Finite Group Databasebeta

← #6556042

Subgroups — #6556042

Order 512.

1431 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#34324C211yesmaximal, characteristic
3#30168C212yesmaximal, complemented
4#29869C211yesmaximal, characteristic, complemented
5#29304C212yesmaximal, complemented
6#29276C211yesmaximal, characteristic, complemented
7#28687C211yesmaximal, characteristic, complemented
8#27612C212yesmaximal, complemented
9#27480C214yesmaximal, complemented
10#27060C214yesmaximal, complemented
11#26297C212yesmaximal
12#25701C212yesmaximal
13#23777C211yesmaximal, characteristic, complemented
14#23762C212yesmaximal, complemented
15#23163C212yesmaximal, complemented
16#22971C212yesmaximal, complemented
17#22839C211yesmaximal, characteristic
18C2×#556C2211yescharacteristic
19C2×#498C2212yes—
20C2×#477C2212yes—
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2571—21no—
24#2563C2212yes—
25#2538—21no—
26#2497C2214yes—
27#2492—22no—
28#2439—22no—
29#2373—22no—
30#2369C2212yes—
31#2367—21no—
32#2365C2211yescharacteristic
33#2362C2212yes—
34#2355—21no—
35#2355—21no—
36#2354—21no—
37#2354C2211yescharacteristic
38Q8×C4⋊C4C2211yescharacteristic
39#2346C2212yes—
40#2334—22no—
41#2326—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C22⋊Q8C2212yes—
45#2296C2212yes—
46#2295C2211yescharacteristic
47#2285C2212yes—
48#2276—21no—
49C42×D4C2211yescharacteristic
50#2100C2212yes—
51#2100C2211yescharacteristic
52#2095C2211yescharacteristic
53#2087C2212yes—
54#2084C2212yes—
55#2084C2212yes—
56#2083C2212yes—
57#2083C2212yes—
58#2077C2214yes—
59#2075C2214yes—
60#2072C2212yes—
61#2072C2212yes—
62#2068C2212yes—
63#2058C2212yes—
64#2057C2211yescharacteristic
65#2035C2214yes—
66#2035C2214yes—
67#2024C2211yescharacteristic
68#1990C2212yes—
69#1988C2212yes—
70#1987C2212yes—
71#1954C2212yes—
72#1948C2214yes—
73#1948C2212yes—
74#1944C2214yes—
75#1943C2214yes—
76#1924C2214yes—
77#1920C2214yes—
78#1920C2214yes—
79#1919C2214yes—
80#1918C2214yes—
81#1865C2214yes—
82#1854C2214yes—
83#1854C2214yes—
84#1854C2214yes—
85#1831C2211yescharacteristic
86#1825C2214yes—
87#1817C2212yes—
88#1776C2214yes—
89C4×#107C2212yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#103C2212yes—
94#1749C2212yes—
95#556—24no—
96#556—24no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×#126—22no—
103C2×#124—21no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8C2312yes—
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×D4D411yescharacteristic
113C2×C4×D4D411yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×#117—22no—
116C2×#117C2312yes—
117C2×#117C2312yes—
118C22×C42C2312yes—
119#498—28no—
120#477—28no—
121C2×#107C2314yes—
122C2×#107C2312yes—
123C2×#107C2312yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2312yes—
129C2×#103C2314yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134C2×#97C2314yes—
135C2×C4×C8C2312yes—
136#400—24no—
137#396—24no—
138#393—24no—
139#391—21no—
140#391—21no—
141#391—21no—
142#391—22no—
143#391—24no—
144#391C2311yescharacteristic
145#391C2311yescharacteristic
146#390—22no—
147#390—21no—
148#389—22no—
149#389—21no—
150#389—21no—
151#389—22no—
152#389—22no—
153#389C2312yes—
154#389C2312yes—
155#389C2311yescharacteristic
156#389C2311yescharacteristic
157#389C2314yes—
158#389C2312yes—
159#389C2312yes—
160#388—22no—
161#388—22no—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—24no—
166#387—21no—
167#387—22no—
168#387—21no—
169#387—22no—
170#387—22no—
171#387—24no—
172#385—22no—
173#385—22no—
174#385—22no—
175#384—21no—
176#384—21no—
177#384—21no—
178#384C2311yescharacteristic
179#384C2311yescharacteristic
180#384C2312yes—
181#384C2312yes—
182#382—22no—
183#382—22no—
184#382—22no—
185#380—22no—
186#380—22no—
187#380C2314yes—
188#380C2314yes—
189#380C2312yes—
190C4×C4⋊C4—21no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—21no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2312yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2314yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C22⋊C4—22no—
210C4×C22⋊C4D412yes—
211C4×C22⋊C4D412yes—
212C4×C22⋊C4C2312yes—
213#376—21no—
214#376—22no—
215C43C2311yescharacteristic
216#340C2312yes—
217#340C2314yes—
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#336C2311yescharacteristic
228#336C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2314yes—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8C2411yescharacteristic
236C22×D4C2×D411yescharacteristic
237C23×C4C2×D412yes—
238C23×C4C2×D412yes—
