Finite Group Databasebeta

← #329992

Subgroups — #329992

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33148C211yesmaximal, characteristic
2#33109C211yesmaximal, characteristic
3#29877C211yesmaximal, characteristic
4#21951C211yesmaximal, characteristic
5#10657C212yesmaximal, complemented
6#10598C212yesmaximal, complemented
7#10589C212yesmaximal, complemented
8#10544C212yesmaximal, complemented
9#9130C211yesmaximal, characteristic
10#9128C211yesmaximal, characteristic
11#8598C211yesmaximal, characteristic
12C2×#498C2211yescharacteristic
13C2×#484—21no—
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16#2857C2211yescharacteristic
17#2857C2211yescharacteristic
18C2×C4×C4⋊C4C2211yescharacteristic
19#2088—22no—
20#2077—22no—
21#1990C2211yescharacteristic
22#1990C2211yescharacteristic
23#1987C2212yes—
24#1981C2212yes—
25#1948C2212yes—
26#1947C2212yes—
27#1939—22no—
28#1939C2212yes—
29#1939C2212yes—
30#1928—22no—
31#1920—22no—
32#1872—21no—
33#1853—21no—
34#1834C2211yescharacteristic
35#1823—21no—
36#1797—22no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C2×#334C2212yes—
40C2×#333C2212yes—
41C2×#332C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C42—21no—
52C22×C42C2311yescharacteristic
53#498—42no—
54#484—44no—
55#431—42no—
56#431—42no—
57C2×#107—22no—
58C2×#106—22no—
59C2×#106—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8D412yes—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×#103—22no—
67C2×#103—22no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8D412yes—
72C2×C4×C8—21no—
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#391—21no—
76#391—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#384C2311yescharacteristic
82#384C2311yescharacteristic
83#380—21no—
84#380—21no—
85#380—21no—
86#380—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4D412yes—
90C4×C4⋊C4C2312yes—
91C2×#95—21no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#336—22no—
100#336—22no—
101#336—21no—
102#336—21no—
103#334—42no—
104#333—42no—
105#332—42no—
106C22×Q8—21no—
107C22×Q8—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4#4111yescharacteristic
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8C2×D412yes—
117C22×C8C2×D412yes—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C22⋊Q8—42no—
123C22⋊Q8—42no—
124C22⋊Q8—82no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
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⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4SD1614yes—
142C2×C4⋊C4C2×D412yes—
143C2×C4⋊C4C2×D412yes—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—82no—
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42C2411yescharacteristic
155#107—44no—
156#106—44no—
157#106—44no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—42no—
165C4⋊C8—82no—
166C4⋊C8—82no—
167#103—44no—
168#103—44no—
169C22⋊C8—42no—
170C22⋊C8—42no—
171C22⋊C8—42no—
172C22⋊C8—42no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—42no—
176#95—22no—
177#95—22no—
178#95—41no—
179#95—41no—
180#95—41no—
181#95—41no—
182C24#12111yescharacteristic
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—42no—
188C2×Q8—81no—
189C2×Q8—81no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—42no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4#12111yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4C2×SD1612yes—
215C22×C4#13512yes—
216C22×C4C22×D411yescharacteristic
217C22×C4C22×D411yescharacteristic
218C22×C4C2×#4111yescharacteristic
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C4⋊C4—22no—
235C4⋊C4—24no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—81no—
251C4⋊C4—81no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4C2×SD1614yes—
255C22⋊C4—82no—
256C22⋊C4—82no—
257C42—21no—
258C42—21no—
259C42—22no—
260C42—22no—
261C42—42no—
262C42—42no—
263C42—82no—
264C42H(5,𝔽2)12yes—
265C422−1+412yes—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23#46512yes—
273C23C2×#12111yescharacteristic
274Q8—82no—
275Q8—82no—
276Q8—162no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—24no—
282C2×C4—24no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—24no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—82no—
298C2×C4—84no—
299C2×C4—84no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4#46512yes—
313C2×C4#53411yescharacteristic
314C2×C4#53611yescharacteristic
315C2×C4#53712yes—
316C2×C4#53912yes—
317C2×C4C22×SD1611yescharacteristic
318C2×C4C2×#13511yescharacteristic
319C2×C4#57512yes—
320C2×C4#57712yes—
321C2×C4#57812yes—
322C8—42no—
323C8—42no—
324C8—82no—
325C8—82no—
326C8—82no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22#261811yescharacteristic
335C22#304211yescharacteristic
336C22#304411yescharacteristic
337C22C2×#46511yescharacteristic
338C22#305411yescharacteristic
339C22#306811yescharacteristic
340C22#306911yescharacteristic
341C4—22no—
342C4—22no—
343C4—41no—
344C4—41no—
345C4—44no—
346C4—82no—
347C4—161no—
348C4—161no—
349C4—161no—
350C4—161no—
351C4#320112yes—
352C4#324212yes—
353C4#324612yes—
354C4#325012yes—
355C2—42no—
356C2#1939111yescharacteristic
357C2#1940211yescharacteristic
358C2#1943111yescharacteristic
359C2#1944211yescharacteristic
360C2#2976411yescharacteristic
361C2#2977411yescharacteristic
362C2#2979311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10657C2semidirect
6. #10598C2semidirect
7. #10589C2semidirect
8. #10544C2semidirect

Maximal subgroups

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

subgroupindex
#331482
#331092
#298772
#219512
#106572
#105982
#105892
#105442
#91302
#91282
#85982