Finite Group Databasebeta

← #499679

Subgroups — #499679

Order 512.

534 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 154 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#21725C211yesmaximal, characteristic
2#21698C211yesmaximal, characteristic
3#13539C212yesmaximal
4#13483C212yesmaximal
5#13409C212yesmaximal
6#13364C212yesmaximal
7#12638C212yesmaximal
8C4×#358C212yesmaximal
9#12592C211yesmaximal, characteristic
10C2×#500C2211yescharacteristic
11C2×#494C2211yescharacteristic
12C2×#441—21no—
13C2×C4×Q16C2212yes—
14#2331—21no—
15#2327—21no—
16C2×#358C2212yes—
17C2×#358C2212yes—
18C2×#358C2212yes—
19C2×#358C2212yes—
20C2×#346C2211yescharacteristic
21C2×C4×C16C2211yescharacteristic
22#1993—21no—
23#1971—21no—
24#1969C2212yes—
25#1960—21no—
26#1943—21no—
27#1942C2212yes—
28#1846C2211yescharacteristic
29#1846C2211yescharacteristic
30#1776—21no—
31C4×#107C2212yes—
32#1726C2211yescharacteristic
33#1380C2212yes—
34#1380C2212yes—
35#1380C2212yes—
36#1380C2212yes—
37#1374C2212yes—
38#1374C2212yes—
39C22×Q16—21no—
40C22×Q16C2312yes—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4×Q8—22no—
44C22×C16C2312yes—
45C22×C16C2312yes—
46#500—24no—
47#494—24no—
48#441—44no—
49C4×Q16—24no—
50C4×Q16—24no—
51C2×#107—21no—
52C2×#107C2312yes—
53C2×#107C2312yes—
54C2×#107C2312yes—
55C2×#106—21no—
56C2×#106—21no—
57C2×#103—22no—
58C2×#103—22no—
59C2×#97D411yescharacteristic
60C2×#97D411yescharacteristic
61C2×#97D411yescharacteristic
62C2×C4×C8C2311yescharacteristic
63C2×C4×C8C2311yescharacteristic
64C2×C4×C8C2311yescharacteristic
65#391—22no—
66#391—42no—
67#389—22no—
68#387—42no—
69#387—42no—
70#387—42no—
71#382—21no—
72#382—21no—
73C4×C4⋊C4—22no—
74#376D411yescharacteristic
75#358—24no—
76#358—24no—
77#358—24no—
78#358—24no—
79#346—22no—
80C4×C16—22no—
81#340—22no—
82#340—22no—
83C22×Q8—22no—
84C22×Q8—41no—
85C2×Q16—24no—
86C2×Q16—22no—
87C2×Q16—22no—
88C2×Q16—22no—
89C2×Q16—22no—
90C2×Q16—22no—
91C2×Q16—42no—
92C2×Q16—42no—
93C2×Q16#4114yes—
94C22×C8C2×D411yescharacteristic
95C22×C8C2×D411yescharacteristic
96C22×C8C2×D411yescharacteristic
97C22×C8C2411yescharacteristic
98C4⋊Q8—24no—
99C4⋊Q8—24no—
100C4⋊Q8—42no—
101C4⋊Q8—42no—
102C4×Q8—44no—
103C4×Q8—44no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—42no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4—42no—
111C2×C42—42no—
112C2×C42—42no—
113C2×C42C2×D411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C2×C42C2×D411yescharacteristic
116C2×C16—22no—
117C2×C16—22no—
118C2×C16—22no—
119C2×C16—22no—
120C2×C16—22no—
121C2×C16—22no—
122#107—24no—
123#107—22no—
124#107—22no—
125#107—22no—
126#107—22no—
127#107—22no—
128#107#4114yes—
129#106—42no—
130#106—42no—
131#103—44no—
132#103—44no—
133#97—41no—
134#97—41no—
135#97SD1614yes—
136C4×C8—22no—
137C4×C8—22no—
138C4×C8#4112yes—
139C4×C8#4112yes—
140#95—42no—
141#95—41no—
142#95—42no—
143#95—41no—
144#95—42no—
145#95—42no—
146#95—42no—
147#95D812yes—
148#95D812yes—
149C2×Q8—24no—
150C2×Q8—44no—
151C2×Q8—42no—
152C2×Q8—42no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—42no—
156C2×Q8—162no—
157C22×C4—42no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4C2×D811yescharacteristic
162C22×C4C2×D811yescharacteristic
163C22×C4C2×D812yes—
164C22×C4C2×D812yes—
165C22×C4C22×D411yescharacteristic
166Q16—24no—
167Q16—24no—
168Q16—48no—
169Q16—42no—
170Q16—42no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8#12412yes—
178C2×C8C2×SD1612yes—
179C2×C8C2×SD1612yes—
180C2×C8C2×#4112yes—
181C2×C8C2×#4112yes—
182C2×C8C2×#4111yescharacteristic
183C2×C8C2×#4111yescharacteristic
184C4⋊C4—24no—
185C4⋊C4—44no—
186C4⋊C4—48no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—84no—
193C4⋊C4—162no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—84no—
199C42—162no—
200C42#12412yes—
201C42C2×SD1612yes—
202C16—42no—
203C16—42no—
204C23C22×D811yescharacteristic
205Q8—44no—
206Q8—44no—
207Q8—82no—
208Q8—82no—
209Q8—164no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
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×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4#48614yes—
232C2×C4#52511yescharacteristic
233C2×C4#52511yescharacteristic
234C2×C4C2×#12411yescharacteristic
235C2×C4C22×SD1611yescharacteristic
236C2×C4#57511yescharacteristic
237C2×C4#57511yescharacteristic
238C8—22no—
239C8—22no—
240C8—22no—
241C8#52912yes—
242C8#52912yes—
243C22#310011yescharacteristic
244C22#310011yescharacteristic
245C22C2×#48611yescharacteristic
246C22#341612yes—
247C22#341712yes—
248C4—24no—
249C4—42no—
250C4—41no—
251C4—41no—
252C4—84no—
253C4—84no—
254C4—162no—
255C4—162no—
256C4#239811yescharacteristic
257C4#310111yescharacteristic
258C4#310311yescharacteristic
259C4#313511yescharacteristic
260C2#2381511yescharacteristic
261C2#3368112yes—
262C2#3368311yescharacteristic
263C2#3368711yescharacteristic
264C2#3370312yes—
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#217252
#216982
#135392
#134832
#134092
#133642
#126382
C4×#3582
#125922