Finite Group Databasebeta

← #134781

Subgroups — #134781

Order 512.

445 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 445 conjugacy classes of subgroups — too many to draw. Only its 80 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#22030C211yesmaximal, characteristic
2#12793C212yesmaximal
3#8616C211yesmaximal, characteristic, complemented
4#8608C211yesmaximal, characteristic, complemented
5#7236C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7C22×C22⋊C8C411yescharacteristic, complemented
8C4×#120C2211yescharacteristic
9C2×AΣL(1,𝔽16)—22no—
10#1955—21no—
11#1924—21no—
12#1923—21no—
13#1869—21no—
14#1869C2211yescharacteristic
15#1850C411yescharacteristic, complemented
16#1807—21no—
17#1800C2212yes—
18#1799C2212yes—
19#1779C411yescharacteristic, complemented
20#1752C411yescharacteristic, complemented
21#1567—21no—
22#1390—22no—
23#1292—21no—
24#1289—21no—
25C24×C4C2×C411yescharacteristic
26C23×C8C2×C411yescharacteristic
27C23×C8C2×C411yescharacteristic
28C23×C8D411yescharacteristic
29C2×#120D411yescharacteristic
30C2×C4×D4—21no—
31C2×C4×D4C2311yescharacteristic
32C8×D4—41no—
33C8×D4—41no—
34C2×#106—41no—
35#423—22no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8D411yescharacteristic
38C2×#100—22no—
39C2×#99—22no—
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—41no—
46C2×C22⋊C8—41no—
47C2×C22⋊C8C2×C411yescharacteristic
48C2×C22⋊C8C2×C411yescharacteristic
49C2×C22⋊C8D411yescharacteristic
50C2×#97—21no—
51C2×C4×C8D411yescharacteristic
52#386—21no—
53#386D411yescharacteristic
54#385—21no—
55#385Q811yescharacteristic
56#380—22no—
57#379Q811yescharacteristic
58C4×C22⋊C4—21no—
59C4×C22⋊C4C2×C411yescharacteristic
60AΣL(1,𝔽16)—44no—
61#343—22no—
62#342—22no—
63#340—41no—
64#336—21no—
65#336—21no—
66#336—21no—
67#335—41no—
68#335—41no—
69C25C4⋊C411yescharacteristic
70C22×D4—21no—
71C22×D4C2×D411yescharacteristic
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—41no—
75C23×C4—41no—
76C23×C4—81no—
77C23×C4—81no—
78C23×C4C4⋊C411yescharacteristic
79C23×C4C22×C411yescharacteristic
80C23×C4C2×D411yescharacteristic
81C2×M16—42no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—41no—
89C22×C8—81no—
90C22×C8—81no—
91C22×C8—81no—
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C22×C8C22⋊C411yescharacteristic
95C22×C8C22⋊C411yescharacteristic
96#120—22no—
97#120—22no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—81no—
101C4×D4—81no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—42no—
104C2×C4⋊C4#4111yescharacteristic
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4C4⋊C411yescharacteristic
111C2×C22⋊C4C4⋊C411yescharacteristic
112C2×C22⋊C4C2×Q811yescharacteristic
113C2×C22⋊C4#4111yescharacteristic
114C2×C42—21no—
115C2×C42C2×D411yescharacteristic
116#106—82no—
117C4⋊C8—41no—
118C4⋊C8—42no—
119C4⋊C8—42no—
120C4⋊C8—81no—
121#100—42no—
122#100—42no—
123#99—42no—
124#99—42no—
125C22⋊C8—41no—
126C22⋊C8—41no—
127C22⋊C8—42no—
128C22⋊C8—41no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131C22⋊C8—81no—
132C22⋊C8—81no—
133C22⋊C8—81no—
134#97—81no—
135C4×C8—21no—
136C4×C8—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—42no—
141C24—21no—
142C24—22no—
143C24—41no—
144C24—42no—
145C24—41no—
146C24—81no—
147C24C2×C4⋊C411yescharacteristic
148C24#12111yescharacteristic
149C2×D4—41no—
150C2×D4—41no—
151C2×D4—42no—
152C2×D4—42no—
153C2×D4—81no—
154C2×D4AGL(1,ℤ/8)12yes—
155C2×D4#13712yes—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—42no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—84no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4C2×C22⋊C411yescharacteristic
183C22×C4#11711yescharacteristic
184C22×C4#12111yescharacteristic
185C22×C4C22⋊Q811yescharacteristic
186C22×C4C22⋊Q811yescharacteristic
187M16—82no—
188M16—82no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—41no—
193C2×C8—42no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—81no—
204C2×C8—81no—
205C2×C8—81no—
206C2×C8—82no—
207C2×C8—82no—
208C2×C8#9911yescharacteristic
209C2×C8#9911yescharacteristic
210C4⋊C4—41no—
211C4⋊C4—81no—
212C4⋊C4—82no—
213C4⋊C4—82no—
214C22⋊C4—22no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—42no—
218C22⋊C4—41no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—81no—
222C22⋊C4—81no—
223C22⋊C4—82no—
224C22⋊C4—82no—
225C42—21no—
226C42—21no—
227C42—41no—
228C42—41no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—22no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—42no—
242C23—42no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—82no—
247C23—84no—
248C23—84no—
249C23—82no—
250C23#38011yescharacteristic
251C23#42812yes—
252C23#46812yes—
253D4—42no—
254D4—42no—
255D4—84no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—42no—
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—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—84no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—82no—
287C2×C4—162no—
288C2×C4—162no—
289C2×C4—162no—
290C2×C4C2×#9911yescharacteristic
291C2×C4#41011yescharacteristic
292C2×C4#46911yescharacteristic
293C2×C4#47011yescharacteristic
294C8—21no—
295C8—21no—
296C8—41no—
297C8—41no—
298C8—41no—
299C8—81no—
300C8—81no—
301C8—162no—
302C22—21no—
303C22—21no—
304C22—22no—
305C22—22no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—84no—
320C22—84no—
321C22—81no—
322C22—81no—
323C22#179911yescharacteristic
324C22#180911yescharacteristic
325C22#181211yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—81no—
333C4—81no—
334C4—81no—
335C4—322no—
336C4#212111yescharacteristic
337C4#212211yescharacteristic
338C2—21no—
339C2—21no—
340C2—42no—
341C2—42no—
342C2—82no—
343C2#861011yescharacteristic
344C2#1279511yescharacteristic
345C2#1280511yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC2×#120D4
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. #8616C2semidirect
4. #8608C2semidirect
5. #7236C2semidirect
7. C22×C22⋊C8C4semidirect
15. #1850C4semidirect
19. #1779C4semidirect
20. #1752C4semidirect

Maximal subgroups

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

subgroupindex
#220302
#127932
#86162
#86082
#72362