Finite Group Databasebeta

← #46820252

Subgroups — #46820252

Order 34432.

698 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#22575501C211yesmaximal, characteristic, complemented
2#31771405C212yesmaximal, complemented
3#31771179C211yesmaximal, characteristic, complemented
4#31770790C214yesmaximal, complemented
5#31770780C214yesmaximal, complemented
6#31770621C212yesmaximal, complemented
7C2×#101×C269C211yesmaximal, characteristic, complemented
8#22575501—24no—
9#22575501—22no—
10#22575501C2214yes—
11C2×#16864403—22no—
12C2×#16864402—22no—
13C2×#16864402C2212yes—
14C2×#16864402C2211yescharacteristic
15C2×Q8×D269—22no—
16#22575486—22no—
17#22575486C2214yescomplemented
18#22575481—22no—
19#22575481C2214yescomplemented
20#22575475—24no—
21#22575451C2212yescomplemented
22#22575442C2212yescomplemented
23#22575436—22no—
24#22575429—22no—
25#22575412C2211yescharacteristic
26#22575379C2218yescomplemented
27C22×Q8×C269C2211yescharacteristic
28C2×M16×C269C2212yes—
29#101×C269C2214yescomplemented
30#16864403—28no—
31#16864403—28no—
32#16864402—28no—
33#16864402—28no—
34#16864402—22no—
35#16864402—22no—
36#16864402—22no—
37#16864402—22no—
38#16864402D414yes—
39#16864402C2314yes—
40C2×Dic538—22no—
41C2×Dic538D412yes—
42C2×Dic538C2311yescharacteristic
43Q8×D269—28no—
44Q8×D269—28no—
45C2×D1076—22no—
46C2×D1076D412yes—
47C2×D1076C2311yescharacteristic
48C2×C22⋊D269—22no—
49C2×C22⋊D269—21no—
50C2×C4×D269—24no—
51C2×C4×D269—22no—
52C2×C4×D269—21no—
53Dic1076—24no—
54#16864391—24no—
55#16864391—24no—
56D2152—24no—
57C4⋊Dic269—21no—
58#16864381—24no—
59C22⋊Dic269—21no—
60#16864379—24no—
61C4×Dic269C2312yes—
62C2×Q8×C269—22no—
63C2×Q8×C269—22no—
64C2×Q8×C269D412yes—
65C2×Q8×C269C2312yes—
66C22×C4×C269D412yes—
67C22×C4×C269C2311yescharacteristic
68M16×C269—22no—
69M16×C269C2314yes—
70C2×C8×C269—22no—
71C22×D269—22no—
72C22×D269—21no—
73Dic538—28no—
74Dic538—22no—
75Dic538—22no—
76Dic538C2×D414yes—
77D1076—28no—
78D1076—22no—
79D1076—22no—
80D1076C2×D414yes—
81C22⋊D269—24no—
82C22⋊D269—24no—
83C22⋊D269—41no—
84C22⋊D269—41no—
85C2×Dic269—22no—
86C2×Dic269—22no—
87C2×Dic269—21no—
88C4×D269—216no—
89C4×D269—24no—
90C4×D269—24no—
91C4×D269—42no—
92C23×C269C2×D411yescharacteristic
93Q8×C269—24no—
94Q8×C269—24no—
95C2×C4×C269—22no—
96C2×C4×C269—22no—
97C2×C4×C269C2×D412yes—
98C2×C4×C269C2×D412yes—
99C2×C4×C269C2×D414yes—
100C2×C4×C269C2×D411yescharacteristic
101C2×C4×C269C2411yescharacteristic
102C8×C269—24no—
103C22×C269—41no—
104C22×C269#12011yescharacteristic
105C22×C269#12011yescharacteristic
106C22×C269C22×D411yescharacteristic
107C2×D269—24no—
108C2×D269—44no—
109C2×D269—42no—
110C2×D269—41no—
111C4×C269—24no—
112C4×C269#12012yes—
113C4×C269C22×D412yes—
114Dic269—24no—
115Dic269—42no—
116Dic269—41no—
117C2×C269—21no—
118C2×C269—21no—
119C2×C269—21no—
120C2×C269C2×#12011yescharacteristic
121D269—44no—
122D269—81no—
123C269#302011yescharacteristic, complemented, Sylow
124#3020—2691nomaximal, Sylow
125C2×2−1+4—2691no—
126#575—2692no—
