Finite Group Databasebeta

← #102352

Subgroups — #102352

Order 512.

460 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 118 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#33209C211yesmaximal, characteristic
2#10864C211yesmaximal, characteristic
3#8144C211yesmaximal, characteristic
4#7688C212yesmaximal
5#7687C212yesmaximal
6C2×#500C411yescharacteristic
7C22×#108—21no—
8C22×C4×C8C2211yescharacteristic
9#2857C2212yes—
10#1990C412yes—
11#1942—21no—
12#1942—21no—
13#1845C411yescharacteristic
14#1808—21no—
15#1808—21no—
16C4×C22⋊C8C2211yescharacteristic
17C4×C22⋊C8C2211yescharacteristic
18C2×#341—21no—
19#1728C2212yes—
20C2×#322—21no—
21#1273—21no—
22#1272—24no—
23#1271—24no—
24C22×M16—22no—
25C23×C8D411yescharacteristic
26C23×C8D411yescharacteristic
27C2×C4⋊Q8C2×C412yes—
28C2×C22⋊Q8—22no—
29C22×C42C2311yescharacteristic
30#500—42no—
31C2×#108—24no—
32C2×#108—42no—
33C2×#108—42no—
34C2×#107—21no—
35C2×#107—21no—
36C2×#107—21no—
37C2×#107—21no—
38C2×C22⋊C8—21no—
39C2×C22⋊C8—21no—
40C2×C4×C8—21no—
41C2×C4×C8—21no—
42C2×C4×C8—42no—
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8C2×C411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8D411yescharacteristic
47#391—22no—
48#389—22no—
49#389—22no—
50#384C2×C412yes—
51#380—22no—
52#380—22no—
53#341—42no—
54#340—22no—
55#340—22no—
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×M16—24no—
66C2×M16—42no—
67C2×M16—42no—
68C2×M16—42no—
69C2×M16—42no—
70C2×M16—42no—
71C22×C8—22no—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8—22no—
75C22×C8—41no—
76C22×C8—41no—
77C22×C8—42no—
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22⋊C411yescharacteristic
82C22×C8SD1612yes—
83C22×C8SD1612yes—
84C22×C8Q1612yes—
85C22×C8Q1612yes—
86C4⋊Q8—42no—
87C4⋊Q8—42no—
88C22⋊Q8—84no—
89C2×C4⋊C4—22no—
90C2×C4⋊C4—22no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C42—41no—
97C2×C42—41no—
98C2×C42C22×C411yescharacteristic
99C2×C42C2×D411yescharacteristic
100C2×C42C2×D411yescharacteristic
101C2×C42C2×D411yescharacteristic
102#108—48no—
103#107—42no—
104#107—42no—
105#107—42no—
106#107—42no—
107C22⋊C8—42no—
108C22⋊C8—42no—
109C4×C8—22no—
110C4×C8—22no—
111C4×C8—41no—
112C4×C8—41no—
113C4×C8—41no—
114C4×C8—41no—
115C4×C8—41no—
116C4×C8—41no—
117#95—42no—
118#95—42no—
119C24#12411yescharacteristic
120C2×Q8—44no—
121C2×Q8—82no—
122C2×Q8—82no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4C2×C22⋊C411yescharacteristic
136C22×C4C4×D411yescharacteristic
137C22×C4C4×D411yescharacteristic
138C22×C4#12111yescharacteristic
139C22×C4#12111yescharacteristic
140C22×C4#12711yescharacteristic
141C22×C4C2×SD1611yescharacteristic
142C22×C4C2×SD1611yescharacteristic
143C22×C4C2×Q1611yescharacteristic
144C22×C4C2×Q1611yescharacteristic
145M16—44no—
146M16—44no—
147M16—82no—
148M16—82no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—24no—
156C2×C8—24no—
157C2×C8—42no—
158C2×C8—42no—
159C2×C8—42no—
160C2×C8—42no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—44no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—82no—
167C2×C8—82no—
168C2×C8#9912yes—
169C2×C8#10312yes—
170C2×C8#10312yes—
171C2×C8#10312yes—
172C2×C8#10312yes—
173C2×C8#10412yes—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—44no—
179C4⋊C4—82no—
180C4⋊C4—82no—
181C4⋊C4—84no—
182C22⋊C4—84no—
183C42—21no—
184C42—21no—
185C42—21no—
186C42—21no—
187C42—41no—
188C42—41no—
189C23—21no—
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23#39011yescharacteristic
196C23#48711yescharacteristic
197C23#48711yescharacteristic
198Q8—164no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
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—21no—
216C2×C4—21no—
217C2×C4—44no—
218C2×C4—44no—
219C2×C4—82no—
220C2×C4—82no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4C2×#9911yescharacteristic
226C2×C4#41011yescharacteristic
227C2×C4C2×#10311yescharacteristic
228C2×C4C2×#10311yescharacteristic
229C2×C4C2×#10411yescharacteristic
230C2×C4#42111yescharacteristic
231C2×C4#44412yes—
232C2×C4#44412yes—
233C2×C4#47212yes—
234C2×C4#47312yes—
235C2×C4#49211yescharacteristic
236C2×C4#49411yescharacteristic
237C8—22no—
238C8—22no—
239C8—22no—
240C8—82no—
241C8—82no—
242C8—84no—
243C8#32814yes—
244C22—21no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22—21no—
249C22—21no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—41no—
254C22#195611yescharacteristic
255C22#196411yescharacteristic
256C22#197511yescharacteristic
257C22#197511yescharacteristic
258C22#197811yescharacteristic
259C22#197911yescharacteristic
260C22#198011yescharacteristic
261C4—22no—
262C4—22no—
263C4—22no—
264C4—22no—
265C4—21no—
266C4—21no—
267C4—162no—
268C4—162no—
269C4C2×#32811yescharacteristic
270C4#150311yescharacteristic
271C4#151811yescharacteristic
272C4#152011yescharacteristic
273C2—41no—
274C2—41no—
275C2#967911yescharacteristic
276C2#972611yescharacteristic
277C2#972711yescharacteristic
278C2#977111yescharacteristic
279C2#981711yescharacteristic
280C2#981811yescharacteristic
281C2#986111yescharacteristic
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
#332092
#108642
#81442
#76882
#76872