Finite Group Databasebeta

← #7459843

Subgroups — #7459843

Order 512.

1443 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46152C211yesmaximal, characteristic, complemented
2#38955C211yesmaximal, characteristic
3#32694C212yesmaximal, complemented
4#32261C212yesmaximal, complemented
5#31710C214yesmaximal, complemented
6#31096C214yesmaximal, complemented
7#30310C212yesmaximal, complemented
8#29578C212yesmaximal, complemented
9#28854C214yesmaximal, complemented
10#26068C214yesmaximal, complemented
11#25850C212yesmaximal, complemented
12#24089C212yesmaximal
13#23273C211yesmaximal, characteristic, complemented
14C2×#554C2211yescharacteristic
15C2×#547C2211yescharacteristic
16C2×#501C2212yes—
17C2×#481C2212yes—
18C2×#474C2212yes—
19C22×#106C2212yes—
20#2862C2211yescharacteristic
21#2859C2212yes—
22#2844—21no—
23#2809—21no—
24#2720—24no—
25#2687—24no—
26#2683—21no—
27#2621—21no—
28#2567C2212yes—
29#2563C2211yescharacteristic
30#2546C2211yescharacteristic
31#2503—21no—
32#2497C2214yes—
33#2487C2212yes—
34#2442C2214yes—
35#2440C2214yes—
36#2432—21no—
37#2420C2214yes—
38#2379—21no—
39#2311—21no—
40C4×#124C2211yescharacteristic
41#2293C2211yescharacteristic
42#2099C2212yes—
43#2093C2212yes—
44#2088C2214yes—
45#2085C2214yes—
46#2085C2214yes—
47#2084C2212yes—
48#2076C2214yes—
49#2076C2214yes—
50#2071C2212yes—
51#2059C2214yes—
52#2058C2214yes—
53#2058C2214yes—
54#2057C2214yes—
55#2034C2214yes—
56#1989C2212yes—
57#1985C2212yes—
58#1944C2214yes—
59#1943C2214yes—
60#1936C2214yes—
61#1924C2214yes—
62#1921C2214yes—
63#1918C2214yes—
64#1893C2212yes—
65#1869C2212yes—
66#1864C2214yes—
67#1862C2214yes—
68#1853C2214yes—
69#1852C2214yes—
70#1847C2214yes—
71#1836C2214yes—
72#1823C2212yes—
73#1808C2214yes—
74#1762C2212yes—
75#1752C2214yes—
76#1745C2212yes—
77C22×#41—21no—
78C23×C8C2312yes—
79#554—42no—
80#554—42no—
81#547—42no—
82#547—42no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2311yescharacteristic
87C2×#125—22no—
88C2×#125C2312yes—
89C2×#124D411yescharacteristic
90C2×#123—22no—
91C2×C22⋊Q8—21no—
92C2×C22⋊Q8—21no—
93C2×C22⋊Q8—22no—
94C2×C22⋊Q8—24no—
95C2×#121C2312yes—
96C2×C4×Q8C2311yescharacteristic
97C2×C4×D4C2311yescharacteristic
98C2×C4×D4C2312yes—
99C2×C4×D4C2311yescharacteristic
100C2×#117—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104#501—44no—
105#481—44no—
106#474—44no—
107C2×#107C2314yes—
108C2×#107C2314yes—
109C2×#107C2314yes—
110C2×#106—44no—
111C2×#106C2314yes—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2312yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×C22⋊C8C2314yes—
118C2×C22⋊C8C2312yes—
119C2×#97C2314yes—
120C2×#97C2312yes—
121#400—22no—
122#400—24no—
123#400—24no—
124#399—22no—
125#398—22no—
126#398—24no—
127#398—24no—
128#397—24no—
129#396—24no—
130#395—22no—
131#395—22no—
132#395—24no—
133#394—24no—
134#393—22no—
135#391—22no—
136#391—21no—
137#391—21no—
138#391—22no—
139#391—21no—
140#391D411yescharacteristic
141#390—21no—
142#390D411yescharacteristic
143#390C2312yes—
144#389—22no—
145#389—22no—
146#389C2314yes—
147#389C2314yes—
148#389C2314yes—
149#389C2312yes—
150#389C2314yes—
151#388—24no—
152#388—22no—
153#388D414yes—
154#387—24no—
155#386—22no—
156#386—24no—
157#385—24no—
158#385—24no—
159#385—22no—
160#384—22no—
161#384C2312yes—
162#384C2312yes—
163#382—21no—
164#382—22no—
165#382D411yescharacteristic
166#382C2314yes—
167#381—24no—
168#380—24no—
169#380C2314yes—
170#380C2314yes—
171#380C2312yes—
172C4×C4⋊C4C2311yescharacteristic
173C4×C4⋊C4C2312yes—
174C4×C22⋊C4—21no—
175C4×C22⋊C4C2314yes—
176C43C2311yescharacteristic
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#336C2314yes—
186#336C2314yes—
187#336C2312yes—
188#336C2312yes—
189C2×#41—44no—
190C2×#41—44no—
191C2×#41—82no—
192C22×Q8—21no—
193C22×Q8—21no—
194C22×Q8—22no—
195C22×Q8C2×D411yescharacteristic
196C22×D4—22no—
197C22×D4C2×D411yescharacteristic
198C22×D4C2411yescharacteristic
199C23×C4—21no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4C2412yes—
