Finite Group Databasebeta

← #329975

Subgroups — #329975

Order 512.

602 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33109C211yesmaximal, characteristic
2#33108C211yesmaximal, characteristic
3#29876C211yesmaximal, characteristic
4#21951C211yesmaximal, characteristic
5#10655C212yesmaximal, complemented
6#10602C212yesmaximal, complemented
7#10594C212yesmaximal, complemented
8#10545C212yesmaximal, complemented
9#9130C211yesmaximal, characteristic
10#9130C211yesmaximal, characteristic
11#8599C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#484—21no—
14C2×#483—21no—
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17#2857C2211yescharacteristic
18#2857C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#2086—22no—
21#2075—22no—
22#1990C2211yescharacteristic
23#1990C2211yescharacteristic
24#1981C2212yes—
25#1981C2212yes—
26#1948C2212yes—
27#1947C2212yes—
28#1939—22no—
29#1939C2212yes—
30#1939C2212yes—
31#1928—22no—
32#1927—22no—
33#1872—21no—
34#1872—21no—
35#1834C2211yescharacteristic
36#1797—22no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C2×#335C2212yes—
40C2×#332C2212yes—
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#500—42no—
54#484—44no—
55#483—44no—
56#431—42no—
57#431—42no—
58C2×#107—22no—
59C2×#107—22no—
60C2×#106—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8D412yes—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8D412yes—
75C2×C4×C8—21no—
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#391—21no—
79#391—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#384C2311yescharacteristic
85#384C2311yescharacteristic
86#380—21no—
87#380—21no—
88#380—21no—
89#380—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4D412yes—
93C4×C4⋊C4C2312yes—
94C2×#95—21no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#336—22no—
101#336—22no—
102#336—21no—
103#336—21no—
104#335—42no—
105#332—42no—
106#332—42no—
107C22×Q8—21no—
108C22×Q8—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4#4111yescharacteristic
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8C2×D412yes—
118C22×C8C2×D412yes—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C22⋊Q8—42no—
124C22⋊Q8—42no—
125C22⋊Q8—42no—
126C22⋊Q8—42no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
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—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4D814yes—
144C2×C4⋊C4C2×D412yes—
145C2×C4⋊C4C2×D412yes—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—82no—
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42C2411yescharacteristic
157#107—44no—
158#107—44no—
159#106—44no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—42no—
167C4⋊C8—82no—
168C4⋊C8—82no—
169#103—44no—
170#103—44no—
171#103—44no—
172#103—44no—
173C22⋊C8—42no—
174C22⋊C8—42no—
175C22⋊C8—42no—
176C22⋊C8—42no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8—42no—
180#95—22no—
181#95—22no—
182#95—41no—
183#95—41no—
184#95—41no—
185#95—41no—
186C24#12111yescharacteristic
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—41no—
193C2×Q8—42no—
194C2×Q8—42no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—42no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—81no—
215C22×C4—81no—
216C22×C4#12111yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4C2×D812yes—
220C22×C4#13512yes—
221C22×C4C22×D411yescharacteristic
222C22×C4C22×D411yescharacteristic
223C22×C4C2×#4111yescharacteristic
224C2×C8—22no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—81no—
257C4⋊C4—81no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4C2×D814yes—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C42—21no—
264C42—21no—
265C42—22no—
266C42—22no—
267C42—42no—
268C42—42no—
269C42—82no—
270C42H(5,𝔽2)12yes—
271C422−1+412yes—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23#46612yes—
279C23C2×#12111yescharacteristic
280Q8—82no—
281Q8—82no—
282Q8—82no—
283Q8—82no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—24no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—82no—
305C2×C4—84no—
306C2×C4—84no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4#46612yes—
320C2×C4#53411yescharacteristic
321C2×C4#53611yescharacteristic
322C2×C4#53712yes—
323C2×C4#53912yes—
324C2×C4C22×D811yescharacteristic
325C2×C4C2×#13511yescharacteristic
326C2×C4#57512yes—
327C2×C4#57712yes—
328C2×C4#57812yes—
329C8—42no—
330C8—42no—
331C8—82no—
332C8—82no—
333C8—82no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22#261811yescharacteristic
342C22#304211yescharacteristic
343C22#304411yescharacteristic
344C22C2×#46611yescharacteristic
345C22#305411yescharacteristic
346C22#306911yescharacteristic
347C22#306911yescharacteristic
348C4—22no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—44no—
353C4—82no—
354C4—161no—
355C4—161no—
356C4—161no—
357C4—161no—
358C4#320112yes—
359C4#324212yes—
360C4#324512yes—
361C4#325012yes—
362C2—42no—
363C2#1939211yescharacteristic
364C2#1940411yescharacteristic
365C2#1943111yescharacteristic
366C2#1944311yescharacteristic
367C2#2976411yescharacteristic
368C2#2977011yescharacteristic
369C2#2979211yescharacteristic
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. #10655C2semidirect
6. #10602C2semidirect
7. #10594C2semidirect
8. #10545C2semidirect

Maximal subgroups

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

subgroupindex
#331092
#331082
#298762
#219512
#106552
#106022
#105942
#105452
#91302
#85992