Finite Group Databasebeta

← #6798412

Subgroups — #6798412

Order 512.

1842 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36180C211yesmaximal, characteristic, complemented
2#36127C211yesmaximal, characteristic
3#33289C211yesmaximal, characteristic, complemented
4#31350C218yesmaximal, complemented
5#29876C212yesmaximal, complemented
6#29139C211yesmaximal, characteristic, complemented
7#28689C211yesmaximal, characteristic, complemented
8#27960C212yesmaximal, complemented
9#26106C214yesmaximal, complemented
10#25673C212yesmaximal, complemented
11#24965C214yesmaximal, complemented
12#21859C212yesmaximal, complemented
13C4×#486C211yesmaximal, characteristic, complemented
14#21264C211yesmaximal, characteristic, complemented
15C2×#554—21no—
16C2×#547—21no—
17C2×#532C2212yes—
18C2×#500C2212yes—
19C2×#486C2211yescharacteristic
20C2×#431C2212yes—
21#2868C2211yescharacteristic
22#2868C2211yescharacteristic
23#2857C2212yes—
24#2857C2212yes—
25#2567—21no—
26#2552—21no—
27#2552C2212yes—
28#2547—21no—
29#2537—21no—
30#2537C2212yes—
31#2502—22no—
32#2440—22no—
33#2440C2214yes—
34#2432—22no—
35#2422—22no—
36#2422C2214yes—
37D4×C4⋊C4—21no—
38D4×C4⋊C4—21no—
39D4×C4⋊C4C2212yes—
40D4×C4⋊C4C2212yes—
41C4×C4⋊Q8C2211yescharacteristic
42C4×#127C2211yescharacteristic
43C2×#384C2212yes—
44C42×D4C2211yescharacteristic
45#2099C2218yes—
46#2086C2218yes—
47#2083C2214yes—
48#2075C2218yes—
49#2033C2218yes—
50#2023C2214yes—
51#1990C2212yes—
52#1990C2212yes—
53#1990C2214yes—
54#1990C2212yes—
55#1990C2211yescharacteristic
56#1957C2212yes—
57#1957C2211yescharacteristic
58#1954C2212yes—
59#1947C2214yes—
60#1939C2212yes—
61#1934C2214yes—
62#1923C2214yes—
63#1874C2218yes—
64#1862C2218yes—
65#1834C2214yes—
66#1834C2211yescharacteristic
67#1834C2212yes—
68#1823C2218yes—
69#1802C2218yes—
70#1802C2214yes—
71C4×C4⋊C8C2211yescharacteristic
72C4×#102C2214yes—
73C4×C22⋊C8C2212yes—
74C42×C8C2211yescharacteristic
75C22×#41—22no—
76#554—48no—
77#547—48no—
78#532—28no—
79#532—44no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8—21no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2312yes—
87C2×#127C2311yescharacteristic
88C2×#125—22no—
89C2×#124—22no—
90C2×#123—24no—
91C2×C22⋊Q8—22no—
92C2×C22⋊Q8—22no—
93C2×C22⋊Q8—24no—
94C2×#121—22no—
95C2×#121—22no—
96C2×#121C2314yes—
97C2×C4×Q8—21no—
98C2×C4×D4—22no—
99C2×C4×D4—21no—
100C2×C4×D4—21no—
101C2×C4×D4—22no—
102C2×C4×D4D411yescharacteristic
103C2×C4×D4D411yescharacteristic
104C2×C4×D4C2312yes—
105C2×C4×D4C2314yes—
106C2×C4×D4C2311yescharacteristic
107C22×C4⋊C4—22no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4C2314yes—
111C22×C42C2312yes—
112#500—44no—
113#486—24no—
114#431—44no—
115C2×#107C2318yes—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2314yes—
118C2×#102C2314yes—
119C2×#102C2318yes—
120C2×#102C2314yes—
121C2×C22⋊C8C2314yes—
122C2×C22⋊C8C2314yes—
123C2×C4×C8C2312yes—
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2312yes—
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2312yes—
128#398—24no—
129#398—24no—
130#397—24no—
131#395—24no—
132#394—24no—
133#394—24no—
134#391—21no—
135#391—22no—
136#391—22no—
137#390—21no—
138#390—21no—
139#390C2312yes—
140#389—22no—
141#389—22no—
142#389—21no—
143#389—22no—
144#389—22no—
145#389C2312yes—
146#389C2314yes—
147#389C2314yes—
148#385—24no—
149#385—24no—
150#385—24no—
151#385—24no—
152#384—22no—
153#384—21no—
154#384—21no—
155#384—21no—
156#384—21no—
157#384—22no—
158#384C2311yescharacteristic
159#384C2311yescharacteristic
160#384C2312yes—
161#384C2314yes—
162#384C2312yes—
163#384C2312yes—
164#384C2312yes—
165#380—22no—
166#380—22no—
167#380—22no—
168#380—22no—
169#380C2314yes—
170#380C2314yes—
171C4×C4⋊C4—21no—
172C4×C4⋊C4D411yescharacteristic
173C4×C4⋊C4D411yescharacteristic
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2311yescharacteristic
176C4×C22⋊C4D412yes—
177C4×C22⋊C4D412yes—
178C4×C22⋊C4C2312yes—
179C43C2311yescharacteristic
180#340C2318yes—
181#340C2318yes—
182#340C2318yes—
183#340C2314yes—
184#340C2314yes—
185#340C2318yes—
186#336C2312yes—
187#336C2314yes—
188C2×#41—48no—
189C2×#41—48no—
190C2×#41—84no—
191C22×Q8—21no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×D4—24no—
195C22×D4—21no—
196C22×D4—21no—
197C22×D4C2×D411yescharacteristic
198C22×D4C2412yes—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4C2×D412yes—
203C23×C4C2×D412yes—
204C23×C4C2412yes—
205C22×C8C2412yes—
206C22×C8C2414yes—
207C22×C8C2412yes—
208C4⋊Q8—24no—
209C4⋊Q8—22no—
210C4⋊Q8—24no—
211C4⋊Q8—22no—
212C4⋊Q8—44no—
213C4⋊Q8—44no—
214C4⋊Q8—44no—
215C4⋊Q8—82no—
216C4⋊Q8—82no—
217#127—22no—
218#127#4114yes—
219#125—48no—
220#124—48no—
221#123—416no—
222C22⋊Q8—48no—
223C22⋊Q8—48no—
