Finite Group Databasebeta

← #331206

Subgroups — #331206

Order 512.

840 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 178 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#33207C211yesmaximal, characteristic, complemented
2#33171C211yesmaximal, characteristic
3#29843C212yesmaximal, complemented
4#29807C212yesmaximal, complemented
5#10683C211yesmaximal, characteristic, complemented
6#10677C211yesmaximal, characteristic, complemented
7#10675C212yesmaximal, complemented
8#10315C211yesmaximal, characteristic, complemented
9#10312C211yesmaximal, characteristic, complemented
10#10305C212yesmaximal, complemented
11#7849C211yesmaximal, characteristic, complemented
12C2×#498C2211yescharacteristic
13C2×#493C2211yescharacteristic
14C2×#492C2212yes—
15C2×#486C2212yes—
16C2×#466—22no—
17C2×#431C2212yes—
18C22×C4×C8C2211yescharacteristic
19#2857C2212yes—
20#2856C2211yescharacteristic
21#2856C2211yescharacteristic
22#2065—24no—
23#2055—24no—
24#2034—24no—
25#2000—24no—
26#1990C2212yes—
27#1990C2212yes—
28#1970—22no—
29#1957C2212yes—
30#1957C2212yes—
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1956C2212yes—
34#1941—22no—
35#1845C2211yescharacteristic
36#1844C2212yes—
37#1844C2211yescharacteristic
38#1834C2212yes—
39#1807—22no—
40#1273C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×SD16—24no—
44C22×D8—22no—
45C23×C8D412yes—
46C2×C4⋊Q8C2312yes—
47C2×#127C2311yescharacteristic
48C2×#127C2311yescharacteristic
49C2×C22⋊Q8—22no—
50C2×#121—21no—
51C2×#121—21no—
52C2×#121—22no—
53C2×#121—22no—
54C22×C42C2311yescharacteristic
55#498—42no—
56#493—42no—
57#492—42no—
58#492—42no—
59#486—42no—
60#486—42no—
61#466—48no—
62#431—42no—
63#431—42no—
64C2×#107—22no—
65C2×#106—22no—
66C2×#106—22no—
67C2×#106—24no—
68C2×C4⋊C8D414yes—
69C2×#103—24no—
70C2×#102—24no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#102—24no—
74C2×#102—24no—
75C2×C22⋊C8D414yes—
76C2×C4×C8—42no—
77C2×C4×C8D412yes—
78C2×C4×C8C2312yes—
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2312yes—
81C2×C4×C8C2311yescharacteristic
82#391—22no—
83#390—22no—
84#390—22no—
85#390—21no—
86#390—21no—
87#389—22no—
88#389—22no—
89#384C2311yescharacteristic
90#384C2311yescharacteristic
91#384C2312yes—
92#380—22no—
93#380—22no—
94#340—24no—
95#340—24no—
96#340—24no—
97C22×Q8—22no—
98C22×D4—21no—
99C22×D4—21no—
100C22×D4—22no—
101C22×D4—22no—
102C22×D4—41no—
103C22×D4—41no—
104C23×C4C2×D412yes—
105C23×C4C2×D411yescharacteristic
106C2×SD16—48no—
107C2×SD16—84no—
108C2×SD16—84no—
109C2×D8—44no—
110C2×D8—44no—
111C2×D8—44no—
112C22×C8—42no—
113C22×C8—42no—
114C22×C8SD1618yescomplemented
115C22×C8C2×D414yes—
116C22×C8C2×D414yes—
117C22×C8C2×D412yes—
118C22×C8C2×D412yes—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123#127—21no—
124#127—22no—
125#127—21no—
126#127—22no—
127#127—21no—
128#127—21no—
129C22⋊Q8—84no—
130#121—44no—
131#121—44no—
132#121—84no—
133#121—84no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C42—22no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D412yes—
154C2×C42C2411yescharacteristic
155#107—44no—
156#106—44no—
157#106—44no—
158#106—84no—
159C4⋊C8—44no—
160#103—84no—
161#102—48no—
