Finite Group Databasebeta

← #224583

Subgroups — #224583

Order 512.

1216 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36579C211yesmaximal, characteristic
2#22510C211yesmaximal, characteristic, complemented
3Q8×#95C211yesmaximal, characteristic, complemented
4#9713C212yesmaximal
5#9711C211yesmaximal, characteristic, complemented
6#9682C212yesmaximal
7#9679C211yesmaximal, characteristic, complemented
8#8578C212yesmaximal, complemented
9#8571C211yesmaximal, characteristic
10#8219C212yesmaximal, complemented
11C4×#327C211yesmaximal, characteristic
12C2×#556C411yescharacteristic, complemented
13C22×C4×Q8—21no—
14C22×C4×Q8C2211yescharacteristic
15C2×#446—21no—
16#2857C2211yescharacteristic
17#2850C2212yes—
18#2592C412yescomplemented
19#2581C412yescomplemented
20#2568C411yescharacteristic, complemented
21#2538C411yescharacteristic, complemented
22#2481—22no—
23#2469—22no—
24#2421—21no—
25#2409—21no—
26#2362—21no—
27#2355C411yescharacteristic, complemented
28#2352—21no—
29C4×C22⋊Q8C2212yes—
30C4×#121C2211yescharacteristic
31#1985—22no—
32#1984—21no—
33#1797C2211yescharacteristic
34#1797C2212yes—
35#1749C2212yes—
36C4×C22⋊C8C2211yescharacteristic
37C2×#328C2212yes—
38C2×#328C2212yes—
39C2×#328C2212yes—
40C2×#328C2212yes—
41C2×#327C2211yescharacteristic
42C2×#327C2211yescharacteristic
43C2×#327C2212yes—
44C4⋊#95—21no—
45C4⋊#95—22no—
46C4⋊#95C2211yescharacteristic
47C4⋊#95C2212yes—
48C4×#95C2211yescharacteristic
49C4×#95C2212yes—
50#1294C2211yescharacteristic
51#1294C2211yescharacteristic
52#1294C2212yes—
53C23×Q8D411yescharacteristic
54#556—44no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#126—22no—
57C2×#124C2×C412yes—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C22⋊Q8C2312yes—
61C2×#121C2311yescharacteristic
62C2×C4×Q8—28no—
63C2×C4×Q8—44no—
64C2×C4×Q8—44no—
65C2×C4×Q8—44no—
66C2×C4×Q8C2×C412yes—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C412yes—
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D412yes—
73C2×C4×D4—21no—
74C2×C4×D4C2×C411yescharacteristic
75C2×#117—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D412yes—
80C22×C42—22no—
81C22×C42—21no—
82C22×C42C2312yes—
83C22×C42C2311yescharacteristic
84#446—44no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—22no—
87C2×C22⋊C8C2311yescharacteristic
88C2×C22⋊C8C2312yes—
89C2×C22⋊C8C2311yescharacteristic
90C2×#97—22no—
91C2×C4×C8—21no—
92#400—22no—
93#400—22no—
94#397—22no—
95#396—22no—
96#395—22no—
97#393—22no—
98#391—21no—
99#391—21no—
100#391C2×C412yes—
101#391C2×C412yes—
102#391C2×C412yes—
103#390C2×C411yescharacteristic
104#390C2×C412yes—
105#389—21no—
106#389—21no—
107#389C2×C411yescharacteristic
108#389C2×C412yes—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—21no—
114#387—22no—
115#387—22no—
116#387—22no—
117#387—22no—
118#386—22no—
119#386—22no—
120#386—21no—
121#385—22no—
122#384C2×C411yescharacteristic
123#382—21no—
124#382C2×C412yes—
125#380—21no—
126#380—21no—
127#380—21no—
128#380C2311yescharacteristic
129C4×C4⋊C4—21no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C412yes—
133C4×C22⋊C4—22no—
134C4×C22⋊C4—21no—
135C2×#95—22no—
136C2×#95—21no—
137C2×#95—22no—
138C2×#95—21no—
139C2×#95C2311yescharacteristic
140C2×#95C2312yes—
141C2×#95C2311yescharacteristic
142#336—22no—
143#336—21no—
144#336—22no—
145#336—21no—
146#328—24no—
147#328—24no—
148#328—42no—
149#328—42no—
150#327—22no—
151#327—22no—
152#327—42no—
153C22×Q8—24no—
154C22×Q8—24no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8C22⋊C412yes—
158C22×Q8C22⋊C411yescharacteristic
159C22×Q8C22⋊C411yescharacteristic
160C22×Q8C22×C412yes—
161C22×D4—21no—
162C22×D4C22×C411yescharacteristic
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4C2×D412yes—
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—22no—
175C4⋊Q8—42no—
176#126—48no—
177#124—44no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—44no—
183C22⋊Q8—44no—
184#121—21no—
185#121—21no—
186#121—21no—
187#121—21no—
188C4×Q8—24no—
189C4×Q8—24no—
190C4×Q8—42no—
191C4×Q8—44no—
192C4×Q8—42no—
193C4×Q8—44no—
194C4×Q8—44no—
195C4×Q8—432no—
196C4×D4—42no—
197C4×D4—42no—
198C4×D4—42no—
199#117—44no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4C22⋊C411yescharacteristic
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22⋊C412yes—
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C22×C411yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C42—28no—
