Finite Group Databasebeta

← #6631988

Subgroups — #6631988

Order 512.

1187 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35323C211yesmaximal, characteristic
2#35283C211yesmaximal, characteristic
3#32857C212yesmaximal, complemented
4#32842C212yesmaximal, complemented
5#32837C212yesmaximal, complemented
6#32828C212yesmaximal, complemented
7#32789C212yesmaximal
8#32771C212yesmaximal
9#32752C214yesmaximal
10#23845C212yesmaximal, complemented
11#23829C212yesmaximal, complemented
12#23783C212yesmaximal, complemented
13#23776C212yesmaximal, complemented
14#23769C214yesmaximal
15#20781C211yesmaximal, characteristic
16#2868C2211yescharacteristic
17#2863C2212yes—
18#2862C2211yescharacteristic
19#2857C2212yes—
20#2857C2211yescharacteristic
21#2851C2211yescharacteristic
22#2647C2212yes—
23#2647C2211yescharacteristic
24#2642C2211yescharacteristic
25#2628C2212yes—
26#2625C2212yes—
27#2590—22no—
28#2577—22no—
29#2533—22no—
30#2530—22no—
31#2511—22no—
32#2492—22no—
33#2451—22no—
34#2423—22no—
35#2331C2212yes—
36#2330C2211yescharacteristic
37#2327C2211yescharacteristic
38#2323C2212yes—
39#2314C2212yes—
40C4×C4⋊Q8C2212yes—
41#2295C2212yes—
42C4×#117C2211yescharacteristic
43#2101C2214yes—
44#2101C2214yes—
45#2099C2212yes—
46#2099C2212yes—
47#2097C2214yes—
48#2096C2212yes—
49#2096C2212yes—
50#2088C2214yes—
51#2088C2214yes—
52#2085C2214yes—
53#2084C2214yes—
54#2084C2214yes—
55#2077C2214yes—
56#2076C2214yes—
57#2074C2214yes—
58#2072C2214yes—
59#2068C2214yes—
60#2066C2214yes—
61#2059C2214yes—
62#2058C2212yes—
63#2058C2214yes—
64#2057C2212yes—
65#2043C2214yes—
66#2035C2214yes—
67#2035C2214yes—
68#2024C2212yes—
69#2024C2212yes—
70#2024C2214yes—
71#1839C2212yes—
72#1832C2212yes—
73#1817C2212yes—
74#1798C2212yes—
75#1776C2214yes—
76C4×#106C2214yes—
77#1767C2211yescharacteristic
78C4×C4⋊C8C2211yescharacteristic
79#1763C2214yes—
80C4×#103C2214yes—
81#1750C2212yes—
82#1746C2212yes—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×#125C2311yescharacteristic
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8C2312yes—
90C2×C4×Q8—22no—
91C2×C4×Q8C2312yes—
92C2×#117C2311yescharacteristic
93C2×#117C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C2×#107C2314yes—
96C2×#106C2314yes—
97C2×#106C2314yes—
98C2×#106C2314yes—
99C2×C4⋊C8C2312yes—
100C2×C4⋊C8C2312yes—
101C2×#103C2314yes—
102C2×#103C2314yes—
103C2×#103C2314yes—
104C2×#103C2314yes—
105C2×C22⋊C8C2314yes—
106C2×#97C2312yes—
107C2×C4×C8C2312yes—
108#400—22no—
109#400—22no—
110#400—22no—
111#400—22no—
112#400—24no—
113#400—22no—
114#399—22no—
115#398—22no—
116#398—22no—
117#398—24no—
118#397—22no—
119#397—22no—
120#395—22no—
121#393—22no—
122#391—22no—
123#391—22no—
124#391—22no—
125#391—22no—
126#391—24no—
127#391C2312yes—
128#391C2312yes—
129#391C2312yes—
130#391C2312yes—
131#389—22no—
132#389—22no—
133#389—22no—
134#389—22no—
135#389—22no—
136#389C2314yes—
137#389C2312yes—
138#389C2312yes—
139#389C2314yes—
140#389C2312yes—
141#389C2312yes—
142#388—24no—
143#388—22no—
144#388—22no—
145#387—24no—
146#387—22no—
147#387—22no—
148#387—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#387—24no—
157#385—22no—
158#384—22no—
159#384C2312yes—
160#384C2312yes—
161#384C2311yescharacteristic
162#383—22no—
163#382C2312yes—
164#382C2311yescharacteristic
165#382C2312yes—
166#380—22no—
167#380—22no—
168#380C2314yes—
169#380C2312yes—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4D412yes—
173C4×C4⋊C4D412yes—
174C4×C4⋊C4C2314yes—
175C4×C4⋊C4C2312yes—
176C4×C22⋊C4D412yes—
177C4×C22⋊C4D412yes—
178#376—22no—
179#376C2311yescharacteristic
180#376C2311yescharacteristic
181C43C2311yescharacteristic
182C43C2311yescharacteristic
