Finite Group Databasebeta

← #6631957

Subgroups — #6631957

Order 512.

1185 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35314C211yesmaximal, characteristic
2#35294C211yesmaximal, characteristic
3#32852C212yesmaximal, complemented
4#32839C212yesmaximal, complemented
5#32837C212yesmaximal, complemented
6#32820C212yesmaximal, complemented
7#32794C212yesmaximal
8#32780C212yesmaximal
9#32755C214yesmaximal
10#23845C212yesmaximal, complemented
11#23831C212yesmaximal, complemented
12#23789C214yesmaximal, complemented
13#23763C212yesmaximal
14#23756C212yesmaximal
15#20784C211yesmaximal, characteristic
16#2868C2211yescharacteristic
17#2864C2211yescharacteristic
18#2863C2212yes—
19#2862C2211yescharacteristic
20#2859C2212yes—
21#2859C2211yescharacteristic
22#2629C2212yes—
23#2628C2212yes—
24#2628C2211yescharacteristic
25#2625C2212yes—
26#2621C2211yescharacteristic
27#2570—22no—
28#2550—22no—
29#2524—22no—
30#2523—22no—
31#2520—22no—
32#2517—22no—
33#2441—22no—
34#2439—22no—
35#2331C2211yescharacteristic
36#2331C2212yes—
37#2330C2212yes—
38#2327C2211yescharacteristic
39#2320C2212yes—
40C4×C22⋊Q8C2212yes—
41#2285C2212yes—
42C4×#117C2211yescharacteristic
43#2101C2214yes—
44#2101C2214yes—
45#2099C2212yes—
46#2099C2212yes—
47#2097C2214yes—
48#2096C2212yes—
49#2096C2212yes—
50#2088C2214yes—
51#2087C2214yes—
52#2085C2214yes—
53#2084C2214yes—
54#2084C2212yes—
55#2083C2212yes—
56#2077C2214yes—
57#2077C2214yes—
58#2072C2214yes—
59#2072C2212yes—
60#2072C2212yes—
61#2066C2214yes—
62#2066C2214yes—
63#2059C2214yes—
64#2058C2214yes—
65#2058C2214yes—
66#2043C2214yes—
67#2036C2214yes—
68#2035C2214yes—
69#2026C2214yes—
70#2024C2214yes—
71#1839C2212yes—
72#1831C2212yes—
73#1818C2212yes—
74#1798C2212yes—
75#1776C2214yes—
76C4×#106C2214yes—
77#1768C2212yes—
78#1763C2214yes—
79C4×#103C2214yes—
80#1749C2211yescharacteristic
81C4×C22⋊C8C2211yescharacteristic
82#1746C2212yes—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2312yes—
87C2×#125—22no—
88C2×#125C2312yes—
89C2×C22⋊Q8C2312yes—
90C2×C4×Q8C2312yes—
91C2×#117—22no—
92C2×#117C2311yescharacteristic
93C2×#117C2311yescharacteristic
94C22×C42C2311yescharacteristic
95C2×#107C2314yes—
96C2×#106C2314yes—
97C2×#106C2314yes—
98C2×#106C2314yes—
99C2×C4⋊C8C2314yes—
100C2×#103C2314yes—
101C2×#103C2314yes—
102C2×#103C2314yes—
103C2×#103C2314yes—
104C2×C22⋊C8C2312yes—
105C2×C22⋊C8C2312yes—
106C2×#97C2312yes—
107C2×C4×C8C2312yes—
108#400—22no—
109#400—22no—
110#400—21no—
111#399—22no—
112#399—21no—
113#398—24no—
114#398—21no—
115#398—21no—
116#398—22no—
117#397—21no—
118#397—21no—
119#397—22no—
120#397—22no—
121#396—24no—
122#395—21no—
123#395—22no—
124#393—22no—
125#393—21no—
126#391—22no—
127#391—24no—
128#391—22no—
129#391C2312yes—
130#391C2312yes—
131#391C2312yes—
132#391C2312yes—
133#389—22no—
134#389—22no—
135#389—22no—
136#389—22no—
137#389—24no—
138#389C2312yes—
139#389C2314yes—
140#389C2312yes—
141#389C2312yes—
142#389C2314yes—
143#389C2312yes—
144#387—22no—
145#387—22no—
146#387—22no—
147#387—22no—
148#387—22no—
149#387—22no—
150#387—24no—
151#387—24no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#387—22no—
157#385—22no—
158#384—22no—
159#384C2312yes—
160#384C2312yes—
161#384C2312yes—
162#383—22no—
163#383—22no—
164#382—22no—
165#382—22no—
166#382—22no—
167#382—22no—
168#382C2311yescharacteristic
169#382C2311yescharacteristic
170#382C2312yes—
171#380—22no—
172#380C2312yes—
173#380C2314yes—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4D412yes—
177C4×C4⋊C4D412yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2314yes—
180C4×C22⋊C4—22no—
181C4×C22⋊C4D412yes—
182C4×C22⋊C4D412yes—
183#376C2311yescharacteristic
184#376C2311yescharacteristic
185C43C2311yescharacteristic
186C43C2311yescharacteristic
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#336C2314yes—
196#336C2314yes—
