Finite Group Databasebeta

← #6798473

Subgroups — #6798473

Order 512.

1558 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36192C211yesmaximal, characteristic, complemented
2#36149C211yesmaximal, characteristic
3#33296C211yesmaximal, characteristic, complemented
4#31361C214yesmaximal, complemented
5#31359C214yesmaximal, complemented
6#29883C212yesmaximal, complemented
7#29149C211yesmaximal, characteristic, complemented
8#28695C211yesmaximal, characteristic, complemented
9#27968C212yesmaximal, complemented
10#26124C214yesmaximal, complemented
11#25676C212yesmaximal, complemented
12#24978C214yesmaximal, complemented
13#21869C212yesmaximal, complemented
14#21656C211yesmaximal, characteristic, complemented
15#21271C211yesmaximal, characteristic, complemented
16C2×#532C2212yes—
17C2×#501C2212yes—
18C2×#490C2211yescharacteristic
19C2×#432C2212yes—
20#2862C2211yescharacteristic
21#2859C2211yescharacteristic
22#2857C2212yes—
23#2851C2212yes—
24#2552C2212yes—
25#2535C2212yes—
26#2522—22no—
27#2522—22no—
28#2454—22no—
29#2447—22no—
30#2441C2212yes—
31#2441C2212yes—
32#2427—22no—
33#2420C2214yes—
34#2407—22no—
35#2370—22no—
36#2354C2212yes—
37#2349C2212yes—
38#2339—22no—
39C4×#127C2211yescharacteristic
40C4×#125C2211yescharacteristic
41#2285C2212yes—
42C42×D4C2211yescharacteristic
43#2100C2214yes—
44#2099C2214yes—
45#2088C2214yes—
46#2087C2214yes—
47#2085C2214yes—
48#2077C2214yes—
49#2075C2214yes—
50#2034C2214yes—
51#2033C2214yes—
52#2025C2212yes—
53#2025C2212yes—
54#1993C2212yes—
55#1993C2212yes—
56#1993C2212yes—
57#1992C2211yescharacteristic
58#1991C2212yes—
59#1991C2212yes—
60#1961C2212yes—
61#1961C2211yescharacteristic
62#1955C2212yes—
63#1949C2214yes—
64#1940C2212yes—
65#1934C2214yes—
66#1923C2214yes—
67#1875C2214yes—
68#1875C2214yes—
69#1864C2214yes—
70#1862C2214yes—
71#1836C2214yes—
72#1836C2211yescharacteristic
73#1836C2212yes—
74#1825C2214yes—
75#1825C2214yes—
76#1802C2214yes—
77#1802C2214yes—
78#1802C2214yes—
79#1767C2211yescharacteristic
80#1762C2214yes—
81#1749C2212yes—
82C4×#97C2211yescharacteristic
83C22×#41—22no—
84#532—28no—
85#532—44no—
86C2×C4⋊Q8C2312yes—
87C2×#127C2311yescharacteristic
88C2×#125—22no—
89C2×#125C2311yescharacteristic
90C2×#125C2311yescharacteristic
91C2×#125C2311yescharacteristic
92C2×#124—22no—
93C2×C22⋊Q8—22no—
94C2×#121—22no—
95C2×#121C2314yes—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4D412yes—
99C2×C4×D4C2312yes—
100C2×C4×D4C2314yes—
101C2×C4×D4C2311yescharacteristic
102C2×#117C2312yes—
103C2×#117C2312yes—
104C22×C42C2312yes—
105#501—44no—
106#490—24no—
107#432—44no—
108C2×#107C2314yes—
109C2×#106C2314yes—
110C2×C4⋊C8C2312yes—
111C2×C4⋊C8C2314yes—
112C2×#102C2314yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×C22⋊C8C2314yes—
117C2×C22⋊C8C2314yes—
118C2×#97C2312yes—
119C2×#97C2311yescharacteristic
120C2×#97C2312yes—
121C2×#97C2311yescharacteristic
122C2×C4×C8C2312yes—
123#399—22no—
124#399—22no—
125#398—22no—
126#398—22no—
127#397—22no—
128#397—24no—
129#396—22no—
130#396—22no—
131#394—24no—
132#392—22no—
133#391—22no—
134#391—22no—
135#391—22no—
136#390—22no—
137#390—22no—
138#390C2312yes—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389C2312yes—
146#389C2312yes—
147#389C2312yes—
148#389C2314yes—
149#388—22no—
150#388—22no—
151#387—22no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#386—24no—
157#386—24no—
158#385—22no—
159#385—22no—
160#385—24no—
161#384C2311yescharacteristic
162#384C2312yes—
163#384C2312yes—
164#384C2312yes—
165#382C2311yescharacteristic
166#382C2312yes—
167#382C2312yes—
168#382C2312yes—
169#381—24no—
170#380—22no—
171#380—22no—
172#380—22no—
173#380—22no—
174#380C2312yes—
175#380C2314yes—
176#380C2312yes—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4D412yes—
182C4×C4⋊C4C2312yes—
183C4×C4⋊C4C2311yescharacteristic
184C4×C22⋊C4D412yes—
185C4×C22⋊C4D412yes—
186C4×C22⋊C4C2312yes—
187C2×#95—22no—
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—
201C2×#41—48no—
202C2×#41—48no—
203C2×#41—84no—
204C22×Q8—22no—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4C2×D411yescharacteristic
208C22×D4C2412yes—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4C2×D412yes—
212C23×C4C2×D412yes—
213C23×C4C2412yes—
214C22×C8C2412yes—
215C22×C8C2414yes—
216C22×C8C2412yes—
217C4⋊Q8—24no—
218C4⋊Q8—24no—
219#127—22no—
220#127#4114yes—
221#125—22no—
