Finite Group Databasebeta

← #125065

Subgroups — #125065

Order 512.

383 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 80 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#33135C211yesmaximal, characteristic
2#12846C211yesmaximal, characteristic, complemented
3C2×#1275C211yesmaximal, characteristic, complemented
4#7126C212yesmaximal, complemented
5#7110C212yesmaximal, complemented
6C2×#466C411yescharacteristic
7C2×#431—21no—
8C22×#97C2211yescharacteristic
9#2854—21no—
10#2112—21no—
11#1972C411yescharacteristic
12#1972C2211yescharacteristic
13#1944C411yescharacteristic
14#1944C2211yescharacteristic
15#1936—21no—
16#1936—21no—
17#1893—21no—
18#1807C411yescharacteristic
19#1377—21no—
20#1377C2212yes—
21#1275—21no—
22#1275C2212yes—
23C22×D8C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×#123—41no—
27C2×#121—21no—
28C22×C42D411yescharacteristic
29C2×M32—22no—
30#466—41no—
31#466—41no—
32#431—22no—
33#431—22no—
34#431—41no—
35#431—41no—
36C2×#107C2×C411yescharacteristic
37C2×#106C2×C411yescharacteristic
38C2×#106C2×C411yescharacteristic
39C2×C4⋊C8—41no—
40C2×#102—21no—
41C2×#102—21no—
42C2×#102—21no—
43C2×C22⋊C8—41no—
44C2×#97—41no—
45C2×#97—41no—
46C2×#97C2×C411yescharacteristic
47C2×#97D411yescharacteristic
48C2×#97D411yescharacteristic
49C2×#97C2311yescharacteristic
50C2×C4×C8—22no—
51#390—21no—
52#390—21no—
53#389—21no—
54#389—21no—
55#388—41no—
56#388—41no—
57#386—41no—
58#380—21no—
59C22⋊C16—41no—
60C22⋊C16—41no—
61#340—21no—
62#340—21no—
63#340—21no—
64C22×D4—21no—
65C22×D4—41no—
66C23×C4C22⋊C411yescharacteristic
67C23×C4D811yescharacteristic
68C23×C4SD1611yescharacteristic
69C2×D8—22no—
70C2×D8—41no—
71C2×D8—41no—
72C22×C8—21no—
73C22×C8—21no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—41no—
79C22×C8—41no—
80C22×C8—42no—
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22⋊C411yescharacteristic
83C22×C8C22×C411yescharacteristic
84C22×C8C2×D411yescharacteristic
85#123—84no—
86#121—41no—
87#121—41no—
88#121—41no—
89#121—41no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4—41no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4—41no—
100C2×C22⋊C4—41no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42C22⋊C411yescharacteristic
106C2×C42D811yescharacteristic
107C2×C42SD1611yescharacteristic
108C2×C42C2×D411yescharacteristic
109M32—42no—
110M32—42no—
111C2×C16—42no—
112#107—41no—
113#106—22no—
114#106—41no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117#102—42no—
118#102—42no—
119#102—42no—
120C22⋊C8—42no—
121C22⋊C8—42no—
122#97—41no—
123#97—41no—
124#97—41no—
125#97—41no—
126C4×C8—24no—
127C4×C8—42no—
128#95—41no—
129#95—41no—
130#95—41no—
131C24—41no—
132C24#10211yescharacteristic
133C2×D4—41no—
134C2×D4—41no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4—41no—
138C2×D4—41no—
139C2×D4—82no—
140C2×D4—82no—
141C2×D4—81no—
142C2×D4—81no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—42no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—81no—
156C22×C4#9911yescharacteristic
157C22×C4#9911yescharacteristic
158C22×C4#10211yescharacteristic
159C22×C4#10211yescharacteristic
160C22×C4#10211yescharacteristic
161C22×C4#10411yescharacteristic
162C22×C4#10411yescharacteristic
163C22×C4C2×C22⋊C411yescharacteristic
164C22×C4C2×D811yescharacteristic
165C22×C4C2×SD1611yescharacteristic
166D8—82no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—44no—
182C2×C8—84no—
183C2×C8#13112yes—
184C2×C8#13512yes—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—82no—
192C4⋊C4—82no—
193C22⋊C4—82no—
194C22⋊C4—82no—
195C22⋊C4—82no—
196C22⋊C4—82no—
197C22⋊C4—82no—
198C22⋊C4—81no—
199C22⋊C4—81no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—21no—
208C16—82no—
209C23—21no—
210C23—22no—
211C23—21no—
212C23—41no—
213C23—81no—
214C23—81no—
215C23—81no—
216C23—81no—
217C23—81no—
218C23—81no—
219C23—81no—
220C23#32711yescharacteristic
221C23#32711yescharacteristic
222C23C2×#10211yescharacteristic
223D4—81no—
224D4—81no—
225D4—81no—
226D4—81no—
227D4—162no—
228D4—162no—
229C2×C4—21no—
230C2×C4—21no—
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—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
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—81no—
259C2×C4—81no—
260C2×C4#32711yescharacteristic
261C2×C4#32711yescharacteristic
262C2×C4#40811yescharacteristic
263C2×C4C2×#9911yescharacteristic
264C2×C4#41311yescharacteristic
265C2×C4#41611yescharacteristic
266C2×C4C2×#10411yescharacteristic
267C2×C4#42111yescharacteristic
268C8—22no—
269C8—22no—
270C8—41no—
271C8—41no—
272C8—84no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—42no—
281C22—42no—
282C22—161no—
283C22—161no—
284C22—161no—
285C22—161no—
286C22—161no—
287C22—161no—
288C22—161no—
289C22#134112yes—
290C22#148011yescharacteristic
291C22C2×#32711yescharacteristic
292C22#154011yescharacteristic
293C4—21no—
294C4—21no—
295C4—41no—
296C4—41no—
297C4—41no—
298C4—41no—
299C4—161no—
300C4—161no—
301C4—161no—
302C4#146811yescharacteristic
303C4#149411yescharacteristic
304C4#151611yescharacteristic
305C4#153211yescharacteristic
306C2—21no—
307C2—21no—
308C2—22no—
309C2—41no—
310C2—321no—
311C2#765411yescharacteristic
312C2#1168511yescharacteristic
313C2C2×#134111yescharacteristic
subgroupG/H
center, socleC22C2×#327
commutatorC22×C8C22×C4
Frattini, cosocleC2×#97C23
prime radicalC2×#466C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G
LECSG ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12846C2semidirect
3. C2×#1275C2semidirect
4. #7126C2semidirect
5. #7110C2semidirect

Maximal subgroups

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

subgroupindex
#331352
#128462
C2×#12752
#71262
#71102