Finite Group Databasebeta

← #6801188

Subgroups — #6801188

Order 512.

1421 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36560C211yesmaximal, characteristic
2#34951C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic
4#32476C214yesmaximal
5#31964C214yesmaximal
6#29996C212yesmaximal
7#29123C212yesmaximal
8#28717C211yesmaximal, characteristic
9#28639C212yesmaximal
10#28283C211yesmaximal, characteristic
11#28283C212yesmaximal
12#26510C212yesmaximal
13#25472C212yesmaximal
14#25276C212yesmaximal
15Q8×C4⋊C8C211yesmaximal, characteristic
16#22112C212yesmaximal
17C4×#475C211yesmaximal, characteristic
18C2×#557—21no—
19C2×#487C2212yes—
20C2×#475C2212yes—
21C2×#475C2211yescharacteristic
22C2×#475C2212yes—
23C2×C8×Q8C2212yes—
24#2868C2211yescharacteristic
25#2859C2212yes—
26#2585—22no—
27#2582C2211yescharacteristic
28#2568—21no—
29#2568C2211yescharacteristic
30#2565—22no—
31#2552C2211yescharacteristic
32#2546—21no—
33#2546C2212yes—
34#2546C2211yescharacteristic
35#2511C2212yes—
36#2509—21no—
37#2506—22no—
38#2506C2214yes—
39#2453—21no—
40#2446C2212yes—
41#2402—22no—
42#2365—21no—
43#2365C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×#125—21no—
49#2295—21no—
50#2295C2212yes—
51C42×Q8C2211yescharacteristic
52C42×Q8C2211yescharacteristic
53#2100C2214yes—
54#2088C2214yes—
55#2088C2214yes—
56#2072C2214yes—
57#2066C2214yes—
58#2066C2214yes—
59#2066C2214yes—
60#2066C2214yes—
61#2066C2214yes—
62#2043C2214yes—
63#1990C2212yes—
64#1988C2211yescharacteristic
65#1988C2212yes—
66#1981C2214yes—
67#1981C2212yes—
68#1948C2211yescharacteristic
69#1948C2212yes—
70#1948C2212yes—
71#1948C2214yes—
72#1948C2212yes—
73#1941C2214yes—
74#1939C2212yes—
75#1939C2214yes—
76#1933C2214yes—
77#1933C2214yes—
78#1933C2214yes—
79#1927C2214yes—
80#1920C2214yes—
81#1919C2212yes—
82#1865C2212yes—
83#1842C2212yes—
84#1842C2211yescharacteristic
85#1803C2212yes—
86#1803C2212yes—
87#1803C2212yes—
88#1803C2211yescharacteristic
89#1803C2212yes—
90#1803C2212yes—
91C4×#106C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×C4⋊C8C2212yes—
94C4×#103C2212yes—
95#557—48no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2312yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125—21no—
104C2×#125—24no—
105C2×#125—22no—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—21no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2311yescharacteristic
116#487—44no—
117#475—28no—
118#475—24no—
119#475—44no—
120C8×Q8—44no—
121C2×#106C2312yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×#106C2312yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2311yescharacteristic
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2311yescharacteristic
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2314yes—
131C2×C4⋊C8C2314yes—
132C2×#103C2312yes—
133C2×#103C2314yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×C4×C8C2312yes—
140C2×C4×C8C2314yes—
141#400—22no—
142#400—22no—
143#400—24no—
144#400—22no—
145#400—22no—
146#398—24no—
147#398—24no—
148#398—22no—
149#398—22no—
150#398—22no—
151#398—22no—
152#391—21no—
153#391—21no—
154#391—22no—
155#391—22no—
156#391—21no—
157#391—22no—
158#391C2312yes—
159#391C2312yes—
160#391C2312yes—
161#391C2312yes—
162#391C2312yes—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389—22no—
169#389—22no—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—22no—
174#389—22no—
175#389C2312yes—
176#389C2314yes—
177#389C2314yes—
178#389C2314yes—
179#389C2312yes—
180#389C2312yes—
181#387—22no—
182#387—22no—
183#387—22no—
184#387—22no—
185#387—22no—
186#387—22no—
187#387—24no—
188#387—24no—
189#384—21no—
190#384—21no—
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2311yescharacteristic
194#384C2312yes—
195#384C2312yes—
196#382—21no—
197#382—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D412yes—
210C4×C4⋊C4D412yes—
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219#376—22no—
220C43—21no—
221C43C2312yes—
222C43C2311yescharacteristic
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#340C2312yes—
231#340C2312yes—
232#336C2312yes—
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2412yes—
238C22×Q8C2411yescharacteristic
239C22×C8C2412yes—
240C22×C8C2412yes—
241C22×C8C2414yes—
242C4⋊Q8—24no—
