Finite Group Databasebeta

← #193992

Subgroups — #193992

Order 512.

1266 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×#3076C211yesmaximal, characteristic
2#34121C211yesmaximal, characteristic
3#21400C211yesmaximal, characteristic, complemented
4#21385C211yesmaximal, characteristic, complemented
5C2×#1476C211yesmaximal, characteristic
6#8984C212yesmaximal, complemented
7#8982C212yesmaximal, complemented
8#8976C212yesmaximal, complemented
9#8974C212yesmaximal, complemented
10#8039C211yesmaximal, characteristic, complemented
11C2×#1279C211yesmaximal, characteristic, complemented
12C2×#533C2211yescharacteristic
13C2×C4×#41C2211yescharacteristic
14#3076—24no—
15#3076C418yescomplemented
16C2×#479—21no—
17C2×#477C2211yescharacteristic
18C2×#475C2211yescharacteristic
19C2×#478—21no—
20C2×#476C2211yescharacteristic
21C2×#474C2211yescharacteristic
22C2×#431C2211yescharacteristic
23C2×#427—21no—
24C22×C4⋊C8C2211yescharacteristic
25C2×#416—21no—
26C22×C4×C8C2211yescharacteristic
27#2357—21no—
28#2356—21no—
29#2355C2212yes—
30#2354C2212yes—
31#2353—22no—
32#2352—22no—
33#2298—22no—
34#2297—21no—
35#2285C2211yescharacteristic
36#2283—21no—
37C4×#117—21no—
38C4×#117—21no—
39C2×C4×C4⋊C4C2211yescharacteristic
40#1990C2212yes—
41#1987C2212yes—
42#1981C2212yes—
43#1956C2212yes—
44#1892—22no—
45#1872—22no—
46C4×#103—21no—
47C4×#103—21no—
48C4×#103—22no—
49#1476—24no—
50C2×Q8⋊C8C2212yes—
51C2×D4⋊C8C2212yes—
52#1296C2212yes—
53#1296C2212yes—
54#1279C2212yes—
55#1279C2212yes—
56C22×#41D411yescharacteristic
57C23×C8—21no—
58C23×C8—21no—
59#533—22no—
60#533—22no—
61#533—22no—
62#533C2×C414yes—
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2311yescharacteristic
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C4×#41—22no—
69C4×#41—22no—
70C4×#41C2×C414yes—
71C4×#41D414yes—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×C4×D4D411yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C2×#117—21no—
81C2×#117—22no—
82C2×#117—21no—
83C2×#117—21no—
84C2×#117D411yescharacteristic
85C22×C4⋊C4D412yes—
86C22×C42—21no—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#479—28no—
90#478—28no—
91#477—22no—
92#477C2×C414yescomplemented
93#476—22no—
94#476C2×C414yescomplemented
95#475—22no—
96#475C2×C414yescomplemented
97#474—22no—
98#474C2×C414yescomplemented
99#431—22no—
100#431—22no—
101#427—28no—
102C2×#107—21no—
103C2×#107—21no—
104C2×#106—21no—
105C2×#106—21no—
106C2×C4⋊C8—21no—
107C2×C4⋊C8—22no—
108C2×C4⋊C8C2×C414yes—
109C2×C4⋊C8C2312yes—
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112#416—28no—
113C2×#103—21no—
114C2×#103—21no—
115C2×#103—22no—
116C2×#103—22no—
117C2×#103—24no—
118C2×#102—21no—
119C2×#102—21no—
120C2×C22⋊C8—21no—
121C2×C4×C8—22no—
122C2×C4×C8—22no—
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8C2312yes—
126#391—22no—
127#390—22no—
128#388—24no—
129#388—24no—
130#387—24no—
131#387—24no—
132#386—24no—
133#384C2312yes—
134#383—22no—
135#382—22no—
136#382—21no—
137#381—22no—
138#380—21no—
139#380—22no—
140#380D412yes—
141C4×C4⋊C4—22no—
142C4×C4⋊C4—22no—
143C4×C4⋊C4—22no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4—21no—
146C4×C4⋊C4—21no—
147C4×C4⋊C4D411yescharacteristic
148C4×C4⋊C4D411yescharacteristic
149C4×C4⋊C4D412yes—
150C4×C4⋊C4C2312yes—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—22no—
153C4×C22⋊C4—22no—
154C4×C22⋊C4—22no—
155#376—22no—
156#376—22no—
157C2×#95—21no—
158C43—22no—
159#340—22no—
160#340—22no—
161#340—24no—
162Q8⋊C8—24no—
163D4⋊C8—24no—
164C2×#41—22no—
165C2×#41—22no—
166C2×#41—22no—
167C2×#41—22no—
168C2×#41—22no—
169C2×#41C22⋊C414yes—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×Q8C2×D411yescharacteristic
173C22×D4—21no—
174C22×D4C2×D411yescharacteristic
175C23×C4—21no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—22no—
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C22×C8—24no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—44no—
188C22×C8—42no—
189C22×C8—42no—
190C22×C8—42no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8C22×C412yes—
198C4⋊Q8C22×C412yes—
199#125—24no—
200#125—42no—
201C22⋊Q8—24no—
202C22⋊Q8—24no—
203C22⋊Q8—42no—
204C22⋊Q8—42no—
205C4×Q8—22no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—42no—
211C4×Q8—42no—
212C4×Q8C22⋊C414yes—
213C4×Q8C22×C412yes—
214C4×Q8C2×D412yes—
215C4×D4—24no—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—22no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4C22⋊C414yes—
223C4×D4C22×C412yes—
224C4×D4C2×D412yes—
225#117—22no—
226#117—24no—
227#117—24no—
228#117—24no—
229#117—44no—
230#117—42no—
231#117—42no—
232#117—42no—
233#117—44no—
234#117C22⋊C414yes—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4C2×D412yes—
249C2×C4⋊C4C2×D411yescharacteristic
250C2×C4⋊C4C2×D411yescharacteristic
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D414yes—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—41no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—24no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—22no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—22no—
