Finite Group Databasebeta

← #46823607

Subgroups — #46823607

Order 34432.

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

normal subgroups only

This group has 372 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#31771329C211yesmaximal, characteristic, complemented
2D4×Dic538C211yesmaximal, characteristic, complemented
3#31770966C212yesmaximal, complemented
4#31770955C211yesmaximal, characteristic, complemented
5#31770898C212yesmaximal, complemented
6#31770886C211yesmaximal, characteristic, complemented
7#31770822C212yesmaximal
8#31770732C211yesmaximal, characteristic, complemented
9#31770706C212yesmaximal, complemented
10C4×#16864387C211yesmaximal, characteristic
11#461×C269C211yesmaximal, characteristic, complemented
12C22×Dic538—21no—
13C2×#16864387C2211yescharacteristic
14#22575453C2211yescharacteristic
15#22575452C2212yes—
16#22575448—22no—
17#22575442C2212yes—
18#22575433—21no—
19#22575432—21no—
20#22575430—22no—
21Q8×Dic269C2212yes—
22C4×Dic538C2211yescharacteristic
23C4×D1076C2211yescharacteristic
24D4×Dic269C2212yes—
25#22575411—21no—
26#22575387C2212yes—
27#22575386C2211yescharacteristic
28#22575382C2212yes—
29#22575377C2212yes—
30#22575377C2212yes—
31#22575375C2212yes—
32#22575374C2211yescharacteristic
33#22575373C2211yescharacteristic
34#22575363C2212yes—
35C4×#14524130C2211yescharacteristic
36C2×SD16×C269C2212yes—
37C4⋊Q8×C269C2211yescharacteristic
38C4×D4×C269C2211yescharacteristic
39C4⋊C8×C269C2211yescharacteristic
40#103×C269C2212yes—
41C2×Dic538—22no—
42C2×Dic538—21no—
43C2×Dic538—44no—
44C2×Dic538D411yescharacteristic
45C2×D1076C2311yescharacteristic
46C22×Dic269—22no—
47C2×C4×D269—21no—
48#16864387—22no—
49C2×#14524130C2311yescharacteristic
50C2×#14524130C2311yescharacteristic
51C4⋊Dic269—22no—
52C4⋊Dic269—21no—
53C4⋊Dic269D411yescharacteristic
54C4⋊Dic269C2311yescharacteristic
55C4⋊Dic269C2312yes—
56#16864381—22no—
57#16864381D412yes—
58C22⋊Dic269—22no—
59#16864379—22no—
60#16864379—22no—
61#16864379—21no—
62C4×Dic269—22no—
63C4×Dic269C2312yes—
64C2×Q8×C269C2312yes—
65C2×D4×C269C2311yescharacteristic
66C22×C4×C269—21no—
67SD16×C269—24no—
68C2×C8×C269C2312yes—
69C4⋊C4×C269—21no—
70C4⋊C4×C269—21no—
71C4⋊C4×C269C2311yescharacteristic
72C22⋊C4×C269—21no—
73C42×C269C2311yescharacteristic
74C22×D269—21no—
75Dic538—22no—
76Dic538—44no—
77Dic538—42no—
78D1076—21no—
79D1076—21no—
80C2×Dic269—21no—
81C2×Dic269—22no—
82C2×Dic269—44no—
83C2×Dic269C2×D412yes—
84C2×Dic269C2×D412yes—
85C4×D269—42no—
86#14524130—21no—
87#14524130—21no—
88C23×C269—21no—
89Q8×C269—22no—
90Q8×C269—22no—
91D4×C269—21no—
92D4×C269#4112yes—
93C2×C4×C269—21no—
94C2×C4×C269—22no—
95C2×C4×C269—42no—
96C2×C4×C269C2×D411yescharacteristic
97C2×C4×C269C2×D411yescharacteristic
98C2×C4×C269C2411yescharacteristic
99C8×C269—22no—
100C22×C269—22no—
101C22×C269—41no—
102C22×C269—41no—
103C22×C269C22×D411yescharacteristic
104C2×D269—41no—
105C2×D269—41no—
106C2×D269—41no—
107C4×C269—21no—
108C4×C269—42no—
109C4×C269—41no—
110C4×C269AGL(1,ℤ/8)12yes—
111C4×C269C2×#4111yescharacteristic
112C4×C2692−1+411yescharacteristic
113Dic269—22no—
114Dic269—22no—
115Dic269—41no—
116Dic269—42no—
117C2×C269—42no—
118C2×C269#54811yescharacteristic
