Finite Group Databasebeta

← #122603

Subgroups — #122603

Order 512.

334 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 70 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#23699C211yesmaximal, characteristic
2#11460C211yesmaximal, characteristic
3#7987C211yesmaximal, characteristic
4#7083C212yesmaximal, complemented
5#7082C212yesmaximal, complemented
6C2×C4×Q16C411yescharacteristic
7#2920—21no—
8C2×#431C2211yescharacteristic
9C4×C22⋊Q8—21no—
10#2163—21no—
11#2086C2211yescharacteristic
12#2075C411yescharacteristic
13#2066—21no—
14#2056C2211yescharacteristic
15#2035—21no—
16#2001C411yescharacteristic
17C4×#106C411yescharacteristic
18C4×#103—21no—
19#1322C2212yes—
20#1314—21no—
21#1290—21no—
22#1290C2212yes—
23C22×Q16C2×C411yescharacteristic
24C2×C22⋊Q8—21no—
25C2×C4×Q8—21no—
26C22×C42D411yescharacteristic
27C4×Q16—41no—
28C4×Q16—41no—
29#432—22no—
30#432—22no—
31#431—41no—
32#431—41no—
33#431—41no—
34#431—41no—
35C2×#107C2×C411yescharacteristic
36C2×#106—21no—
37#423—41no—
38C2×C4⋊C8D411yescharacteristic
39C2×C4⋊C8C2311yescharacteristic
40C2×#103—21no—
41C2×#103—21no—
42C2×#103—21no—
43#407—41no—
44C2×C22⋊C8C2×C411yescharacteristic
45C2×C22⋊C8D411yescharacteristic
46C4×M16—22no—
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8D411yescharacteristic
49#391—21no—
50#389—21no—
51#387—41no—
52#385—41no—
53#380—21no—
54C4×C4⋊C4—21no—
55C4×C4⋊C4—41no—
56C4×C22⋊C4—41no—
57C4⋊C16—41no—
58C22⋊C16—22no—
59#346—41no—
60#340—21no—
61#340—21no—
62#340C2×C411yescharacteristic
63#340C2×C411yescharacteristic
64C22×Q8—21no—
65C23×C4C22⋊C411yescharacteristic
66C23×C4D811yescharacteristic
67C23×C4SD1611yescharacteristic
68C2×Q16—41no—
69C2×Q16—42no—
70C2×M16—42no—
71C22×C8C22⋊C411yescharacteristic
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22×C411yescharacteristic
74C22×C8C2×D411yescharacteristic
75C22⋊Q8—42no—
76C22⋊Q8—42no—
77C4×Q8—41no—
78C4×Q8—41no—
79C4×Q8—41no—
80C4×Q8—41no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C42—21no—
90C2×C42—21no—
91C2×C42—21no—
92C2×C42—21no—
93C2×C42—41no—
94C2×C42—41no—
95C2×C42C22⋊C411yescharacteristic
96C2×C42D811yescharacteristic
97C2×C42SD1611yescharacteristic
98C2×C42C2×D411yescharacteristic
99C2×C16—42no—
100#107—41no—
101#106—42no—
102C4⋊C8—41no—
103C4⋊C8—41no—
104C4⋊C8—42no—
105C4⋊C8—42no—
106#103—42no—
107#103—42no—
108#103—42no—
109C22⋊C8—41no—
110C22⋊C8—41no—
111C22⋊C8—42no—
112C22⋊C8—42no—
113#97—24no—
114C4×C8—21no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8—21no—
118C4×C8—42no—
119#95—41no—
120#95—41no—
121#95—41no—
122#95—41no—
123#95—41no—
124C24#10211yescharacteristic
125C2×Q8—22no—
126C2×Q8—41no—
127C2×Q8—41no—
128C2×Q8—41no—
129C2×Q8—41no—
130C2×Q8—41no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—41no—
141C22×C4—42no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4#10211yescharacteristic
147C22×C4#10211yescharacteristic
148C22×C4#10211yescharacteristic
149C22×C4C2×C22⋊C411yescharacteristic
150C22×C4C2×D811yescharacteristic
151C22×C4C2×SD1611yescharacteristic
152Q16—42no—
153Q16—42no—
154M16—84no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—41no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—44no—
164C4⋊C4—22no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—44no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—82no—
172C4⋊C4—81no—
173C4⋊C4—81no—
174C22⋊C4—44no—
175C22⋊C4—82no—
176C42—21no—
177C42—21no—
178C42—21no—
179C42—21no—
180C42—21no—
181C42—21no—
182C42—82no—
183C42—81no—
184C42—81no—
185C42#9911yescharacteristic
186C42#9911yescharacteristic
187C42#10411yescharacteristic
188C42#10411yescharacteristic
189C16—82no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—41no—
194C23—41no—
195C23C2×#10211yescharacteristic
196Q8—42no—
197Q8—42no—
198Q8—81no—
199Q8—81no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4#32711yescharacteristic
232C2×C4#32711yescharacteristic
233C2×C4#32711yescharacteristic
234C2×C4#32711yescharacteristic
235C2×C4C2×#9911yescharacteristic
236C2×C4C2×#10411yescharacteristic
237C8—41no—
238C8—41no—
239C8—41no—
240C8—41no—
241C8—84no—
242C22—21no—
243C22—21no—
244C22—42no—
245C22—41no—
246C22—41no—
247C22—41no—
248C22—81no—
249C22C2×#32711yescharacteristic
250C22#214311yescharacteristic
251C22#214811yescharacteristic
252C4—21no—
253C4—21no—
254C4—21no—
255C4—21no—
256C4—41no—
257C4—41no—
258C4—41no—
259C4—82no—
260C4—82no—
261C4—161no—
262C4—161no—
263C4#134012yes—
264C4#212011yescharacteristic
265C4#213211yescharacteristic
266C2—21no—
267C2—21no—
268C2—81no—
269C2C2×#134011yescharacteristic
270C2#1169711yescharacteristic
271C2#1171011yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4⋊C8C23
prime radicalC24#102
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#431 ◁ G
LECSG ▷ C2×C4⋊C8 ▷ C22×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7083C2semidirect
5. #7082C2semidirect

Maximal subgroups

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

subgroupindex
#236992
#114602
#79872
#70832
#70822