127#536—2691no—
128#456—2694no—
129#454—2694no—
130#421—2692no—
131C2×#101D26911yescharacteristic, complemented
1322−1+4—2694no—
1332−1+4—5384no—
1342−1+4—5382no—
135C2×#41—2692no—
136C2×#41—2691no—
137C2×#41—5382no—
138C2×#41—5382no—
139C22×Q8—5382no—
140C22×Q8C2×D26911yescharacteristic
141#137—2694no—
142#137—5382no—
143AGL(1,ℤ/8)—2694no—
144AGL(1,ℤ/8)—5382no—
145#135—5384no—
146C2×M16C2×D26912yes—
147C4⋊Q8—2692no—
148#124—2692no—
149#123—5382no—
150C22⋊Q8—5382no—
151#117—2691no—
152#104—2698no—
153#101C2×D26914yescomplemented
154#41—2694no—
155#41—2694no—
156#41—5388no—
157#41—5388no—
158#41—5388no—
159#41—5382no—
160#41—5382no—
161#41—5388no—
162#41—5382no—
163#41—5382no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—2692no—
167C2×Q8—2691no—
168C2×Q8—5382no—
169C2×Q8—5388no—
170C2×Q8—5388no—
171C2×Q8D107612yes—
172C2×Q8C22×D26912yes—
173C2×D4—2692no—
174C2×D4—2691no—
175C2×D4—5382no—
176C2×D4—5382no—
177C2×D4—5381no—
178C22×C4—5384no—
179C22×C4—5382no—
180C22×C4—5381no—
181C22×C4D107612yes—
182C22×C4C22×D26911yescharacteristic
183Q16—5384no—
184SD16—5384no—
185SD16—5384no—
186D8—5384no—
187M16—22no—
188M16C22×D26914yes—
189C2×C8—22no—
190C4⋊C4—5381no—
191C4⋊C4—5384no—
192C22⋊C4—5381no—
193C22⋊C4—5384no—
194C42—2692no—
195C23—5382no—
196C23—5381no—
197C23C2×D107611yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—2694no—
201Q8—5388no—
202Q8—5382no—
203Q8—5382no—
204D4—2694no—
205D4—5388no—
206D4—5384no—
207D4—5384no—
208D4—5382no—
209D4—5382no—
210D4—10761no—
211D4—10761no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—53816no—
215C2×C4—5382no—
216C2×C4—5384no—
217C2×C4—5384no—
218C2×C4—5382no—
219C2×C4—5381no—
220C2×C4—10762no—
221C2×C4D4×D26912yes—
222C2×C4D4×D26912yes—
223C2×C4C2×D107614yes—
224C2×C4C2×D107611yescharacteristic
225C2×C4C23×D26911yescharacteristic
226C8—24no—
227C22—41no—
228C22—5384no—
229C22—10764no—
230C22—10762no—
231C22—10761no—
232C22#2257542311yescharacteristic
233C22#2257542311yescharacteristic
234C22C22×D107611yescharacteristic
235C4—24no—
236C4—5384no—
237C4—10762no—
238C4—10761no—
239C4#2257542312yes—
240C4C2×D4×D26912yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—10764no—
245C2—21521no—
246C2C2×#2257542311yescharacteristic
subgroupG/H
centerC2C2×#22575423
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×#101×C269C2
socleC2×C269C2×#120
hypercenterC2×#101D269
hypocenterC269#3020
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
1. C2×#22575501C2semidirect
2. #31771405C2semidirect
3. #31771179C2semidirect
4. #31770790C2semidirect
5. #31770780C2semidirect
6. #31770621C2semidirect
7. C2×#101×C269C2semidirect
17. #22575486C22semidirect
19. #22575481C22semidirect
21. #22575451C22semidirect
22. #22575442C22semidirect
26. #22575379C22semidirect
29. #101×C269C22semidirect
123. C269#3020semidirect
131. C2×#101D269semidirect
153. #101C2×D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225755012
#317714052
#317711792
#317707902
#317707802
#317706212
C2×#101×C2692
#3020269