Finite Group Databasebeta

← #223166

Subgroups — #223166

Order 512.

833 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36209C211yesmaximal, characteristic
2#22748C212yesmaximal
3#9901C212yesmaximal, complemented
4#9128C212yesmaximal, complemented
5#9127C212yesmaximal, complemented
6#8726C212yesmaximal, complemented
7#8594C211yesmaximal, characteristic, complemented
8C4×#333C211yesmaximal, characteristic, complemented
9#8039C212yesmaximal, complemented
10C2×#431C2212yes—
11C22×C4⋊C8C2212yes—
12#2859C2211yescharacteristic
13#2857C2212yes—
14#2536C412yes—
15#2508—22no—
16#2495—22no—
17#2489C414yes—
18#2485—22no—
19#2356C412yes—
20#2346—22no—
21C4×C4⋊Q8C2211yescharacteristic
22#2285C2212yes—
23C42×D4C2211yescharacteristic
24#1923—22no—
25#1916—22no—
26#1869—22no—
27#1831C2212yes—
28#1797C2212yes—
29#1797C2212yes—
30C4×C4⋊C8C2212yes—
31C8×C22⋊C4—22no—
32C2×#333C2212yes—
33C2×#333C2211yescharacteristic
34C2×#333C2211yescharacteristic
35#1296C2211yescharacteristic
36#1296C2212yes—
37#1296C2212yes—
38#1296C2212yes—
39#1296C2211yescharacteristic
40#1296C2212yes—
41#1296C2212yes—
42C23×C8—22no—
43C2×C4⋊Q8C2311yescharacteristic
44C2×#125C2×C412yes—
45C2×C22⋊Q8C2×C414yes—
46C2×C4×Q8C2×C412yes—
47C2×C4×D4C2×C411yescharacteristic
48C2×C4×D4D412yes—
49C2×#117—22no—
50C22×C42C2312yes—
51#431—24no—
52#431—42no—
53C2×C4⋊C8—24no—
54C2×C4⋊C8—22no—
55C2×C4⋊C8—42no—
56C2×C4⋊C8C2312yes—
57C2×C4⋊C8C2312yes—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×C22⋊C8—22no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—22no—
63C2×C4×C8—22no—
64C2×C4×C8—22no—
65#400—24no—
66#398—22no—
67#397—24no—
68#394—22no—
69#391C2×C412yes—
70#391C2×C414yes—
71#389—22no—
72#389—22no—
73#389—22no—
74#389—22no—
75#389—22no—
76#388—24no—
77#388—24no—
78#387—24no—
79#384C2311yescharacteristic
80#384C2312yes—
81#383—22no—
82#382—22no—
83#382—24no—
84#382C2×C412yes—
85#380—22no—
86#380—22no—
87#380—22no—
88#380C2×C414yes—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4C2×C414yes—
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C4⋊C4D412yes—
94C4×C22⋊C4C2×C412yes—
95C4×C22⋊C4D412yes—
96C4×C22⋊C4D412yes—
97#376—22no—
98C43C2311yescharacteristic
99#336—22no—
100#336—22no—
101#336—22no—
102#336—22no—
103#336—22no—
104#333—24no—
105#333—41no—
106#333—41no—
107C22×Q8C22×C412yes—
108C22×D4C22⋊C411yescharacteristic
109C23×C4C22×C412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D412yes—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—44no—
117C22×C8—44no—
118C22×C8—82no—
119C4⋊Q8—22no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122#125—82no—
123C22⋊Q8—84no—
124C4×Q8—44no—
125C4×D4—24no—
126C4×D4—24no—
127C4×D4—42no—
128#117—44no—
129#117—44no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4C22⋊C412yes—
138C2×C4⋊C4C22⋊C411yescharacteristic
139C2×C4⋊C4C22×C412yes—
140C2×C4⋊C4C22×C414yes—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4C22⋊C412yes—
144C2×C22⋊C4C22⋊C412yes—
145C2×C22⋊C4C22⋊C412yes—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—24no—
151C2×C42—22no—
152C2×C42C22×C411yescharacteristic
153C2×C42C22×C412yes—
154C2×C42C22×C411yescharacteristic
155C2×C42C2×D412yes—
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D412yes—
159C2×C42C2411yescharacteristic
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—44no—
168C4⋊C8#4114yes—
169C22⋊C8—44no—
170C22⋊C8—44no—
171C22⋊C8—82no—
172C4×C8—44no—
173C4×C8—82no—
174#95—24no—
175#95—22no—
176#95—24no—
177#95—24no—
178#95—24no—
179#95—24no—
180#95—22no—
181#95C22⋊C411yescharacteristic
182#95C22⋊C412yes—
183#95C22⋊C412yes—
184#95C22⋊C411yescharacteristic
185C24C2×C22⋊C412yes—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×D4—24no—
191C2×D4—41no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4C2×C22⋊C411yescharacteristic
214C22×C4C2×C22⋊C412yes—
215C22×C4#13514yes—
216C22×C4#13514yes—
217C22×C4C23×C411yescharacteristic
218C22×C4C22×D412yes—
219C2×C8—24no—
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—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—88no—
232C4⋊C4—24no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—84no—
246C4⋊C4—84no—
247C22⋊C4—24no—
248C22⋊C4—24no—
249C22⋊C4—24no—
250C22⋊C4—48no—
251C22⋊C4—84no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—22no—
262C42—22no—
263C42—41no—
264C42—44no—
265C42—48no—
266C42C2×#4112yes—
267C42H(5,𝔽2)11yescharacteristic
268C422−1+411yescharacteristic
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23#41614yes—
276C23C22×C22⋊C411yescharacteristic
277Q8—82no—
278Q8—82no—
279D4—44no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4#41312yes—
315C2×C4#41614yes—
316C2×C4C4×#4111yescharacteristic
317C2×C4#52011yescharacteristic
318C2×C4#54612yes—
319C2×C4#54712yes—
320C2×C4C2×#13512yes—
321C2×C4C2×#13512yes—
322C2×C4#57712yes—
323C2×C4#57812yes—
324C8—44no—
325C8—82no—
326C8—82no—
327C8—82no—
328C22—24no—
329C22—24no—
330C22—24no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22#233911yescharacteristic
336C22C2×#41311yescharacteristic
337C22#288811yescharacteristic
338C22C2×#41612yes—
339C22#289812yes—
340C4—22no—
341C4—21no—
342C4—21no—
343C4—24no—
344C4—41no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—82no—
349C4—84no—
350C4—82no—
351C4#329612yes—
352C4#330012yes—
353C4#333612yes—
354C4#333712yes—
355C2—44no—
356C2#1770011yescharacteristic
357C2#1772612yes—
358C2#1778011yescharacteristic
359C2#2248311yescharacteristic
360C2#2248612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9901C2semidirect
4. #9128C2semidirect
5. #9127C2semidirect
6. #8726C2semidirect
7. #8594C2semidirect
8. C4×#333C2semidirect
9. #8039C2semidirect

Maximal subgroups

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

subgroupindex
#362092
#227482
#99012
#91282
#91272
#87262
#85942
C4×#3332
#80392