Finite Group Databasebeta

← #103133

Subgroups — #103133

Order 512.

648 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 134 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#33310C211yesmaximal, characteristic, complemented
2#28509C211yesmaximal, characteristic
3#28438C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic, complemented
5#9896C211yesmaximal, characteristic, complemented
6#9638C211yesmaximal, characteristic
7#7015C211yesmaximal, characteristic, complemented
8C2×#498—22no—
9C2×#492—21no—
10C2×#476—22no—
11C2×#461—22no—
12#2868C2211yescharacteristic
13#2868C2211yescharacteristic
14C4×#127C2211yescharacteristic
15#2094—22no—
16#2013—22no—
17#1957—21no—
18#1956—21no—
19#1844—22no—
20#1844—22no—
21#1844—21no—
22#1842C2211yescharacteristic
23C4×C4⋊C8C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C42×C8C2211yescharacteristic
26C2×D4⋊C8—22no—
27#1296—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#498—48no—
41#492—42no—
42#492—42no—
43#476—44no—
44#476—44no—
45#461—44no—
46#461—44no—
47C2×#106—24no—
48C2×#106—28no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8D412yes—
53C2×C4⋊C8D412yes—
54C2×#103—24no—
55C2×#102—24no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58C2×C4×C8—21no—
59C2×C4×C8—21no—
60C2×C4×C8D412yes—
61#390—21no—
62#390—21no—
63#390—21no—
64#384—21no—
65#384—21no—
66#384—21no—
67#384—21no—
68#384—21no—
69#384—21no—
70#384—21no—
71#384D411yescharacteristic
72#384D412yes—
73C43C2311yescharacteristic
74#336—21no—
75#336—21no—
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—44no—
92C4⋊Q8—42no—
93C4⋊Q8—42no—
94C4⋊Q8—41no—
95C4⋊Q8—41no—
96C4⋊Q8—42no—
97C4⋊Q8—42no—
98C4⋊Q8—41no—
99C4⋊Q8—41no—
100C4⋊Q8—82no—
101#127—21no—
102#127—21no—
103#127—21no—
104#127SD1612yes—
105C4×D4—42no—
106C4×D4—42no—
107C4×D4—42no—
108C4×D4—42no—
109C4×D4—42no—
110C4×D4—42no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—41no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C42#4111yescharacteristic
136#106—48no—
137#106—416no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—42no—
141C4⋊C8—42no—
142C4⋊C8—42no—
143C4⋊C8SD1614yes—
144C4⋊C8SD1614yescomplemented
145#103—48no—
146#102—48no—
147C4×C8—24no—
148C4×C8—24no—
149C4×C8—22no—
150C4×C8—41no—
151C4×C8—41no—
152C4×C8—41no—
153C4×C8—41no—
154C4×C8SD1614yescomplemented
155C24—41no—
156C2×Q8—44no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—44no—
160C2×Q8—82no—
161C2×Q8—82no—
162C2×Q8—82no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—41no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—41no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—41no—
174C2×D4—42no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4#12011yescharacteristic
185C22×C4#12011yescharacteristic
186C22×C4#12011yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4#12711yescharacteristic
191SD16—816no—
192C2×C8—24no—
193C2×C8—24no—
194C2×C8—24no—
195C2×C8—24no—
196C2×C8—24no—
197C2×C8—42no—
198C2×C8—42no—
199C2×C8—42no—
200C2×C8—42no—
201C2×C8—81no—
202C2×C8—81no—
203C2×C8—81no—
204C4⋊C4—44no—
205C4⋊C4—44no—
206C4⋊C4—44no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C4⋊C4—82no—
211C4⋊C4—82no—
212C4⋊C4—81no—
213C4⋊C4—81no—
214C4⋊C4—82no—
215C4⋊C4—82no—
216C4⋊C4—81no—
217C4⋊C4—81no—
218C4⋊C4—82no—
219C4⋊C4—81no—
220C4⋊C4—81no—
221C4⋊C4—81no—
222C4⋊C4—81no—
223C22⋊C4—82no—
224C22⋊C4—82no—
225C22⋊C4—82no—
226C42—22no—
227C42—22no—
228C42—22no—
229C42C2×SD1611yescharacteristic
230C42C2×SD1612yes—
231C42C2×SD1612yes—
232C42C2×SD1612yes—
233C42#13511yescharacteristic
234C42AGL(1,ℤ/8)11yescharacteristic
235C42AGL(1,ℤ/8)12yes—
236C42AGL(1,ℤ/8)12yes—
237C42#13711yescharacteristic
238C42#13712yes—
239C23—42no—
240C23—81no—
241C23—81no—
242C23—81no—
243C23—81no—
244C23—81no—
245C23—81no—
246C23#39211yescharacteristic
247Q8—84no—
248Q8—84no—
249Q8—164no—
250D4—44no—
251D4—44no—
252D4—44no—
253D4—82no—
254D4—82no—
255D4—82no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—161no—
272C2×C4—161no—
273C2×C4—161no—
274C2×C4—161no—
275C2×C4#44812yes—
276C2×C4#44811yescharacteristic
277C2×C4#44911yescharacteristic
278C2×C4#45011yescharacteristic
279C2×C4#45012yes—
280C2×C4#45012yes—
281C2×C4#45211yescharacteristic
282C2×C4#45312yes—
283C2×C4#45612yes—
284C2×C4#46012yes—
285C2×C4#46012yes—
286C2×C4#46112yes—
287C2×C4#46511yescharacteristic
288C2×C4#46811yescharacteristic
289C2×C4#47112yes—
290C2×C4#49211yescharacteristic
291C2×C4#49211yescharacteristic
292C2×C4#49211yescharacteristic
293C2×C4#49611yescharacteristic
294C2×C4#49611yescharacteristic
295C2×C4#49711yescharacteristic
296C8—28no—
297C8—44no—
298C8—44no—
299C8—161no—
300C22—82no—
301C22—82no—
302C22—82no—
303C22—161no—
304C22—161no—
305C22—161no—
306C22—161no—
307C22#200011yescharacteristic
308C22#200211yescharacteristic
309C22#200511yescharacteristic
310C22#201311yescharacteristic
311C22#201411yescharacteristic
312C22#201411yescharacteristic
313C22#201911yescharacteristic
314C4—22no—
315C4—22no—
316C4—22no—
317C4—162no—
318C4—161no—
319C4—162no—
320C4#162912yes—
321C4#163112yes—
322C4#165112yes—
323C4#165412yes—
324C4#167112yes—
325C4#169112yes—
326C4#170511yescharacteristic
327C4#171411yescharacteristic
328C4#171411yescharacteristic
329C4#171511yescharacteristic
330C2—162no—
331C2#1005411yescharacteristic
332C2#1006811yescharacteristic
333C2#1014611yescharacteristic
334C2#1017211yescharacteristic
335C2#1027111yescharacteristic
336C2#1028311yescharacteristic
337C2#1031511yescharacteristic
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. #33310C2semidirect
4. #9898C2semidirect
5. #9896C2semidirect
7. #7015C2semidirect
144. C4⋊C8SD16semidirect
154. C4×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#333102
#285092
#284382
#98982
#98962
#96382
#70152