Finite Group Databasebeta

← #139166

Subgroups — #139166

Order 512.

285 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 82 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#33498C211yesmaximal, characteristic
2#19262C211yesmaximal, characteristic
3#11863C211yesmaximal, characteristic
4#11653C211yesmaximal, characteristic
5#8661C211yesmaximal, characteristic
6#7291C211yesmaximal, characteristic
7#7223C211yesmaximal, characteristic
8#3170C2211yescharacteristic
9#3078—21no—
10C2×C8×Q8C2211yescharacteristic
11#2183—21no—
12#2183—22no—
13Q8×C16—21no—
14#2163—21no—
15#2163C2211yescharacteristic
16#2162—21no—
17#2162C2211yescharacteristic
18#1837C2211yescharacteristic
19#1684—21no—
20#1684—21no—
21#1676C411yescharacteristic
22#1676C411yescharacteristic
23#1670C411yescharacteristic
24#1670C411yescharacteristic
25#1567C2211yescharacteristic
26#1564C2211yescharacteristic
27#1550—21no—
28#1487—21no—
29#1350—21no—
30#1350—21no—
31#1334—22no—
32#1301—21no—
33#533—21no—
34C2×C4×Q8D411yescharacteristic
35#500C2×C411yescharacteristic
36#494C2×C411yescharacteristic
37#487—21no—
38#479—41no—
39#479—41no—
40#477—21no—
41#477—21no—
42#475—21no—
43#475—21no—
44#467—21no—
45#467—21no—
46C8×Q8—21no—
47C8×Q8—21no—
48C8×Q8C2×C411yescharacteristic
49C8×Q8C2×C411yescharacteristic
50C8×Q8D411yescharacteristic
51C8×Q8D411yescharacteristic
52C4×Q16—21no—
53C4×Q16—21no—
54#431—21no—
55#428—41no—
56#423—21no—
57#423—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8D411yescharacteristic
60#419—41no—
61C2×C4×C8—21no—
62C2×C4×C8C2311yescharacteristic
63C4⋊C16—22no—
64C4⋊C16—22no—
65C4⋊C16—21no—
66C4⋊C16—21no—
67C22⋊C16—21no—
68C22⋊C16D411yescharacteristic
69C22⋊C16D411yescharacteristic
70#346—22no—
71#346D412yes—
72C4×C16—22no—
73C4×C16—21no—
74#341—21no—
75#341—21no—
76#335—21no—
77#335—21no—
78#335C2×C411yescharacteristic
79#335C2×C411yescharacteristic
80#334—21no—
81#333—21no—
82#333—21no—
83#332—21no—
84#332—21no—
85Q8⋊C8—41no—
86Q8⋊C8—41no—
87C22×Q8SD1611yescharacteristic
88C2×Q16—21no—
89C2×Q16—21no—
90C2×M16—41no—
91C2×M16—41no—
92C22×C8—21no—
93C22×C8C2×D411yescharacteristic
94C22×C8#4111yescharacteristic
95C4⋊Q8—21no—
96C4⋊Q8—21no—
97#125—41no—
98C22⋊Q8—41no—
99C22⋊Q8—41no—
100C4×Q8—21no—
101C4×Q8—21no—
102C4×Q8—41no—
103C4×Q8—41no—
104C4×Q8C22⋊C411yescharacteristic
105C4×Q8C22⋊C411yescharacteristic
106C4×Q8D811yescharacteristic
107C4×Q8D811yescharacteristic
108#117—41no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4SD1611yescharacteristic
111C2×C42—21no—
112C2×C42C2×D411yescharacteristic
113C2×C16—21no—
114C2×C16—21no—
115C2×C16—22no—
116#107—21no—
117#107—21no—
118#107—41no—
119#107—41no—
120#106—41no—
121#106—41no—
122C4⋊C8—21no—
123C4⋊C8—21no—
124C4⋊C8—21no—
125C4⋊C8—21no—
126C4⋊C8—21no—
127C4⋊C8—21no—
128C4⋊C8—21no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8C22⋊C411yescharacteristic
134C4⋊C8C22⋊C411yescharacteristic
135#103—41no—
136#103—41no—
137#103—41no—
138#103—41no—
139#103—41no—
140#103—41no—
141C22⋊C8—41no—
142C4×C8—21no—
143C4×C8—21no—
144C4×C8—21no—
145C4×C8—21no—
146C4×C8—41no—
147C4×C8C22×C411yescharacteristic
148C4×C8C2×D411yescharacteristic
149C4×C8C2×D411yescharacteristic
150C4×C8#4111yescharacteristic
151C2×Q8—21no—
152C2×Q8—22no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8—41no—
156C2×Q8#10211yescharacteristic
157C2×Q8#10211yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4#12111yescharacteristic
161C22×C4C2×SD1611yescharacteristic
162C22×C4#13511yescharacteristic
163Q16—42no—
164Q16—81no—
165M16—82no—
166M16—81no—
167M16—81no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—81no—
175C2×C8—81no—
176C2×C8—81no—
177C2×C8C4×D411yescharacteristic
178C2×C8C4×D411yescharacteristic
179C2×C8#12011yescharacteristic
180C2×C8#12311yescharacteristic
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—81no—
191C4⋊C4—81no—
192C4⋊C4#10211yescharacteristic
193C4⋊C4#10211yescharacteristic
194C22⋊C4—81no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—81no—
200C42C2×C22⋊C411yescharacteristic
201C42#12111yescharacteristic
202C42C2×D811yescharacteristic
203C42#13511yescharacteristic
204C16—41no—
205C16—41no—
206C16—42no—
207C23#46511yescharacteristic
208Q8—21no—
209Q8—21no—
210Q8—21no—
211Q8—42no—
212Q8—81no—
213Q8—81no—
214Q8—161no—
215Q8#36212yes—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4#38611yescharacteristic
230C2×C4C2×#10211yescharacteristic
231C2×C4#41611yescharacteristic
232C2×C4#46511yescharacteristic
233C2×C4#46611yescharacteristic
234C2×C4#46611yescharacteristic
235C8—21no—
236C8—21no—
237C8—21no—
238C8—21no—
239C8—41no—
240C8—41no—
241C8—81no—
242C8—81no—
243C8—161no—
244C8—161no—
245C22—21no—
246C22—21no—
247C22—21no—
248C22#189311yescharacteristic
249C4—21no—
250C4—22no—
251C4—21no—
252C4—21no—
253C4—21no—
254C4—41no—
255C4—41no—
256C4—41no—
257C4—161no—
258C4—161no—
259C4#188111yescharacteristic
260C4#190511yescharacteristic
261C4C2×#36211yescharacteristic
262C4#214811yescharacteristic
263C2—41no—
264C2#930611yescharacteristic
265C2#1306611yescharacteristic
266C2#1307111yescharacteristic
subgroupG/H
center, socleC22#1893
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#465
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#334982
#192622
#118632
#116532
#86612
#72912
#72232