Finite Group Databasebeta

← #103101

Subgroups — #103101

Order 512.

620 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 130 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#28508C211yesmaximal, characteristic
3#28438C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic, complemented
5#9692C211yesmaximal, characteristic, complemented
6#9638C211yesmaximal, characteristic
7#7015C211yesmaximal, characteristic, complemented
8C2×#492—21no—
9C2×#474—22no—
10C2×#461—22no—
11#2868C2211yescharacteristic
12#2868C2211yescharacteristic
13C4×#127C2211yescharacteristic
14#2093—22no—
15#2013—22no—
16#1990—22no—
17#1990—22no—
18#1957—21no—
19#1956—21no—
20#1845—21no—
21#1842C2211yescharacteristic
22#1834—22no—
23C4×C4⋊C8C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C42×C8C2211yescharacteristic
26C2×#332—22no—
27C2×D4⋊C8—22no—
28#1296—22no—
29C22×SD16—24no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8D412yes—
36C2×#127D411yescharacteristic
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39C2×C4×D4—21no—
40#492—42no—
41#492—42no—
42#474—44no—
43#474—44no—
44#461—44no—
45#461—44no—
46C2×#107—24no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8D412yes—
51C2×C4⋊C8D412yes—
52C2×#103—24no—
53C2×#102—24no—
54C2×C4×C8—21no—
55C2×C4×C8—21no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58C2×C4×C8D412yes—
59#390—21no—
60#390—21no—
61#390—21no—
62#384—21no—
63#384—21no—
64#384—21no—
65#384—21no—
66#384—21no—
67#384—21no—
68#384—21no—
69#384D411yescharacteristic
70#384D412yes—
71C43C2311yescharacteristic
72#340—28no—
73#336—21no—
74#336—21no—
75#332—44no—
76D4⋊C8—44no—
77C22×Q8—22no—
78C22×Q8—41no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—21no—
82C23×C4—41no—
83C2×SD16—48no—
84C2×SD16—48no—
85C2×SD16—84no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—41no—
90C4⋊Q8—44no—
91C4⋊Q8—42no—
92C4⋊Q8—42no—
93C4⋊Q8—42no—
94C4⋊Q8—41no—
95C4⋊Q8—42no—
96C4⋊Q8—41no—
97C4⋊Q8—42no—
98C4⋊Q8—42no—
99C4⋊Q8—42no—
100C4⋊Q8—42no—
101C4⋊Q8—41no—
102C4⋊Q8—41no—
103#127—21no—
104#127—21no—
105#127—21no—
106#127SD1612yes—
107C4×D4—42no—
108C4×D4—42no—
109C4×D4—42no—
110C4×D4—42no—
111C4×D4—42no—
112C4×D4—42no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D411yescharacteristic
137C2×C42#4111yescharacteristic
138#107—48no—
139C4⋊C8—22no—
140C4⋊C8—22no—
141C4⋊C8—42no—
142C4⋊C8—42no—
143C4⋊C8—42no—
144C4⋊C8D814yes—
145C4⋊C8SD1614yescomplemented
146#103—48no—
147#102—48no—
148C4×C8—24no—
149C4×C8—22no—
150C4×C8—22no—
151C4×C8—42no—
152C4×C8—41no—
153C4×C8—41no—
154C4×C8—41no—
155C4×C8—41no—
156C24—41no—
157C2×Q8—44no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—44no—
161C2×Q8—81no—
162C2×Q8—81no—
163C2×Q8—81no—
164C2×Q8—81no—
165C2×Q8—81no—
166C2×Q8—81no—
167C2×D4—22no—
168C2×D4—22no—
169C2×D4—22no—
170C2×D4—42no—
171C2×D4—41no—
172C2×D4—42no—
173C2×D4—41no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—41no—
178C2×D4—42no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4#12011yescharacteristic
189C22×C4#12011yescharacteristic
190C22×C4#12011yescharacteristic
