Finite Group Databasebeta

← #46840005

Subgroups — #46840005

Order 34688.

800 conjugacy classes of subgroups; only the 193 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 193 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
1C22×Q8×D271C211yesmaximal, characteristic
2C2×#22576158C211yesmaximal, characteristic, complemented
3C2×#22576136C211yesmaximal, characteristic
4#31773817C214yesmaximal, complemented
5#31773804C211yesmaximal, characteristic, complemented
6#31773803C211yesmaximal, characteristic, complemented
7#31773768C214yesmaximal
8C4×(C2×D271)⋊C4C211yesmaximal, characteristic
9C2×C4⋊Q8×C271C211yesmaximal, characteristic, complemented
10C22×Dic542C2212yes—
11C22×Dic542C2211yescharacteristic
12C2×Q8×D271—24no—
13C2×Q8×D271—24no—
14C2×Q8×D271—24no—
15C22×C4×D271C2212yes—
16C22×C4×D271C2211yescharacteristic
17#22576158—24no—
18#22576136—24no—
19C2×C4⋊Dic271C2211yescharacteristic
20C2×Dic271⋊C4C2214yes—
21C2×Dic271⋊C4C2212yes—
22C2×(C2×D271)⋊C4C2212yes—
23C2×(C2×D271)⋊C4C2212yes—
24C2×C4×Dic271C2211yescharacteristic
25C271⋊#95C2212yes—
26C271⋊#95C2214yes—
27C271⋊#95C2214yes—
28C22×Q8×C271C2211yescharacteristic
29C22×Q8×C271C2211yescharacteristic
30C4⋊Q8×C271—24no—
31C2×C4⋊C4×C271C2214yes—
32C2×C42×C271C2211yescharacteristic
33C23×D271C2311yescharacteristic
34C2×Dic542—24no—
35C2×Dic542—24no—
36C2×Dic542—22no—
37C2×Dic542—24no—
38C2×Dic542—22no—
39C2×Dic542D414yes—
40Q8×D271—232no—
41C22×Dic271C2312yes—
42C22×Dic271C2312yes—
43C22×Dic271C2311yescharacteristic
44C22×Dic271C2312yes—
45C2×C4×D271—24no—
46C2×C4×D271—24no—
47C2×C4×D271—22no—
48C2×C4×D271—24no—
49C2×C4×D271—22no—
50C2×C4×D271D414yes—
51C4⋊Dic271—22no—
52Dic271⋊C4—24no—
53Dic271⋊C4—28no—
54(C2×D271)⋊C4—24no—
55(C2×D271)⋊C4Q818yescomplemented
56C4×Dic271—22no—
57C2×Q8×C271—22no—
58C2×Q8×C271—22no—
59C2×Q8×C271—22no—
60C2×Q8×C271—22no—
61C2×Q8×C271—22no—
62C2×Q8×C271D414yes—
63C22×C4×C271C2312yes—
64C22×C4×C271C2312yes—
65C22×C4×C271C2312yes—
66C22×C4×C271C2311yescharacteristic
67C4⋊C4×C271—28no—
68C42×C271—22no—
69C22×D271—21no—
70C22×D271—21no—
71C22×D271—22no—
72C22×D271C2×D412yes—
73C22×D271C2×Q814yes—
74Dic542—216no—
75Dic542—28no—
76C2×Dic271—22no—
77C2×Dic271—22no—
78C2×Dic271—21no—
79C2×Dic271—22no—
80C2×Dic271—21no—
81C2×Dic271—22no—
82C2×Dic271—22no—
83C2×Dic271—24no—
84C2×Dic271C2×D414yes—
85C2×Dic271C2×D412yes—
86C2×Dic271C2×Q814yescomplemented
87C4×D271—216no—
88C4×D271—28no—
89C23×C271C2411yescharacteristic
90Q8×C271—28no—
91Q8×C271—44no—
92C2×C4×C271—22no—
93C2×C4×C271—22no—
94C2×C4×C271—22no—
95C2×C4×C271—22no—
96C2×C4×C271—22no—
97C2×C4×C271—22no—
98C2×C4×C271—22no—
99C2×C4×C271C2×D412yes—
100C2×C4×C271C2×D412yes—
101C2×C4×C271C2×D414yes—
102C2×C4×C271C2×D412yes—
103C2×C4×C271C2×Q814yescomplemented
104C22×C271C22×D411yescharacteristic
105C22×C271C22×D411yescharacteristic
106C22×C271C22×D411yescharacteristic
107C22×C271C22×Q812yes—
108C22×C2712−1+412yes—
109C2×D271—22no—
110C2×D271—22no—
111C2×D271—24no—
112C2×D271—24no—
113C2×D271C4⋊Q814yescomplemented
114C4×C271—24no—
115C4×C271—24no—
116C4×C271—42no—
117C4×C271#12014yes—
118Dic271—24no—
119Dic271—22no—
120Dic271—24no—
121Dic271—42no—
122C2×C271C2×#12011yescharacteristic
123C2×C271C2×C4⋊Q811yescharacteristic