197#336C2314yes—
198#336C2312yes—
199#336C2312yes—
200C22×Q8—22no—
201C22×Q8—21no—
202C22×Q8C2412yes—
203C23×C4C2×D412yes—
204C23×C4C2411yescharacteristic
205C22×C8C2414yes—
206C22×C8C2414yes—
207C4⋊Q8—42no—
208C4⋊Q8—42no—
209C4⋊Q8—42no—
210C4⋊Q8—42no—
211C4⋊Q8—84no—
212#125—44no—
213#125—84no—
214C22⋊Q8—42no—
215C22⋊Q8—42no—
216C4×Q8—28no—
217#117—42no—
218#117—42no—
219#117—84no—
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—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2412yes—
242C2×C4⋊C4C2412yes—
243C2×C4⋊C4C2414yes—
244C2×C4⋊C4C2414yes—
245C2×C4⋊C4C2412yes—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4C2×D411yescharacteristic
250C2×C22⋊C4C2×D412yes—
251C2×C22⋊C4C2×D411yescharacteristic
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D412yes—
260C2×C42C2×D412yes—
261C2×C42C2411yescharacteristic
262C2×C42C2411yescharacteristic
263C2×C42C2411yescharacteristic
264C2×C42C2411yescharacteristic
265C2×C42C2411yescharacteristic
266C2×C42C2411yescharacteristic
267#107—44no—
268#106—44no—
269#106—44no—
270#106—44no—
271C4⋊C8—44no—
272#103—44no—
273#103—44no—
274#103—44no—
275#103—44no—
276C22⋊C8—42no—
277C22⋊C8—42no—
278#97—24no—
279C4×C8—24no—
280#95—22no—
281#95—22no—
282#95—24no—
283#95—22no—
284#95—22no—
285#95—24no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—24no—
291#95—24no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—22no—
296#95—21no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95C2×D412yes—
305#95C2×D412yes—
306#95C2×D412yes—
307#95C2×D412yes—
308C24C22×D411yescharacteristic
309C2×Q8—24no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—42no—
314C2×Q8—44no—
315C2×Q8—84no—
316C2×Q8—82no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—42no—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C22×D412yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D412yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×D412yes—
337C22×C4C22×D412yes—
338C22×C4C2511yescharacteristic
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—41no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—48no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—44no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C4⋊C4—84no—
374C22⋊C4—41no—
375C22⋊C4—42no—
376C22⋊C4—41no—
377C22⋊C4—44no—
378C22⋊C4—82no—
379C22⋊C4—84no—
380C42—24no—
381C42—24no—
382C42—24no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—42no—
389C42—42no—
390C42—44no—
391C42—82no—
392C42—84no—
393C422−1+414yes—
394C23—22no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—22no—
399C23C23×D411yescharacteristic
400Q8—44no—
401Q8—44no—
402Q8—164no—
403Q8—162no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—48no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4#53614yes—
446C2×C4#57714yes—
447C2×C4C2×2−1+412yes—
448C2×C4#58512yes—
449C2×C4#58512yes—
450C8—44no—
451C8—44no—
452C22—42no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—42no—
457C22C2×#53611yescharacteristic
458C22#344911yescharacteristic
459C22#344911yescharacteristic
460C22C2×#57712yes—
461C22#358612yes—
462C4—24no—
463C4—24no—
464C4—42no—
465C4—42no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—82no—
472C4—84no—
473C4—84no—
474C4—82no—
475C4—82no—
476C4—82no—
477C4#353114yes—
478C2—81no—
479C2#3669911yescharacteristic
480C2#6087212yes—
481C2#6099212yes—
482C2#6101211yescharacteristic
483C2#6101811yescharacteristic
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. #32852C2semidirect
4. #32839C2semidirect
5. #32837C2semidirect
6. #32820C2semidirect
10. #23845C2semidirect
11. #23831C2semidirect
12. #23789C2semidirect

Maximal subgroups

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

subgroupindex
#353142
#352942
#328522
#328392
#328372
#328202
#327942
#327802
#327552
#238452
#238312
#237892
#237632
#237562
#207842