Finite Group Databasebeta

← #360838

Subgroups — #360838

Order 512.

1484 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#427C211yesmaximal, characteristic
2C2×#1921C212yesmaximal, complemented
3#13951C211yesmaximal, characteristic
4#13801C211yesmaximal, characteristic
5#11379C212yesmaximal, complemented
6#11220C212yesmaximal, complemented
7#10908C212yesmaximal, complemented
8#10731C211yesmaximal, characteristic
9#10731C211yesmaximal, characteristic
10#10721C211yesmaximal, characteristic
11#10714C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C2×#427—28no—
16C2×#427C418yescomplemented
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#106C2211yescharacteristic
20C22×#106C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#389C2212yes—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1921—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1439—22no—
37#1437—22no—
38C4⋊#95—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41#1434—21no—
42#1434—21no—
43C4×#95—21no—
44C24×C4C2311yescharacteristic
45C23×C8—22no—
46C23×C8—22no—
47C23×C8—24no—
48C23×C8—24no—
49C23×C8C2×C414yes—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—42no—
58C2×#117—42no—
59C2×#117C2×C414yes—
60C2×#117C2×C414yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4D412yes—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42—22no—
76C22×C42C2312yes—
77#427—232no—
78C2×#107—24no—
79C2×#107C2×C414yes—
80C2×#107C2×C414yes—
81C2×#107D418yes—
82C2×#106—24no—
83C2×#106—24no—
84C2×#106C2×C414yes—
85C2×#106C2×C414yes—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89#389—24no—
90#389—42no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380Q814yes—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—42no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95D411yescharacteristic
108C2×#95D411yescharacteristic
109#340—24no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—42no—
114#340—42no—
115#336—24no—
116#336—24no—
117C25C2×D411yescharacteristic
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4C22×C412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×Q812yes—
133C23×C4#4112yes—
134C23×C4#4112yes—
135C23×C4#4112yes—
136C23×C4#4112yes—
137C23×C4C2411yescharacteristic
138C22×C8—216no—
139C22×C8—216no—
140C22×C8—28no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—28no—
148C22×C8—28no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8C22×C412yes—
157C22×C8C22×C412yes—
158C22×C8C2×D414yes—
159C22×C8#4114yes—
160#117—28no—
161#117—28no—
162#117—44no—
163#117—44no—
164#117—48no—
165#117—48no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4C2×D414yes—
196C2×C4⋊C4C2×Q814yes—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
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—44no—
220C2×C42—44no—
221C2×C42—44no—
222C2×C42—82no—
223#107—28no—
224#107C22⋊C4116yescomplemented
225#106—28no—
226#106—28no—
227C22⋊C8—44no—
228#95—24no—
229#95—22no—
230#95—22no—
231#95—44no—
232#95—44no—
233#95—42no—
234#95—44no—
235#95—44no—
236#95—82no—
237#95—82no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24C4×D412yes—
247C24C22⋊Q812yes—
248C24#12312yes—
249C24C22×D411yescharacteristic
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
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—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4#11714yes—
314C22×C4C4×D412yes—
315C22×C4C4×Q814yes—
316C22×C4C22⋊Q812yes—
317C22×C4#12312yes—
318C22×C4#12514yes—
319C22×C4#12614yes—
320C22×C4#13514yes—
321C22×C4C23×C411yescharacteristic
322C22×C4C22×D412yes—
323C22×C4C22×Q811yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4112yes—
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4H(5,𝔽2)12yes—
330C22×C42−1+412yes—
331C2×C8—232no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8C2×C22⋊C414yes—
348C2×C8#11714yes—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—88no—
368C4⋊C4C2×C22⋊C414yes—
369C4⋊C4C2×C22⋊C414yes—
370C22⋊C4—48no—
371C22⋊C4—48no—
372C22⋊C4—48no—
373C22⋊C4—84no—
374C22⋊C4—84no—
375C42—48no—
376C42—48no—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—42no—
381C42—88no—
382C23—24no—
383C23—22no—
384C23—22no—
385C23—24no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—24no—
395C23—24no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23#38714yes—
405C23C2×C4×D411yescharacteristic
406C23C2×C22⋊Q811yescharacteristic
407C23C2×#12311yescharacteristic
408C23D4212yes—
409C23#54712yes—
410C23C2×#13512yes—
411C23#57512yes—
412C23#57611yescharacteristic
413C23#57711yescharacteristic
414C23#57711yescharacteristic
415C23#57811yescharacteristic
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—28no—
424C2×C4—28no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4#38714yes—
475C2×C4C22×C22⋊C411yescharacteristic
476C2×C4C2×#11711yescharacteristic
477C2×C4C2×#11711yescharacteristic
478C2×C4C2×C4×Q811yescharacteristic
479C2×C4#51811yescharacteristic
480C2×C4#51911yescharacteristic
481C2×C4#52512yes—
482C2×C4#52512yes—
483C2×C4C2×#12511yescharacteristic
484C2×C4C2×#12611yescharacteristic
485C2×C4#53912yes—
486C2×C4#53912yes—
487C2×C4#54612yes—
488C2×C4#54812yes—
489C2×C4D4×Q812yes—
490C2×C4#55212yes—
491C2×C4#55312yes—
492C2×C4#55412yes—
493C8—24no—
494C8—24no—
495C8—82no—
496C22—24no—
497C22—24no—
498C22—24no—
499C22—44no—
500C22—44no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—44no—
507C22C2×#38711yescharacteristic
508C22D4×C22⋊C411yescharacteristic
509C22#234411yescharacteristic
510C22#244312yes—
511C22#244912yes—
512C22C4×#13511yescharacteristic
513C22#294311yescharacteristic
514C22#295111yescharacteristic
515C22#295211yescharacteristic
516C22#307612yes—
517C22#308111yescharacteristic
518C22#308211yescharacteristic
519C22#309112yes—
520C22#309611yescharacteristic
521C22#309711yescharacteristic
522C22#326712yes—
523C22#326712yes—
524C22#328512yes—
525C22#328712yes—
526C22#333712yes—
527C22#334012yes—
528C22#334712yes—
529C22#334912yes—
530C4—24no—
531C4—44no—
532C4—44no—
533C4—44no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4—84no—
538C4—162no—
539C4#228011yescharacteristic
540C4#231311yescharacteristic
541C4Q8×C22⋊C411yescharacteristic
542C4#234511yescharacteristic
543C4#247312yes—
544C4#248112yes—
545C2—44no—
546C2#1388211yescharacteristic
547C2#2103211yescharacteristic
548C2#2103911yescharacteristic
549C2#2243811yescharacteristic
550C2#2245111yescharacteristic
551C2#2261011yescharacteristic
552C2#2261611yescharacteristic
553C2#2612912yes—
554C2#2613712yes—
555C2#2633912yes—
556C2#2634712yes—
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
2. C2×#1921C2semidirect
5. #11379C2semidirect
6. #11220C2semidirect
7. #10908C2semidirect
16. C2×#427C4semidirect
224. #107C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4272
C2×#19212
#139512
#138012
#113792
#112202
#109082
#107312
#107212
#107142