Finite Group Databasebeta

← #46821280

Subgroups — #46821280

Order 34432.

424 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 125 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#31771156C211yesmaximal, characteristic
2C4⋊Q8×D269C211yesmaximal, characteristic, complemented
3#31770966C214yesmaximal, complemented
4#31770955C211yesmaximal, characteristic, complemented
5#31770953C211yesmaximal, characteristic, complemented
6#31770952C211yesmaximal, characteristic, complemented
7#31770605C214yesmaximal, complemented
8C4×#16864385C211yesmaximal, characteristic, complemented
9#487×C269C211yesmaximal, characteristic, complemented
10C2×#16864403C2212yes—
11C2×Q8×D269—21no—
12C2×#16864385C2212yes—
13#22575453C2211yescharacteristic
14#22575452—21no—
15#22575452C2212yes—
16#22575451—21no—
17#22575450C2211yescharacteristic
18#22575442C2212yes—
19#22575428—21no—
20#22575428—21no—
21#22575415—21no—
22#22575415—21no—
23C4⋊C4×D269—21no—
24C4⋊C4×D269C2212yes—
25C42×D269C2211yescharacteristic
26#22575377C2214yes—
27#22575375C2214yes—
28#22575373C2214yes—
29#22575363C2212yes—
30C4×#14524130C2211yescharacteristic
31C4⋊Q8×C269C2211yescharacteristic
32C4⋊Q8×C269C2211yescharacteristic
33#103×C269C2214yes—
34C4×C8×C269C2211yescharacteristic
35#16864403—28no—
36C2×Dic538—22no—
37C2×Dic538—21no—
38Q8×D269—44no—
39C2×D1076—22no—
40C2×D1076—21no—
41C2×D1076—21no—
42C2×D1076C2312yes—
43C2×C4×D269—22no—
44C2×C4×D269—22no—
45C2×C4×D269D411yescharacteristic
46C2×C4×D269D411yescharacteristic
47C2×C4×D269C2311yescharacteristic
48#16864385—24no—
49C2×#14524130C2312yes—
50C4⋊Dic269—21no—
51C4⋊Dic269C2312yes—
52#16864381—22no—
53#16864381—22no—
54#16864379—22no—
55#16864379—22no—
56C4×Dic269D411yescharacteristic
57C4×Dic269D411yescharacteristic
58C4×Dic269C2311yescharacteristic
59C2×Q8×C269—21no—
60C2×Q8×C269C2312yes—
61C2×C8×C269C2312yes—
62C4⋊C4×C269—21no—
63C4⋊C4×C269—21no—
64C4⋊C4×C269—21no—
65C4⋊C4×C269C2312yes—
66C42×C269C2311yescharacteristic
67C22×D269—22no—
68C22×D269C2×D411yescharacteristic
69Dic538—44no—
70Dic538—42no—
71D1076—24no—
72D1076—44no—
73D1076—42no—
74D1076—42no—
75C2×Dic269—22no—
76C2×Dic269C2×D411yescharacteristic
77C2×Dic269C2×D411yescharacteristic
78C2×Dic269C2×D411yescharacteristic
79C4×D269—22no—
80C4×D269—22no—
81C4×D269—44no—
82C4×D269—44no—
83C4×D269#4114yes—
84#14524130—22no—
85Q8×C269—24no—
86Q8×C269—42no—
87C2×C4×C269—22no—
88C2×C4×C269—22no—
89C2×C4×C269C2×D411yescharacteristic
90C2×C4×C269C2×D411yescharacteristic
91C2×C4×C269C2411yescharacteristic
92C8×C269—22no—
93C22×C269C22×D411yescharacteristic
94C2×D269—21no—
95C2×D269—21no—
96C2×D269—42no—
97C2×D269—42no—
98C2×D269—42no—
99C2×D269#12412yes—
100C4×C269—42no—
101C4×C269—42no—
102C4×C269AGL(1,ℤ/8)12yes—
103C4×C269#13712yes—
104C4×C269C2×#4112yes—
105Dic269—21no—
106Dic269—21no—
107Dic269—42no—
108Dic269#12412yes—
109C2×C269C2×#12411yescharacteristic
110C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
