Finite Group Databasebeta

← #46837542

Subgroups — #46837542

Order 34688.

278 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 63 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#31774523C211yesmaximal, characteristic, complemented
2C2×#22576074C211yesmaximal, characteristic
3C2×#22576074C211yesmaximal, characteristic
4C2×#22576074C211yesmaximal, characteristic
5#31773664C211yesmaximal, characteristic, complemented
6#31773663C211yesmaximal, characteristic, complemented
7#342×C271C211yesmaximal, characteristic, complemented
8C22×C22⋊D271C2211yescharacteristic
9#22576139—22no—
10#22576137—22no—
11#22576137—22no—
12C2×C22⋊Dic271C2211yescharacteristic
13C2×C22⋊Dic271C2211yescharacteristic
14C2×C22⋊Dic271C2211yescharacteristic
15#22576074—22no—
16#22576074—22no—
17#22576074—22no—
18C2×C22⋊C4×C271C2211yescharacteristic
19C2×C22⋊C4×C271C2211yescharacteristic
20C2×C22⋊C4×C271C2211yescharacteristic
21C23×D271Q811yescharacteristic, complemented
22C2×Dic542—22no—
23C2×C22⋊D271—21no—
24C2×C22⋊D271—21no—
25C2×C22⋊D271—21no—
26C2×C22⋊D271—21no—
27C2×C22⋊D271—21no—
28C2×C22⋊D271—21no—
29C22×Dic271—21no—
30C22×Dic271—21no—
31C22×Dic271—21no—
32C22×Dic271Q811yescharacteristic, complemented
33C4⋊Dic271—22no—
34Dic271⋊C4—24no—
35C22⋊Dic271—21no—
36C22⋊Dic271—21no—
37C22⋊Dic271—21no—
38C22⋊Dic271—21no—
39C22⋊Dic271D412yes—
40C22⋊Dic271D412yes—
41C24×C271C2311yescharacteristic
42C22×C4×C271—21no—
43C22×C4×C271—21no—
44C22×C4×C271—21no—
45C22⋊C4×C271—21no—
46C22⋊C4×C271—21no—
47C22⋊C4×C271—21no—
48C22⋊C4×C271—21no—
49C22⋊C4×C271—21no—
50C22⋊C4×C271D412yes—
51C22×D271—21no—
52C22×D271—21no—
53C22×D271—21no—
54C22×D271—81no—
55Dic542—82no—
56C22⋊D271—41no—
57C22⋊D271—41no—
58C22⋊D271—41no—
59C22⋊D271—41no—
60C2×Dic271—22no—
61C2×Dic271—22no—
62C2×Dic271—21no—
63C2×Dic271—21no—
64C2×Dic271—21no—
65C2×Dic271—41no—
66C2×Dic271—81no—
67C2×Dic271—81no—
68C2×Dic271—81no—
69C23×C271—81no—
70C23×C271C2×D411yescharacteristic
71C23×C271C2×D411yescharacteristic
72C23×C271C2×D411yescharacteristic
73C23×C271C2×Q811yescharacteristic
74C23×C271#4111yescharacteristic
75C23×C271#4111yescharacteristic
76C23×C271#4111yescharacteristic
77C2×C4×C271—22no—
78C2×C4×C271—41no—
79C2×C4×C271—41no—
80C2×C4×C271—81no—
81C2×C4×C271—81no—
82C2×C4×C271—81no—
83C22×C271—41no—
84C22×C271—41no—
85C22×C271—41no—
86C22×C271—41no—
87C22×C271—41no—
88C22×C271—41no—
89C22×C271—41no—
90C22×C271#12011yescharacteristic
91C22×C271C22⋊Q811yescharacteristic
92C22×C271C22⋊Q811yescharacteristic
93C22×C271C22⋊Q811yescharacteristic
94C22×C271#12411yescharacteristic
95C22×C271#12411yescharacteristic
96C22×C271#12411yescharacteristic
97C2×D271—41no—
98C2×D271—81no—
99C2×D271—81no—
100C2×D271—81no—
101C4×C271—42no—
102C4×C271—81no—
103C4×C271—81no—
104Dic271—42no—
105Dic271—42no—
106Dic271—41no—
107Dic271—81no—
108C2×C271—21no—
109C2×C271—21no—
110C2×C271—21no—
111C2×C271—21no—
112C2×C271—21no—
113C2×C271—21no—
114C2×C271—21no—
