Finite Group Databasebeta

← #46836396

Subgroups — #46836396

Order 34688.

316 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16864637C211yesmaximal, characteristic
2C2×#22576093C211yesmaximal, characteristic, complemented
3C2×#22576091C211yesmaximal, characteristic, complemented
4#31773754C211yesmaximal, characteristic
5#31773754C211yesmaximal, characteristic
6#31773660C211yesmaximal, characteristic, complemented
7#340×C271C211yesmaximal, characteristic, complemented
8C22×C4×D271C2211yescharacteristic
9C2×#16864637—22no—
10C2×#16864637—22no—
11C2×#16864637C414yescomplemented
12C22×C271⋊C8C2211yescharacteristic
13C2×C4⋊Dic271C2211yescharacteristic
14C2×C4⋊Dic271C2211yescharacteristic
15C2×(C2×D271)⋊C4—21no—
16C2×(C2×D271)⋊C4—21no—
17#22576093—22no—
18#22576091—22no—
19C271⋊#95—21no—
20C271⋊#95—21no—
21C22×C8×C271C2211yescharacteristic
22C2×C4⋊C4×C271C2211yescharacteristic
23C2×C4⋊C4×C271C2211yescharacteristic
24C23×D271D411yescharacteristic
25C22×Dic271—21no—
26C22×Dic271—21no—
27C22×Dic271D411yescharacteristic
28C2×C4×D271—21no—
29C2×C4×D271—21no—
30C2×C4×D271—21no—
31C2×C4×D271—21no—
32C2×C4×D271C2×C412yes—
33C2×C4×D271Q812yes—
34#16864637—28no—
35C2×C271⋊C8C2×C412yes—
36C2×C271⋊C8D412yes—
37C2×C271⋊C8D412yes—
38C4⋊Dic271—21no—
39C4⋊Dic271—21no—
40C4⋊Dic271—21no—
41C4⋊Dic271—21no—
42(C2×D271)⋊C4—42no—
43(C2×D271)⋊C4—42no—
44C22×C4×C271—21no—
45C22×C4×C271—21no—
46C22×C4×C271C2311yescharacteristic
47C2×C8×C271—21no—
48C2×C8×C271—21no—
49C2×C8×C271C2×C412yes—
50C4⋊C4×C271—21no—
51C4⋊C4×C271—21no—
52C4⋊C4×C271—21no—
53C4⋊C4×C271—21no—
54C22×D271—21no—
55C22×D271—21no—
56C22×D271—21no—
57C22×D271—21no—
58C22×D271—21no—
59C22×D271—21no—
60C22×D271C4⋊C412yes—
61C2×Dic271—21no—
62C2×Dic271—21no—
63C2×Dic271—41no—
64C2×Dic271—41no—
65C2×Dic271—41no—
66C2×Dic271—41no—
67C2×Dic271—41no—
68C2×Dic271—41no—
69C2×Dic271C4⋊C412yes—
70C4×D271—22no—
71C4×D271—22no—
72C4×D271—22no—
73C4×D271—22no—
74C271⋊C8C22⋊C414yescomplemented
75C23×C271C2×D411yescharacteristic
76C2×C4×C271—41no—
77C2×C4×C271—41no—
78C2×C4×C271—41no—
79C2×C4×C271—41no—
80C2×C4×C271—41no—
81C2×C4×C271—41no—
82C2×C4×C271C22×C411yescharacteristic
83C2×C4×C271C2×D411yescharacteristic
84C2×C4×C271C2×D411yescharacteristic
85C2×C4×C271C2×Q811yescharacteristic
86C2×C4×C271#4111yescharacteristic
87C2×C4×C271#4111yescharacteristic
88C8×C271—22no—
89C22×C271C2×C4⋊C411yescharacteristic
90C22×C271#12111yescharacteristic
91C22×C271#12111yescharacteristic
92C22×C271AGL(1,ℤ/8)11yescharacteristic
93C22×C271AGL(1,ℤ/8)11yescharacteristic
94C22×C271#13711yescharacteristic
95C22×C271#13711yescharacteristic
96C2×D271—22no—
97C2×D271—22no—
98C2×D271—22no—
99C2×D271—41no—
100C2×D271—41no—
101C2×D271—41no—
102C2×D271—41no—
103C4×C271—81no—
104C4×C271—81no—
105C4×C271C2×C22⋊C411yescharacteristic
106C4×C271#11711yescharacteristic
107C4×C271C22⋊Q811yescharacteristic
108C4×C271C22⋊Q811yescharacteristic
109Dic271—22no—
110Dic271—81no—
111Dic271—81no—
112C2×C271#38011yescharacteristic
113C2×C271#42811yescharacteristic
114C2×C271#42811yescharacteristic
115C2×C271#46811yescharacteristic
116C2×C271#46811yescharacteristic
117C2×C271#46911yescharacteristic
118C2×C271#47011yescharacteristic
119D271—42no—
