Finite Group Databasebeta

← #46835285

Subgroups — #46835285

Order 34688.

316 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 51 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#31774522C211yesmaximal, characteristic, complemented
2#31774048C211yesmaximal, characteristic, complemented
3#31773805C211yesmaximal, characteristic
4#31773635C211yesmaximal, characteristic, complemented
5#31773634C211yesmaximal, characteristic, complemented
6#31773620C211yesmaximal, characteristic, complemented
7#450×C271C211yesmaximal, characteristic, complemented
8C22×C22⋊D271—21no—
9C2×#16864642—21no—
10#22576140—21no—
11#22576139C2211yescharacteristic
12#22576138—21no—
13#22576138—21no—
14#22576137—21no—
15#22576136—21no—
16C2×C22⋊Dic271—21no—
17C2×(C2×D271)⋊C4C2211yescharacteristic
18C22⋊C4×D271—21no—
19#22576081—21no—
20C22⋊(C271⋊C8)C2211yescharacteristic
21C271⋊#95C2211yescharacteristic
22C22×D4×C271C2211yescharacteristic
23C2×SD16×C271—21no—
24C22⋊Q8×C271C2211yescharacteristic
25#102×C271—21no—
26C22⋊C8×C271C2211yescharacteristic
27C23×D271D411yescharacteristic
28C2×Dic542—21no—
29C2×Dic542D411yescharacteristic, complemented
30C2×C22⋊D271—42no—
31C2×C22⋊D271—42no—
32C2×C22⋊D271—42no—
33C22×Dic271—21no—
34C22×Dic271D411yescharacteristic
35C2×C4×D271—21no—
36C2×C4×D271—21no—
37#16864642—42no—
38C2×C271⋊C8—21no—
39C4⋊Dic271—21no—
40C4⋊Dic271D411yescharacteristic, complemented
41Dic271⋊C4—21no—
42Dic271⋊C4—21no—
43Dic271⋊C4—21no—
44C22⋊Dic271—21no—
45C22⋊Dic271—42no—
46(C2×D271)⋊C4—21no—
47(C2×D271)⋊C4—21no—
48(C2×D271)⋊C4—21no—
49(C2×D271)⋊C4—21no—
50C24×C271—21no—
51C2×Q8×C271D411yescharacteristic, complemented
52C2×D4×C271—21no—
53C2×D4×C271—21no—
54C2×D4×C271—42no—
55C22×C4×C271C2311yescharacteristic
56SD16×C271—42no—
57C2×C8×C271—21no—
58C4⋊C4×C271—21no—
59C4⋊C4×C271D411yescharacteristic, complemented
60C22⋊C4×C271—21no—
61C22×D271—21no—
62C22×D271—21no—
63C22×D271—41no—
64C22×D271—41no—
65C22×D271SD1612yescomplemented
66Dic542—41no—
67Dic542—81no—
68C22⋊D271—48no—
69C2×Dic271—22no—
70C2×Dic271—21no—
71C2×Dic271—21no—
72C2×Dic271—21no—
73C2×Dic271—41no—
74C2×Dic271—41no—
75C2×Dic271—41no—
76C4×D271—81no—
77C4×D271—81no—
78C271⋊C8—41no—
79C23×C271—22no—
80C23×C271—41no—
81C23×C271—41no—
82C23×C271—41no—
83C23×C271—41no—
84C23×C271—41no—
85C23×C271—41no—
86C23×C271C2×D411yescharacteristic
87Q8×C271—41no—
88D4×C271—22no—
89D4×C271—22no—
90D4×C271—41no—
91D4×C271—41no—
92C2×C4×C271—21no—
93C2×C4×C271—21no—
94C2×C4×C271—41no—
95C2×C4×C271C2×D411yescharacteristic
96C2×C4×C271C2×D411yescharacteristic
97C8×C271—41no—
98C22×C271—21no—
99C22×C271—21no—
100C22×C271—42no—
101C22×C271—42no—
102C22×C271—42no—
103C22×C271—81no—
104C22×C271—81no—
105C22×C271—81no—
106C22×C271—81no—
107C22×C271#12011yescharacteristic
108C22×C271C2×SD1611yescharacteristic
109C22×C271#13711yescharacteristic
110C2×D271—22no—
111C2×D271—41no—
112C2×D271—41no—
113C2×D271—42no—
114C2×D271—81no—
115C4×C271—21no—
116C4×C271—21no—
117C4×C271—41no—
118C4×C271—41no—
119Dic271—42no—
120Dic271—41no—
121Dic271—41no—
122Dic271—41no—
123C2×C271—21no—
124C2×C271—21no—
125C2×C271—82no—
126C2×C271#44811yescharacteristic
127C2×C271#45311yescharacteristic
128C2×C271B(4,𝔽2)11yescharacteristic
