Finite Group Databasebeta

← #6434721

Subgroups — #6434721

Order 512.

2920 conjugacy classes of subgroups; only the 612 normal subgroups are shown.

normal subgroups only

This group has 2920 conjugacy classes of subgroups — too many to draw. Only its 612 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×#3458C211yesmaximal, characteristic
2#35311C213yesmaximal
3#29659C218yesmaximal, complemented
4#29657C218yesmaximal
5#23566C214yesmaximal, complemented
6#23470C214yesmaximal, complemented
7#20688C213yesmaximal
8#3458—26no—
9#3458—26no—
10#3458C2214yes—
11#3458C2214yes—
12C2×#544—23no—
13C2×#544—23no—
14C2×#543C2214yes—
15C2×#533—26no—
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117C2213yes—
18#2855—26no—
19#2625—26no—
20#2503—23no—
21#2502—23no—
22#2499—26no—
23#2484C22112yes—
24#2483C22112yes—
25#2317C2216yes—
26#2310C2216yes—
27#2280C2213yes—
28#2062C22132yes—
29#2030C22132yes—
30#1820C22112yes—
31#1819C22112yes—
32#1740C22112yes—
33C23×Q8C2311yescharacteristic
34C24×C4C2311yescharacteristic
35#544—224no—
36#544—224no—
37#544—212no—
38#544—212no—
39#544—212no—
40#544—212no—
41#543—224no—
42#543C23116yes—
43#533—224no—
44#533—224no—
45#533—224no—
46#533—224no—
47C2×C4⋊Q8—26no—
48C2×C4⋊Q8—26no—
49C2×#125—212no—
50C2×C22⋊Q8—212no—
51C2×C22⋊Q8—212no—
52C2×C22⋊Q8—23no—
53C2×C22⋊Q8—23no—
54C2×C22⋊Q8—23no—
55C2×C22⋊Q8—23no—
56C2×C4×Q8—24no—
57C2×C4×Q8—24no—
58C2×#117—26no—
59C2×#117—26no—
60C2×#117—26no—
61C2×#117—26no—
62C2×#117D416yes—
63C2×#117D416yes—
64C2×#117C2316yes—
65C2×#117C2316yes—
66C2×#117C2316yes—
67C22×C4⋊C4D413yes—
68C22×C4⋊C4D413yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4C2312yes—
72C22×C22⋊C4C2316yes—
73C22×C42D416yes—
74#400—212no—
75#398—212no—
76#397—224no—
77#391—26no—
78#391—26no—
79#391—212no—
80#389—212no—
81#387—224no—
82#385—212no—
83#385C23124yes—
84#384—26no—
85#382—26no—
86#381—212no—
87#381C23124yes—
88#381C2316yes—
89#380—23no—
90#380—23no—
91#379C2316yes—
92C4×C22⋊C4—23no—
93C4×C22⋊C4—23no—
94#342C23148yes—
95C25C2411yescharacteristic
96C22×Q8—24no—
97C22×Q8—24no—
98C22×Q8—23no—
99C22×Q8—23no—
100C22×Q8—28no—
101C23×C4—28no—
102C23×C4C2×D413yes—
103C23×C4C2×D413yes—
104C23×C4C2×D413yes—
105C23×C4C2×D413yes—
106C23×C4C2×D416yes—
107C23×C4C2×D416yes—
108C23×C4C2412yes—
109C23×C4C2411yescharacteristic
110C23×C4C2412yes—
111C23×C4C2411yescharacteristic
112C4⋊Q8—224no—
113C4⋊Q8—224no—
114C4⋊Q8—224no—
115C4⋊Q8—224no—
116#125—248no—
117#125—248no—
118C22⋊Q8—248no—
119C22⋊Q8—248no—
120C22⋊Q8—248no—
121C22⋊Q8—248no—
122C22⋊Q8—41no—
123C22⋊Q8—43no—
124C22⋊Q8—43no—
125C22⋊Q8—43no—
126C22⋊Q8—46no—
127C4×Q8—216no—
128C4×Q8—216no—
129C4×Q8—216no—
130C4×Q8—216no—
131#117—212no—
132#117—212no—
133#117—212no—
134#117—212no—
135#117—212no—
136#117—212no—
137#117—212no—
138#117—212no—
139#117—224no—
140C2×C4⋊C4—212no—
141C2×C4⋊C4—212no—
142C2×C4⋊C4—26no—
143C2×C4⋊C4—26no—
144C2×C4⋊C4—26no—
145C2×C4⋊C4—26no—
146C2×C4⋊C4—23no—
147C2×C4⋊C4—26no—
148C2×C4⋊C4—23no—
149C2×C4⋊C4—26no—
150C2×C4⋊C4—26no—
151C2×C4⋊C4—224no—
152C2×C4⋊C4C2×D416yes—
153C2×C4⋊C4C2×D416yes—
154C2×C4⋊C4C2×D416yes—