222#125—22no—
223#125—42no—
224#125—42no—
225#125—48no—
226#124—48no—
227C22⋊Q8—84no—
228#121—28no—
229#121—44no—
230#121—48no—
231C4×D4—28no—
232C4×D4—24no—
233C4×D4—28no—
234C4×D4—44no—
235C4×D4—48no—
236C4×D4—44no—
237C4×D4—84no—
238#117—44no—
239#117—44no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
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⋊C4C2×D412yes—
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2412yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2412yes—
262C2×C4⋊C4C2412yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4C2×D412yes—
273C2×C22⋊C4C2×D412yes—
274C2×C22⋊C4C2×D412yes—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—22no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D412yes—
283C2×C42C2×D412yes—
284C2×C42C2412yes—
285C2×C42C2411yescharacteristic
286C2×C42C2411yescharacteristic
287C2×C42C2411yescharacteristic
288#107—44no—
289#106—44no—
290C4⋊C8—24no—
291C4⋊C8—44no—
292#102—28no—
293#102—44no—
294#102—44no—
295#102—44no—
296C22⋊C8—44no—
297C22⋊C8—44no—
298#97—22no—
299#97—22no—
300#97—24no—
301#97—24no—
302C4×C8—42no—
303#95—24no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95C2×D411yescharacteristic
318#95C2×D411yescharacteristic
319#95C2×D412yes—
320#95C2×D412yes—
321C24—22no—
322C24C22×D412yes—
323#41—832no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×D4—22no—
328C2×D4—28no—
329C2×D4—22no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—42no—
333C2×D4—44no—
334C2×D4—44no—
335C2×D4—48no—
336C2×D4—48no—
337C2×D4—84no—
338C2×D4C2×#4114yes—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—44no—
358C22×C4—84no—
359C22×C4—84no—
360C22×C4C22×D412yes—
361C22×C4C22×D412yes—
362C22×C4C22×D412yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C2511yescharacteristic
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—24no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—24no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—24no—
377C4⋊C4—22no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—84no—
400C4⋊C4—82no—
401C4⋊C4—82no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—42no—
405C22⋊C4—48no—
406C22⋊C4—48no—
407C22⋊C4—44no—
408C22⋊C4—84no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—84no—
414C42—24no—
415C42—24no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—42no—
420C42—42no—
421C42—44no—
422C42—82no—
423C42—82no—
424C42C2×#4112yes—
425C42H(5,𝔽2)12yes—
426C422−1+414yes—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—44no—
436C23—42no—
437C23—42no—
438C23—44no—
439C23C23×D411yescharacteristic
440Q8—88no—
441D4—28no—
442D4—44no—
443D4—416no—
444D4—44no—
445D4—816no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
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—48no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—84no—
482C2×C4—816no—
483C2×C4—164no—
484C2×C4#53612yes—
485C2×C4#53612yes—
486C2×C4#54614yes—
487C2×C4#57514yes—
488C2×C4C22×#4111yescharacteristic
489C2×C4C2×H(5,𝔽2)11yescharacteristic
490C2×C4C2×2−1+412yes—
491C2×C4#58512yes—
492C8—42no—
493C8—42no—
494C8—44no—
495C22—42no—
496C22—44no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—48no—
503C22—42no—
504C22—84no—
505C22—82no—
506C22—82no—
507C22C2×#53611yescharacteristic
508C22C2×#54611yescharacteristic
509C22#350011yescharacteristic
510C22C2×#57512yes—
511C22#358512yes—
512C4—22no—
513C4—22no—
514C4—24no—
515C4—24no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—82no—
520C4—82no—
521C4—82no—
522C4—82no—
523C4—82no—
524C4—84no—
525C4#351512yes—
526C4#354712yes—
527C2—82no—
528C2—84no—
529C2#3895511yescharacteristic
530C2#6086311yescharacteristic
531C2#6088311yescharacteristic
532C2#6115612yes—
533C2#6258612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36149C2
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
1. #36192C2semidirect
3. #33296C2semidirect
4. #31361C2semidirect
5. #31359C2semidirect
6. #29883C2semidirect
7. #29149C2semidirect
8. #28695C2semidirect
9. #27968C2semidirect
10. #26124C2semidirect
11. #25676C2semidirect
12. #24978C2semidirect
13. #21869C2semidirect
14. #21656C2semidirect
15. #21271C2semidirect

Maximal subgroups

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

subgroupindex
#361922
#361492
#332962
#313612
#313592
#298832
#291492
#286952
#279682
#261242
#256762
#249782
#218692
#216562
#212712