Finite Group Databasebeta

← #393460

Subgroups — #393460

Order 512.

1494 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33996C211yesmaximal, characteristic
2#20846C212yesmaximal
3C2×#1822C211yesmaximal, characteristic
4#13939C211yesmaximal, characteristic
5#10899C214yesmaximal, complemented
6#10899C214yesmaximal, complemented
7#10733C212yesmaximal
8C22×#117C2212yes—
9C22×#117C2211yescharacteristic
10C2×#428C2214yes—
11C2×#407C2212yes—
12C2×#380C2211yescharacteristic
13#2274—22no—
14#2274—22no—
15#2274C2212yes—
16#2274C2212yes—
17#2273—22no—
18C4×#117—22no—
19C4×#117—24no—
20C2×C4×C22⋊C4C2211yescharacteristic
21#1854—24no—
22#1853—22no—
23#1852—22no—
24#1822C418yescomplemented
25C2×#343C2212yes—
26C2×#343C2212yes—
27#1736C2214yes—
28C2×#339C2214yes—
29C2×#339C2214yes—
30#1440—22no—
31C4⋊#95C2212yes—
32C4⋊#95C2212yes—
33#1434—22no—
34C24×C4C2311yescharacteristic
35C22×M16C2314yes—
36C2×#117—22no—
37C2×#117—22no—
38C2×#117—22no—
39C2×#117—22no—
40C2×#117—24no—
41C2×#117—28no—
42C2×#117—28no—
43C2×#117C2×C414yes—
44C2×#117D414yes—
45C2×#117Q814yes—
46C2×#117C2314yes—
47C22×C4⋊C4D412yes—
48C22×C4⋊C4D412yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C22⋊C4—22no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C22⋊C4D412yes—
56C22×C42—22no—
57C22×C42D412yes—
58C22×C42C2312yes—
59#428—28no—
60#428—28no—
61C2×#107—24no—
62C2×#106—24no—
63#407—24no—
64#407C2×C418yes—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67#380—22no—
68#380—22no—
69C4×C4⋊C4—22no—
70C4×C4⋊C4—24no—
71C4×C4⋊C4—22no—
72C4×C4⋊C4—24no—
73C4×C4⋊C4—24no—
74C4×C4⋊C4—24no—
75C4×C4⋊C4—24no—
76C4×C22⋊C4—28no—
77C4×C22⋊C4—28no—
78C4×C22⋊C4—24no—
79C4×C22⋊C4—24no—
80C4×C22⋊C4—24no—
81#376—24no—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—24no—
85C2×#95—24no—
86C43—24no—
87#343C2×C418yescomplemented
88#343C2×C418yescomplemented
89#340—24no—
90#340—24no—
91#339—28no—
92#339—28no—
93C25C2×D411yescharacteristic
94C23×C4—24no—
95C23×C4—24no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4C22×C412yes—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D412yes—
103C23×C4C2×D412yes—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×Q812yes—
107C23×C4#4114yes—
108C23×C4C2411yescharacteristic
109C2×M16—24no—
110C2×M16—24no—
111C2×M16—24no—
112C2×M16—24no—
113C2×M16—24no—
114C2×M16C22×C418yes—
115C22×C8—24no—
116#117—28no—
117#117—28no—
118#117—216no—
119#117—28no—
120#117—28no—
121#117—28no—
122#117—28no—
123#117—24no—
124#117—24no—
125#117—28no—
126#117—28no—
127#117#4118yes—
128#117#4118yes—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—28no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4C22×C412yes—
142C2×C4⋊C4C22×C412yes—
143C2×C4⋊C4C2×D414yes—
144C2×C4⋊C4C2×D414yes—
145C2×C4⋊C4C2×Q812yes—
146C2×C4⋊C4C2×Q812yes—
147C2×C22⋊C4—28no—
148C2×C22⋊C4—28no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—82no—
157C2×C22⋊C4C4⋊C418yes—
158C2×C42—216no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—24no—
162C2×C42—24no—
163C2×C42—24no—
164C2×C42—24no—
165C2×C42—24no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—28no—
169C2×C42—24no—
170C2×C42—24no—
171C2×C42C4⋊C418yes—
172#107—48no—
173#106—48no—
174C22⋊C8—28no—
175C22⋊C8—28no—
176#95—24no—
177#95—24no—
178#95—28no—
179#95—48no—
180#95—41no—
