Finite Group Databasebeta

← #46823534

Subgroups — #46823534

Order 34432.

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

normal subgroups only

This group has 350 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#31771328C211yesmaximal, characteristic, complemented
2#31770962C212yesmaximal, complemented
3#31770957C211yesmaximal, characteristic, complemented
4#31770909C212yesmaximal, complemented
5#31770892C211yesmaximal, characteristic, complemented
6#31770820C212yesmaximal
7#31770810C211yesmaximal, characteristic, complemented
8Q8×#14524130C211yesmaximal, characteristic, complemented
9C4×#16864390C212yesmaximal, complemented
10C4×#16864387C211yesmaximal, characteristic
11#554×C269C211yesmaximal, characteristic, complemented
12C2×#16864390C2212yes—
13C2×#16864387C2211yescharacteristic
14#22575453C2211yescharacteristic
15#22575448—22no—
16#22575441C2212yes—
17#22575429—21no—
18C4×Dic538C2212yes—
19C4×D1076C2211yescharacteristic
20#22575411—21no—
21#22575386C2211yescharacteristic
22#22575384C2212yes—
23C4⋊#14524130C2211yescharacteristic
24C4⋊#14524130C2212yes—
25#22575376C2212yes—
26#22575376C2212yes—
27#22575376C2212yes—
28#22575374C2211yescharacteristic
29#22575373C2212yes—
30#22575373C2211yescharacteristic
31C4×#14524130C2211yescharacteristic
32C4×#14524130C2212yes—
33C4⋊Q8×C269C2211yescharacteristic
34#125×C269C2212yes—
35C22⋊Q8×C269—21no—
36C22⋊Q8×C269—22no—
37C4×Q8×C269C2211yescharacteristic
38C4×D4×C269C2211yescharacteristic
39C4×D4×C269C2212yes—
40C2×C4⋊C4×C269—21no—
41C2×Dic538C2312yes—
42C2×D1076C2311yescharacteristic
43C2×C4×D269—21no—
44#16864390—24no—
45#16864387—22no—
46C2×#14524130C2311yescharacteristic
47C2×#14524130C2312yes—
48C2×#14524130C2311yescharacteristic
49C4⋊Dic269—21no—
50C4⋊Dic269—21no—
51C4⋊Dic269C2311yescharacteristic
52#16864381—22no—
53#16864379—22no—
54#16864379—21no—
55#16864379—22no—
56C4×Dic269—22no—
57C2×Q8×C269—21no—
58C2×Q8×C269D411yescharacteristic
59C2×D4×C269C2311yescharacteristic
60C22×C4×C269—21no—
61C22×C4×C269—22no—
62C4⋊C4×C269—22no—
63C4⋊C4×C269—21no—
64C4⋊C4×C269—22no—
65C4⋊C4×C269D411yescharacteristic
66C4⋊C4×C269D412yes—
67C4⋊C4×C269C2312yes—
68C4⋊C4×C269C2311yescharacteristic
69C22⋊C4×C269—21no—
70C22⋊C4×C269—22no—
71C42×C269C2311yescharacteristic
72C42×C269C2312yes—
73C22×D269—21no—
74Dic538—22no—
75Dic538—22no—
76D1076—21no—
77D1076—21no—
78C2×Dic269—21no—
79C2×Dic269—22no—
80C4×D269—42no—
81#14524130—21no—
82#14524130—22no—
83#14524130—21no—
84C23×C269—21no—
85Q8×C269—22no—
86Q8×C269—81no—
87D4×C269—21no—
88D4×C269#4112yes—
89C2×C4×C269—21no—
90C2×C4×C269—22no—
91C2×C4×C269—42no—
92C2×C4×C269—82no—
93C2×C4×C269C2×D412yes—
94C2×C4×C269C2×D411yescharacteristic
95C2×C4×C269C2×D412yes—
96C2×C4×C269C2×D411yescharacteristic
97C2×C4×C269C2411yescharacteristic
98C22×C269—22no—
99C22×C269—41no—
100C22×C269—41no—
101C22×C269C22×D411yescharacteristic
102C2×D269—41no—
103C2×D269—41no—
104C2×D269—41no—
105C4×C269—22no—
106C4×C269—22no—
107C4×C269—21no—
108C4×C269—41no—
109C4×C269—42no—
110C4×C269#13512yes—
111C4×C269C2×#4111yescharacteristic
112C4×C2692−1+411yescharacteristic
113Dic269—42no—
114Dic269—41no—
115C2×C269—42no—
116C2×C269#54811yescharacteristic
