Finite Group Databasebeta

← #46849081

Subgroups — #46849081

Order 34688.

372 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 115 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#31774617C211yesmaximal, characteristic, complemented
2#31774562C211yesmaximal, characteristic, complemented
3#31774255C212yesmaximal, complemented
4#31774249C211yesmaximal, characteristic, complemented
5#31774191C212yesmaximal, complemented
6#31774111C212yesmaximal
7#31774103C211yesmaximal, characteristic, complemented
8#31774024C211yesmaximal, characteristic, complemented
9#31773996C212yesmaximal, complemented
10C4×#16864639C211yesmaximal, characteristic
11#474×C271C211yesmaximal, characteristic, complemented
12C2×Q8×D271—21no—
13C2×#16864643C2212yes—
14C2×#16864639C2211yescharacteristic
15#22576160C2211yescharacteristic
16#22576159C2212yes—
17#22576153—22no—
18#22576147C2212yes—
19#22576139—21no—
20#22576138—22no—
21#22576136—21no—
22C4×Dic542C2211yescharacteristic
23#22576125C2212yes—
24C4×D1084C2211yescharacteristic
25#22576122C2212yes—
26#22576119—21no—
27#22576093C2211yescharacteristic
28Dic271⋊C8C2212yes—
29#22576085C2212yes—
30#22576083C2212yes—
31#22576083C2212yes—
32#22576081C2211yescharacteristic
33#22576080C2211yescharacteristic
34#22576070C2212yes—
35C4×C271⋊C8C2211yescharacteristic
36C4⋊Q8×C271C2211yescharacteristic
37C4×D4×C271C2211yescharacteristic
38#107×C271C2212yes—
39C4⋊C8×C271C2211yescharacteristic
40#102×C271C2212yes—
41C2×Dic542—22no—
42C2×Dic542D411yescharacteristic
43C2×Dic542C2312yes—
44Q8×D271—44no—
45C2×D1084C2311yescharacteristic
46C2×C4×D271—22no—
47C2×C4×D271—21no—
48#16864643—24no—
49#16864639—22no—
50C2×C271⋊C8C2311yescharacteristic
51C2×C271⋊C8C2311yescharacteristic
52C4⋊Dic271—21no—
53C4⋊Dic271—21no—
54C4⋊Dic271D411yescharacteristic
55C4⋊Dic271C2311yescharacteristic
56Dic271⋊C4—22no—
57Dic271⋊C4—22no—
58Dic271⋊C4D412yes—
59C22⋊Dic271—22no—
60C22⋊Dic271—22no—
61(C2×D271)⋊C4—22no—
62(C2×D271)⋊C4—21no—
63C4×Dic271—22no—
64C4×Dic271C2312yes—
65C2×Q8×C271—21no—
66C2×D4×C271C2311yescharacteristic
67C22×C4×C271—21no—
68C2×C8×C271C2312yes—
69C4⋊C4×C271—21no—
70C4⋊C4×C271C2312yes—
71C4⋊C4×C271C2311yescharacteristic
72C22⋊C4×C271—21no—
73C42×C271C2311yescharacteristic
74C22×D271—21no—
75Dic542—22no—
76Dic542—22no—
77Dic542—22no—
78Dic542—44no—
79D1084—21no—
80D1084#4112yes—
81C2×Dic271—22no—
82C2×Dic271—21no—
83C2×Dic271C2×D412yes—
84C2×Dic271C2×D412yes—
85C4×D271—44no—
86C4×D271—42no—
87C271⋊C8—21no—
88C271⋊C8—21no—
89C23×C271—21no—
90Q8×C271—42no—
91D4×C271—21no—
92D4×C271—21no—
93C2×C4×C271—21no—
94C2×C4×C271—22no—
95C2×C4×C271—42no—
96C2×C4×C271C2×D411yescharacteristic
97C2×C4×C271C2×D411yescharacteristic
98C2×C4×C271C2411yescharacteristic
99C8×C271—22no—
100C22×C271—41no—
101C22×C271—41no—
102C22×C271—41no—
103C22×C271C22×D411yescharacteristic
104C2×D271—22no—
105C2×D271—41no—
106C2×D271—41no—
107C4×C271—21no—
108C4×C271—41no—
109C4×C271—42no—
110C4×C271AGL(1,ℤ/8)12yes—
111C4×C271C2×#4111yescharacteristic
112C4×C2712−1+411yescharacteristic
113Dic271—22no—
114Dic271—22no—
115Dic271—42no—
116Dic271—41no—
117C2×C271—81no—
118C2×C271#54811yescharacteristic