181#95—41no—
182#95—42no—
183#95—84no—
184C24—22no—
185C24—22no—
186C24—42no—
187C24—41no—
188C24—41no—
189C24C2×C4⋊C412yes—
190C24#12012yes—
191C24#12312yes—
192C24C22×D411yescharacteristic
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—24no—
198C22×C4—24no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—28no—
202C22×C4—28no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—28no—
210C22×C4—28no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—44no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4C2×C4⋊C412yes—
233C22×C4C2×C4⋊C414yes—
234C22×C4C2×C4⋊C414yes—
235C22×C4C4×D414yes—
236C22×C4C4×D414yes—
237C22×C4#12011yescharacteristic
238C22×C4#12011yescharacteristic
239C22×C4C22⋊Q814yes—
240C22×C4C22⋊Q814yes—
241C22×C4#12312yes—
242C22×C4C23×C411yescharacteristic
243C22×C4C22×D412yes—
244C22×C4C22×Q811yescharacteristic
245C22×C4C2×#4112yes—
246C22×C4C2×#4114yes—
247C22×C4C2×#4114yes—
248M16—28no—
249M16—28no—
250M16—44no—
251M16—44no—
252C2×C8—28no—
253C2×C8—44no—
254C2×C8—44no—
255C4⋊C4—28no—
256C4⋊C4—28no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—44no—
264C4⋊C4—44no—
265C4⋊C4#12318yes—
266C4⋊C4#12318yes—
267C22⋊C4—216no—
268C22⋊C4—28no—
269C22⋊C4—28no—
270C22⋊C4—216no—
271C22⋊C4—48no—
272C22⋊C4—84no—
273C42—216no—
274C42—28no—
275C42—28no—
276C42—28no—
277C42—28no—
278C42—44no—
279C42—44no—
280C42—88no—
281C42—84no—
282C42—84no—
283C23—22no—
284C23—21no—
285C23—21no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—24no—
293C23—22no—
294C23—22no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—82no—
300C23—82no—
301C23—82no—
302C23—82no—
303C23#38514yes—
304C23C22×C4⋊C411yescharacteristic
305C23C2×#12011yescharacteristic
306C23C2×#12311yescharacteristic
307C23#52512yes—
308C23#52512yes—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—28no—
315C2×C4—28no—
316C2×C4—28no—
317C2×C4—28no—
318C2×C4—28no—
319C2×C4—28no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—88no—
337C2×C4—88no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4#38512yes—
348C2×C4#38512yes—
349C2×C4C2×C4×D412yes—
350C2×C4C4×#4112yes—
351C2×C4C4×#4112yes—
352C2×C4C2×C22⋊Q812yes—
353C2×C4C2×#12314yes—
354C2×C4#52514yes—
355C2×C4#52914yes—
356C2×C4#52914yes—
357C2×C4#53312yes—
358C2×C4#53312yes—
359C8—48no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—24no—
369C22—24no—
370C22—24no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—84no—
376C22—82no—
377C22—82no—
378C22C2×#38511yescharacteristic
379C22#229611yescharacteristic
380C22#229611yescharacteristic
381C22#284912yes—
382C22#285312yes—
383C22#301812yes—
384C22#302111yescharacteristic
385C22#302211yescharacteristic
386C4—22no—
387C4—22no—
388C4—44no—
389C4—44no—
390C4—48no—
391C4—48no—
392C4—84no—
393C4—162no—
394C4C4×#12314yes—
395C4#285514yes—
396C2—22no—
397C2—21no—
398C2—21no—
399C2—44no—
400C2#1470511yescharacteristic
401C2#2085511yescharacteristic
402C2#2085811yescharacteristic
403C2#3299812yes—
404C2#3300112yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10899C2semidirect
6. #10899C2semidirect
24. #1822C4semidirect
87. #343C2×C4semidirect
88. #343C2×C4semidirect

Maximal subgroups

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

subgroupindex
#339962
#208462
C2×#18222
#139392
#108992
#107332