Finite Group Databasebeta

← #127336

Subgroups — #127336

Order 512.

709 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 709 conjugacy classes of subgroups — too many to draw. Only its 180 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#21847C211yesmaximal, characteristic
2#20632C212yesmaximal
3#7237C214yesmaximal, complemented
4C2×C4×#41C2211yescharacteristic
5C2×#431C411yescharacteristic, complemented
6C22×C4⋊C8C411yescharacteristic, complemented
7C2×#410—22no—
8#2285—22no—
9#1984C412yescomplemented
10#1954C2212yes—
11#1869—22no—
12#1799—22no—
13#1799—22no—
14#1799C414yescomplemented
15#1799C2214yes—
16#1797—22no—
17#1740C414yescomplemented
18C22×#41D411yescharacteristic
19C23×C8—21no—
20C4×#41—42no—
21C4×#41—42no—
22C2×C4×Q8C2×C411yescharacteristic
23C2×C4×Q8Q811yescharacteristic
24C2×C4×D4C2×C412yes—
25C2×C4×D4D412yes—
26C2×C4×D4D411yescharacteristic
27C2×C4×D4C2311yescharacteristic
28C2×#117—24no—
29C2×#117C2×C412yes—
30C2×#117Q811yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C42D412yes—
33#431—42no—
34C2×C4⋊C8—21no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—42no—
37C2×C4⋊C8C2×C412yes—
38#410—44no—
39#410—44no—
40C2×#99—24no—
41C2×#99—24no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—24no—
44C2×C4×C8C2×C412yes—
45#384—42no—
46#382—42no—
47#380—24no—
48#380—24no—
49C4×C4⋊C4—42no—
50C4×C4⋊C4—42no—
51#342—24no—
52#342C2×C418yes—
53#336—24no—
54C2×#41—42no—
55C2×#41—42no—
56C2×#41D812yes—
57C2×#41SD1614yes—
58C2×#41Q1612yes—
59C22×Q8C4⋊C411yescharacteristic
60C22×D4C22⋊C412yes—
61C22×D4C2×D411yescharacteristic
62C23×C4C4211yescharacteristic
63C23×C4C22⋊C412yes—
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C412yes—
66C23×C4C4⋊C411yescharacteristic
67C23×C4C2×D412yes—
68C22×C8—22no—
69C22×C8—22no—
70C22×C8—42no—
71C22×C8—82no—
72C4×Q8—41no—
73C4×Q8—41no—
74C4×Q8—42no—
75C4×D4—42no—
76C4×D4—42no—
77C4×D4—42no—
78C4×D4—42no—
79C4×D4—42no—
80C4×D4—41no—
81C4×D4—41no—
82#117—42no—
83#117—42no—
84#117—42no—
85#117—48no—
86#117—48no—
87C2×C4⋊C4—44no—
88C2×C4⋊C4—44no—
89C2×C4⋊C4—44no—
90C2×C4⋊C4C4212yes—
91C2×C4⋊C4C4⋊C412yes—
92C2×C4⋊C4C4⋊C411yescharacteristic
93C2×C4⋊C4C2×Q811yescharacteristic
94C2×C22⋊C4—28no—
95C2×C22⋊C4C22⋊C412yes—
96C2×C22⋊C4C4⋊C412yes—
97C2×C22⋊C4C22×C412yes—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42—22no—
104C2×C42—44no—
105C2×C42C4211yescharacteristic
106C2×C42C22⋊C412yes—
107C2×C42C22⋊C411yescharacteristic
108C2×C42C4⋊C412yes—
109C2×C42C4⋊C412yes—
110C2×C42C4⋊C411yescharacteristic
111C2×C42C22×C411yescharacteristic
112C4⋊C8—42no—
113C4⋊C8—42no—
114C4⋊C8—42no—
115#99—48no—
116#99—44no—
117#99—44no—
118C22⋊C8—42no—
119C22⋊C8—84no—
120C4×C8—42no—
121#95—44no—
122#95—44no—
123C24#9511yescharacteristic
124C24C2×C22⋊C412yes—
125#41—44no—
126#41—44no—
127#41—44no—
128#41—44no—
129C2×Q8—41no—
130C2×Q8—41no—
131C2×Q8#10612yes—
132C2×Q8#10712yes—
133C2×D4—42no—
