Finite Group Databasebeta

← #134753

Subgroups — #134753

Order 512.

452 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 88 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#12793C212yesmaximal
3#8616C211yesmaximal, characteristic, complemented
4#8609C211yesmaximal, characteristic, complemented
5#7236C212yesmaximal, complemented
6C2×C8×D4C2211yescharacteristic
7C22×C22⋊C8C411yescharacteristic, complemented
8C4×#120C2211yescharacteristic
9C2×AΣL(1,𝔽16)—22no—
10#1954—21no—
11#1923—21no—
12#1923—21no—
13#1869—21no—
14#1869C2211yescharacteristic
15#1850C411yescharacteristic, complemented
16#1808—21no—
17#1800C2212yes—
18#1799C2212yes—
19#1779C411yescharacteristic, complemented
20C8×C22⋊C4C411yescharacteristic, 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×#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×C4×C8—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—21no—
97#120—21no—
98#120D812yes—
99#120Q1612yes—
100C4×D4—41no—
101C4×D4—41no—
102C4×D4—81no—
103C4×D4—81no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—42no—
106C2×C4⋊C4#4111yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—42no—
112C2×C22⋊C4C4⋊C411yescharacteristic
113C2×C22⋊C4C4⋊C411yescharacteristic
114C2×C22⋊C4C2×Q811yescharacteristic
115C2×C22⋊C4#4111yescharacteristic
116C2×C42—21no—
117C2×C42C2×D411yescharacteristic
118#107—82no—
119C4⋊C8—41no—
120C4⋊C8—42no—
121C4⋊C8—42no—
122C4⋊C8—81no—
123#100—42no—
124#100—42no—
125#99—42no—
126#99—42no—
127C22⋊C8—41no—
128C22⋊C8—41no—
129C22⋊C8—42no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—81no—
134C22⋊C8—81no—
135C22⋊C8—81no—
136C4×C8—21no—
137C4×C8—21no—
138C4×C8—81no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—42no—
143C24—21no—
144C24—21no—
145C24—41no—
146C24—41no—
147C24—81no—
148C24—81no—
149C24#10712yescomplemented
150C24C2×C4⋊C411yescharacteristic
151C24#12111yescharacteristic
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—81no—
157C2×D4—81no—
158C2×D4C2×D811yescharacteristic
159C2×D4C2×Q1611yescharacteristic
160C2×D4#13511yescharacteristic
161C2×D4#13511yescharacteristic
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—42no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4—84no—
185C22×C4—82no—
186C22×C4—82no—
187C22×C4—82no—
188C22×C4C2×C22⋊C411yescharacteristic
189C22×C4#11711yescharacteristic
190C22×C4#12111yescharacteristic
191C22×C4C22⋊Q811yescharacteristic
192C22×C4C22⋊Q811yescharacteristic
193M16—82no—
194M16—82no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—41no—
199C2×C8—42no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—41no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—81no—
210C2×C8—81no—
211C2×C8—81no—
212C2×C8—82no—
213C2×C8—82no—
214C2×C8#9911yescharacteristic
215C2×C8#9911yescharacteristic
216C4⋊C4—41no—
217C4⋊C4—81no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C22⋊C4—21no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—81no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C22⋊C4#10712yescomplemented
232C42—21no—
233C42—21no—
234C42—41no—
235C42—41no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—22no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—42no—
249C23—41no—
250C23—42no—
251C23—41no—
252C23—41no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23—82no—
257C23—84no—
258C23#38011yescharacteristic
259C23C2×#10711yescharacteristic
260C23#42711yescharacteristic
261C23#46611yescharacteristic
262C23#46711yescharacteristic
263D4—41no—
264D4—41no—
265D4—41no—
266D4—41no—
267D4—82no—
268D4—82no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—84no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—162no—
301C2×C4—162no—
302C2×C4—162no—
303C2×C4C2×#9911yescharacteristic
304C2×C4#41011yescharacteristic
305C2×C4#46611yescharacteristic
306C2×C4#46711yescharacteristic
307C8—21no—
308C8—21no—
309C8—41no—
310C8—41no—
311C8—41no—
312C8—81no—
313C8—81no—
314C8—162no—
315C22—21no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—42no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—84no—
337C22—84no—
338C22—81no—
339C22—81no—
340C22#179911yescharacteristic
341C22#180811yescharacteristic
342C22#181211yescharacteristic
343C4—21no—
344C4—21no—
345C4—21no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—322no—
353C4#212111yescharacteristic
354C4#212211yescharacteristic
355C2—21no—
356C2—21no—
357C2—41no—
358C2—41no—
359C2—41no—
360C2—41no—
361C2—82no—
362C2#860911yescharacteristic
363C2#1279511yescharacteristic
364C2#1280611yescharacteristic
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. #8609C2semidirect
5. #7236C2semidirect
7. C22×C22⋊C8C4semidirect
15. #1850C4semidirect
19. #1779C4semidirect
20. C8×C22⋊C4C4semidirect
149. C24#107semidirect
231. C22⋊C4#107semidirect

Maximal subgroups

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

subgroupindex
#220282
#127932
#86162
#86092
#72362