Finite Group Databasebeta

← #6434726

Subgroups — #6434726

Order 512.

3168 conjugacy classes of subgroups; only the 610 normal subgroups are shown.

normal subgroups only

This group has 3168 conjugacy classes of subgroups — too many to draw. Only its 610 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#62843C211yesmaximal, characteristic, complemented
2#35351C213yesmaximal
3#29659C214yesmaximal, complemented
4#29658C214yesmaximal, complemented
5#29657C214yesmaximal, complemented
6#29656C214yesmaximal, complemented
7C2×#2062C214yesmaximal, complemented
8#23468C214yesmaximal, complemented
9#20689C213yesmaximal, complemented
10#3501—212no—
11#3501—212no—
12#3458C2214yes—
13C2×#543C2214yes—
14C2×#543C2214yes—
15C22×#121C2211yescharacteristic
16C2×#518C2213yes—
17#2484C2216yes—
18#2483C2216yes—
19#2482C2216yes—
20#2444—212no—
21#2443—212no—
22#2405C2216yes—
23#2308C2216yes—
24#2308C2213yes—
25C2×#385C2216yes—
26#2062C22116yes—
27#2062C22116yes—
28#2030C22116yes—
29#2029C22116yes—
30#1820C22112yes—
31#1819C22112yes—
32C2×#342C22112yes—
33C23×D4C2311yescharacteristic
34C23×D4C2311yescharacteristic
35C23×D4C2311yescharacteristic
36C24×C4C2311yescharacteristic
37#554—248no—
38#554—248no—
39D4×Q8—216no—
40D4×Q8—216no—
41#544—212no—
42#544—212no—
43#543—224no—
44#543—224no—
45#543C23116yes—
46#533—224no—
47C2×#123—26no—
48C2×C22⋊Q8—212no—
49C2×C22⋊Q8—212no—
50C2×C22⋊Q8—212no—
51C2×C22⋊Q8—212no—
52C2×#121—212no—
53#518—224no—
54C2×C4×D4D4112yes—
55C2×C4×D4D4112yes—
56C2×#117C2316yes—
57C22×C4⋊C4—26no—
58C22×C4⋊C4C2311yescharacteristic
59C22×C22⋊C4C2311yescharacteristic
60C22×C22⋊C4C2311yescharacteristic
61C22×C22⋊C4C2316yes—
62C22×C22⋊C4C2316yes—
63C22×C22⋊C4C2316yes—
64#400—26no—
65#399—26no—
66#398—26no—
67#397—26no—
68#397—212no—
69#394—212no—
70#388—212no—
71#387—212no—
72#386—212no—
73#386—26no—
74#386—26no—
75#386—26no—
76#386—212no—
77#385—26no—
78#385—212no—
79#385C23112yes—
80#385C23112yes—
81#381—26no—
82#381C2316yes—
83#381C23112yes—
84#379C2316yes—
85#379C23112yes—
86C2×#95—26no—
87#342C23148yes—
88C25C2411yescharacteristic
89C25C2411yescharacteristic
90C25C2411yescharacteristic
91C22×Q8—24no—
92C22×Q8—24no—
93C22×Q8—24no—
94C22×Q8—24no—
95C22×D4—28no—
96C22×D4—26no—
97C22×D4—28no—
98C22×D4—26no—
99C22×D4—82no—
100C22×D4C2×D4112yes—
101C23×C4—26no—
102C23×C4—26no—
103C23×C4—82no—
104C23×C4C2×D416yes—
105C23×C4C2×D416yes—
106C23×C4C2×D416yes—
107C23×C4C2×D416yes—
108C23×C4C2411yescharacteristic
109C23×C4C2411yescharacteristic
110C23×C4C2411yescharacteristic
111C23×C4C2411yescharacteristic
112C4⋊Q8—224no—
113C4⋊Q8—224no—
114#125—248no—
115#123—812no—
116C22⋊Q8—248no—
117C22⋊Q8—248no—
118C22⋊Q8—248no—
119C22⋊Q8—248no—
120C22⋊Q8—248no—
121C22⋊Q8—248no—
122#121—416no—
123C4×Q8—216no—
124C4×Q8—216no—
125C4×D4—248no—
126C4×D4—248no—
127#117—224no—
128C2×C4⋊C4—212no—
129C2×C4⋊C4—212no—
130C2×C4⋊C4—212no—
131C2×C4⋊C4—212no—
132C2×C4⋊C4—212no—
133C2×C4⋊C4—26no—
134C2×C4⋊C4—212no—
135C2×C4⋊C4—812no—
136C2×C4⋊C4C2×D416yes—
137C2×C4⋊C4C2×D416yes—
138C2×C22⋊C4—224no—
139C2×C22⋊C4—224no—
140C2×C22⋊C4—224no—
141C2×C22⋊C4—26no—
142C2×C22⋊C4—26no—
143C2×C22⋊C4—26no—
144C2×C22⋊C4—26no—
145C2×C22⋊C4—26no—
