Finite Group Databasebeta

← #365661

Subgroups — #365661

Order 512.

1122 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#34214C211yesmaximal, characteristic
2C2×#1995C211yesmaximal, characteristic
3#21323C211yesmaximal, characteristic
4#21172C211yesmaximal, characteristic
5#14348C211yesmaximal, characteristic
6#11276C214yesmaximal
7#10899C214yesmaximal
8#10857C211yesmaximal, characteristic
9#10701C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#428C2212yes—
12C2×#423C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2360—24no—
15#2355—22no—
16Q8×C4⋊C4—22no—
17#2330—22no—
18#2317C2212yes—
19C4×C4⋊Q8—22no—
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22#2274C2212yes—
23C4×#117C2211yescharacteristic
24C2×#376C2211yescharacteristic
25#1995C418yescomplemented
26#1993—22no—
27#1991—22no—
28#1949—22no—
29#1949—22no—
30C2×#343C2214yes—
31#1736C2212yes—
32#1736C2214yes—
33C2×#339C2212yes—
34C2×#339C2212yes—
35C2×#339C2214yes—
36#1442—24no—
37C4⋊#95C2214yes—
38C22×M16C2312yes—
39C22×M16C2312yes—
40#533—22no—
41#533—22no—
42#533C2×C414yes—
43#533Q814yes—
44C2×C4⋊Q8D412yes—
45C2×#125D412yes—
46C2×C22⋊Q8—22no—
47C2×C4×Q8—24no—
48C2×C4×Q8—22no—
49C2×#117C2311yescharacteristic
50C2×#117C2312yes—
51C2×#117C2311yescharacteristic
52C22×C4⋊C4D414yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C42C2312yes—
56C22×C42C2311yescharacteristic
57#428—28no—
58C2×#107—22no—
59C2×#106—22no—
60#423—22no—
61#423C2×C414yes—
62C2×C4⋊C8—22no—
63C4×M16—22no—
64C4×M16—22no—
65C2×#97—21no—
66C2×C4×C8—21no—
67#391—24no—
68#389—24no—
69#389—28no—
70#389—24no—
71#387—28no—
72#385—24no—
73#384—22no—
74#384—21no—
75#384—21no—
76#384—22no—
77#382—22no—
78#382—22no—
79#382—21no—
80#382—21no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—24no—
83C4×C4⋊C4—24no—
84C4×C4⋊C4D412yes—
85C4×C4⋊C4D412yes—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—24no—
88#376—21no—
89#376—22no—
90#376C2×C414yes—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—28no—
94C43—21no—
95#343—28no—
96#340—22no—
97#340—22no—
98#340—28no—
99#339—24no—
100#339—24no—
101#339C2×C4116yes—
102C22×Q8—22no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4C2×D414yes—
106C23×C4C2×D412yes—
107C23×C4C2411yescharacteristic
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—24no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—24no—
115C2×M16—22no—
116C2×M16C22×C414yes—
117C2×M16#4114yes—
118C22×C8—22no—
119C22×C8—22no—
120C4⋊Q8—24no—
121C4⋊Q8C4⋊C418yes—
122#125—24no—
123#125C4⋊C418yes—
124C22⋊Q8—28no—
125C22⋊Q8—28no—
126C4×Q8—28no—
127C4×Q8—28no—
128C4×Q8—88no—
129#117—22no—
130#117—22no—
131#117—22no—
132#117—22no—
133#117—22no—
134#117C22×C414yes—
135#117C2×Q814yes—
136#117#4114yes—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—84no—
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C4⋊C4C2×D414yes—
154C2×C4⋊C4C2×D412yes—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C42—22no—
159C2×C42—21no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—21no—
167C2×C42—22no—
168C2×C42—28no—
169C2×C42C22×C414yes—
170C2×C42C22×C412yes—
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D412yes—
174C2×C42C2×Q812yes—
175C2×C42#4112yes—
176C2×C42#4112yes—
177#107—44no—
178#106—44no—
179C4⋊C8—28no—
180#97—42no—
181C4×C8—42no—
182#95—28no—
183#95—28no—
184#95—28no—
185#95—22no—
186#95—22no—
187#95—42no—