117C2×C269C2×#13511yescharacteristic
118C2×C269#57711yescharacteristic
119D269—81no—
120C269#336011yescharacteristic, complemented, Sylow
121#3360—2691nomaximal, Sylow
122#556—2691no—
123#554D26911yescharacteristic, complemented
124#488—2692no—
125#487—2691no—
126#479—2692no—
127#476—2691no—
128#463—2692no—
129#461—2691no—
130C8×Q8—2691no—
131C4×SD16—2692no—
132C4×D8—2691no—
133C2×SD16—2692no—
134C2×D8—2691no—
135C4⋊Q8—2691no—
136C4⋊Q8C2×D26911yescharacteristic
137#126—5382no—
138#125C2×D26912yes—
139#124—2692no—
140C22⋊Q8—21no—
141C22⋊Q8—22no—
142C22⋊Q8—5381no—
143C4×Q8—2692no—
144C4×Q8C2×D26911yescharacteristic
145C4×D4—2691no—
146C4×D4C2×D26911yescharacteristic
147C4×D4C2×D26912yes—
148#117—5381no—
149C2×C4⋊C4—21no—
150#107—2691no—
151#106—2692no—
152C4⋊C8—2691no—
153C4⋊C8—2692no—
154#103—2692no—
155#103—2692no—
156#103—2692no—
157#102—2691no—
158#102—2691no—
159#102—2692no—
160C4×C8—2691no—
161C4×C8—2692no—
162C2×Q8—21no—
163C2×Q8—2692no—
164C2×Q8C22⋊D26911yescharacteristic
165C2×D4—2691no—
166C2×D4C22×D26911yescharacteristic
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—5381no—
170SD16—5384no—
171D8—5382no—
172C2×C8—2691no—
173C2×C8—2692no—
174C2×C8—2691no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—2691no—
179C4⋊C4—5381no—
180C4⋊C4—5382no—
181C4⋊C4—5381no—
182C4⋊C4C22⋊D26911yescharacteristic
183C4⋊C4C22⋊D26912yes—
184C4⋊C4C22×D26912yes—
185C4⋊C4C22×D26911yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—22no—
188C22⋊C4—5382no—
189C22⋊C4—5381no—
190C22⋊C4—5382no—
191C42—5382no—
192C42C22×D26911yescharacteristic
193C42C22×D26912yes—
194C23—21no—
195C23—5381no—
196Q8—22no—
197Q8—81no—
198Q8—5382no—
199Q8—5382no—
200D4—21no—
201D4—5381no—
202D4—5381no—
203D4#1686440312yes—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—82no—
208C2×C4—5381no—
209C2×C4—5382no—
210C2×C4—10762no—
211C2×C4C2×C22⋊D26912yes—
212C2×C4C2×C22⋊D26912yes—
213C2×C4C2×C22⋊D26911yescharacteristic
214C2×C4C2×C22⋊D26911yescharacteristic
215C2×C4C23×D26911yescharacteristic
216C8—5381no—
217C8—5382no—
218C8—5381no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—10761no—
223C22—10761no—
224C22—10761no—
225C22C22×C22⋊D26911yescharacteristic
226C4—22no—
227C4—21no—
228C4—22no—
229C4—42no—
230C4—41no—
231C4—10762no—
232C4—10761no—
233C4#2257547312yes—
234C4C2×#1686440311yescharacteristic
235C4#2257550111yescharacteristic
236C2—42no—
237C2—21521no—
238C2#3177126611yescharacteristic
239C2C2×#2257547311yescharacteristic
240C2#3177141611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#554×C269C2
socleC22×C269C22×D4
prime radicalC2×#16864387C22
hypercenter#554D269
hypocenterC269#3360
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #31771328C2semidirect
2. #31770962C2semidirect
3. #31770957C2semidirect
4. #31770909C2semidirect
5. #31770892C2semidirect
7. #31770810C2semidirect
8. Q8×#14524130C2semidirect
9. C4×#16864390C2semidirect
11. #554×C269C2semidirect
120. C269#3360semidirect
123. #554D269semidirect

Maximal subgroups

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

subgroupindex
#317713282
#317709622
#317709572
#317709092
#317708922
#317708202
#317708102
Q8×#145241302
C4×#168643902
C4×#168643872
#554×C2692
#3360269