Finite Group Databasebeta

← #28403288

Subgroups — #28403288

Order 14464.

372 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 123 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
1Q8×D452C211yesmaximal, characteristic, complemented
2#20298957C212yesmaximal, complemented
3#20298948C211yesmaximal, characteristic, complemented
4#20298893C212yesmaximal, complemented
5#20298878C211yesmaximal, characteristic, complemented
6#20298810C212yesmaximal
7#20298800C211yesmaximal, characteristic, complemented
8#20298729C211yesmaximal, characteristic, complemented
9#20298696C212yesmaximal, complemented
10C4×#14268632C211yesmaximal, characteristic
11#556×C113C211yesmaximal, characteristic, complemented
12C2×Q8×D113—21no—
13C2×#14268634C2212yes—
14C2×#14268632C2211yescharacteristic
15#16813622C2211yescharacteristic
16#16813622C2212yes—
17#16813602—21no—
18#16813602—22no—
19C4×D452C2211yescharacteristic
20C4×D452C2212yes—
21#16813555C2212yes—
22#16813555C2211yescharacteristic
23C4⋊#11744751C2211yescharacteristic
24C4⋊#11744751C2212yes—
25#16813546C2212yes—
26#16813546C2212yes—
27#16813546C2212yes—
28#16813543C2212yes—
29#16813543C2211yescharacteristic
30#16813542C2211yescharacteristic
31#16813533C2212yes—
32C4×#11744751C2211yescharacteristic
33C4⋊Q8×C113C2211yescharacteristic
34#126×C113—22no—
35#124×C113C2212yes—
36C22⋊Q8×C113—21no—
37C4×Q8×C113C2212yes—
38C4×Q8×C113C2211yescharacteristic
39C4×D4×C113C2211yescharacteristic
40#117×C113—21no—
41C2×Dic226—22no—
42C2×Dic226—21no—
43Q8×D113—44no—
44C2×D452C2311yescharacteristic
45C2×C4×D113—22no—
46C2×C4×D113—21no—
47#14268634—24no—
48#14268632—22no—
49C2×#11744751C2311yescharacteristic
50C2×#11744751C2312yes—
51C2×#11744751C2311yescharacteristic
52C4⋊Dic113—22no—
53C4⋊Dic113—21no—
54C4⋊Dic113C2311yescharacteristic
55C4⋊Dic113C2312yes—
56#14268624—22no—
57#14268624—21no—
58C2×Q8×C113D411yescharacteristic
59C2×Q8×C113C2312yes—
60C2×D4×C113C2311yescharacteristic
61C22×C4×C113—21no—
62C4⋊C4×C113—21no—
63C4⋊C4×C113—22no—
64C4⋊C4×C113—21no—
65C4⋊C4×C113D411yescharacteristic
66C4⋊C4×C113D412yes—
67C4⋊C4×C113C2311yescharacteristic
68C22⋊C4×C113—22no—
69C22⋊C4×C113—21no—
70C22⋊C4×C113—22no—
71C42×C113—22no—
72C42×C113C2311yescharacteristic
73C42×C113C2312yes—
74C23×C113—21no—
75C22×D113—21no—
76Q8×C113—22no—
77Q8×C113—22no—
78Q8×C113—22no—
79D4×C113—21no—
80D4×C113—21no—
81C2×C4×C113—21no—
82C2×C4×C113—22no—
83C2×C4×C113—42no—
84C2×C4×C113C2×D412yes—
85C2×C4×C113C2×D411yescharacteristic
86C2×C4×C113C2×D412yes—
87C2×C4×C113C2×D411yescharacteristic
88C2×C4×C113C2411yescharacteristic
89C2×Dic113—21no—
90C2×Dic113—22no—
91D452—21no—
92D452#4112yes—
93C4×D113—44no—
94C4×D113—42no—
95Dic226—44no—
96Dic226—42no—
97#11744751—21no—
98#11744751—22no—
99#11744751—21no—
100C22×C113—41no—
101C22×C113—41no—
102C22×C113—41no—
103C22×C113C22×D411yescharacteristic
104C2×D113—22no—
105C2×D113—41no—
106C2×D113—41no—
107C4×C113—22no—
108C4×C113—22no—
109C4×C113—21no—
110C4×C113—42no—
111C4×C113—41no—
112C4×C113AGL(1,ℤ/8)12yes—
113C4×C113C2×#4111yescharacteristic
114C4×C1132−1+411yescharacteristic
115Dic113—41no—
116Dic113—42no—
117C2×C113—81no—
118C2×C113#54811yescharacteristic
