Finite Group Databasebeta

← #46849017

Subgroups — #46849017

Order 34688.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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#31774621C211yesmaximal, characteristic, complemented
2#31774604C211yesmaximal, characteristic, complemented
3#31774253C212yesmaximal, complemented
4#31774247C211yesmaximal, characteristic, complemented
5#31774202C212yesmaximal, complemented
6#31774188C211yesmaximal, characteristic, complemented
7#31774114C212yesmaximal
8#31774021C211yesmaximal, characteristic, complemented
9SD16×Dic271C212yesmaximal, complemented
10C4×#16864639C211yesmaximal, characteristic
11#461×C271C211yesmaximal, characteristic, complemented
12C2×#16864639C2211yescharacteristic
13#22576160C2211yescharacteristic
14#22576155—22no—
15#22576151C2212yes—
16#22576149C2212yes—
17#22576140—21no—
18#22576139—21no—
19#22576137—22no—
20Q8×Dic271C2212yes—
21C4×Dic542C2211yescharacteristic
22C4×D1084C2211yescharacteristic
23D4×Dic271C2212yes—
24#22576118—21no—
25C2×C4⋊Dic271—21no—
26#22576093C2211yescharacteristic
27#22576092C2212yes—
28Dic271⋊C8C2212yes—
29#22576084C2212yes—
30#22576084C2212yes—
31#22576082C2212yes—
32#22576081C2211yescharacteristic
33#22576080C2211yescharacteristic
34C4×C271⋊C8C2211yescharacteristic
35C8×Dic271C2212yes—
36C2×SD16×C271C2212yes—
37C4⋊Q8×C271C2211yescharacteristic
38C4×D4×C271C2211yescharacteristic
39C4⋊C8×C271C2211yescharacteristic
40#103×C271C2212yes—
41C2×Dic542—21no—
42C2×Dic542D411yescharacteristic
43C2×D1084C2311yescharacteristic
44C22×Dic271—22no—
45C2×C4×D271—21no—
46#16864639—22no—
47C2×C271⋊C8C2311yescharacteristic
48C2×C271⋊C8C2311yescharacteristic
49C4⋊Dic271—22no—
50C4⋊Dic271—22no—
51C4⋊Dic271—21no—
52C4⋊Dic271D411yescharacteristic
53C4⋊Dic271C2312yes—
54C4⋊Dic271C2311yescharacteristic
55Dic271⋊C4—22no—
56Dic271⋊C4D412yes—
57C22⋊Dic271—22no—
58(C2×D271)⋊C4—22no—
59(C2×D271)⋊C4—22no—
60(C2×D271)⋊C4—21no—
61C4×Dic271—22no—
62C4×Dic271C2312yes—
63C2×Q8×C271C2312yes—
64C2×D4×C271C2311yescharacteristic
65C22×C4×C271—21no—
66SD16×C271—24no—
67C2×C8×C271C2312yes—
68C4⋊C4×C271—21no—
69C4⋊C4×C271—21no—
70C4⋊C4×C271C2311yescharacteristic
71C22⋊C4×C271—21no—
72C42×C271C2311yescharacteristic
73C22×D271—21no—
74Dic542—22no—
75Dic542—81no—
76D1084—21no—
77D1084—21no—
78C2×Dic271—21no—
79C2×Dic271—22no—
80C2×Dic271—82no—
81C2×Dic271C2×D412yes—
82C2×Dic271C2×D412yes—
83C4×D271—42no—
84C271⋊C8—21no—
85C271⋊C8—21no—
86C23×C271—21no—
87Q8×C271—22no—
88Q8×C271—22no—
89D4×C271—21no—
90D4×C271#4112yes—
91C2×C4×C271—21no—
92C2×C4×C271—22no—
93C2×C4×C271—42no—
94C2×C4×C271C2×D411yescharacteristic
95C2×C4×C271C2×D411yescharacteristic
96C2×C4×C271C2411yescharacteristic
97C8×C271—22no—
98C22×C271—22no—
99C22×C271—41no—
100C22×C271—41no—
101C22×C271C22×D411yescharacteristic
102C2×D271—41no—
103C2×D271—41no—
104C2×D271—41no—
105C4×C271—21no—
106C4×C271—42no—
107C4×C271—41no—
108C4×C271#13512yes—
109C4×C271C2×#4111yescharacteristic
110C4×C2712−1+411yescharacteristic
111Dic271—22no—
112Dic271—22no—
113Dic271—41no—
114Dic271—42no—
115C2×C271—42no—
116C2×C271#54811yescharacteristic