238C2×C42—24no—
239C2×C42—22no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—42no—
243C2×C42—44no—
244C2×C42—44no—
245C2×C42—44no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C412yes—
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C412yes—
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2×D411yescharacteristic
256C4⋊C8—42no—
257C4⋊C8—44no—
258C22⋊C8—21no—
259C22⋊C8—21no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—21no—
263C22⋊C8#4112yes—
264#97—44no—
265C4×C8—42no—
266#95—22no—
267#95—22no—
268#95—21no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—21no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—22no—
282#95—22no—
283#95—21no—
284#95—44no—
285#95—42no—
286#95—82no—
287#95—81no—
288#95#4112yes—
289C24—21no—
290C24C22×D411yescharacteristic
291C2×Q8—216no—
292C2×Q8—216no—
293C2×Q8—24no—
294C2×Q8—22no—
295C2×Q8—24no—
296C2×Q8—22no—
297C2×Q8—22no—
298C2×Q8—24no—
299C2×Q8—22no—
300C2×Q8—22no—
301C2×Q8—22no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8#9914yes—
305C2×Q8#10414yes—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C412yes—
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4AGL(1,ℤ/8)12yes—
370C22×C4#13712yes—
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4C2×#4111yescharacteristic
375C22×C4H(5,𝔽2)11yescharacteristic
376C22×C42−1+411yescharacteristic
377C2×C8—22no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—41no—
383C2×C8—42no—
384C2×C8—41no—
385C2×C8—42no—
386C4⋊C4—22no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—24no—
390C4⋊C4—21no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—416no—
400C4⋊C4—48no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—44no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—81no—
410C22⋊C4—44no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—44no—
419C22⋊C4—42no—
420C22⋊C4—44no—
421C22⋊C4—81no—
422C22⋊C4—82no—
423C42—22no—
424C42—24no—
425C42—22no—
426C42—24no—
427C42—41no—
428C42—42no—
429C42—42no—
430C42—41no—
431C42—416no—
432C42—48no—
433C42—44no—
434C42—42no—
435C42—82no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—41no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23C22×C22⋊C411yescharacteristic
448C23#54811yescharacteristic
449C23#55011yescharacteristic
450C23C2×AGL(1,ℤ/8)11yescharacteristic
451C23C2×#13711yescharacteristic
452C23#57711yescharacteristic
453C23#57811yescharacteristic
454Q8—216no—
455Q8—28no—
456Q8—28no—
457Q8—82no—
458Q8—82no—
459D4—81no—
460D4—81no—
461D4—81no—
462D4—81no—
463D4—81no—
464D4—81no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—82no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—82no—
535C2×C4—84no—
536C2×C4C2×#9914yes—
537C2×C4C2×#9912yes—
538C2×C4#41712yes—
539C2×C4#41712yes—
540C2×C4C2×#10414yes—
541C2×C4C2×#10412yes—
542C2×C4C4×#4111yescharacteristic
543C2×C4#52011yescharacteristic
544C2×C4#54811yescharacteristic
545C2×C4#55011yescharacteristic
546C8—42no—
547C8—81no—
548C8—82no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22—81no—
567C22—81no—
568C22#234111yescharacteristic
569C22C22×#9911yescharacteristic
570C22#288411yescharacteristic
571C22C2×#41711yescharacteristic
572C22#289811yescharacteristic
573C22C22×#10411yescharacteristic
574C22#290111yescharacteristic
575C22#336711yescharacteristic
576C22#336911yescharacteristic
577C22#339711yescharacteristic
578C22#339811yescharacteristic
579C4—21no—
580C4—21no—
581C4—21no—
582C4—21no—
583C4—24no—
584C4—22no—
585C4—24no—
586C4—44no—
587C4—42no—
588C4—41no—
589C4—41no—
590C4—42no—
591C4—84no—
592C4—82no—
593C4—82no—
594C4—81no—
595C4—82no—
596C4—81no—
597C4—81no—
598C4#150414yes—
599C2—21no—
600C2—21no—
601C2—21no—
602C2—21no—
603C2—161no—
604C2C2×#150411yescharacteristic
605C2#1754911yescharacteristic
606C2#1760611yescharacteristic
607C2#1762811yescharacteristic
608C2#2253411yescharacteristic
609C2#2254411yescharacteristic
610C2#2255011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22510C2semidirect
3. Q8×#95C2semidirect
5. #9711C2semidirect
7. #9679C2semidirect
8. #8578C2semidirect
10. #8219C2semidirect
12. C2×#556C4semidirect
18. #2592C4semidirect
19. #2581C4semidirect
20. #2568C4semidirect
21. #2538C4semidirect
27. #2355C4semidirect

Maximal subgroups

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

subgroupindex
#365792
#225102
Q8×#952
#97132
#97112
#96822
#96792
#85782
#85712
#82192
C4×#3272