Finite Group Databasebeta

← #326534

Subgroups — #326534

Order 512.

408 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 138 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#33264C211yesmaximal, characteristic
2#21734C211yesmaximal, characteristic
3#10369C212yesmaximal
4#10367C212yesmaximal
5#9759C212yesmaximal
6#8794C212yesmaximal
7#8677C212yesmaximal
8#8675C212yesmaximal
9#8269C211yesmaximal, characteristic
10#3160C2211yescharacteristic
11#2960C2212yes—
12#2864C2212yes—
13#1994C2212yes—
14#1946—21no—
15#1946—21no—
16#1944—21no—
17#1944—21no—
18#1943—21no—
19#1943—21no—
20#1848C2211yescharacteristic
21#1846C2211yescharacteristic
22#1846C2211yescharacteristic
23#1835C2212yes—
24#1773—21no—
25#1773—21no—
26C4×#97C2211yescharacteristic
27#1564C2212yes—
28#1518C2212yes—
29#1518C2212yes—
30#1518C2212yes—
31#1518C2212yes—
32#1452C2212yes—
33#1286C2212yes—
34#1286C2212yes—
35#1286C2212yes—
36#1286C2212yes—
37#1276C2211yescharacteristic
38#1276C2211yescharacteristic
39#533C2312yes—
40#500Q812yes—
41#499—21no—
42#499—21no—
43#498Q811yescharacteristic
44#498Q811yescharacteristic
45#446—22no—
46#446—22no—
47C8×Q8—22no—
48C8×Q8—22no—
49#431C2312yes—
50#431C2312yes—
51C2×#108—21no—
52C2×#108—21no—
53#427—21no—
54#427—21no—
55C2×#107—21no—
56C2×#107—21no—
57C2×#106—21no—
58C2×#106—21no—
59#424—22no—
60#423C2312yes—
61#405—22no—
62C2×#97D411yescharacteristic
63C2×#97D411yescharacteristic
64C2×#97D411yescharacteristic
65C2×#97D411yescharacteristic
66C2×C4×C8C2311yescharacteristic
67C2×C4×C8C2311yescharacteristic
68#389—22no—
69#389—22no—
70#387—42no—
71#387—42no—
72#382C2312yes—
73#382C2312yes—
74C43C2311yescharacteristic
75#340—22no—
76#340—22no—
77#339—22no—
78#339—22no—
79#335—22no—
80#334—22no—
81#328—42no—
82#328—42no—
83#328—42no—
84#328—42no—
85#322—22no—
86#322—22no—
87C2×M16—22no—
88C22×C8C2×D411yescharacteristic
89C22×C8C2×D411yescharacteristic
90C22×C8C2×D411yescharacteristic
91C22×C8C2×D411yescharacteristic
92C4⋊Q8C2×Q812yes—
93C4⋊Q8C2×Q812yes—
94#125—22no—
95C22⋊Q8—22no—
96C22⋊Q8—22no—
97C4×Q8—22no—
98C4×Q8—22no—
99#117—22no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—42no—
103C2×C4⋊C4—42no—
104C2×C42—22no—
105C2×C42C2×D411yescharacteristic
106C2×C42C2×D411yescharacteristic
107C2×C42C2411yescharacteristic
108#108—42no—
109#108—42no—
110#107—22no—
111#107—22no—
112#107—42no—
113#107—42no—
114#106—21no—
115#106—21no—
116#106—21no—
117#106—21no—
118#106—42no—
119#106—42no—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—42no—
125C4⋊C8—42no—
126C22⋊C8—22no—
127C22⋊C8—22no—
128#97—22no—
129#97—22no—
130#97—41no—
131#97—41no—
132#97—42no—
133C4×C8—22no—
134C4×C8—22no—
135C4×C8—42no—
136C4×C8C2×Q811yescharacteristic
137C4×C8C2×Q811yescharacteristic
138C4×C8#4111yescharacteristic
139C4×C8#4111yescharacteristic
140C4×C8#4112yes—
141C4×C8#4112yes—
142#95—22no—
143#95—22no—
144#95—22no—
145#95—22no—
146#95—42no—
147#95—42no—
148C2×Q8—22no—
149C2×Q8—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—42no—
153C22×C4—42no—
154C22×C4C22×D411yescharacteristic
155C22×C4H(5,𝔽2)12yes—
156M16—42no—
157M16—42no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8C22⋊Q811yescharacteristic
168C2×C8C22⋊Q811yescharacteristic
169C2×C8C22⋊Q811yescharacteristic
170C2×C8C22⋊Q811yescharacteristic
171C2×C8#12411yescharacteristic
172C2×C8#12411yescharacteristic
173C2×C8#12412yes—
174C2×C8#13712yes—
175C2×C8#13712yes—
176C4⋊C4—22no—
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—84no—
187C4⋊C4—84no—
188C22⋊C4—42no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—42no—
194C42—82no—
195C42C22×Q811yescharacteristic
196C42C2×#4111yescharacteristic
197C42C2×#4112yes—
198C23#53511yescharacteristic
199Q8—82no—
200Q8—82no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—82no—
214C2×C4C2×C22⋊Q811yescharacteristic
215C2×C4C2×#12411yescharacteristic
216C2×C4#55412yes—
217C2×C4#55512yes—
218C2×C4C2×#13711yescharacteristic
219C2×C4C2×#13711yescharacteristic
220C2×C4#57511yescharacteristic
221C2×C4#57511yescharacteristic
222C8—21no—
223C8—21no—
224C8—21no—
225C8—21no—
226C8—21no—
227C8—21no—
228C8—82no—
229C8—82no—
230C8—82no—
231C8#49114yes—
232C22—22no—
233C22#256311yescharacteristic
234C22#321611yescharacteristic
235C22#321611yescharacteristic
236C4—22no—
237C4—22no—
238C4—81no—
239C4—81no—
240C4—82no—
241C4—82no—
242C4—162no—
243C4—162no—
244C4#307811yescharacteristic
245C4#307811yescharacteristic
246C4C2×#49111yescharacteristic
247C4#314211yescharacteristic
248C2—21no—
249C2—21no—
250C2#2915011yescharacteristic
251C2#2915512yes—
subgroupG/H
center, socleC22#2563
commutator, Frattini, cosocleC2×C42C24
prime radicalC23#535
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#332642
#217342
#103692
#103672
#97592
#87942
#86772
#86752
#82692