Finite Group Databasebeta

← #364790

Subgroups — #364790

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#446C211yesmaximal, characteristic
2C2×#2581C211yesmaximal, characteristic
3C2×#1805C211yesmaximal, characteristic
4C2×C4×#103C211yesmaximal, characteristic
5#13925C211yesmaximal, characteristic
6#11228C212yesmaximal
7#11077C212yesmaximal
8#10928C212yesmaximal
9#10881C212yesmaximal
10#10850C211yesmaximal, characteristic
11#10842C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#446—24no—
15C2×#446—24no—
16C2×#446—24no—
17C22×C4⋊C8C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×#103C2212yes—
21C22×#103C2211yescharacteristic
22C22×#97C2212yes—
23C22×C4×C8C2211yescharacteristic
24#2581—24no—
25#2581C418yescomplemented
26C2×#400—21no—
27C2×#395—21no—
28C2×#391C2212yes—
29C2×#389C2212yes—
30C2×#387—22no—
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1805—24no—
35C4×#103—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1446—21no—
44#1445—21no—
45#1436—22no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×Q8D411yescharacteristic
49C23×Q8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2311yescharacteristic
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8C2×C414yes—
63C2×C4×Q8C2×C414yes—
64C2×C4×Q8D414yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D412yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C42—21no—
75C22×C42—21no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78#446—232no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8D414yes—
85C2×#103—22no—
86C2×#103—22no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—42no—
93C2×#103Q814yescomplemented
94C2×#97—24no—
95C2×#97—24no—
96C2×#97—24no—
97C2×C4×C8—22no—
98C2×C4×C8—22no—
99C2×C4×C8D414yes—
100#400—28no—
101#395—28no—
102#391—24no—
103#391C2×C418yes—
104#389—24no—
105#389C2×C418yes—
106#387—216no—
107#382—22no—
108#382—22no—
109#382—22no—
110#382C2×C414yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125#340—24no—
126#340—22no—
127#340—22no—
128#340—24no—
129#340—24no—
130#340—42no—
131#340—42no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×Q8—21no—
140C22×Q8—21no—
141C22×Q8—21no—
142C22×Q8—21no—
143C22×Q8—21no—
144C22×Q8—21no—
145C22×Q8—42no—
146C22×Q8—42no—
147C22×Q8C22⋊C414yes—
148C22×Q8C22×C412yes—
149C22×Q8C2×Q812yes—
150C22×Q8#4112yes—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—21no—
177C22×C8—22no—
178C22×C8C2×D412yes—
179C22×C8C2×D414yes—
180C22×C8C2×D412yes—
181C22×C8C2×Q812yes—
182C22×C8#4112yes—
183C4×Q8—28no—
184C4×Q8—28no—
185C4×Q8—28no—
186C4×Q8—28no—
187C4×Q8—28no—
188C4×Q8—28no—
189C4×Q8—44no—
190C4×Q8—44no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4C22⋊C418yes—
228C2×C4⋊C4C22⋊C414yes—
229C2×C4⋊C4C22×C412yes—
230C2×C4⋊C4C22×C414yes—
231C2×C4⋊C4C2×Q812yes—
232C2×C4⋊C4#4112yes—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42—82no—
242C2×C42C22×C414yes—
243C2×C42C22×C412yes—
244C2×C42C2×D414yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D412yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250C4⋊C8—216no—
251C4⋊C8—28no—
252#103—28no—
253#103—28no—
254#103—48no—
255#97—216no—
256C4×C8—28no—
257#95—24no—
258#95—24no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—44no—
263#95—44no—
264#95—42no—
265#95—42no—
266#95—42no—
267#95—82no—
268#95—82no—
269C24C22×D411yescharacteristic
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—28no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—48no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8—48no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8C4×Q814yes—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4C2×C22⋊C414yes—
356C22×C4C2×C22⋊C414yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C412yes—
