Finite Group Databasebeta

← #516396

Subgroups — #516396

Order 512.

5986 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#117C211yesmaximal, characteristic
2C2×#2315C212yesmaximal
3C22×#381C211yesmaximal, characteristic
4C22×#381C211yesmaximal, characteristic
5#14705C214yesmaximal, complemented
6#14263C214yesmaximal, complemented
7#14128C218yesmaximal, complemented
8#13878C218yesmaximal, complemented
9#13794C212yesmaximal
10C25×C4C2211yescharacteristic
11C22×#117—28no—
12C22×#117—216no—
13C22×#117C2214yes—
14C22×#117C2214yes—
15C23×C4⋊C4C2212yes—
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2212yes—
19C23×C22⋊C4C2212yes—
20C23×C42C2212yes—
21#2315—216no—
22C2×#385C2218yes—
23C2×#385C2218yes—
24C2×#381—28no—
25C2×#381C4116yescomplemented
26C2×#381C2214yes—
27C2×#381C2212yes—
28C2×#381C2212yes—
29C2×#381C2214yes—
30C2×#381C2212yes—
31C2×#381C2212yes—
32C2×#380C2218yes—
33C22×#95C2214yes—
34C22×#95C2214yes—
35#1444C22116yescomplemented
36#1442C22116yescomplemented
37#1436C22132yescomplemented
38C22⋊#95C2218yes—
39C22⋊#95C2218yes—
40C22⋊#95C2218yes—
41C26C2311yescharacteristic
42C24×C4—28no—
43C24×C4—28no—
44C24×C4C2×C418yes—
45C24×C4D418yes—
46C24×C4Q818yes—
47C24×C4C2311yescharacteristic
48C24×C4C2311yescharacteristic
49C24×C4C2312yes—
50C24×C4C2312yes—
51C24×C4C2312yes—
52C24×C4C2312yes—
53C24×C4C2314yes—
54C24×C4C2314yes—
55C2×#117—2128no—
56C2×#117—216no—
57C2×#117—216no—
58C2×#117—232no—
59C2×#117—232no—
60C22×C4⋊C4—216no—
61C22×C4⋊C4—28no—
62C22×C4⋊C4C2318yes—
63C22×C4⋊C4C23116yes—
64C22×C22⋊C4—28no—
65C22×C22⋊C4—28no—
66C22×C22⋊C4—28no—
67C22×C22⋊C4—28no—
68C22×C22⋊C4—28no—
69C22×C22⋊C4—24no—
70C22×C22⋊C4—24no—
71C22×C22⋊C4—28no—
72C22×C22⋊C4C2×C4116yes—
73C22×C22⋊C4D4116yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2312yes—
76C22×C22⋊C4C2312yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C42—216no—
83C22×C42—28no—
84C22×C42C2318yes—
85#385—232no—
86#385—232no—
87#381—28no—
88#381—216no—
89#381—28no—
90#381C2×C4132yescomplemented
91#381C2×C4116yescomplemented
92#381C2×C4116yescomplemented
93#380—232no—
94C2×#95—216no—
95C2×#95C2×C4132yescomplemented
96C2×#95C2318yes—
97C2×#95C23116yes—
98C2×#95C2318yes—
99C2×#95C23132yes—
100C2×#95C2318yes—
101C2×#95C2318yes—
102C25—48no—
103C25C22×C414yes—
104C25C2×D414yes—
105C25C2×D418yes—
106C25C2×Q814yes—
107C25#4118yes—
108C25C2411yescharacteristic
109C25C2411yescharacteristic
110C25C2411yescharacteristic
111C23×C4—264no—
112C23×C4—232no—
113C23×C4—264no—
114C23×C4—232no—
115C23×C4—24no—
116C23×C4—28no—
117C23×C4—28no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—28no—
122C23×C4—28no—
123C23×C4—28no—
124C23×C4—28no—