243C4⋊Q8—24no—
244C4⋊Q8—22no—
245C4⋊Q8—22no—
246C4⋊Q8—24no—
247C4⋊Q8—22no—
248C4⋊Q8—42no—
249C4⋊Q8—44no—
250C4⋊Q8—48no—
251C4⋊Q8—44no—
252C4⋊Q8#4114yes—
253#125—44no—
254#125—416no—
255#125—48no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—24no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8—24no—
265C4×Q8—42no—
266C4×Q8—41no—
267C4×Q8—48no—
268C4×Q8—48no—
269C4×Q8—44no—
270C4×Q8SD1618yes—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D412yes—
300C2×C4⋊C4C2×D412yes—
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D412yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2411yescharacteristic
307C2×C4⋊C4C2414yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—22no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D412yes—
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D412yes—
323C2×C42C2×D411yescharacteristic
324C2×C42C2×D412yes—
325C2×C42C2×D412yes—
326C2×C42C2×D411yescharacteristic
327C2×C42C2411yescharacteristic
328C2×C42C2412yes—
329C2×C42C2412yes—
330C2×C42C2411yescharacteristic
331C2×C42C2411yescharacteristic
332#106—24no—
333#106—28no—
334#106—42no—
335#106—44no—
336C4⋊C8—24no—
337C4⋊C8—22no—
338C4⋊C8—24no—
339C4⋊C8—22no—
340C4⋊C8—42no—
341C4⋊C8—44no—
342C4⋊C8—44no—
343#103—24no—
344#103—24no—
345#103—28no—
346#103—44no—
347#103—44no—
348#103—44no—
349#103—44no—
350C4×C8—24no—
351C4×C8—44no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—24no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376C2×Q8—22no—
377C2×Q8—24no—
378C2×Q8—22no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—21no—
383C2×Q8—22no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—44no—
391C2×Q8—42no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8C2×SD1614yes—
396C2×Q82−1+412yes—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C2511yescharacteristic
417C2×C8—22no—
418C2×C8—24no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—24no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—22no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—82no—
467C4⋊C4—81no—
468C4⋊C4—82no—
469C4⋊C4C2×SD1614yes—
470C4⋊C4C2×SD1618yes—
471C4⋊C4C2×#4114yes—
472C42—22no—
473C42—22no—
474C42—24no—
475C42—22no—
476C42—22no—
477C42—21no—
478C42—22no—
479C42—21no—
480C42—22no—
481C42—41no—
482C42—42no—
483C42—42no—
484C42—42no—
485C42—42no—
486C42—42no—
487C42—44no—
488C42—48no—
489C42—44no—
490C42—42no—
491C42—42no—
492C42—81no—
493C42C2×SD1618yes—
494C42C2×SD1614yes—
495C42C2×#4112yes—
496C42H(5,𝔽2)12yes—
497C422−1+412yes—
498C422−1+412yes—
499C23C23×D411yescharacteristic
500Q8—28no—
501Q8—22no—
502Q8—22no—
503Q8—48no—
504Q8—41no—
505Q8—41no—
506Q8—42no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510Q8—88no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4#53712yes—
558C2×C4#53712yes—
559C2×C4#54812yes—
560C2×C4#54812yes—
561C2×C4C22×SD1614yes—
562C2×C4C22×SD1612yes—
563C2×C4C22×SD1614yes—
564C2×C4C22×SD1612yes—
565C2×C4C22×SD1612yes—
566C2×C4#57512yes—
567C2×C4#57812yes—
568C2×C4C22×#4111yescharacteristic
569C2×C4C2×H(5,𝔽2)11yescharacteristic
570C2×C4C2×2−1+411yescharacteristic
571C2×C4C2×2−1+411yescharacteristic
572C2×C4C2×2−1+411yescharacteristic
573C2×C4#58511yescharacteristic
574C8—42no—
575C8—44no—
576C8—42no—
577C22C2×#53711yescharacteristic
578C22C2×#54811yescharacteristic
579C22#348911yescharacteristic
580C22C23×SD1611yescharacteristic
581C22C2×#57511yescharacteristic
582C22C2×#57811yescharacteristic
583C22#358611yescharacteristic
584C4—22no—
585C4—22no—
586C4—24no—
587C4—24no—
588C4—22no—
589C4—44no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—81no—
595C4—82no—
596C4—82no—
597C4—81no—
598C4—82no—
599C4—84no—
600C4—81no—
601C4#324614yes—
602C4#335914yes—
603C4#351711yescharacteristic
604C4#352311yescharacteristic
605C4#355211yescharacteristic
606C4#355511yescharacteristic
607C2#3912711yescharacteristic
608C2C2×#324611yescharacteristic
609C2#6092511yescharacteristic
610C2C2×#335911yescharacteristic
611C2#6132011yescharacteristic
612C2#6209911yescharacteristic
613C2#6214611yescharacteristic
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
#365602
#349512
#331962
#324762
#319642
#299962
#291232
#287172
#286392
#282832
#265102
#254722
#252762
Q8×C4⋊C82
#221122
C4×#4752