Finite Group Databasebeta

← #125020

Subgroups — #125020

Order 512.

416 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C4×#466C211yesmaximal, characteristic
2#12844C211yesmaximal, characteristic, complemented
3#8596C211yesmaximal, characteristic, complemented
4#7120C212yesmaximal, complemented
5#7112C212yesmaximal, complemented
6C2×#466C411yescharacteristic
7C2×C4×D8C411yescharacteristic
8C22×C4×C8C2211yescharacteristic
9C4×#121—21no—
10C2×C22⋊C16—21no—
11#1970C2211yescharacteristic
12#1942C2211yescharacteristic
13#1934—21no—
14#1893—21no—
15#1808C411yescharacteristic
16#1797—21no—
17C4×#107C411yescharacteristic
18C4×#102—21no—
19C2×#335—21no—
20#1374—21no—
21#1374C2212yes—
22#1277C2212yes—
23C22×D8C2×C411yescharacteristic
24C23×C8C2×C411yescharacteristic
25C23×C8D411yescharacteristic
26C2×#121—21no—
27C2×C4×D4—21no—
28C2×C4×D4—41no—
29C22×C42D411yescharacteristic
30C22×C16—22no—
31#466—22no—
32#466—22no—
33C4×D8—42no—
34C2×#107C2×C411yescharacteristic
35C2×#107C2×C411yescharacteristic
36C2×#107C2×C411yescharacteristic
37C2×C4⋊C8—22no—
38C2×#102—21no—
39C2×#102—21no—
40C2×#102—21no—
41C2×C22⋊C8—41no—
42C2×C4×C8—42no—
43C2×C4×C8C2×C411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8C2311yescharacteristic
47#390—21no—
48#389—21no—
49#388—41no—
50#386—41no—
51#380—21no—
52C4×C4⋊C4—21no—
53C4×C22⋊C4—41no—
54C22⋊C16—42no—
55#340—21no—
56#340—21no—
57#340—21no—
58#336—41no—
59#335—42no—
60C22×D4—21no—
61C22×D4—41no—
62C23×C4—41no—
63C23×C4C22⋊C411yescharacteristic
64C23×C4D811yescharacteristic
65C23×C4SD1611yescharacteristic
66C2×D8—21no—
67C2×D8—41no—
68C2×D8—41no—
69C2×D8M1612yes—
70C22×C8—22no—
71C22×C8—21no—
72C22×C8—21no—
73C22×C8—22no—
74C22×C8—41no—
75C22×C8—41no—
76C22×C8—41no—
77C22×C8—42no—
78C22×C8C22⋊C411yescharacteristic
79C22×C8C22⋊C411yescharacteristic
80C22×C8C22×C411yescharacteristic
81C22×C8C2×D411yescharacteristic
82#121—24no—
83#121—42no—
84C4×D4—42no—
85C4×D4—42no—
86C4×D4—82no—
87C4×D4—82no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C22⋊C4—41no—
97C2×C42—41no—
98C2×C42—41no—
99C2×C42—41no—
100C2×C42C22⋊C411yescharacteristic
101C2×C42D811yescharacteristic
102C2×C42SD1611yescharacteristic
103C2×C42C2×D411yescharacteristic
104C2×C16—42no—
105C2×C16—42no—
106C2×C16—42no—
107#107—21no—
108#107—41no—
109#107—41no—
110#107M1612yes—
111C4⋊C8—42no—
112C4⋊C8—42no—
113#102—24no—
114#102—42no—
115#102—42no—
116C22⋊C8—82no—
117C4×C8—22no—
118C4×C8—22no—
119C4×C8—41no—
120C4×C8—41no—
121#95—41no—
122#95—41no—
123#95—41no—
124C24—41no—
125C24#10211yescharacteristic
126C2×D4—22no—
127C2×D4—44no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—82no—
134C2×D4—82no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—41no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—82no—
150C22×C4—82no—
151C22×C4—82no—
152C22×C4#10211yescharacteristic
153C22×C4#10211yescharacteristic
154C22×C4#10211yescharacteristic
155C22×C4D1612yes—
156C22×C4SD3212yes—
157C22×C4C2×C22⋊C411yescharacteristic
158C22×C4C2×D811yescharacteristic
159C22×C4C2×SD1611yescharacteristic
160D8—42no—
161D8—42no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—44no—
172C2×C8—42no—
173C2×C8—42no—
174C2×C8—82no—
175C2×C8—82no—
176C2×C8—82no—
177C2×C8#9912yes—
178C2×C8C2×M1611yescharacteristic
179C2×C8C2×M1611yescharacteristic
180C2×C8#13512yes—
181C4⋊C4—22no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—81no—
188C4⋊C4—81no—
189C22⋊C4—44no—
190C22⋊C4—82no—
191C22⋊C4—82no—
192C22⋊C4—82no—
193C22⋊C4—81no—
194C22⋊C4—81no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—41no—
200C42—41no—
201C42—82no—
202C16—82no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—21no—
208C23—21no—
209C23—42no—
210C23—81no—
211C23—81no—
212C23—81no—
213C23—81no—
214C23—81no—
215C23—81no—
216C23#35712yes—
217C23C2×#10211yescharacteristic
218D4—42no—
219D4—42no—
220D4—88no—
221D4—81no—
222D4—81no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
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—164no—
253C2×C4#35712yes—
254C2×C4#40711yescharacteristic
255C2×C4C2×#9911yescharacteristic
256C2×C4#41311yescharacteristic
257C2×C4#41611yescharacteristic
258C2×C4#42111yescharacteristic
259C2×C4#42111yescharacteristic
260C2×C4C2×D1611yescharacteristic
261C2×C4C2×SD3211yescharacteristic
262C8—21no—
263C8—21no—
264C8—22no—
265C8—22no—
266C8—41no—
267C8—162no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—161no—
279C22—161no—
280C22—161no—
281C22—161no—
282C22#148611yescharacteristic
283C22#151911yescharacteristic
284C22#154011yescharacteristic
285C22C2×#35711yescharacteristic
286C22#214311yescharacteristic
287C22#214511yescharacteristic
288C22#214811yescharacteristic
289C4—21no—
290C4—22no—
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—82no—
296C4—161no—
297C4—161no—
298C4#146511yescharacteristic
299C4#148811yescharacteristic
300C4#151611yescharacteristic
301C4#153511yescharacteristic
302C2—42no—
303C2—162no—
304C2#764711yescharacteristic
305C2#1165711yescharacteristic
306C2#1166511yescharacteristic
307C2#1188611yescharacteristic
308C2#1189411yescharacteristic
309C2#1194611yescharacteristic
310C2#1194711yescharacteristic
subgroupG/H
center, socleC23C2×#102
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC2×#466C4
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #12844C2semidirect
3. #8596C2semidirect
4. #7120C2semidirect
5. #7112C2semidirect

Maximal subgroups

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

subgroupindex
C4×#4662
#128442
#85962
#71202
#71122