Finite Group Databasebeta

← #7228784

Subgroups — #7228784

Order 512.

1475 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46152C211yesmaximal, characteristic, complemented
2#36164C211yesmaximal, characteristic
3#33068C212yesmaximal, complemented
4#32821C212yesmaximal, complemented
5#32645C212yesmaximal, complemented
6#32268C212yesmaximal, complemented
7#32215C214yesmaximal, complemented
8#30801C214yesmaximal, complemented
9#27544C214yesmaximal, complemented
10#25970C214yesmaximal, complemented
11#25850C212yesmaximal, complemented
12#24055C212yesmaximal
13#23353C211yesmaximal, characteristic, complemented
14C2×#554C2211yescharacteristic
15C2×#547C2211yescharacteristic
16C2×#501C2212yes—
17C2×#483C2212yes—
18C2×#474C2212yes—
19C2×#417C2212yes—
20#2862C2211yescharacteristic
21#2859C2212yes—
22#2850—21no—
23#2844—21no—
24#2809—21no—
25#2720—24no—
26#2683—21no—
27#2568C2211yescharacteristic
28#2567C2212yes—
29#2563C2211yescharacteristic
30#2504—21no—
31#2503—21no—
32#2497C2214yes—
33#2489C2214yes—
34#2488—24no—
35#2487C2212yes—
36#2440C2214yes—
37#2379—21no—
38#2339C2214yes—
39C4×#124C2211yescharacteristic
40C2×#382—21no—
41C42×D4C2211yescharacteristic
42#2095C2214yes—
43#2077C2214yes—
44#2075C2214yes—
45#2072C2214yes—
46#2068C2214yes—
47#2068C2214yes—
48#2066C2214yes—
49#2059C2214yes—
50#2057C2214yes—
51#2043C2214yes—
52#2035C2214yes—
53#2024C2214yes—
54#1989C2212yes—
55#1985C2212yes—
56#1948C2212yes—
57#1948C2212yes—
58#1935C2214yes—
59#1927C2214yes—
60#1924C2214yes—
61#1920C2214yes—
62#1920C2214yes—
63#1894C2212yes—
64#1870C2212yes—
65#1864C2214yes—
66#1864C2214yes—
67#1862C2214yes—
68#1852C2214yes—
69#1851C2214yes—
70#1831C2212yes—
71#1809C2214yes—
72#1802C2212yes—
73#1802C2212yes—
74#1788C2214yes—
75#1788C2212yes—
76#1762C2212yes—
77C22×#41C2312yes—
78C22×M16C2312yes—
79#554—42no—
80#554—42no—
81#547—42no—
82#547—42no—
83C2×C4⋊Q8C2311yescharacteristic
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×#125—22no—
87C2×#125C2312yes—
88C2×#124D411yescharacteristic
89C2×#123—22no—
90C2×C22⋊Q8—21no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8C2312yes—
94C2×#121—21no—
95C2×C4×Q8C2311yescharacteristic
96C2×C4×D4C2311yescharacteristic
97C2×C4×D4C2312yes—
98C2×C4×D4C2312yes—
99C2×C4×D4C2311yescharacteristic
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4C2312yes—
103C22×C42—21no—
104#501—44no—
105#483—44no—
106#474—44no—
107C2×#107C2314yes—
108C2×#107C2314yes—
109C2×#106C2314yes—
110C2×#106C2314yes—
111C2×C4⋊C8C2312yes—
112C2×C4⋊C8C2312yes—
113C2×C4⋊C8C2312yes—
114#417—44no—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×C22⋊C8C2312yes—
122C2×C22⋊C8C2314yes—
123C2×#97C2312yes—
124#400—22no—
125#400—22no—
126#400—24no—
127#400—24no—
128#399—22no—
129#398—24no—
130#397—24no—
131#396—24no—
132#395—22no—
133#395—22no—
134#395—24no—
135#393—22no—
136#391—22no—
137#391—21no—
138#391—22no—
139#391D411yescharacteristic
140#391C2314yes—
141#391C2314yes—
142#390—21no—
143#390D411yescharacteristic
144#389—22no—
145#389—22no—
146#389C2314yes—
147#389C2314yes—
148#389C2314yes—
149#388—24no—
150#388—24no—
151#388D414yes—
152#387—24no—
153#386—22no—
154#386—24no—
155#386—24no—
156#385—24no—
157#385—22no—
158#384—22no—
159#384C2312yes—
160#384C2312yes—
161#382—21no—
162#382—21no—
163#382—22no—
164#382—21no—
165#382—21no—
166#382—24no—
167#382D411yescharacteristic
168#381—24no—
169#380C2314yes—
170#380C2314yes—
171#380C2312yes—
172C4×C4⋊C4C2311yescharacteristic
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2312yes—
175C4×C22⋊C4—22no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4C2314yes—
178C2×#95—24no—
179C43C2311yescharacteristic
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#336C2314yes—
187#336C2314yes—
188#336C2314yes—
189C2×#41—24no—
190C2×#41—28no—
191C2×#41—42no—
192C2×#41—44no—
193C22×Q8—21no—
194C22×Q8—22no—
195C22×Q8C2×D411yescharacteristic
196C22×Q8C2412yes—
197C22×D4—22no—
198C22×D4C2×D411yescharacteristic
199C22×D4C2411yescharacteristic
200C23×C4—21no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4C2412yes—
206C2×M16—44no—
207C2×M16—42no—
208C22×C8C2414yes—
209C22×C8C2414yes—
210C4⋊Q8—41no—
211C4⋊Q8—41no—
212C4⋊Q8—42no—
213C4⋊Q8—42no—
214C4⋊Q8—41no—
215C4⋊Q8—41no—
216#125—24no—
217#125—42no—
218#125—84no—
219#124—24no—
220#123—48no—
221C22⋊Q8—24no—
222C22⋊Q8—48no—