117C2×C271C2×#13511yescharacteristic
118C2×C271#57711yescharacteristic
119D271—81no—
120C271#336011yescharacteristic, complemented, Sylow
121#3360—2711nomaximal, Sylow
122#556—2711no—
123#554—2711no—
124#488—2712no—
125#487—2711no—
126#479—2712no—
127#476—2711no—
128#463—2712no—
129#461D27111yescharacteristic, complemented
130C8×Q8—2711no—
131C4×SD16—2712no—
132C4×D8—2711no—
133C2×SD16C2×D27112yes—
134C2×D8—2711no—
135C4⋊Q8—2711no—
136C4⋊Q8C2×D27111yescharacteristic
137#126—5422no—
138#125—2712no—
139#124—2712no—
140C22⋊Q8—5422no—
141C22⋊Q8—5421no—
142C22⋊Q8—5421no—
143C4×Q8—2711no—
144C4×Q8—2712no—
145C4×D4—2712no—
146C4×D4—2711no—
147C4×D4C2×D27111yescharacteristic
148#117—5421no—
149C2×C4⋊C4—5421no—
150#107—2711no—
151#106—2712no—
152C4⋊C8—2712no—
153C4⋊C8C2×D27111yescharacteristic
154#103—2712no—
155#103—2712no—
156#103C2×D27112yes—
157#102—2711no—
158#102—2712no—
159#102—2711no—
160C4×C8—2712no—
161C4×C8—2711no—
162C2×Q8—2711no—
163C2×Q8—5421no—
164C2×Q8C22×D27112yes—
165C2×D4—2711no—
166C2×D4C22×D27111yescharacteristic
167C22×C4—21no—
168C22×C4—5422no—
169C22×C4—5421no—
170SD16—24no—
171D8—5422no—
172C2×C8—2711no—
173C2×C8—2711no—
174C2×C8C22×D27112yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—2711no—
178C4⋊C4—2712no—
179C4⋊C4—2712no—
180C4⋊C4—2711no—
181C4⋊C4—5422no—
182C4⋊C4—5422no—
183C4⋊C4—5422no—
184C4⋊C4—5421no—
185C4⋊C4C22×D27111yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—5422no—
188C22⋊C4—5422no—
189C22⋊C4—5422no—
190C22⋊C4—5421no—
191C42—2712no—
192C42—5422no—
193C42C22×D27111yescharacteristic
194C23—21no—
195C23—5421no—
196Q8—22no—
197Q8—22no—
198Q8—5422no—
199Q8—21681no—
200D4—21no—
201D4—5421no—
202D4—5421no—
203D4#1686465312yes—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—2712no—
208C2×C4—2712no—
209C2×C4—5421no—
210C2×C4—5422no—
211C2×C4—10842no—
212C2×C4—21682no—
213C2×C4D4×D27111yescharacteristic
214C2×C4D4×D27111yescharacteristic
215C2×C4C23×D27111yescharacteristic
216C8—22no—
217C8—5421no—
218C8—5421no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—10841no—
223C22—10841no—
224C22—10841no—
225C22C2×D4×D27111yescharacteristic
226C4—21no—
227C4—42no—
228C4—41no—
229C4—5422no—
230C4—5422no—
231C4—10841no—
232C4—10842no—
233C4#2257618112yes—
234C4C2×#1686465311yescharacteristic
235C4#2257620711yescharacteristic
236C2—42no—
237C2—21681no—
238C2#3177455311yescharacteristic
239C2C2×#2257618111yescharacteristic
240C2#3177470611yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#461×C271C2
socleC22×C271C22×D4
prime radicalC2×#16864639C22
hypercenter#461D271
hypocenterC271#3360
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #31774621C2semidirect
2. #31774604C2semidirect
3. #31774253C2semidirect
4. #31774247C2semidirect
5. #31774202C2semidirect
6. #31774188C2semidirect
8. #31774021C2semidirect
9. SD16×Dic271C2semidirect
11. #461×C271C2semidirect
120. C271#3360semidirect
129. #461D271semidirect

Maximal subgroups

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

subgroupindex
#317746212
#317746042
#317742532
#317742472
#317742022
#317741882
#317741142
#317740212
SD16×Dic2712
C4×#168646392
#461×C2712
#3360271