119C2×C113C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C113#57811yescharacteristic
121D113—42no—
122#3367—1131nomaximal, Sylow
123C113#336711yescharacteristic, complemented, Sylow
124#556D11311yescharacteristic, complemented
125D4×Q8—1131no—
126#489—1132no—
127#487—1131no—
128#477—1132no—
129#474—1131no—
130#463—1132no—
131#461—1131no—
132#446—1131no—
133#440—1132no—
134C4×D8—1131no—
135C22×Q8—2261no—
136C2×SD16—1132no—
137C2×D8—1131no—
138C4⋊Q8—1131no—
139C4⋊Q8—1132no—
140C4⋊Q8C2×D11311yescharacteristic
141#126—22no—
142#124C2×D11312yes—
143C22⋊Q8—21no—
144C22⋊Q8—2261no—
145C22⋊Q8—2262no—
146C4×Q8C2×D11312yes—
147C4×Q8C2×D11311yescharacteristic
148C4×D4—1131no—
149C4×D4—1132no—
150C4×D4C2×D11311yescharacteristic
151#117—21no—
152#107—1131no—
153#107—1132no—
154C4⋊C8—1131no—
155C4⋊C8—1132no—
156#103—1132no—
157#103—1132no—
158#103—1132no—
159#102—1131no—
160#102—1132no—
161#102—1131no—
162#97—1132no—
163C4×C8—1131no—
164C2×Q8—4524no—
165C2×Q8—2262no—
166C2×Q8—2261no—
167C2×Q8C22⋊D11311yescharacteristic
168C2×Q8C22×D11312yes—
169C2×D4—1131no—
170C2×D4C22×D11311yescharacteristic
171C22×C4—21no—
172C22×C4—2262no—
173C22×C4—2261no—
174SD16—2264no—
175D8—2262no—
176C2×C8—1131no—
177C2×C8—1131no—
178C2×C8—1132no—
179C4⋊C4—21no—
180C4⋊C4—22no—
181C4⋊C4—21no—
182C4⋊C4—1131no—
183C4⋊C4—1132no—
184C4⋊C4—2262no—
185C4⋊C4—2261no—
186C4⋊C4C22⋊D11311yescharacteristic
187C4⋊C4C22⋊D11312yes—
188C4⋊C4C22×D11311yescharacteristic
189C22⋊C4—22no—
190C22⋊C4—21no—
191C22⋊C4—22no—
192C22⋊C4—2262no—
193C22⋊C4—2261no—
194C42—22no—
195C42C22×D11311yescharacteristic
196C42C22×D11312yes—
197C23—21no—
198C23—2261no—
199Q8—22no—
200Q8—22no—
201Q8—22no—
202Q8—4524no—
203Q8—4522no—
204D4—21no—
205D4—21no—
206D4—1132no—
207D4—2261no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—4524no—
212C2×C4—4522no—
213C2×C4—2262no—
214C2×C4—2261no—
215C2×C4C2×C22⋊D11312yes—
216C2×C4C2×C22⋊D11311yescharacteristic
217C2×C4C2×C22⋊D11312yes—
218C2×C4C2×C22⋊D11311yescharacteristic
219C2×C4C23×D11311yescharacteristic
220C8—2261no—
221C8—2262no—
222C8—2261no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—4521no—
227C22—4521no—
228C22—2262no—
229C22C22×C22⋊D11311yescharacteristic
230C4—22no—
231C4—22no—
232C4—21no—
233C4—41no—
234C4—42no—
235C4—4522no—
236C4—4521no—
237C4#1681364712yes—
238C4#41×D11311yescharacteristic
239C4#1681366911yescharacteristic
240C2—81no—
241C2—4522no—
242C2#2029925511yescharacteristic
243C2C2×#1681364711yescharacteristic
244C2#2029941311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
Fitting#556×C113C2
socleC22×C113C22×D4
prime radicalC2×#14268632C22
hypercenter#556D113
hypocenterC113#3367
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. Q8×D452C2semidirect
2. #20298957C2semidirect
3. #20298948C2semidirect
4. #20298893C2semidirect
5. #20298878C2semidirect
7. #20298800C2semidirect
8. #20298729C2semidirect
9. #20298696C2semidirect
11. #556×C113C2semidirect
123. C113#3367semidirect
124. #556D113semidirect

Maximal subgroups

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

subgroupindex
Q8×D4522
#202989572
#202989482
#202988932
#202988782
#202988102
#202988002
#202987292
#202986962
C4×#142686322
#556×C1132
#3367113