188#95—42no—
189#95—88no—
190C24C22×D411yescharacteristic
191C2×Q8—28no—
192C2×Q8—44no—
193C2×Q8—44no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—21no—
198C22×C4—24no—
199C22×C4—21no—
200C22×C4—28no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—48no—
208C22×C4—44no—
209C22×C4—44no—
210C22×C4—42no—
211C22×C4—44no—
212C22×C4C4×D414yes—
213C22×C4#12012yes—
214C22×C4#12012yes—
215C22×C4C22⋊Q814yes—
216C22×C4C23×C411yescharacteristic
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D411yescharacteristic
219C22×C4C22×Q811yescharacteristic
220C22×C4C2×#4111yescharacteristic
221C22×C4C2×#4112yes—
222C22×C4C2×#4112yes—
223C22×C4C2×#4111yescharacteristic
224C22×C42−1+412yes—
225C22×C42−1+412yes—
226M16—28no—
227M16—24no—
228M16—24no—
229M16—44no—
230M16—42no—
231M16—42no—
232C2×C8—24no—
233C2×C8—24no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C4⋊C4—24no—
239C4⋊C4—22no—
240C4⋊C4—216no—
241C4⋊C4—28no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—48no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—44no—
253C4⋊C4—84no—
254C4⋊C4—88no—
255C4⋊C4C2×C4⋊C418yes—
256C4⋊C4#12414yes—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—24no—
260C22⋊C4—42no—
261C42—24no—
262C42—24no—
263C42—24no—
264C42—24no—
265C42—24no—
266C42—44no—
267C42—44no—
268C42—42no—
269C42—42no—
270C42—42no—
271C42—88no—
272C42C2×C4⋊C412yes—
273C42C2×C4⋊C412yes—
274C42C4×D414yes—
275C42C22⋊Q814yes—
276C42#12312yes—
277C42#12312yes—
278C23—22no—
279C23—22no—
280C23C2×C4×D411yescharacteristic
281C23C2×#12011yescharacteristic
282C23C2×C22⋊Q811yescharacteristic
283C23#54712yes—
284C23#54812yes—
285Q8—416no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—28no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—48no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—48no—
305C2×C4—42no—
306C2×C4—84no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4—88no—
310C2×C4—88no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4#38514yes—
315C2×C4#38514yes—
316C2×C4C22×C4⋊C411yescharacteristic
317C2×C4C2×C4×D411yescharacteristic
318C2×C4C4×#4112yes—
319C2×C4#51912yes—
320C2×C4C2×C22⋊Q811yescharacteristic
321C2×C4C2×#12311yescharacteristic
322C2×C4C2×#12411yescharacteristic
323C2×C4#52912yes—
324C2×C4#53211yescharacteristic
325C2×C4#53611yescharacteristic
326C2×C4#53611yescharacteristic
327C2×C4#54112yes—
328C2×C4#54812yes—
329C2×C4#54812yes—
330C2×C4#55212yes—
331C2×C4#55612yes—
332C2×C4#55712yes—
333C2×C4#55712yes—
334C8—44no—
335C8—44no—
336C22—24no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22C2×#38511yescharacteristic
342C22#236212yes—
343C22#243211yescharacteristic
344C22#243611yescharacteristic
345C22#245312yes—
346C22#301912yes—
347C22#302012yes—
348C4—24no—
349C4—24no—
350C4—44no—
351C4—48no—
352C4—84no—
353C4—81no—
354C4—84no—
355C4—81no—
356C4—162no—
357C4C4×#12411yescharacteristic
358C4#232711yescharacteristic
359C4#235112yes—
360C4#242112yes—
361C4#256511yescharacteristic
362C4#256811yescharacteristic
363C2—22no—
364C2—22no—
365C2#1397811yescharacteristic
366C2#2086412yes—
367C2#2569011yescharacteristic
368C2#2569111yescharacteristic
369C2#2592612yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #1995C4semidirect

Maximal subgroups

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

subgroupindex
#342142
C2×#19952
#213232
#211722
#143482
#112762
#108992
#108572
#107012