Finite Group Databasebeta

← #6436297

Subgroups — #6436297

Order 512.

1355 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic
2#34281C211yesmaximal, characteristic
3#30183C214yesmaximal
4#30098C212yesmaximal, complemented
5#30079C212yesmaximal, complemented
6#29984C212yesmaximal, complemented
7#29948C212yesmaximal
8#29871C212yesmaximal, complemented
9#29701C212yesmaximal
10#23844C212yesmaximal, complemented
11#23829C212yesmaximal, complemented
12#23762C212yesmaximal
13#23757C212yesmaximal
14#21621C214yesmaximal, complemented
15#20784C211yesmaximal, characteristic
16C2×#541C2211yescharacteristic
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#2373—22no—
28#2373—22no—
29#2371—22no—
30#2370—22no—
31#2331C2212yes—
32#2330C2211yescharacteristic
33#2330C2212yes—
34#2327C2211yescharacteristic
35#2323C2212yes—
36#2314C2211yescharacteristic
37C4×C4⋊Q8C2211yescharacteristic
38C4×#125C2211yescharacteristic
39C4×C22⋊Q8C2212yes—
40#2295C2212yes—
41#2285C2212yes—
42#2276C2212yes—
43C4×#117C2211yescharacteristic
44#2101C2214yes—
45#2100C2212yes—
46#2100C2212yes—
47#2095C2212yes—
48#2095C2212yes—
49#2085C2214yes—
50#2084C2212yes—
51#2083C2214yes—
52#2083C2212yes—
53#2076C2214yes—
54#2075C2214yes—
55#2074C2214yes—
56#2072C2212yes—
57#2072C2212yes—
58#2059C2214yes—
59#2057C2214yes—
60#2043C2214yes—
61#2036C2214yes—
62#2035C2214yes—
63#2024C2214yes—
64#1989C2214yes—
65#1947C2214yes—
66#1919C2214yes—
67#1918C2214yes—
68#1863C2214yes—
69#1863C2214yes—
70#1839C2212yes—
71#1831C2212yes—
72#1818C2212yes—
73#1803C2214yes—
74#1798C2212yes—
75#1776C2214yes—
76C4×#107C2214yes—
77#1768C2212yes—
78#1763C2214yes—
79C4×#103C2214yes—
80#1749C2211yescharacteristic
81C4×C22⋊C8C2211yescharacteristic
82#1746C2212yes—
83#541—28no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C2311yescharacteristic
86C2×C4⋊Q8C2312yes—
87C2×#126—22no—
88C2×#125—22no—
89C2×#125—22no—
90C2×#125C2311yescharacteristic
91C2×#125C2312yes—
92C2×C22⋊Q8C2312yes—
93C2×C4×Q8C2312yes—
94C2×C4×Q8C2312yes—
95C2×C4×Q8C2312yes—
96C2×#117—22no—
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C22×C42C2311yescharacteristic
100#479—216no—
101C2×#107C2314yes—
102C2×#107C2314yes—
103C2×#107C2314yes—
104C2×#106C2314yes—
105C2×C4⋊C8C2314yes—
106C2×#103C2314yes—
107C2×#103C2314yes—
108C2×#103C2314yes—
109C2×#103C2314yes—
110C2×C22⋊C8C2312yes—
111C2×C22⋊C8C2312yes—
112C2×#97C2312yes—
113C2×C4×C8C2312yes—
114#400—22no—
115#399—22no—
116#398—24no—
117#396—24no—
118#395—22no—
119#393—22no—
120#391—24no—
121#391C2312yes—
122#391C2312yes—
123#389—22no—
124#389—22no—
125#389—22no—
126#389—22no—
127#389—22no—
128#389—22no—
129#389—22no—
130#389—24no—
131#389C2312yes—
132#389C2312yes—
133#389C2314yes—
134#389C2314yes—
135#388—22no—
136#388—22no—
137#388—22no—
138#387—22no—
139#387—22no—
140#387—22no—
141#387—22no—
142#387—22no—
143#387—22no—
144#387—24no—
145#387—24no—
146#387—22no—
147#387—22no—
148#387—22no—
149#385—22no—
150#384—22no—
151#384C2311yescharacteristic
152#384C2312yes—
153#384C2312yes—
154#384C2312yes—
155#382—22no—
156#382—22no—
157#382C2311yescharacteristic
158#382C2312yes—
159#380—22no—
160#380C2312yes—
161#380C2314yes—
162C4×C4⋊C4—22no—
163C4×C4⋊C4—22no—
164C4×C4⋊C4—22no—
165C4×C4⋊C4—22no—
166C4×C4⋊C4—22no—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—22no—
169C4×C4⋊C4—22no—
170C4×C4⋊C4D412yes—
171C4×C4⋊C4D412yes—
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2314yes—
175C4×C4⋊C4C2312yes—
176C4×C22⋊C4—22no—
177C4×C22⋊C4D412yes—
178C4×C22⋊C4D412yes—
179#376—22no—
180#376—22no—
181#376C2311yescharacteristic
182#376C2311yescharacteristic
183C43C2311yescharacteristic
184C43C2311yescharacteristic
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#336C2314yes—
194#336C2314yes—
195#336C2312yes—
196#336C2312yes—
197#336C2314yes—
198C22×Q8—22no—
199C22×Q8C2412yes—
200C23×C4C2×D412yes—
