Finite Group Databasebeta

← #125276

Subgroups — #125276

Order 512.

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

normal subgroups only

This group has 443 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
1#33129C211yesmaximal, characteristic
2#12544C211yesmaximal, characteristic, complemented
3#7874C211yesmaximal, characteristic, complemented
4#7121C212yesmaximal, complemented
5#7112C212yesmaximal, complemented
6C22×M32—21no—
7C2×#466C411yescharacteristic
8C2×#423—21no—
9C22×C4×C8C2211yescharacteristic
10#2854—21no—
11#1970C411yescharacteristic
12#1970C2211yescharacteristic
13#1942C411yescharacteristic
14#1942C2211yescharacteristic
15#1934—21no—
16#1934—21no—
17#1893—21no—
18#1808C411yescharacteristic
19#1374—21no—
20#1374C2212yes—
21#1277—21no—
22#1277C2212yes—
23C22×D8C2×C411yescharacteristic
24C22×M16—41no—
25C23×C8C2×C411yescharacteristic
26C23×C8D411yescharacteristic
27C2×#123—41no—
28C2×#121—21no—
29C22×C42D411yescharacteristic
30C2×M32—24no—
31C2×M32—42no—
32C2×M32—42no—
33C22×C16—22no—
34#466—42no—
35C2×#107C2×C411yescharacteristic
36C2×#107C2×C411yescharacteristic
37C2×#107C2×C411yescharacteristic
38#423—24no—
39#423—42no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—41no—
42C2×#102—21no—
43C2×#102—21no—
44C2×#102—21no—
45C2×C4×C8—42no—
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8C2311yescharacteristic
50#390—21no—
51#390—21no—
52#389—21no—
53#389—21no—
54#388—41no—
55#388—41no—
56#386—41no—
57#380—21no—
58#340—21no—
59#340—21no—
60#340—21no—
61C22×D4—21no—
62C22×D4—41no—
63C23×C4C22⋊C411yescharacteristic
64C23×C4D811yescharacteristic
65C23×C4SD1611yescharacteristic
66C2×D8—22no—
67C2×D8—41no—
68C2×D8—41no—
69C2×M16—44no—
70C2×M16—82no—
71C2×M16—82no—
72C22×C8—22no—
73C22×C8—22no—
74C22×C8—21no—
75C22×C8—21no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—41no—
79C22×C8—42no—
80C22×C8C22⋊C411yescharacteristic
81C22×C8C22⋊C411yescharacteristic
82C22×C8C22×C411yescharacteristic
83C22×C8C2×D411yescharacteristic
84#123—84no—
85#121—42no—
86#121—42no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—41no—
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×C22⋊C4—41no—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42C22⋊C411yescharacteristic
101C2×C42D811yescharacteristic
102C2×C42SD1611yescharacteristic
103C2×C42C2×D411yescharacteristic
104M32—48no—
105C2×C16—24no—
106C2×C16—42no—
107C2×C16—42no—
108#107—22no—
109#107—41no—
110#107—41no—
111C4⋊C8—24no—
112C4⋊C8—42no—
113C4⋊C8—44no—
114#102—42no—
115#102—42no—
116#102—41no—
117#102—41no—
118C4×C8—41no—
119C4×C8—41no—
120C4×C8—41no—
121C4×C8—41no—
122#95—41no—
123#95—41no—
124#95—41no—
125C24—41no—
126C24#10211yescharacteristic
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—82no—
134C2×D4—82no—
135C2×D4—82no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4#9912yes—
151C22×C4#10211yescharacteristic
152C22×C4#10211yescharacteristic
153C22×C4#10211yescharacteristic
154C22×C4#10412yes—
155C22×C4C2×C22⋊C411yescharacteristic
156C22×C4C2×D811yescharacteristic
157C22×C4C2×SD1611yescharacteristic
158D8—82no—
159M16—88no—
160C2×C8—24no—
161C2×C8—24no—
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—44no—
173C2×C8—82no—
174C2×C8—82no—
175C2×C8#10412yes—
176C2×C8#13112yes—
177C2×C8AGL(1,ℤ/8)11yescharacteristic
178C2×C8#13711yescharacteristic
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—82no—
186C4⋊C4—82no—
187C22⋊C4—82no—
188C22⋊C4—82no—
189C22⋊C4—82no—
190C22⋊C4—82no—
191C22⋊C4—82no—
192C22⋊C4—81no—
193C22⋊C4—81no—
194C42—22no—
195C42—22no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C16—44no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—81no—
208C23—81no—
209C23—81no—
210C23—81no—
211C23—81no—
212C23—81no—
213C23—81no—
214C23#32712yes—
215C23C2×#10211yescharacteristic
216D4—81no—
217D4—81no—
218D4—81no—
219D4—81no—
220D4—164no—
221C2×C4—24no—
222C2×C4—24no—
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—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—44no—
239C2×C4—81no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4#32712yes—
249C2×C4#36012yes—
250C2×C4#36112yes—
251C2×C4#40811yescharacteristic
252C2×C4C2×#9911yescharacteristic
253C2×C4#41311yescharacteristic
254C2×C4#41711yescharacteristic
255C2×C4C2×#10411yescharacteristic
256C2×C4C2×#10411yescharacteristic
257C2×C4#50911yescharacteristic
258C2×C4#51011yescharacteristic
259C8—22no—
260C8—21no—
261C8—22no—
262C8—21no—
263C8—41no—
264C8—84no—
265C22—22no—
266C22—22no—
267C22—22no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—161no—
273C22—161no—
274C22—161no—
275C22—161no—
276C22—161no—
277C22—161no—
278C22—161no—
279C22#149211yescharacteristic
280C22C2×#32711yescharacteristic
281C22#154711yescharacteristic
282C22#213911yescharacteristic
283C22#214311yescharacteristic
284C22C2×#36011yescharacteristic
285C22C2×#36111yescharacteristic
286C4—21no—
287C4—22no—
288C4—21no—
289C4—22no—
290C4—22no—
291C4—41no—
292C4—161no—
293C4—161no—
294C4—161no—
295C4#146811yescharacteristic
296C4#148311yescharacteristic
297C4#150411yescharacteristic
298C4#152411yescharacteristic
299C2—42no—
300C2—321no—
301C2#767811yescharacteristic
302C2#1167411yescharacteristic
303C2#1168011yescharacteristic
304C2#1170411yescharacteristic
305C2#1171111yescharacteristic
306C2#1198011yescharacteristic
307C2#1198611yescharacteristic
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. #12544C2semidirect
3. #7874C2semidirect
4. #7121C2semidirect
5. #7112C2semidirect

Maximal subgroups

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

subgroupindex
#331292
#125442
#78742
#71212
#71122