162#102—44no—
163#102—44no—
164#102—84no—
165#102—84no—
166C22⋊C8—44no—
167C4×C8—22no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—42no—
174#95—42no—
175#95—42no—
176C24—41no—
177C24—41no—
178C24#12711yescharacteristic
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—44no—
182C2×Q8—44no—
183C2×D4—42no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—42no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—44no—
196C2×D4—44no—
197C2×D4—44no—
198C2×D4—44no—
199C2×D4—84no—
200C2×D4—84no—
201C2×D4—82no—
202C2×D4—82no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4#12712yes—
215C22×C4#12711yescharacteristic
216C22×C4C2×SD1614yes—
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D412yes—
219SD16—816no—
220D8—88no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—24no—
226C2×C8—24no—
227C2×C8—24no—
228C2×C8—24no—
229C2×C8—24no—
230C2×C8—24no—
231C2×C8—48no—
232C2×C8C2×SD1614yes—
233C2×C8C2×SD1614yes—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—44no—
241C4⋊C4—44no—
242C4⋊C4—44no—
243C4⋊C4—44no—
244C4⋊C4—44no—
245C4⋊C4—84no—
246C22⋊C4—84no—
247C22⋊C4—84no—
248C22⋊C4—84no—
249C22⋊C4—82no—
250C22⋊C4—82no—
251C42—21no—
252C42—22no—
253C42—22no—
254C42—21no—
255C42H(5,𝔽2)12yes—
256C42H(5,𝔽2)11yescharacteristic
257C422−1+411yescharacteristic
258C23—21no—
259C23—22no—
260C23—22no—
261C23—21no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23—82no—
267C23—82no—
268C23—82no—
269C23—82no—
270C23—81no—
271C23—81no—
272C23#49212yes—
273C23C2×#12711yescharacteristic
274Q8—84no—
275Q8—84no—
276D4—82no—
277D4—82no—
278D4—84no—
279D4—84no—
280D4—82no—
281D4—82no—
282D4—84no—
283D4—84no—
284D4—164no—
285D4—164no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4#49212yes—
306C2×C4#53411yescharacteristic
307C2×C4#53412yes—
308C2×C4#53711yescharacteristic
309C2×C4#53712yes—
310C2×C4C22×SD1612yes—
311C2×C4#57512yes—
312C2×C4#57614yes—
313C2×C4#57714yes—
314C8—24no—
315C8—24no—
316C8—44no—
317C8—44no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—42no—
322C22—42no—
323C22—161no—
324C22—161no—
325C22—162no—
326C22—162no—
327C22—161no—
328C22—161no—
329C22—161no—
330C22—161no—
331C22—162no—
332C22—162no—
333C22#265611yescharacteristic
334C22C2×#49211yescharacteristic
335C22#315011yescharacteristic
336C22#315412yes—
337C22#315512yes—
338C4—22no—
339C4—42no—
340C4—162no—
341C4—161no—
342C4—161no—
343C4—162no—
344C4#318712yes—
345C4#319814yes—
346C4#324612yes—
347C4#325714yes—
348C2—42no—
349C2—321no—
350C2—321no—
351C2#1965211yescharacteristic
352C2#1965411yescharacteristic
353C2#1970812yes—
354C2#3024011yescharacteristic
355C2#3025112yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#33171C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33207C2semidirect
3. #29843C2semidirect
4. #29807C2semidirect
5. #10683C2semidirect
6. #10677C2semidirect
7. #10675C2semidirect
8. #10315C2semidirect
9. #10312C2semidirect
10. #10305C2semidirect
11. #7849C2semidirect
114. C22×C8SD16semidirect

Maximal subgroups

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

subgroupindex
#332072
#331712
#298432
#298072
#106832
#106772
#106752
#103152
#103122
#103052
#78492