Finite Group Databasebeta

← #6498153

Subgroups — #6498153

Order 512.

1396 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic
2#34254C211yesmaximal, characteristic
3#33157C212yesmaximal
4#32967C212yesmaximal
5#29983C212yesmaximal, complemented
6#29871C212yesmaximal, complemented
7#29581C212yesmaximal, complemented
8#29575C212yesmaximal, complemented
9#29403C214yesmaximal, complemented
10#29311C212yesmaximal
11#29118C212yesmaximal
12#27615C212yesmaximal
13#27583C212yesmaximal
14#22128C214yesmaximal, complemented
15#20780C211yesmaximal, characteristic
16C2×#519C2211yescharacteristic
17C2×#479C2214yes—
18#2868C2211yescharacteristic
19#2859C2212yes—
20#2859C2211yescharacteristic
21#2629C2212yes—
22#2628C2212yes—
23#2570—22no—
24#2550—22no—
25#2441—22no—
26#2439—22no—
27#2356—22no—
28#2355—22no—
29#2343—22no—
30#2343—22no—
31#2331C2212yes—
32#2330C2212yes—
33#2320C2212yes—
34#2320C2212yes—
35#2317C2211yescharacteristic
36#2310C2211yescharacteristic
37#2296C2211yescharacteristic
38#2295C2212yes—
39#2295C2212yes—
40C2×#391C2211yescharacteristic
41#2285C2212yes—
42#2274C2211yescharacteristic
43C4×#117C2211yescharacteristic
44#2099C2212yes—
45#2099C2212yes—
46#2096C2212yes—
47#2096C2212yes—
48#2085C2214yes—
49#2084C2212yes—
50#2083C2212yes—
51#2074C2214yes—
52#2072C2212yes—
53#2072C2212yes—
54#2058C2214yes—
55#2024C2214yes—
56#1989C2214yes—
57#1948C2214yes—
58#1943C2214yes—
59#1941C2214yes—
60#1935C2214yes—
61#1933C2214yes—
62#1927C2214yes—
63#1927C2214yes—
64#1921C2214yes—
65#1920C2214yes—
66#1919C2214yes—
67#1919C2214yes—
68#1865C2214yes—
69#1863C2214yes—
70#1854C2214yes—
71#1853C2214yes—
72#1840C2212yes—
73#1831C2212yes—
74#1825C2214yes—
75#1817C2212yes—
76#1803C2214yes—
77#1798C2212yes—
78#1787C2214yes—
79#1768C2212yes—
80#1750C2212yes—
81#1745C2211yescharacteristic
82#1744C2211yescharacteristic
83C23×Q8C2311yescharacteristic
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C2312yes—
86C2×#125—22no—
87C2×#125C2312yes—
88#519—28no—
89C2×C4×Q8—22no—
90C2×C4×Q8C2312yes—
91C2×C4×Q8C2312yes—
92C2×#117—22no—
93C2×#117—22no—
94C2×#117C2311yescharacteristic
95C2×#117C2311yescharacteristic
96C2×#117C2311yescharacteristic
97C2×#117C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#479—216no—
101C2×#107C2314yes—
102C2×#106C2314yes—
103C2×#106C2314yes—
104C2×#106C2314yes—
105C2×C4⋊C8C2314yes—
106C2×#103C2314yes—
107C2×#103C2314yes—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×C22⋊C8C2314yes—
111C2×#97C2312yes—
112C2×C4×C8C2312yes—
113#400—22no—
114#399—22no—
115#398—24no—
116#396—24no—
117#395—22no—
118#393—22no—
119#391—21no—
120#391—21no—
121#391—22no—
122#391—24no—
123#391C2312yes—
124#391C2312yes—
125#391C2312yes—
126#391C2312yes—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—22no—
131#389—24no—
132#389C2312yes—
133#389C2312yes—
134#389C2312yes—
135#389C2314yes—
136#389C2314yes—
137#389C2312yes—
138#387—22no—
139#387—22no—
140#387—22no—
141#387—22no—
142#387—24no—
143#387—24no—
144#387—22no—
145#387—22no—
146#387—22no—
147#387—22no—
148#385—22no—
149#385—21no—
150#385—21no—
151#384—21no—
152#384—22no—
153#384C2312yes—
154#384C2312yes—
155#384C2312yes—
156#383—22no—
157#383—22no—
158#382—21no—
159#382—22no—
160#382—22no—
161#382—22no—
162#382—22no—
163#382C2312yes—
164#381—21no—
165#381—21no—
166#380—22no—
167#380C2314yes—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4D412yes—
174C4×C4⋊C4D412yes—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2314yes—
178C4×C22⋊C4—22no—
179C4×C22⋊C4D412yes—
180C4×C22⋊C4D412yes—
181#376—22no—
182#376—22no—
183#376C2311yescharacteristic
184#376C2311yescharacteristic
185C2×#95—22no—
186C43C2311yescharacteristic
187C43C2311yescharacteristic
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2312yes—
197#336C2312yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2314yes—
202C22×Q8—24no—
203C22×Q8—21no—
204C22×Q8—22no—
205C22×Q8—24no—
206C22×Q8—42no—
207C22×Q8C2412yes—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4C2×D412yes—
