Finite Group Databasebeta

← #369003

Subgroups — #369003

Order 512.

1562 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2546C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic
3C2×#1993C211yesmaximal, characteristic
4C2×#1987C211yesmaximal, characteristic
5C2×#1985C211yesmaximal, characteristic
6#11208C212yesmaximal
7#11207C212yesmaximal
8#10885C212yesmaximal
9#10884C212yesmaximal
10#10850C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4×Q8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17#2546—24no—
18#2546C418yescomplemented
19C2×#398—21no—
20C2×#398—21no—
21#2355—24no—
22#2355—24no—
23C2×#391—21no—
24C2×#391—21no—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#387—21no—
28C2×#387—21no—
29C2×#384C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×C4×C4⋊C4—21no—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35C2×C4×C4⋊C4C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1993—24no—
38#1987—24no—
39#1985—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47C2×#340C2212yes—
48C2×#336C2212yes—
49C2×#336C2212yes—
50C23×Q8—21no—
51C23×Q8D411yescharacteristic
52C23×C8C2312yes—
53C23×C8C2312yes—
54C2×C4⋊Q8—28no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—44no—
61C2×C4⋊Q8—44no—
62C2×C4⋊Q8C2×C414yes—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8C2×C414yes—
68C2×C4×Q8D414yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C42—21no—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C22×C42C2311yescharacteristic
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×#97—22no—
94C2×#97—22no—
95C2×#97—22no—
96#398—28no—
97#398—28no—
98#391—28no—
99#391—28no—
100#389—24no—
101#389—24no—
102#389C2×C418yes—
103#389C2×C418yes—
104#387—28no—
105#387—28no—
106#384—22no—
107#384—22no—
108#382—22no—
109#382—22no—
110#382—22no—
111#382D414yes—
112C4×C4⋊C4—28no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C414yes—
120C4×C4⋊C4D414yes—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136#340—24no—
137#340—24no—
138#340Q818yescomplemented
139#336—24no—
140#336—24no—
141C22×Q8—24no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—41no—
147C22×Q8—41no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×Q8C22⋊C414yes—
155C23×C4—22no—
156C23×C4—21no—
157C23×C4—22no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2411yescharacteristic
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8C2×Q814yes—
182C4⋊Q8—28no—
183C4⋊Q8—28no—
184C4⋊Q8—432no—
185C4⋊Q8—44no—
186C4⋊Q8—44no—
187C4×Q8—28no—
188C4×Q8—28no—
189C4×Q8—28no—
190C4×Q8—28no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—82no—
230C2×C4⋊C4C22⋊C414yes—
231C2×C4⋊C4C22⋊C414yes—
232C2×C4⋊C4C22⋊C414yes—
233C2×C4⋊C4C22×C414yes—
234C2×C4⋊C4C22×C414yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×Q814yes—
240C2×C4⋊C4C2×Q814yes—
241C2×C4⋊C4#4114yes—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—42no—
257C2×C42—42no—
258C2×C42—82no—
259C2×C42—82no—
260C2×C42C22×C412yes—
261C2×C42C22×C412yes—
262C2×C42C22×C412yes—
263C2×C42C2×D412yes—
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42#4112yes—
269C2×C42#4112yes—
270C4⋊C8—44no—
271#97—44no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—28no—
278#95—28no—
279#95—24no—
280#95—24no—
281#95—24no—
282C24C22×D411yescharacteristic
283C2×Q8—28no—
284C2×Q8—28no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8—44no—
290C2×Q8—48no—
291C2×Q8—48no—
292C2×Q8—48no—
293C2×Q8—84no—
294C2×Q8—84no—
295C2×Q8—84no—
296C2×Q8—84no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C412yes—
361C22×C4#12012yes—
362C22×C4#12012yes—
363C22×C4#12312yes—
364C22×C4#12312yes—
365C22×C4AGL(1,ℤ/8)12yes—
366C22×C4AGL(1,ℤ/8)12yes—
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×Q812yes—
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4112yes—
375C22×C42−1+412yes—
376C22×C42−1+412yes—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—28no—
397C4⋊C4—28no—
398C4⋊C4—28no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—48no—
406C4⋊C4—416no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—48no—
415C4⋊C4—416no—
416C4⋊C4—84no—
417C4⋊C4C4×Q818yes—
418C4⋊C4C4⋊Q814yescomplemented
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—48no—
428C42—48no—
429C42—84no—
430C42—84no—
431C42C4×D414yes—
432C42C4×D414yes—
433C42#12114yes—
434C42#12114yes—
435C23C22×C22⋊C411yescharacteristic
436C23C2×#12011yescharacteristic
437C23C2×#12311yescharacteristic
438C23#54812yes—
439C23D4×Q812yes—
440C23C2×AGL(1,ℤ/8)11yescharacteristic
441C23C2×AGL(1,ℤ/8)11yescharacteristic
442C23#57511yescharacteristic
443C23#57511yescharacteristic
444C23#57712yes—
445C23#57812yes—
446Q8—416no—
447Q8—816no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—28no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4#38614yes—
516C2×C4#38614yes—
517C2×C4#41814yes—
518C2×C4C2×C4×D411yescharacteristic
519C2×C4C2×C4×D411yescharacteristic
520C2×C4C2×C4×Q812yes—
521C2×C4#51912yes—
522C2×C4C2×#12111yescharacteristic
523C2×C4C2×#12111yescharacteristic
524C2×C4C2×C4⋊Q811yescharacteristic
525C2×C4#53211yescharacteristic
526C2×C4#53312yes—
527C2×C4#53611yescharacteristic
528C2×C4#53611yescharacteristic
529C2×C4#54112yes—
530C2×C4#54712yes—
531C2×C4#54812yes—
532C2×C4D4×Q812yes—
533C2×C4D4×Q812yes—
534C2×C4#55612yes—
535C2×C4#55712yes—
536C8—44no—
537C8—82no—
538C22C2×#38611yescharacteristic
539C22Q8×C22⋊C412yes—
540C22#243211yescharacteristic
541C22#243411yescharacteristic
542C22#244612yes—
543C22C2×#41811yescharacteristic
544C22#289611yescharacteristic
545C22#289812yes—
546C22#300611yescharacteristic
547C22#300611yescharacteristic
548C22#300911yescharacteristic
549C22#301111yescharacteristic
550C22#309411yescharacteristic
551C22#309411yescharacteristic
552C22#309712yes—
553C22#336712yes—
554C22#337112yes—
555C22#337312yes—
556C22#337712yes—
557C22#338412yes—
558C22#338612yes—
559C22#338812yes—
560C22#339012yes—
561C4—24no—
562C4—24no—
563C4—24no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—48no—
568C4—44no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4—84no—
573C4C4×C4⋊Q811yescharacteristic
574C4#232711yescharacteristic
575C4#236512yes—
576C4#242912yes—
577C4#254511yescharacteristic
578C4#254611yescharacteristic
579C2#1402411yescharacteristic
580C2#2091211yescharacteristic
581C2#2091311yescharacteristic
582C2#2252312yes—
583C2#2252712yes—
584C2#2565911yescharacteristic
585C2#2566111yescharacteristic
586C2#2584911yescharacteristic
587C2#2585111yescharacteristic
588C2#2629212yes—
589C2#2629412yes—
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
18. #2546C4semidirect
138. #340Q8semidirect
418. C4⋊C4C4⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25462
C2×#23552
C2×#19932
C2×#19872
C2×#19852
#112082
#112072
#108852
#108842
#108502
#107032