Finite Group Databasebeta

← #46839633

Subgroups — #46839633

Order 34688.

528 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 155 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×#22576125C211yesmaximal, characteristic, complemented
2C2×#22576116C211yesmaximal, characteristic
3#31773835C211yesmaximal, characteristic, complemented
4#31773828C212yesmaximal, complemented
5#31773820C212yesmaximal
6#31773818C211yesmaximal, characteristic, complemented
7#31773809C211yesmaximal, characteristic
8C4⋊C4×Dic271C212yesmaximal, complemented
9#31773748C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic271C212yesmaximal
11#399×C271C211yesmaximal, characteristic
12C22×Dic542C2211yescharacteristic
13C23×Dic271C2211yescharacteristic
14#22576125—24no—
15#22576116—24no—
16C2×C4⋊Dic271C2212yes—
17C2×Dic271⋊C4C2212yes—
18C2×Dic271⋊C4C2211yescharacteristic
19C2×Dic271⋊C4C2212yes—
20C2×Dic271⋊C4C2211yescharacteristic
21C2×C22⋊Dic271C2212yes—
22C2×C22⋊Dic271C2211yescharacteristic
23C2×C4×Dic271C2212yes—
24C2×C4×Dic271C2212yes—
25C2×C4×Dic271C2211yescharacteristic
26C2×C4×Dic271C2211yescharacteristic
27C271⋊#95C2212yes—
28C271⋊#95C2212yes—
29C271⋊#95C2211yescharacteristic
30C271⋊#95C2212yes—
31C271⋊#95C2212yes—
32C2×C4⋊C4×C271C2212yes—
33C2×C4⋊C4×C271C2211yescharacteristic
34C2×C22⋊C4×C271C2212yes—
35C2×C22⋊C4×C271C2211yescharacteristic
36#95×C271C2211yescharacteristic
37C2×Dic542—24no—
38C2×Dic542—22no—
39C22×Dic271—22no—
40C22×Dic271—22no—
41C22×Dic271Q814yescomplemented
42C22×Dic271C2312yes—
43C22×Dic271C2311yescharacteristic
44C22×Dic271C2311yescharacteristic
45C22×Dic271C2311yescharacteristic
46C22×Dic271C2312yes—
47C22×Dic271C2311yescharacteristic
48C4⋊Dic271—24no—
49Dic271⋊C4—22no—
50Dic271⋊C4—24no—
51Dic271⋊C4—22no—
52Dic271⋊C4—24no—
53C22⋊Dic271—22no—
54C22⋊Dic271—24no—
55C4×Dic271—22no—
56C4×Dic271—24no—
57C4×Dic271—24no—
58C4×Dic271D414yes—
59C24×C271C2311yescharacteristic
60C22×C4×C271C2311yescharacteristic
61C22×C4×C271C2312yes—
62C22×C4×C271C2311yescharacteristic
63C22×C4×C271C2312yes—
64C4⋊C4×C271—22no—
65C4⋊C4×C271—24no—
66C22⋊C4×C271—22no—
67C22⋊C4×C271—24no—
68Dic542—28no—
69C2×Dic271—22no—
70C2×Dic271—21no—
71C2×Dic271—22no—
72C2×Dic271—21no—
73C2×Dic271—22no—
74C2×Dic271—28no—
75C2×Dic271—21no—
76C2×Dic271—22no—
77C2×Dic271—22no—
78C2×Dic271—22no—
79C2×Dic271—22no—
80C2×Dic271C2×D412yes—
81C2×Dic271C2×D412yes—
82C2×Dic271C2×Q812yes—
83C2×Dic271C2×Q812yes—
84C2×Dic271#4114yes—
85C2×Dic271#4112yes—
86C23×C271—21no—
87C23×C271—22no—
88C23×C271—22no—
89C23×C271—21no—
90C23×C271C2×Q812yes—
91C23×C271C2411yescharacteristic
92C2×C4×C271—22no—
93C2×C4×C271—22no—
94C2×C4×C271—22no—
95C2×C4×C271—21no—
96C2×C4×C271—22no—
97C2×C4×C271—22no—
98C2×C4×C271—22no—
99C2×C4×C271—22no—
100C2×C4×C271C2×D412yes—
101C2×C4×C271#4114yes—
102C22×C271—22no—
103C22×C271—24no—
104C22×C271—41no—
105C22×C271—42no—
106C22×C271—41no—
107C22×C271C22×D411yescharacteristic
108C22×C271C22×Q811yescharacteristic
109C22×C271C2×#4111yescharacteristic
110C22×C271C2×#4112yes—
111C22×C271C2×#4112yes—
112C4×C271—22no—
113C4×C271—24no—
114C4×C271—41no—
115C4×C271—42no—
116Dic271—22no—
117Dic271—24no—
118Dic271—22no—
119Dic271—41no—
120Dic271—42no—
121Dic271C22⋊Q814yes—
122C2×C271—42no—
123C2×C271C2×C22⋊Q811yescharacteristic