223C22⋊Q8—42no—
224C22⋊Q8—44no—
225C22⋊Q8—88no—
226#121—44no—
227C4×Q8—41no—
228C4×Q8—41no—
229C4×D4—22no—
230C4×D4—22no—
231C4×D4—24no—
232C4×D4—24no—
233C4×D4—42no—
234C4×D4—42no—
235C4×D4—41no—
236C4×D4—41no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2414yes—
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2414yes—
263C2×C4⋊C4C2411yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4C2×D414yes—
272C2×C22⋊C4C2×D412yes—
273C2×C22⋊C4C2×D414yes—
274C2×C42—22no—
275C2×C42—21no—
276C2×C42—22no—
277C2×C42—44no—
278C2×C42C2×D412yes—
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2411yescharacteristic
282C2×C42C2412yes—
283C2×C42C2412yes—
284#107—44no—
285#107—44no—
286#106—28no—
287#106—44no—
288C4⋊C8—42no—
289C4⋊C8—42no—
290C4⋊C8—42no—
291#103—44no—
292#103—44no—
293#103—44no—
294#103—44no—
295#102—28no—
296#102—28no—
297C22⋊C8—44no—
298C22⋊C8—42no—
299#97—42no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—21no—
304#95—22no—
305#95—22no—
306#95—24no—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95C2×D414yes—
314#95C2×D414yes—
315C24—21no—
316C24C22×D412yes—
317#41—48no—
318#41—84no—
319#41—88no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—24no—
324C2×Q8—24no—
325C2×Q8—41no—
326C2×Q8—41no—
327C2×Q8—42no—
328C2×Q8—84no—
329C2×Q8—82no—
330C2×Q8—82no—
331C2×D4—22no—
332C2×D4—21no—
333C2×D4—22no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—44no—
337C2×D4—48no—
338C2×D4H(5,𝔽2)12yes—
339C2×D42−1+414yes—
340C22×C4—24no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—48no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—82no—
359C22×C4—84no—
360C22×C4—82no—
361C22×C4—84no—
362C22×C4C22×D412yes—
363C22×C4C22×D412yes—
364C22×C4C22×D412yes—
365C22×C4C22×D412yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D412yes—
368C22×C4C22×D411yescharacteristic
369C22×C4C2511yescharacteristic
370M16—84no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C4⋊C4—21no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—22no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—44no—
395C4⋊C4—48no—
396C4⋊C4—42no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—84no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C4⋊C4—84no—
403C4⋊C4—82no—
404C4⋊C4H(5,𝔽2)12yes—
405C22⋊C4—24no—
406C22⋊C4—28no—
407C22⋊C4—42no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—84no—
413C22⋊C4—82no—
414C22⋊C4—84no—
415C42—22no—
416C42—22no—
417C42—21no—
418C42—21no—
419C42—22no—
420C42—22no—
421C42—41no—
422C42—42no—
423C42—44no—
424C42—42no—
425C42—82no—
426C42#13714yes—
427C23—22no—
428C23—24no—
429C23—24no—
430C23—22no—
431C23—22no—
432C23—24no—
433C23—22no—
434C23—42no—
435C23—41no—
436C23—41no—
437C23C23×D411yescharacteristic
438Q8—42no—
439Q8—42no—
440Q8—44no—
441Q8—82no—
442Q8—162no—
443Q8—164no—
444D4—22no—
445D4—24no—
446D4—82no—
447D4—88no—
448D4#56314yes—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—48no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—88no—
486C2×C4—88no—
487C2×C4—164no—
488C2×C4—164no—
489C2×C4C2×#13714yes—
490C2×C4C2×#13712yes—
491C2×C4C2×H(5,𝔽2)11yescharacteristic
492C2×C4C2×H(5,𝔽2)11yescharacteristic
493C2×C4C2×2−1+412yes—
494C2×C4#58512yes—
495C8—44no—
496C8—44no—
497C22—24no—
498C22—44no—
499C22—42no—
500C22—44no—
501C22—42no—
502C22—44no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—42no—
507C22#348411yescharacteristic
508C22#348912yes—
509C22C22×#13711yescharacteristic
510C22#358512yes—
511C22#358611yescharacteristic
512C4—22no—
513C4—24no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—41no—
520C4—84no—
521C4—82no—
522C4—82no—
523C4—84no—
524C4—82no—
525C4C2×#56311yescharacteristic
526C4#354511yescharacteristic
527C4#356112yes—
528C2—44no—
529C2—82no—
530C2#4613611yescharacteristic
531C2#6158611yescharacteristic
532C2#6167311yescharacteristic
533C2#6210812yes—
534C2#6223812yes—
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. #33068C2semidirect
4. #32821C2semidirect
5. #32645C2semidirect
6. #32268C2semidirect
7. #32215C2semidirect
8. #30801C2semidirect
9. #27544C2semidirect
10. #25970C2semidirect
11. #25850C2semidirect
13. #23353C2semidirect

Maximal subgroups

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

subgroupindex
#461522
#361642
#330682
#328212
#326452
#322682
#322152
#308012
#275442
#259702
#258502
#240552
#233532