Finite Group Databasebeta

← #135352

Subgroups — #135352

Order 512.

335 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 335 conjugacy classes of subgroups — too many to draw. Only its 76 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#20782C211yesmaximal, characteristic
2#10824C211yesmaximal, characteristic
3#8731C211yesmaximal, characteristic
4#7253C211yesmaximal, characteristic
5#7253C211yesmaximal, characteristic
6#7252C211yesmaximal, characteristic
7#7252C211yesmaximal, characteristic
8C4×#117C2211yescharacteristic
9#1840—21no—
10#1832C411yescharacteristic, complemented
11#1832C411yescharacteristic, complemented
12#1818—21no—
13#1798C2211yescharacteristic
14#1798C2211yescharacteristic
15C4×C4⋊C8C411yescharacteristic, complemented
16C4×C4⋊C8C411yescharacteristic, complemented
17C4×C22⋊C8—21no—
18#1744—21no—
19#1742C2211yescharacteristic
20#1740C2211yescharacteristic
21C2×#339—21no—
22C2×H(3,ℤ/4)—21no—
23#1458C2211yescharacteristic
24#1458C2211yescharacteristic
25#1276—21no—
26#1274—21no—
27C22×M16—21no—
28C2×#117—21no—
29C2×#117D411yescharacteristic
30C2×#117C2311yescharacteristic
31C22×C42D411yescharacteristic
32#431—21no—
33#431—21no—
34C2×C4⋊C8C2×C411yescharacteristic
35C2×C4⋊C8C2×C411yescharacteristic
36C2×C4⋊C8C2×C411yescharacteristic
37C2×C4⋊C8C2×C411yescharacteristic
38C2×C22⋊C8—21no—
39C2×C22⋊C8—21no—
40C2×C4×C8—21no—
41C2×C4×C8—21no—
42C4×C4⋊C4—21no—
43C4×C4⋊C4—21no—
44C4×C22⋊C4—21no—
45C4×C22⋊C4—21no—
46#376C2×C411yescharacteristic
47#376D411yescharacteristic
48C43C2×C411yescharacteristic
49C43D411yescharacteristic
50#343—21no—
51#343—21no—
52#342—21no—
53#342—21no—
54#339—24no—
55H(3,ℤ/4)—42no—
56#336—21no—
57#336—21no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#324—21no—
65#324—21no—
66#323—21no—
67#323—21no—
68C23×C4—21no—
69C23×C4C2×D411yescharacteristic
70C2×M16—22no—
71C2×M16—41no—
72C2×M16—41no—
73C2×M16—41no—
74C2×M16—41no—
75C2×M16—41no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—41no—
81#117—21no—
82#117—21no—
83#117—21no—
84#117—21no—
85#117—22no—
86#117—41no—
87#117—41no—
88#117—41no—
89#117—41no—
90#117—41no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4#4111yescharacteristic
94C2×C4⋊C4#4111yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—41no—
97C2×C22⋊C4#4111yescharacteristic
98C2×C22⋊C4#4111yescharacteristic
99C2×C42—22no—
100C2×C42—21no—
101C2×C42—21no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—41no—
106C2×C42—41no—
107C2×C42C22⋊C411yescharacteristic
108C2×C42C22⋊C411yescharacteristic
109C2×C42C22⋊C411yescharacteristic
110C2×C42C22⋊C411yescharacteristic
111C2×C42C22×C411yescharacteristic
112C2×C42C2×D411yescharacteristic
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C22⋊C8—41no—
120C22⋊C8—41no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125C4×C8—42no—
126C4×C8—41no—
127C4×C8—41no—
128C4×C8—42no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133C24#12311yescharacteristic
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—42no—
143C22×C4—42no—
144C22×C4—42no—
145C22×C4—81no—
146C22×C4C2×C22⋊C411yescharacteristic
147C22×C4#11711yescharacteristic
148C22×C4#11711yescharacteristic
149C22×C4#12311yescharacteristic
150C22×C4#12311yescharacteristic
151C22×C4#12311yescharacteristic
152C22×C4#13511yescharacteristic
153C22×C4#13511yescharacteristic
154C22×C4AGL(1,ℤ/8)11yescharacteristic
155C22×C4#13711yescharacteristic
156M16—42no—
157M16—42no—
158M16—81no—
159M16—81no—
160C2×C8—42no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—41no—
164C2×C8—41no—
165C2×C8—41no—
166C2×C8—41no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—81no—
174C2×C8—81no—
175C2×C8—81no—
176C2×C8—81no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—42no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C4⋊C4—81no—
183C22⋊C4—41no—
184C22⋊C4—41no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C22⋊C4—42no—
188C22⋊C4—81no—
189C42—21no—
190C42—21no—
191C42—21no—
192C42—21no—
193C42—21no—
194C42—21no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—42no—
201C42—42no—
202C42—42no—
203C42—42no—
204C42—42no—
205C42—42no—
206C42—81no—
207C42#10412yes—
208C23—21no—
209C23—21no—
210C23—41no—
211C23—41no—
212C23#38111yescharacteristic
213C23#48211yescharacteristic
214C23#48511yescharacteristic
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—21no—
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—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—84no—
246C2×C4—82no—
247C2×C4—161no—
248C2×C4—161no—
249C2×C4#41011yescharacteristic
250C2×C4#41011yescharacteristic
251C2×C4#41611yescharacteristic
252C2×C4#41711yescharacteristic
253C2×C4C2×#10411yescharacteristic
254C2×C4#42111yescharacteristic
255C2×C4#48211yescharacteristic
256C2×C4#48511yescharacteristic
257C8—82no—
258C8—81no—
259C8—81no—
260C8—81no—
261C8—81no—
262C8—161no—
263C8—161no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—81no—
273C22#182011yescharacteristic
274C22#182511yescharacteristic
275C22#182611yescharacteristic
276C4—24no—
277C4—21no—
278C4—21no—
279C4—41no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—81no—
284C4—81no—
285C4—81no—
286C4—162no—
287C4#150011yescharacteristic
288C4#150911yescharacteristic
289C4#151211yescharacteristic
290C4#152111yescharacteristic
291C2—21no—
292C2—21no—
293C2—41no—
294C2—41no—
295C2#871411yescharacteristic
296C2#1286411yescharacteristic
297C2#1286411yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
socleC22#1820
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #1832C4semidirect
11. #1832C4semidirect
15. C4×C4⋊C8C4semidirect
16. C4×C4⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
#207822
#108242
#87312
#72532
#72522