Finite Group Databasebeta

← #7598440

Subgroups — #7598440

Order 512.

1516 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59828C211yesmaximal, characteristic
2#59828C211yesmaximal, characteristic
3#31964C214yesmaximal
4#29013C214yesmaximal
5#28283C211yesmaximal, characteristic
6#28283C211yesmaximal, characteristic
7#27749C212yesmaximal
8#26108C212yesmaximal
9#25647C212yesmaximal
10#25472C212yesmaximal
11#25276C212yesmaximal
12#24994C212yesmaximal
13#22964C212yesmaximal
14#22542C212yesmaximal
15C8×C4⋊Q8C211yesmaximal, characteristic
16C4×#487C211yesmaximal, characteristic
17C4×#487C211yesmaximal, characteristic
18C2×#557—21no—
19C2×#557C2212yes—
20C2×#487C2211yescharacteristic
21C2×#487C2211yescharacteristic
22C2×#475C2212yes—
23C2×#475C2212yes—
24C2×C8×Q8C2212yes—
25#2726—22no—
26#2726—22no—
27#2582—21no—
28#2582C2212yes—
29#2552—21no—
30#2552C2212yes—
31#2546C2212yes—
32#2546C2212yes—
33#2506—22no—
34#2506C2214yes—
35#2453—22no—
36#2453—22no—
37#2446—22no—
38#2446C2214yes—
39#2365—21no—
40#2365C2212yes—
41Q8×C4⋊C4C2212yes—
42Q8×C4⋊C4C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48#2099C2214yes—
49#2088C2214yes—
50#2072C2214yes—
51#2066C2214yes—
52#2066C2214yes—
53#2066C2214yes—
54#2043C2214yes—
55#2043C2214yes—
56#1990C2211yescharacteristic
57#1990C2211yescharacteristic
58#1984C2212yes—
59#1981C2212yes—
60#1981C2214yes—
61#1981C2212yes—
62#1956C2211yescharacteristic
63#1956C2214yes—
64#1956C2211yescharacteristic
65#1948C2212yes—
66#1948C2212yes—
67#1941C2214yes—
68#1939C2212yes—
69#1939C2212yes—
70#1933C2212yes—
71#1933C2212yes—
72#1933C2214yes—
73#1933C2212yes—
74#1933C2212yes—
75#1927C2214yes—
76#1920C2214yes—
77#1865C2214yes—
78#1865C2214yes—
79#1842C2211yescharacteristic
80#1834C2211yescharacteristic
81#1834C2211yescharacteristic
82#1803C2212yes—
83#1803C2214yes—
84#1803C2212yes—
85C8×C4⋊C4C2212yes—
86C8×C4⋊C4C2212yes—
87C4×#103C2214yes—
88C4×#103C2212yes—
89C4×#103C2212yes—
90C4×#103C2212yes—
91C4×#103C2212yes—
92C42×C8C2211yescharacteristic
93#557—28no—
94#557—44no—
95#557—48no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8D411yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2314yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2311yescharacteristic
103C2×#125—22no—
104C2×#125—24no—
105C2×#125—22no—
106C2×#125—24no—
107C2×C4×Q8—22no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8—21no—
111C2×C4×Q8—21no—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2312yes—
114C2×C4×Q8C2314yes—
115C2×C4×Q8C2312yes—
116#487—24no—
117#487—24no—
118#475—28no—
119#475—44no—
120C8×Q8—44no—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2314yes—
126C2×C4⋊C8C2312yes—
127C2×#103C2314yes—
128C2×#103C2312yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×#103C2312yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2312yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×C4×C8C2311yescharacteristic
140C2×C4×C8C2311yescharacteristic
141C2×C4×C8C2311yescharacteristic
142C2×C4×C8C2311yescharacteristic
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145C2×C4×C8C2314yes—
146#400—24no—
147#400—24no—
148#398—24no—
149#398—24no—
150#398—24no—
151#398—24no—
152#398—24no—
153#398—24no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391—21no—
158#391—21no—
159#391D412yes—
160#391C2312yes—
161#391C2312yes—
162#391C2314yes—
163#389—22no—
164#389—22no—
165#389—22no—
166#389—24no—
167#389—22no—
168#389—22no—
169#389—21no—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—24no—
176#389D412yes—
177#389D412yes—
178#389C2312yes—
179#389C2312yes—
180#389C2314yes—
181#389C2314yes—
182#389C2314yes—
183#387—24no—
184#387—24no—
185#387—24no—
186#387—24no—
187#384—22no—
188#384D411yescharacteristic
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2311yescharacteristic
193#384C2314yes—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4C2312yes—
202C4×C4⋊C4C2312yes—
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C43C2311yescharacteristic
207#340C2314yes—
208#340C2312yes—
209#340C2314yes—
210#340C2312yes—
211#340C2312yes—
212#340C2312yes—
213#340C2312yes—
214#340C2312yes—
215#340C2312yes—
216#340C2312yes—
217#336C2314yes—
218#336C2314yes—
219C22×Q8—21no—
220C22×Q8—21no—
221C22×Q8C2×D412yes—
222C22×Q8C2412yes—
223C22×Q8C2412yes—
224C22×C8C2411yescharacteristic
225C22×C8C2411yescharacteristic
226C22×C8C2414yes—
227C22×C8C2411yescharacteristic
228C22×C8C2411yescharacteristic
229C4⋊Q8—28no—
