Finite Group Databasebeta

← #508337

Subgroups — #508337

Order 512.

579 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 579 conjugacy classes of subgroups — too many to draw. Only its 142 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#33649C212yesmaximal
2#33172C211yesmaximal, characteristic
3C2×#1945C211yesmaximal, characteristic
4#13508C212yesmaximal, complemented
5#13479C214yesmaximal, complemented
6#13415C212yesmaximal, complemented
7#13003C212yesmaximal
8#12788C211yesmaximal, characteristic
9C2×#494C2211yescharacteristic
10C2×#467—21no—
11C2×#467C2212yes—
12C22×#108C2211yescharacteristic
13C2×#427C2211yescharacteristic
14C2×#424—21no—
15C22×C4×C8C2211yescharacteristic
16#2857—21no—
17#2237—24no—
18#2236—22no—
19#2235—22no—
20#2159C2214yes—
21C2×#364C2212yes—
22C2×#362C2214yes—
23#2141C2214yes—
24#2112C2212yes—
25#1969—21no—
26#1969C2212yes—
27#1956—21no—
28#1945—21no—
29#1945—21no—
30#1945C2212yes—
31#1945C2212yes—
32#1845C2211yescharacteristic
33C2×#339—21no—
34#1387C2214yes—
35#1376C2212yes—
36C22×Q16—21no—
37C22×Q16C2312yes—
38C22×M16—21no—
39C23×C8D411yescharacteristic
40C23×C8C2311yescharacteristic
41C2×C4⋊Q8—21no—
42C2×C22⋊Q8—22no—
43C2×#117—21no—
44C22×C42D411yescharacteristic
45C2×M32C2314yes—
46#494—42no—
47#467—22no—
48#467—22no—
49#467—22no—
50#467—22no—
51#467—42no—
52#467—42no—
53C2×#108—21no—
54C2×#108—42no—
55C2×#108C2312yes—
56#427—21no—
57#427—21no—
58#427D412yes—
59#427C2312yes—
60C2×#107—21no—
61C2×#107D411yescharacteristic
62C2×#107D411yescharacteristic
63C2×#107C2312yes—
64C2×#106—21no—
65#424—22no—
66#424—22no—
67#424—42no—
68C2×C4⋊C8—42no—
69C2×#103—22no—
70C2×#103—22no—
71C2×C4×C8—42no—
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#391—22no—
77#389—42no—
78#384—21no—
79#380—42no—
80#367—24no—
81#366—24no—
82#364—42no—
83#362—44no—
84#358—24no—
85#358—24no—
86C22⋊C16—22no—
87C22⋊C16—22no—
88#339—22no—
89#339—22no—
90C22×Q8—22no—
91C23×C4—21no—
92C23×C4C2×D411yescharacteristic
93C2×Q16—22no—
94C2×Q16—22no—
95C2×Q16—24no—
96C2×Q16—24no—
97C2×Q16—42no—
98C2×Q16—42no—
99C2×Q16—42no—
100C2×M16—22no—
101C2×M16—41no—
102C2×M16—42no—
103C2×M16—42no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—41no—
107C22×C8—41no—
108C22×C8—41no—
109C22×C8D812yes—
110C22×C8D812yes—
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8#4111yescharacteristic
117C22×C8#4111yescharacteristic
118C22×C8C2411yescharacteristic
119C4⋊Q8—42no—
120C4⋊Q8—42no—
121C22⋊Q8—42no—
122C22⋊Q8—42no—
123C22⋊Q8—42no—
124C22⋊Q8—42no—
125#117—22no—
126#117—22no—
127#117—41no—
128#117—41no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—42no—
137C2×C42—22no—
138C2×C42—21no—
139C2×C42—41no—
140C2×C42—41no—
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143M32—24no—
144M32—24no—
145C2×C16—24no—
146#108—24no—
147#108—42no—
148#107—21no—
149#107—21no—
150#107—22no—
151#107—22no—
152#107—42no—
153#107C2×D412yes—
154#107C2×D412yes—
155#106—22no—
156#106—22no—
157C4⋊C8—44no—
158C4⋊C8—44no—
159#103—42no—
160#103—42no—
161#103—44no—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—41no—
165C4×C8—41no—
166#95—44no—
167C24AGL(1,ℤ/8)11yescharacteristic
168C2×Q8—42no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—42no—
172C2×Q8—42no—
173C2×Q8—42no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—24no—
180C22×C4—21no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4—41no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4C2×D811yescharacteristic
190C22×C4C2×D811yescharacteristic
191C22×C4AGL(1,ℤ/8)11yescharacteristic
192C22×C4C22×D411yescharacteristic
193Q16—44no—
194Q16—44no—
195Q16—84no—
196M16—42no—
197M16—42no—
198M16—82no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—82no—
213C2×C8#12111yescharacteristic
214C2×C8#12111yescharacteristic
215C2×C8C2×D812yes—
216C2×C8C2×D812yes—
217C2×C8C22×D411yescharacteristic
218C2×C8C2×#4111yescharacteristic
219C2×C8H(5,𝔽2)12yes—
220C2×C82−1+412yes—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—84no—
228C4⋊C4—84no—
229C4⋊C4—82no—
230C4⋊C4—82no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—82no—
234C22⋊C4—82no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—42no—
239C42—42no—
240C42—42no—
241C42#13712yes—
242C16—44no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—41no—
247C23#45911yescharacteristic
248C23#45911yescharacteristic
249C23C2×AGL(1,ℤ/8)11yescharacteristic
250Q8—82no—
251Q8—82no—
252Q8—82no—
253Q8—82no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—48no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—84no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4#45911yescharacteristic
281C2×C4#45911yescharacteristic
282C2×C4C2×#12111yescharacteristic
283C2×C4#53411yescharacteristic
284C2×C4#53611yescharacteristic
285C2×C4C22×D811yescharacteristic
286C2×C4C2×#13711yescharacteristic
287C2×C4#57511yescharacteristic
288C8—21no—
289C8—21no—
290C8—22no—
291C8—22no—
292C8—82no—
293C8#53712yes—
294C8#53912yes—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—42no—
304C22#221412yes—
305C22C2×#45911yescharacteristic
306C22#319311yescharacteristic
307C22#323411yescharacteristic
308C4—24no—
309C4—21no—
310C4—21no—
311C4—42no—
312C4—82no—
313C4—82no—
314C4—162no—
315C4—162no—
316C4#261811yescharacteristic
317C4#303711yescharacteristic
318C4#318611yescharacteristic
319C4#322611yescharacteristic
320C2—21no—
321C2—21no—
322C2—22no—
323C2—41no—
324C2#2975411yescharacteristic
325C2C2×#221411yescharacteristic
326C2#3359411yescharacteristic
subgroupG/H
center, socleC22C2×#459
commutator, Frattini, cosocleC22×C8C24
prime radicalC24AGL(1,ℤ/8)
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13508C2semidirect
5. #13479C2semidirect
6. #13415C2semidirect

Maximal subgroups

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

subgroupindex
#336492
#331722
C2×#19452
#135082
#134792
#134152
#130032
#127882