125C23×C4—28no—
126C23×C4—28no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—28no—
130C23×C4—28no—
131C23×C4—416no—
132C23×C4—416no—
133C23×C4—48no—
134C23×C4—48no—
135C23×C4C4⋊C4164yescomplemented
136C23×C4C22×C418yes—
137C23×C4C22×C414yes—
138C23×C4C22×C414yes—
139C23×C4C22×C418yes—
140C23×C4C22×C4116yes—
141C23×C4C2×D414yes—
142C23×C4C2×D418yes—
143C23×C4C2×D418yes—
144C23×C4C2×D414yes—
145C23×C4C2×D4116yes—
146C23×C4C2×Q814yes—
147C23×C4C2×Q814yes—
148C23×C4C2×Q818yes—
149C23×C4C2×Q818yes—
150C23×C4C2414yes—
151C23×C4C2414yes—
152C23×C4C2418yes—
153C23×C4C2418yes—
154C23×C4C2412yes—
155C23×C4C2412yes—
156#117—2128no—
157#117—2128no—
158C2×C4⋊C4—264no—
159C2×C4⋊C4—232no—
160C2×C4⋊C4—216no—
161C2×C4⋊C4—232no—
162C2×C4⋊C4—432no—
163C2×C22⋊C4—264no—
164C2×C22⋊C4—264no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—416no—
182C2×C22⋊C4—416no—
183C2×C22⋊C4C22×C4116yes—
184C2×C22⋊C4C22×C4116yes—
185C2×C22⋊C4C2×D4116yes—
186C2×C22⋊C4C2×D4116yes—
187C2×C42—264no—
188C2×C42—232no—
189C2×C42—216no—
190#95—216no—
191#95—216no—
192#95—232no—
193#95—432no—
194#95C22×C4132yes—
195#95C22×C4132yes—
196C24—28no—
197C24—28no—
198C24—28no—
199C24—416no—
200C24—416no—
201C24—432no—
202C24—416no—
203C24—432no—
204C24C2×C4⋊C4116yes—
205C24C4×D4132yes—
206C24#120116yes—
207C24C22⋊Q8132yes—
208C24#123116yes—
209C24C23×C412yes—
210C24C23×C412yes—
211C24C23×C412yes—
212C24C22×D412yes—
213C24C22×D412yes—
214C24C22×D412yes—
215C24C22×D414yes—
216C24C22×D414yes—
217C24C22×D414yes—
218C24C22×Q812yes—
219C24C22×Q812yes—
220C24C22×Q812yes—
221C24C2×#4114yes—
222C24C2×#4114yes—
223C24C2×#4114yes—
224C24C2511yescharacteristic
225C22×C4—216no—
226C22×C4—232no—
227C22×C4—216no—
228C22×C4—232no—
229C22×C4—28no—
230C22×C4—28no—
231C22×C4—216no—
232C22×C4—216no—
233C22×C4—216no—
234C22×C4—232no—
235C22×C4—232no—
236C22×C4—216no—
237C22×C4—28no—
238C22×C4—216no—
239C22×C4—28no—
240C22×C4—216no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—2256no—
246C22×C4—232no—
247C22×C4—28no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—232no—
252C22×C4—232no—
253C22×C4—232no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—28no—
257C22×C4—28no—
258C22×C4—216no—
259C22×C4—216no—
260C22×C4—432no—
261C22×C4—432no—
262C22×C4—432no—
263C22×C4—432no—
264C22×C4—416no—
265C22×C4—416no—
266C22×C4—48no—
267C22×C4—48no—
268C22×C4—816no—
269C22×C4—816no—
270C22×C4—816no—
271C22×C4—816no—
272C22×C4C2×C4⋊C4132yes—
273C22×C4C2×C4⋊C4116yes—
274C22×C4C2×C4⋊C4132yes—
275C22×C4C2×C4⋊C4116yes—
276C22×C4C23×C4116yes—
277C22×C4C23×C414yes—
278C22×C4C23×C414yes—
279C22×C4C22×D4116yes—
280C22×C4C22×D414yes—
281C22×C4C22×D414yes—
282C22×C4C22×Q814yes—