120C271#180911yescharacteristic, complemented, Sylow
121#1809—2711nomaximal, Sylow
122C22×M16—2711no—
123C2×#107—2711no—
124C2×#106—2711no—
125#380—2711no—
126#380—2711no—
127#340—2711no—
128#340D27111yescharacteristic, complemented
129C23×C4—2711no—
130C2×M16—2714no—
131C2×M16—5422no—
132C2×M16—5422no—
133C22×C8—2711no—
134C22×C8C2×D27111yescharacteristic
135C2×C4⋊C4—2711no—
136C2×C4⋊C4—2711no—
137C2×C4⋊C4C2×D27111yescharacteristic
138C2×C4⋊C4C2×D27111yescharacteristic
139C2×C22⋊C4—5421no—
140C2×C22⋊C4—5421no—
141#107—5422no—
142#106—5422no—
143#95—5421no—
144#95—5421no—
145C24—2711no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—2711no—
149C22×C4—2712no—
150C22×C4—2712no—
151C22×C4—5421no—
152C22×C4—5421no—
153C22×C4—5421no—
154C22×C4—5421no—
155C22×C4—5421no—
156C22×C4—5421no—
157C22×C4C22×D27111yescharacteristic
158M16—5428no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—2712no—
162C2×C8—2712no—
163C2×C8—2712no—
164C2×C8C4×D27112yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—5421no—
170C4⋊C4—5421no—
171C4⋊C4—5421no—
172C4⋊C4—5421no—
173C22⋊C4—10842no—
174C22⋊C4—10842no—
175C23—2712no—
176C23—5421no—
177C23—5421no—
178C23—5421no—
179C23—5421no—
180C23—5421no—
181C23—5421no—
182C23D4×D27111yescharacteristic
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—2712no—
190C2×C4—5421no—
191C2×C4—5421no—
192C2×C4—5422no—
193C2×C4—5422no—
194C2×C4—5422no—
195C2×C4—5422no—
196C2×C4—10841no—
197C2×C4—10841no—
198C2×C4—10841no—
199C2×C4—10841no—
200C2×C4—10841no—
201C2×C4—10841no—
202C2×C4C2×C4×D27111yescharacteristic
203C2×C4D4×D27111yescharacteristic
204C2×C4D4×D27111yescharacteristic
205C2×C4Q8×D27111yescharacteristic
206C2×C4#1686465411yescharacteristic
207C2×C4#1686465411yescharacteristic
208C8—22no—
209C8—2714no—
210C22—5422no—
211C22—5422no—
212C22—5422no—
213C22—10841no—
214C22—10841no—
215C22—10841no—
216C22—10841no—
217C22C4⋊C4×D27111yescharacteristic
218C22#2257613311yescharacteristic
219C22#2257613311yescharacteristic
220C22#2257618411yescharacteristic
221C22#2257618711yescharacteristic
222C22#2257618911yescharacteristic
223C22#2257619211yescharacteristic
224C4—81no—
225C4—81no—
226C4—5422no—
227C4—21681no—
228C4—21681no—
229C4C22⋊C4×D27111yescharacteristic
230C4#2257611711yescharacteristic
231C4#2257613811yescharacteristic
232C4#2257613811yescharacteristic
233C2—10842no—
234C2#3177375611yescharacteristic
235C2#3177393611yescharacteristic
236C2#3177393811yescharacteristic
237C2#3177414211yescharacteristic
238C2#3177414611yescharacteristic
239C2#3177414911yescharacteristic
240C2#3177415411yescharacteristic
subgroupG/H
centerC23D4×D271
commutatorC2×C4×C271C22×C4
FrattiniC22×C4C22×D271
Fitting#340×C271C2
socleC23×C271C2×D4
prime radicalC23×D271D4
cosocleC22×C4×C271C23
hypercenter#340D271
hypocenterC271#1809
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#22576093C2semidirect
3. C2×#22576091C2semidirect
6. #31773660C2semidirect
7. #340×C271C2semidirect
11. C2×#16864637C4semidirect
74. C271⋊C8C22⋊C4semidirect
120. C271#1809semidirect
128. #340D271semidirect

Maximal subgroups

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

subgroupindex
C22×#168646372
C2×#225760932
C2×#225760912
#317737542
#317736602
#340×C2712
#1809271