134C2×D4—41no—
135C2×D4—42no—
136C2×D4—41no—
137C2×D4#10214yes—
138C2×D4#10314yes—
139C2×D4C2×D811yescharacteristic
140C2×D4C2×SD1612yes—
141C2×D4C2×Q1611yescharacteristic
142C22×C4—24no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—42no—
157C22×C4—42no—
158C22×C4—48no—
159C22×C4—82no—
160C22×C4—81no—
161C22×C4#9512yes—
162C22×C4#9511yescharacteristic
163C22×C4#9512yes—
164C22×C4#9511yescharacteristic
165C22×C4#9511yescharacteristic
166C22×C4#10214yes—
167C22×C4#10314yes—
168C22×C4#10612yes—
169C22×C4#10712yes—
170C22×C4C2×C4211yescharacteristic
171C22×C4C2×C22⋊C411yescharacteristic
172C22×C4C2×C4⋊C411yescharacteristic
173C22×C4C2×C4⋊C412yes—
174C2×C8—42no—
175C2×C8—44no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—84no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—41no—
183C4⋊C4—41no—
184C4⋊C4—88no—
185C4⋊C4—88no—
186C4⋊C4—84no—
187C4⋊C4—84no—
188C22⋊C4—42no—
189C22⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—48no—
192C22⋊C4—48no—
193C42—22no—
194C42—22no—
195C42—22no—
196C42—22no—
197C42—22no—
198C42—22no—
199C42—21no—
200C42—21no—
201C42—22no—
202C42—22no—
203C42—88no—
204C42#9912yes—
205C42#9912yes—
206C23—21no—
207C23—22no—
208C23—21no—
209C23—22no—
210C23—41no—
211C23—41no—
212C23—42no—
213C23—42no—
214C23#34012yes—
215C23C2×#9511yescharacteristic
216C23C2×#10212yes—
217C23C2×#10312yes—
218Q8—42no—
219Q8—42no—
220D4—44no—
221D4—44no—
222D4—41no—
223D4—42no—
224D4—41no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—24no—
230C2×C4—24no—
231C2×C4—24no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—44no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—41no—
252C2×C4—82no—
253C2×C4—82no—
254C2×C4—82no—
255C2×C4—88no—
256C2×C4—88no—
257C2×C4—88no—
258C2×C4H(3,ℤ/4)12yes—
259C2×C4#34014yes—
260C2×C4#34012yes—
261C2×C4#34212yes—
262C2×C4#34212yes—
263C2×C4C2×#9911yescharacteristic
264C2×C4C2×#9911yescharacteristic
265C2×C4C2×#10611yescharacteristic
266C2×C4C2×#10711yescharacteristic
267C8—82no—
268C8—82no—
269C22—21no—
270C22—21no—
271C22—22no—
272C22—22no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—41no—
280C22—41no—
281C22—42no—
282C22—82no—
283C22C2×H(3,ℤ/4)11yescharacteristic
284C22C2×#34011yescharacteristic
285C22C2×#34211yescharacteristic
286C4—22no—
287C4—21no—
288C4—24no—
289C4—21no—
290C4—21no—
291C4—42no—
292C4—41no—
293C4—41no—
294C4—42no—
295C4—168no—
296C4#128914yes—
297C4#212811yescharacteristic
298C4#212911yescharacteristic
299C2—21no—
300C2—21no—
301C2—42no—
302C2—42no—
303C2—42no—
304C2C2×#128911yescharacteristic
305C2#1257711yescharacteristic
306C2#1258311yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #7237C2semidirect
5. C2×#431C4semidirect
6. C22×C4⋊C8C4semidirect
9. #1984C4semidirect
14. #1799C4semidirect
17. #1740C4semidirect

Maximal subgroups

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

subgroupindex
#218472
#206322
#72372