Finite Group Databasebeta

← #46835305

Subgroups — #46835305

Order 34688.

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

normal subgroups only

This group has 266 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#31774526C211yesmaximal, characteristic, complemented
2#31774050C211yesmaximal, characteristic
3#31774047C211yesmaximal, characteristic
4#31773815C211yesmaximal, characteristic
5#31773633C211yesmaximal, characteristic, complemented
6#31773622C211yesmaximal, characteristic, complemented
7#328×C271C211yesmaximal, characteristic, complemented
8C22×D1084C2211yescharacteristic
9C2×#16864643—21no—
10C2×#16864641—21no—
11#22576159—21no—
12#22576158—21no—
13#22576150—21no—
14#22576150—21no—
15#22576139C2211yescharacteristic
16#22576137—21no—
17#22576116—21no—
18C2×Dic271⋊C4C2211yescharacteristic
19C2×(C2×D271)⋊C4—21no—
20C2×(C2×D271)⋊C4—21no—
21#22576082—21no—
22#22576080—21no—
23C22⋊(C271⋊C8)C2211yescharacteristic
24C22⋊Q8×C271C2211yescharacteristic
25C22⋊C8×C271C2211yescharacteristic
26#95×C271C2211yescharacteristic
27C23×D271—21no—
28C2×Dic542—21no—
29C2×Dic542D411yescharacteristic
30C2×D1084—21no—
31C2×D1084—21no—
32C2×D1084—42no—
33C22×Dic271D411yescharacteristic
34C22×Dic271D411yescharacteristic
35#16864643—42no—
36#16864641—42no—
37C2×C271⋊C8—21no—
38C4⋊Dic271—21no—
39C4⋊Dic271D411yescharacteristic
40Dic271⋊C4—21no—
41Dic271⋊C4—21no—
42Dic271⋊C4—21no—
43Dic271⋊C4—21no—
44Dic271⋊C4—21no—
45Dic271⋊C4—21no—
46Dic271⋊C4—21no—
47C22⋊Dic271—21no—
48(C2×D271)⋊C4—42no—
49(C2×D271)⋊C4—42no—
50C4×Dic271—21no—
51C4×Dic271—21no—
52C2×Q8×C271D411yescharacteristic
53C22×C4×C271—21no—
54C22×C4×C271C2311yescharacteristic
55C2×C8×C271—21no—
56C4⋊C4×C271—21no—
57C4⋊C4×C271D411yescharacteristic
58C22⋊C4×C271—21no—
59C22×D271—22no—
60C22×D271—41no—
61C22×D271—41no—
62C22×D271—41no—
63C22×D271—41no—
64C22×D271—41no—
65C22×D271—41no—
66Dic542—41no—
67Dic542—81no—
68D1084—22no—
69D1084—22no—
70D1084—41no—
71D1084—41no—
72C2×Dic271—21no—
73C2×Dic271—21no—
74C2×Dic271—21no—
75C2×Dic271—21no—
76C2×Dic271—21no—
77C2×Dic271—41no—
78C2×Dic271SD1612yescomplemented
79C271⋊C8—41no—
80C23×C271C2×D411yescharacteristic
81Q8×C271—41no—
82C2×C4×C271—21no—
83C2×C4×C271—21no—
84C2×C4×C271—41no—
85C2×C4×C271—41no—
86C2×C4×C271—41no—
87C2×C4×C271—41no—
88C2×C4×C271C2×D411yescharacteristic
89C2×C4×C271C2×D411yescharacteristic
90C8×C271—41no—
91C22×C271—21no—
92C22×C271—21no—
93C22×C271#12011yescharacteristic
94C22×C271C2×SD1611yescharacteristic
95C22×C271#13711yescharacteristic
96C2×D271—42no—
97C2×D271—42no—
98C2×D271—42no—
99C2×D271—81no—
100C2×D271—81no—
101C2×D271—81no—
102C2×D271—81no—
103C4×C271—21no—
104C4×C271—21no—
105C4×C271—41no—
106C4×C271—41no—
107C4×C271—81no—
108Dic271—22no—
109Dic271—41no—
110Dic271—41no—
111Dic271—41no—
112C2×C271—21no—
113C2×C271—21no—
114C2×C271#44811yescharacteristic
115C2×C271#45411yescharacteristic