204C22×C8—28no—
205C22×C8—42no—
206C22×C8C2414yes—
207C22×C8C2414yes—
208C4⋊Q8—41no—
209C4⋊Q8—41no—
210C4⋊Q8—41no—
211C4⋊Q8—41no—
212C4⋊Q8—42no—
213C4⋊Q8—42no—
214C4⋊Q8—82no—
215#125—24no—
216#125—42no—
217#125—84no—
218#124—24no—
219#123—48no—
220C22⋊Q8—48no—
221C22⋊Q8—44no—
222C22⋊Q8—44no—
223C22⋊Q8—88no—
224#121—24no—
225#121—42no—
226C4×Q8—41no—
227C4×Q8—41no—
228C4×D4—22no—
229C4×D4—22no—
230C4×D4—24no—
231C4×D4—42no—
232C4×D4—41no—
233C4×D4—41no—
234#117—82no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2411yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4C2×D414yes—
274C2×C22⋊C4C2×D412yes—
275C2×C22⋊C4C2×D414yes—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D412yes—
281C2×C42C2411yescharacteristic
282C2×C42C2412yes—
283C2×C42C2412yes—
284#107—28no—
285#107—44no—
286#107—44no—
287#106—28no—
288#106—44no—
289#106—44no—
290C4⋊C8—42no—
291#102—28no—
292#102—28no—
293C22⋊C8—28no—
294C22⋊C8—42no—
295#97—28no—
296#97—42no—
297#95—22no—
298#95—21no—
299#95—22no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95C2×D414yes—
310#95C2×D414yes—
311C24—21no—
312C24C22×D412yes—
313#41—816no—
314C2×Q8—22no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—84no—
322C2×Q8—82no—
323C2×Q8—82no—
324C2×D4—21no—
325C2×D4—24no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—44no—
329C2×D4—44no—
330C2×D4—82no—
331C2×D4#13714yes—
332C2×D4H(5,𝔽2)12yes—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—84no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—84no—
353C22×C4C22×D412yes—
354C22×C4C22×D412yes—
355C22×C4C22×D412yes—
356C22×C4C22×D412yes—
357C22×C4C22×D412yes—
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C2511yescharacteristic
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—48no—
367C2×C82−1+418yes—
368C4⋊C4—21no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—48no—
387C4⋊C4—42no—
388C4⋊C4—82no—
389C4⋊C4—82no—
390C4⋊C4—82no—
391C4⋊C4—84no—
392C4⋊C4—82no—
393C4⋊C4—82no—
394C4⋊C4—84no—
395C4⋊C4—82no—
396C4⋊C4—84no—
397C4⋊C4H(5,𝔽2)12yes—
398C22⋊C4—24no—
399C22⋊C4—28no—
400C22⋊C4—42no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—82no—
405C22⋊C4—82no—
406C22⋊C4—82no—
407C22⋊C4—84no—
408C22⋊C4—84no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—21no—
413C42—21no—
414C42—24no—
415C42—41no—
416C42—42no—
417C42—42no—
418C42—82no—
419C23—24no—
420C23—22no—
421C23—22no—
422C23—24no—
423C23—22no—
424C23—41no—
425C23—42no—
426C23—42no—
427C23—41no—
428C23C2×#13714yes—
429C23C23×D411yescharacteristic
430Q8—82no—
431Q8—84no—
432Q8—162no—
433Q8—164no—
434D4—24no—
435D4—44no—
436D4—42no—
437D4—88no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—48no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—88no—
477C2×C4—164no—
478C2×C4—164no—
479C2×C4C2×#13712yes—
480C2×C4C2×H(5,𝔽2)11yescharacteristic
481C2×C4C2×H(5,𝔽2)11yescharacteristic
482C2×C4C2×2−1+414yes—
483C8—28no—
484C8—44no—
485C22—28no—
486C22—24no—
487C22—44no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—82no—
494C22—82no—
495C22#348411yescharacteristic
496C22#349912yes—
497C22C22×#13711yescharacteristic
498C22#358511yescharacteristic
499C22#358512yes—
500C4—22no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—41no—
508C4—82no—
509C4—84no—
510C4—82no—
511C4—84no—
512C4—82no—
513C4#353812yes—
514C4#354412yes—
515C2—44no—
516C2—82no—
517C2#5154211yescharacteristic
518C2#6157011yescharacteristic
519C2#6167311yescharacteristic
520C2#6250112yes—
521C2#6250412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #46152C2semidirect
3. #32694C2semidirect
4. #32261C2semidirect
5. #31710C2semidirect
6. #31096C2semidirect
7. #30310C2semidirect
8. #29578C2semidirect
9. #28854C2semidirect
10. #26068C2semidirect
11. #25850C2semidirect
13. #23273C2semidirect

Maximal subgroups

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

subgroupindex
#461522
#389552
#326942
#322612
#317102
#310962
#303102
#295782
#288542
#260682
#258502
#240892
#232732