191C22×C4#12111yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4#12711yescharacteristic
195SD16—816no—
196C2×C8—24no—
197C2×C8—24no—
198C2×C8—24no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—81no—
206C2×C8—81no—
207C2×C8—81no—
208C4⋊C4—44no—
209C4⋊C4—42no—
210C4⋊C4—44no—
211C4⋊C4—44no—
212C4⋊C4—42no—
213C4⋊C4—44no—
214C4⋊C4—42no—
215C4⋊C4—82no—
216C4⋊C4—82no—
217C4⋊C4—81no—
218C4⋊C4—81no—
219C4⋊C4—82no—
220C4⋊C4—82no—
221C4⋊C4—82no—
222C4⋊C4—81no—
223C4⋊C4—81no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C22⋊C4—82no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C42—22no—
230C42—22no—
231C42—22no—
232C42C2×D812yes—
233C42C2×SD1611yescharacteristic
234C42C2×SD1612yes—
235C42#13511yescharacteristic
236C42#13512yes—
237C42AGL(1,ℤ/8)12yes—
238C42AGL(1,ℤ/8)11yescharacteristic
239C42AGL(1,ℤ/8)12yes—
240C42AGL(1,ℤ/8)12yes—
241C42#13711yescharacteristic
242C23—42no—
243C23—81no—
244C23—81no—
245C23—81no—
246C23—81no—
247C23—81no—
248C23—81no—
249C23#39211yescharacteristic
250Q8—84no—
251Q8—84no—
252Q8—161no—
253Q8—161no—
254Q8—161no—
255Q8—161no—
256D4—44no—
257D4—44no—
258D4—44no—
259D4—82no—
260D4—82no—
261D4—82no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—81no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—81no—
276C2×C4—82no—
277C2×C4—161no—
278C2×C4—161no—
279C2×C4—161no—
280C2×C4—161no—
281C2×C4#44712yes—
282C2×C4#44811yescharacteristic
283C2×C4#44912yes—
284C2×C4#44911yescharacteristic
285C2×C4#45011yescharacteristic
286C2×C4#45012yes—
287C2×C4#45211yescharacteristic
288C2×C4#45512yes—
289C2×C4#45612yes—
290C2×C4#45912yes—
291C2×C4#46012yes—
292C2×C4#46312yes—
293C2×C4#46511yescharacteristic
294C2×C4#46811yescharacteristic
295C2×C4#47312yes—
296C2×C4#49211yescharacteristic
297C2×C4#49311yescharacteristic
298C2×C4#49511yescharacteristic
299C2×C4#49611yescharacteristic
300C2×C4#49611yescharacteristic
301C2×C4#49611yescharacteristic
302C8—44no—
303C8—44no—
304C8—44no—
305C8—161no—
306C22—82no—
307C22—82no—
308C22—82no—
309C22—161no—
310C22—161no—
311C22—161no—
312C22—161no—
313C22#200011yescharacteristic
314C22#200411yescharacteristic
315C22#200511yescharacteristic
316C22#201111yescharacteristic
317C22#201211yescharacteristic
318C22#201411yescharacteristic
319C22#202011yescharacteristic
320C4—22no—
321C4—22no—
322C4—22no—
323C4—161no—
324C4—162no—
325C4—162no—
326C4#163912yes—
327C4#164112yes—
328C4#165112yes—
329C4#165412yes—
330C4#167912yes—
331C4#169912yes—
332C4#170611yescharacteristic
333C4#171511yescharacteristic
334C4#171711yescharacteristic
335C4#172111yescharacteristic
336C2—162no—
337C2#1006811yescharacteristic
338C2#1007011yescharacteristic
339C2#1017011yescharacteristic
340C2#1017411yescharacteristic
341C2#1027011yescharacteristic
342C2#1028811yescharacteristic
343C2#1030511yescharacteristic
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. #9898C2semidirect
5. #9692C2semidirect
7. #7015C2semidirect
145. C4⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#332892
#285082
#284382
#98982
#96922
#96382
#70152