119C2×C271C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C271#57811yescharacteristic
121D271—42no—
122C271#336711yescharacteristic, complemented, Sylow
123#3367—2711nomaximal, Sylow
124#556—2711no—
125D4×Q8—2711no—
126#489—2712no—
127#487—2711no—
128#477—2712no—
129#474D27111yescharacteristic, complemented
130#463—2712no—
131#461—2711no—
132#446—2711no—
133#440—2712no—
134C4×D8—2711no—
135C22×Q8—5421no—
136C2×SD16—2712no—
137C2×D8—2711no—
138C4⋊Q8—2712no—
139C4⋊Q8—2711no—
140C4⋊Q8C2×D27111yescharacteristic
141#126—5422no—
142#124—2712no—
143C22⋊Q8—5422no—
144C22⋊Q8—5421no—
145C22⋊Q8—5421no—
146C4×Q8—2711no—
147C4×Q8—2712no—
148C4×D4—2712no—
149C4×D4—2711no—
150C4×D4C2×D27111yescharacteristic
151#117—5421no—
152#107—2711no—
153#107C2×D27112yes—
154C4⋊C8—2712no—
155C4⋊C8C2×D27111yescharacteristic
156#103—2712no—
157#103—2712no—
158#103—2712no—
159#102—2711no—
160#102—2711no—
161#102C2×D27112yes—
162#97—2712no—
163C4×C8—2711no—
164C2×Q8—21no—
165C2×Q8—2711no—
166C2×Q8—2712no—
167C2×Q8—5422no—
168C2×Q8—10844no—
169C2×D4—2711no—
170C2×D4C22×D27111yescharacteristic
171C22×C4—21no—
172C22×C4—5422no—
173C22×C4—5421no—
174SD16—5424no—
175D8—5422no—
176C2×C8—2711no—
177C2×C8—2711no—
178C2×C8C22×D27112yes—
179C4⋊C4—21no—
180C4⋊C4—2711no—
181C4⋊C4—2712no—
182C4⋊C4—2711no—
183C4⋊C4—5422no—
184C4⋊C4—5422no—
185C4⋊C4—5421no—
186C4⋊C4—5421no—
187C4⋊C4C22×D27111yescharacteristic
188C4⋊C4C22×D27112yes—
189C22⋊C4—21no—
190C22⋊C4—5422no—
191C22⋊C4—5422no—
192C22⋊C4—5422no—
193C22⋊C4—5421no—
194C42—2712no—
195C42—5422no—
196C42C22×D27111yescharacteristic
197C23—21no—
198C23—5421no—
199Q8—42no—
200Q8—5422no—
201Q8—5422no—
202Q8—5422no—
203Q8—10844no—
204D4—21no—
205D4—21no—
206D4—2712no—
207D4—5421no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—42no—
211C2×C4—2712no—
212C2×C4—2712no—
213C2×C4—5422no—
214C2×C4—5421no—
215C2×C4—10844no—
216C2×C4—10842no—
217C2×C4D4×D27111yescharacteristic
218C2×C4D4×D27111yescharacteristic
219C2×C4C23×D27111yescharacteristic
220C8—22no—
221C8—5421no—
222C8—5421no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—5422no—
227C22—10841no—
228C22—10841no—
229C22C2×D4×D27111yescharacteristic
230C4—21no—
231C4—41no—
232C4—42no—
233C4—5422no—
234C4—5422no—
235C4—10842no—
236C4—10841no—
237C4#2257618412yes—
238C4#41×D27111yescharacteristic
239C4#2257620811yescharacteristic
240C2—81no—
241C2—10842no—
242C2#3177455711yescharacteristic
243C2C2×#2257618411yescharacteristic
244C2#3177471311yescharacteristic
subgroupG/H
centerC22C2×D4×D271
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#474×C271C2
socleC22×C271C22×D4
prime radicalC2×#16864639C22
hypercenter#474D271
hypocenterC271#3367
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #31774617C2semidirect
2. #31774562C2semidirect
3. #31774255C2semidirect
4. #31774249C2semidirect
5. #31774191C2semidirect
7. #31774103C2semidirect
8. #31774024C2semidirect
9. #31773996C2semidirect
11. #474×C271C2semidirect
122. C271#3367semidirect
129. #474D271semidirect

Maximal subgroups

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

subgroupindex
#317746172
#317745622
#317742552
#317742492
#317741912
#317741112
#317741032
#317740242
#317739962
C4×#168646392
#474×C2712
#3367271