Finite Group Databasebeta

← #134785

Subgroups — #134785

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#8614C211yesmaximal, characteristic, complemented
4#8609C211yesmaximal, 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#1808—21no—
17#1800C2212yes—
18#1799C2212yes—
19#1779C411yescharacteristic, complemented
20#1752C411yescharacteristic, complemented
21#1566—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×#107—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#334—41no—
68#334—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#107—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—41no—
153C2×D4—41no—
154C2×D4—81no—
155C2×D4AGL(1,ℤ/8)12yes—
156C2×D4#13712yes—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4—84no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4—82no—
183C22×C4C2×C22⋊C411yescharacteristic
184C22×C4#11711yescharacteristic
185C22×C4#12111yescharacteristic
186C22×C4C22⋊Q811yescharacteristic
187C22×C4C22⋊Q811yescharacteristic
188M16—82no—
189M16—82no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—41no—
194C2×C8—42no—
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—41no—
204C2×C8—81no—
205C2×C8—81no—
206C2×C8—81no—
207C2×C8—82no—
208C2×C8—82no—
209C2×C8#9911yescharacteristic
210C2×C8#9911yescharacteristic
211C4⋊C4—41no—
212C4⋊C4—81no—
213C4⋊C4—82no—
214C4⋊C4—82no—
215C22⋊C4—22no—
216C22⋊C4—41no—
217C22⋊C4—41no—
218C22⋊C4—42no—
219C22⋊C4—42no—
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#46911yescharacteristic
253C23#47011yescharacteristic
254D4—42no—
255D4—42no—
256D4—82no—
257D4—82no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—42no—
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—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—84no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—162no—
290C2×C4—162no—
291C2×C4—162no—
292C2×C4C2×#9911yescharacteristic
293C2×C4#41011yescharacteristic
294C2×C4#46812yes—
295C8—21no—
296C8—21no—
297C8—41no—
298C8—41no—
299C8—41no—
300C8—81no—
301C8—81no—
302C8—162no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—41no—
309C22—41no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—42no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—84no—
323C22—84no—
324C22—81no—
325C22—81no—
326C22#179911yescharacteristic
327C22#180911yescharacteristic
328C22#181211yescharacteristic
329C4—22no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—322no—
338C4#212111yescharacteristic
339C4#212211yescharacteristic
340C2—21no—
341C2—21no—
342C2—41no—
343C2—42no—
344C2—41no—
345C2—82no—
346C2#861011yescharacteristic
347C2#1279511yescharacteristic
348C2#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. #8614C2semidirect
4. #8609C2semidirect
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
#86142
#86092
#72362