Finite Group Databasebeta

← #135331

Subgroups — #135331

Order 512.

744 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 100 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#13852C211yesmaximal, characteristic
2#10726C211yesmaximal, characteristic
3#8300C211yesmaximal, characteristic
4#7250C214yesmaximal
5C22×#117C2211yescharacteristic
6C2×#381—21no—
7C2×C4×C4⋊C4C411yescharacteristic, complemented
8C2×C4×C4⋊C4C411yescharacteristic, complemented
9C2×C4×C22⋊C4—21no—
10#1817—21no—
11C2×#342—21no—
12C2×#342C2212yes—
13C2×#324—21no—
14C2×#323C2212yes—
15#1441—21no—
16#1441C412yescomplemented
17#1439—21no—
18#1439C2212yes—
19C4×#95—21no—
20C24×C4D411yescharacteristic
21C2×#117—24no—
22C2×#117—42no—
23C2×#117—42no—
24C22×C4⋊C4—21no—
25C22×C4⋊C4C2×C411yescharacteristic
26C22×C4⋊C4C2×C412yes—
27C22×C4⋊C4D411yescharacteristic
28C22×C22⋊C4—22no—
29C22×C22⋊C4—22no—
30C22×C22⋊C4D411yescharacteristic
31C22×C22⋊C4C2311yescharacteristic
32C22×C42—21no—
33C22×C42—21no—
34C22×C42C2×C411yescharacteristic
35C22×C42D411yescharacteristic
36C2×C22⋊C8—22no—
37#381—24no—
38#381—42no—
39C4×C4⋊C4—42no—
40C4×C4⋊C4—42no—
41C4×C22⋊C4—44no—
42C2×#95—22no—
43C2×#95—21no—
44C2×#95—22no—
45C2×#95—22no—
46C2×#95—42no—
47C2×#95C2×C412yes—
48#342—24no—
49#342—42no—
50#336—42no—
51#324—42no—
52#323—24no—
53C25C2×D411yescharacteristic
54C23×C4—22no—
55C23×C4—22no—
56C23×C4—22no—
57C23×C4—22no—
58C23×C4—22no—
59C23×C4—22no—
60C23×C4—41no—
61C23×C4—41no—
62C23×C4—42no—
63C23×C4C22⋊C411yescharacteristic
64C23×C4C22⋊C411yescharacteristic
65C23×C4C22⋊C411yescharacteristic
66C23×C4C22⋊C411yescharacteristic
67C23×C4SD1612yes—
68C23×C4Q1612yes—
69C23×C4C22×C411yescharacteristic
70C23×C4C2×D411yescharacteristic
71C22×C8—42no—
72#117—416no—
73C2×C4⋊C4—22no—
74C2×C4⋊C4—22no—
75C2×C4⋊C4—22no—
76C2×C4⋊C4—41no—
77C2×C4⋊C4—41no—
78C2×C4⋊C4—41no—
79C2×C4⋊C4—42no—
80C2×C4⋊C4—44no—
81C2×C4⋊C4—42no—
82C2×C4⋊C4—44no—
83C2×C22⋊C4—22no—
84C2×C22⋊C4—22no—
85C2×C22⋊C4—24no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—44no—
90C2×C22⋊C4—44no—
91C2×C22⋊C4—44no—
92C2×C22⋊C4—42no—
93C2×C22⋊C4—42no—
94C2×C22⋊C4—42no—
95C2×C22⋊C4—42no—
96C2×C22⋊C4—42no—
97C2×C22⋊C4#4114yes—
98C2×C42—22no—
99C2×C42—22no—
100C2×C42—22no—
101C2×C42—22no—
102C2×C42—22no—
103C2×C42—41no—
104C2×C42—42no—
105C2×C42—42no—
106C2×C42—42no—
107C2×C42—42no—
108C2×C42—42no—
109C2×C42—42no—
110C22⋊C8—42no—
111C22⋊C8—42no—
112#95—42no—
113#95—44no—
114#95—44no—
115#95—44no—
116#95—42no—
117#95—48no—
118C24—21no—
119C24—21no—
120C24—21no—
121C24—21no—
122C24—42no—
123C24—42no—
124C24C2×C22⋊C411yescharacteristic
125C24#12312yes—
126C24C2×SD1611yescharacteristic
127C24C2×Q1611yescharacteristic
128C24AGL(1,ℤ/8)12yes—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—21no—
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—21no—
143C22×C4—21no—
144C22×C4—24no—
145C22×C4—24no—
146C22×C4—21no—
147C22×C4—24no—
148C22×C4—24no—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—42no—
155C22×C4—42no—
156C22×C4—44no—
157C22×C4—44no—
158C22×C4—44no—
159C22×C4—44no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—42no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—44no—
177C22×C4—82no—
178C22×C4—82no—
179C22×C4—84no—
180C22×C4—84no—
181C22×C4#9912yes—
182C22×C4#9912yes—
183C22×C4#10312yes—
184C22×C4#10312yes—
185C22×C4#10412yes—
186C22×C4#11712yes—
187C22×C4#12312yes—
188C2×C8—82no—
189C2×C8—84no—
190C4⋊C4—44no—
191C4⋊C4—44no—
192C4⋊C4—44no—
193C4⋊C4—44no—
194C4⋊C4—48no—
195C22⋊C4—44no—
196C22⋊C4—44no—
197C22⋊C4—416no—
198C22⋊C4—44no—
199C22⋊C4—44no—
200C22⋊C4—82no—
201C22⋊C4—82no—
202C42—28no—
203C42—44no—
204C42—84no—
205C42—84no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—21no—
210C23—21no—
211C23—22no—
212C23—22no—
213C23—22no—
214C23—21no—
215C23—21no—
216C23—44no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—82no—
224C23—81no—
225C23—81no—
226C23—82no—
227C23#38111yescharacteristic
228C23C2×#9911yescharacteristic
229C23C2×#9911yescharacteristic
230C23C2×#10311yescharacteristic
231C23#41811yescharacteristic
232C23C2×#10411yescharacteristic
233C23#42111yescharacteristic
234C23#48112yes—
235C23#48412yes—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—48no—
249C2×C4—48no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—44no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—81no—
281C2×C4—84no—
282C2×C4—88no—
283C2×C4—164no—
284C2×C4—162no—
285C2×C4—162no—
286C2×C4#32814yes—
287C8—162no—
288C22—22no—
289C22—22no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22—22no—
298C22—42no—
299C22—42no—
300C22—44no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22—82no—
307C22—81no—
308C22—81no—
309C22C2×#32811yescharacteristic
310C22#150711yescharacteristic
311C22#151011yescharacteristic
312C22#152011yescharacteristic
313C22#181911yescharacteristic
314C22#182411yescharacteristic
315C22#182611yescharacteristic
316C22#212012yes—
317C22#212612yes—
318C4—24no—
319C4—42no—
320C4—42no—
321C4—42no—
322C4—42no—
323C4—84no—
324C4—84no—
325C4—81no—
326C4—164no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—42no—
332C2—42no—
333C2#871311yescharacteristic
334C2#1171512yes—
335C2#1179812yes—
336C2#1285311yescharacteristic
337C2#1285711yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC23×C4C22×C4
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×C4×C4⋊C4C4semidirect
8. C2×C4×C4⋊C4C4semidirect
16. #1441C4semidirect

Maximal subgroups

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

subgroupindex
#138522
#107262
#83002
#72502