Finite Group Databasebeta

← #360895

Subgroups — #360895

Order 512.

1304 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1735C211yesmaximal, characteristic
2#13951C211yesmaximal, characteristic
3#13801C211yesmaximal, characteristic
4#11315C212yesmaximal, complemented
5#11293C212yesmaximal, complemented
6#10885C212yesmaximal, complemented
7#10806C212yesmaximal, complemented
8#10731C211yesmaximal, characteristic
9#10731C211yesmaximal, characteristic
10#10721C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C2×#389C2212yes—
17C2×#380C2211yescharacteristic
18C2×#380C2211yescharacteristic
19C2×C4×C4⋊C4C2212yes—
20C2×C4×C22⋊C4C2211yescharacteristic
21#1735C418yescomplemented
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#336C2212yes—
33C2×#336C2212yes—
34#1439—22no—
35#1437—22no—
36C4⋊#95—21no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39#1434—21no—
40#1434—21no—
41C4×#95—21no—
42C24×C4C2311yescharacteristic
43C23×C8—22no—
44C23×C8—22no—
45C23×C8—42no—
46C23×C8—42no—
47C23×C8C2×C414yes—
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—24no—
54C2×#117—24no—
55C2×#117—42no—
56C2×#117—42no—
57C2×#117C2×C414yes—
58C2×#117C2×C414yes—
59C22×C4⋊C4—22no—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4D412yes—
71C22×C42—21no—
72C22×C42—21no—
73C22×C42—22no—
74C22×C42C2312yes—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78#389—24no—
79#389—42no—
80#380—22no—
81#380—22no—
82#380—22no—
83#380Q814yes—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—42no—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—22no—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95D411yescharacteristic
97C2×#95D411yescharacteristic
98#340—24no—
99#340—24no—
100#340—42no—
101#340—42no—
102#340—42no—
103#340—42no—
104#340C2×C414yes—
105#340C2×C414yes—
106#340C2×C414yes—
107#340C2×C414yes—
108#336—42no—
109#336—42no—
110C25C2×D411yescharacteristic
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4C22×C412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D411yescharacteristic
125C23×C4C2×Q812yes—
126C23×C4#4112yes—
127C23×C4#4112yes—
128C23×C4#4112yes—
129C23×C4#4112yes—
130C23×C4C2411yescharacteristic
131C22×C8—28no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—44no—
148C22×C8—44no—
149C22×C8—48no—
150C22×C8—48no—
151C22×C8C22×C412yes—
152C22×C8C22×C412yes—
153#117—28no—
154#117—28no—
155#117—44no—
156#117—44no—
157#117—48no—
158#117—48no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C2×Q814yes—
188C2×C4⋊C4#4114yes—
189C2×C4⋊C4#4114yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—81no—
200C2×C22⋊C4—81no—
201C2×C22⋊C4—81no—
202C2×C22⋊C4—81no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—44no—
212C2×C42—44no—
213C2×C42—44no—
214C2×C42—44no—
215C2×C42—82no—
216C22⋊C8—44no—
217#95—24no—
218#95—22no—
219#95—22no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—42no—
224#95—44no—
225#95—82no—
226#95—82no—
227C24—22no—
228C24—22no—
229C24—22no—
230C24—22no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24C4×D412yes—
236C24C22⋊Q812yes—
237C24#12312yes—
238C24C22×D411yescharacteristic
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4#11714yes—
303C22×C4C4×D412yes—
304C22×C4C4×Q814yes—
305C22×C4C22⋊Q812yes—
306C22×C4#12312yes—
307C22×C4#12514yes—
308C22×C4#12614yes—
309C22×C4#13514yes—
310C22×C4C23×C411yescharacteristic
311C22×C4C22×Q811yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4112yes—
315C22×C4C2×#4112yes—
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4H(5,𝔽2)12yes—
319C22×C4H(5,𝔽2)12yes—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—416no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—88no—
357C4⋊C4#11714yes—
358C4⋊C4#11714yes—
359C22⋊C4—48no—
360C22⋊C4—48no—
361C22⋊C4—48no—
362C22⋊C4—84no—
363C22⋊C4—84no—
364C42—48no—
365C42—48no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—88no—
371C23—24no—
372C23—22no—
373C23—22no—
374C23—24no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—24no—
384C23—24no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23#38714yes—
394C23C2×C4×D411yescharacteristic
395C23C2×C22⋊Q811yescharacteristic
396C23C2×#12311yescharacteristic
397C23#54612yes—
398C23#54612yes—
399C23C2×#13512yes—
400C23#57512yes—
401C23#57611yescharacteristic
402C23#57711yescharacteristic
403C23#57711yescharacteristic
404C23#57811yescharacteristic
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4#38714yes—
464C2×C4C2×#11711yescharacteristic
465C2×C4C2×#11711yescharacteristic
466C2×C4C2×#11711yescharacteristic
467C2×C4C2×C4×Q811yescharacteristic
468C2×C4#51811yescharacteristic
469C2×C4#51811yescharacteristic
470C2×C4C2×#12511yescharacteristic
471C2×C4C2×#12611yescharacteristic
472C2×C4#52912yes—
473C2×C4#52912yes—
474C2×C4#53812yes—
475C2×C4#53812yes—
476C2×C4#55112yes—
477C2×C4#55112yes—
478C2×C4#55312yes—
479C2×C4#55312yes—
480C2×C4#55412yes—
481C2×C4#55412yes—
482C8—42no—
483C8—42no—
484C8—82no—
485C22—24no—
486C22—24no—
487C22—24no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—44no—
493C22—44no—
494C22—44no—
495C22—44no—
496C22C2×#38711yescharacteristic
497C22#234211yescharacteristic
498C22#234211yescharacteristic
499C22#244412yes—
500C22#245012yes—
501C22C4×#13511yescharacteristic
502C22#294311yescharacteristic
503C22#295111yescharacteristic
504C22#295211yescharacteristic
505C22#307612yes—
506C22#308111yescharacteristic
507C22#308211yescharacteristic
508C22#309112yes—
509C22#309611yescharacteristic
510C22#309711yescharacteristic
511C22#329612yes—
512C22#330012yes—
513C22#330312yes—
514C22#330512yes—
515C22#330712yes—
516C22#330912yes—
517C22#331112yes—
518C22#331312yes—
519C4—24no—
520C4—44no—
521C4—44no—
522C4—44no—
523C4—82no—
524C4—82no—
525C4—84no—
526C4—82no—
527C4—162no—
528C4#228311yescharacteristic
529C4#231211yescharacteristic
530C4#234511yescharacteristic
531C4#234511yescharacteristic
532C4#250512yes—
533C4#251012yes—
534C2—44no—
535C2#1388311yescharacteristic
536C2#2103211yescharacteristic
537C2#2103911yescharacteristic
538C2#2256811yescharacteristic
539C2#2256911yescharacteristic
540C2#2257211yescharacteristic
541C2#2257311yescharacteristic
542C2#2620512yes—
543C2#2621312yes—
544C2#2641512yes—
545C2#2642312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #11315C2semidirect
5. #11293C2semidirect
6. #10885C2semidirect
7. #10806C2semidirect
21. #1735C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17352
#139512
#138012
#113152
#112932
#108852
#108062
#107312
#107212
#107142