230C4⋊Q8—22no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233C4⋊Q8—22no—
234C4⋊Q8—22no—
235C4⋊Q8—22no—
236C4⋊Q8—44no—
237C4⋊Q8—48no—
238C4⋊Q8SD1618yes—
239C4⋊Q8#4114yes—
240C4⋊Q8#4114yes—
241#125—416no—
242#125—48no—
243#125—416no—
244#125—48no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×Q8—28no—
248C4×Q8—24no—
249C4×Q8—24no—
250C4×Q8—24no—
251C4×Q8—24no—
252C4×Q8—44no—
253C4×Q8—48no—
254C4×Q8—48no—
255C4×Q8—48no—
256C4×Q8—44no—
257C4×Q8—44no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4C2×D412yes—
285C2×C4⋊C4C2×D414yes—
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2414yes—
307C2×C42C2411yescharacteristic
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310#106—28no—
311#106—44no—
312C4⋊C8—24no—
313C4⋊C8—24no—
314C4⋊C8—42no—
315C4⋊C8—44no—
316#103—24no—
317#103—24no—
318#103—24no—
319#103—24no—
320#103—28no—
321#103—24no—
322#103—24no—
323#103—42no—
324#103—42no—
325#103—44no—
326#103—44no—
327#103—44no—
328C4×C8—22no—
329C4×C8—22no—
330C4×C8—22no—
331C4×C8—22no—
332C4×C8—22no—
333C4×C8—41no—
334C4×C8—44no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—24no—
342#95—22no—
343#95—24no—
344#95—22no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95C2×D414yes—
350#95C2×D414yes—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—22no—
364C2×Q8—22no—
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—42no—
368C2×Q8—42no—
369C2×Q8—42no—
370C2×Q8C2×SD1618yes—
371C2×Q8C2×#4114yes—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D414yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D414yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C2511yescharacteristic
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—24no—
396C2×C8—21no—
397C2×C8—24no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—21no—
402C2×C8—24no—
403C2×C82−1+412yes—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—28no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—42no—
422C4⋊C4—48no—
423C4⋊C4—48no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C4⋊C4—42no—
427C4⋊C4—44no—
428C4⋊C4—42no—
429C4⋊C4—42no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—48no—
433C4⋊C4—44no—
434C4⋊C4—42no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—81no—
440C4⋊C4—81no—
441C4⋊C4—81no—
442C4⋊C4—81no—
443C4⋊C4—84no—
444C4⋊C4—84no—
445C4⋊C4C2×SD1618yes—
446C4⋊C4C2×SD1618yes—
447C4⋊C4C2×#4114yes—
448C42—22no—
449C42—22no—
450C42—24no—
451C42—24no—
452C42—21no—
453C42—21no—
454C42—41no—
455C42—48no—
456C42—44no—
457C42—44no—
458C42—48no—
459C42—44no—
460C42—44no—
461C42C2×SD1614yes—
462C42C2×#4112yes—
463C42C2×#4112yes—
464C42H(5,𝔽2)12yes—
465C422−1+412yes—
466C23C23×D411yescharacteristic
467Q8—216no—
468Q8—24no—
469Q8—24no—
470Q8—48no—
471Q8—42no—
472Q8—42no—
473Q8—42no—
474Q8—42no—
475Q8—42no—
476Q8—42no—
477Q8—84no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—24no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—24no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4#54612yes—
517C2×C4#54612yes—
518C2×C4#54812yes—
519C2×C4#54812yes—
520C2×C4C22×SD1618yes—
521C2×C4C22×SD1612yes—
522C2×C4C22×SD1612yes—
523C2×C4C22×SD1612yes—
524C2×C4#57812yes—
525C2×C4#57812yes—
526C2×C4C22×#4111yescharacteristic
527C2×C4C22×#4111yescharacteristic
528C2×C4C2×H(5,𝔽2)11yescharacteristic
529C2×C4C2×2−1+411yescharacteristic
530C2×C4C2×2−1+411yescharacteristic
531C2×C4#58511yescharacteristic
532C8—22no—
533C8—41no—
534C8—41no—
535C8—41no—
536C8—44no—
537C22C2×#54611yescharacteristic
538C22C2×#54811yescharacteristic
539C22#349711yescharacteristic
540C22C23×SD1611yescharacteristic
541C22C2×#57811yescharacteristic
542C22C2×#57811yescharacteristic
543C22#358411yescharacteristic
544C4—22no—
545C4—22no—
546C4—22no—
547C4—28no—
548C4—44no—
549C4—44no—
550C4—44no—
551C4—41no—
552C4—82no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4—82no—
557C4—82no—
558C4#329814yes—
559C4#335914yes—
560C4#348211yescharacteristic
561C4#351511yescharacteristic
562C4#351711yescharacteristic
563C4#355211yescharacteristic
564C2#5941811yescharacteristic
565C2C2×#329811yescharacteristic
566C2#6121411yescharacteristic
567C2C2×#335911yescharacteristic
568C2#6132811yescharacteristic
569C2#6235711yescharacteristic
570C2#6237411yescharacteristic
subgroupG/H
center, socle, prime radicalC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊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
#598282
#319642
#290132
#282832
#277492
#261082
#256472
#254722
#252762
#249942
#229642
#225422
C8×C4⋊Q82
C4×#4872