Finite Group Databasebeta

← #103105

Subgroups — #103105

Order 512.

558 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 126 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#33289C211yesmaximal, characteristic, complemented
2#29139C211yesmaximal, characteristic
3#28508C211yesmaximal, characteristic
4#9692C211yesmaximal, characteristic, complemented
5#9691C211yesmaximal, characteristic, complemented
6#9638C211yesmaximal, characteristic
7#7015C211yesmaximal, characteristic, complemented
8C2×#474—22no—
9#2868C2211yescharacteristic
10#2868C2211yescharacteristic
11C4×#127C2211yescharacteristic
12#2093—22no—
13#2023—22no—
14#1990—22no—
15#1990—22no—
16#1957—21no—
17#1957—21no—
18#1845—21no—
19#1842C2211yescharacteristic
20#1834—22no—
21#1834—21no—
22#1802—22no—
23C4×C4⋊C8C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C42×C8C2211yescharacteristic
26C2×#333—22no—
27C2×#332—22no—
28C2×D4⋊C8—22no—
29C2×C4⋊Q8—21no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8D412yes—
35C2×#127D411yescharacteristic
36C2×C4×D4—21no—
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39#474—44no—
40#474—44no—
41C2×#107—24no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8D412yes—
46C2×C4⋊C8D412yes—
47C2×#102—24no—
48C2×#102—24no—
49C2×C4×C8—21no—
50C2×C4×C8—21no—
51C2×C4×C8—21no—
52C2×C4×C8—21no—
53C2×C4×C8D412yes—
54#390—21no—
55#390—21no—
56#390—21no—
57#384—21no—
58#384—21no—
59#384—21no—
60#384—21no—
61#384—21no—
62#384—21no—
63#384—21no—
64#384D411yescharacteristic
65#384D412yes—
66C43C2311yescharacteristic
67#340—24no—
68#340—28no—
69#336—21no—
70#336—21no—
71#333—44no—
72#332—44no—
73D4⋊C8—44no—
74C22×Q8—22no—
75C22×Q8—41no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—21no—
79C23×C4—41no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—41no—
84C4⋊Q8—44no—
85C4⋊Q8—42no—
86C4⋊Q8—42no—
87C4⋊Q8—42no—
88C4⋊Q8—42no—
89C4⋊Q8—42no—
90C4⋊Q8—42no—
91C4⋊Q8—42no—
92C4⋊Q8—42no—
93C4⋊Q8—82no—
94#127—21no—
95#127—21no—
96#127—21no—
97#127Q1612yes—
98C4×D4—42no—
99C4×D4—42no—
100C4×D4—42no—
101C4×D4—42no—
102C4×D4—42no—
103C4×D4—42no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C22⋊C4—41no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—41no—
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D411yescharacteristic
125C2×C42C2×D411yescharacteristic
126C2×C42C2×D411yescharacteristic
127C2×C42C2×D411yescharacteristic
128C2×C42#4111yescharacteristic
129#107—48no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132C4⋊C8—22no—
133C4⋊C8—42no—
134C4⋊C8—42no—
135C4⋊C8—42no—
136C4⋊C8D814yes—
137#102—48no—
138#102—84no—
139C4×C8—24no—
140C4×C8—22no—
141C4×C8—22no—
142C4×C8—42no—
143C4×C8—41no—
144C4×C8—41no—
145C4×C8—41no—
146C4×C8—41no—
147C24—41no—
148C2×Q8—44no—
149C2×Q8—82no—
150C2×Q8—82no—
151C2×Q8—81no—
152C2×Q8—81no—
153C2×Q8—81no—
154C2×Q8—81no—
155C2×Q8—81no—
156C2×Q8—81no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4—41no—
168C2×D4—42no—
169C22×C4—41no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4#12011yescharacteristic
179C22×C4#12011yescharacteristic
180C22×C4#12011yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4#12111yescharacteristic