124C2×C271#53611yescharacteristic
125C2×C271D4×Q812yes—
126C2×C271D4×Q812yes—
127D271—24no—
128C271#243411yescharacteristic, complemented, Sylow
129#2434—2711nomaximal, Sylow
130C23×Q8—2711no—
131C2×C4⋊Q8—2711no—
132C2×C4⋊Q8D27111yescharacteristic, complemented
133C2×C22⋊Q8—2711no—
134#395—2714no—
135#391—2711no—
136#391—2711no—
137#385—2714no—
138C4×C22⋊C4—2711no—
139C22×Q8—2711no—
140C22×Q8—2712no—
141C22×Q8—5424no—
142C22×Q8—5424no—
143C22×Q8—5424no—
144C22×Q8C2×D27111yescharacteristic
145C22×Q8C2×D27111yescharacteristic
146C23×C4—2711no—
147C23×C4—2712no—
148C4⋊Q8—24no—
149C4⋊Q8—5424no—
150C22⋊Q8—5424no—
151C2×C4⋊C4—2711no—
152C2×C4⋊C4—2712no—
153C2×C4⋊C4—2714no—
154C2×C4⋊C4C2×D27114yes—
155C2×C22⋊C4—2712no—
156C2×C22⋊C4—2712no—
157C2×C42—2711no—
158C2×C42C2×D27111yescharacteristic
159#95—2712no—
160#95—2714no—
161#95—2714no—
162C24—2711no—
163C2×Q8—22no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—2714no—
169C2×Q8—5424no—
170C2×Q8—5424no—
171C2×Q8—54232no—
172C2×Q8—5422no—
173C2×Q8—5424no—
174C2×Q8—5422no—
175C2×Q8C22⋊D27114yes—
176C22×C4—2712no—
177C22×C4—2711no—
178C22×C4—2712no—
179C22×C4—2714no—
180C22×C4—2712no—
181C22×C4—5424no—
182C22×C4—5424no—
183C22×C4—5422no—
184C22×C4—5424no—
185C22×C4—5422no—
186C22×C4C22×D27112yes—
187C22×C4C22×D27112yes—
188C22×C4C22×D27112yes—
189C22×C4C22×D27111yescharacteristic
190C4⋊C4—28no—
191C4⋊C4—5424no—
192C4⋊C4—5428no—
193C4⋊C4—5422no—
194C22⋊C4—2718no—
195C22⋊C4—5424no—
196C42—22no—
197C42—5422no—
198C23—2712no—
199C23—2714no—
200C23—5421no—
201C23—5421no—
202C23—5422no—
203C23C23×D27111yescharacteristic
204Q8—28no—
205Q8—44no—
206Q8—54216no—
207Q8—5428no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—2712no—
216C2×C4—2714no—
217C2×C4—2714no—
218C2×C4—54216no—
219C2×C4—5422no—
220C2×C4—5422no—
221C2×C4—5421no—
222C2×C4—5421no—
223C2×C4—5422no—
224C2×C4—5422no—
225C2×C4—5428no—
226C2×C4—5422no—
227C2×C4—5424no—
228C2×C4D4×D27112yes—
229C2×C4D4×D27112yes—
230C2×C4C2×C22⋊D27114yes—
231C2×C4C2×C22⋊D27112yes—
232C2×C4Q8×D27114yescomplemented
233C22—2714no—
234C22—5422no—
235C22—5424no—
236C22—5424no—
237C22—5422no—
238C22C2×D4×D27111yescharacteristic
239C22C2×D4×D27111yescharacteristic
240C22C22×C22⋊D27111yescharacteristic
241C22C2×Q8×D27112yes—
242C22#2257620812yes—
243C4—24no—
244C4—24no—
245C4—42no—
246C4—5424no—
247C4—5422no—
248C4—5424no—
249C4—10842no—
250C4#2257612814yes—
251C2—5424no—
252C2C2×#2257612811yescharacteristic
253C2C4⋊Q8×D27111yescharacteristic
254C2#3177447211yescharacteristic
255C2Q8×C22⋊D27112yes—
256C2#3177456212yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
FittingC2×C4⋊Q8×C271C2
prime radicalC23×D271C23
hypercenterC2×C4⋊Q8D271
hypocenterC271#2434
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#22576158C2semidirect
4. #31773817C2semidirect
5. #31773804C2semidirect
6. #31773803C2semidirect
9. C2×C4⋊Q8×C271C2semidirect
55. (C2×D271)⋊C4Q8semidirect
86. C2×Dic271C2×Q8semidirect
103. C2×C4×C271C2×Q8semidirect
113. C2×D271C4⋊Q8semidirect
128. C271#2434semidirect
132. C2×C4⋊Q8D271semidirect
232. C2×C4Q8×D271semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D2712
C2×#225761582
C2×#225761362
#317738172
#317738042
#317738032
#317737682
C4×(C2×D271)⋊C42
C2×C4⋊Q8×C2712
#2434271