Finite Group Databasebeta

← #103766

Subgroups — #103766

Order 512.

539 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 539 conjugacy classes of subgroups — too many to draw. Only its 110 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#29218C211yesmaximal, characteristic, complemented
2#29139C211yesmaximal, characteristic
3#28252C211yesmaximal, characteristic
4#9694C211yesmaximal, characteristic, complemented
5#9674C211yesmaximal, characteristic
6#7570C211yesmaximal, characteristic, complemented
7#7031C211yesmaximal, characteristic, complemented
8C2×#460—21no—
9C2×#461—21no—
10#2868C2211yescharacteristic
11C4×C4⋊Q8C2211yescharacteristic
12C4×#127C2211yescharacteristic
13#2100—22no—
14#2055—22no—
15#2023—22no—
16#1990—21no—
17#1987—22no—
18#1957—21no—
19#1842C2211yescharacteristic
20#1834—21no—
21#1834—21no—
22#1803—21no—
23#1802—21no—
24#1802—22no—
25C4×C4⋊C8C2211yescharacteristic
26C4×C4⋊C8C2211yescharacteristic
27C4×C4⋊C8C2211yescharacteristic
28C2×#332—22no—
29C2×D4⋊C8—22no—
30#1271—22no—
31C22×SD16—22no—
32C2×C4⋊Q8—21no—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8D411yescharacteristic
36C2×#127D411yescharacteristic
37C2×C4×Q8—21no—
38C2×C4×D4—21no—
39C2×C4×D4—21no—
40C2×C4×D4—21no—
41#461—42no—
42#461—42no—
43#460—42no—
44#460—42no—
45C2×C4⋊C8—21no—
46C2×C4⋊C8—21no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8D412yes—
50C2×C4⋊C8D412yes—
51C2×#103—22no—
52C2×#102—22no—
53C2×#102—24no—
54C2×C4×C8—21no—
55C2×C4×C8—21no—
56C2×C4×C8—21no—
57#391—21no—
58#390—21no—
59#390—21no—
60#390—21no—
61#389—21no—
62#389—21no—
63#384—21no—
64#384—21no—
65#384—21no—
66#384D411yescharacteristic
67#384D411yescharacteristic
68#384D412yes—
69C4×C4⋊C4—21no—
70C4×C4⋊C4—21no—
71C43C2311yescharacteristic
72#340—22no—
73#340—24no—
74#340—24no—
75#340—24no—
76#336—21no—
77#336—21no—
78#336D412yes—
79#332—44no—
80D4⋊C8—44no—
81C22×Q8—21no—
82C22×Q8—41no—
83C22×D4—21no—
84C22×D4—21no—
85C22×D4—21no—
86C23×C4—41no—
87C2×SD16—44no—
88C2×SD16—44no—
89C2×SD16—82no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—41no—
94C4⋊Q8—21no—
95C4⋊Q8—21no—
96C4⋊Q8—21no—
97C4⋊Q8—42no—
98C4⋊Q8—42no—
99C4⋊Q8—42no—
100C4⋊Q8—42no—
101C4⋊Q8—82no—
102C4⋊Q8SD1612yes—
103#127—21no—
104#127—21no—
105#127—21no—
106#127SD1612yes—
107C4×Q8—41no—
108C4×Q8—41no—
109C4×Q8—41no—
110C4×Q8—41no—
111C4×D4—44no—
112C4×D4—41no—
113C4×D4—41no—
114C4×D4—42no—
115C4×D4—42no—
116C4×D4—41no—
117C4×D4—41no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C42—41no—
137C2×C42C2×D411yescharacteristic
138C2×C42C2×D411yescharacteristic
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42#4111yescharacteristic
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—42no—
148C4⋊C8—42no—
149C4⋊C8—41no—
150C4⋊C8—41no—
151C4⋊C8—41no—
152C4⋊C8—41no—
153C4⋊C8—41no—
154#103—44no—
155#102—44no—
156#102—84no—
157C4×C8—41no—
158C4×C8—41no—
159C4×C8—41no—
160C4×C8—41no—
161C4×C8—42no—
162#95—41no—
163#95—41no—
164C24—41no—
165C2×Q8—22no—
166C2×Q8—41no—
167C2×Q8—41no—
168C2×Q8—41no—
169C2×Q8—41no—
170C2×Q8—41no—
171C2×Q8—82no—
172C2×Q8—82no—
173C2×Q8—81no—
174C2×Q8—81no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—42no—
187C2×D4—41no—
