Finite Group Databasebeta

← #316172

Subgroups — #316172

Order 512.

500 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 134 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#33046C211yesmaximal, characteristic
2#20807C211yesmaximal, characteristic
3#10633C212yesmaximal, complemented
4#10632C212yesmaximal, complemented
5#9775C212yesmaximal, complemented
6#9195C211yesmaximal, characteristic
7#9195C211yesmaximal, characteristic
8#9193C211yesmaximal, characteristic
9#9191C211yesmaximal, characteristic
10#8794C212yesmaximal, complemented
11#8157C211yesmaximal, characteristic
12C2×#487C2211yescharacteristic
13C2×#430—21no—
14C22×#108—21no—
15#2905C2211yescharacteristic
16#2905C2211yescharacteristic
17C2×#405C2211yescharacteristic
18#2628C2211yescharacteristic
19#2621C2211yescharacteristic
20#1993C2211yescharacteristic
21#1960C2211yescharacteristic
22#1935—21no—
23#1933—21no—
24#1894—21no—
25#1892—21no—
26#1849—21no—
27#1849C2212yes—
28#1848—21no—
29#1848C2212yes—
30#1835C2211yescharacteristic
31#1549C2212yes—
32#1548C2212yes—
33#1546C2212yes—
34#1546C2212yes—
35#1458C2211yescharacteristic
36#1458C2211yescharacteristic
37#1452C2212yes—
38#1286C2212yes—
39#1286C2212yes—
40#1286C2212yes—
41#1286C2212yes—
42#1281C2212yes—
43C22×M16—21no—
44C22×M16—21no—
45C22×M16D411yescharacteristic
46C23×C8D411yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×C4⋊Q8C2311yescharacteristic
49C2×C22⋊Q8—21no—
50C2×#117C2311yescharacteristic
51#487—24no—
52#431C2312yes—
53#431C2312yes—
54#430—44no—
55C2×#108—22no—
56C2×#108—22no—
57C2×#108—22no—
58C2×#108—22no—
59C2×#108—44no—
60#424—42no—
61#424—42no—
62#423—21no—
63#423—21no—
64#423C2312yes—
65#423C2312yes—
66C2×#103—21no—
67C2×#103—21no—
68C2×#103—21no—
69C2×#103—21no—
70#405—24no—
71C2×#97D411yescharacteristic
72C2×#97C2311yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#400—21no—
76#399—21no—
77#398—21no—
78#398—21no—
79#397—21no—
80#397—21no—
81#395—21no—
82#393—21no—
83#391—21no—
84#389—21no—
85#382C2311yescharacteristic
86#382C2311yescharacteristic
87#380—21no—
88#340—21no—
89#340—21no—
90#340—21no—
91#340—21no—
92#333—42no—
93#332—42no—
94#332—42no—
95#332—42no—
96#324—21no—
97#324—21no—
98#323—21no—
99#323—21no—
100#322—24no—
101C22×Q8—21no—
102C22×Q8—21no—
103C23×C4C2×D411yescharacteristic
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—41no—
110C2×M16—41no—
111C2×M16—42no—
112C2×M16—42no—
113C2×M16—42no—
114C2×M16—42no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—41no—
119C22×C8—41no—
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C22⋊Q8—42no—
129C22⋊Q8—42no—
130#117—22no—
131#117—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4#4111yescharacteristic
141C2×C4⋊C4#4111yescharacteristic
142C2×C22⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4#4111yescharacteristic
145C2×C22⋊C4#4111yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42C2411yescharacteristic
148#108—44no—
149#108—44no—
150#108—44no—
151#108—44no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—42no—
159C4⋊C8—42no—
160#103—42no—
161#103—42no—
162#103—42no—
163#103—42no—
164C22⋊C8—22no—
165C22⋊C8—22no—
166#97—22no—
167#97—22no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8#4114yes—
172#95—21no—
173#95—21no—
174#95—21no—
175#95—21no—
176#95—41no—
177#95—41no—
178#95—41no—
179#95—41no—
180C24#12411yescharacteristic
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—42no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—161no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4#12411yescharacteristic
198C22×C4#12411yescharacteristic
199C22×C4#12411yescharacteristic
200C22×C4AGL(1,ℤ/8)12yes—
201C22×C4#13712yes—
202C22×C4C22×D411yescharacteristic
203C22×C4C2×#4111yescharacteristic
204C22×C4C2×#4111yescharacteristic
205M16—28no—
206M16—42no—
207M16—42no—
208M16—42no—
209M16—42no—
210M16—82no—
211M16—82no—
212C2×C8—28no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—82no—
226C2×C8—82no—
227C2×C8#12312yes—
228C2×C8#12312yes—
229C2×C8AGL(1,ℤ/8)12yes—
230C2×C8#13712yes—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—82no—
243C4⋊C4—161no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—82no—
247C22⋊C4—161no—
248C42—21no—
249C42—21no—
250C42C2×#4112yes—
251C42H(5,𝔽2)12yes—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23#48912yes—
257C23C2×#12411yescharacteristic
258Q8—81no—
259Q8—81no—
260Q8—81no—
261Q8—81no—
262Q8—162no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4#48912yes—
288C2×C4C2×#12311yescharacteristic
289C2×C4#53511yescharacteristic
290C2×C4#55312yes—
291C2×C4#55312yes—
292C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
293C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
294C2×C4C2×#13711yescharacteristic
295C2×C4C2×#13711yescharacteristic
296C8—22no—
297C8—22no—
298C8—22no—
299C8—22no—
300C8—82no—
301C8—82no—
302C8—82no—
303C8—82no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—42no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22#250411yescharacteristic
312C22C2×#48911yescharacteristic
313C22#314111yescharacteristic
314C4—22no—
315C4—22no—
316C4—22no—
317C4—41no—
318C4—41no—
319C4—161no—
320C4—161no—
321C4—161no—
322C4—161no—
323C4#309211yescharacteristic
324C4#309211yescharacteristic
325C4#321711yescharacteristic
326C4#321811yescharacteristic
327C2—21no—
328C2—21no—
329C2—42no—
330C2#2797811yescharacteristic
331C2#2798711yescharacteristic
332C2#2798811yescharacteristic
subgroupG/H
center, socleC22#2504
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#124
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #10633C2semidirect
4. #10632C2semidirect
5. #9775C2semidirect
10. #8794C2semidirect

Maximal subgroups

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

subgroupindex
#330462
#208072
#106332
#106322
#97752
#91952
#91932
#91912
#87942
#81572