Finite Group Databasebeta

← #298630

Subgroups — #298630

Order 512.

1522 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1522 conjugacy classes of subgroups — too many to draw. Only its 222 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
1C2×#3147C211yesmaximal, characteristic, complemented
2#33171C211yesmaximal, characteristic, complemented
3#25599C214yesmaximal, complemented
4C2×#1712C211yesmaximal, characteristic, complemented
5C2×#1709C211yesmaximal, characteristic, complemented
6#10305C214yesmaximal, complemented
7#9825C211yesmaximal, characteristic, complemented
8#9824C211yesmaximal, characteristic, complemented
9C8×C22⋊C8C211yesmaximal, characteristic, complemented
10C22×#135—22no—
11C2×#532C2212yes—
12#3147—24no—
13#3147—24no—
14C2×#495—22no—
15C2×#494C2211yescharacteristic
16C2×#493C2211yescharacteristic
17C2×#493C2214yes—
18C2×#493C2211yescharacteristic
19C2×#492C2212yes—
20C2×#486C2214yes—
21C2×#466—22no—
22C22×C4×C8C2211yescharacteristic
23#2856C2212yes—
24#1999—28no—
25#1990C2214yes—
26#1970—22no—
27#1957C2212yes—
28#1957C2214yes—
29#1879—28no—
30#1845C2211yescharacteristic
31C4×C22⋊C8C2212yes—
32#1712—24no—
33#1709—24no—
34C2×C82C2211yescharacteristic
35C2×C82C2211yescharacteristic
36#1273C2212yes—
37C22×#41—22no—
38C2×#135—216no—
39C2×#135—48no—
40C2×#135—48no—
41C22×Q16—22no—
42C22×SD16—24no—
43C22×D8—22no—
44C22×D8—22no—
45C22×D8—28no—
46C23×C8D412yes—
47#532—24no—
48#532—24no—
49#532—44no—
50C2×C4⋊Q8C2312yes—
51C2×#127C2312yes—
52C2×#127C2312yes—
53C2×#124—22no—
54C2×#121—22no—
55C2×#121—24no—
56C2×#121—24no—
57C2×C4×D4—24no—
58C22×C42C2311yescharacteristic
59#495—216no—
60#494—22no—
61#494D414yescomplemented
62#493—28no—
63#493—22no—
64#493—22no—
65#493—22no—
66#493—44no—
67#493D414yescomplemented
68#492—24no—
69#492—24no—
70#486—28no—
71#486—44no—
72#466—48no—
73C2×#107—22no—
74C2×#102—28no—
75C2×#102—24no—
76C2×#102—28no—
77C2×C22⋊C8D414yes—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80C2×C4×C8D412yes—
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2312yes—
83C2×C4×C8C2312yes—
84C2×C4×C8C2311yescharacteristic
85#390—24no—
86#390—22no—
87#384C2312yes—
88#340—28no—
89#336D414yes—
90C82—22no—
91C82—22no—
92C2×#41—28no—
93C2×#41—44no—
94C2×#41—48no—
95C2×#41—48no—
96C22×Q8—22no—
97C22×D4—24no—
98C22×D4—22no—
99C22×D4—22no—
100C22×D4—24no—
101C22×D4—42no—
102C22×D4—42no—
103C23×C4—42no—
104C23×C4C2×D412yes—
105C23×C4C2×D411yescharacteristic
106#135—464no—
107C2×Q16—28no—
108C2×Q16—44no—
109C2×Q16—44no—
110C2×SD16—216no—
111C2×SD16—48no—
112C2×SD16—48no—
113C2×D8—28no—
114C2×D8—416no—
115C2×D8—44no—
116C2×D8—44no—
117C2×D8—44no—
118C2×D8—416no—
119C2×D8—44no—
120C2×D8—44no—
121C2×D8—88no—
122C22×C8—24no—
123C22×C8—24no—
124C22×C8D818yescomplemented
125C22×C8C2×D414yes—
126C22×C8C2×D412yes—
127C22×C8C2×D414yes—
128C22×C8C2×D412yes—
129C4⋊Q8—22no—
130C4⋊Q8—24no—
131C4⋊Q8C2×D414yes—
132#127—22no—
133#127—22no—
134#127—22no—
135#127—24no—
136#127—24no—
137#127C2×D414yes—
138#124—28no—
139#124—44no—
140#121—48no—
141#121—48no—
142#121—48no—
143#121—88no—
144C4×D4—48no—
145C4×D4—48no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C22⋊C4—44no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—42no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—22no—
155C2×C42C2×D412yes—
156C2×C42C2×D411yescharacteristic
157C2×C42C2411yescharacteristic
158#107—44no—
159#102—416no—
160#102—48no—