116C2×C271B(4,𝔽2)11yescharacteristic
117D271—82no—
118C271#159911yescharacteristic, complemented, Sylow
119#1599—2711nomaximal, Sylow
120#544—2711no—
121#450—2711no—
122#450—2711no—
123#394—2711no—
124#328—2711no—
125#328D27111yescharacteristic, complemented
126#324—2711no—
127C22×D4—2711no—
128C2×SD16—5421no—
129C2×SD16—5421no—
130C4⋊Q8—5421no—
131C4⋊Q8—5421no—
132#125—5421no—
133#125—5421no—
134C22⋊Q8—2711no—
135C22⋊Q8—5421no—
136C22⋊Q8C2×D27111yescharacteristic
137#117—5421no—
138C2×C4⋊C4—2711no—
139C2×C22⋊C4—5421no—
140C2×C22⋊C4—5421no—
141#102—5421no—
142#102—5421no—
143C22⋊C8—2711no—
144C22⋊C8C2×D27111yescharacteristic
145#95C2×D27111yescharacteristic
146C24—5421no—
147C2×Q8—2711no—
148C2×Q8—5421no—
149C2×Q8D108411yescharacteristic
150C2×D4—5421no—
151C2×D4—5421no—
152C2×D4—10842no—
153C22×C4—21no—
154C22×C4—2711no—
155C22×C4—2711no—
156C22×C4C22×D27111yescharacteristic
157SD16—10842no—
158SD16—10842no—
159C2×C8—21no—
160C2×C8—5421no—
161C4⋊C4—21no—
162C4⋊C4—2711no—
163C4⋊C4—5421no—
164C4⋊C4—5421no—
165C4⋊C4—5421no—
166C4⋊C4—5421no—
167C4⋊C4—5421no—
168C4⋊C4—5421no—
169C4⋊C4—5421no—
170C4⋊C4—5421no—
171C4⋊C4D108411yescharacteristic
172C22⋊C4—21no—
173C22⋊C4—5421no—
174C22⋊C4—10842no—
175C22⋊C4—10842no—
176C42—5421no—
177C42—5421no—
178C23—5422no—
179C23—10841no—
180C23—10841no—
181C23—10841no—
182C23—10841no—
183C23—10841no—
184C23—10841no—
185C23D4×D27111yescharacteristic
186Q8—41no—
187Q8—10841no—
188Q8—21681no—
189D4—5422no—
190D4—5422no—
191D4—10841no—
192D4—10841no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—2712no—
200C2×C4—5421no—
201C2×C4—5421no—
202C2×C4—5421no—
203C2×C4—5421no—
204C2×C4—5421no—
205C2×C4—10841no—
206C2×C4D4×D27111yescharacteristic
207C2×C4C2×D108411yescharacteristic
208C8—41no—
209C8—10841no—
210C22—21no—
211C22—21no—
212C22—10842no—
213C22—10842no—
214C22—10842no—
215C22—21681no—
216C22—21681no—
217C22—21681no—
218C22—21681no—
219C22#2257613011yescharacteristic
220C22SD16×D27111yescharacteristic
221C22#2257619211yescharacteristic
222C4—21no—
223C4—21no—
224C4—41no—
225C4—41no—
226C4—81no—
227C4—5422no—
228C4—10841no—
229C4—10841no—
230C4—10841no—
231C2—21no—
232C2—21no—
233C2—21682no—
234C2#3177403711yescharacteristic
235C2#3177407111yescharacteristic
236C2#3177408511yescharacteristic
subgroupG/H
centerC22#22576130
commutator, cosocleC22×C4×C271C23
FrattiniC22×C4C22×D271
Fitting#328×C271C2
socleC22×C271#120
prime radicalC22×D1084C22
hypercenter#328D271
hypocenterC271#1599
LCSG ▷ C22×C4×C271 ▷ C22×C271 ▷ C271
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
1. #31774526C2semidirect
5. #31773633C2semidirect
6. #31773622C2semidirect
7. #328×C271C2semidirect
78. C2×Dic271SD16semidirect
118. C271#1599semidirect
125. #328D271semidirect

Maximal subgroups

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

subgroupindex
#317745262
#317740502
#317740472
#317738152
#317736332
#317736222
#328×C2712
#1599271