129D271—42no—
130C271#159611yescharacteristic, complemented, Sylow
131#1596—2711nomaximal, Sylow
132#543—2711no—
133#450—2711no—
134#450D27111yescharacteristic, complemented
135#392—2711no—
136#328—2711no—
137#328—2711no—
138#323—2711no—
139C22×D4—5421no—
140C22×D4C2×D27111yescharacteristic
141C2×SD16—21no—
142C2×SD16—5421no—
143C22⋊Q8—2711no—
144C22⋊Q8—5421no—
145C22⋊Q8—5421no—
146C22⋊Q8—5421no—
147C22⋊Q8—5421no—
148C22⋊Q8—5421no—
149C22⋊Q8C2×D27111yescharacteristic
150C2×C22⋊C4—2711no—
151C2×C22⋊C4—5421no—
152C2×C22⋊C4—5421no—
153#102—21no—
154#102—5421no—
155C22⋊C8—2711no—
156C22⋊C8C2×D27111yescharacteristic
157#95—2711no—
158C24—21no—
159C24—2711no—
160C2×Q8—2711no—
161C2×Q8—5421no—
162C2×Q8C22⋊D27111yescharacteristic, complemented
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—42no—
166C2×D4—10842no—
167C2×D4—10842no—
168C2×D4—10842no—
169C22×C4—2711no—
170C22×C4—5421no—
171C22×C4—5421no—
172C22×C4—5421no—
173C22×C4C22×D27111yescharacteristic
174SD16—42no—
175SD16—10842no—
176C2×C8—21no—
177C2×C8—5421no—
178C4⋊C4—21no—
179C4⋊C4—2711no—
180C4⋊C4—5421no—
181C4⋊C4—5421no—
182C4⋊C4—5421no—
183C4⋊C4—5421no—
184C4⋊C4C22⋊D27111yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—5421no—
187C22⋊C4—5421no—
188C22⋊C4—5421no—
189C22⋊C4—5421no—
190C22⋊C4—5421no—
191C22⋊C4—10842no—
192C23—22no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—2712no—
200C23—5421no—
201C23—5421no—
202C23—10841no—
203C23—10841no—
204C23D4×D27111yescharacteristic
205Q8—41no—
206Q8—10841no—
207Q8—21681no—
208D4—22no—
209D4—22no—
210D4—41no—
211D4—41no—
212D4—10848no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—5422no—
217C2×C4—5421no—
218C2×C4—5421no—
219C2×C4—5421no—
220C2×C4—10841no—
221C2×C4—10841no—
222C2×C4—10841no—
223C2×C4—21681no—
224C2×C4—21681no—
225C2×C4D4×D27111yescharacteristic
226C2×C4C2×C22⋊D27111yescharacteristic
227C8—41no—
228C8—10841no—
229C22—21no—
230C22—21no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22—5422no—
239C22—10842no—
240C22—10841no—
241C22—10841no—
242C22—21681no—
243C22#2257612811yescharacteristic
244C22SD16×D27111yescharacteristic
245C22#2257618911yescharacteristic
246C4—21no—
247C4—21no—
248C4—41no—
249C4—41no—
250C4—10842no—
251C4—10841no—
252C4—10841no—
253C4—10841no—
254C2—21no—
255C2—21no—
256C2—82no—
257C2—10842no—
258C2#3177403411yescharacteristic
259C2#3177406311yescharacteristic
260C2#3177408311yescharacteristic
subgroupG/H
centerC22#22576128
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#450×C271C2
socleC22×C271#120
prime radical#31773805C2
hypercenter#450D271
hypocenterC271#1596
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #31774522C2semidirect
2. #31774048C2semidirect
4. #31773635C2semidirect
5. #31773634C2semidirect
6. #31773620C2semidirect
7. #450×C271C2semidirect
29. C2×Dic542D4semidirect
40. C4⋊Dic271D4semidirect
51. C2×Q8×C271D4semidirect
59. C4⋊C4×C271D4semidirect
65. C22×D271SD16semidirect
130. C271#1596semidirect
134. #450D271semidirect
162. C2×Q8C22⋊D271semidirect
184. C4⋊C4C22⋊D271semidirect

Maximal subgroups

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

subgroupindex
#317745222
#317740482
#317738052
#317736352
#317736342
#317736202
#450×C2712
#1596271