283C22×C4C22×Q814yes—
284C4⋊C4—264no—
285C4⋊C4—832no—
286C22⋊C4—232no—
287C22⋊C4—232no—
288C22⋊C4—232no—
289C22⋊C4—232no—
290C22⋊C4—48no—
291C22⋊C4—48no—
292C22⋊C4—48no—
293C22⋊C4—48no—
294C42—264no—
295C23—28no—
296C23—28no—
297C23—28no—
298C23—28no—
299C23—28no—
300C23—28no—
301C23—28no—
302C23—28no—
303C23—28no—
304C23—216no—
305C23—216no—
306C23—216no—
307C23—216no—
308C23—216no—
309C23—216no—
310C23—432no—
311C23—464no—
312C23—432no—
313C23—464no—
314C23—432no—
315C23#385164yes—
316C23C22×C4⋊C414yes—
317C23C22×C4⋊C414yes—
318C23C22×C4⋊C414yes—
319C23C2×C4×D418yes—
320C23C2×C4×D418yes—
321C23C2×C4×D418yes—
322C23C2×#12014yes—
323C23C2×#12014yes—
324C23C2×#12014yes—
325C23C2×C22⋊Q818yes—
326C23C2×C22⋊Q818yes—
327C23C2×C22⋊Q818yes—
328C23C2×#12314yes—
329C23C2×#12314yes—
330C23C2×#12314yes—
331C23C24×C411yescharacteristic
332C23C23×D411yescharacteristic
333C23C23×D412yes—
334C23C23×Q811yescharacteristic
335C23C22×#4112yes—
336C23#58518yes—
337C2×C4—28no—
338C2×C4—28no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—232no—
346C2×C4—232no—
347C2×C4—264no—
348C2×C4—264no—
349C2×C4—216no—
350C2×C4—232no—
351C2×C4—216no—
352C2×C4—232no—
353C2×C4—464no—
354C2×C4—416no—
355C2×C4—416no—
356C2×C4—416no—
357C2×C4—416no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—48no—
361C2×C4—48no—
362C2×C4—416no—
363C2×C4—416no—
364C2×C4—832no—
365C2×C4—816no—
366C2×C4—816no—
367C2×C4C22×C4⋊C418yes—
368C2×C4C22×C4⋊C418yes—
369C22—28no—
370C22—28no—
371C22—28no—
372C22—28no—
373C22—28no—
374C22—28no—
375C22—216no—
376C22—216no—
377C22—216no—
378C22—28no—
379C22—28no—
380C22—28no—
381C22—216no—
382C22—216no—
383C22—216no—
384C22—464no—
385C22C2×#38518yes—
386C22C2×#38518yes—
387C22C2×#38518yes—
388C22C23×C4⋊C411yescharacteristic
389C22C22×C4×D412yes—
390C22#342814yes—
391C22C22×#12011yescharacteristic
392C22C22×C22⋊Q812yes—
393C22C22×#12311yescharacteristic
394C22#344914yes—
395C22#345814yes—
396C22#348518yes—
397C22#349118yes—
398C4—28no—
399C4—28no—
400C4—416no—
401C4—416no—
402C4—84no—
403C4—84no—
404C2—28no—
405C2—28no—
406C2—28no—
407C2C22×#38511yescharacteristic
408C2#3405212yes—
409C2#3419914yes—
410C2#3508812yes—
411C2#3509014yes—
412C2#3509712yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #14705C2semidirect
6. #14263C2semidirect
7. #14128C2semidirect
8. #13878C2semidirect
25. C2×#381C4semidirect
35. #1444C22semidirect
36. #1442C22semidirect
37. #1436C22semidirect
90. #381C2×C4semidirect
91. #381C2×C4semidirect
92. #381C2×C4semidirect
95. C2×#95C2×C4semidirect
135. C23×C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C23×#1172
C2×#23152
C22×#3812
#147052
#142632
#141282
#138782
#137942