201C23×C4C2411yescharacteristic
202C22×C8C2414yes—
203C22×C8C2414yes—
204C4⋊Q8—24no—
205C4⋊Q8—44no—
206C4⋊Q8—84no—
207#126—216no—
208#125—216no—
209#125—24no—
210#125—28no—
211#125—84no—
212C22⋊Q8—24no—
213C22⋊Q8—24no—
214C4×Q8—24no—
215C4×Q8—28no—
216C4×Q8—44no—
217C4×Q8#4118yes—
218#117—24no—
219#117—42no—
220#117—84no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4C2×D412yes—
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2412yes—
243C2×C4⋊C4C2412yes—
244C2×C4⋊C4C2414yes—
245C2×C4⋊C4C2414yes—
246C2×C4⋊C4C2412yes—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4C2×D412yes—
251C2×C22⋊C4C2×D411yescharacteristic
252C2×C22⋊C4C2×D411yescharacteristic
253C2×C42—22no—
254C2×C42—24no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42C2×D412yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2411yescharacteristic
267C2×C42C2411yescharacteristic
268C2×C42C2411yescharacteristic
269C2×C42C2411yescharacteristic
270C2×C42C2411yescharacteristic
271C2×C42C2411yescharacteristic
272#107—28no—
273#107—44no—
274#107—44no—
275#106—28no—
276C4⋊C8—28no—
277#103—28no—
278#103—28no—
279#103—44no—
280#103—44no—
281C22⋊C8—42no—
282C22⋊C8—42no—
283#97—24no—
284C4×C8—24no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—24no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—22no—
301#95—22no—
302#95C2×D412yes—
303#95C2×D412yes—
304#95C2×D412yes—
305#95C2×D412yes—
306C24C22×D411yescharacteristic
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—44no—
311C2×Q8—84no—
312C2×Q8C2×#4114yes—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—42no—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D412yes—
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D412yes—
332C22×C4C22×D412yes—
333C22×C4C22×D412yes—
334C22×C4C2511yescharacteristic
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C4⋊C4—28no—
342C4⋊C4—28no—
343C4⋊C4—28no—
344C4⋊C4—28no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—48no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C4⋊C4—84no—
367C4⋊C4—84no—
368C4⋊C4—84no—
369C4⋊C4C2×#4114yes—
370C22⋊C4—28no—
371C22⋊C4—24no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—82no—
375C22⋊C4—84no—
376C42—28no—
377C42—24no—
378C42—24no—
379C42—24no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—42no—
385C42—42no—
386C42—44no—
387C42—44no—
388C42—82no—
389C42—82no—
390C42—82no—
391C42—84no—
392C42C2×#4114yes—
393C422−1+414yes—
394C23—22no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—22no—
399C23C23×D411yescharacteristic
400Q8—28no—
401Q8—44no—
402Q8—164no—
403C2×C4—22no—
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—22no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—48no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4#52514yes—
444C2×C4#53614yes—
445C2×C4#57814yes—
446C2×C4C22×#4112yes—
447C2×C4C2×2−1+412yes—
448C2×C4#58512yes—
449C8—44no—
450C8—44no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—42no—
455C22—42no—
456C22C2×#52511yescharacteristic
457C22C2×#53611yescharacteristic
458C22#344911yescharacteristic
459C22C2×#57812yes—
460C22#358512yes—
461C4—24no—
462C4—24no—
463C4—24no—
464C4—41no—
465C4—44no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—42no—
471C4—44no—
472C4—82no—
473C4—84no—
474C4—84no—
475C4—82no—
476C4#346114yes—
477C2—81no—
478C2#3462711yescharacteristic
479C2#6060012yes—
480C2#6087412yes—
481C2#6101311yescharacteristic
482C2#6101911yescharacteristic
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
4. #30098C2semidirect
5. #30079C2semidirect
6. #29984C2semidirect
8. #29871C2semidirect
10. #23844C2semidirect
11. #23829C2semidirect
14. #21621C2semidirect

Maximal subgroups

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

subgroupindex
#352942
#342812
#301832
#300982
#300792
#299842
#299482
#298712
#297012
#238442
#238292
#237622
#237572
#216212
#207842