224C22⋊Q8—416no—
225#121—28no—
226#121—44no—
227#121—48no—
228#121—48no—
229C4×Q8—44no—
230C4×D4—28no—
231C4×D4—24no—
232C4×D4—24no—
233C4×D4—28no—
234C4×D4—44no—
235C4×D4—48no—
236C4×D4—48no—
237C4×D4—44no—
238C4×D4—44no—
239C4×D4D818yes—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—48no—
262C2×C4⋊C4—48no—
263C2×C4⋊C4—48no—
264C2×C4⋊C4—84no—
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D411yescharacteristic
267C2×C4⋊C4C2×D411yescharacteristic
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4C2×D412yes—
281C2×C22⋊C4C2×D412yes—
282C2×C22⋊C4C2×D412yes—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2412yes—
293C2×C42C2411yescharacteristic
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296#107—48no—
297C4⋊C8—24no—
298C4⋊C8—44no—
299#102—28no—
300#102—44no—
301#102—48no—
302C22⋊C8—44no—
303C22⋊C8—44no—
304C4×C8—22no—
305C4×C8—22no—
306C4×C8—24no—
307C4×C8—24no—
308C4×C8—42no—
309#95—22no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95C2×D412yes—
316#95C2×D412yes—
317#95C2×D412yes—
318C24—22no—
319C24C22×D412yes—
320#41—832no—
321C2×Q8—44no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×Q8—42no—
328C2×Q8—82no—
329C2×Q8—82no—
330C2×D4—22no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—24no—
336C2×D4—24no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—42no—
342C2×D4—42no—
343C2×D4—84no—
344C2×D4C2×D814yes—
345C2×D4C2×#4114yes—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—48no—
361C22×C4—48no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—48no—
365C22×C4—44no—
366C22×C4—84no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4C2×D818yes—
370C22×C4C22×D412yes—
371C22×C4C22×D412yes—
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C2511yescharacteristic
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—24no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—24no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—28no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—44no—
402C4⋊C4—48no—
403C4⋊C4—48no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—48no—
409C4⋊C4—44no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—84no—
414C4⋊C4—84no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C4⋊C4—82no—
419C4⋊C4C2×D814yes—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4C2×D818yes—
430C42—24no—
431C42—24no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—44no—
438C42C2×D814yes—
439C42#13714yes—
440C42C2×#4112yes—
441C42H(5,𝔽2)12yes—
442C422−1+414yes—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23C22×D814yes—
457C23C23×D411yescharacteristic
458Q8—84no—
459Q8—88no—
460Q8—164no—
461D4—28no—
462D4—28no—
463D4—48no—
464D4—44no—
465D4—48no—
466D4—816no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—216no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—48no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—816no—
501C2×C4—88no—
502C2×C4—168no—
503C2×C4#53612yes—
504C2×C4#53612yes—
505C2×C4#54612yes—
506C2×C4#54612yes—
507C2×C4C22×D812yes—
508C2×C4C22×D814yes—
509C2×C4C22×D812yes—
510C2×C4C22×D812yes—
511C2×C4C2×#13712yes—
512C2×C4C2×#13712yes—
513C2×C4C2×#13712yes—
514C2×C4#57712yes—
515C2×C4#57712yes—
516C2×C4C22×#4111yescharacteristic
517C2×C4C2×H(5,𝔽2)11yescharacteristic
518C2×C4C2×2−1+412yes—
519C2×C4#58512yes—
520C8—42no—
521C8—42no—
522C8—44no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22C2×#53611yescharacteristic
538C22C2×#54611yescharacteristic
539C22#350011yescharacteristic
540C22C23×D811yescharacteristic
541C22C22×#13711yescharacteristic
542C22C2×#57711yescharacteristic
543C22C2×#57711yescharacteristic
544C4—22no—
545C4—22no—
546C4—24no—
547C4—22no—
548C4—42no—
549C4—42no—
550C4—82no—
551C4—84no—
552C4—82no—
553C4—84no—
554C4—82no—
555C4#322614yes—
556C4#329414yes—
557C4#333114yes—
558C4#351512yes—
559C4#354712yes—
560C2—44no—
561C2—84no—
562C2#3895511yescharacteristic
563C2C2×#322611yescharacteristic
564C2#6088111yescharacteristic
565C2C2×#329411yescharacteristic
566C2C2×#333111yescharacteristic
567C2#6252911yescharacteristic
568C2#6254011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
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. #36180C2semidirect
3. #33289C2semidirect
4. #31350C2semidirect
5. #29876C2semidirect
6. #29139C2semidirect
7. #28689C2semidirect
8. #27960C2semidirect
9. #26106C2semidirect
10. #25673C2semidirect
11. #24965C2semidirect
12. #21859C2semidirect
13. C4×#486C2semidirect
14. #21264C2semidirect

Maximal subgroups

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

subgroupindex
#361802
#361272
#332892
#313502
#298762
#291392
#286892
#279602
#261062
#256732
#249652
#218592
C4×#4862
#212642