115C2×C271#39311yescharacteristic
116D271—81no—
117C271#203211yescharacteristic, complemented, Sylow
118#2032—2711nomaximal, Sylow
119#543—2711no—
120C2×#99—2711no—
121C2×#99—2711no—
122C2×#99—2711no—
123#342—2711no—
124#342—2711no—
125#342D27111yescharacteristic, complemented
126C22×D4—2711no—
127C22⋊Q8—5422no—
128C22⋊Q8—5422no—
129C22⋊Q8—5422no—
130C2×C22⋊C4—2711no—
131C2×C22⋊C4—2711no—
132C2×C22⋊C4—2711no—
133C2×C22⋊C4C2×D27111yescharacteristic
134C2×C22⋊C4C2×D27111yescharacteristic
135C2×C22⋊C4C2×D27111yescharacteristic
136#99—5422no—
137#99—5422no—
138#99—5422no—
139C24—2711no—
140C24C22×D27111yescharacteristic
141C2×Q8—5422no—
142C2×D4—5421no—
143C2×D4—5421no—
144C2×D4—5421no—
145C2×D4—5421no—
146C2×D4—5421no—
147C2×D4—5421no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—2711no—
152C22×C4—5421no—
153C22×C4—5421no—
154C22×C4—5421no—
155C4⋊C4—5424no—
156C4⋊C4—5422no—
157C22⋊C4—21no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—21no—
161C22⋊C4—21no—
162C22⋊C4—2712no—
163C22⋊C4—2712no—
164C22⋊C4—5421no—
165C22⋊C4—5421no—
166C22⋊C4—5421no—
167C22⋊C4—5421no—
168C22⋊C4D108412yes—
169C23—81no—
170C23—5421no—
171C23—5421no—
172C23—5421no—
173C23—21681no—
174C23D4×D27111yescharacteristic
175C23D4×D27111yescharacteristic
176C23C2×D108411yescharacteristic
177C23Q8×D27111yescharacteristic
178C23#1686465311yescharacteristic
179C23#1686465411yescharacteristic
180C23#1686465411yescharacteristic
181Q8—21682no—
182D4—10841no—
183D4—10841no—
184D4—10841no—
185D4—10841no—
186C2×C4—22no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4—81no—
192C2×C4—5422no—
193C2×C4—5422no—
194C2×C4—5421no—
195C2×C4—5421no—
196C2×C4—5421no—
197C2×C4—10841no—
198C2×C4—21681no—
199C2×C4—21681no—
200C2×C4—21681no—
201C22—41no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—10841no—
209C22—21681no—
210C22—21681no—
211C22—21681no—
212C22#2257613011yescharacteristic
213C22#2257613811yescharacteristic
214C22#2257613811yescharacteristic
215C22#2257614011yescharacteristic
216C22#2257614611yescharacteristic
217C22#2257614611yescharacteristic
218C22#2257614811yescharacteristic
219C4—42no—
220C4—81no—
221C4—81no—
222C4—10842no—
223C4—10842no—
224C4—10841no—
225C4—21681no—
226C2—21no—
227C2—21no—
228C2—21no—
229C2—21no—
230C2—21no—
231C2—21no—
232C2—21no—
233C2—21681no—
234C2#3177381011yescharacteristic
subgroupG/H
centerC2#31773810
commutator, cosocleC24×C271C23
FrattiniC24C22×D271
Fitting#342×C271C2
socleC2×C271#393
prime radicalC22×C22⋊D271C22
hypercenter#342D271
hypocenterC271#2032
LCSG ▷ C24×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #31774523C2semidirect
5. #31773664C2semidirect
6. #31773663C2semidirect
7. #342×C271C2semidirect
21. C23×D271Q8semidirect
32. C22×Dic271Q8semidirect
117. C271#2032semidirect
125. #342D271semidirect

Maximal subgroups

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

subgroupindex
#317745232
C2×#225760742
#317736642
#317736632
#342×C2712
#2032271