161#102—88no—
162C22⋊C8—28no—
163C4×C8—24no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—21no—
167C4×C8—21no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8C2×D412yes—
171C4×C8C2×D412yes—
172C24—42no—
173C24—42no—
174C24#12711yescharacteristic
175#41—432no—
176#41—816no—
177C2×Q8—24no—
178C2×Q8—42no—
179C2×Q8—44no—
180C2×Q8—44no—
181C2×D4—24no—
182C2×D4—28no—
183C2×D4—48no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—44no—
188C2×D4—44no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—44no—
192C2×D4—48no—
193C2×D4—48no—
194C2×D4—44no—
195C2×D4—44no—
196C2×D4—48no—
197C2×D4—48no—
198C2×D4—84no—
199C2×D4—84no—
200C2×D4—88no—
201C2×D4—84no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—48no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—84no—
213C22×C4—84no—
214C22×C4#12712yes—
215C22×C4#12711yescharacteristic
216C22×C4C2×D814yes—
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D412yes—
219Q16—416no—
220SD16—432no—
221D8—416no—
222D8—832no—
223D8—88no—
224C2×C8—22no—
225C2×C8—24no—
226C2×C8—22no—
227C2×C8—24no—
228C2×C8—216no—
229C2×C8—24no—
230C2×C8—24no—
231C2×C8—24no—
232C2×C8#12014yes—
233C2×C8C2×D814yes—
234C2×C8C2×D814yes—
235C2×C8#13518yescomplemented
236C2×C8AGL(1,ℤ/8)14yes—
237C4⋊C4—28no—
238C4⋊C4—44no—
239C4⋊C4—44no—
240C4⋊C4—48no—
241C22⋊C4—416no—
242C22⋊C4—88no—
243C22⋊C4—84no—
244C42—21no—
245C42—21no—
246C42—22no—
247C42—22no—
248C42C22×D411yescharacteristic
249C42C22×D411yescharacteristic
250C42H(5,𝔽2)12yes—
251C23—22no—
252C23—21no—
253C23—22no—
254C23—21no—
255C23—44no—
256C23—82no—
257C23—84no—
258C23—82no—
259C23—84no—
260C23—82no—
261C23—84no—
262C23—82no—
263C23—84no—
264C23—82no—
265C23#49312yes—
266C23C2×#12711yescharacteristic
267Q8—48no—
268Q8—84no—
269D4—48no—
270D4—416no—
271D4—816no—
272D4—84no—
273D4—84no—
274D4—84no—
275D4—88no—
276D4—88no—
277D4—88no—
278D4—168no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—816no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—84no—
294C2×C4#49312yes—
295C2×C4C2×#12011yescharacteristic
296C2×C4#53511yescharacteristic
297C2×C4D4212yes—
298C2×C4D4212yes—
299C2×C4C22×D812yes—
300C2×C4C2×#13514yes—
301C2×C4C2×AGL(1,ℤ/8)12yes—
302C2×C4#57614yes—
303C8—24no—
304C8—28no—
305C8—44no—
306C8#44718yes—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—42no—
311C22—42no—
312C22—84no—
313C22—88no—
314C22—162no—
315C22—164no—
316C22—162no—
317C22—162no—
318C22—162no—
319C22—164no—
320C22—164no—
321C22—162no—
322C22#243311yescharacteristic
323C22C2×#49311yescharacteristic
324C22#314712yes—
325C22#315111yescharacteristic
326C22#315512yes—
327C4—22no—
328C4—42no—
329C4—84no—
330C4—162no—
331C4C2×#44712yes—
332C4#320512yes—
333C4#328014yes—
334C4#328714yes—
335C2—42no—
336C2—164no—
337C2—322no—
338C2#1960311yescharacteristic
339C2#1964712yes—
340C2#1966111yescharacteristic
341C2#2572111yescharacteristic
342C2#2577712yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3147C2semidirect
2. #33171C2semidirect
3. #25599C2semidirect
4. C2×#1712C2semidirect
5. C2×#1709C2semidirect
6. #10305C2semidirect
7. #9825C2semidirect
8. #9824C2semidirect
9. C8×C22⋊C8C2semidirect
61. #494D4semidirect
67. #493D4semidirect
124. C22×C8D8semidirect
235. C2×C8#135semidirect

Maximal subgroups

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

subgroupindex
C2×#31472
#331712
#255992
C2×#17122
C2×#17092
#103052
#98252
#98242
C8×C22⋊C82