Finite Group Databasebeta

← #103084

Subgroups — #103084

Order 512.

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

normal subgroups only

This group has 646 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#33224C211yesmaximal, characteristic
2#29218C211yesmaximal, characteristic, complemented
3#28441C211yesmaximal, characteristic
4#9896C211yesmaximal, characteristic, complemented
5#9870C211yesmaximal, characteristic, complemented
6#9695C211yesmaximal, characteristic
7#7015C211yesmaximal, characteristic, complemented
8C2×#492—21no—
9C2×#486—22no—
10C2×#460—22no—
11#2868C2211yescharacteristic
12#2867C2211yescharacteristic
13C4×C4⋊Q8C2211yescharacteristic
14#2100—22no—
15#2014—22no—
16#1990—21no—
17#1987—22no—
18#1957—22no—
19#1957—21no—
20#1957—22no—
21#1842C2211yescharacteristic
22#1834—21no—
23C4×C4⋊C8C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C42×C8C2211yescharacteristic
26C2×#331—22no—
27#1296—22no—
28#1279—22no—
29C22×SD16—24no—
30C2×C4⋊Q8—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8D411yescharacteristic
34C2×#127—21no—
35C2×#127—21no—
36C2×#127—21no—
37C2×#127—21no—
38C2×#127—21no—
39C2×#127—21no—
40C2×#127D412yes—
41C2×C4×Q8—21no—
42#492—42no—
43#492—42no—
44#486—48no—
45#460—44no—
46#460—44no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8D412yes—
51C2×C4⋊C8D412yes—
52C2×#102—28no—
53C2×#102—24no—
54C2×C4×C8—21no—
55C2×C4×C8—21no—
56C2×C4×C8—21no—
57C2×C4×C8—21no—
58C2×C4×C8D412yes—
59#391—21no—
60#389—21no—
61#389—21no—
62#384—21no—
63#384—21no—
64#384—21no—
65#384D411yescharacteristic
66#384D412yes—
67C4×C4⋊C4—21no—
68C4×C4⋊C4—21no—
69C43C2311yescharacteristic
70#340—24no—
71#340—24no—
72#336—21no—
73#336—21no—
74#331—44no—
75C22×Q8—21no—
76C22×Q8—41no—
77C22×D4—22no—
78C22×D4—22no—
79C22×D4—22no—
80C22×D4—41no—
81C22×D4—41no—
82C22×D4—41no—
83C22×D4—41no—
84C2×SD16—48no—
85C2×SD16—48no—
86C2×SD16—84no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—41no—
91C4⋊Q8—21no—
92C4⋊Q8—21no—
93C4⋊Q8—21no—
94C4⋊Q8—44no—
95C4⋊Q8—42no—
96C4⋊Q8—42no—
97C4⋊Q8—82no—
98C4⋊Q8SD1612yes—
99#127—44no—
100#127—44no—
101#127—42no—
102#127—41no—
103#127—41no—
104#127—42no—
105#127—42no—
106#127—42no—
107#127—42no—
108#127—41no—
109#127—41no—
110#127—42no—
111#127—82no—
112C4×Q8—42no—
113C4×Q8—42no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C42—41no—
127C2×C42C2×D411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42C2×D411yescharacteristic
133C2×C42#4111yescharacteristic
134C4⋊C8—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—42no—
138C4⋊C8—42no—
139C4⋊C8—42no—
140C4⋊C8SD1614yes—
141#102—48no—
142#102—416no—
143C4×C8—24no—
144C4×C8—22no—
145C4×C8—22no—
146C4×C8—42no—
147C4×C8—41no—
148C4×C8—41no—
149C4×C8—41no—
150C4×C8—41no—
151#95—41no—
152#95—41no—
153C24—42no—
154C2×Q8—22no—
155C2×Q8—41no—
156C2×Q8—42no—
157C2×Q8—42no—
158C2×Q8—82no—
159C2×Q8—82no—
160C2×Q8—82no—
161C2×D4—44no—
162C2×D4—44no—
163C2×D4—44no—
164C2×D4—42no—
165C2×D4—44no—
166C2×D4—42no—
167C2×D4—44no—
168C2×D4—44no—
169C2×D4—82no—
170C2×D4—82no—
171C2×D4—82no—
172C2×D4—82no—
173C2×D4—81no—
174C2×D4—82no—
175C2×D4—81no—
176C2×D4—82no—
177C2×D4—82no—
178C2×D4—82no—
179C2×D4—81no—
180C2×D4—81no—
181C2×D4—82no—
182C2×D4—81no—
183C2×D4—81no—
184C2×D4—81no—
185C2×D4—81no—