124C2×C271#52511yescharacteristic
125C2×C271#52912yes—
126C2×C271#53211yescharacteristic
127C2×C271#53312yes—
128C271#239611yescharacteristic, complemented, Sylow
129#2396—2711nomaximal, Sylow
130C2×C4×Q8—2711no—
131C2×#117—2711no—
132#400—2711no—
133#399D27111yescharacteristic
134#398—2712no—
135#396—2712no—
136#395—2711no—
137#393—2711no—
138C4×C4⋊C4—2712no—
139C4×C4⋊C4—2711no—
140C4×C22⋊C4—2712no—
141C22×Q8—2711no—
142C23×C4—2711no—
143C4×Q8—5424no—
144#117—5424no—
145C2×C4⋊C4—2711no—
146C2×C4⋊C4—2711no—
147C2×C4⋊C4—2712no—
148C2×C4⋊C4—2712no—
149C2×C4⋊C4—2712no—
150C2×C4⋊C4C2×D27111yescharacteristic
151C2×C4⋊C4C2×D27112yes—
152C2×C22⋊C4—2711no—
153C2×C22⋊C4—2712no—
154C2×C22⋊C4C2×D27111yescharacteristic
155C2×C22⋊C4C2×D27112yes—
156C2×C42—2711no—
157C2×C42—2711no—
158C2×C42—2712no—
159C2×C42—2712no—
160#95—2712no—
161#95—2712no—
162#95—2712no—
163#95—2711no—
164#95—2712no—
165#95C2×D27111yescharacteristic
166C24C22×D27111yescharacteristic
167C2×Q8—5424no—
168C2×Q8—5422no—
169C22×C4—2711no—
170C22×C4—2714no—
171C22×C4—2711no—
172C22×C4—2712no—
173C22×C4—2711no—
174C22×C4—2712no—
175C22×C4—2711no—
176C22×C4—5422no—
177C22×C4—5422no—
178C22×C4C22×D27111yescharacteristic
179C22×C4C22×D27112yes—
180C22×C4C22×D27111yescharacteristic
181C22×C4C22×D27112yes—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—5422no—
185C4⋊C4—5424no—
186C4⋊C4—5424no—
187C4⋊C4—5422no—
188C4⋊C4—5424no—
189C22⋊C4—22no—
190C22⋊C4—24no—
191C22⋊C4—5422no—
192C22⋊C4—5424no—
193C42—2714no—
194C42—5422no—
195C42—5424no—
196C42—5424no—
197C23—21no—
198C23—22no—
199C23—21no—
200C23—22no—
201C23Q8×D27112yes—
202C23C23×D27111yescharacteristic
203Q8—5428no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—2712no—
213C2×C4—2712no—
214C2×C4—2712no—
215C2×C4—2712no—
216C2×C4—2712no—
217C2×C4—2714no—
218C2×C4—5428no—
219C2×C4—5422no—
220C2×C4—5421no—
221C2×C4—5421no—
222C2×C4—5422no—
223C2×C4—5422no—
224C2×C4—5421no—
225C2×C4—5422no—
226C2×C4—5422no—
227C2×C4—5422no—
228C2×C4—5422no—
229C2×C4D4×D27112yes—
230C2×C4#1686465314yes—
231C22—22no—
232C22—24no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22C2×D4×D27111yescharacteristic
237C22C2×Q8×D27111yescharacteristic
238C22#41×D27111yescharacteristic
239C22#41×D27112yes—
240C22C2×#1686465312yes—
241C4—22no—
242C4—24no—
243C4—41no—
244C4—42no—
245C4—2714no—
246C4—5422no—
247C4—5424no—
248C4—5422no—
249C4—10841no—
250C4—10842no—
251C2—42no—
252C2C22⋊Q8×D27111yescharacteristic
253C2#3177440511yescharacteristic
254C2#3177442412yes—
255C2#3177444511yescharacteristic
256C2#3177444912yes—
subgroupG/H
center, FrattiniC23C23×D271
commutator, socle, cosocleC23×C271C24
Fitting#399×C271C2
prime radicalC24×C271C23
hypercenter#399D271
hypocenterC271#2396
LCSG ▷ C23×C271 ▷ C271
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#22576125C2semidirect
3. #31773835C2semidirect
4. #31773828C2semidirect
6. #31773818C2semidirect
8. C4⋊C4×Dic271C2semidirect
9. #31773748C2semidirect
41. C22×Dic271Q8semidirect
128. C271#2396semidirect

Maximal subgroups

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

subgroupindex
C2×#225761252
C2×#225761162
#317738352
#317738282
#317738202
#317738182
#317738092
C4⋊C4×Dic2712
#317737482
C22⋊C4×Dic2712
#399×C2712
#2396271