184C22×C4#12711yescharacteristic
185C2×C8—24no—
186C2×C8—24no—
187C2×C8—42no—
188C2×C8—42no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—42no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—81no—
195C2×C8—81no—
196C2×C8—81no—
197C4⋊C4—44no—
198C4⋊C4—44no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—44no—
202C4⋊C4—44no—
203C4⋊C4—42no—
204C4⋊C4—44no—
205C4⋊C4—82no—
206C4⋊C4—82no—
207C4⋊C4—82no—
208C4⋊C4—82no—
209C4⋊C4—82no—
210C4⋊C4—82no—
211C4⋊C4—81no—
212C4⋊C4—81no—
213C22⋊C4—82no—
214C22⋊C4—82no—
215C22⋊C4—82no—
216C42—22no—
217C42—22no—
218C42—22no—
219C42C2×D812yes—
220C42C2×Q1611yescharacteristic
221C42#13512yes—
222C42#13512yes—
223C42#13511yescharacteristic
224C42AGL(1,ℤ/8)12yes—
225C42AGL(1,ℤ/8)12yes—
226C42AGL(1,ℤ/8)12yes—
227C42#13711yescharacteristic
228C42#13711yescharacteristic
229C23—42no—
230C23—81no—
231C23—81no—
232C23—81no—
233C23—81no—
234C23—81no—
235C23—81no—
236C23#39211yescharacteristic
237Q8—164no—
238Q8—161no—
239Q8—161no—
240Q8—161no—
241Q8—161no—
242D4—44no—
243D4—44no—
244D4—44no—
245D4—82no—
246D4—82no—
247D4—82no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—82no—
255C2×C4—81no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—82no—
261C2×C4—82no—
262C2×C4—82no—
263C2×C4—161no—
264C2×C4—161no—
265C2×C4—161no—
266C2×C4—161no—
267C2×C4#44712yes—
268C2×C4#44912yes—
269C2×C4#44912yes—
270C2×C4#45111yescharacteristic
271C2×C4#45111yescharacteristic
272C2×C4#45211yescharacteristic
273C2×C4#45211yescharacteristic
274C2×C4#45512yes—
275C2×C4#45612yes—
276C2×C4#45912yes—
277C2×C4#46312yes—
278C2×C4#46312yes—
279C2×C4#46711yescharacteristic
280C2×C4#47011yescharacteristic
281C2×C4#47112yes—
282C2×C4#49311yescharacteristic
283C2×C4#49511yescharacteristic
284C2×C4#49511yescharacteristic
285C2×C4#49611yescharacteristic
286C2×C4#49611yescharacteristic
287C2×C4#49611yescharacteristic
288C8—44no—
289C8—44no—
290C8—82no—
291C8—161no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22—161no—
296C22—161no—
297C22—161no—
298C22—161no—
299C22#200111yescharacteristic
300C22#200411yescharacteristic
301C22#200511yescharacteristic
302C22#201111yescharacteristic
303C22#201211yescharacteristic
304C22#201211yescharacteristic
305C22#201911yescharacteristic
306C4—22no—
307C4—22no—
308C4—22no—
309C4—161no—
310C4—162no—
311C4—162no—
312C4#164412yes—
313C4#164612yes—
314C4#164712yes—
315C4#164912yes—
316C4#168312yes—
317C4#170312yes—
318C4#170911yescharacteristic
319C4#171711yescharacteristic
320C4#171711yescharacteristic
321C4#172111yescharacteristic
322C2—162no—
323C2#1007211yescharacteristic
324C2#1007611yescharacteristic
325C2#1016611yescharacteristic
326C2#1016811yescharacteristic
327C2#1027511yescharacteristic
328C2#1028811yescharacteristic
329C2#1030711yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC43C23
prime radicalC2×#127D4
LCSG ▷ C43 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33289C2semidirect
4. #9692C2semidirect
5. #9691C2semidirect
7. #7015C2semidirect

Maximal subgroups

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

subgroupindex
#332892
#291392
#285082
#96922
#96912
#96382
#70152