Finite Group Databasebeta

← #6357308

Subgroups — #6357308

Order 512.

1844 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60943C212yesmaximal
2C2×#3229C212yesmaximal
3#60663C211yesmaximal, characteristic
4C2×#3139C211yesmaximal, characteristic
5#19706C218yesmaximal, complemented
6#19681C218yesmaximal, complemented
7#19680C218yesmaximal, complemented
8#17319C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C2×#533C2212yes—
11C22×C4⋊Q8C2211yescharacteristic
12#3249—24no—
13#3230C2214yes—
14#3230C2214yes—
15#3229—24no—
16#3229—24no—
17#3229C2218yes—
18#3229C2218yes—
19#3209C2214yes—
20#3209C2214yes—
21#3189—24no—
22#3170C2218yes—
23#3166—24no—
24#3165C2214yes—
25#3165C2214yes—
26#3162—22no—
27#3160C2214yes—
28C2×#498C2214yes—
29#3139—24no—
30#3139C2218yes—
31C2×#491—22no—
32C2×#487C2214yes—
33C2×#487C2214yes—
34#3104—24no—
35#3104—24no—
36C2×#483—24no—
37C2×#475—24no—
38C2×#431C2212yes—
39#2911—22no—
40C2×#419—22no—
41C2×C4×M16C2211yescharacteristic
42#1723C22116yescomplemented
43#1718C22116yescomplemented
44#1718C22116yescomplemented
45#1716C22116yescomplemented
46#1452C2214yes—
47#1451C2218yes—
48C23×Q8—21no—
49C22×M16D412yes—
50#533—22no—
51#533—22no—
52#533—22no—
53#533—22no—
54#533—22no—
55#533—22no—
56#533—22no—
57#533—22no—
58#533—22no—
59#533C2314yes—
60#533C2314yes—
61#533C2314yes—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—21no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8C2312yes—
68C2×C4⋊Q8C2312yes—
69C2×C4⋊Q8C2312yes—
70C2×C4⋊Q8C2314yes—
71C2×C4⋊Q8C2312yes—
72C2×#125—21no—
73C2×#125—22no—
74C2×C22⋊Q8—22no—
75C2×C22⋊Q8—22no—
76C2×C22⋊Q8—22no—
77C2×C4×Q8—22no—
78C2×C4×Q8—22no—
79C2×C4×Q8—22no—
80C2×#117—21no—
81C2×#117—22no—
82C22×C4⋊C4—22no—
83C22×C42C2311yescharacteristic
84#501—28no—
85#501—28no—
86#500C2318yes—
87#500C2318yes—
88#499—28no—
89#498—28no—
90#498C23116yes—
91#498C2314yes—
92#498C2314yes—
93#494C2318yes—
94#491—216no—
95#487—28no—
96#487—28no—
97#487C23116yes—
98#487C23116yes—
99#487C2318yes—
100#487C2318yes—
101#484—28no—
102#484—28no—
103#483—216no—
104#483—216no—
105#477—28no—
106#477—28no—
107#475—216no—
108#475—216no—
109#467—216no—
110#462—28no—
111#462—28no—
112#451—28no—
113#451—28no—
114C4×Q16—216no—
115#431D414yes—
116#431D414yes—
117#431C2314yes—
118#431C2314yes—
119#428—28no—
120#428—28no—
121#427—28no—
122C2×#106—28no—
123C2×C4⋊C8D414yes—
124#419—216no—
125C2×#103—28no—
126C2×#103—28no—
127C2×C22⋊C8D414yes—
128C4×M16D414yes—
129C4×M16C2314yes—
130C2×#97D412yes—
131C2×C4×C8C2312yes—
132C2×C4×C8C2314yes—
133#322C23116yes—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—24no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—42no—
140C22×Q8—41no—
141C22×Q8—48no—
142C23×C4—41no—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C2×Q16—216no—
146C2×M16—28no—
147C2×M16C2×D418yes—
148C22×C8C2×D414yes—
149C22×C8C2×D418yes—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—24no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—41no—
165C4⋊Q8—42no—
166C4⋊Q8—41no—
167C4⋊Q8—42no—
168C4⋊Q8C2414yes—
169C4⋊Q8C2414yes—
170C4⋊Q8C2414yes—
171C4⋊Q8C2414yes—
172#125—24no—
173#125—24no—
174#125—24no—
175#125—41no—
176#125—41no—
177#125—42no—
178#125—42no—
179C22⋊Q8—24no—
