Finite Group Databasebeta

← #196480

Subgroups — #196480

Order 512.

1300 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2924C211yesmaximal, characteristic
2#35311C211yesmaximal, characteristic
3C2×#1520C211yesmaximal, characteristic
4C2×#1508C211yesmaximal, characteristic
5#13939C211yesmaximal, characteristic
6#9116C214yesmaximal, complemented
7#9112C214yesmaximal, complemented
8#8016C211yesmaximal, characteristic
9#8012C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C22×#117C2211yescharacteristic
12#2924—24no—
13#2924—24no—
14C2×#433—22no—
15C2×#432C2212yes—
16C2×#431C2212yes—
17C2×#424—21no—
18C2×#407C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C2×#405—21no—
21#2855C2214yes—
22#2625C414yescomplemented
23#2503—21no—
24#2502—21no—
25#2499—21no—
26#2499—21no—
27#2484—22no—
28#2483—22no—
29#2317C411yescharacteristic, complemented
30#2317C411yescharacteristic, complemented
31#2310C411yescharacteristic, complemented
32#2310C411yescharacteristic, complemented
33#2280C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36#1520—24no—
37#1508—24no—
38C2×#328C2214yes—
39C2×#328C2212yes—
40C2×#328C2212yes—
41#1440—22no—
42C4⋊#95—21no—
43C4⋊#95—21no—
44#1434C2214yes—
45#1294C2212yes—
46#1294C2212yes—
47#1294C2214yes—
48C23×Q8C2×C411yescharacteristic
49C24×C4C2311yescharacteristic
50C22×M16—21no—
51C23×C8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8—22no—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8C2×C412yes—
59C2×C22⋊Q8C2312yes—
60C2×#117—22no—
61C2×#117—22no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117C2×C411yescharacteristic
65C2×#117C2×C411yescharacteristic
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C2×#117C2×C412yes—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C2×#117D412yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4D412yes—
81C22×C42—21no—
82C22×C42C2312yes—
83#433—216no—
84#432—24no—
85#432—24no—
86#431—24no—
87#431—24no—
88#424—28no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91#407—22no—
92#407D414yes—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8D414yes—
97C2×C22⋊C8C2311yescharacteristic
98C2×C22⋊C8C2311yescharacteristic
99C2×C22⋊C8C2312yes—
100#405—28no—
101C2×#97—22no—
102C2×C4×C8—22no—
103#400—24no—
104#398—24no—
105#397—24no—
106#397—24no—
107#391—21no—
108#391—21no—
109#391—21no—
110#391—21no—
111#391C2×C414yes—
112#391C2×C414yes—
113#389C2×C414yes—
114#389C2×C414yes—
115#387—24no—
116#387—24no—
117#385—24no—
118#385—22no—
119#385—22no—
120#384—21no—
121#384—21no—
122#382—21no—
123#382—21no—
124#381—24no—
125#381—21no—
126#381—22no—
127#381—22no—
128#380—22no—
129#380—22no—
130#380C2×C412yes—
131#380C2312yes—
132#379—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C2×#95—24no—
138C2×#95—24no—
139C2×#95C2312yes—
140C2×#95C2312yes—
141#328—28no—
142#328—24no—
143#328—24no—
144C25C2×D411yescharacteristic
145C22×Q8—21no—
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8C22×C412yes—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—42no—
156C23×C4—42no—
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D412yes—
165C23×C4C2×D412yes—
166C23×C4#4112yes—
167C23×C4#4112yes—
168C23×C4C2411yescharacteristic
169C2×M16—24no—
170C2×M16—42no—
171C2×M16—42no—
172C22×C8—24no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—42no—
177C22×C8—42no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C22⋊Q8—22no—
181C22⋊Q8—22no—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—42no—
185C22⋊Q8—42no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188#117—22no—
189#117—22no—
190#117—22no—
191#117—22no—
192#117—24no—
193#117—24no—
194#117—22no—
195#117—22no—
196#117—24no—
197#117—22no—
198#117—22no—
199#117—24no—
200#117—42no—
201#117—42no—
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—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22⋊C411yescharacteristic
