Finite Group Databasebeta

← #6434681

Subgroups — #6434681

Order 512.

2888 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3458C211yesmaximal, characteristic
2C2×#3095C212yesmaximal
3C2×#2625C211yesmaximal, characteristic
4#29649C218yesmaximal, complemented
5#29648C218yesmaximal, complemented
6#23554C214yesmaximal, complemented
7#23455C214yesmaximal, complemented
8#20702C212yesmaximal
9#20683C211yesmaximal, characteristic
10#3458—216no—
11C2×#544—21no—
12C2×#544—21no—
13C2×#544C2214yes—
14C2×#544C2214yes—
15C2×#543—22no—
16C2×#533—24no—
17C2×#533C2214yes—
18C22×C22⋊Q8C2211yescharacteristic
19C22×#117C2212yes—
20C22×#117C2211yescharacteristic
21#3095—216no—
22C2×#484C2214yes—
23C2×#485C2218yes—
24C2×#483C2214yes—
25C2×#428C2214yes—
26C2×#419C2214yes—
27C2×#407C2212yes—
28#2625—24no—
29#2625C2218yes—
30C2×#400—21no—
31C2×#398—21no—
32C2×#397—22no—
33C2×#391C2212yes—
34C2×#389C2212yes—
35C2×#380C2211yescharacteristic
36#2059C22116yescomplemented
37#2058C2218yescomplemented
38#2057C2218yescomplemented
39#2026C22116yescomplemented
40#2024C2218yescomplemented
41#2024C2218yescomplemented
42#1825C2218yes—
43#1824C2214yes—
44#1823C2214yes—
45#1818C2218yes—
46#1736C2218yes—
47#1728C2214yes—
48C23×Q8C2311yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16C2314yes—
51#544—216no—
52#544—216no—
53#544—216no—
54#544—216no—
55#544—216no—
56#544—216no—
57#543—232no—
58#533—264no—
59#533—232no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—24no—
62C2×C4⋊Q8C2314yes—
63C2×C4⋊Q8C2314yes—
64C2×#125—24no—
65C2×#125—24no—
66C2×#125C2318yes—
67C2×C22⋊Q8—24no—
68C2×C22⋊Q8—24no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—28no—
71C2×C22⋊Q8—28no—
72C2×C22⋊Q8—28no—
73C2×C22⋊Q8C2314yes—
74C2×C4×Q8—24no—
75C2×C4×Q8—24no—
76C2×#117—22no—
77C2×#117—22no—
78C2×#117—28no—
79C2×#117—28no—
80C2×#117D4116yes—
81C2×#117D414yes—
82C2×#117C2314yes—
83C2×#117C2314yes—
84C2×#117C2314yes—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4D411yescharacteristic
88C22×C4⋊C4D411yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C22⋊C4—22no—
94C22×C22⋊C4—21no—
95C22×C22⋊C4—21no—
96C22×C22⋊C4D412yes—
97C22×C42—22no—
98C22×C42C2312yes—
99#485—232no—
100#484—216no—
101#483—216no—
102#428—216no—
103C2×#107C2318yes—
104C2×#106C2318yes—
105#419—216no—
106C2×#103C2318yes—
107C2×#103C2318yes—
108#407—28no—
109C2×C22⋊C8C2314yes—
110C2×C22⋊C8C2314yes—
111#400—28no—
112#398—28no—
113#397—216no—
114#391—24no—
115#391C2318yes—
116#389—24no—
117#389C2318yes—
118#380—22no—
119#380C2314yes—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—21no—
124#340C23116yes—
125#340C23116yes—
126#336C2318yes—
127#336C2318yes—
128C25C2×D411yescharacteristic
129C22×Q8—24no—
130C22×Q8—24no—
131C22×Q8—21no—
132C22×Q8—24no—
133C22×Q8—28no—
134C22×Q8C2412yes—
135C23×C4—28no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D418yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C23×C4C2412yes—
147C23×C4C2411yescharacteristic
148C2×M16—28no—
149C2×M16—216no—
150C22×C8C2418yes—
151C4⋊Q8—232no—
152C4⋊Q8—232no—
153C4⋊Q8—216no—
154C4⋊Q8—216no—
155#125—232no—
156#125—232no—
157#125—216no—
158#125—216no—
159C22⋊Q8—232no—
160C22⋊Q8—232no—
161C22⋊Q8—264no—
162C22⋊Q8—264no—
163C22⋊Q8—28no—
164C22⋊Q8—28no—
165C22⋊Q8—48no—
166C4×Q8—232no—
167C4×Q8—232no—
168#117—216no—
169#117—216no—
170#117—216no—
171#117—28no—
172#117—28no—
173#117—28no—
174#117—28no—
175#117C2×D4116yes—
176#117C2×D4116yes—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—84no—