188C2×D4—42no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4—82no—
196C22×C4—82no—
197C22×C4#12011yescharacteristic
198C22×C4#12011yescharacteristic
199C22×C4#12011yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4#12711yescharacteristic
204SD16—88no—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—81no—
216C2×C8—81no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—44no—
220C4⋊C4—42no—
221C4⋊C4—44no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—44no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—81no—
232C4⋊C4—82no—
233C4⋊C4—81no—
234C4⋊C4—81no—
235C4⋊C4—81no—
236C4⋊C4—81no—
237C4⋊C4—82no—
238C22⋊C4—82no—
239C22⋊C4—81no—
240C22⋊C4—81no—
241C22⋊C4—81no—
242C22⋊C4—81no—
243C42—21no—
244C42—22no—
245C42—21no—
246C42—22no—
247C42—41no—
248C42—81no—
249C42—81no—
250C42C2×SD1611yescharacteristic
251C42C2×SD1611yescharacteristic
252C42#13511yescharacteristic
253C42#13511yescharacteristic
254C42#13512yes—
255C42AGL(1,ℤ/8)11yescharacteristic
256C42AGL(1,ℤ/8)11yescharacteristic
257C42AGL(1,ℤ/8)12yes—
258C42#13711yescharacteristic
259C42#13711yescharacteristic
260C23—42no—
261C23—81no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23#39211yescharacteristic
268Q8—42no—
269Q8—42no—
270Q8—81no—
271Q8—81no—
272Q8—162no—
273Q8—162no—
274D4—44no—
275D4—42no—
276D4—42no—
277D4—44no—
278D4—82no—
279D4—81no—
280D4—81no—
281D4—82no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—162no—
304C2×C4—162no—
305C2×C4#44811yescharacteristic
306C2×C4#44811yescharacteristic
307C2×C4#44911yescharacteristic
308C2×C4#44912yes—
309C2×C4#44911yescharacteristic
310C2×C4#45011yescharacteristic
311C2×C4#45011yescharacteristic
312C2×C4#45211yescharacteristic
313C2×C4#45211yescharacteristic
314C2×C4#45311yescharacteristic
315C2×C4#45412yes—
316C2×C4#45411yescharacteristic
317C2×C4#46312yes—
318C2×C4#46511yescharacteristic
319C2×C4#46511yescharacteristic
320C2×C4#46811yescharacteristic
321C2×C4#46811yescharacteristic
322C2×C4#47312yes—
323C2×C4#49511yescharacteristic
324C2×C4#49611yescharacteristic
325C8—82no—
326C8—82no—
327C8—82no—
328C8—82no—
329C22—82no—
330C22—82no—
331C22—82no—
332C22—161no—
333C22—161no—
334C22—161no—
335C22—161no—
336C22#200011yescharacteristic
337C22#200011yescharacteristic
338C22#200311yescharacteristic
339C22#200611yescharacteristic
340C22#200811yescharacteristic
341C22#201211yescharacteristic
342C22#202011yescharacteristic
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—82no—
351C4—161no—
352C4—161no—
353C4—162no—
354C4#162612yes—
355C4#162812yes—
356C4#162911yescharacteristic
357C4#163411yescharacteristic
358C4#163511yescharacteristic
359C4#163511yescharacteristic
360C4#167912yes—
361C4#169912yes—
362C2—162no—
363C2#1005311yescharacteristic
364C2#1010911yescharacteristic
365C2#1018111yescharacteristic
366C2#1019311yescharacteristic
367C2#1023511yescharacteristic
368C2#1024111yescharacteristic
369C2#1027011yescharacteristic
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. #29218C2semidirect
4. #9694C2semidirect
6. #7570C2semidirect
7. #7031C2semidirect

Maximal subgroups

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

subgroupindex
#292182
#291392
#282522
#96942
#96742
#75702
#70312