Finite Group Databasebeta

← #6374367

Subgroups — #6374367

Order 512.

3026 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#544C211yesmaximal, characteristic
2C2×#2921C211yesmaximal, characteristic
3#34048C211yesmaximal, characteristic
4#21843C2116yesmaximal, complemented
5#21325C212yesmaximal, complemented
6#21324C212yesmaximal, complemented
7#21314C212yesmaximal, complemented
8#21313C212yesmaximal, complemented
9#20701C212yesmaximal
10#20700C212yesmaximal
11C2×#544—28no—
12C2×#544—24no—
13C2×#544C4116yescomplemented
14C2×#544C2218yes—
15C22×C4⋊Q8—21no—
16C22×C4⋊Q8C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#125—21no—
19C22×#125C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C22×C22⋊Q8—21no—
22C22×C22⋊Q8—21no—
23C22×#117C2211yescharacteristic
24C22×#117C2211yescharacteristic
25C23×C4⋊C4C2211yescharacteristic
26#2921—28no—
27C2×#433C2218yes—
28C2×#423C2212yes—
29C2×#407C2212yes—
30C2×C4×M16C2212yes—
31#2314C2218yes—
32#2313—21no—
33#2313—21no—
34#2312—21no—
35#2312—21no—
36#2298—24no—
37#2285C2212yes—
38#2280C2212yes—
39C4×#117C2212yes—
40#1993C2218yescomplemented
41#1992C2218yescomplemented
42#1991C2218yescomplemented
43#1990C2218yescomplemented
44#1989C2218yescomplemented
45#1989C2218yescomplemented
46#1988C2218yescomplemented
47#1987C2218yescomplemented
48#1825C2218yes—
49#1825C2218yes—
50#1824C2218yes—
51#1823C2218yes—
52#1736C2214yes—
53#1736C2214yes—
54#1736C2214yes—
55#1736C2214yes—
56C23×Q8—21no—
57C23×Q8—21no—
58C24×C4—21no—
59C24×C4C2311yescharacteristic
60C22×M16C2314yes—
61#544—264no—
62#544—232no—
63#544C2×C4164yes—
64C2×C4⋊Q8—216no—
65C2×C4⋊Q8—28no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8—22no—
68C2×C4⋊Q8—24no—
69C2×C4⋊Q8—24no—
70C2×C4⋊Q8C2×C418yes—
71C2×C4⋊Q8C2×C418yes—
72C2×C4⋊Q8C2314yes—
73C2×C4⋊Q8C2314yes—
74C2×#125—216no—
75C2×#125—28no—
76C2×#125—22no—
77C2×#125—22no—
78C2×#125—24no—
79C2×#125—24no—
80C2×#125C2×C418yes—
81C2×#125C2×C418yes—
82C2×#125C2314yes—
83C2×#125C2314yes—
84C2×C22⋊Q8—216no—
85C2×C22⋊Q8—216no—
86C2×C22⋊Q8—28no—
87C2×C22⋊Q8—28no—
88C2×#117—24no—
89C2×#117—22no—
90C2×#117C2×C418yes—
91C2×#117C2×C418yes—
92C2×#117D418yes—
93C2×#117D414yes—
94C2×#117C2314yes—
95C2×#117C2314yes—
96C2×#117C2314yes—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—22no—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4C2×C418yes—
107C22×C4⋊C4D411yescharacteristic
108C22×C4⋊C4D411yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4—21no—
115C22×C22⋊C4D411yescharacteristic
116C22×C22⋊C4D411yescharacteristic
117C22×C42—21no—
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120#433—232no—
121#423—28no—
122C2×C4⋊C8C2314yes—
123C2×C4⋊C8C2314yes—
124#407—28no—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2314yes—
127C4×M16—28no—
128C2×#97C2314yes—
129C2×C4×C8C2314yes—
130#388—28no—
131#388—28no—