146C2×C22⋊C4—212no—
147C2×C22⋊C4—212no—
148C2×C22⋊C4C2×D4112yes—
149C2×C22⋊C4C2×D4112yes—
150C2×C22⋊C4C24112yes—
151C2×C22⋊C4C24112yes—
152C2×C42C2×D4112yes—
153#95—212no—
154#95—212no—
155#95—212no—
156#95—212no—
157#95—26no—
158#95—26no—
159C24—24no—
160C24—23no—
161C24—24no—
162C24—81no—
163C24—81no—
164C24—81no—
165C24—81no—
166C24—81no—
167C24—81no—
168C24C22×D416yes—
169C24C22×D416yes—
170C242−1+418yes—
171C24C2511yescharacteristic
172C2×Q8—216no—
173C2×Q8—216no—
174C2×Q8—164no—
175C2×Q8—164no—
176C2×Q8—164no—
177C2×Q8—164no—
178C2×D4—224no—
179C2×D4—224no—
180C2×D4—412no—
181C2×D4—44no—
182C2×D4—44no—
183C2×D4—412no—
184C2×D4—412no—
185C2×D4—86no—
186C2×D4#120116yes—
187C22×C4—24no—
188C22×C4—23no—
189C22×C4—212no—
190C22×C4—212no—
191C22×C4—212no—
192C22×C4—212no—
193C22×C4—23no—
194C22×C4—212no—
195C22×C4—23no—
196C22×C4—24no—
197C22×C4—42no—
198C22×C4—412no—
199C22×C4—44no—
200C22×C4—412no—
201C22×C4—81no—
202C22×C4—86no—
203C22×C4—86no—
204C22×C4—86no—
205C22×C4—86no—
206C22×C4—86no—
207C22×C4—86no—
208C22×C4—86no—
209C22×C4C22×D416yes—
210C22×C4C22×D4112yes—
211C22×C4C22×D4112yes—
212C4⋊C4—248no—
213C4⋊C4—248no—
214C4⋊C4—212no—
215C4⋊C4—212no—
216C4⋊C4—424no—
217C4⋊C4—44no—
218C4⋊C4—812no—
219C22⋊C4—224no—
220C22⋊C4—224no—
221C22⋊C4—224no—
222C22⋊C4—224no—
223C22⋊C4—44no—
224C22⋊C4—412no—
225C22⋊C4—44no—
226C22⋊C4—812no—
227C22⋊C4—86no—
228C22⋊C4—812no—
229C22⋊C4—86no—
230C42—224no—
231C23—212no—
232C23—46no—
233C23—41no—
234C23—46no—
235C23—41no—
236C23—46no—
237C23—46no—
238C23—46no—
239C23—46no—
240C23—41no—
241C23—46no—
242C23—88no—
243C23—164no—
244C23—164no—
245C23C2×#12014yes—
246C23C2×#12014yes—
247C23#536112yes—
248C23#536112yes—
249C23C23×D413yes—
250C23C2×2−1+414yes—
251Q8—816no—
252Q8—816no—
253D4—232no—
254D4—88no—
255D4—88no—
256C2×C4—212no—
257C2×C4—26no—
258C2×C4—212no—
259C2×C4—212no—
260C2×C4—43no—
261C2×C4—41no—
262C2×C4—412no—
263C2×C4—412no—
264C2×C4—412no—
265C2×C4—412no—
266C2×C4—46no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—412no—
270C2×C4—46no—
271C2×C4—812no—
272C2×C4—812no—
273C2×C4—812no—
274C2×C4—812no—
275C2×C4—88no—
276C2×C4—824no—
277C2×C4C2×#12014yes—
278C22—23no—
279C22—24no—
280C22—23no—
281C22—24no—
282C22—24no—
283C22—24no—
284C22—24no—
285C22—412no—
286C22—44no—
287C22—412no—
288C22—812no—
289C22—812no—
290C22—81no—
291C22—81no—
292C22—81no—
293C22#261418yes—
294C22C22×#12011yescharacteristic
295C22C2×#53616yes—
296C4—24no—
297C4—42no—
298C4—412no—
299C4—412no—
300C4—812no—
301C4—81no—
302C4—81no—
303C2—26no—
304C2—28no—
305C2—44no—
306C2—161no—
307C2C2×#261411yescharacteristic
308C2#6039911yescharacteristic
309C2#6040111yescharacteristic
subgroupG/H
center, socleC22C22×#120
commutator, Frattini, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×D4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #62843C2semidirect
3. #29659C2semidirect
4. #29658C2semidirect
5. #29657C2semidirect
6. #29656C2semidirect
7. C2×#2062C2semidirect
8. #23468C2semidirect
9. #20689C2semidirect

Maximal subgroups

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

subgroupindex
#628432
#353512
#296592
#296582
#296572
#296562
C2×#20622
#234682
#206892