186C2×D4—81no—
187C2×D4—81no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—42no—
191C22×C4#12011yescharacteristic
192C22×C4#12011yescharacteristic
193C22×C4#12011yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12711yescharacteristic
198SD16—816no—
199C2×C8—24no—
200C2×C8—24no—
201C2×C8—42no—
202C2×C8—42no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—81no—
209C2×C8—81no—
210C2×C8—81no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—44no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—44no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—82no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C4⋊C4—82no—
224C4⋊C4—81no—
225C4⋊C4—81no—
226C42—22no—
227C42—22no—
228C42—22no—
229C42—81no—
230C42—81no—
231C42C2×SD1611yescharacteristic
232C42C2×SD1612yes—
233C42#13511yescharacteristic
234C42#13512yes—
235C42#13512yes—
236C42AGL(1,ℤ/8)11yescharacteristic
237C42AGL(1,ℤ/8)12yes—
238C42AGL(1,ℤ/8)12yes—
239C42#13711yescharacteristic
240C42#13712yes—
241C23—82no—
242C23—82no—
243C23—82no—
244C23—82no—
245C23—82no—
246C23—82no—
247C23—82no—
248C23#39211yescharacteristic
249Q8—44no—
250Q8—82no—
251Q8—164no—
252D4—88no—
253D4—84no—
254D4—84no—
255D4—164no—
256D4—164no—
257D4—162no—
258D4—162no—
259D4—162no—
260D4—162no—
261D4—161no—
262D4—161no—
263D4—161no—
264D4—161no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—81no—
276C2×C4—82no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4#44811yescharacteristic
281C2×C4#44912yes—
282C2×C4#44911yescharacteristic
283C2×C4#45011yescharacteristic
284C2×C4#45012yes—
285C2×C4#45212yes—
286C2×C4#45211yescharacteristic
287C2×C4#45312yes—
288C2×C4#45412yes—
289C2×C4#46012yes—
290C2×C4#46312yes—
291C2×C4#46412yes—
292C2×C4#46511yescharacteristic
293C2×C4#46811yescharacteristic
294C2×C4#47212yes—
295C2×C4#49211yescharacteristic
296C2×C4#49511yescharacteristic
297C2×C4#49511yescharacteristic
298C2×C4#49611yescharacteristic
299C2×C4#49611yescharacteristic
300C2×C4#49711yescharacteristic
301C8—44no—
302C8—44no—
303C8—82no—
304C8—161no—
305C22—162no—
306C22—162no—
307C22—162no—
308C22—162no—
309C22—162no—
310C22—162no—
311C22—162no—
312C22#200011yescharacteristic
313C22#200211yescharacteristic
314C22#200311yescharacteristic
315C22#201211yescharacteristic
316C22#201411yescharacteristic
317C22#201511yescharacteristic
318C22#201811yescharacteristic
319C4—22no—
320C4—22no—
321C4—22no—
322C4—82no—
323C4—162no—
324C4—161no—
325C4#162612yes—
326C4#162812yes—
327C4#162912yes—
328C4#163112yes—
329C4#167812yes—
330C4#169812yes—
331C4#170811yescharacteristic
332C4#171411yescharacteristic
333C4#171711yescharacteristic
334C4#171911yescharacteristic
335C2—322no—
336C2#1005311yescharacteristic
337C2#1005411yescharacteristic
338C2#1014511yescharacteristic
339C2#1014811yescharacteristic
340C2#1026911yescharacteristic
341C2#1029211yescharacteristic
342C2#1031011yescharacteristic
subgroupG/H
center, socleC23#392
commutator, Frattini, cosocleC43C23
prime radical#2867C22
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
2. #29218C2semidirect
4. #9896C2semidirect
5. #9870C2semidirect
7. #7015C2semidirect

Maximal subgroups

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

subgroupindex
#332242
#292182
#284412
#98962
#98702
#96952
#70152