Finite Group Databasebeta

← #244334

Subgroups — #244334

Order 512.

850 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 170 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#22268C212yesmaximal, complemented
2#22220C212yesmaximal, complemented
3#21727C211yesmaximal, characteristic, complemented
4#21690C211yesmaximal, characteristic
5#10684C212yesmaximal, complemented
6#10673C211yesmaximal, characteristic, complemented
7#10671C211yesmaximal, characteristic, complemented
8#10315C211yesmaximal, characteristic, complemented
9#10312C211yesmaximal, characteristic, complemented
10#10305C212yesmaximal
11#8134C211yesmaximal, characteristic, complemented
12C2×#498C2211yescharacteristic
13C2×#493C2211yescharacteristic
14C2×#492C2212yes—
15C2×#486C2212yes—
16C2×#442—22no—
17C4×C4⋊Q8C2212yes—
18C4×#127C2211yescharacteristic
19C4×#127C2211yescharacteristic
20#1990C2212yes—
21#1990C2212yes—
22#1972—22no—
23#1957C2212yes—
24#1957C2212yes—
25#1957C2211yescharacteristic
26#1957C2211yescharacteristic
27#1956C2212yes—
28#1944—22no—
29#1925—24no—
30#1880—24no—
31#1877—24no—
32#1862—24no—
33#1845C2211yescharacteristic
34#1845C2212yes—
35#1844C2211yescharacteristic
36#1834C2212yes—
37#1776—22no—
38#1744C2212yes—
39C4×#97C2211yescharacteristic
40#1273C2212yes—
41#1273C2211yescharacteristic
42#1273C2211yescharacteristic
43C22×SD16—24no—
44C22×D8—22no—
45C2×C4⋊Q8C2312yes—
46C2×#127C2311yescharacteristic
47C2×#127C2311yescharacteristic
48C2×C4×Q8—22no—
49C2×C4×D4—21no—
50C2×C4×D4—21no—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53#498—24no—
54#493—22no—
55#493—22no—
56#492—24no—
57#492—24no—
58#486—24no—
59#486—24no—
60#442—48no—
61C2×#107—22no—
62C2×#107—24no—
63C2×#106—22no—
64C2×#106—22no—
65C2×#103—24no—
66C2×#102—24no—
67C2×#102—22no—
68C2×#102—22no—
69C2×#102—24no—
70C2×#102—24no—
71C2×#97D412yes—
72C2×#97D412yes—
73C2×C4×C8C2312yes—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2312yes—
76C2×C4×C8C2311yescharacteristic
77#391—22no—
78#390—22no—
79#390—22no—
80#390—21no—
81#390—21no—
82#389—22no—
83#389—22no—
84#384C2311yescharacteristic
85#384C2311yescharacteristic
86#384C2312yes—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C43C2311yescharacteristic
90#340—24no—
91#340—24no—
92#340—24no—
93#336D414yes—
94#336D414yes—
95C22×Q8—22no—
96C22×D4—21no—
97C22×D4—21no—
98C22×D4—22no—
99C22×D4—22no—
100C23×C4—41no—
101C23×C4—41no—
102C2×SD16—216no—
103C2×SD16—84no—
104C2×SD16—84no—
105C2×D8—28no—
106C2×D8—44no—
107C2×D8—44no—
108C22×C8C2×D414yes—
109C22×C8C2×D414yes—
110C22×C8C2×D412yes—
111C22×C8C2×D412yes—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8#4114yes—
116#127—22no—
117#127—22no—
118#127—21no—
119#127—21no—
120#127#4112yes—
121#127#4112yes—
122C4×Q8—84no—
123C4×D4—44no—
124C4×D4—44no—
125C4×D4—84no—
126C4×D4—84no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—42no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C42—42no—
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D412yes—
144C2×C42C2×D412yes—
145C2×C42C2×D411yescharacteristic
146C2×C42C2411yescharacteristic
147#107—44no—
