Finite Group Databasebeta

← #366934

Subgroups — #366934

Order 512.

1442 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic
4C2×#1940C211yesmaximal, characteristic
5C2×#1805C211yesmaximal, characteristic
6C2×#1763C211yesmaximal, characteristic
7#13804C211yesmaximal, characteristic
8#11357C212yesmaximal
9#11077C212yesmaximal
10#10932C212yesmaximal
11#10849C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×#97C2211yescharacteristic
20#2567—24no—
21#2567C418yescomplemented
22C2×#395—21no—
23C2×#395—21no—
24C2×#391C2212yes—
25C2×#389—22no—
26C2×#384C2212yes—
27C2×#382C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1993—24no—
32#1988—24no—
33#1940—24no—
34#1805—24no—
35#1763—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2211yescharacteristic
43C2×#336C2212yes—
44#1436—22no—
45C4⋊#95C2212yes—
46#1434—21no—
47C4×#95—21no—
48C23×Q8—21no—
49C23×Q8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8C2×C414yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D412yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C42—21no—
76C22×C42—21no—
77C22×C42—22no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80C2×C4⋊C8—24no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—24no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—42no—
86C2×#103—22no—
87C2×#103—22no—
88C2×#103—22no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—42no—
94C2×#103Q814yescomplemented
95C2×#97—22no—
96C2×#97—22no—
97C2×#97—22no—
98#395—28no—
99#395—28no—
100#391—24no—
101#391C2×C418yes—
102#389—216no—
103#384—24no—
104#384C2×C418yes—
105#382—22no—
106#382D414yes—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—42no—
110C4×C4⋊C4C2×C414yes—
111C4×C4⋊C4D414yes—
112C4×C4⋊C4D414yes—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95D411yescharacteristic
122C2×#95D411yescharacteristic
123#340—24no—
124#340—24no—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—22no—
129#340—22no—
130#336—24no—
131C22×Q8—24no—
132C22×Q8—21no—
133C22×Q8—21no—
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—42no—
144C22×Q8—42no—
145C22×Q8—82no—
146C22×Q8—82no—
147C22×Q8C22×C412yes—
148C22×Q8C2×Q812yes—
149C22×Q8#4112yes—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2411yescharacteristic
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8C2×Q812yes—
181C22×C8#4112yes—
182C4⋊Q8—28no—
183C4⋊Q8—28no—
184C4⋊Q8—44no—
185C4⋊Q8—44no—
186C4×Q8—28no—
187C4×Q8—28no—
188C4×Q8—44no—
189C4×Q8—44no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—28no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4—82no—
227C2×C4⋊C4C22⋊C418yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C22×C414yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×Q812yes—
234C2×C4⋊C4#4112yes—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—42no—
242C2×C42—42no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42—82no—
246C2×C42—82no—
247C2×C42C22×C414yes—
248C2×C42C22×C412yes—
249C2×C42C2×D414yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42#4112yes—
254C2×C42#4112yes—
255C4⋊C8—48no—
256C4⋊C8—44no—
257#103—28no—
258#103—28no—
259#103—48no—
260#97—44no—
261#95—28no—
262#95—28no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—44no—
267#95—44no—
268#95—82no—
269#95C22⋊C414yes—
270#95C22⋊C414yes—
271C24C22×D411yescharacteristic
272C2×Q8—22no—
273C2×Q8—22no—
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—22no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
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—48no—
300C2×Q8—88no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×Q8—88no—
304C2×Q8C4×Q814yes—
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—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
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⋊C412yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C414yes—
