Finite Group Databasebeta

← #6798810

Subgroups — #6798810

Order 512.

1492 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36211C211yesmaximal, characteristic
2#36132C211yesmaximal, characteristic, complemented
3#33295C211yesmaximal, characteristic, complemented
4#31753C214yesmaximal, complemented
5#31751C214yesmaximal, complemented
6#29851C212yesmaximal, complemented
7#29214C211yesmaximal, characteristic, complemented
8#28724C211yesmaximal, characteristic, complemented
9#27880C212yesmaximal, complemented
10#26036C214yesmaximal, complemented
11#25911C212yesmaximal, complemented
12#24978C214yesmaximal, complemented
13#21839C212yesmaximal, complemented
14#21656C211yesmaximal, characteristic, complemented
15#21271C211yesmaximal, characteristic, complemented
16C2×#533C2212yes—
17C2×#490C2211yescharacteristic
18C2×#490C2212yes—
19C2×#432C2212yes—
20#2859C2211yescharacteristic
21#2859C2211yescharacteristic
22#2856C2212yes—
23#2851C2212yes—
24#2551C2212yes—
25#2535C2212yes—
26#2513—22no—
27#2490C2214yes—
28#2465—22no—
29#2465—22no—
30#2452—22no—
31#2439C2212yes—
32#2439C2212yes—
33#2420—22no—
34#2414—22no—
35#2365—22no—
36#2363—22no—
37#2357C2212yes—
38#2354C2212yes—
39C4×#127C2211yescharacteristic
40C4×#125C2211yescharacteristic
41#2285C2212yes—
42C42×D4C2211yescharacteristic
43#2077C2214yes—
44#2075C2214yes—
45#2073C2212yes—
46#2073C2212yes—
47#2067C2214yes—
48#2065C2214yes—
49#2059C2214yes—
50#2034C2214yes—
51#2033C2214yes—
52#2023C2214yes—
53#2023C2214yes—
54#1992C2212yes—
55#1992C2212yes—
56#1992C2212yes—
57#1992C2212yes—
58#1992C2211yescharacteristic
59#1991C2212yes—
60#1991C2212yes—
61#1988C2214yes—
62#1987C2214yes—
63#1961C2211yescharacteristic
64#1955C2212yes—
65#1949C2214yes—
66#1940C2212yes—
67#1934C2214yes—
68#1923C2214yes—
69#1921C2214yes—
70#1919C2214yes—
71#1875C2214yes—
72#1875C2214yes—
73#1847C2212yes—
74#1836C2214yes—
75#1836C2211yescharacteristic
76#1825C2214yes—
77#1825C2214yes—
78#1802C2214yes—
79#1767C2211yescharacteristic
80#1762C2214yes—
81#1749C2212yes—
82C4×#97C2211yescharacteristic
83#533—28no—
84#533—44no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×#127C2311yescharacteristic
89C2×#125C2312yes—
90C2×#125C2311yescharacteristic
91C2×#125C2312yes—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—22no—
94C2×#121—22no—
95C2×#121—22no—
96C2×#121C2314yes—
97C2×C4×Q8—22no—
98C2×C4×Q8—22no—
99C2×C4×D4—22no—
100C2×C4×D4D412yes—
101C2×C4×D4C2312yes—
102C2×C4×D4C2311yescharacteristic
103C2×#117—22no—
104C2×#117C2312yes—
105C2×#117C2312yes—
106C22×C42C2312yes—
107#490—24no—
108#490—44no—
109#432—44no—
110C2×#107C2314yes—
111C2×#106C2314yes—
112C2×C4⋊C8C2312yes—
113C2×C4⋊C8C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×C22⋊C8C2314yes—
119C2×C22⋊C8C2314yes—
120C2×#97C2312yes—
121C2×#97C2311yescharacteristic
122C2×#97C2312yes—
123C2×#97C2311yescharacteristic
124C2×C4×C8C2312yes—
125#397—22no—
126#397—24no—
127#396—22no—
128#396—22no—
129#395—22no—
130#395—22no—
131#394—24no—
132#393—22no—
133#393—22no—
134#392—22no—
135#391—22no—
136#391—22no—
137#390—22no—
138#390—24no—
139#390—22no—
140#390—22no—
141#390—22no—
142#390C2312yes—
143#390C2314yes—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389C2312yes—
150#389C2312yes—
151#389C2312yes—
152#388—24no—
153#388—22no—
154#388—22no—
155#388—24no—
156#387—22no—
157#387—22no—
158#387—22no—
159#387—22no—
160#387—22no—
161#386—24no—
162#385—22no—
163#385—22no—
164#384C2311yescharacteristic
165#384C2312yes—
166#384C2312yes—
167#382—22no—
168#382C2311yescharacteristic
169#382C2311yescharacteristic
170#382C2311yescharacteristic
171#382C2312yes—
172#381—22no—
173#380—22no—
174#380C2312yes—
175#380C2314yes—
176#380C2312yes—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4D412yes—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2314yes—
183C4×C4⋊C4C2311yescharacteristic
184C4×C22⋊C4—22no—
185C4×C22⋊C4D412yes—
186C4×C22⋊C4D412yes—
187C4×C22⋊C4C2312yes—
188C43C2311yescharacteristic
189#340C2314yes—
190#340C2314yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#336C2312yes—
200#336C2314yes—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4C2×D411yescharacteristic
206C22×D4C2412yes—
207C23×C4—22no—
208C23×C4C2×D412yes—
209C23×C4C2×D412yes—
210C23×C4C2412yes—
211C22×C8C2412yes—
212C22×C8C2414yes—
213C22×C8C2412yes—
214C4⋊Q8—24no—
