Finite Group Databasebeta

← #6857036

Subgroups — #6857036

Order 512.

1331 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#36598C211yesmaximal, characteristic
2#34951C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic
4#32482C214yesmaximal
5#31812C214yesmaximal
6#29831C212yesmaximal
7#29349C212yesmaximal
8#29288C212yesmaximal
9#29219C211yesmaximal, characteristic
10#28283C211yesmaximal, characteristic
11#28187C212yesmaximal
12#26027C212yesmaximal
13#25472C212yesmaximal
14#24683C212yesmaximal
15Q8×C4⋊C8C211yesmaximal, characteristic
16#22112C212yesmaximal
17C4×#462C211yesmaximal, characteristic
18C2×#475C2212yes—
19C2×#475C2212yes—
20C2×#462C2211yescharacteristic
21C2×C8×Q8C2212yes—
22C2×C4×Q16C2212yes—
23#2868C2211yescharacteristic
24#2862C2212yes—
25#2585—22no—
26#2582C2211yescharacteristic
27#2582C2212yes—
28#2571—21no—
29#2570—22no—
30#2568—21no—
31#2568C2211yescharacteristic
32#2552C2211yescharacteristic
33#2546C2211yescharacteristic
34#2511—21no—
35#2511C2214yes—
36#2509C2212yes—
37#2453C2212yes—
38#2446—22no—
39#2446—21no—
40#2402—22no—
41#2365—21no—
42#2355—21no—
43#2355C2212yes—
44C4×C4⋊Q8—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×#125C2212yes—
49#2295—21no—
50#2295C2212yes—
51C42×Q8C2211yescharacteristic
52C42×Q8C2211yescharacteristic
53#2099C2214yes—
54#2088C2214yes—
55#2087C2214yes—
56#2074C2214yes—
57#2066C2214yes—
58#2066C2214yes—
59#2066C2214yes—
60#2056C2214yes—
61#2056C2214yes—
62#2026C2214yes—
63#1990C2212yes—
64#1989C2214yes—
65#1988C2212yes—
66#1988C2212yes—
67#1988C2212yes—
68#1981C2212yes—
69#1981C2211yescharacteristic
70#1949C2214yes—
71#1948C2212yes—
72#1948C2212yes—
73#1948C2212yes—
74#1947C2211yescharacteristic
75#1941C2214yes—
76#1939C2214yes—
77#1939C2212yes—
78#1933C2214yes—
79#1933C2214yes—
80#1933C2214yes—
81#1928C2212yes—
82#1927C2214yes—
83#1920C2214yes—
84#1878C2212yes—
85#1842C2212yes—
86#1842C2211yescharacteristic
87#1834C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90#1803C2212yes—
91#1803C2211yescharacteristic
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#103C2212yes—
95C22×Q16C2312yes—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#125C2312yes—
103C2×C4×Q8—21no—
104C2×C4×Q8—22no—
105C2×C4×Q8D411yescharacteristic
106C2×C4×Q8D411yescharacteristic
107C2×C4×Q8C2312yes—
108C2×C4×Q8C2311yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112#475—28no—
113#475—44no—
114#462—24no—
115C8×Q8—44no—
116C4×Q16—44no—
117C2×#107C2312yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120C2×C4⋊C8C2312yes—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2314yes—
125C2×C4⋊C8C2314yes—
126C2×C4⋊C8C2312yes—
127C2×#103C2314yes—
128C2×#103C2312yes—
129C2×#103C2312yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×C4×C8C2312yes—
135C2×C4×C8C2314yes—
136#400—24no—
137#400—22no—
138#400—22no—
139#400—24no—
140#400—22no—
141#400—22no—
142#398—24no—
143#398—22no—
144#398—22no—
145#398—22no—
146#398—22no—
147#391—21no—
148#391—22no—
149#391—21no—
150#391—21no—
151#391—21no—
152#391—22no—
153#391—21no—
154#391—22no—
155#391—22no—
156#391C2312yes—
157#391C2312yes—
158#391C2312yes—
159#391C2314yes—
160#391C2314yes—
161#389—22no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389—21no—
169#389—21no—
170#389C2312yes—
171#389C2312yes—
172#389C2312yes—
173#389C2314yes—
174#389C2312yes—
175#389C2312yes—
176#387—24no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—24no—
182#387—24no—
183#387—22no—
184#387—22no—
185#384—21no—
186#384—21no—
187#384—21no—
188#384C2311yescharacteristic
189#384C2311yescharacteristic
190#384C2312yes—
191#384C2312yes—
192#382—22no—
193#382—22no—
194#382—22no—
195#382C2312yes—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D412yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2312yes—
218#376—22no—
219C43—21no—
220C43C2312yes—
221C43C2311yescharacteristic
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2314yes—
231#336C2314yes—
232#336C2312yes—
233C22×Q8—22no—
234C22×Q8—21no—
235C22×Q8C2×D411yescharacteristic
236C22×Q8C2412yes—
237C22×Q8C2411yescharacteristic
238C2×Q16—24no—
239C2×Q16—42no—
240C2×Q16—42no—
241C22×C8C2412yes—
242C22×C8C2412yes—
243C22×C8C2414yes—
244C4⋊Q8—24no—
