Finite Group Databasebeta

← #134809

Subgroups — #134809

Order 512.

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

normal subgroups only

This group has 417 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#22028C211yesmaximal, characteristic
2#11525C212yesmaximal
3#8622C211yesmaximal, characteristic, complemented
4#8612C211yesmaximal, characteristic, complemented
5#7247C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7C22×C22⋊C8C411yescharacteristic
8C4×#120C2211yescharacteristic
9#1954—21no—
10#1923—21no—
11#1923—21no—
12#1869—21no—
13#1869C2211yescharacteristic
14#1850C411yescharacteristic
15#1810—21no—
16#1779C411yescharacteristic
17C8×C22⋊C4C411yescharacteristic
18#1568—21no—
19#1466C2212yes—
20#1465C2212yes—
21#1326—22no—
22#1316—22no—
23#1293—21no—
24#1290—21no—
25C24×C4C2×C411yescharacteristic
26C23×C8C2×C411yescharacteristic
27C23×C8C2×C411yescharacteristic
28C23×C8D411yescharacteristic
29C2×#120Q811yescharacteristic
30C2×C4×D4—21no—
31C2×C4×D4C2311yescharacteristic
32C8×D4—41no—
33C8×D4—41no—
34#432—22no—
35C2×#108—41no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8D411yescharacteristic
38#407—22no—
39C2×C22⋊C8—21no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—41no—
45C2×C22⋊C8—41no—
46C2×C22⋊C8C2×C411yescharacteristic
47C2×C22⋊C8C2×C411yescharacteristic
48C2×C22⋊C8D411yescharacteristic
49C2×C4×C8—21no—
50C2×C4×C8D411yescharacteristic
51#386—21no—
52#386Q811yescharacteristic
53#385—21no—
54#385D411yescharacteristic
55#379D411yescharacteristic
56C4×C22⋊C4—21no—
57C4×C22⋊C4C2×C411yescharacteristic
58#341—41no—
59#336—21no—
60#336—21no—
61#336—21no—
62#335—41no—
63#334—41no—
64#324—22no—
65#323—22no—
66#323—22no—
67#323—22no—
68C25C4⋊C411yescharacteristic
69C22×D4—21no—
70C22×D4C2×Q811yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—41no—
74C23×C4—41no—
75C23×C4—81no—
76C23×C4—81no—
77C23×C4C4⋊C411yescharacteristic
78C23×C4C22×C411yescharacteristic
79C23×C4C2×D411yescharacteristic
80C2×M16—42no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—41no—
88C22×C8—81no—
89C22×C8—81no—
90C22×C8—81no—
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C22×C8C22⋊C411yescharacteristic
95#120—22no—
96#120—22no—
97C4×D4—41no—
98C4×D4—41no—
99C4×D4—81no—
100C4×D4—81no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4#4111yescharacteristic
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4C4⋊C411yescharacteristic
108C2×C22⋊C4C4⋊C411yescharacteristic
109C2×C22⋊C4C2×D411yescharacteristic
110C2×C22⋊C4#4111yescharacteristic
111C2×C42—21no—
112C2×C42C2×D411yescharacteristic
113#108—82no—
114C4⋊C8—41no—
115C4⋊C8—42no—
116C4⋊C8—81no—
117C22⋊C8—41no—
118C22⋊C8—41no—
119C22⋊C8—42no—
120C22⋊C8—41no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126C22⋊C8—81no—
127C22⋊C8—81no—
128C22⋊C8—81no—
129#97—42no—
130C4×C8—21no—
131C4×C8—21no—
132C4×C8—81no—
133#95—21no—
134#95—21no—
135#95—21no—
136C24—21no—
137C24—21no—
138C24—41no—
139C24—41no—
140C24—81no—
141C24—81no—
142C24#10812yescomplemented
143C24C2×C4⋊C411yescharacteristic
144C24C22⋊Q811yescharacteristic
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—24no—
148C2×D4—41no—
149C2×D4—41no—
150C2×D4—81no—
151C2×D4—81no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—42no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—84no—
175C22×C4—82no—
176C22×C4—82no—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4#11711yescharacteristic
179C22×C4#12111yescharacteristic
180C22×C4#12111yescharacteristic
181C22×C4C22⋊Q811yescharacteristic
182M16—82no—
183M16—82no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—41no—
188C2×C8—42no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—81no—
199C2×C8—81no—
200C2×C8—81no—
201C2×C8—82no—
202C2×C8—82no—
203C2×C8—82no—
204C2×C8#9911yescharacteristic
205C2×C8#9911yescharacteristic
206C4⋊C4—41no—
207C4⋊C4—81no—
208C22⋊C4—24no—
209C22⋊C4—21no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—41no—
213C22⋊C4—81no—
214C22⋊C4—81no—
215C22⋊C4—81no—
216C22⋊C4#10812yescomplemented
217C42—21no—
218C42—21no—
219C42—41no—
220C42—41no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23—42no—
235C23—41no—
236C23—42no—
237C23—41no—
238C23—41no—
239C23—82no—
240C23—82no—
241C23—82no—
242C23—82no—
243C23—84no—
244C23#38011yescharacteristic
245C23C2×#10811yescharacteristic
246C23C2×#10811yescharacteristic
247D4—42no—
248D4—42no—
249D4—84no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—22no—
258C2×C4—21no—
259C2×C4—42no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—84no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4C2×#9911yescharacteristic
283C2×C4#41011yescharacteristic
284C8—21no—
285C8—21no—
286C8—41no—
287C8—41no—
288C8—41no—
289C8—81no—
290C8—81no—
291C8—162no—
292C8—162no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—41no—
298C22—41no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—42no—
308C22—41no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—84no—
313C22—84no—
314C22—81no—
315C22—81no—
316C22#179911yescharacteristic
317C22#181011yescharacteristic
318C22#181311yescharacteristic
319C4—21no—
320C4—42no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—81no—
326C4—81no—
327C4#212111yescharacteristic
328C4#212211yescharacteristic
329C2—21no—
330C2—21no—
331C2—41no—
332C2—41no—
333C2—81no—
334C2—82no—
335C2#861211yescharacteristic
336C2#1279511yescharacteristic
337C2#1280611yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC2×#120Q8
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8622C2semidirect
4. #8612C2semidirect
5. #7247C2semidirect
142. C24#108semidirect
216. C22⋊C4#108semidirect

Maximal subgroups

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

subgroupindex
#220282
#115252
#86222
#86122
#72472