111C2×C269C2×#13711yescharacteristic
112D269—22no—
113D269—82no—
114C269#314111yescharacteristic, complemented, Sylow
115#3141—2691nomaximal, Sylow
116#532—2691no—
117C2×C4⋊Q8—2691no—
118#489—2694no—
119#487—2691no—
120#487D26911yescharacteristic, complemented
121#486—2691no—
122#486—2691no—
123#417—2694no—
124C4×M16—2691no—
125C2×#41—2692no—
126C22×Q8—5381no—
127C2×M16—2692no—
128C4⋊Q8—2692no—
129C4⋊Q8—2691no—
130C4⋊Q8—5381no—
131C4⋊Q8—5381no—
132C4⋊Q8C2×D26911yescharacteristic
133C4⋊Q8C2×D26911yescharacteristic
134#127—2691no—
135#124—2692no—
136#121—5381no—
137#121—5381no—
138C4×D4—5381no—
139C4×D4—5381no—
140C2×C4⋊C4—2692no—
141C2×C4⋊C4—5381no—
142C2×C42—2691no—
143#103—2694no—
144#103C2×D26914yes—
145#102—2694no—
146#102—2694no—
147#97—2692no—
148C4×C8—2691no—
149C4×C8C2×D26911yescharacteristic
150#41—5388no—
151C2×Q8—21no—
152C2×Q8—5382no—
153C2×Q8—5381no—
154C2×Q8—10764no—
155C2×Q8C22×D26912yes—
156C2×D4—2692no—
157C2×D4—5382no—
158C2×D4—5381no—
159C2×D4—5381no—
160C22×C4—2691no—
161C22×C4—2691no—
162C22×C4—2691no—
163C22×C4—5382no—
164C22×C4—5382no—
165M16—5384no—
166C2×C8—2692no—
167C2×C8C22×D26912yes—
168C4⋊C4—21no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—2692no—
172C4⋊C4—5382no—
173C4⋊C4—5382no—
174C4⋊C4—5381no—
175C4⋊C4C22×D26912yes—
176C22⋊C4—5382no—
177C22⋊C4—5382no—
178C42—2691no—
179C42—2691no—
180C42—2691no—
181C42C22×D26911yescharacteristic
182C23—2691no—
183C23—5382no—
184Q8—24no—
185Q8—42no—
186Q8—10764no—
187Q8—10762no—
188D4—5384no—
189D4—10764no—
190D4—10762no—
191D4—10762no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—2691no—
195C2×C4—2694no—
196C2×C4—2691no—
197C2×C4—2691no—
198C2×C4—5382no—
199C2×C4—5382no—
200C2×C4—5382no—
201C2×C4—10764no—
202C2×C4—10764no—
203C2×C4D4×D26911yescharacteristic
204C2×C4D4×D26911yescharacteristic
205C2×C4C23×D26911yescharacteristic
206C8—22no—
207C8—5382no—
208C22—2692no—
209C22—5381no—
210C22—5381no—
211C22—10762no—
212C22—10762no—
213C22—10762no—
214C22C2×D4×D26911yescharacteristic
215C4—42no—
216C4—42no—
217C4—2692no—
218C4—5381no—
219C4—5381no—
220C4—10762no—
221C4#2257548012yes—
222C4#2257548512yes—
223C4#41×D26912yes—
224C2—5382no—
225C2—21522no—
226C2#124×D26911yescharacteristic
227C2C2×#2257548011yescharacteristic
228C2C2×#2257548511yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#487×C269C2
socleC22×C269C22×D4
prime radical#31771156C2
hypercenter#487D269
hypocenterC269#3141
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D269C2semidirect
3. #31770966C2semidirect
4. #31770955C2semidirect
5. #31770953C2semidirect
6. #31770952C2semidirect
7. #31770605C2semidirect
8. C4×#16864385C2semidirect
9. #487×C269C2semidirect
114. C269#3141semidirect
120. #487D269semidirect

Maximal subgroups

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

subgroupindex
#317711562
C4⋊Q8×D2692
#317709662
#317709552
#317709532
#317709522
#317706052
C4×#168643852
#487×C2692
#3141269