—
126#418—2692no—
127C4×M16—2691no—
128C22×Q8—2691no—
129C22×D4—2691no—
130C2×M16—2692no—
131C4⋊Q8—2692no—
132#125—2691no—
133#125C2×D26911yescharacteristic, complemented
134#124—2691no—
135#124—2691no—
136#124—2691no—
137#124—5382no—
138#124C2×D26911yescharacteristic
139C22⋊Q8—5382no—
140C4×Q8—5382no—
141#117—2692no—
142C2×C22⋊C4—5382no—
143C2×C42—2691no—
144#103—2692no—
145#103—2692no—
146#103—2692no—
147#103C2×D26912yescomplemented
148#102—2692no—
149#102—2692no—
150#102—2692no—
151#102C2×D26912yes—
152#97—2692no—
153C4×C8—2691no—
154C4×C8C2×D26911yescharacteristic, complemented
155C24—5381no—
156C2×Q8—2691no—
157C2×Q8—5382no—
158C2×Q8—5381no—
159C2×Q8—5384no—
160C2×Q8C22×D26911yescharacteristic
161C2×D4—2691no—
162C2×D4—5384no—
163C2×D4—5381no—
164C2×D4C22×D26911yescharacteristic
165C22×C4—2691no—
166C22×C4—2692no—
167C22×C4—5381no—
168M16—5384no—
169C2×C8—2692no—
170C2×C8C22×D26912yes—
171C4⋊C4—22no—
172C4⋊C4—2692no—
173C4⋊C4—5382no—
174C4⋊C4—5382no—
175C4⋊C4C22×D26912yes—
176C22⋊C4—22no—
177C22⋊C4—5382no—
178C22⋊C4—5382no—
179C22⋊C4—5382no—
180C22⋊C4—5382no—
181C42—2692no—
182C42—2691no—
183C42—5382no—
184C42C22×D26911yescharacteristic
185C23—21no—
186C23—2691no—
187C23—5381no—
188C23—10762no—
189C23—10764no—
190Q8—22no—
191Q8—5382no—
192Q8—10762no—
193Q8—10764no—
194D4—22no—
195D4—5382no—
196D4—10762no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—2691no—
200C2×C4—2694no—
201C2×C4—2692no—
202C2×C4—5382no—
203C2×C4—5381no—
204C2×C4—10762no—
205C2×C4—10762no—
206C2×C4D4×D26912yes—
207C2×C4C23×D26911yescharacteristic
208C8—22no—
209C8—5382no—
210C22—42no—
211C22—41no—
212C22—2692no—
213C22—5382no—
214C22—10762no—
215C22—10761no—
216C22—21524no—
217C22C2×D4×D26911yescharacteristic
218C4—22no—
219C4—42no—
220C4—41no—
221C4—2692no—
222C4—5382no—
223C4—10762no—
224C4—10761no—
225C4#41×D26912yes—
226C2—81no—
227C2—5382no—
228C2—21521no—
229C2#124×D26911yescharacteristic
230C2#3177140212yes—
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#488×C269C2
socleC22×C269C22×D4
prime radicalC2×D4×D269C22
hypercenter#488D269
hypocenterC269#3140
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #31771161C2semidirect
3. #31770966C2semidirect
4. #31770963C2semidirect
5. #31770962C2semidirect
6. #31770960C2semidirect
7. #31770959C2semidirect
8. #31770612C2semidirect
10. C4×#16864385C2semidirect
11. #488×C269C2semidirect
15. #22575452C22semidirect
16. #22575445C22semidirect
26. #22575378C22semidirect
27. #22575377C22semidirect
28. #22575376C22semidirect
32. #22575363C22semidirect
33. C4×#14524130C22semidirect
34. #125×C269C22semidirect
36. #103×C269C22semidirect
38. C4×C8×C269C22semidirect
115. C269#3140semidirect
124. #488D269semidirect
133. #125C2×D269semidirect
147. #103C2×D269semidirect
154. C4×C8C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317711612
#124×D2692
#317709662
#317709632
#317709622
#317709602
#317709592
#317706122
#317706082
C4×#168643852
#488×C2692
#3140269