Finite Group Databasebeta

← #360337

Subgroups — #360337

Order 512.

1318 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1825C211yesmaximal, characteristic
2C2×#1808C211yesmaximal, characteristic
3C2×#1735C211yesmaximal, characteristic
4#13939C211yesmaximal, characteristic
5#13801C211yesmaximal, characteristic
6#11316C212yesmaximal, complemented
7#11256C212yesmaximal, complemented
8#10907C212yesmaximal, complemented
9#10806C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#107C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C2×#380C2211yescharacteristic
18C2×#380C2211yescharacteristic
19C2×C4×C4⋊C4C2212yes—
20C2×C4×C22⋊C4C2211yescharacteristic
21#1825—24no—
22#1808—24no—
23#1735C418yescomplemented
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1440—22no—
36#1437—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40#1434—21no—
41#1434—21no—
42#1434—21no—
43C4×#95—21no—
44C24×C4C2311yescharacteristic
45C23×C8—22no—
46C23×C8—22no—
47C23×C8—24no—
48C23×C8—42no—
49C23×C8C2×C414yes—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—24no—
60C2×#117—42no—
61C2×#117C2×C414yes—
62C2×#117C2×C414yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4D412yes—
74C22×C42—21no—
75C22×C42—21no—
76C22×C42—22no—
77C22×C42C2312yes—
78C2×#107—24no—
79C2×#107—24no—
80C2×#107—42no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84#380—22no—
85#380—22no—
86#380—22no—
87#380D414yes—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—42no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95D411yescharacteristic
102C2×#95D411yescharacteristic
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340C2×C414yes—
109#340C2×C414yes—
110#340C2×C414yes—
111#340C2×C414yes—
112#336—24no—
113#336—42no—
114C25C2×D411yescharacteristic
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4C22×C412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4#4112yes—
131C23×C4#4112yes—
132C23×C4#4112yes—
133C23×C4#4112yes—
134C23×C4C2411yescharacteristic
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—28no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—28no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—44no—
152C22×C8—48no—
153C22×C8—48no—
154C22×C8C22×C412yes—
155C22×C8C22×C412yes—
156C22×C8#4114yes—
157#117—28no—
158#117—28no—
159#117—28no—
160#117—44no—
161#117—48no—
162#117—44no—
163#117—44no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4#4114yes—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—81no—
207C2×C22⋊C4—81no—
208C2×C22⋊C4—81no—
209C2×C22⋊C4—81no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—44no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42—44no—
222C2×C42—44no—
223C2×C42—82no—
224#107—48no—
225C22⋊C8—44no—
226#95—24no—
227#95—22no—
228#95—22no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—42no—
233#95—44no—
234#95—82no—
235#95—82no—
236#95—82no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24C4×D412yes—
246C24#12112yes—
247C24#12312yes—
248C24C22×D411yescharacteristic
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
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—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4#11714yes—
315C22×C4C4×D414yes—
316C22×C4C4×D412yes—
317C22×C4#12112yes—
318C22×C4#12312yes—
319C22×C4#12414yes—
320C22×C4#12614yes—
321C22×C4#13514yes—
322C22×C4C23×C411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4112yes—
327C22×C4C2×#4112yes—
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4H(5,𝔽2)12yes—
331C22×C42−1+412yes—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—416no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8#11714yes—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4#11714yes—
372C22⋊C4—48no—
373C22⋊C4—48no—
374C22⋊C4—48no—
375C22⋊C4—84no—
376C22⋊C4—84no—
377C42—48no—
378C42—42no—
379C42—42no—
380C42—42no—
381C42—42no—
382C42—48no—
383C42—88no—
384C23—24no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—24no—
390C23—24no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—24no—
396C23—22no—
397C23—22no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23#38814yes—
407C23C2×C4×D411yescharacteristic
408C23C2×#12111yescharacteristic
409C23C2×#12311yescharacteristic
410C23#54612yes—
411C23#54712yes—
412C23C2×#13512yes—
413C23#57512yes—
414C23#57611yescharacteristic
415C23#57611yescharacteristic
416C23#57811yescharacteristic
417C23#57811yescharacteristic
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—28no—
426C2×C4—28no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—82no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4#38814yes—
479C2×C4C2×#11711yescharacteristic
480C2×C4C2×#11711yescharacteristic
481C2×C4C2×#11711yescharacteristic
482C2×C4C2×C4×D411yescharacteristic
483C2×C4#51811yescharacteristic
484C2×C4#51911yescharacteristic
485C2×C4C2×#12411yescharacteristic
486C2×C4C2×#12611yescharacteristic
487C2×C4#52912yes—
488C2×C4#52912yes—
489C2×C4#53912yes—
490C2×C4#53912yes—
491C2×C4#54612yes—
492C2×C4#54712yes—
493C2×C4#55112yes—
494C2×C4#55112yes—
495C2×C4#55212yes—
496C2×C4#55212yes—
497C8—24no—
498C8—42no—
499C8—82no—
500C22—24no—
501C22—24no—
502C22—24no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—44no—
511C22C2×#38811yescharacteristic
512C22#234211yescharacteristic
513C22#234411yescharacteristic
514C22#241912yes—
515C22#242512yes—
516C22C4×#13511yescharacteristic
517C22#294311yescharacteristic
518C22#295111yescharacteristic
519C22#295111yescharacteristic
520C22#303512yes—
521C22#304111yescharacteristic
522C22#304411yescharacteristic
523C22#309112yes—
524C22#309811yescharacteristic
525C22#309811yescharacteristic
526C22#329712yes—
527C22#330212yes—
528C22#330912yes—
529C22#331312yes—
530C22#333612yes—
531C22#334112yes—
532C22#334712yes—
533C22#334912yes—
534C4—24no—
535C4—44no—
536C4—44no—
537C4—82no—
538C4—82no—
539C4—82no—
540C4—82no—
541C4—84no—
542C4—162no—
543C4#228311yescharacteristic
544C4#231311yescharacteristic
545C4#234211yescharacteristic
546C4#234411yescharacteristic
547C4#248812yes—
548C4#249312yes—
549C2—44no—
550C2#1387411yescharacteristic
551C2#2105811yescharacteristic
552C2#2106711yescharacteristic
553C2#2256911yescharacteristic
554C2#2257611yescharacteristic
555C2#2261311yescharacteristic
556C2#2261611yescharacteristic
557C2#2494012yes—
558C2#2494812yes—
559C2#2520712yes—
560C2#2521512yes—
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
6. #11316C2semidirect
7. #11256C2semidirect
8. #10907C2semidirect
9. #10806C2semidirect
23. #1735C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18252
C2×#18082
C2×#17352
#139392
#138012
#113162
#112562
#109072
#108062
#107312
#107142