245C4⋊Q8—22no—
246C4⋊Q8—22no—
247C4⋊Q8—22no—
248C4⋊Q8—24no—
249C4⋊Q8—22no—
250C4⋊Q8—44no—
251C4⋊Q8—44no—
252C4⋊Q8—82no—
253#125—24no—
254#125—42no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×Q8—22no—
262C4×Q8—24no—
263C4×Q8—22no—
264C4×Q8—24no—
265C4×Q8—41no—
266C4×Q8—44no—
267C4×Q8—84no—
268C4×Q8#4114yes—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D412yes—
296C2×C4⋊C4C2×D412yes—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2411yescharacteristic
306C2×C42—21no—
307C2×C42—22no—
308C2×C42—21no—
309C2×C42—21no—
310C2×C42—22no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—24no—
314C2×C42—21no—
315C2×C42C2×D412yes—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D412yes—
322C2×C42C2×D411yescharacteristic
323C2×C42C2411yescharacteristic
324C2×C42C2412yes—
325C2×C42C2412yes—
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328#107—42no—
329#106—28no—
330#106—44no—
331C4⋊C8—22no—
332C4⋊C8—24no—
333C4⋊C8—24no—
334C4⋊C8—22no—
335C4⋊C8—42no—
336C4⋊C8—44no—
337C4⋊C8—44no—
338#103—28no—
339#103—24no—
340#103—24no—
341#103—44no—
342#103—42no—
343#103—42no—
344#103—44no—
345C4×C8—24no—
346C4×C8—44no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—21no—
352#95—21no—
353#95—21no—
354#95—24no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—22no—
365#95—24no—
366#95—22no—
367#95—24no—
368#95—24no—
369#95—24no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—22no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—22no—
386C2×Q8—42no—
387C2×Q8—44no—
388C2×Q8—42no—
389C2×Q8—44no—
390C2×Q8—42no—
391C2×Q8—44no—
392C2×Q8#13714yes—
393C2×Q8C2×#4112yes—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D412yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C2511yescharacteristic
414Q16—48no—
415C2×C8—24no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—24no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—22no—
424C4⋊C4—24no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—42no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—81no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—84no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4C2×#4112yes—
467C4⋊C42−1+412yes—
468C42—24no—
469C42—24no—
470C42—24no—
471C42—22no—
472C42—24no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—21no—
477C42—22no—
478C42—21no—
479C42—22no—
480C42—41no—
481C42—42no—
482C42—42no—
483C42—42no—
484C42—44no—
485C42—44no—
486C42—42no—
487C42—42no—
488C42—81no—
489C42—84no—
490C42#13514yes—
491C42C2×#4112yes—
492C42H(5,𝔽2)12yes—
493C422−1+412yes—
494C422−1+412yes—
495C23C23×D411yescharacteristic
496Q8—28no—
497Q8—22no—
498Q8—22no—
499Q8—41no—
500Q8—41no—
501Q8—42no—
502Q8—42no—
503Q8—42no—
504Q8—42no—
505Q8—88no—
506Q8—84no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4#53712yes—
555C2×C4#53712yes—
556C2×C4#54812yes—
557C2×C4#54812yes—
558C2×C4C2×#13512yes—
559C2×C4C2×#13512yes—
560C2×C4C2×#13512yes—
561C2×C4C2×#13714yes—
562C2×C4C2×#13712yes—
563C2×C4#57712yes—
564C2×C4C22×#4111yescharacteristic
565C2×C4C2×H(5,𝔽2)11yescharacteristic
566C2×C4C2×2−1+411yescharacteristic
567C2×C4C2×2−1+411yescharacteristic
568C2×C4C2×2−1+411yescharacteristic
569C2×C4#58511yescharacteristic
570C8—44no—
571C8—42no—
572C8—42no—
573C22C2×#53711yescharacteristic
574C22C2×#54811yescharacteristic
575C22#348911yescharacteristic
576C22C22×#13511yescharacteristic
577C22C22×#13711yescharacteristic
578C22C2×#57711yescharacteristic
579C22#358611yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—24no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—41no—
590C4—42no—
591C4—81no—
592C4—82no—
593C4—82no—
594C4—84no—
595C4—82no—
596C4—82no—
597C4#324814yes—
598C4#336514yes—
599C4#350611yescharacteristic
600C4#350911yescharacteristic
601C4#355011yescharacteristic
602C4#355511yescharacteristic
603C2#3972711yescharacteristic
604C2C2×#324811yescharacteristic
605C2#6093111yescharacteristic
606C2C2×#336511yescharacteristic
607C2#6132211yescharacteristic
608C2#6210111yescharacteristic
609C2#6214711yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#365982
#349512
#331962
#324822
#318122
#298312
#293492
#292882
#292192
#282832
#281872
#260272
#254722
#246832
Q8×C4⋊C82
#221122
C4×#4622