Finite Group Databasebeta

← #46837379

Subgroups — #46837379

Order 34688.

478 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 75 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×#22576186C211yesmaximal, characteristic, complemented
2#31774464C211yesmaximal, characteristic, complemented
3C2×#22576088C211yesmaximal, characteristic, complemented
4#31773902C211yesmaximal, characteristic, complemented
5C2×#22576077C211yesmaximal, characteristic, complemented
6#31773657C211yesmaximal, characteristic, complemented
7C2×#127×C271C211yesmaximal, characteristic, complemented
8C2×#16864654C2211yescharacteristic
9#22576186—22no—
10#22576186—22no—
11C2×#16864642—21no—
12C2×#16864639—21no—
13C2×#16864638C2211yescharacteristic
14C2×#16864638C2211yescharacteristic
15#22576156—21no—
16#22576147—21no—
17#22576131—22no—
18#22576129—22no—
19#22576119C2211yescharacteristic
20#22576088—22no—
21#22576081—22no—
22#22576077—22no—
23C22×D4×C271—22no—
24C22×D4×C271C2211yescharacteristic
25C22×D4×C271C2211yescharacteristic
26#127×C271—22no—
27#127×C271—22no—
28C2×C42×C271C2211yescharacteristic
29#16864654—21no—
30#16864654—21no—
31#16864654—21no—
32#16864654—21no—
33C2×Dic542D411yescharacteristic, complemented
34C2×D1084D411yescharacteristic, complemented
35C2×C22⋊D271—24no—
36C2×C22⋊D271—21no—
37C2×C4×D271—21no—
38#16864642—24no—
39#16864639—24no—
40#16864638—21no—
41#16864638—21no—
42#16864638—21no—
43#16864638D412yescomplemented
44C2×C271⋊C8—21no—
45C2×C271⋊C8—21no—
46C4⋊Dic271D412yescomplemented
47C22⋊Dic271—24no—
48C22⋊Dic271—21no—
49C4×Dic271—21no—
50C24×C271—21no—
51C24×C271D412yes—
52C2×D4×C271—28no—
53C2×D4×C271—21no—
54C2×D4×C271—21no—
55C2×D4×C271—21no—
56C2×D4×C271—21no—
57C2×D4×C271—21no—
58C2×D4×C271—21no—
59C2×D4×C271—21no—
60C2×D4×C271—21no—
61C2×D4×C271—21no—
62C2×D4×C271—21no—
63C2×D4×C271—21no—
64C2×D4×C271—48no—
65C2×D4×C271D412yescomplemented
66C22×C4×C271—21no—
67C22×C4×C271C2311yescharacteristic
68C42×C271—21no—
69C42×C271D412yescomplemented
70C22×D271—21no—
71Dic542—21no—
72Dic542—21no—
73D1084—21no—
74D1084—21no—
75C22⋊D271—41no—
76C22⋊D271—41no—
77C22⋊D271—84no—
78C2×Dic271—22no—
79C2×Dic271—21no—
80C4×D271—41no—
81C4×D271—41no—
82C271⋊C8—22no—
83C271⋊C8—41no—
84C23×C271—22no—
85C23×C271—22no—
86C23×C271—22no—
87C23×C271—22no—
88C23×C271—44no—
89C23×C271—44no—
90C23×C271—41no—
91C23×C271—41no—
92C23×C271C2×D411yescharacteristic
93D4×C271—22no—
94D4×C271—22no—
95D4×C271—416no—
96D4×C271—41no—
97D4×C271—41no—
98D4×C271—41no—
99D4×C271—41no—
100D4×C271—41no—
101D4×C271—41no—
102C2×C4×C271—24no—
103C2×C4×C271—22no—
104C2×C4×C271C2×D411yescharacteristic
105C2×C4×C271C2×D411yescharacteristic
106C2×C4×C271C2×D411yescharacteristic
107C2×C4×C271C2×D411yescharacteristic
108C2×C4×C271C2×D411yescharacteristic
109C2×C4×C271#4111yescharacteristic
110C22×C271—22no—
111C22×C271—44no—
112C22×C271—44no—
113C22×C271—42no—
114C22×C271—42no—
115C22×C271—44no—
116C22×C271—82no—
117C22×C271—82no—
118C22×C271—82no—
119C22×C271#12011yescharacteristic
120C22×C271#12011yescharacteristic
121C22×C271#12111yescharacteristic
122C2×D271—41no—
123C2×D271—42no—
124C4×C271—24no—
125C4×C271#12011yescharacteristic
126C4×C271#12111yescharacteristic
127C4×C271#12111yescharacteristic
128C4×C271#12711yescharacteristic
129Dic271—41no—
130Dic271—42no—
131C2×C271—21no—
132C2×C271—21no—
133C2×C271—82no—
134C2×C271—82no—
135C2×C271#39211yescharacteristic
