Finite Group Databasebeta

← #329976

Subgroups — #329976

Order 512.

566 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33149C211yesmaximal, characteristic
2#33148C211yesmaximal, characteristic
3#29878C211yesmaximal, characteristic
4#21951C211yesmaximal, characteristic
5#10656C212yesmaximal, complemented
6#10601C212yesmaximal, complemented
7#10593C212yesmaximal, complemented
8#10543C212yesmaximal, complemented
9#9128C211yesmaximal, characteristic
10#9128C211yesmaximal, characteristic
11#8599C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2857C2211yescharacteristic
16#2857C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18#2087—22no—
19#2076—22no—
20#1990C2211yescharacteristic
21#1990C2211yescharacteristic
22#1987C2212yes—
23#1987C2212yes—
24#1948C2212yes—
25#1947C2212yes—
26#1939—22no—
27#1939C2212yes—
28#1939C2212yes—
29#1921—22no—
30#1920—22no—
31#1853—21no—
32#1852—21no—
33#1834C2211yescharacteristic
34#1824—21no—
35#1823—21no—
36#1797—22no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C2×#335C2212yes—
40C2×#333C2212yes—
41C2×#333C2212yes—
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#500—42no—
54#431—42no—
55#431—42no—
56C2×#107—22no—
57C2×#107—22no—
58C2×#106—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8D412yes—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8D412yes—
69C2×C4×C8—21no—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#391—21no—
73#391—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#384C2311yescharacteristic
79#384C2311yescharacteristic
80#380—21no—
81#380—21no—
82#380—21no—
83#380—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4D412yes—
87C4×C4⋊C4C2312yes—
88C2×#95—21no—
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#336—22no—
99#336—22no—
100#336—21no—
101#336—21no—
102#335—42no—
103#333—42no—
104#333—42no—
105C22×Q8—21no—
106C22×Q8—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4#4111yescharacteristic
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8C2×D412yes—
116C22×C8C2×D412yes—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C22⋊Q8—82no—
122C22⋊Q8—82no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
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—42no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4Q1614yes—
140C2×C4⋊C4C2×D412yes—
141C2×C4⋊C4C2×D412yes—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—82no—
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42#4111yescharacteristic
152C2×C42C2411yescharacteristic
153#107—44no—
154#107—44no—
155#106—44no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—42no—
163C4⋊C8—82no—
164C4⋊C8—82no—
165C22⋊C8—42no—
166C22⋊C8—42no—
167C22⋊C8—42no—
168C22⋊C8—42no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8—42no—
172#95—22no—
173#95—22no—
174#95—41no—
175#95—41no—
176#95—41no—
177#95—41no—
178C24#12111yescharacteristic
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—81no—
182C2×Q8—81no—
183C2×Q8—81no—
184C2×Q8—81no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—24no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4—81no—
203C22×C4—81no—
204C22×C4—81no—
205C22×C4—81no—
206C22×C4#12111yescharacteristic
207C22×C4#12111yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4C2×Q1612yes—
210C22×C4#13512yes—
211C22×C4C22×D411yescharacteristic
212C22×C4C22×D411yescharacteristic
213C22×C4C2×#4111yescharacteristic
214C2×C8—22no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C4⋊C4—22no—
230C4⋊C4—24no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—81no—
245C4⋊C4—81no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C4⋊C4C2×Q1614yes—
249C22⋊C4—82no—
250C22⋊C4—82no—
251C42—21no—
252C42—21no—
253C42—22no—
254C42—22no—
255C42—42no—
256C42—42no—
257C42—82no—
258C42H(5,𝔽2)12yes—
259C422−1+412yes—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23#46712yes—
267C23C2×#12111yescharacteristic
268Q8—162no—
269Q8—162no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—24no—
275C2×C4—24no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—82no—
291C2×C4—84no—
292C2×C4—84no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4#46712yes—
306C2×C4#53411yescharacteristic
307C2×C4#53611yescharacteristic
308C2×C4#53712yes—
309C2×C4#53912yes—
310C2×C4C22×Q1611yescharacteristic
311C2×C4C2×#13511yescharacteristic
312C2×C4#57512yes—
313C2×C4#57712yes—
314C2×C4#57812yes—
315C8—42no—
316C8—42no—
317C8—82no—
318C8—82no—
319C8—82no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22#261811yescharacteristic
328C22#304211yescharacteristic
329C22#304411yescharacteristic
330C22C2×#46711yescharacteristic
331C22#305411yescharacteristic
332C22#306811yescharacteristic
333C22#306811yescharacteristic
334C4—22no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—44no—
339C4—82no—
340C4—161no—
341C4—161no—
342C4—161no—
343C4—161no—
344C4#320112yes—
345C4#324212yes—
346C4#324912yes—
347C4#325012yes—
348C2—42no—
349C2#1939311yescharacteristic
350C2#1940511yescharacteristic
351C2#1943011yescharacteristic
352C2#1944211yescharacteristic
353C2#2976411yescharacteristic
354C2#2977111yescharacteristic
355C2#2979111yescharacteristic
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. #10656C2semidirect
6. #10601C2semidirect
7. #10593C2semidirect
8. #10543C2semidirect

Maximal subgroups

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

subgroupindex
#331492
#331482
#298782
#219512
#106562
#106012
#105932
#105432
#91282
#85992