195C2×C4⋊C4C2×D414yes—
196C2×C4⋊C4C2×D414yes—
197C2×C4⋊C4C2×D418yes—
198C2×C4⋊C4C2×D418yes—
199C2×C4⋊C4C2412yes—
200C2×C4⋊C4C2414yes—
201C2×C4⋊C4C2414yes—
202C2×C4⋊C4C2414yes—
203C2×C22⋊C4—216no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—82no—
211C2×C22⋊C4C2×D418yes—
212C2×C42—216no—
213C2×C42—28no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42C2×D414yes—
217C2×C42C2414yes—
218#107—216no—
219#106—216no—
220#103—216no—
221#103—216no—
222C22⋊C8—28no—
223C22⋊C8—28no—
224#95—24no—
225#95—28no—
226#95—28no—
227#95—28no—
228C24—24no—
229C24—41no—
230C24—41no—
231C24—42no—
232C24#12014yes—
233C24C22×D412yes—
234C24C22×D411yescharacteristic
235C2×Q8—216no—
236C2×Q8—216no—
237C2×Q8—24no—
238C2×Q8—28no—
239C2×Q8—42no—
240C2×Q8—44no—
241C2×Q8—416no—
242C2×Q8—84no—
243C2×Q8—88no—
244C2×Q8—84no—
245C22×C4—24no—
246C22×C4—232no—
247C22×C4—28no—
248C22×C4—28no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—82no—
270C22×C4—84no—
271C22×C4—82no—
272C22×C4#12018yes—
273C22×C4#12014yes—
274C22×C4C22×D414yes—
275C22×C4C22×D414yes—
276C22×C4C22×D412yes—
277C22×C4C22×D414yes—
278C22×C4C22×D414yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D414yes—
281C22×C42−1+418yes—
282C22×C4C2511yescharacteristic
283M16—416no—
284C2×C8—28no—
285C2×C8—216no—
286C4⋊C4—216no—
287C4⋊C4—216no—
288C4⋊C4—232no—
289C4⋊C4—232no—
290C4⋊C4—28no—
291C4⋊C4—28no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—28no—
298C4⋊C4—42no—
299C4⋊C4—48no—
300C4⋊C4—44no—
301C4⋊C4—88no—
302C4⋊C4C22×D418yes—
303C4⋊C4C22×D418yes—
304C22⋊C4—216no—
305C22⋊C4—216no—
306C22⋊C4—216no—
307C22⋊C4—48no—
308C22⋊C4—84no—
309C42—216no—
310C42—216no—
311C42—28no—
312C42—28no—
313C23—21no—
314C23—21no—
315C23—22no—
316C23—22no—
317C23—24no—
318C23—22no—
319C23—24no—
320C23—28no—
321C23—44no—
322C23—44no—
323C23—84no—
324C23—84no—
325C23C2×#12014yes—
326C23C2×#12012yes—
327C23#53614yes—
328C23C23×D411yescharacteristic
329Q8—48no—
330Q8—816no—
331Q8—816no—
332Q8—84no—
333C2×C4—28no—
334C2×C4—24no—
335C2×C4—216no—
336C2×C4—28no—
337C2×C4—28no—
338C2×C4—24no—
339C2×C4—28no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—44no—
346C2×C4—48no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—88no—
359C2×C4—88no—
360C2×C4—88no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4C2×#12014yes—
364C2×C4C2×#12012yes—
365C2×C4#53614yes—
366C2×C4#536116yes—
367C2×C4C23×D412yes—
368C2×C4C2×2−1+414yes—
369C8—48no—
370C22—24no—
371C22—24no—
372C22—22no—
373C22—22no—
374C22—24no—
375C22—24no—
376C22—24no—
377C22—44no—
378C22—44no—
379C22—88no—
380C22#261414yes—
381C22C22×#12011yescharacteristic
382C22C2×#53612yes—
383C22#358511yescharacteristic
384C22#358611yescharacteristic
385C22#358612yes—
386C4—24no—
387C4—44no—
388C4—48no—
389C4—44no—
390C4—48no—
391C4—82no—
392C4—84no—
393C4—82no—
394C4#261414yes—
395C4C2×#53614yes—
396C2—24no—
397C2—44no—
398C2C2×#261411yescharacteristic
399C2#6037011yescharacteristic
400C2#6037111yescharacteristic
401C2#6086912yes—
402C2#6087012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×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. #29649C2semidirect
5. #29648C2semidirect
6. #23554C2semidirect
7. #23455C2semidirect
36. #2059C22semidirect
37. #2058C22semidirect
38. #2057C22semidirect
39. #2026C22semidirect
40. #2024C22semidirect
41. #2024C22semidirect

Maximal subgroups

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

subgroupindex
C2×#34582
C2×#30952
C2×#26252
#296492
#296482
#235542
#234552
#207022
#206832