Finite Group Databasebeta

← #6806146

Subgroups — #6806146

Order 512.

1574 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36602C211yesmaximal, characteristic, complemented
2#36559C211yesmaximal, characteristic, complemented
3#33230C211yesmaximal, characteristic, complemented
4#31641C214yesmaximal, complemented
5#31639C214yesmaximal, complemented
6#30217C212yesmaximal, complemented
7#29225C211yesmaximal, characteristic, complemented
8#29146C212yesmaximal, complemented
9#28618C212yesmaximal, complemented
10#28445C211yesmaximal, characteristic
11#25940C214yesmaximal, complemented
12#25335C214yesmaximal, complemented
13#22272C212yesmaximal, complemented
14#21656C211yesmaximal, characteristic, complemented
15#21346C211yesmaximal, characteristic, complemented
16C2×#554C2212yes—
17C2×#496C2212yes—
18C2×#490C2211yescharacteristic
19C2×#489C2214yes—
20C2×#463C2214yes—
21#2868C2211yescharacteristic
22#2861C2212yes—
23#2859C2212yes—
24#2858C2211yescharacteristic
25#2597—22no—
26#2593—22no—
27#2592C2212yes—
28#2566C2214yes—
29#2563C2212yes—
30#2538—22no—
31#2458—22no—
32#2452—22no—
33#2452—22no—
34#2446C2212yes—
35#2445C2212yes—
36#2374—22no—
37#2367C2212yes—
38D4×C4⋊C4—22no—
39C4×#127C2212yes—
40C4×#127C2211yescharacteristic
41C4×#125C2211yescharacteristic
42C4×#124C2212yes—
43C42×Q8C2211yescharacteristic
44#2100C2214yes—
45#2074C2214yes—
46#2073C2214yes—
47#2055C2214yes—
48#2054C2214yes—
49#2026C2214yes—
50#2026C2214yes—
51#2025C2214yes—
52#2025C2214yes—
53#2013C2212yes—
54#2011C2212yes—
55#1992C2211yescharacteristic
56#1992C2212yes—
57#1988C2214yes—
58#1985C2212yes—
59#1961C2211yescharacteristic
60#1959C2212yes—
61#1959C2212yes—
62#1959C2212yes—
63#1957C2212yes—
64#1956C2212yes—
65#1940C2212yes—
66#1934C2214yes—
67#1916C2214yes—
68#1880C2214yes—
69#1879C2214yes—
70#1879C2214yes—
71#1879C2214yes—
72#1877C2214yes—
73#1877C2214yes—
74#1836C2212yes—
75#1836C2211yescharacteristic
76#1805C2214yes—
77#1804C2214yes—
78#1767C2211yescharacteristic
79#1767C2212yes—
80#1762C2214yes—
81#1745C2212yes—
82C4×#97C2211yescharacteristic
83C22×SD16C2314yes—
84C22×D8C2314yes—
85#554—48no—
86C2×C4⋊Q8—22no—
87C2×C4⋊Q8C2311yescharacteristic
88C2×C4⋊Q8C2312yes—
89C2×#127C2311yescharacteristic
90C2×#127C2311yescharacteristic
91C2×#125C2311yescharacteristic
92C2×#125C2314yes—
93C2×#124C2312yes—
94C2×#124C2312yes—
95C2×#123—22no—
96C2×C22⋊Q8—22no—
97C2×C22⋊Q8—24no—
98C2×C4×Q8D412yes—
99C2×C4×Q8C2312yes—
100C2×C4×Q8C2311yescharacteristic
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4—22no—
104C2×C4×D4C2312yes—
105C2×C4×D4C2314yes—
106C2×C4×D4C2312yes—
107C22×C4⋊C4—22no—
108#496—28no—
109#490—24no—
110#489—48no—
111#463—48no—
112C2×C4⋊C8C2312yes—
113C2×C4⋊C8C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#103C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#97C2311yescharacteristic
123C2×#97C2312yes—
124C2×#97C2312yes—
125C2×#97C2311yescharacteristic
126C2×C4×C8C2312yes—
127#400—22no—
128#398—22no—
129#397—22no—
130#396—24no—
131#396—24no—
132#394—22no—
133#394—22no—
134#393—22no—
135#393—22no—
136#393—22no—
137#391C2312yes—
138#391C2312yes—
139#391C2312yes—
140#390—22no—
141#390—22no—
142#390—22no—
143#390—22no—
144#390—22no—
145#390C2314yes—
146#390C2312yes—
147#390C2312yes—
148#390C2312yes—
149#390C2312yes—
150#389—22no—
151#389—22no—
152#389—22no—
153#389—22no—
154#389C2312yes—
155#388—22no—
156#388—22no—
157#388—22no—
158#388—22no—
159#388—22no—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#386—22no—
169#385—22no—
170#385—24no—
171#385—24no—
172#384—22no—
173#384C2311yescharacteristic
174#384C2312yes—
175#384C2312yes—
176#382C2311yescharacteristic
177#382C2314yes—
178#380—24no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4D412yes—
181C4×C4⋊C4D412yes—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2311yescharacteristic
185C4×C4⋊C4C2312yes—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—22no—
189C43C2311yescharacteristic
190C43C2312yes—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#336C2314yes—
198#336C2312yes—
199#336C2314yes—
200#336C2314yes—
201C22×Q8—22no—
202C22×Q8C2×D411yescharacteristic
203C22×Q8C2412yes—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4C2412yes—
207C22×D4C2412yes—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—22no—
211C2×SD16—28no—
212C2×SD16—44no—
213C2×SD16—44no—
214C2×D8—28no—
215C2×D8—44no—
216C2×D8—44no—
217C22×C8C2414yes—
218C22×C8C2412yes—