155C2×C4⋊C4C2×D416yes—
156C2×C22⋊C4—212no—
157C2×C22⋊C4—212no—
158C2×C22⋊C4—26no—
159C2×C22⋊C4—26no—
160C2×C22⋊C4—26no—
161C2×C22⋊C4—26no—
162C2×C22⋊C4C2×D4112yes—
163C2×C22⋊C4C24112yes—
164C2×C22⋊C4C24112yes—
165C2×C42—26no—
166C2×C42—26no—
167C2×C42—26no—
168C2×C42—26no—
169C2×C42—26no—
170C2×C42C2×D4112yes—
171C2×C42C2×D4112yes—
172#95—26no—
173#95—26no—
174#95—224no—
175#95—224no—
176#95—224no—
177C24—24no—
178C24—42no—
179C24—81no—
180C24C22×D413yes—
181C24C22×D413yes—
182C24C2511yescharacteristic
183C2×Q8—216no—
184C2×Q8—216no—
185C2×Q8—41no—
186C2×Q8—43no—
187C2×Q8—412no—
188C2×Q8—412no—
189C2×Q8—88no—
190C2×Q8—88no—
191C22×C4—24no—
192C22×C4—212no—
193C22×C4—212no—
194C22×C4—212no—
195C22×C4—212no—
196C22×C4—26no—
197C22×C4—26no—
198C22×C4—26no—
199C22×C4—26no—
200C22×C4—212no—
201C22×C4—26no—
202C22×C4—26no—
203C22×C4—28no—
204C22×C4—412no—
205C22×C4—82no—
206C22×C4—86no—
207C22×C4#12011yescharacteristic
208C22×C4#12013yes—
209C22×C4#12013yes—
210C22×C4#12013yes—
211C22×C4#12016yes—
212C22×C4C22×D413yes—
213C22×C4C22×D413yes—
214C22×C4C22×D4112yes—
215C22×C4C22×D4112yes—
216C22×C4C22×D413yes—
217C22×C4C22×D413yes—
218C22×C42−1+418yes—
219C4⋊C4—248no—
220C4⋊C4—248no—
221C4⋊C4—212no—
222C4⋊C4—212no—
223C4⋊C4—212no—
224C4⋊C4—212no—
225C4⋊C4—41no—
226C4⋊C4—43no—
227C4⋊C4—42no—
228C4⋊C4—46no—
229C4⋊C4—412no—
230C4⋊C4—824no—
231C22⋊C4—224no—
232C22⋊C4—224no—
233C22⋊C4—224no—
234C22⋊C4—412no—
235C22⋊C4—46no—
236C22⋊C4—42no—
237C22⋊C4—86no—
238C42—212no—
239C42—212no—
240C42—212no—
241C42—212no—
242C42—412no—
243C23—26no—
244C23—26no—
245C23—26no—
246C23—43no—
247C23—41no—
248C23—41no—
249C23—43no—
250C23—43no—
251C23—88no—
252C23C2×#12013yes—
253C23C2×#12011yescharacteristic
254C23C23×D413yes—
255C23C2×2−1+414yes—
256Q8—816no—
257Q8—816no—
258Q8—82no—
259Q8—86no—
260C2×C4—26no—
261C2×C4—26no—
262C2×C4—22no—
263C2×C4—26no—
264C2×C4—26no—
265C2×C4—26no—
266C2×C4—212no—
267C2×C4—212no—
268C2×C4—212no—
269C2×C4—412no—
270C2×C4—412no—
271C2×C4—412no—
272C2×C4—412no—
273C2×C4—412no—
274C2×C4—412no—
275C2×C4—412no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—412no—
279C2×C4—412no—
280C2×C4—86no—
281C2×C4—86no—
282C2×C4C2×#12011yescharacteristic
283C2×C4C2×#12013yes—
284C2×C4C2×#12011yescharacteristic
285C2×C4C2×#12013yes—
286C2×C4#536112yes—
287C2×C4#536112yes—
288C22—21no—
289C22—23no—
290C22—21no—
291C22—23no—
292C22—23no—
293C22—23no—
294C22—23no—
295C22—44no—
296C22—412no—
297C22—86no—
298C22#302812yes—
299C22C22×#12011yescharacteristic
300C22C2×#53616yes—
301C4—21no—
302C4—23no—
303C4—412no—
304C4—412no—
305C4—812no—
306C4—82no—
307C4—82no—
308C4#261412yes—
309C4#261416yes—
310C2—26no—
311C2—22no—
312C2—43no—
313C2C2×#261411yescharacteristic
314C2C2×#302811yescharacteristic
315C2#6039811yescharacteristic
subgroupG/H
center, socleC22C22×#120
commutator, Frattini, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29659C2semidirect
5. #23566C2semidirect
6. #23470C2semidirect

Maximal subgroups

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

subgroupindex
C2×#34582
#353112
#296592
#296572
#235662
#234702
#206882