Finite Group Databasebeta

← #501×D271 #46846879

Subgroups — #501×D271 #46846879

Order 34688.

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

normal subgroups only

This group has 384 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
1C4⋊Q8×D271C211yesmaximal, characteristic
2#125×D271C211yesmaximal, characteristic
3#31774307C211yesmaximal, characteristic, complemented
4#31774306C211yesmaximal, characteristic, complemented
5#31774305C212yesmaximal, complemented
6#31774304C212yesmaximal, complemented
7#31774303C211yesmaximal, characteristic, complemented
8#107×D271C212yesmaximal
9#106×D271C212yesmaximal
10#97×D271C211yesmaximal, characteristic
11#501×C271C211yesmaximal, characteristic, complemented
12C2×Q8×D271—21no—
13C2×C8×D271C2212yes—
14#22576160C2211yescharacteristic
15#22576159—21no—
16#22576159C2212yes—
17#22576158—21no—
18#22576152C2211yescharacteristic
19#22576151C2211yescharacteristic
20#22576151C2211yescharacteristic
21#22576150—21no—
22#22576150—21no—
23C4⋊C4×D271—21no—
24C4⋊C4×D271—21no—
25C4⋊C4×D271—21no—
26C4⋊C4×D271C2211yescharacteristic
27C4⋊C4×D271C2211yescharacteristic
28C4⋊C4×D271C2212yes—
29C42×D271C2211yescharacteristic
30#22576094C2212yes—
31#22576093C2212yes—
32#22576093C2212yes—
33#22576092C2212yes—
34#22576091C2212yes—
35#22576091C2212yes—
36#22576071C2211yescharacteristic
37#22576070C2212yes—
38C4⋊Q8×C271C2211yescharacteristic
39#125×C271C2211yescharacteristic
40#107×C271C2212yes—
41#106×C271C2212yes—
42#97×C271C2211yescharacteristic
43C2×Dic542—22no—
44C2×Dic542—21no—
45Q8×D271—44no—
46C2×C4×D271—22no—
47C2×C4×D271—21no—
48C2×C4×D271—21no—
49C2×C4×D271D411yescharacteristic
50C2×C4×D271D411yescharacteristic
51C2×C4×D271C2311yescharacteristic
52C2×C271⋊C8C2312yes—
53C8×D271Q818yescomplemented
54C4⋊Dic271—21no—
55C4⋊Dic271—21no—
56C4⋊Dic271—21no—
57C4⋊Dic271C2311yescharacteristic
58C4⋊Dic271C2311yescharacteristic
59C4⋊Dic271C2312yes—
60Dic271⋊C4—21no—
61Dic271⋊C4—21no—
62Dic271⋊C4—21no—
63Dic271⋊C4—21no—
64Dic271⋊C4—22no—
65Dic271⋊C4—22no—
66Dic271⋊C4—21no—
67Dic271⋊C4—21no—
68C4×Dic271D411yescharacteristic
69C4×Dic271D411yescharacteristic
70C4×Dic271C2311yescharacteristic
71C2×Q8×C271—21no—
72C2×C8×C271C2312yes—
73C4⋊C4×C271—21no—
74C4⋊C4×C271—21no—
75C4⋊C4×C271—21no—
76C4⋊C4×C271C2311yescharacteristic
77C4⋊C4×C271C2311yescharacteristic
78C4⋊C4×C271C2312yes—
79C42×C271C2311yescharacteristic
80C22×D271C2×D411yescharacteristic
81Dic542—44no—
82Dic542—42no—
83C2×Dic271—21no—
84C2×Dic271—21no—
85C2×Dic271—22no—
86C2×Dic271C2×D411yescharacteristic
87C2×Dic271C2×D411yescharacteristic
88C2×Dic271C2×D411yescharacteristic
89C4×D271—22no—
90C4×D271—22no—
91C4×D271—42no—
92C4×D271—44no—
93C4×D271—42no—
94C4×D271C2×Q814yes—
95C271⋊C8C2×Q814yescomplemented
96Q8×C271—42no—
97C2×C4×C271—21no—
98C2×C4×C271—21no—
99C2×C4×C271—22no—
100C2×C4×C271C2×D411yescharacteristic
101C2×C4×C271C2×D411yescharacteristic
102C2×C4×C271C2411yescharacteristic
103C8×C271C2×Q814yescomplemented
104C22×C271C22×D411yescharacteristic
105C2×D271C4⋊Q812yes—
106C2×D271AGL(1,ℤ/8)12yes—
107C2×D271#13712yes—
108C4×C271—21no—
109C4×C271—21no—
110C4×C271—42no—
111C4×C271—41no—
112C4×C271—41no—
113C4×C271C22×Q812yes—
114Dic271—21no—
115Dic271—21no—
116Dic271—41no—
117Dic271—42no—
118Dic271—41no—
119Dic271C4⋊Q812yes—
120C2×C271C2×C4⋊Q811yescharacteristic