148#107—84no—
149#106—28no—
150#106—44no—
151#103—84no—
152#102—216no—
153#102—44no—
154#102—44no—
155#102—84no—
156#102—84no—
157#97—24no—
158#97—24no—
159C4×C8—22no—
160C4×C8—22no—
161C4×C8—22no—
162C4×C8—22no—
163C4×C8—21no—
164C4×C8—21no—
165C4×C8—22no—
166#95—42no—
167#95—42no—
168C24—41no—
169C24—41no—
170C2×Q8—24no—
171C2×Q8—42no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×D4—24no—
175C2×D4—22no—
176C2×D4—22no—
177C2×D4—24no—
178C2×D4—41no—
179C2×D4—41no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—44no—
187C2×D4—44no—
188C2×D4—44no—
189C2×D4—44no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—82no—
195C22×C4—82no—
196C22×C4—84no—
197C22×C4—84no—
198C22×C4#12711yescharacteristic
199C22×C4#12712yes—
200C22×C4#12711yescharacteristic
201C22×C4C22×D411yescharacteristic
202C22×C4C22×D412yes—
203SD16—816no—
204D8—416no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—24no—
210C2×C8—24no—
211C2×C8—24no—
212C2×C8—24no—
213C2×C8—24no—
214C2×C8—24no—
215C2×C8AGL(1,ℤ/8)14yes—
216C2×C8#13714yes—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—44no—
227C4⋊C4—44no—
228C4⋊C4—84no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C22⋊C4—84no—
232C22⋊C4—84no—
233C42—22no—
234C42—22no—
235C42—24no—
236C42—24no—
237C42—84no—
238C42C2×#4112yes—
239C42C2×#4111yescharacteristic
240C42C2×#4111yescharacteristic
241C23—42no—
242C23—42no—
243C23—81no—
244C23—81no—
245C23—81no—
246C23—81no—
247C23—82no—
248C23—82no—
249C23—82no—
250C23—82no—
251C23C2×#12711yescharacteristic
252Q8—48no—
253Q8—84no—
254D4—44no—
255D4—44no—
256D4—48no—
257D4—48no—
258D4—82no—
259D4—82no—
260D4—84no—
261D4—84no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—44no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—164no—
280C2×C4—164no—
281C2×C4#52511yescharacteristic
282C2×C4#52511yescharacteristic
283C2×C4#52512yes—
284C2×C4#52512yes—
285C2×C4C2×AGL(1,ℤ/8)12yes—
286C2×C4C2×#13712yes—
287C2×C4#57614yes—
288C2×C4#57714yes—
289C8—24no—
290C8—24no—
291C8—44no—
292C8—44no—
293C22—84no—
294C22—84no—
295C22—82no—
296C22—82no—
297C22—161no—
298C22—161no—
299C22—162no—
300C22—162no—
301C22—161no—
302C22—161no—
303C22#239711yescharacteristic
304C22#315111yescharacteristic
305C22#315211yescharacteristic
306C22#315412yes—
307C22#315512yes—
308C4—41no—
309C4—42no—
310C4—41no—
311C4—84no—
312C4—161no—
313C4—162no—
314C4—161no—
315C4#311012yes—
316C4#311212yes—
317C4#311814yes—
318C4#311914yes—
319C2—162no—
320C2—162no—
321C2#1971311yescharacteristic
322C2#1972612yes—
323C2#1972711yescharacteristic
324C2#2380311yescharacteristic
325C2#2380812yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#493C22
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. #22268C2semidirect
2. #22220C2semidirect
3. #21727C2semidirect
5. #10684C2semidirect
6. #10673C2semidirect
7. #10671C2semidirect
8. #10315C2semidirect
9. #10312C2semidirect
11. #8134C2semidirect

Maximal subgroups

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

subgroupindex
#222682
#222202
#217272
#216902
#106842
#106732
#106712
#103152
#103122
#103052
#81342