215C4⋊Q8—24no—
216C4⋊Q8—42no—
217C4⋊Q8—42no—
218C4⋊Q8—48no—
219#127—22no—
220#127—22no—
221#125—24no—
222#125—22no—
223#125—24no—
224#125—42no—
225#125—42no—
226#125#4114yes—
227C22⋊Q8—48no—
228C22⋊Q8—48no—
229#121—48no—
230#121—44no—
231#121—44no—
232#121—84no—
233C4×Q8—48no—
234C4×Q8—48no—
235C4×D4—24no—
236C4×D4—28no—
237C4×D4—44no—
238C4×D4—84no—
239#117—24no—
240#117—24no—
241#117—42no—
242#117—42no—
243#117—84no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2412yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4C2×D412yes—
277C2×C22⋊C4C2×D412yes—
278C2×C22⋊C4C2×D412yes—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—24no—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2411yescharacteristic
290C2×C42C2412yes—
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293#107—44no—
294#106—44no—
295C4⋊C8—24no—
296C4⋊C8—44no—
297#102—28no—
298#102—44no—
299#102—44no—
300#102—44no—
301C22⋊C8—44no—
302C22⋊C8—44no—
303#97—22no—
304#97—22no—
305#97—24no—
306#97—24no—
307C4×C8—42no—
308#95—22no—
309#95—24no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—24no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95C2×D411yescharacteristic
322#95C2×D411yescharacteristic
323#95C2×D412yes—
324#95C2×D412yes—
325C24—22no—
326C24C22×D412yes—
327C2×Q8—44no—
328C2×Q8—44no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—82no—
332C2×Q8—82no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—42no—
339C2×D4—44no—
340C2×D4—44no—
341C2×D4—48no—
342C2×D4—44no—
343C2×D4—84no—
344C2×D4—84no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—44no—
362C22×C4—84no—
363C22×C4C22×D412yes—
364C22×C4C22×D412yes—
365C22×C4C22×D412yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D412yes—
369C22×C4C22×D412yes—
370C22×C4C2511yescharacteristic
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—24no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—24no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—24no—
380C4⋊C4—22no—
381C4⋊C4—28no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4—84no—
405C4⋊C4C2×#4114yes—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—42no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—48no—
413C22⋊C4—84no—
414C22⋊C4—84no—
415C22⋊C4—84no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—84no—
419C42—24no—
420C42—24no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—42no—
425C42—42no—
426C42—44no—
427C42—44no—
428C42—44no—
429C42—84no—
430C42C2×#4112yes—
431C42H(5,𝔽2)12yes—
432C422−1+414yes—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—44no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—44no—
445C23C23×D411yescharacteristic
446Q8—88no—
447Q8—88no—
448D4—44no—
449D4—44no—
450D4—44no—
451D4—88no—
452D4—168no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—48no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—84no—
489C2×C4—164no—
490C2×C4#53612yes—
491C2×C4#53612yes—
492C2×C4#54614yes—
493C2×C4#57514yes—
494C2×C4C22×#4111yescharacteristic
495C2×C4C2×H(5,𝔽2)11yescharacteristic
496C2×C4C2×2−1+412yes—
497C2×C4#58512yes—
498C8—42no—
499C8—42no—
500C8—44no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22—82no—
509C22—84no—
510C22—82no—
511C22—82no—
512C22—84no—
513C22C2×#53611yescharacteristic
514C22C2×#54611yescharacteristic
515C22#350011yescharacteristic
516C22C2×#57512yes—
517C22#358512yes—
518C4—22no—
519C4—22no—
520C4—24no—
521C4—24no—
522C4—42no—
523C4—42no—
524C4—42no—
525C4—44no—
526C4—82no—
527C4—84no—
528C4—82no—
529C4—82no—
530C4—82no—
531C4#351812yes—
532C4#354712yes—
533C2—82no—
534C2—162no—
535C2#3895811yescharacteristic
536C2#6086311yescharacteristic
537C2#6088311yescharacteristic
538C2#6116412yes—
539C2#6258012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36211C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36132C2semidirect
3. #33295C2semidirect
4. #31753C2semidirect
5. #31751C2semidirect
6. #29851C2semidirect
7. #29214C2semidirect
8. #28724C2semidirect
9. #27880C2semidirect
10. #26036C2semidirect
11. #25911C2semidirect
12. #24978C2semidirect
13. #21839C2semidirect
14. #21656C2semidirect
15. #21271C2semidirect

Maximal subgroups

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

subgroupindex
#362112
#361322
#332952
#317532
#317512
#298512
#292142
#287242
#278802
#260362
#259112
#249782
#218392
#216562
#212712