270C2×C42—24no—
271C2×C42—41no—
272C2×C42—42no—
273C2×C42—42no—
274C2×C42C22×C412yes—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D412yes—
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42#4112yes—
280C2×C42#4112yes—
281C2×C42C2411yescharacteristic
282#107—24no—
283#107—24no—
284#106—24no—
285#106—24no—
286C4⋊C8—24no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—22no—
290C4⋊C8—22no—
291C4⋊C8C22×C412yes—
292C4⋊C8C22×C412yes—
293#103—24no—
294#103—24no—
295#103—44no—
296#103—48no—
297#103—44no—
298#102—24no—
299#102—24no—
300C22⋊C8—42no—
301C4×C8—22no—
302C4×C8—22no—
303C4×C8—42no—
304C4×C8—42no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—21no—
310#95—21no—
311#95—22no—
312#95—22no—
313#95—22no—
314C24—21no—
315C24#12011yescharacteristic
316#41—28no—
317#41—28no—
318#41—44no—
319#41—44no—
320C2×Q8—21no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—44no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8C2×C22⋊C412yes—
333C2×D4—24no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4C2×C22⋊C412yes—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—44no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4C2×C22⋊C412yes—
366C22×C4C4×D412yes—
367C22×C4C4×D412yes—
368C22×C4#12011yescharacteristic
369C22×C4#12011yescharacteristic
370C22×C4#12011yescharacteristic
371C22×C4#12112yes—
372C22×C4#12112yes—
373C22×C4#12312yes—
374C22×C4#13512yes—
375C22×C4#13512yes—
376C22×C4#13712yes—
377C22×C4#13712yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C2×C8—24no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—42no—
385C2×C8—48no—
386C2×C8—42no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—48no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4C2×C22⋊C412yes—
420C4⋊C4C2×C22⋊C412yes—
421C4⋊C4#13514yes—
422C22⋊C4—24no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—24no—
426C22⋊C4—48no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—84no—
430C42—22no—
431C42—22no—
432C42—24no—
433C42—24no—
434C42—48no—
435C42—42no—
436C42—42no—
437C42—44no—
438C42—42no—
439C42—42no—
440C42—44no—
441C42—84no—
442C42C2×C22⋊C412yes—
443C42C4×D412yes—
444C42C4×D412yes—
445C42#12112yes—
446C42#12112yes—
447C42#12312yes—
448C42C23×C411yescharacteristic
449C42C22×D411yescharacteristic
450C42C2×#4111yescharacteristic
451C42C2×#4111yescharacteristic
452C23—22no—
453C23—22no—
454C23—22no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—42no—
459C23#38612yes—
460C23#45212yes—
461C23#45212yes—
462C23C2×#12011yescharacteristic
463Q8—22no—
464Q8—22no—
465Q8—44no—
466Q8—41no—
467Q8—41no—
468Q8—42no—
469Q8—42no—
470Q8—82no—
471Q8—82no—
472D4—22no—
473D4—22no—
474D4—48no—
475D4—41no—
476D4—41no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—48no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—84no—
508C2×C4—88no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4#38612yes—
512C2×C4#38612yes—
513C2×C4#38612yes—
514C2×C4#41614yes—
515C2×C4#41914yes—
516C2×C4#45212yes—
517C2×C4#45212yes—
518C2×C4#48514yes—
519C2×C4#48514yes—
520C2×C4C22×C22⋊C411yescharacteristic
521C2×C4C2×C4×D411yescharacteristic
522C2×C4C2×C4×D411yescharacteristic
523C2×C4C2×#12111yescharacteristic
524C2×C4C2×#12111yescharacteristic
525C2×C4C2×#12311yescharacteristic
526C2×C4C2×#13511yescharacteristic
527C2×C4C2×#13511yescharacteristic
528C2×C4C2×#13512yes—
529C2×C4C2×#13711yescharacteristic
530C2×C4C2×#13711yescharacteristic
531C2×C4#57512yes—
532C8—44no—
533C8—42no—
534C8—42no—
535C8—82no—
536C22—24no—
537C22—24no—
538C22—22no—
539C22—22no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—82no—
544C22—82no—
545C22#187314yes—
546C22C2×#38611yescharacteristic
547C22C2×#45211yescharacteristic
548C22C2×#45211yescharacteristic
549C22#300212yes—
550C22#301812yes—
551C4—22no—
552C4—21no—
553C4—22no—
554C4—21no—
555C4—44no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—84no—
560C4—82no—
561C4—82no—
562C4—82no—
563C4#187314yes—
564C4C2×#41611yescharacteristic
565C4C2×#41911yescharacteristic
566C4C4×#13511yescharacteristic
567C4#294311yescharacteristic
568C4#303512yes—
569C4C2×#48511yescharacteristic
570C4C2×#48511yescharacteristic
571C2—24no—
572C2—82no—
573C2#1852112yes—
574C2#1852312yes—
575C2C2×#187311yescharacteristic
576C2#2092411yescharacteristic
577C2#2094211yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×#120
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21400C2semidirect
4. #21385C2semidirect
6. #8984C2semidirect
7. #8982C2semidirect
8. #8976C2semidirect
9. #8974C2semidirect
10. #8039C2semidirect
11. C2×#1279C2semidirect
15. #3076C4semidirect
92. #477C2×C4semidirect
94. #476C2×C4semidirect
96. #475C2×C4semidirect
98. #474C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30762
#341212
#214002
#213852
C2×#14762
#89842
#89822
#89762
#89742
#80392
C2×#12792