Finite Group Databasebeta

← #149918

Subgroups — #149918

Order 512.

302 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 54 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#27908C211yesmaximal, characteristic
2#11513C211yesmaximal, characteristic
3#11447C211yesmaximal, characteristic
4#7371C212yesmaximal, complemented
5#7355C212yesmaximal, complemented
6C2×#485—21no—
7C2×#485—21no—
8#2857C411yescharacteristic
9#2285C2211yescharacteristic
10#2097—21no—
11#2097—21no—
12#1956C411yescharacteristic
13#1956C2211yescharacteristic
14#1845C411yescharacteristic
15#1844C2211yescharacteristic
16C4×C22⋊C8C411yescharacteristic
17#1464—21no—
18#1458—21no—
19#1325C2212yes—
20#1325C2212yes—
21#1324—21no—
22#1324—21no—
23C2×C4⋊Q8C2×C411yescharacteristic
24C2×C22⋊Q8—21no—
25C2×#117D411yescharacteristic
26C2×#117D411yescharacteristic
27C22×C42C2×C411yescharacteristic
28#485—42no—
29#485—42no—
30#485—42no—
31#485—42no—
32#432—41no—
33#431—41no—
34C2×#107—21no—
35C2×#107—21no—
36C2×#106—21no—
37C2×#106—21no—
38C2×#103—21no—
39C2×#103—21no—
40C2×#103—21no—
41C2×#103—21no—
42C2×C22⋊C8—21no—
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8C2×C411yescharacteristic
45#391—21no—
46#389—41no—
47#384C2×C411yescharacteristic
48#384C2×C411yescharacteristic
49#384C2311yescharacteristic
50#382D411yescharacteristic
51#382D411yescharacteristic
52#380—41no—
53#380—41no—
54C4×C4⋊C4—41no—
55#336—21no—
56#324—22no—
57#324—22no—
58#323—41no—
59#323—41no—
60C22×Q8—21no—
61C23×C4—21no—
62C23×C4C22⋊C411yescharacteristic
63C22×C8—21no—
64C22×C8—21no—
65C4⋊Q8—22no—
66C4⋊Q8—41no—
67C22⋊Q8—41no—
68C22⋊Q8—41no—
69C22⋊Q8—41no—
70C22⋊Q8—41no—
71#117—41no—
72#117—41no—
73#117—41no—
74#117—41no—
75C2×C4⋊C4—21no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—41no—
80C2×C4⋊C4—41no—
81C2×C4⋊C4C22⋊C411yescharacteristic
82C2×C4⋊C4C22⋊C411yescharacteristic
83C2×C4⋊C4C2×D411yescharacteristic
84C2×C4⋊C4C2×D411yescharacteristic
85C2×C22⋊C4—21no—
86C2×C22⋊C4—21no—
87C2×C22⋊C4—41no—
88C2×C42—22no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—41no—
93C2×C42C22⋊C411yescharacteristic
94C2×C42C22×C411yescharacteristic
95#107—42no—
96#107—42no—
97#106—42no—
98#106—42no—
99C4⋊C8—81no—
100C4⋊C8—81no—
101#103—42no—
102#103—42no—
103#103—42no—
104#103—42no—
105C22⋊C8—22no—
106C22⋊C8—22no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109#97—81no—
110C4×C8—22no—
111C4×C8—41no—
112C4×C8—81no—
113#95—41no—
114#95—41no—
115#95—42no—
116C24#9911yescharacteristic
117C2×Q8—41no—
118C2×Q8—41no—
119C2×Q8—41no—
120C2×Q8—41no—
121C2×Q8—41no—
122C2×Q8—41no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—41no—
134C22×C4—42no—
135C22×C4—42no—
136C22×C4—42no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4#9911yescharacteristic
140C22×C4C2×C22⋊C411yescharacteristic
141C2×C8—22no—
142C2×C8—22no—
143C2×C8—41no—
144C2×C8—41no—
145C2×C8—41no—
146C2×C8—41no—
147C2×C8—82no—
148C2×C8—82no—
149C4⋊C4—41no—
150C4⋊C4—41no—
151C4⋊C4—41no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—82no—
156C4⋊C4—82no—
157C4⋊C4—82no—
158C4⋊C4—82no—
159C4⋊C4—81no—
160C4⋊C4—81no—
161C22⋊C4—81no—
162C22⋊C4—81no—
163C22⋊C4—81no—
164C22⋊C4—81no—
165C42—21no—
166C42—21no—
167C42—21no—
168C42—42no—
169C42—41no—
170C42—41no—
171C42—41no—
172C42—41no—
173C42#13112yes—
174C23—21no—
175C23—21no—
176C23—21no—
177C23—41no—
178C23—41no—
179C23C2×#9911yescharacteristic
180Q8—81no—
181Q8—81no—
182Q8—81no—
183Q8—81no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—84no—
203C2×C4—81no—
204C2×C4—81no—
205C2×C4—81no—
206C2×C4—81no—
207C2×C4—82no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4—81no—
211C2×C4—81no—
212C2×C4#35311yescharacteristic
213C2×C4#35311yescharacteristic
214C2×C4#40811yescharacteristic
215C2×C4#41311yescharacteristic
216C8—42no—
217C8—42no—
218C8—162no—
219C8—162no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—81no—
226C22—81no—
227C22#146811yescharacteristic
228C22C2×#35311yescharacteristic
229C22#213311yescharacteristic
230C4—24no—
231C4—21no—
232C4—21no—
233C4—21no—
234C4—21no—
235C4—81no—
236C4—81no—
237C4—81no—
238C4—81no—
239C4—81no—
240C4—161no—
241C4—161no—
242C2—21no—
243C2—21no—
244C2—81no—
245C2#1154911yescharacteristic
246C2#1358611yescharacteristic
247C2#1358911yescharacteristic
subgroupG/H
center, socleC22C2×#353
commutatorC2×C42C22×C4
Frattini, cosocle#384C23
prime radicalC24#99
LCSG ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G
LECSG ▷ #384 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7371C2semidirect
5. #7355C2semidirect

Maximal subgroups

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

subgroupindex
#279082
#115132
#114472
#73712
#73552