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#336C2314yes—
192#336C2312yes—
193#336C2312yes—
194#336C2314yes—
195#336C2314yes—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8C2412yes—
199C23×C4C2×D412yes—
200C23×C4C2411yescharacteristic
201C22×C8C2414yes—
202C22×C8C2414yes—
203C4⋊Q8—44no—
204C4⋊Q8—44no—
205C4⋊Q8—42no—
206C4⋊Q8—84no—
207#125—42no—
208C22⋊Q8—44no—
209C22⋊Q8—84no—
210C4×Q8—24no—
211C4×Q8—24no—
212C4×Q8—84no—
213#117—22no—
214#117—22no—
215#117—42no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2412yes—
236C2×C4⋊C4C2412yes—
237C2×C4⋊C4C2412yes—
238C2×C4⋊C4C2414yes—
239C2×C4⋊C4C2414yes—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4C2×D412yes—
244C2×C22⋊C4C2×D411yescharacteristic
245C2×C22⋊C4C2×D411yescharacteristic
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—42no—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2411yescharacteristic
256C2×C42C2411yescharacteristic
257C2×C42C2411yescharacteristic
258C2×C42C2411yescharacteristic
259C2×C42C2411yescharacteristic
260C2×C42C2411yescharacteristic
261#107—44no—
262#106—44no—
263#106—44no—
264#106—44no—
265C4⋊C8—42no—
266C4⋊C8—42no—
267#103—44no—
268#103—44no—
269#103—44no—
270#103—44no—
271C22⋊C8—44no—
272#97—24no—
273C4×C8—24no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—24no—
281#95—22no—
282#95—24no—
283#95—24no—
284#95—22no—
285#95—24no—
286#95—22no—
287#95—22no—
288#95—24no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95C2×D412yes—
297#95C2×D412yes—
298#95C2×D412yes—
299#95C2×D412yes—
300C24C22×D411yescharacteristic
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—44no—
306C2×Q8—44no—
307C2×Q8—44no—
308C2×Q8—44no—
309C2×Q8—84no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D412yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D412yes—
330C22×C4C22×D412yes—
331C22×C4C22×D412yes—
332C22×C4C2511yescharacteristic
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—44no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—84no—
364C4⋊C4—84no—
365C4⋊C4—84no—
366C22⋊C4—24no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—44no—
370C22⋊C4—82no—
371C22⋊C4—84no—
372C42—24no—
373C42—24no—
374C42—21no—
375C42—21no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—44no—
384C42—82no—
385C42—84no—
386C422−1+412yes—
387C422−1+412yes—
388C23—22no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—22no—
393C23C23×D411yescharacteristic
394Q8—44no—
395Q8—44no—
396Q8—88no—
397Q8—164no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—84no—
440C2×C4#53614yes—
441C2×C4#57714yes—
442C2×C4C2×2−1+411yescharacteristic
443C2×C4C2×2−1+411yescharacteristic
444C2×C4#58512yes—
445C2×C4#58512yes—
446C8—44no—
447C8—44no—
448C22—42no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—42no—
453C22C2×#53611yescharacteristic
454C22#344911yescharacteristic
455C22#344911yescharacteristic
456C22C2×#57712yes—
457C22#358612yes—
458C4—24no—
459C4—24no—
460C4—42no—
461C4—41no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—42no—
467C4—82no—
468C4—84no—
469C4—84no—
470C4—82no—
471C4—82no—
472C4—82no—
473C4#353112yes—
474C4#353112yes—
475C2—81no—
476C2#3669911yescharacteristic
477C2#6088912yes—
478C2#6097712yes—
479C2#6101211yescharacteristic
480C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32857C2semidirect
4. #32842C2semidirect
5. #32837C2semidirect
6. #32828C2semidirect
10. #23845C2semidirect
11. #23829C2semidirect
12. #23783C2semidirect
13. #23776C2semidirect

Maximal subgroups

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

subgroupindex
#353232
#352832
#328572
#328422
#328372
#328282
#327892
#327712
#327522
#238452
#238292
#237832
#237762
#237692
#207812