119C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C269#57811yescharacteristic
121D269—81no—
122C269#336711yescharacteristic, complemented, Sylow
123#3367—2691nomaximal, Sylow
124#556—2691no—
125D4×Q8—2691no—
126#489—2692no—
127#487—2691no—
128#477—2692no—
129#474—2691no—
130#463—2692no—
131#461D26911yescharacteristic, complemented
132#446—2691no—
133#440—2692no—
134C4×D8—2691no—
135C22×Q8—5381no—
136C2×SD16C2×D26912yes—
137C2×D8—2691no—
138C4⋊Q8—2692no—
139C4⋊Q8—2691no—
140C4⋊Q8C2×D26911yescharacteristic
141#126—5382no—
142#124—2692no—
143C22⋊Q8—5382no—
144C22⋊Q8—5381no—
145C22⋊Q8—5381no—
146C4×Q8—2691no—
147C4×Q8—2692no—
148C4×D4—2692no—
149C4×D4—2691no—
150C4×D4C2×D26911yescharacteristic
151#117—5381no—
152#107—2692no—
153#107—2691no—
154C4⋊C8—2692no—
155C4⋊C8C2×D26911yescharacteristic
156#103—2692no—
157#103—2692no—
158#103C2×D26912yes—
159#102—2691no—
160#102—2692no—
161#102—2691no—
162#97—2692no—
163C4×C8—2691no—
164C2×Q8—2691no—
165C2×Q8—5382no—
166C2×Q8—5381no—
167C2×Q8—10764no—
168C2×Q8C22×D26912yes—
169C2×D4—2691no—
170C2×D4C22×D26911yescharacteristic
171C22×C4—21no—
172C22×C4—5382no—
173C22×C4—5381no—
174SD16—24no—
175D8—5382no—
176C2×C8—2691no—
177C2×C8—2691no—
178C2×C8C22×D26912yes—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—2691no—
182C4⋊C4—2692no—
183C4⋊C4—2692no—
184C4⋊C4—2691no—
185C4⋊C4—5382no—
186C4⋊C4—5382no—
187C4⋊C4—5381no—
188C4⋊C4C22×D26911yescharacteristic
189C22⋊C4—21no—
190C22⋊C4—5382no—
191C22⋊C4—5382no—
192C22⋊C4—5382no—
193C22⋊C4—5381no—
194C42—2692no—
195C42—5382no—
196C42C22×D26911yescharacteristic
197C23—21no—
198C23—5381no—
199Q8—22no—
200Q8—22no—
201Q8—5382no—
202Q8—10764no—
203Q8—10762no—
204D4—21no—
205D4—5381no—
206D4—5381no—
207D4#1686440112yes—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—2692no—
212C2×C4—2692no—
213C2×C4—5381no—
214C2×C4—5382no—
215C2×C4—10764no—
216C2×C4—10762no—
217C2×C4D4×D26911yescharacteristic
218C2×C4D4×D26911yescharacteristic
219C2×C4C23×D26911yescharacteristic
220C8—22no—
221C8—5381no—
222C8—5381no—
223C22—22no—
224C22—41no—
225C22—41no—
226C22—10761no—
227C22—10761no—
228C22—10761no—
229C22C2×D4×D26911yescharacteristic
230C4—21no—
231C4—41no—
232C4—42no—
233C4—5382no—
234C4—5382no—
235C4—10762no—
236C4—10761no—
237C4#2257548012yes—
238C4C2×#1686440111yescharacteristic
239C4#2257550011yescharacteristic
240C2—42no—
241C2—21521no—
242C2#3177126111yescharacteristic
243C2C2×#2257548011yescharacteristic
244C2#3177142111yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#461×C269C2
socleC22×C269C22×D4
prime radicalC2×#16864387C22
hypercenter#461D269
hypocenterC269#3367
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #31771329C2semidirect
2. D4×Dic538C2semidirect
3. #31770966C2semidirect
4. #31770955C2semidirect
5. #31770898C2semidirect
6. #31770886C2semidirect
8. #31770732C2semidirect
9. #31770706C2semidirect
11. #461×C269C2semidirect
122. C269#3367semidirect
131. #461D269semidirect

Maximal subgroups

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

subgroupindex
#317713292
D4×Dic5382
#317709662
#317709552
#317708982
#317708862
#317708222
#317707322
#317707062
C4×#168643872
#461×C2692
#3367269