Finite Group Databasebeta

← #102159

Subgroups — #102159

Order 512.

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

normal subgroups only

This group has 502 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#10820C211yesmaximal, characteristic
3#9884C212yesmaximal
4#9679C212yesmaximal
5#8145C211yesmaximal, characteristic
6C2×#500C411yescharacteristic, complemented
7C22×C4×C8C2211yescharacteristic
8#2857C2212yes—
9#1990C412yescomplemented
10#1942—21no—
11#1942—21no—
12#1845C411yescharacteristic, complemented
13#1808—21no—
14#1808—21no—
15C4×C22⋊C8C2211yescharacteristic
16C4×C22⋊C8C2211yescharacteristic
17C2×#340—21no—
18C2×#340—21no—
19C2×#328—24no—
20C2×#322—21no—
21C4⋊#95C2212yes—
22#1294—24no—
23#1273—21no—
24C23×C8D411yescharacteristic
25C23×C8D411yescharacteristic
26C2×C4⋊Q8C2×C412yes—
27C2×C22⋊Q8—22no—
28C22×C4⋊C4—22no—
29C22×C42C2311yescharacteristic
30#500—42no—
31C2×#107—21no—
32C2×#107—21no—
33C2×#107—21no—
34C2×#107—21no—
35C2×C22⋊C8D412yes—
36C2×C22⋊C8D412yes—
37C2×C4×C8—21no—
38C2×C4×C8—21no—
39C2×C4×C8—42no—
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44#391—22no—
45#389—22no—
46#389—22no—
47#384C2×C412yes—
48#380—22no—
49#380—22no—
50C2×#95—22no—
51C2×#95—22no—
52#340—24no—
53#340—22no—
54#340—22no—
55#340—42no—
56#336—21no—
57#336—21no—
58#328—48no—
59#322—42no—
60C22×Q8—22no—
61C23×C4—42no—
62C23×C4C2×D411yescharacteristic
63C23×C4C2×D411yescharacteristic
64C23×C4C2×D411yescharacteristic
65C22×C8—22no—
66C22×C8—22no—
67C22×C8—22no—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—22no—
71C22×C8—41no—
72C22×C8—41no—
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22⋊C411yescharacteristic
75C22×C8C22⋊C411yescharacteristic
76C22×C8C22⋊C411yescharacteristic
77C4⋊Q8—42no—
78C4⋊Q8—42no—
79C22⋊Q8—44no—
80C22⋊Q8—44no—
81C2×C4⋊C4—24no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4—22no—
84C2×C4⋊C4—22no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—42no—
88C2×C4⋊C4—42no—
89C2×C4⋊C4—42no—
90C2×C4⋊C4—42no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C42—41no—
95C2×C42—41no—
96C2×C42C22×C411yescharacteristic
97C2×C42C2×D411yescharacteristic
98C2×C42#4111yescharacteristic
99C2×C42#4111yescharacteristic
100#107—42no—
101#107—42no—
102#107—42no—
103#107—42no—
104C22⋊C8—22no—
105C22⋊C8—24no—
106C22⋊C8D814yes—
107C4×C8—22no—
108C4×C8—22no—
109C4×C8—41no—
110C4×C8—41no—
111C4×C8—41no—
112C4×C8—41no—
113C4×C8—41no—
114C4×C8—41no—
115#95—42no—
116#95—42no—
117#95—44no—
118#95—44no—
119C24#12711yescharacteristic
120C2×Q8—44no—
121C2×Q8—82no—
122C2×Q8—82no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—41no—
128C22×C4—41no—
129C22×C4—41no—
130C22×C4—41no—
131C22×C4—44no—
132C22×C4—42no—
133C22×C4—42no—
134C22×C4—82no—
135C22×C4—82no—
136C22×C4—82no—
137C22×C4—82no—
138C22×C4—82no—
139C22×C4—82no—
140C22×C4C2×C22⋊C411yescharacteristic
141C22×C4C4×D411yescharacteristic
142C22×C4C4×D411yescharacteristic
143C22×C4#12111yescharacteristic
144C22×C4#12111yescharacteristic
145C22×C4#12411yescharacteristic
146C22×C4C2×D812yes—
147C22×C4#13512yes—
148C22×C4#13712yes—
149C22×C4#13712yes—
150C22×C4#13714yes—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—21no—
161C2×C8—24no—
162C2×C8—44no—
163C2×C8—44no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—42no—
169C2×C8#9912yes—
170C2×C8#10412yes—
171C4⋊C4—44no—
172C4⋊C4—44no—
173C4⋊C4—42no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—44no—
178C4⋊C4—82no—
179C4⋊C4—82no—
180C4⋊C4—82no—
181C4⋊C4—82no—
182C4⋊C4—84no—
183C22⋊C4—82no—
184C22⋊C4—82no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—41no—
190C42—41no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—21no—
195C23#39011yescharacteristic
196C23#49311yescharacteristic
197C23#49511yescharacteristic
198C23#49712yes—
199C23#49712yes—
200Q8—164no—
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—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—44no—
218C2×C4—84no—
219C2×C4—84no—
220C2×C4—84no—
221C2×C4—82no—
222C2×C4—82no—
223C2×C4—82no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4—82no—
227C2×C4—162no—
228C2×C4—162no—
229C2×C4—162no—
230C2×C4—162no—
231C2×C4C2×#9911yescharacteristic
232C2×C4#41011yescharacteristic
233C2×C4#41911yescharacteristic
234C2×C4#41911yescharacteristic
235C2×C4C2×#10411yescharacteristic
236C2×C4#42111yescharacteristic
237C2×C4C4×D812yes—
238C2×C4#44112yes—
239C2×C4#46612yes—
240C2×C4#47012yes—
241C2×C4#49112yes—
242C8—22no—
243C8—22no—
244C8—44no—
245C8—41no—
246C8—41no—
247C8—82no—
248C22—22no—
249C22—21no—
250C22—21no—
251C22—22no—
252C22—22no—
253C22—21no—
254C22—21no—
255C22—22no—
256C22#171812yes—
257C22#172212yes—
258C22#196011yescharacteristic
259C22#196411yescharacteristic
260C22#197011yescharacteristic
261C22#197111yescharacteristic
262C22#197611yescharacteristic
263C22#197611yescharacteristic
264C22#198011yescharacteristic
265C4—22no—
266C4—22no—
267C4—22no—
268C4—22no—
269C4—22no—
270C4—164no—
271C4—162no—
272C4—162no—
273C4#150611yescharacteristic
274C4#150811yescharacteristic
275C4#151111yescharacteristic
276C4#151511yescharacteristic
277C2—22no—
278C2—22no—
279C2#968311yescharacteristic
280C2#971911yescharacteristic
281C2#972011yescharacteristic
282C2#975911yescharacteristic
283C2#978311yescharacteristic
284C2#978411yescharacteristic
285C2#983111yescharacteristic
subgroupG/H
center, socleC23#390
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. C2×#500C4semidirect
9. #1990C4semidirect
12. #1845C4semidirect

Maximal subgroups

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

subgroupindex
#332092
#108202
#98842
#96792
#81452