180C22⋊Q8—24no—
181C22⋊Q8—24no—
182C22⋊Q8—24no—
183C22⋊Q8—24no—
184C22⋊Q8—24no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188C22⋊Q8—42no—
189C22⋊Q8—42no—
190C22⋊Q8—42no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203#117—24no—
204#117—24no—
205#117—24no—
206#117—42no—
207#117—42no—
208#117—42no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—42no—
226C2×C42—41no—
227C2×C42—42no—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D412yes—
233C2×C42C2412yes—
234C2×C42C2411yescharacteristic
235#107—216no—
236#107—216no—
237#106—232no—
238#106—28no—
239#106—28no—
240C4⋊C8C2×D418yes—
241C4⋊C8C2×D418yes—
242#103—232no—
243#103—232no—
244#103—216no—
245#103—216no—
246C22⋊C8C2×D418yes—
247C22⋊C8C2×D418yes—
248#97C2×D418yes—
249C4×C8C2×D414yes—
250C4×C8C2×D418yes—
251C4×C8C2414yes—
252C4×C8C2418yes—
253C24#12711yescharacteristic
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—48no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—44no—
264C2×Q8—48no—
265C2×Q8—416no—
266C2×Q8—84no—
267C2×Q8—84no—
268C2×Q8—88no—
269C2×Q8—84no—
270C22×C4—44no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—84no—
277C22×C4#12712yes—
278C22×C4#12712yes—
279C22×C4#12712yes—
280C22×C4#12712yes—
281C22×C4#12711yescharacteristic
282C22×C4#13718yes—
283C22×C4C22×D412yes—
284C22×C4C22×D412yes—
285C22×C4C22×D412yes—
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D412yes—
288Q16—816no—
289M16—216no—
290C2×C8—28no—
291C2×C8—48no—
292C2×C8C22×D418yes—
293C2×C8C22×D4116yes—
294C4⋊C4—24no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—44no—
319C42—44no—
320C42—44no—
321C42—44no—
322C42#12711yescharacteristic
323C42#12712yes—
324C42#12711yescharacteristic
325C42#12712yes—
326C42C22×D412yes—
327C42C22×D411yescharacteristic
328C42C22×D412yes—
329C42C22×D412yes—
330C42C22×D412yes—
331C42C2511yescharacteristic
332C23—42no—
333C23#49714yes—
334C23C2×#12711yescharacteristic
335C23C2×#12712yes—
336Q8—48no—
337Q8—48no—
338Q8—48no—
339Q8—48no—
340Q8—84no—
341Q8—84no—
342Q8—84no—
343Q8—84no—
344Q8—88no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4—88no—
360C2×C4#49718yes—
361C2×C4#49714yes—
362C2×C4C2×#12712yes—
363C2×C4C2×#12712yes—
364C2×C4C2×#12712yes—
365C2×C4C2×#12712yes—
366C2×C4C2×#12711yescharacteristic
367C2×C4C2×#13714yes—
368C2×C4C2×#13718yes—
369C2×C4C23×D412yes—
370C2×C4C23×D411yescharacteristic
371C8—216no—
372C8—28no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—44no—
377C22C2×#49712yes—
378C22C2×#49714yes—
379C22C22×#12711yescharacteristic
380C4—22no—
381C4—22no—
382C4—22no—
383C4—84no—
384C4—84no—
385C4—84no—
386C4—84no—
387C4C2×#49712yes—
388C4C2×#49714yes—
389C4C22×#13712yes—
390C4#358614yes—
391C2—24no—
392C2—22no—
393C2C22×#49711yescharacteristic
394C2#6064512yes—
subgroupG/H
center, socleC22C22×#127
commutator, Frattini, cosocleC42C25
prime radicalC24#127
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #19706C2semidirect
6. #19681C2semidirect
7. #19680C2semidirect
42. #1723C22semidirect
43. #1718C22semidirect
44. #1718C22semidirect
45. #1716C22semidirect

Maximal subgroups

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

subgroupindex
#609432
C2×#32292
#606632
C2×#31392
#197062
#196812
#196802
#173192