Finite Group Databasebeta

← #102348

Subgroups — #102348

Order 512.

482 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 98 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#33172C211yesmaximal, characteristic
2#10822C211yesmaximal, characteristic
3#8145C211yesmaximal, characteristic
4#7685C212yesmaximal
5#7685C212yesmaximal
6C2×#494C411yescharacteristic
7C2×#467—21no—
8C2×#467—21no—
9C22×C4×C8C2211yescharacteristic
10#2857C2212yes—
11#1969—21no—
12#1969—21no—
13#1956C412yes—
14#1845C411yescharacteristic
15C4×C22⋊C8C2211yescharacteristic
16C4×C22⋊C8C2211yescharacteristic
17C2×#341—21no—
18C2×#341—21no—
19#1728C2212yes—
20C2×#322—21no—
21#1273—21no—
22#1271—24no—
23#1271—24no—
24C22×Q16—21no—
25C22×Q16—21no—
26C22×M16—22no—
27C23×C8D411yescharacteristic
28C23×C8D411yescharacteristic
29C2×C4⋊Q8C2×C412yes—
30C2×C22⋊Q8—22no—
31C22×C42C2311yescharacteristic
32#494—42no—
33#467—44no—
34#467—44no—
35C2×#107—21no—
36C2×#107—21no—
37C2×#103—22no—
38C2×#103—22no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41C2×C4×C8—21no—
42C2×C4×C8—21no—
43C2×C4×C8—42no—
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8D411yescharacteristic
47C2×C4×C8D411yescharacteristic
48#391—22no—
49#389—22no—
50#389—22no—
51#384C2×C412yes—
52#380—22no—
53#380—22no—
54#341—24no—
55#341—42no—
56#336—22no—
57#336—22no—
58#336D412yes—
59#336D412yes—
60#322—42no—
61C22×Q8—22no—
62C23×C4C2×D411yescharacteristic
63C23×C4#4111yescharacteristic
64C23×C4#4111yescharacteristic
65C2×Q16—42no—
66C2×Q16—42no—
67C2×Q16—42no—
68C2×Q16—42no—
69C2×Q16—42no—
70C2×Q16—42no—
71C2×M16—24no—
72C2×M16—42no—
73C2×M16—42no—
74C2×M16—42no—
75C2×M16—42no—
76C2×M16—42no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—42no—
86C22×C8C22⋊C411yescharacteristic
87C22×C8C22⋊C411yescharacteristic
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90C4⋊Q8—42no—
91C4⋊Q8—42no—
92C22⋊Q8—44no—
93C22⋊Q8—44no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—22no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—42no—
100C2×C22⋊C4—42no—
101C2×C42—41no—
102C2×C42—41no—
103C2×C42C22×C411yescharacteristic
104C2×C42C2×D411yescharacteristic
105C2×C42C2×D411yescharacteristic
106C2×C42C2×D411yescharacteristic
107#107—42no—
108#107—42no—
109#103—44no—
110#103—44no—
111C22⋊C8—24no—
112C22⋊C8—42no—
113C4×C8—22no—
114C4×C8—22no—
115C4×C8—41no—
116C4×C8—41no—
117C4×C8—41no—
118C4×C8—41no—
119C4×C8—41no—
120C4×C8—41no—
121#95—42no—
122#95—42no—
123C24#12411yescharacteristic
124C2×Q8—42no—
125C2×Q8—42no—
126C2×Q8—42no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8—42no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—42no—
141C22×C4—42no—
142C22×C4C2×C22⋊C411yescharacteristic
143C22×C4C4×D411yescharacteristic
144C22×C4C4×D411yescharacteristic
145C22×C4#12111yescharacteristic
146C22×C4#12111yescharacteristic
147C22×C4#12711yescharacteristic
148C22×C4AGL(1,ℤ/8)12yes—
149C22×C4AGL(1,ℤ/8)12yes—
150Q16—84no—
151Q16—84no—
152M16—44no—
153M16—44no—
154M16—82no—
155M16—82no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—24no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—44no—
168C2×C8—44no—
169C2×C8—42no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—44no—
174C2×C8—42no—
175C2×C8—82no—
176C2×C8—82no—
177C2×C8#9912yes—
178C2×C8#10412yes—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—44no—
182C4⋊C4—44no—
183C4⋊C4—82no—
184C4⋊C4—82no—
185C4⋊C4—84no—
186C22⋊C4—84no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—41no—
192C42—41no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—21no—
197C23—21no—
198C23—21no—
199C23#39011yescharacteristic
200C23#49012yes—
201Q8—82no—
202Q8—82no—
203Q8—82no—
204Q8—82no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—44no—
220C2×C4—44no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4C2×#9911yescharacteristic
228C2×C4#41011yescharacteristic
229C2×C4#41811yescharacteristic
230C2×C4#41811yescharacteristic
231C2×C4C2×#10411yescharacteristic
232C2×C4#42111yescharacteristic
233C2×C4#44312yes—
234C2×C4#44412yes—
235C2×C4#47312yes—
236C2×C4#47312yes—
237C2×C4#49612yes—
238C8—22no—
239C8—22no—
240C8—44no—
241C8—41no—
242C8—41no—
243C8—82no—
244C8—84no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—41no—
251C22—41no—
252C22#196111yescharacteristic
253C22#196411yescharacteristic
254C22#197411yescharacteristic
255C22#197511yescharacteristic
256C22#197911yescharacteristic
257C22#197911yescharacteristic
258C22#198011yescharacteristic
259C4—22no—
260C4—22no—
261C4—22no—
262C4—22no—
263C4—22no—
264C4—162no—
265C4—162no—
266C4#150511yescharacteristic
267C4#150711yescharacteristic
268C4#151011yescharacteristic
269C4#151411yescharacteristic
270C2—42no—
271C2#968411yescharacteristic
272C2#972511yescharacteristic
273C2#972711yescharacteristic
274C2#977911yescharacteristic
275C2#981411yescharacteristic
276C2#981711yescharacteristic
277C2#986011yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#331722
#108222
#81452
#76852