136C2×C271#45312yes—
137D271—81no—
138C271#200211yescharacteristic, complemented, Sylow
139#2002—2711nomaximal, Sylow
140C2×AGL(1,ℤ/8)—2711no—
141#535—2711no—
142C2×#127D27111yescharacteristic, complemented
143C2×#104—2711no—
144#418—2711no—
145C2×#100—2711no—
146#339—2711no—
147C2×#41—2711no—
148C22×D4—22no—
149C22×D4C2×D27111yescharacteristic
150C22×D4C2×D27111yescharacteristic
151AGL(1,ℤ/8)—5422no—
152AGL(1,ℤ/8)—5422no—
153C2×SD16—5421no—
154C2×D8—5421no—
155C2×M16—2711no—
156C2×M16—2711no—
157#127—22no—
158#127—22no—
159#127—5421no—
160#124—5421no—
161#121—5422no—
162#120—5422no—
163#117—2711no—
164C2×C42C2×D27111yescharacteristic
165#104—5422no—
166#102—5422no—
167#100—5422no—
168C24—21no—
169C24C22⋊D27112yes—
170#41—5421no—
171#41—5421no—
172#41—5421no—
173#41—5421no—
174C2×Q8—2711no—
175C2×D4—28no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—48no—
188C2×D4—2711no—
189C2×D4—5421no—
190C2×D4—5424no—
191C2×D4C22⋊D27112yescomplemented
192C22×C4—21no—
193C22×C4—5421no—
194C22×C4C22×D27111yescharacteristic
195SD16—5424no—
196D8—5424no—
197M16—2712no—
198M16—5421no—
199M16—5421no—
200M16—5421no—
201C2×C8—5421no—
202C2×C8—5421no—
203C4⋊C4—2712no—
204C22⋊C4—5424no—
205C22⋊C4—5421no—
206C42—21no—
207C42—5421no—
208C42C22⋊D27112yescomplemented
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—44no—
214C23—44no—
215C23—41no—
216C23—41no—
217C23—5421no—
218C23C2×C22⋊D27111yescharacteristic
219Q8—5421no—
220Q8—5421no—
221D4—22no—
222D4—22no—
223D4—416no—
224D4—41no—
225D4—41no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—41no—
230D4—5421no—
231D4—5421no—
232D4—10841no—
233D4—10841no—
234D4—21684no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—5422no—
238C2×C4—5421no—
239C2×C4—10841no—
240C2×C4—10841no—
241C2×C4D4×D27111yescharacteristic
242C2×C4D4×D27111yescharacteristic
243C2×C4D4×D27111yescharacteristic
244C2×C4C2×C22⋊D27111yescharacteristic
245C2×C4C2×C22⋊D27111yescharacteristic
246C2×C4#1686465311yescharacteristic
247C8—5422no—
248C8—10841no—
249C22—22no—
250C22—44no—
251C22—44no—
252C22—42no—
253C22—42no—
254C22—44no—
255C22—82no—
256C22—82no—
257C22—82no—
258C22—10841no—
259C22—10842no—
260C22#2257612811yescharacteristic
261C22#2257612911yescharacteristic
262C22#2257613211yescharacteristic
263C4—24no—
264C4—10841no—
265C4—10842no—
266C4#2257612811yescharacteristic
267C4#2257613111yescharacteristic
268C4#2257613211yescharacteristic
269C4#2257615611yescharacteristic
270C2—21no—
271C2—21no—
272C2—82no—
273C2—82no—
274C2—21681no—
275C2#3177380511yescharacteristic
276C2#3177406412yes—
subgroupG/H
centerC22#22576128
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
FittingC2×#127×C271C2
socleC22×C271#120
hypercenterC2×#127D271
hypocenterC271#2002
LCSG ▷ C22×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×#22576186C2semidirect
2. #31774464C2semidirect
3. C2×#22576088C2semidirect
4. #31773902C2semidirect
5. C2×#22576077C2semidirect
6. #31773657C2semidirect
7. C2×#127×C271C2semidirect
33. C2×Dic542D4semidirect
34. C2×D1084D4semidirect
43. #16864638D4semidirect
46. C4⋊Dic271D4semidirect
65. C2×D4×C271D4semidirect
69. C42×C271D4semidirect
138. C271#2002semidirect
142. C2×#127D271semidirect
191. C2×D4C22⋊D271semidirect
208. C42C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
C2×#225761862
#317744642
C2×#225760882
#317739022
C2×#225760772
#317736572
C2×#127×C2712
#2002271