Finite Group Databasebeta

← #144591

Subgroups — #144591

Order 512.

365 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 365 conjugacy classes of subgroups — too many to draw. Only its 74 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#28769C211yesmaximal, characteristic, complemented
2#11886C211yesmaximal, characteristic
3C2×#1347C211yesmaximal, characteristic
4#9783C211yesmaximal, characteristic
5#7258C211yesmaximal, characteristic, complemented
6#7230C211yesmaximal, characteristic, complemented
7#7199C211yesmaximal, characteristic, complemented
8C2×#466C2211yescharacteristic
9#2440—21no—
10C2×C22⋊C16C2211yescharacteristic
11C2×C22⋊C16C2211yescharacteristic
12#2083—21no—
13#2057—21no—
14#1990—21no—
15#1964—21no—
16#1942—21no—
17#1852—21no—
18#1808C2211yescharacteristic
19#1808C2211yescharacteristic
20C8×C22⋊C4C2211yescharacteristic
21C2×#340C2211yescharacteristic
22#1519—21no—
23#1347—21no—
24#1347—21no—
25#1347—21no—
26#1347—21no—
27C22×D8Q811yescharacteristic, complemented
28C23×C8C2311yescharacteristic
29C2×C4⋊Q8—21no—
30C2×C22⋊Q8—41no—
31C2×#121—21no—
32C22×C4⋊C4—21no—
33C22×C16—21no—
34C22×C16—21no—
35#466—21no—
36#466—21no—
37#466—21no—
38#466—21no—
39C2×#107—21no—
40C2×#107—21no—
41C2×#107—21no—
42C2×#107—21no—
43C2×#107Q811yescharacteristic, complemented
44C2×#102—21no—
45C2×#99—41no—
46C2×#99—41no—
47C2×C22⋊C8D411yescharacteristic
48C2×C4×C8—21no—
49#398—41no—
50#395—41no—
51#389—21no—
52#389—21no—
53#385—41no—
54#384—21no—
55#380—21no—
56#380—21no—
57C4×C22⋊C4D411yescharacteristic
58C22⋊C16—21no—
59C22⋊C16—21no—
60C22⋊C16—21no—
61C22⋊C16—21no—
62#340—21no—
63#340—21no—
64#340—21no—
65#340—21no—
66#340D411yescharacteristic
67#340D411yescharacteristic
68#340D411yescharacteristic
69#340D411yescharacteristic
70#336—21no—
71#327—42no—
72C22×Q8—41no—
73C22×D4—21no—
74C22×D4—41no—
75C23×C4—41no—
76C23×C4C2×D411yescharacteristic
77C2×D8—21no—
78C2×D8—21no—
79C2×D8—81no—
80C22×C8—21no—
81C22×C8—81no—
82C22×C8C2×D411yescharacteristic
83C22×C8C2×D411yescharacteristic
84C22×C8C2×Q811yescharacteristic
85C22×C8#4111yescharacteristic
86C22×C8#4111yescharacteristic
87C22×C8#4111yescharacteristic
88C4⋊Q8—42no—
89C4⋊Q8—81no—
90C22⋊Q8—82no—
91C22⋊Q8—82no—
92#121—41no—
93#121—41no—
94#121—41no—
95#121—41no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—42no—
102C2×C4⋊C4—41no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4—41no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—81no—
110C2×C22⋊C4—41no—
111C2×C22⋊C4—41no—
112C2×C22⋊C4—41no—
113C2×C22⋊C4D811yescharacteristic
114C2×C22⋊C4D811yescharacteristic
115C2×C42—21no—
116C2×C16—41no—
117C2×C16—41no—
118C2×C16—81no—
119C2×C16—81no—
120#107—21no—
121#107—21no—
122#107—42no—
123#107—81no—
124#107—81no—
125#107—81no—
126#102—41no—
127#102—41no—
128#99—82no—
129#99—82no—
130C22⋊C8—21no—
131C22⋊C8SD1612yes—
132C4×C8—81no—
133#95—21no—
134#95—41no—
135#95—41no—
136#95—41no—
137#95—41no—
138C24—41no—
139C24C2×D811yescharacteristic
140C2×Q8—82no—
141C2×Q8—82no—
142C2×Q8—161no—
143C2×D4—41no—
144C2×D4—41no—
145C2×D4—81no—
146C2×D4—81no—
147C2×D4—81no—
