Finite Group Databasebeta

← #135372

Subgroups — #135372

Order 512.

354 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 84 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#20813C211yesmaximal, characteristic
2#10824C211yesmaximal, characteristic
3#8731C211yesmaximal, characteristic
4#7253C211yesmaximal, characteristic
5#7253C211yesmaximal, characteristic
6#7252C211yesmaximal, characteristic
7#7252C211yesmaximal, characteristic
8C2×#433—21no—
9C4×#117C2211yescharacteristic
10#1843C411yescharacteristic, complemented
11#1843C411yescharacteristic, complemented
12#1831—21no—
13#1798C2211yescharacteristic
14#1798C2211yescharacteristic
15#1797—21no—
16#1768—21no—
17#1767C411yescharacteristic, complemented
18C4×C4⋊C8C411yescharacteristic, complemented
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#433—42no—
33#433—42no—
34#431—21no—
35#431—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8C2×C411yescharacteristic
41C2×C4⋊C8C2×C411yescharacteristic
42C2×C4⋊C8C2×C411yescharacteristic
43C2×C4⋊C8C2×C411yescharacteristic
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×#97—21no—
47C2×C4×C8—21no—
48C4×C4⋊C4—21no—
49C4×C4⋊C4—21no—
50C4×C22⋊C4—21no—
51C4×C22⋊C4—21no—
52#376C2×C411yescharacteristic
53#376D411yescharacteristic
54C43C2×C411yescharacteristic
55C43D411yescharacteristic
56#343—21no—
57#343—21no—
58#342—21no—
59#342—21no—
60#339—24no—
61H(3,ℤ/4)—42no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#324—21no—
67#324—21no—
68#323—21no—
69#323—21no—
70C23×C4—21no—
71C23×C4C2×D411yescharacteristic
72C2×M16—22no—
73C2×M16—41no—
74C2×M16—41no—
75C2×M16—41no—
76C2×M16—41no—
77C2×M16—41no—
78C22×C8—21no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—41no—
83#117—21no—
84#117—21no—
85#117—22no—
86#117—41no—
87#117—41no—
88#117—41no—
89#117—41no—
90#117—41no—
91#117D812yes—
92#117SD1612yes—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4#4111yescharacteristic
96C2×C4⋊C4#4111yescharacteristic
97C2×C22⋊C4—21no—
98C2×C22⋊C4—41no—
99C2×C22⋊C4#4111yescharacteristic
100C2×C22⋊C4#4111yescharacteristic
101C2×C42—22no—
102C2×C42—21no—
103C2×C42—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—41no—
108C2×C42—41no—
109C2×C42C22⋊C411yescharacteristic
110C2×C42C22⋊C411yescharacteristic
111C2×C42C22⋊C411yescharacteristic
112C2×C42C22⋊C411yescharacteristic
113C2×C42C22×C411yescharacteristic
114C2×C42C2×D411yescharacteristic
115C4⋊C8—41no—
116C4⋊C8—41no—
117C4⋊C8—41no—
118C4⋊C8—41no—
119C4⋊C8—42no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C4⋊C8—42no—
123C4⋊C8—42no—
124C4⋊C8—42no—
125C22⋊C8—41no—
126C22⋊C8—41no—
127C22⋊C8—41no—
128C22⋊C8—41no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131#97—42no—
132C4×C8—41no—
133C4×C8—41no—
134C4×C8—42no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139C24#12311yescharacteristic
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—42no—
149C22×C4—42no—
150C22×C4—81no—
151C22×C4—81no—
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4#11711yescharacteristic
154C22×C4#11711yescharacteristic
155C22×C4#12311yescharacteristic
156C22×C4#12311yescharacteristic
157C22×C4#12311yescharacteristic
158C22×C4C2×D811yescharacteristic
159C22×C4C2×SD1611yescharacteristic
160C22×C4#13711yescharacteristic
161C22×C4#13711yescharacteristic
162M16—42no—
163M16—42no—
164M16—81no—
165M16—81no—
166C2×C8—42no—
167C2×C8—41no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—41no—
172C2×C8—41no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—41no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—81no—
180C2×C8—81no—
181C2×C8—81no—
182C2×C8—81no—
183C4⋊C4—22no—
184C4⋊C4—22no—
185C4⋊C4—41no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C4⋊C4—81no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—41no—
192C22⋊C4—41no—
193C22⋊C4—42no—
194C22⋊C4—81no—
195C42—21no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—21no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—42no—
205C42—42no—
206C42—42no—
207C42—42no—
208C42—81no—
209C42—81no—
210C42—81no—
211C42#10212yes—
212C42#10212yes—
213C42#10412yes—
214C23—21no—
215C23—21no—
216C23—41no—
217C23—41no—
218C23#38111yescharacteristic
219C23#48011yescharacteristic
220C23#48311yescharacteristic
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—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—84no—
252C2×C4—82no—
253C2×C4—161no—
254C2×C4—161no—
255C2×C4#41011yescharacteristic
256C2×C4#41011yescharacteristic
257C2×C4C2×#10211yescharacteristic
258C2×C4#41911yescharacteristic
259C2×C4C2×#10411yescharacteristic
260C2×C4#42111yescharacteristic
261C2×C4#48011yescharacteristic
262C2×C4#48311yescharacteristic
263C8—82no—
264C8—81no—
265C8—81no—
266C8—81no—
267C8—81no—
268C8—161no—
269C8—161no—
270C22—21no—
271C22—21no—
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—41no—
278C22—81no—
279C22#182011yescharacteristic
280C22#182311yescharacteristic
281C22#182611yescharacteristic
282C4—21no—
283C4—21no—
284C4—42no—
285C4—42no—
286C4—42no—
287C4—42no—
288C4—41no—
289C4—41no—
290C4—41no—
291C4—41no—
292C4—162no—
293C4#150211yescharacteristic
294C4#150611yescharacteristic
295C4#151511yescharacteristic
296C4#151711yescharacteristic
297C2—21no—
298C2—21no—
299C2—41no—
300C2—41no—
301C2#871611yescharacteristic
302C2#1286411yescharacteristic
303C2#1286511yescharacteristic
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. #1843C4semidirect
11. #1843C4semidirect
17. #1767C4semidirect
18. C4×C4⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
#208132
#108242
#87312
#72532
#72522