239C23×C4C2412yes—
240C22×C8C2412yes—
241C22×C8C2414yes—
242C22×C8C2412yes—
243C4⋊Q8—22no—
244C4⋊Q8—24no—
245C4⋊Q8—24no—
246C4⋊Q8—42no—
247C4⋊Q8—82no—
248C4⋊Q8—82no—
249C4⋊Q8#4114yes—
250#126—216no—
251#124—28no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254C22⋊Q8—84no—
255C4×Q8—28no—
256C4×Q8—22no—
257C4×Q8—44no—
258C4×Q8—44no—
259C4×Q8—44no—
260C4×Q8—44no—
261C4×Q8#4114yes—
262C4×D4—24no—
263C4×D4—22no—
264C4×D4—22no—
265C4×D4—42no—
266#117—24no—
267#117—24no—
268#117—44no—
269#117—84no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2412yes—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D412yes—
310C2×C22⋊C4C2×D412yes—
311C2×C42—22no—
312C2×C42—21no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—44no—
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2×D411yescharacteristic
327C2×C42C2412yes—
328C2×C42C2411yescharacteristic
329C2×C42C2411yescharacteristic
330C2×C42C2411yescharacteristic
331#107—24no—
332#107—28no—
333#107—42no—
334#106—28no—
335#106—28no—
336#106—44no—
337C4⋊C8—24no—
338C4⋊C8—24no—
339#103—28no—
340#103—24no—
341#103—42no—
342C22⋊C8—28no—
343C22⋊C8—44no—
344#97—44no—
345C4×C8—24no—
346#95—21no—
347#95—22no—
348#95—21no—
349#95—21no—
350#95—22no—
351#95—21no—
352#95—21no—
353#95—22no—
354#95—24no—
355#95—24no—
356#95—22no—
357#95—21no—
358#95—22no—
359#95—24no—
360#95—22no—
361#95—22no—
362#95—21no—
363#95—22no—
364#95—24no—
365#95C2×D412yes—
366#95C2×D412yes—
367#95C2×D412yes—
368C24C22×D412yes—
369C2×Q8—24no—
370C2×Q8—21no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8—44no—
376C2×Q8—42no—
377C2×Q8—42no—
378C2×Q8—42no—
379C2×Q8—82no—
380C2×Q8—82no—
381C2×Q8C2×#4112yes—
382C2×D4—22no—
383C2×D4—22no—
384C2×D4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C2511yescharacteristic
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—22no—
420C2×C8—24no—
421C2×C8—22no—
422C2×C82−1+414yes—
423C4⋊C4—28no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—22no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—42no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—42no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—82no—
454C4⋊C4—82no—
455C4⋊C4—84no—
456C4⋊C4—82no—
457C4⋊C4—82no—
458C4⋊C4C2×#4114yes—
459C4⋊C4C2×#4112yes—
460C22⋊C4—28no—
461C22⋊C4—28no—
462C22⋊C4—24no—
463C22⋊C4—24no—
464C22⋊C4—42no—
465C22⋊C4—82no—
466C22⋊C4—82no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C42—28no—
470C42—24no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—24no—
475C42—22no—
476C42—42no—
477C42—42no—
478C42—41no—
479C42—44no—
480C42—44no—
481C42—42no—
482C42—44no—
483C42—82no—
484C42—84no—
485C42—82no—
486C42#13714yes—
487C42C2×#4112yes—
488C42C2×#4112yes—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23C23×D411yescharacteristic
497Q8—24no—
498Q8—48no—
499Q8—42no—
500Q8—84no—
501Q8—164no—
502D4—44no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—48no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—44no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—84no—
554C2×C4#52512yes—
555C2×C4#52512yes—
556C2×C4C2×#13712yes—
557C2×C4C2×#13712yes—
558C2×C4C2×#13712yes—
559C2×C4#57512yes—
560C2×C4#57712yes—
561C2×C4C22×#4111yescharacteristic
562C2×C4C22×#4111yescharacteristic
563C2×C4C2×2−1+412yes—
564C2×C4#58512yes—
565C8—24no—
566C8—44no—
567C8—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22C2×#52511yescharacteristic
576C22#348511yescharacteristic
577C22#349911yescharacteristic
578C22C22×#13711yescharacteristic
579C22C2×#57511yescharacteristic
580C22C2×#57711yescharacteristic
581C22#358511yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—42no—
590C4—41no—
591C4—44no—
592C4—42no—
593C4—82no—
594C4—81no—
595C4—82no—
596C4—82no—
597C4—81no—
598C4—84no—
599C4#311614yes—
600C4#350712yes—
601C4#351612yes—
602C2—82no—
603C2#3584611yescharacteristic
604C2C2×#311611yescharacteristic
605C2#6059311yescharacteristic
606C2#6171911yescharacteristic
607C2#6173111yescharacteristic
608C2#6247811yescharacteristic
609C2#6250811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #30168C2semidirect
4. #29869C2semidirect
5. #29304C2semidirect
6. #29276C2semidirect
7. #28687C2semidirect
8. #27612C2semidirect
9. #27480C2semidirect
10. #27060C2semidirect
13. #23777C2semidirect
14. #23762C2semidirect
15. #23163C2semidirect
16. #22971C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#343242
#301682
#298692
#293042
#292762
#286872
#276122
#274802
#270602
#262972
#257012
#237772
#237622
#231632
#229712
#228392