Finite Group Databasebeta

← #224739

Subgroups — #224739

Order 512.

785 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36609C211yesmaximal, characteristic
2#22823C212yesmaximal, complemented
3#9710C212yesmaximal
4#9708C212yesmaximal, complemented
5#9696C212yesmaximal
6#8593C212yesmaximal, complemented
7#8585C211yesmaximal, characteristic
8#8236C212yesmaximal, complemented
9#8231C211yesmaximal, characteristic
10#2862C2211yescharacteristic
11#2858C2212yes—
12#2592—22no—
13#2580C412yes—
14#2513C414yes—
15#2448—22no—
16#2447—22no—
17#2367—22no—
18#2357C412yes—
19C4×C4⋊Q8C2212yes—
20C4×#127C2211yescharacteristic
21C42×Q8C2211yescharacteristic
22#1985—22no—
23#1917—22no—
24#1916—22no—
25#1842C2212yes—
26#1831C2212yes—
27#1831C2212yes—
28#1771—22no—
29#1767C2212yes—
30#1767C2212yes—
31C4×C4⋊C8C2212yes—
32C2×#332C2212yes—
33C2×#332C2212yes—
34C2×#332C2212yes—
35C2×#332C2212yes—
36C2×#331C2212yes—
37C2×#331C2211yescharacteristic
38C2×#331C2211yescharacteristic
39#1296C2211yescharacteristic
40#1296C2211yescharacteristic
41#1296C2212yes—
42C2×C4⋊Q8C2312yes—
43C2×#127C2311yescharacteristic
44C2×#124—22no—
45C2×#124—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8C2×C411yescharacteristic
48C2×C4×Q8D412yes—
49C2×C4×D4—22no—
50C2×C4×D4C2×C412yes—
51C2×C4⋊C8—22no—
52C2×C4⋊C8—22no—
53C2×C4⋊C8C2312yes—
54C2×C4⋊C8C2312yes—
55C2×C4⋊C8C2312yes—
56C2×C4⋊C8C2312yes—
57C2×#97—22no—
58C2×#97—22no—
59C2×C4×C8—22no—
60#399—22no—
61#397—24no—
62#397—24no—
63#393—22no—
64#391—22no—
65#391—22no—
66#391—22no—
67#390—22no—
68#390—22no—
69#390—22no—
70#390C2×C412yes—
71#390C2×C414yes—
72#390C2×C414yes—
73#389—22no—
74#389C2×C414yes—
75#388—24no—
76#388—24no—
77#388—24no—
78#385—24no—
79#384C2311yescharacteristic
80#382C2×C412yes—
81#382C2×C412yes—
82#381—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4C2×C414yes—
85C4×C4⋊C4C2×C411yescharacteristic
86C4×C4⋊C4C2×C412yes—
87C4×C4⋊C4D412yes—
88C4×C4⋊C4D412yes—
89C4×C4⋊C4D412yes—
90C4×C22⋊C4—22no—
91C43C2311yescharacteristic
92C43C2312yes—
93#336—22no—
94#336—22no—
95#336—22no—
96#336—22no—
97#336—22no—
98#336—22no—
99#336—22no—
100#332—24no—
101#332—24no—
102#332—42no—
103#332—42no—
104#331—24no—
105#331—41no—
106#331—41no—
107C22×Q8—22no—
108C22×Q8C22⋊C411yescharacteristic
109C22×D4—22no—
110C22×D4C22×C412yes—
111C23×C4—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120#127—22no—
121#127—21no—
122#127—21no—
123#124—44no—
124#124—44no—
125#124—82no—
126#124—82no—
127C4×Q8—42no—
128C4×Q8—42no—
129C4×Q8—42no—
130C4×Q8—82no—
131C4×Q8—82no—
132C4×D4—44no—
133C4×D4—82no—
134C4×D4—82no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4C22⋊C412yes—
139C2×C4⋊C4C22⋊C411yescharacteristic
140C2×C4⋊C4C22⋊C412yes—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22×C414yes—
144C2×C4⋊C4C22×C412yes—
145C2×C22⋊C4—22no—
146C2×C22⋊C4—24no—
147C2×C22⋊C4—24no—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C42—24no—
152C2×C42C22×C411yescharacteristic
153C2×C42C22×C412yes—
154C2×C42C22×C412yes—
155C2×C42C22×C411yescharacteristic
156C2×C42C2×D412yes—
157C2×C42C2×D412yes—
158C2×C42C2×D412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C2×C42C2×D412yes—
162C2×C42C2411yescharacteristic
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—44no—
171C4⋊C8—82no—
172C4⋊C8#4114yes—
173#97—44no—
174#97—82no—
175C4×C8—82no—
176#95—22no—
177#95—24no—
178#95—24no—
179#95—22no—
180#95—24no—
181#95C22⋊C411yescharacteristic
182#95C22⋊C411yescharacteristic
183#95C22⋊C412yes—
184#95C22⋊C412yes—
185C24—22no—
186C2×Q8—41no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—44no—
207C22×C4—84no—
208C22×C4C2×C22⋊C412yes—
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C411yescharacteristic
211C22×C4C2×C22⋊C412yes—
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C22⋊C412yes—
214C22×C4C2×C22⋊C411yescharacteristic
215C22×C4C23×C411yescharacteristic
216C22×C4C22×D412yes—
217C2×C8—24no—
218C2×C8—42no—
219C2×C8—42no—
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—24no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—84no—
242C22⋊C4—44no—
243C22⋊C4—48no—
244C22⋊C4—48no—
245C22⋊C4—84no—
246C22⋊C4—84no—
247C22⋊C4—84no—
248C42—22no—
249C42—22no—
250C42—22no—
251C42—22no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—22no—
260C42—22no—
261C42—42no—
262C42—41no—
263C42—42no—
264C42—84no—
265C42C2×#4112yes—
266C42H(5,𝔽2)11yescharacteristic
267C422−1+411yescharacteristic
268C23—42no—
269C23—42no—
270C23—42no—
271C23—44no—
272C23—44no—
273C23C22×C22⋊C411yescharacteristic
274Q8—44no—
275Q8—82no—
276Q8—82no—
277Q8—82no—
278Q8—82no—
279D4—82no—
280D4—82no—
281D4—82no—
282D4—82no—
283D4—82no—
284D4—82no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
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—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—164no—
311C2×C4#41312yes—
312C2×C4C4×#4111yescharacteristic
313C2×C4#52011yescharacteristic
314C2×C4#54812yes—
315C2×C4#55012yes—
316C2×C4#57512yes—
317C2×C4#57512yes—
318C2×C4#57612yes—
319C2×C4#57712yes—
320C8—44no—
321C8—82no—
322C8—82no—
323C8—82no—
324C22—82no—
325C22—84no—
326C22—82no—
327C22—84no—
328C22—82no—
329C22#234111yescharacteristic
330C22C2×#41311yescharacteristic
331C22#288911yescharacteristic
332C22#289612yes—
333C22#289712yes—
334C4—21no—
335C4—21no—
336C4—22no—
337C4—41no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—42no—
342C4—42no—
343C4—84no—
344C4—82no—
345C4#337012yes—
346C4#337112yes—
347C4#340112yes—
348C4#340312yes—
349C2—162no—
350C2#1775711yescharacteristic
351C2#1776912yes—
352C2#1777111yescharacteristic
353C2#2253711yescharacteristic
354C2#2254912yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22823C2semidirect
4. #9708C2semidirect
6. #8593C2semidirect
8. #8236C2semidirect

Maximal subgroups

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

subgroupindex
#366092
#228232
#97102
#97082
#96962
#85932
#85852
#82362
#82312