132#387—28no—
133#387—28no—
134#384C2314yes—
135#382C2314yes—
136#381—22no—
137#380—22no—
138#380—22no—
139#380C2314yes—
140#380C2314yes—
141#379—22no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4C2314yes—
145C4×C4⋊C4C2314yes—
146C4×C22⋊C4—22no—
147C4×C22⋊C4—22no—
148C4×C22⋊C4—22no—
149C4×C22⋊C4—22no—
150#376—22no—
151C43—22no—
152#340C23116yes—
153#340C23116yes—
154#340C23116yes—
155#340C23116yes—
156C25C2×D411yescharacteristic
157C22×Q8—28no—
158C22×Q8—28no—
159C22×Q8—24no—
160C22×Q8—24no—
161C22×Q8—84no—
162C22×Q8—84no—
163C23×C4—28no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4—28no—
169C23×C4—24no—
170C23×C4—84no—
171C23×C4C22×C414yes—
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D412yes—
174C23×C4C2×D414yes—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2412yes—
180C23×C4C2411yescharacteristic
181C2×M16—28no—
182C2×M16—216no—
183C22×C8C2418yes—
184C4⋊Q8—264no—
185C4⋊Q8—216no—
186C4⋊Q8—216no—
187C4⋊Q8—28no—
188C4⋊Q8—28no—
189C4⋊Q8C22×C4116yes—
190C4⋊Q8C22×C4116yes—
191#125—264no—
192#125—216no—
193#125—216no—
194#125—28no—
195#125—28no—
196#125C22×C4116yes—
197#125C22×C4116yes—
198C22⋊Q8—264no—
199C22⋊Q8—264no—
200#117—216no—
201#117—28no—
202#117—216no—
203#117C22⋊C4132yes—
204#117C22×C4116yes—
205#117C22×C4116yes—
206#117C2×D4116yes—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—216no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4C22⋊C418yes—
234C2×C4⋊C4C22⋊C418yes—
235C2×C4⋊C4C22×C414yes—
236C2×C4⋊C4C22×C414yes—
237C2×C4⋊C4C22×C4116yes—
238C2×C4⋊C4C22×C414yes—
239C2×C4⋊C4C22×C414yes—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4C2×D414yes—
242C2×C4⋊C4C2418yes—
243C2×C4⋊C4C2412yes—
244C2×C4⋊C4C2412yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4C22⋊C418yes—
253C2×C22⋊C4C22⋊C418yes—
254C2×C22⋊C4C2×D414yes—
255C2×C22⋊C4C2×D414yes—
256C2×C42—28no—
257C2×C42—24no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42C22×C414yes—
262C2×C42C22×C414yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D414yes—
266C2×C42C2×D414yes—
267C2×C42C2412yes—
268C2×C42C2412yes—
269C4⋊C8—28no—
270C4⋊C8—28no—
271C22⋊C8—28no—
272C22⋊C8—28no—
273#97—28no—
274C4×C8—28no—
275#95—24no—
276#95—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281C24—24no—
282C24—42no—
283C24—42no—
284C24C2×C22⋊C414yes—
285C24C22×D412yes—
286C24C22×D411yescharacteristic
287C2×Q8—216no—
288C2×Q8—216no—
289C2×Q8—88no—
290C2×Q8—88no—
291C2×Q8—816no—
292C2×Q8—816no—
293C22×C4—24no—
294C22×C4—216no—
295C22×C4—216no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—216no—
307C22×C4—28no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—88no—
317C22×C4—816no—
318C22×C4C2×C22⋊C414yes—
319C22×C4C2×C22⋊C418yes—
320C22×C4C2×C22⋊C414yes—
321C22×C4C2×C22⋊C414yes—
322C22×C4C2×C22⋊C414yes—
323C22×C4C2×C22⋊C414yes—
324C22×C4C23×C414yes—