221C2×C4⋊C4C22⋊C411yescharacteristic
222C2×C4⋊C4C22⋊C412yes—
223C2×C4⋊C4C22⋊C412yes—
224C2×C4⋊C4C22×C412yes—
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C412yes—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—82no—
241C2×C22⋊C4C22⋊C412yes—
242C2×C22⋊C4C22⋊C412yes—
243C2×C22⋊C4C22⋊C412yes—
244C2×C22⋊C4C22⋊C412yes—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42C22×C412yes—
255C2×C42C22×C412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C4⋊C8—28no—
259C4⋊C8—28no—
260C22⋊C8—24no—
261C22⋊C8—24no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—22no—
265C22⋊C8C2×D412yes—
266C22⋊C8C2×D412yes—
267C22⋊C8C2×D414yes—
268#97—28no—
269C4×C8—28no—
270#95—22no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—22no—
278#95—22no—
279#95—22no—
280#95—22no—
281#95—22no—
282#95—48no—
283#95—84no—
284C24—22no—
285C24—22no—
286C24—42no—
287C24—41no—
288C24—41no—
289C24C2×C22⋊C411yescharacteristic
290C24C2×C22⋊C411yescharacteristic
291C24#12112yes—
292C24#12112yes—
293C24C22×D411yescharacteristic
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—84no—
302C2×Q8—84no—
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—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—82no—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C4×D412yes—
354C22×C4C4×D412yes—
355C22×C4C4×D412yes—
356C22×C4C4×D412yes—
357C22×C4#12012yes—
358C22×C4#12012yes—
359C22×C4#12112yes—
360C22×C4#12112yes—
361C22×C4#12312yes—
362C22×C4#12312yes—
363C22×C4C23×C411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C2×#4111yescharacteristic
368C22×C4C2×#4111yescharacteristic
369M16—48no—
370C2×C8—22no—
371C2×C8—24no—
372C2×C8—22no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—48no—
378C2×C8—42no—
379C2×C8—42no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C4⋊C4—84no—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—48no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—44no—
409C22⋊C4—82no—
410C22⋊C4—84no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—42no—
420C42—42no—
421C42—84no—
422C23—21no—
423C23—21no—
424C23—22no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—24no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—82no—
441C23—82no—
442C23—82no—
443C23—82no—
444C23#38612yes—
445C23#38612yes—
446C23C22×C22⋊C411yescharacteristic
447C23C2×#12111yescharacteristic
448C23C2×#12111yescharacteristic
449C23#57712yes—
450C23#57812yes—
451Q8—82no—
452Q8—82no—
453Q8—84no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—84no—
505C2×C4—84no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—84no—
510C2×C4—84no—
511C2×C4—82no—
512C2×C4#38612yes—
513C2×C4#38612yes—
514C2×C4#41012yes—
515C2×C4#41012yes—
516C2×C4C2×C4×D411yescharacteristic
517C2×C4C2×C4×D411yescharacteristic
518C2×C4C2×#12011yescharacteristic
519C2×C4C2×#12311yescharacteristic
520C8—42no—
521C8—42no—
522C8—44no—
523C22—21no—
524C22—21no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—84no—
542C22—82no—
543C22—82no—
544C22C2×#38611yescharacteristic
545C22C2×#41011yescharacteristic
546C22C2×#41011yescharacteristic
547C22#289812yes—
548C22#290112yes—
549C22#305711yescharacteristic
550C22#305911yescharacteristic
551C22#306312yes—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—22no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—84no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4—82no—
566C4—162no—
567C4#189914yes—
568C2—22no—
569C2—21no—
570C2—21no—
571C2—42no—
572C2—42no—
573C2#1760512yes—
574C2#1761212yes—
575C2#2090311yescharacteristic
576C2#2097411yescharacteristic
577C2C2×#189911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9116C2semidirect
7. #9112C2semidirect
22. #2625C4semidirect
29. #2317C4semidirect
30. #2317C4semidirect
31. #2310C4semidirect
32. #2310C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29242
#353112
C2×#15202
C2×#15082
#139392
#91162
#91122
#80162
#80122