Finite Group Databasebeta

← #125016

Subgroups — #125016

Order 512.

416 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 90 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#7121C212yesmaximal, 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—84no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C22⋊C4—41no—
93C2×C22⋊C4—41no—
94C2×C22⋊C4—41no—
95C2×C22⋊C4—41no—
96C2×C42—41no—
97C2×C42—41no—
98C2×C42—41no—
99C2×C42C22⋊C411yescharacteristic
100C2×C42D811yescharacteristic
101C2×C42SD1611yescharacteristic
102C2×C42C2×D411yescharacteristic
103C2×C16—42no—
104C2×C16—42no—
105C2×C16—42no—
106#107—21no—
107#107—41no—
108#107—41no—
109#107M1612yes—
110C4⋊C8—42no—
111C4⋊C8—42no—
112#102—24no—
113#102—42no—
114#102—42no—
115C22⋊C8—82no—
116C4×C8—22no—
117C4×C8—22no—
118C4×C8—41no—
119C4×C8—41no—
120#95—41no—
121#95—41no—
122#95—41no—
123C24—41no—
124C24#10211yescharacteristic
125C2×D4—22no—
126C2×D4—44no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—82no—
133C2×D4—82no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—82no—
151C22×C4—82no—
152C22×C4—82no—
153C22×C4#10211yescharacteristic
154C22×C4#10211yescharacteristic
155C22×C4#10211yescharacteristic
156C22×C4C2×C22⋊C411yescharacteristic
157C22×C4C2×D811yescharacteristic
158C22×C4C2×SD1611yescharacteristic
159D8—42no—
160D8—42no—
161C2×C8—21no—
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—44no—
171C2×C8—42no—
172C2×C8—42no—
173C2×C8—82no—
174C2×C8—82no—
175C2×C8—82no—
176C2×C8#9912yes—
177C2×C8C2×M1611yescharacteristic
178C2×C8C2×M1611yescharacteristic
179C2×C8#13512yes—
180C4⋊C4—22no—
181C4⋊C4—41no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—82no—
187C22⋊C4—44no—
188C22⋊C4—82no—
189C22⋊C4—82no—
190C22⋊C4—82no—
191C22⋊C4—81no—
192C22⋊C4—81no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—41no—
198C42—41no—
199C42—82no—
200C16—82no—
201C23—21no—
202C23—21no—
203C23—21no—
204C23—21no—
205C23—21no—
206C23—21no—
207C23—42no—
208C23—81no—
209C23—81no—
210C23—81no—
211C23—81no—
212C23—81no—
213C23—81no—
214C23#36112yes—
215C23C2×#10211yescharacteristic
216D4—42no—
217D4—42no—
218D4—88no—
219D4—81no—
220D4—81no—
221C2×C4—21no—
222C2×C4—21no—
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—44no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—81no—
250C2×C4—164no—
251C2×C4#35912yes—
252C2×C4#40711yescharacteristic
253C2×C4C2×#9911yescharacteristic
254C2×C4#41311yescharacteristic
255C2×C4#41611yescharacteristic
256C2×C4#42111yescharacteristic
257C2×C4#42111yescharacteristic
258C2×C4#50911yescharacteristic
259C2×C4#51011yescharacteristic
260C8—21no—
261C8—21no—
262C8—22no—
263C8—22no—
264C8—41no—
265C8—162no—
266C22—22no—
267C22—22no—
268C22—22no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—82no—
274C22—82no—
275C22—82no—
276C22—161no—
277C22—161no—
278C22—161no—
279C22—161no—
280C22#148611yescharacteristic
281C22#151911yescharacteristic
282C22#154011yescharacteristic
283C22#213911yescharacteristic
284C22C2×#35911yescharacteristic
285C22#214511yescharacteristic
286C22C2×#36111yescharacteristic
287C4—21no—
288C4—21no—
289C4—22no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—82no—
294C4—161no—
295C4—161no—
296C4#146511yescharacteristic
297C4#148811yescharacteristic
298C4#151611yescharacteristic
299C4#153511yescharacteristic
300C2—42no—
301C2—162no—
302C2#764711yescharacteristic
303C2#1166011yescharacteristic
304C2#1166711yescharacteristic
305C2#1189011yescharacteristic
306C2#1189111yescharacteristic
307C2#1194311yescharacteristic
308C2#1195211yescharacteristic
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. #7121C2semidirect
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
#71212
#71122