325C22×C4C23×C412yes—
326C22×C4C22×D412yes—
327C22×C4C22×D412yes—
328C22×C4C22×D412yes—
329C22×C4C22×D412yes—
330C22×C4C22×D412yes—
331C22×C4C22×D414yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D412yes—
334C22×C42−1+418yes—
335C22×C4C2511yescharacteristic
336M16—416no—
337C2×C8—28no—
338C2×C8—216no—
339C4⋊C4—216no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—216no—
343C4⋊C4—216no—
344C4⋊C4—216no—
345C4⋊C4—216no—
346C4⋊C4—216no—
347C4⋊C4—216no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—216no—
355C4⋊C4—84no—
356C4⋊C4—84no—
357C4⋊C4C2×C22⋊C4116yes—
358C4⋊C4C2×C22⋊C4116yes—
359C4⋊C4C23×C414yes—
360C4⋊C4C23×C414yes—
361C4⋊C4C23×C414yes—
362C4⋊C4C23×C414yes—
363C22⋊C4—216no—
364C22⋊C4—48no—
365C22⋊C4—84no—
366C22⋊C4—84no—
367C22⋊C4C2×C22⋊C4116yes—
368C22⋊C4C2×C22⋊C4116yes—
369C42—216no—
370C42—48no—
371C42—84no—
372C42—84no—
373C42C2×C22⋊C418yes—
374C42C2×C22⋊C418yes—
375C42C23×C414yes—
376C42C23×C414yes—
377C42C22×D414yes—
378C42C22×D414yes—
379C23—22no—
380C23—22no—
381C23—24no—
382C23—24no—
383C23—28no—
384C23—24no—
385C23—44no—
386C23—44no—
387C23—416no—
388C23C22×C22⋊C414yes—
389C23C22×C22⋊C412yes—
390C23#53614yes—
391C23C23×D411yescharacteristic
392Q8—816no—
393Q8—816no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—28no—
399C2×C4—24no—
400C2×C4—216no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—48no—
412C2×C4—48no—
413C2×C4—48no—
414C2×C4—48no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—816no—
423C2×C4C22×C22⋊C414yes—
424C2×C4C22×C22⋊C414yes—
425C2×C4C22×C22⋊C414yes—
426C2×C4C22×C22⋊C414yes—
427C2×C4C22×C22⋊C414yes—
428C2×C4C22×C22⋊C412yes—
429C2×C4#51918yes—
430C2×C4#53614yes—
431C2×C4#53618yes—
432C2×C4C24×C411yescharacteristic
433C2×C4C23×D411yescharacteristic
434C2×C4C2×2−1+414yes—
435C8—48no—
436C22—22no—
437C22—22no—
438C22—28no—
439C22—24no—
440C22—24no—
441C22—24no—
442C22—416no—
443C22—48no—
444C22#231014yes—
445C22C23×C22⋊C411yescharacteristic
446C22C2×#53612yes—
447C22#358512yes—
448C22#358612yes—
449C4—24no—
450C4—24no—
451C4—24no—
452C4—24no—
453C4—24no—
454C4—44no—
455C4—44no—
456C4—44no—
457C4—44no—
458C4—88no—
459C4#231014yes—
460C4C2×#51912yes—
461C4C2×#53612yes—
462C2—24no—
463C2—44no—
464C2C2×#231011yescharacteristic
465C2#6008711yescharacteristic
466C2#6008811yescharacteristic
467C2#6086714yes—
subgroupG/H
center, socleC23C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #21843C2semidirect
5. #21325C2semidirect
6. #21324C2semidirect
7. #21314C2semidirect
8. #21313C2semidirect
13. C2×#544C4semidirect
40. #1993C22semidirect
41. #1992C22semidirect
42. #1991C22semidirect
43. #1990C22semidirect
44. #1989C22semidirect
45. #1989C22semidirect
46. #1988C22semidirect
47. #1987C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5442
C2×#29212
#340482
#218432
#213252
#213242
#213142
#213132
#207012
#207002