219C22×C8C2412yes—
220C4⋊Q8—28no—
221C4⋊Q8—42no—
222C4⋊Q8—84no—
223#127—22no—
224#127—42no—
225#127#4114yes—
226#125—22no—
227#125—22no—
228#125—44no—
229#125—44no—
230#124—24no—
231#124—28no—
232#124—24no—
233#123—48no—
234C22⋊Q8—416no—
235C22⋊Q8—48no—
236C4×Q8—24no—
237C4×Q8—44no—
238C4×Q8—44no—
239C4×D4—28no—
240C4×D4—44no—
241C4×D4—44no—
242C4×D4—48no—
243C4×D4—44no—
244C4×D4—84no—
245C4×D4—84no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D412yes—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2412yes—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2×D412yes—
291C2×C42C2412yes—
292C2×C42C2412yes—
293C2×C42C2411yescharacteristic
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296C4⋊C8—24no—
297C4⋊C8—44no—
298#103—44no—
299#103—44no—
300#103—44no—
301#102—28no—
302#102—44no—
303#102—44no—
304#102—44no—
305#102—44no—
306#97—22no—
307#97—24no—
308#97—24no—
309#97—22no—
310C4×C8—42no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—24no—
318#95—24no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95C2×D411yescharacteristic
323#95C2×D412yes—
324#95C2×D412yes—
325#95C2×D411yescharacteristic
326C24—22no—
327C24—22no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—24no—
331C2×Q8—22no—
332C2×Q8—24no—
333C2×Q8—42no—
334C2×Q8—44no—
335C2×Q8—84no—
336C2×D4—28no—
337C2×D4—22no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—44no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—82no—
348C2×D4—82no—
349C2×D4C2×#4114yes—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—82no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4—82no—
363C22×C4C22×D412yes—
364C22×C4C22×D412yes—
365C22×C4C22×D412yes—
366C22×C4C22×D412yes—
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D412yes—
370C22×C4C22×D412yes—
371C22×C4C2511yescharacteristic
372SD16—88no—
373D8—88no—
374C2×C8—22no—
375C2×C8—24no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—24no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—24no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—42no—
394C4⋊C4—48no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—48no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—82no—
403C4⋊C4—82no—
404C4⋊C4—84no—
405C22⋊C4—48no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—48no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C22⋊C4—84no—
420C22⋊C4—84no—
421C42—24no—
422C42—24no—
423C42—22no—
424C42—24no—
425C42—42no—
426C42—42no—
427C42—42no—
428C42—42no—
429C42—42no—
430C42—82no—
431C42—82no—
432C42C2×#4112yes—
433C42H(5,𝔽2)14yes—
434C422−1+412yes—
435C23—24no—
436C23—44no—
437C23—42no—
438C23—42no—
439C23—44no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23C23×D411yescharacteristic
448Q8—44no—
449Q8—44no—
450Q8—44no—
451Q8—164no—
452D4—28no—
453D4—416no—
454D4—44no—
455D4—44no—
456D4—44no—
457D4—164no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—88no—
491C2×C4—164no—
492C2×C4—168no—
493C2×C4#53512yes—
494C2×C4#53512yes—
495C2×C4#54814yes—
496C2×C4#57514yes—
497C2×C4C22×#4111yescharacteristic
498C2×C4C2×H(5,𝔽2)12yes—
499C2×C4C2×2−1+411yescharacteristic
500C2×C4#58512yes—
501C8—42no—
502C8—44no—
503C8—42no—
504C22—44no—
505C22—48no—
506C22—84no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22—82no—
511C22—82no—
512C22—82no—
513C22—82no—
514C22—82no—
515C22—82no—
516C22C2×#53511yescharacteristic
517C22C2×#54811yescharacteristic
518C22#350011yescharacteristic
519C22C2×#57512yes—
520C22#358512yes—
521C4—24no—
522C4—22no—
523C4—24no—
524C4—22no—
525C4—42no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—82no—
530C4—82no—
531C4—82no—
532C4—82no—
533C4—84no—
534C4#351412yes—
535C4#355412yes—
536C2—84no—
537C2—162no—
538C2#3917111yescharacteristic
539C2#6080111yescharacteristic
540C2#6083011yescharacteristic
541C2#6131812yes—
542C2#6258412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#28445C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36602C2semidirect
2. #36559C2semidirect
3. #33230C2semidirect
4. #31641C2semidirect
5. #31639C2semidirect
6. #30217C2semidirect
7. #29225C2semidirect
8. #29146C2semidirect
9. #28618C2semidirect
11. #25940C2semidirect
12. #25335C2semidirect
13. #22272C2semidirect
14. #21656C2semidirect
15. #21346C2semidirect

Maximal subgroups

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

subgroupindex
#366022
#365592
#332302
#316412
#316392
#302172
#292252
#291462
#286182
#284452
#259402
#253352
#222722
#216562
#213462