Finite Group Databasebeta

← #134337

Subgroups — #134337

Order 512.

400 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 86 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#21051C211yesmaximal, characteristic
2#11431C212yesmaximal
3#8602C211yesmaximal, characteristic, complemented
4#8601C211yesmaximal, characteristic, complemented
5#7188C212yesmaximal, complemented
6C2×#423C411yescharacteristic
7C2×#408C411yescharacteristic
8C2×#408C411yescharacteristic
9C2×#405C2211yescharacteristic
10#2297C2211yescharacteristic
11#1924—21no—
12#1924—21no—
13#1923—21no—
14#1923—21no—
15#1729C2211yescharacteristic
16C2×#336C411yescharacteristic
17#1564—21no—
18#1564—21no—
19#1458C2212yes—
20#1458C2212yes—
21#1319—22no—
22#1318—22no—
23#1275—21no—
24#1275—21no—
25C22×#41C2×C411yescharacteristic
26C22×M16C2×C411yescharacteristic
27C22×M16C2×C411yescharacteristic
28C23×C8C2×C411yescharacteristic
29C23×C8C2×C411yescharacteristic
30C2×#117C2311yescharacteristic
31C22×C42C2×C411yescharacteristic
32#431—22no—
33#431—22no—
34#423—42no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37#408—41no—
38#408—41no—
39#408—42no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44#405—41no—
45#405—41no—
46C2×#97D411yescharacteristic
47C2×#97D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50#391D411yescharacteristic
51#391Q811yescharacteristic
52#390D411yescharacteristic
53#390Q811yescharacteristic
54#386—21no—
55#386—21no—
56C4×C22⋊C4—21no—
57C4×C22⋊C4—21no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62#335—41no—
63#335—41no—
64#334—41no—
65#334—41no—
66#324—22no—
67#324—22no—
68#323—22no—
69#323—22no—
70C2×#41—22no—
71C2×#41—22no—
72C2×#41—41no—
73C2×#41—41no—
74C2×#41—41no—
75C2×#41—41no—
76C22×Q8C4⋊C411yescharacteristic
77C22×D4—21no—
78C22×D4C4⋊C411yescharacteristic
79C23×C4—21no—
80C23×C4—21no—
81C23×C4C22×C411yescharacteristic
82C2×M16—22no—
83C2×M16—22no—
84C2×M16—22no—
85C2×M16—41no—
86C2×M16—41no—
87C2×M16—41no—
88C22×C8—22no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—22no—
94C22×C8—41no—
95C22×C8—41no—
96C22×C8—41no—
97C22×C8C22⋊C411yescharacteristic
98C22×C8C22⋊C411yescharacteristic
99C22×C8C22⋊C411yescharacteristic
100C22×C8C22⋊C411yescharacteristic
101#117—41no—
102#117—41no—
103C2×C4⋊C4C2×D411yescharacteristic
104C2×C4⋊C4C2×Q811yescharacteristic
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4#4111yescharacteristic
110C2×C22⋊C4#4111yescharacteristic
111C2×C42—21no—
112C2×C42—41no—
113C2×C42—41no—
114C2×C42C4⋊C411yescharacteristic
115C2×C42C4⋊C411yescharacteristic
116C2×C42C2×D411yescharacteristic
117C2×C42C2×D411yescharacteristic
118C4⋊C8—42no—
119C4⋊C8—42no—
120C4⋊C8—42no—
121C4⋊C8—42no—
122C22⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—41no—
125C22⋊C8—41no—
126C22⋊C8—42no—
127C22⋊C8—42no—
128C22⋊C8—42no—
129#97—41no—
130#97—41no—
131C4×C8—21no—
132C4×C8—21no—
133C4×C8—21no—
134C4×C8—21no—
135C4×C8—42no—
136C4×C8—42no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141C24—21no—
142C24#11711yescharacteristic
143#41—82no—
144#41—82no—
145#41—82no—
146#41—82no—
147C2×Q8—22no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×D4—22no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—42no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—81no—
172C22×C4C2×C22⋊C411yescharacteristic
173C22×C4C2×C4⋊C411yescharacteristic
174C22×C4#12111yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4C22⋊Q811yescharacteristic
177C22×C4C22⋊Q811yescharacteristic
178C22×C4#13112yes—
179C22×C4#13112yes—
180M16—44no—
181M16—82no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—82no—
201C2×C8—82no—
202C2×C8—82no—
203C2×C8#9911yescharacteristic
204C2×C8#9911yescharacteristic
205C2×C8#10012yes—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—41no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C42—21no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—41no—
219C42—41no—
220C42—41no—
221C42—41no—
222C23—21no—
223C23—21no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23#38011yescharacteristic
234C23#43312yes—
235Q8—82no—
236D4—82no—
237D4—84no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—84no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4#40811yescharacteristic
267C2×C4#40811yescharacteristic
268C2×C4C2×#9911yescharacteristic
269C2×C4#41011yescharacteristic
270C2×C4C2×#10011yescharacteristic
271C2×C4#41311yescharacteristic
272C2×C4#42412yes—
273C8—21no—
274C8—21no—
275C8—22no—
276C8—41no—
277C8—41no—
278C8—81no—
279C8—81no—
280C8—162no—
281C8—162no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—81no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22—81no—
296C22#179811yescharacteristic
297C22#179911yescharacteristic
298C22#180011yescharacteristic
299C4—22no—
300C4—42no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—41no—
305C4—81no—
306C4—81no—
307C4#146811yescharacteristic
308C4#146911yescharacteristic
309C4#147111yescharacteristic
310C4#147311yescharacteristic
311C2—21no—
312C2—21no—
313C2—42no—
314C2—161no—
315C2#856111yescharacteristic
316C2#1280811yescharacteristic
317C2#1280911yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC23×C4C22×C4
Frattini, cosocleC2×#117C23
socleC22#1799
prime radicalC22×#41C2×C4
LCSG ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8602C2semidirect
4. #8601C2semidirect
5. #7188C2semidirect

Maximal subgroups

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

subgroupindex
#210512
#114312
#86022
#86012
#71882