Finite Group Databasebeta

← #125913

Subgroups — #125913

Order 512.

367 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 64 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#31193C211yesmaximal, characteristic
2#13451C211yesmaximal, characteristic, complemented
3#13364C211yesmaximal, characteristic, complemented
4#10264C211yesmaximal, characteristic
5#7120C211yesmaximal, characteristic, complemented
6#7113C211yesmaximal, characteristic, complemented
7#7102C211yesmaximal, characteristic
8C2×#500C2211yescharacteristic
9C2×#485—21no—
10C2×#461—21no—
11#2415—21no—
12C2×#366—21no—
13C2×#358—21no—
14#2066—21no—
15#2036—21no—
16#2001—21no—
17#1969C2211yescharacteristic
18#1954C2211yescharacteristic
19#1942C2211yescharacteristic
20#1854—21no—
21C2×#332—21no—
22#1380—21no—
23#1380—21no—
24#1374C2211yescharacteristic
25#1374C2211yescharacteristic
26#1277C2211yescharacteristic
27C22×Q16D411yescharacteristic
28C22×SD16—41no—
29C2×C4⋊Q8—21no—
30C2×C22⋊Q8—21no—
31C2×C22⋊Q8—21no—
32C2×C4×D4D411yescharacteristic
33C2×#117—21no—
34C22×C16—21no—
35C22×C16—21no—
36#500—42no—
37#485—42no—
38#485—42no—
39#461—42no—
40#461—42no—
41C2×#107—21no—
42C2×#107—21no—
43C2×#107D411yescharacteristic
44C2×#107D411yescharacteristic
45C2×#107D411yescharacteristic
46C2×#106—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8D411yescharacteristic
49C2×#103—21no—
50C2×#103—21no—
51C2×#103—21no—
52C2×#103—41no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C4×C8C2311yescharacteristic
56#397—41no—
57#394—41no—
58#391—21no—
59#389—21no—
60#389—41no—
61#388—41no—
62#380—21no—
63#366—42no—
64#358—42no—
65#340—21no—
66#340—21no—
67#332—42no—
68C22×Q8—21no—
69C22×Q8—41no—
70C22×D4D811yescharacteristic
71C23×C4—21no—
72C2×Q16—21no—
73C2×Q16—21no—
74C2×Q16—41no—
75C2×Q16—41no—
76C2×SD16—44no—
77C2×SD16—82no—
78C2×SD16—82no—
79C22×C8—21no—
80C22×C8—41no—
81C22×C8C2×D411yescharacteristic
82C22×C8C2×D411yescharacteristic
83C4⋊Q8—22no—
84C4⋊Q8—41no—
85C4⋊Q8—41no—
86C4⋊Q8—41no—
87C22⋊Q8—42no—
88C22⋊Q8—81no—
89C22⋊Q8—81no—
90C22⋊Q8—81no—
91C4×D4—41no—
92C4×D4—41no—
93#117—42no—
94#117—81no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4—41no—
100C2×C4⋊C4—41no—
101C2×C4⋊C4—41no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4D811yescharacteristic
104C2×C22⋊C4—21no—
105C2×C22⋊C4—41no—
106C2×C22⋊C4—41no—
107C2×C22⋊C4—41no—
108C2×C22⋊C4—41no—
109C2×C42—41no—
110C2×C42C2×D411yescharacteristic
111C2×C16—22no—
112C2×C16—41no—
113C2×C16—41no—
114C2×C16—41no—
115C2×C16—41no—
116C2×C16—41no—
117#107—21no—
118#107—22no—
119#107—22no—
120#107—41no—
121#107—41no—
122#107—41no—
123#107—41no—
124#107SD1612yescomplemented
125#106—42no—
126C4⋊C8—22no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129#103—42no—
130#103—44no—
131#103—41no—
132#103—41no—
133#103—81no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137C22⋊C8—41no—
138C4×C8—41no—
139#95—41no—
140#95—41no—
141#95—41no—
142#95—41no—
143C24—21no—
144C2×Q8—42no—
145C2×Q8—41no—
146C2×Q8—41no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—81no—
152C2×Q8—81no—
153C2×Q8—81no—
154C2×Q8—81no—
155C2×Q8—81no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—41no—
159C2×D4—41no—
160C22×C4—21no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—81no—
172C22×C4#12011yescharacteristic
173C22×C4C2×D811yescharacteristic
174C22×C4AGL(1,ℤ/8)11yescharacteristic
175Q16—42no—
176Q16—81no—
177SD16—88no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—42no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—81no—
187C2×C8—81no—
188C2×C8C2×SD1611yescharacteristic
189C2×C8#13511yescharacteristic
190C2×C8AGL(1,ℤ/8)11yescharacteristic
191C2×C8#13711yescharacteristic
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—41no—
201C4⋊C4—81no—
202C4⋊C4—82no—
203C4⋊C4—82no—
204C4⋊C4—81no—
205C4⋊C4—81no—
206C4⋊C4—81no—
207C22⋊C4—81no—
208C22⋊C4—82no—
209C22⋊C4—82no—
210C22⋊C4—82no—
211C22⋊C4—82no—
212C42—21no—
213C42—21no—
214C42—82no—
215C16—42no—
216C16—81no—
217C23—41no—
218C23—41no—
219C23—41no—
220C23—41no—
221C23—41no—
222C23—41no—
223C23—41no—
224C23#44711yescharacteristic
225Q8—82no—
226Q8—82no—
227Q8—81no—
228Q8—81no—
229Q8—81no—
230Q8—81no—
231Q8—161no—
232Q8—161no—
233D4—42no—
234D4—81no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—81no—
251C2×C4—81no—
252C2×C4—82no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—161no—
261C2×C4—161no—
262C2×C4#45011yescharacteristic
263C2×C4#45211yescharacteristic
264C2×C4#45311yescharacteristic
265C2×C4#45511yescharacteristic
266C2×C4#45611yescharacteristic
267C2×C4B(4,𝔽2)11yescharacteristic
268C2×C4#50811yescharacteristic
269C2×C4#50911yescharacteristic
270C8—21no—
271C8—21no—
272C8—21no—
273C8—21no—
274C8—82no—
275C8—81no—
276C22—81no—
277C22—81no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—81no—
282C22—81no—
283C22#159511yescharacteristic
284C22#163911yescharacteristic
285C22#164711yescharacteristic
286C22#218611yescharacteristic
287C22#219011yescharacteristic
288C22#219511yescharacteristic
289C22#219511yescharacteristic
290C4—21no—
291C4—21no—
292C4—41no—
293C4—81no—
294C4—82no—
295C4—161no—
296C4—161no—
297C4—162no—
298C4—161no—
299C4#160411yescharacteristic
300C4#160611yescharacteristic
301C4#163111yescharacteristic
302C4#166111yescharacteristic
303C2—161no—
304C2#775211yescharacteristic
305C2#1210411yescharacteristic
306C2#1211211yescharacteristic
307C2#1237111yescharacteristic
308C2#1237911yescharacteristic
309C2#1239311yescharacteristic
310C2#1240311yescharacteristic
subgroupG/H
center, socleC23#447
commutator, Frattini, cosocleC2×C4×C8C23
prime radicalC22×D4D8
LCSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1954 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #13451C2semidirect
3. #13364C2semidirect
5. #7120C2semidirect
6. #7113C2semidirect
124. #107SD16semidirect

Maximal subgroups

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

subgroupindex
#311932
#134512
#133642
#102642
#71202
#71132
#71022