148C2×D4—81no—
149C2×D4—81no—
150C2×D4—81no—
151C2×D4—81no—
152C2×D4—81no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4—41no—
160C22×C4—41no—
161C22×C4—41no—
162C22×C4—82no—
163C22×C4—81no—
164C22×C4—81no—
165C22×C4—81no—
166C22×C4—161no—
167C22×C4#12011yescharacteristic
168C22×C4C22⋊Q811yescharacteristic
169C22×C4#12411yescharacteristic
170C22×C4C2×SD1611yescharacteristic
171C22×C4AGL(1,ℤ/8)11yescharacteristic
172C22×C4#13711yescharacteristic
173D8—81no—
174D8—81no—
175C2×C8—22no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8C22⋊Q811yescharacteristic
185C2×C8C22⋊Q811yescharacteristic
186C2×C8#12411yescharacteristic
187C2×C8#12411yescharacteristic
188C4⋊C4—22no—
189C4⋊C4—41no—
190C4⋊C4—41no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—82no—
199C4⋊C4—82no—
200C4⋊C4—82no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C4⋊C4—82no—
204C4⋊C4—82no—
205C4⋊C4—161no—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—81no—
209C22⋊C4—81no—
210C22⋊C4—81no—
211C22⋊C4—81no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C22⋊C4D1612yes—
215C22⋊C4SD3212yes—
216C42—41no—
217C42—41no—
218C16—81no—
219C16—81no—
220C23—41no—
221C23—41no—
222C23—81no—
223C23—81no—
224C23—81no—
225C23—161no—
226C23—161no—
227C23#44711yescharacteristic
228C23#47411yescharacteristic
229C23#48611yescharacteristic
230C23C2×D1611yescharacteristic
231C23C2×SD3211yescharacteristic
232C23#50911yescharacteristic
233C23#51011yescharacteristic
234Q8—164no—
235D4—81no—
236D4—81no—
237D4—161no—
238D4—161no—
239D4—161no—
240D4—161no—
241D4—161no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—81no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—162no—
273C2×C4—162no—
274C2×C4—161no—
275C2×C4—161no—
276C2×C4—161no—
277C2×C4#39311yescharacteristic
278C2×C4#44811yescharacteristic
279C2×C4#47611yescharacteristic
280C2×C4#48911yescharacteristic
281C8—21no—
282C8—21no—
283C8—21no—
284C8—21no—
285C8—42no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—161no—
297C22—161no—
298C22—161no—
299C22—321no—
300C22—321no—
301C22#202311yescharacteristic
302C22#218411yescharacteristic
303C22#218511yescharacteristic
304C22#222411yescharacteristic
305C22#222611yescharacteristic
306C22#224011yescharacteristic
307C22#224311yescharacteristic
308C4—21no—
309C4—21no—
310C4—21no—
311C4—42no—
312C4—42no—
313C4—82no—
314C4—161no—
315C4—161no—
316C4—161no—
317C4—161no—
318C4—321no—
319C4#203111yescharacteristic
320C4#203211yescharacteristic
321C2—21no—
322C2—21no—
323C2—21no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—321no—
328C2#1037411yescharacteristic
329C2#1336311yescharacteristic
330C2#1337411yescharacteristic
subgroupG/H
center, socleC22#2023
commutator, Frattini, cosocleC23×C8C23
prime radicalC2×#466C22
LCSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #28769C2semidirect
5. #7258C2semidirect
6. #7230C2semidirect
7. #7199C2semidirect
27. C22×D8Q8semidirect
43. C2×#107Q8semidirect

Maximal subgroups

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

subgroupindex
#287692
#118862
C2×#13472
#97832
#72582
#72302
#71992