Finite Group Databasebeta

← #330004

Subgroups — #330004

Order 512.

574 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33148C211yesmaximal, characteristic
2#33147C211yesmaximal, characteristic
3#29844C211yesmaximal, characteristic
4#21951C211yesmaximal, characteristic
5#10652C212yesmaximal, complemented
6#10601C212yesmaximal, complemented
7#10600C212yesmaximal, complemented
8#10541C212yesmaximal, complemented
9#9132C211yesmaximal, characteristic
10#9128C211yesmaximal, characteristic
11#8599C211yesmaximal, characteristic
12C2×#487C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2857C2211yescharacteristic
16#2857C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18#2068—22no—
19#2036—22no—
20#1990C2211yescharacteristic
21#1988C2212yes—
22#1987C2212yes—
23#1956C2211yescharacteristic
24#1948C2212yes—
25#1948C2212yes—
26#1939—22no—
27#1939C2212yes—
28#1939C2212yes—
29#1920—22no—
30#1919—22no—
31#1853—21no—
32#1853—21no—
33#1845C2211yescharacteristic
34#1824—21no—
35#1823—21no—
36#1797—22no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C2×#335C2212yes—
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#487—42no—
54#431—42no—
55#431—42no—
56C2×#107—22no—
57C2×#106—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×#103—22no—
66C2×#103—22no—
67C2×C22⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8D412yes—
71C2×C4×C8—21no—
72C2×C4×C8C2311yescharacteristic
73C2×C4×C8C2311yescharacteristic
74#391—21no—
75#391—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#384C2311yescharacteristic
81#384C2311yescharacteristic
82#380—21no—
83#380—21no—
84#380—21no—
85#380—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4D412yes—
89C4×C4⋊C4C2312yes—
90C2×#95—21no—
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#332—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—82no—
154#106—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—
165#103—44no—
166#103—82no—
167C22⋊C8—42no—
168C22⋊C8—42no—
169C22⋊C8—42no—
170C22⋊C8—42no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—42no—
174#95—22no—
175#95—22no—
176#95—41no—
177#95—41no—
178#95—41no—
179#95—41no—
180C24#12111yescharacteristic
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—81no—
187C2×Q8—81no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—24no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—81no—
206C22×C4—81no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4#12111yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4C2×Q1612yes—
213C22×C4#13512yes—
214C22×C4C22×D411yescharacteristic
215C22×C4C22×D411yescharacteristic
216C22×C4C2×#4111yescharacteristic
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C4⋊C4—22no—
233C4⋊C4—24no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—42no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—81no—
247C4⋊C4—81no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4C2×Q1614yes—
251C22⋊C4—82no—
252C22⋊C4—82no—
253C42—21no—
254C42—21no—
255C42—22no—
256C42—22no—
257C42—42no—
258C42—42no—
259C42—82no—
260C42H(5,𝔽2)12yes—
261C422−1+412yes—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23#46712yes—
269C23C2×#12111yescharacteristic
270Q8—82no—
271Q8—82no—
272Q8—162no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—24no—
278C2×C4—24no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—24no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—82no—
294C2×C4—84no—
295C2×C4—84no—
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—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4#46712yes—
309C2×C4#53411yescharacteristic
310C2×C4#53611yescharacteristic
311C2×C4#53712yes—
312C2×C4#53912yes—
313C2×C4C22×Q1611yescharacteristic
314C2×C4C2×#13511yescharacteristic
315C2×C4#57512yes—
316C2×C4#57712yes—
317C2×C4#57712yes—
318C8—42no—
319C8—42no—
320C8—82no—
321C8—82no—
322C8—82no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22#261811yescharacteristic
331C22#304211yescharacteristic
332C22#304311yescharacteristic
333C22C2×#46711yescharacteristic
334C22#305411yescharacteristic
335C22#306811yescharacteristic
336C22#306911yescharacteristic
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—44no—
342C4—82no—
343C4—161no—
344C4—161no—
345C4—161no—
346C4—161no—
347C4#320412yes—
348C4#324412yes—
349C4#324912yes—
350C4#325012yes—
351C2—42no—
352C2#1939311yescharacteristic
353C2#1939911yescharacteristic
354C2#1943111yescharacteristic
355C2#1943711yescharacteristic
356C2#2976811yescharacteristic
357C2#2977111yescharacteristic
358C2#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. #10652C2semidirect
6. #10601C2semidirect
7. #10600C2semidirect
8. #10541C2semidirect

Maximal subgroups

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

subgroupindex
#331482
#331472
#298442
#219512
#106522
#106012
#106002
#105412
#91322
#91282
#85992