359C22×C4#12112yes—
360C22×C4#12112yes—
361C22×C4#12112yes—
362C22×C4#12112yes—
363C22×C4C23×C411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×Q811yescharacteristic
368C22×C4C2×#4111yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4H(5,𝔽2)11yescharacteristic
374C22×C42−1+411yescharacteristic
375C22×C42−1+411yescharacteristic
376C22×C42−1+411yescharacteristic
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—24no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—42no—
395C2×C8—41no—
396C2×C8—42no—
397C2×C8C22⋊Q814yes—
398C4⋊C4—28no—
399C4⋊C4—216no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—416no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—48no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C42—24no—
423C42—24no—
424C42—24no—
425C42—24no—
426C42—48no—
427C42—84no—
428C42C4×D414yes—
429C42C4×D414yes—
430C42#12014yes—
431C42#12314yes—
432C23C22×C22⋊C411yescharacteristic
433C23C2×#12111yescharacteristic
434C23C2×#12111yescharacteristic
435C23#54811yescharacteristic
436C23#54811yescharacteristic
437C23D4×Q811yescharacteristic
438C23#55011yescharacteristic
439C23#57512yes—
440C23#57512yes—
441C23#57712yes—
442C23#57812yes—
443Q8—24no—
444Q8—24no—
445Q8—416no—
446Q8—42no—
447Q8—42no—
448Q8—42no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4#38614yes—
511C2×C4#38614yes—
512C2×C4C2×C4×D411yescharacteristic
513C2×C4C2×C4×D411yescharacteristic
514C2×C4C2×C4×Q811yescharacteristic
515C2×C4C4×#4111yescharacteristic
516C2×C4#51911yescharacteristic
517C2×C4#52011yescharacteristic
518C2×C4C2×#12011yescharacteristic
519C2×C4C2×C22⋊Q811yescharacteristic
520C2×C4C2×#12311yescharacteristic
521C2×C4#52511yescharacteristic
522C2×C4#52911yescharacteristic
523C2×C4#53311yescharacteristic
524C2×C4#53611yescharacteristic
525C2×C4#53711yescharacteristic
526C2×C4#54111yescharacteristic
527C2×C4#54211yescharacteristic
528C2×C4#54611yescharacteristic
529C2×C4#54711yescharacteristic
530C2×C4#54811yescharacteristic
531C2×C4#54811yescharacteristic
532C2×C4#54811yescharacteristic
533C2×C4D4×Q811yescharacteristic
534C2×C4D4×Q811yescharacteristic
535C2×C4#55011yescharacteristic
536C2×C4#55211yescharacteristic
537C2×C4#55211yescharacteristic
538C2×C4#55411yescharacteristic
539C2×C4#55611yescharacteristic
540C8—24no—
541C8—42no—
542C8—44no—
543C22C2×#38611yescharacteristic
544C22Q8×C22⋊C411yescharacteristic
545C22#234111yescharacteristic
546C22#242611yescharacteristic
547C22#242911yescharacteristic
548C22#253611yescharacteristic
549C22#253811yescharacteristic
550C22#289612yes—
551C22#289812yes—
552C22#305411yescharacteristic
553C22#305411yescharacteristic
554C22#305412yes—
555C22#305711yescharacteristic
556C22#305911yescharacteristic
557C22#306312yes—
558C22#337112yes—
559C22#337612yes—
560C22#337712yes—
561C22#337712yes—
562C22#338812yes—
563C22#339012yes—
564C22#340112yes—
565C22#340212yes—
566C4—24no—
567C4—24no—
568C4—44no—
569C4—44no—
570C4—44no—
571C4—84no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—162no—
576C4C4×C22⋊Q811yescharacteristic
577C4#232311yescharacteristic
578C4#236011yescharacteristic
579C4#236211yescharacteristic
580C4#240711yescharacteristic
581C4#240911yescharacteristic
582C4#242111yescharacteristic
583C4#242311yescharacteristic
584C2#1396911yescharacteristic
585C2#2097011yescharacteristic
586C2#2097411yescharacteristic
587C2#2252712yes—
588C2#2254812yes—
589C2#2529912yes—
590C2#2547612yes—
591C2#2819211yescharacteristic
592C2#2819311yescharacteristic
593C2#2827011yescharacteristic
594C2#2827111yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
25. #2581C4semidirect
93. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#4462
C2×#25812
C2×#18052
C2×C4×#1032
#139252
#112282
#110772
#109282
#108812
#108502
#108422