Finite Group Databasebeta

← #325606

Subgroups — #325606

Order 512.

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

normal subgroups only

This group has 585 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#33227C211yesmaximal, characteristic
2#29357C211yesmaximal, characteristic, complemented
3#29356C211yesmaximal, characteristic
4C8×#125C211yesmaximal, characteristic, complemented
5#10541C212yesmaximal, complemented
6#10540C212yesmaximal
7#9736C212yesmaximal, complemented
8#9735C212yesmaximal
9#9710C211yesmaximal, characteristic
10#9709C211yesmaximal, characteristic, complemented
11#8273C211yesmaximal, characteristic, complemented
12C2×#487C2211yescharacteristic
13C2×#487C2211yescharacteristic
14C2×#486C2211yescharacteristic
15C2×#486C2211yescharacteristic
16#2862C2211yescharacteristic
17#2858C2211yescharacteristic
18C4×#125C2211yescharacteristic
19#2074—21no—
20#2073—21no—
21#2026—21no—
22#2025—21no—
23#1988C2212yes—
24#1987C2212yes—
25#1982C2212yes—
26#1981C2212yes—
27#1958—22no—
28#1958—22no—
29#1939C2212yes—
30#1936—22no—
31#1935—22no—
32#1916—22no—
33#1831C2211yescharacteristic
34C8×C4⋊C4—22no—
35C8×C4⋊C4C2212yes—
36#1767C2211yescharacteristic
37#1767C2211yescharacteristic
38C42×C8C2211yescharacteristic
39C2×#334C2212yes—
40C2×#333C2212yes—
41C2×#332C2212yes—
42C2×#332C2212yes—
43C2×#331C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×#127C2311yescharacteristic
49C2×#125D411yescharacteristic
50C2×#124—22no—
51#487—24no—
52#487—42no—
53#486—24no—
54#486—42no—
55C2×C4⋊C8—22no—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×#103—22no—
60C2×#103—21no—
61C2×#103—21no—
62C2×#103—22no—
63C2×#103—22no—
64C2×#102—22no—
65C2×#102—21no—
66C2×#102—21no—
67C2×#102—22no—
68C2×#102—22no—
69C2×#97—21no—
70C2×#97—21no—
71C2×C4×C8—22no—
72C2×C4×C8D412yes—
73C2×C4×C8D412yes—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#391—21no—
77#391—21no—
78#391—22no—
79#390—21no—
80#390—21no—
81#390—22no—
82#389D412yes—
83#387—22no—
84#382—22no—
85#382D411yescharacteristic
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2312yes—
88C43C2311yescharacteristic
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#336—22no—
94#336—22no—
95#336—21no—
96#336—21no—
97#336—22no—
98#334—24no—
99#333—42no—
100#332—42no—
101#332—42no—
102#331—42no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×D4—21no—
107C22×D4—21no—
108C22×D4—21no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8C2×D412yes—
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121#127—22no—
122#127—22no—
123#125—24no—
124#124—44no—
125#124—44no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4C2×D412yes—
135C2×C4⋊C4C2×D412yes—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C42—22no—
139C2×C42C2×D412yes—
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42#4112yes—
143C2×C42C2411yescharacteristic
144C4⋊C8—22no—
145C4⋊C8—22no—
146C4⋊C8—22no—
147C4⋊C8—22no—
148C4⋊C8—22no—
149C4⋊C8—22no—
150C4⋊C8—44no—
151#103—44no—
152#103—42no—
153#103—42no—
154#103—44no—
155#103—82no—
156#102—44no—
157#102—42no—
158#102—42no—
159#102—44no—
160#102—82no—
161#97—81no—
162#97—81no—
163C4×C8—22no—
164C4×C8—42no—
165C4×C8—42no—
166C4×C8—44no—
167C4×C8#4114yes—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—42no—
172#95—42no—
173C24—41no—
174C2×Q8—41no—
175C2×Q8—41no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—82no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—82no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4#12112yes—
200C22×C4#12112yes—
201C22×C4C22×D411yescharacteristic
202C22×C4C22×D411yescharacteristic
203C22×C4C2×#4111yescharacteristic
204C2×C8—22no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—42no—
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#12412yes—
219C2×C8#12412yes—
220C4⋊C4—28no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—44no—
225C4⋊C4—41no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4#13514yes—
234C22⋊C4—84no—
235C22⋊C4—84no—
236C42—22no—
237C42—22no—
238C42—22no—
239C42—21no—
240C42—21no—
241C42—42no—
242C42—42no—
243C42—42no—
244C42C2×#4112yes—
245C42H(5,𝔽2)12yes—
246C23—82no—
247C23—81no—
248C23—81no—
249C23—81no—
250C23—82no—
251C23C2×#12111yescharacteristic
252Q8—81no—
253Q8—82no—
254Q8—82no—
255Q8—81no—
256Q8—82no—
257Q8—162no—
258D4—81no—
259D4—82no—
260D4—82no—
261D4—81no—
262D4—82no—
263D4—162no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4C2×#12411yescharacteristic
285C2×C4#53511yescharacteristic
286C2×C4#55012yes—
287C2×C4#55912yes—
288C2×C4C2×#13512yes—
289C2×C4#57512yes—
290C2×C4#57512yes—
291C2×C4#57711yescharacteristic
292C2×C4#57711yescharacteristic
293C8—22no—
294C8—22no—
295C8—42no—
296C8—82no—
297C8—82no—
298C8—82no—
299C22—161no—
300C22—162no—
301C22—161no—
302C22—161no—
303C22—162no—
304C22#254811yescharacteristic
305C22#304012yes—
306C22#305011yescharacteristic
307C22#306011yescharacteristic
308C22#306712yes—
309C4—22no—
310C4—42no—
311C4—44no—
312C4—42no—
313C4—82no—
314C4—161no—
315C4—161no—
316C4—161no—
317C4#313511yescharacteristic
318C4#314211yescharacteristic
319C4#321912yes—
320C4#339412yes—
321C4#339412yes—
322C2—321no—
323C2#1927812yes—
324C2#1936612yes—
325C2#2873911yescharacteristic
326C2#2875711yescharacteristic
327C2#2876611yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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
2. #29357C2semidirect
4. C8×#125C2semidirect
5. #10541C2semidirect
7. #9736C2semidirect
10. #9709C2semidirect
11. #8273C2semidirect

Maximal subgroups

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

subgroupindex
#332272
#293572
#293562
C8×#1252
#105412
#105402
#97362
#97352
#97102
#97092
#82732