Finite Group Databasebeta

← #138023

Subgroups — #138023

Order 512.

458 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C2×#2163C211yesmaximal, characteristic
2#33130C211yesmaximal, characteristic
3C2×#1348C212yesmaximal
4#10820C211yesmaximal, characteristic
5#7258C212yesmaximal
6C2×#467C2211yescharacteristic
7C22×C4×C8C2211yescharacteristic
8#2855—21no—
9#2163—24no—
10C2×C4⋊C16—21no—
11C2×C22⋊C16C2212yes—
12C2×#346—21no—
13#1969C411yescharacteristic, complemented
14#1969C411yescharacteristic, complemented
15#1942C411yescharacteristic, complemented
16#1942C411yescharacteristic, complemented
17#1933—21no—
18#1933—21no—
19#1892—21no—
20#1808C2211yescharacteristic
21C2×#340—21no—
22C2×#340C2212yes—
23C4⋊#95—21no—
24#1348—24no—
25C22×Q16C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C23×C8C2311yescharacteristic
28C2×C22⋊Q8—21no—
29C22×C4⋊C4—22no—
30C22×C42D411yescharacteristic
31C22×C16—22no—
32#467—22no—
33#467—22no—
34C2×#107C2×C411yescharacteristic
35C2×#107C2×C411yescharacteristic
36C2×#107C2×C411yescharacteristic
37C2×#103—21no—
38C2×#103—21no—
39C2×#103—21no—
40C2×C4×C8—22no—
41C2×C4×C8—22no—
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46#391—21no—
47#391—21no—
48#389—21no—
49#389—21no—
50#387—41no—
51#387—41no—
52#385—41no—
53#381—41no—
54#380—21no—
55C2×#95—41no—
56C2×#95—41no—
57C4⋊C16—24no—
58C22⋊C16—22no—
59C22⋊C16D414yes—
60#346—24no—
61#340—24no—
62#340—21no—
63#340—21no—
64#340—21no—
65#340—22no—
66#340—22no—
67C22×Q8—21no—
68C23×C4—42no—
69C23×C4D811yescharacteristic
70C23×C4D811yescharacteristic
71C23×C4C2×D411yescharacteristic
72C2×Q16—22no—
73C2×Q16—41no—
74C2×Q16—41no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—41no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22×C411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C22×C8C2×D412yes—
85C22×C8#4112yes—
86C22⋊Q8—42no—
87C22⋊Q8—42no—
88C2×C4⋊C4—24no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4—42no—
96C2×C4⋊C4—42no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C22⋊C4—41no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42C22⋊C411yescharacteristic
105C2×C42C22⋊C411yescharacteristic
106C2×C42SD1611yescharacteristic
107C2×C42SD1611yescharacteristic
108C2×C16—24no—
109C2×C16—42no—
110C2×C16—42no—
111#107—22no—
112#107—41no—
113#107—41no—
114#103—42no—
115#103—42no—
116#103—41no—
117#103—41no—
118C4×C8—22no—
119C4×C8—22no—
120C4×C8—21no—
121C4×C8—21no—
122C4×C8—21no—
123C4×C8—21no—
124#95—41no—
125#95—41no—
126#95—41no—
127#95—41no—
128#95—41no—
129#95—41no—
130#95—41no—
131#95—82no—
132#95—82no—
133C24C2×D811yescharacteristic
134C2×Q8—41no—
135C2×Q8—41no—
136C2×Q8—41no—
137C2×Q8—41no—
138C2×Q8—41no—
139C2×Q8—41no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—41no—
151C22×C4—44no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—82no—
157C22×C4—82no—
158C22×C4—82no—
159C22×C4—82no—
160C22×C4—82no—
161C22×C4—82no—
162C22×C4#10211yescharacteristic
163C22×C4#10211yescharacteristic
164C22×C4#10211yescharacteristic
165C22×C4#10211yescharacteristic
166C22×C4C2×C22⋊C411yescharacteristic
167C22×C4#12112yes—
168C22×C4C2×SD1611yescharacteristic
169C22×C4#13512yes—
170Q16—82no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—44no—
183C2×C8#10412yes—
184C2×C8C4×D412yes—
185C2×C8#12011yescharacteristic
186C2×C8#12311yescharacteristic
187C4⋊C4—44no—
188C4⋊C4—44no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—82no—
196C4⋊C4—82no—
197C4⋊C4—82no—
198C4⋊C4—82no—
199C4⋊C4—82no—
200C22⋊C4—82no—
201C22⋊C4—81no—
202C22⋊C4—81no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—22no—
208C42—22no—
209C16—44no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23C2×#10211yescharacteristic
215C23#46612yes—
216C23#50911yescharacteristic
217C23#51011yescharacteristic
218C23#51012yes—
219Q8—81no—
220Q8—81no—
221Q8—81no—
222Q8—81no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—44no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—84no—
244C2×C4—84no—
245C2×C4—84no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—81no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—162no—
259C2×C4—162no—
260C2×C4—162no—
261C2×C4—162no—
262C2×C4#36012yes—
263C2×C4#38611yescharacteristic
264C2×C4#41011yescharacteristic
265C2×C4#41011yescharacteristic
266C2×C4#41611yescharacteristic
267C2×C4C2×#10411yescharacteristic
268C2×C4#42111yescharacteristic
269C2×C4#46512yes—
270C8—22no—
271C8—21no—
272C8—22no—
273C8—21no—
274C8—41no—
275C22—22no—
276C22—22no—
277C22—21no—
278C22—21no—
279C22—22no—
280C22—22no—
281C22—21no—
282C22—21no—
283C22#150211yescharacteristic
284C22#151711yescharacteristic
285C22#189311yescharacteristic
286C22#213911yescharacteristic
287C22#214111yescharacteristic
288C22C2×#36011yescharacteristic
289C22#214511yescharacteristic
290C22#221912yes—
291C22#222012yes—
292C4—22no—
293C4—22no—
294C4—41no—
295C4—41no—
296C4—41no—
297C4—164no—
298C4—161no—
299C4—161no—
300C4—161no—
301C4—161no—
302C4#150011yescharacteristic
303C4#152111yescharacteristic
304C4#187111yescharacteristic
305C4#189911yescharacteristic
306C2—22no—
307C2—22no—
308C2#911411yescharacteristic
309C2#1174511yescharacteristic
310C2#1174611yescharacteristic
311C2#1185911yescharacteristic
312C2#1186011yescharacteristic
313C2#1306211yescharacteristic
314C2#1306511yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #1969C4semidirect
14. #1969C4semidirect
15. #1942C4semidirect
16. #1942C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21632
#331302
C2×#13482
#108202
#72582