Finite Group Databasebeta

← #330534

Subgroups — #330534

Order 512.

575 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 150 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#33294C211yesmaximal, characteristic, complemented
2#29994C211yesmaximal, characteristic
3#29798C211yesmaximal, characteristic
4#21268C211yesmaximal, characteristic
5#10594C212yesmaximal, complemented
6#10590C212yesmaximal, complemented
7#10477C212yesmaximal
8#10476C212yesmaximal
9#9899C211yesmaximal, characteristic, complemented
10#9692C211yesmaximal, characteristic, complemented
11#8808C211yesmaximal, characteristic, complemented
12C2×#461C2211yescharacteristic
13C2×#461C2211yescharacteristic
14C2×#433—21no—
15#2868C2211yescharacteristic
16#2859C2211yescharacteristic
17#2293C2211yescharacteristic
18#2086—22no—
19#2067—22no—
20#2043—22no—
21#2013—22no—
22#1992—21no—
23#1992—21no—
24#1990C2212yes—
25#1990C2212yes—
26#1981C2211yescharacteristic
27#1981C2211yescharacteristic
28#1954C2212yes—
29#1947C2212yes—
30#1939C2212yes—
31#1924—22no—
32#1842C2211yescharacteristic
33#1836—22no—
34#1831C2211yescharacteristic
35#1802C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37#1750—21no—
38C4×#97C2211yescharacteristic
39C2×#335C2212yes—
40C2×#334C2212yes—
41C2×#332C2212yes—
42C2×#332C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C22×SD16—21no—
46C22×SD16—21no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2311yescharacteristic
50C2×#125—21no—
51C2×C4×D4C2311yescharacteristic
52C2×#117—21no—
53#461—42no—
54#461—42no—
55#433—44no—
56C2×#107—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2311yescharacteristic
63C2×#103—21no—
64C2×#103—21no—
65C2×#103—22no—
66C2×#103—22no—
67C2×#102—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×#97D412yes—
71C2×#97D411yescharacteristic
72C2×#97D411yescharacteristic
73C2×C4×C8—21no—
74C2×C4×C8C2312yes—
75#390D412yes—
76#389—22no—
77#389—22no—
78#389D412yes—
79#388—22no—
80#384—21no—
81#384—21no—
82#384—22no—
83#384C2312yes—
84#382—21no—
85C4×C4⋊C4C2311yescharacteristic
86C4×C22⋊C4—21no—
87C43C2311yescharacteristic
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#336—22no—
95#336—22no—
96#336—21no—
97#335—42no—
98#334—42no—
99#332—42no—
100#332—42no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×D4C2×D411yescharacteristic
104C23×C4—21no—
105C2×SD16—42no—
106C2×SD16—42no—
107C2×SD16—82no—
108C2×SD16—82no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8C2×D412yes—
114C22×C8C2×D412yes—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—82no—
120C4⋊Q8—82no—
121#125—42no—
122#125—42no—
123C4×D4—22no—
124C4×D4—22no—
125#117—42no—
126#117—42no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4C2×D411yescharacteristic
139C2×C4⋊C4C2×D412yes—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—22no—
143C2×C42—21no—
144C2×C42C2×D412yes—
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42#4112yes—
148C2×C42C2411yescharacteristic
149#107—82no—
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—44no—
157C4⋊C8—82no—
158#103—42no—
159#103—42no—
160#103—82no—
161#103—82no—
162#102—82no—
163C22⋊C8—44no—
164C22⋊C8—44no—
165#97—24no—
166#97—22no—
167#97—22no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—42no—
171#95—22no—
172#95—21no—
173#95—42no—
174C24—21no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8—41no—
178C2×Q8—41no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—22no—
186C2×D4—22no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—42no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4#12112yes—
197C22×C4#12112yes—
198C22×C4C22×D411yescharacteristic
199C22×C4C22×D411yescharacteristic
200C22×C4C2×#4111yescharacteristic
201SD16—84no—
202SD16—84no—
203C2×C8—22no—
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8#13714yes—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—44no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—82no—
233C4⋊C4—84no—
234C4⋊C4—84no—
235C4⋊C4—84no—
236C22⋊C4—42no—
237C22⋊C4—44no—
238C22⋊C4—82no—
239C42—22no—
240C42—22no—
241C42—22no—
242C42—42no—
243C42—41no—
244C42—41no—
245C42#13512yes—
246C42AGL(1,ℤ/8)12yes—
247C42H(5,𝔽2)12yes—
248C422−1+412yes—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—42no—
253C23—42no—
254C23C2×#12111yescharacteristic
255Q8—81no—
256Q8—81no—
257Q8—81no—
258Q8—81no—
259Q8—82no—
260Q8—82no—
261D4—41no—
262D4—41no—
263D4—42no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4#46314yes—
286C2×C4#53511yescharacteristic
287C2×C4#53611yescharacteristic
288C2×C4#53712yes—
289C2×C4#54212yes—
290C2×C4C2×#13511yescharacteristic
291C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
292C2×C4C2×#13712yes—
293C2×C4#57612yes—
294C2×C4#57712yes—
295C8—24no—
296C8—42no—
297C8—82no—
298C8—81no—
299C8—81no—
300C8—82no—
301C22—82no—
302C22—81no—
303C22—81no—
304C22—81no—
305C22—82no—
306C22#262011yescharacteristic
307C22C2×#46311yescharacteristic
308C22#303911yescharacteristic
309C22#306212yes—
310C22#306912yes—
311C4—24no—
312C4—22no—
313C4—42no—
314C4—82no—
315C4—81no—
316C4—162no—
317C4—161no—
318C4—161no—
319C4#322011yescharacteristic
320C4#322211yescharacteristic
321C4#324112yes—
322C4#324812yes—
323C4#325412yes—
324C2—161no—
325C2#1923712yes—
326C2#1937812yes—
327C2#2982911yescharacteristic
328C2#2985511yescharacteristic
329C2#2986211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33294C2semidirect
5. #10594C2semidirect
6. #10590C2semidirect
9. #9899C2semidirect
10. #9692C2semidirect
11. #8808C2semidirect

Maximal subgroups

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

subgroupindex
#332942
#299942
#297982
#212682
#105942
#105902
#104772
#104762
#98992
#96922
#88082