211C23×C4C2411yescharacteristic
212C22×C8C2414yes—
213C22×C8C2414yes—
214C4⋊Q8—42no—
215C4⋊Q8—42no—
216C4⋊Q8—84no—
217#125—28no—
218#125—84no—
219C4×Q8—216no—
220C4×Q8—24no—
221C4×Q8—24no—
222C4×Q8—24no—
223C4×Q8#4118yes—
224#117—216no—
225#117—24no—
226#117—24no—
227#117—24no—
228#117—42no—
229#117—84no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4C2×D412yes—
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2412yes—
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2414yes—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2412yes—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4C2×D412yes—
259C2×C22⋊C4C2×D411yescharacteristic
260C2×C22⋊C4C2×D411yescharacteristic
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—21no—
264C2×C42—24no—
265C2×C42—21no—
266C2×C42—24no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42C2×D412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276C2×C42C2411yescharacteristic
277C2×C42C2411yescharacteristic
278C2×C42C2411yescharacteristic
279C2×C42C2411yescharacteristic
280#107—28no—
281#106—28no—
282#106—44no—
283#106—44no—
284C4⋊C8—28no—
285#103—28no—
286#103—28no—
287#103—44no—
288#103—44no—
289C22⋊C8—28no—
290#97—42no—
291C4×C8—42no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—24no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—22no—
311#95—22no—
312#95C2×D412yes—
313#95C2×D412yes—
314#95C2×D412yes—
315#95C2×D412yes—
316C24C22×D411yescharacteristic
317C2×Q8—24no—
318C2×Q8—22no—
319C2×Q8—24no—
320C2×Q8—24no—
321C2×Q8—416no—
322C2×Q8—42no—
323C2×Q8—44no—
324C2×Q8—42no—
325C2×Q8—84no—
326C2×Q8C2×#4114yes—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—82no—
343C22×C4C22×D411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C22×D412yes—
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C22×D412yes—
349C22×C4C22×D412yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D412yes—
352C22×C42−1+414yes—
353C22×C4C2511yescharacteristic
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C4⋊C4—28no—
361C4⋊C4—28no—
362C4⋊C4—24no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—44no—
373C4⋊C4—42no—
374C4⋊C4—44no—
375C4⋊C4—48no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4C2×#4114yes—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—42no—
392C22⋊C4—84no—
393C42—24no—
394C42—28no—
395C42—28no—
396C42—24no—
397C42—24no—
398C42—24no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—42no—
405C42—42no—
406C42—42no—
407C42—82no—
408C42—82no—
409C42—84no—
410C42C2×#4114yes—
411C23—22no—
412C23—21no—
413C23—21no—
414C23—22no—
415C23#53612yes—
416C23C23×D411yescharacteristic
417Q8—28no—
418Q8—48no—
419Q8—44no—
420Q8—164no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—28no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—48no—
445C2×C4—48no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—162no—
463C2×C4#52514yes—
464C2×C4#53612yes—
465C2×C4#57714yes—
466C2×C4C22×#4112yes—
467C2×C4C2×2−1+412yes—
468C2×C4#58512yes—
469C8—44no—
470C8—44no—
471C22—24no—
472C22—22no—
473C22—41no—
474C22—41no—
475C22—42no—
476C22C2×#52511yescharacteristic
477C22C2×#53611yescharacteristic
478C22#344911yescharacteristic
479C22C2×#57712yes—
480C22#358612yes—
481C4—24no—
482C4—24no—
483C4—22no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—41no—
488C4—42no—
489C4—44no—
490C4—42no—
491C4—44no—
492C4—84no—
493C4—82no—
494C4—84no—
495C4—82no—
496C4#349514yes—
497C2—42no—
498C2#3532511yescharacteristic
499C2#6059912yes—
500C2#6087212yes—
501C2#6101211yescharacteristic
502C2#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
5. #29983C2semidirect
6. #29871C2semidirect
7. #29581C2semidirect
8. #29575C2semidirect
9. #29403C2semidirect
14. #22128C2semidirect

Maximal subgroups

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

subgroupindex
#352942
#342542
#331572
#329672
#299832
#298712
#295812
#295752
#294032
#293112
#291182
#276152
#275832
#221282
#207802