121C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
122C2×C271C2×#13711yescharacteristic
123D271#50114yesdirect product factor
124C271C2×#50111yescharacteristic, complemented, Sylow
125C2×#501—2711nomaximal, Sylow
126C2×C4⋊Q8—2711no—
127C2×#125—2711no—
128#501—2711no—
129#501—2711no—
130#501—2712no—
131#501—2712no—
132#501—2711no—
133#501D27111yescharacteristic, direct product factor
134C2×#107—2712no—
135C2×#106—2712no—
136C2×#97—2711no—
137C22×Q8—5421no—
138C22×C8—2712no—
139C4⋊Q8—2712no—
140C4⋊Q8—2711no—
141C4⋊Q8—5421no—
142C4⋊Q8—5421no—
143C4⋊Q8C2×D27111yescharacteristic
144#125—2711no—
145#125—2711no—
146#125—2711no—
147#125—5421no—
148#125—5421no—
149#125C2×D27111yescharacteristic
150C2×C4⋊C4—2711no—
151C2×C4⋊C4—2711no—
152C2×C4⋊C4—2712no—
153C2×C4⋊C4—5421no—
154C2×C4⋊C4—5421no—
155C2×C4⋊C4—5421no—
156C2×C42—2711no—
157#107—2712no—
158#107—2712no—
159#107—2712no—
160#107C2×D27112yes—
161#106—2712no—
162#106—2712no—
163#106—2712no—
164#106C2×D27112yes—
165#97—2712no—
166#97—2711no—
167#97C2×D27111yescharacteristic
168C2×Q8—21no—
169C2×Q8—5422no—
170C2×Q8—5421no—
171C2×Q8—10844no—
172C22×C4—2711no—
173C22×C4—2711no—
174C22×C4—2711no—
175C22×C4—5421no—
176C22×C4—5421no—
177C22×C4—5422no—
178C2×C8—2712no—
179C2×C8—2718no—
180C2×C8C22×D27112yes—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—2711no—
185C4⋊C4—2712no—
186C4⋊C4—2711no—
187C4⋊C4—5421no—
188C4⋊C4—5421no—
189C4⋊C4—5422no—
190C4⋊C4—5421no—
191C4⋊C4—5421no—
192C4⋊C4—5422no—
193C4⋊C4—5421no—
194C4⋊C4—5421no—
195C4⋊C4—5421no—
196C4⋊C4—5421no—
197C4⋊C4—5421no—
198C4⋊C4C22×D27111yescharacteristic
199C4⋊C4C22×D27112yes—
200C4⋊C4C22×D27111yescharacteristic
201C42—2711no—
202C42—2711no—
203C42—2711no—
204C42C22×D27111yescharacteristic
205C23—2711no—
206Q8—42no—
207Q8—10842no—
208Q8—10844no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—2711no—
213C2×C4—2714no—
214C2×C4—2711no—
215C2×C4—2711no—
216C2×C4—5422no—
217C2×C4—5422no—
218C2×C4—5421no—
219C2×C4—5421no—
220C2×C4—5422no—
221C2×C4—10844no—
222C2×C4—10842no—
223C2×C4—10842no—
224C2×C4D4×D27111yescharacteristic
225C2×C4D4×D27111yescharacteristic
226C2×C4C23×D27111yescharacteristic
227C8—2714no—
228C8Q8×D27114yescomplemented
229C22—2712no—
230C22—2712no—
231C22—2712no—
232C22C2×D4×D27111yescharacteristic
233C4—21no—
234C4—21no—
235C4—42no—
236C4—41no—
237C4—41no—
238C4—2712no—
239C4—5421no—
240C4—5421no—
241C4—10842no—
242C4—10841no—
243C4—10841no—
244C4C2×Q8×D27112yes—
245C2—2714no—
246C2C4⋊Q8×D27111yescharacteristic
247C2AGL(1,ℤ/8)×D27111yescharacteristic
248C2#137×D27111yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#501×C271C2
socleC22×C271C22×D4
prime radicalC22×D271C2×D4
hypercenter#501D271
hypocenterC271C2×#501
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #501

Complements

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

NH ≅ G/Ntype
3. #31774307C2semidirect
4. #31774306C2semidirect
5. #31774305C2semidirect
6. #31774304C2semidirect
7. #31774303C2semidirect
11. #501×C271C2semidirect
53. C8×D271Q8semidirect
95. C271⋊C8C2×Q8semidirect
103. C8×C271C2×Q8semidirect
123. D271#501direct (N × H)
124. C271C2×#501semidirect
133. #501D271direct (N × H)
228. C8Q8×D271semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D2712
#125×D2712
#317743072
#317743062
#317743052
#317743042
#317743032
#107×D2712
#106×D2712
#97×D2712
#501×C2712
C2×#501271