359C22×C4#12112yes—
360C22×C4#12112yes—
361C22×C4#12112yes—
362C22×C4#12112yes—
363C22×C4#13714yes—
364C22×C4#13714yes—
365C22×C4C23×C411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×Q811yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4H(5,𝔽2)11yescharacteristic
376C22×C42−1+411yescharacteristic
377C22×C42−1+411yescharacteristic
378C22×C42−1+411yescharacteristic
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—42no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—216no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—88no—
423C4⋊C4—88no—
424C4⋊C4C22⋊Q814yes—
425C42—24no—
426C42—24no—
427C42—24no—
428C42—24no—
429C42—48no—
430C42—84no—
431C42—88no—
432C42C4×D414yes—
433C42C4×D414yes—
434C42#12414yes—
435C42#12714yes—
436C23C22×C22⋊C411yescharacteristic
437C23C2×#12111yescharacteristic
438C23C2×#12111yescharacteristic
439C23#54611yescharacteristic
440C23#54711yescharacteristic
441C23#54811yescharacteristic
442C23D4×Q811yescharacteristic
443C23C2×#13712yes—
444C23C2×#13712yes—
445C23#57612yes—
446C23#57712yes—
447Q8—24no—
448Q8—24no—
449Q8—42no—
450Q8—42no—
451Q8—42no—
452Q8—42no—
453Q8—42no—
454Q8—42no—
455Q8—816no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—81no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4#39014yes—
514C2×C4#39014yes—
515C2×C4#41918yes—
516C2×C4C2×C4×D411yescharacteristic
517C2×C4C2×C4×D411yescharacteristic
518C2×C4C2×C4×Q811yescharacteristic
519C2×C4C4×#4111yescharacteristic
520C2×C4#51911yescharacteristic
521C2×C4#52011yescharacteristic
522C2×C4C2×C22⋊Q811yescharacteristic
523C2×C4#52511yescharacteristic
524C2×C4C2×#12411yescharacteristic
525C2×C4#52911yescharacteristic
526C2×C4C2×#12711yescharacteristic
527C2×C4#53311yescharacteristic
528C2×C4#53611yescharacteristic
529C2×C4#53711yescharacteristic
530C2×C4#54111yescharacteristic
531C2×C4#54211yescharacteristic
532C2×C4#54611yescharacteristic
533C2×C4#54611yescharacteristic
534C2×C4#54711yescharacteristic
535C2×C4#54711yescharacteristic
536C2×C4#54811yescharacteristic
537C2×C4#54811yescharacteristic
538C2×C4D4×Q811yescharacteristic
539C2×C4D4×Q811yescharacteristic
540C2×C4#55211yescharacteristic
541C2×C4#55211yescharacteristic
542C2×C4#55411yescharacteristic
543C2×C4#55611yescharacteristic
544C8—42no—
545C8—42no—
546C8—82no—
547C22C2×#39011yescharacteristic
548C22#233911yescharacteristic
549C22Q8×C22⋊C411yescharacteristic
550C22#241511yescharacteristic
551C22#241711yescharacteristic
552C22#242711yescharacteristic
553C22#242911yescharacteristic
554C22C2×#41912yes—
555C22#289712yes—
556C22#303911yescharacteristic
557C22#303911yescharacteristic
558C22#303912yes—
559C22#304111yescharacteristic
560C22#304311yescharacteristic
561C22#304512yes—
562C22#331412yes—
563C22#331612yes—
564C22#334412yes—
565C22#334612yes—
566C22#336912yes—
567C22#337512yes—
568C22#338512yes—
569C22#338712yes—
570C4—24no—
571C4—24no—
572C4—28no—
573C4—44no—
574C4—44no—
575C4—44no—
576C4—82no—
577C4—84no—
578C4—82no—
579C4—84no—
580C4C4×C22⋊Q811yescharacteristic
581C4#232311yescharacteristic
582C4#236011yescharacteristic
583C4#236211yescharacteristic
584C4#248911yescharacteristic
585C4#249211yescharacteristic
586C4#254011yescharacteristic
587C4#254211yescharacteristic
588C2#1399711yescharacteristic
589C2#2125911yescharacteristic
590C2#2126311yescharacteristic
591C2#2249712yes—
592C2#2252512yes—
593C2#2469611yescharacteristic
594C2#2469811yescharacteristic
595C2#2483111yescharacteristic
596C2#2483311yescharacteristic
597C2#2533612yes—
598C2#2546712yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #2567C4semidirect
94. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#19932
C2×#19882
C2×#19402
C2×#18052
C2×#17632
#138042
#113572
#110772
#109322
#108492