Finite Group Databasebeta

← #99604

Subgroups — #99604

Order 512.

488 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 136 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#23247C211yesmaximal, characteristic
2C4×#494C211yesmaximal, characteristic
3#9864C212yesmaximal
4#7571C211yesmaximal, characteristic
5#7015C212yesmaximal
6C2×#494C411yescharacteristic
7C2×C4×Q16—22no—
8C4×C4⋊Q8C2211yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10#1984—22no—
11#1969—22no—
12#1956C411yescharacteristic
13#1956C411yescharacteristic
14#1939—22no—
15#1845C411yescharacteristic
16C8×C4⋊C4—22no—
17C4×C4⋊C8C2212yes—
18C4×C4⋊C8C2212yes—
19C42×C8C2211yescharacteristic
20#1279—22no—
21#1279—22no—
22#1272—22no—
23C22×Q16—22no—
24C2×C4⋊Q8C2×C411yescharacteristic
25C2×C4⋊Q8C2×C411yescharacteristic
26C2×C4×Q8—21no—
27C2×C4×Q8D412yes—
28#494—22no—
29#494C814yescomplemented
30C4×Q16—44no—
31C4×Q16—44no—
32C2×#107—22no—
33C2×C4⋊C8—22no—
34C2×C4⋊C8—22no—
35C2×C4⋊C8D412yes—
36C2×C4⋊C8D412yes—
37C2×#103—22no—
38C2×#103—22no—
39C2×C4×C8—21no—
40C2×C4×C8—22no—
41C2×C4×C8—21no—
42C2×C4×C8—22no—
43C2×C4×C8—22no—
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8C2×C411yescharacteristic
46#391—21no—
47#391D412yes—
48#389—22no—
49#389—22no—
50#384C2×C411yescharacteristic
51#384C2×C411yescharacteristic
52C4×C4⋊C4—22no—
53C4×C4⋊C4—22no—
54C43C2311yescharacteristic
55#336—24no—
56#336—22no—
57#336—22no—
58C22×Q8—21no—
59C22×Q8C22⋊C412yes—
60C2×Q16—28no—
61C2×Q16—44no—
62C2×Q16—44no—
63C22×C8—21no—
64C22×C8—21no—
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—42no—
68C4⋊Q8—21no—
69C4⋊Q8—22no—
70C4⋊Q8—22no—
71C4⋊Q8C2×C812yes—
72C4⋊Q8C2×C812yes—
73C4⋊Q8M1612yes—
74C4×Q8—42no—
75C4×Q8—42no—
76C4×Q8—42no—
77C4×Q8—42no—
78C2×C4⋊C4—22no—
79C2×C4⋊C4—22no—
80C2×C4⋊C4—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4C22⋊C412yes—
86C2×C42—22no—
87C2×C42—41no—
88C2×C42C22×C411yescharacteristic
89C2×C42C2×D411yescharacteristic
90C2×C42C2×D411yescharacteristic
91C2×C42C2×D412yes—
92C2×C42#4112yes—
93#107—44no—
94C4⋊C8—24no—
95C4⋊C8—22no—
96C4⋊C8—22no—
97C4⋊C8—22no—
98C4⋊C8—42no—
99C4⋊C8—82no—
100C4⋊C8Q1614yescomplemented
101#103—44no—
102#103—44no—
103C4×C8—22no—
104C4×C8—22no—
105C4×C8—21no—
106C4×C8—22no—
107C4×C8—42no—
108C4×C8—42no—
109C4×C8—42no—
110C4×C8—42no—
111C4×C8—42no—
112C4×C8—82no—
113C4×C8C2×C812yes—
114#95—24no—
115#95—42no—
116C2×Q8—22no—
117C2×Q8—22no—
118C2×Q8—42no—
119C2×Q8—42no—
120C2×Q8—42no—
121C2×Q8—42no—
122C2×Q8—41no—
123C2×Q8C22⋊C814yes—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—41no—
127C22×C4—41no—
128C22×C4C2×C22⋊C411yescharacteristic
129C22×C4C4×D412yes—
130C22×C4#12011yescharacteristic
131C22×C4#12112yes—
132C22×C4#12311yescharacteristic
133Q16—416no—
134C2×C8—22no—
135C2×C8—22no—
136C2×C8—22no—
137C2×C8—22no—
138C2×C8—24no—
139C2×C8—41no—
140C2×C8—41no—
141C2×C8—42no—
142C2×C8—42no—
143C2×C8—44no—
144C2×C8—42no—
145C2×C8—42no—
146C2×C8—42no—
147C2×C8—82no—
148C2×C8—82no—
149C4⋊C4—24no—
150C4⋊C4—24no—
151C4⋊C4—24no—
152C4⋊C4—42no—
153C4⋊C4—42no—
154C4⋊C4—42no—
155C4⋊C4—42no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—82no—
160C4⋊C4—82no—
161C42—22no—
162C42—22no—
163C42—21no—
164C42—21no—
165C42—41no—
166C42—42no—
167C42—42no—
168C42—81no—
169C42—81no—
170C42C22×C811yescharacteristic
171C42C2×M1611yescharacteristic
172C42#13112yes—
173C42C2×Q1612yes—
174C42#13512yes—
175C42#13712yes—
176C42#13712yes—
177C23#38611yescharacteristic
178Q8—44no—
179Q8—44no—
180Q8—82no—
181Q8—82no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—24no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—42no—
192C2×C4—44no—
193C2×C4—42no—
194C2×C4—81no—
195C2×C4—81no—
196C2×C4—82no—
197C2×C4—81no—
198C2×C4C2×C22⋊C811yescharacteristic
199C2×C4#40811yescharacteristic
200C2×C4C8×D412yes—
201C2×C4#43512yes—
202C2×C4C4×Q1612yes—
203C2×C4#44112yes—
204C2×C4#44312yes—
205C2×C4#44412yes—
206C2×C4#45112yes—
207C2×C4#45212yes—
208C2×C4#45311yescharacteristic
209C2×C4#45311yescharacteristic
210C2×C4#46212yes—
211C2×C4#46412yes—
212C8—24no—
213C8—42no—
214C8—44no—
215C8—84no—
216C8—82no—
217C22#186911yescharacteristic
218C22#187612yes—
219C22#187812yes—
220C22#188411yescharacteristic
221C22#188711yescharacteristic
222C4—22no—
223C4—22no—
224C4—21no—
225C4—21no—
226C4—41no—
227C4—44no—
228C4—82no—
229C4—82no—
230C4—161no—
231C4C8×Q1612yes—
232C4#157912yes—
233C4#158412yes—
234C4#158712yes—
235C4#163212yes—
236C4#163311yescharacteristic
237C4#163411yescharacteristic
238C2#894612yes—
239C2#895911yescharacteristic
240C2#905612yes—
241C2#906611yescharacteristic
242C2#906911yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC42C22×C8
Frattini, cosocleC43C23
socle, prime radicalC23#386
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
29. #494C8semidirect
100. C4⋊C8Q16semidirect

Maximal subgroups

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

subgroupindex
#232472
C4×#4942
#98642
#75712
#70152