Finite Group Databasebeta

← #181184

Subgroups — #181184

Order 512.

1300 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60277C211yesmaximal, characteristic
2#60125C211yesmaximal, characteristic
3#34189C211yesmaximal, characteristic
4#7962C212yesmaximal, complemented
5#7961C212yesmaximal, complemented
6#7960C212yesmaximal, complemented
7#7959C212yesmaximal, complemented
8#7958C212yesmaximal, complemented
9#7956C212yesmaximal, complemented
10C2×C4×#41C2211yescharacteristic
11#2989—24no—
12#2983—21no—
13#2975—21no—
14#2973—21no—
15#2973—21no—
16#2965C418yescomplemented
17#2925—24no—
18#2925C418yescomplemented
19#2921—21no—
20#2921—21no—
21#2920C2212yes—
22#2920C2212yes—
23#2920C2212yes—
24#2920C2212yes—
25C2×#431C2212yes—
26C2×#423C2212yes—
27#2879—21no—
28#2873—21no—
29#2357—21no—
30#2357C412yes—
31#2356—21no—
32#2356C412yes—
33#2353—21no—
34#2353—21no—
35#2353C412yes—
36#2353C412yes—
37#2314—21no—
38#2314—21no—
39#2311—21no—
40#2285C2211yescharacteristic
41#2283C2211yescharacteristic
42#2283C2211yescharacteristic
43#2283C2211yescharacteristic
44#2283C2211yescharacteristic
45C2×#382C2211yescharacteristic
46#2274—21no—
47#1286C2214yes—
48#1286C2214yes—
49#1286C2214yes—
50#1286C2214yes—
51C22×#41D411yescharacteristic
52C22×M16—22no—
53C4×#41C2×C414yes—
54C4×#41C2×C414yes—
55C4×#41D414yes—
56C4×#41Q814yes—
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×#117—21no—
66C2×#117D411yescharacteristic
67C2×#117D411yescharacteristic
68C2×#117D411yescharacteristic
69C22×C4⋊C4—21no—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73#446—28no—
74#436—28no—
75#435—28no—
76#435—28no—
77#433—28no—
78#433—28no—
79#432—22no—
80#432—22no—
81#432—22no—
82#432—22no—
83#432—22no—
84#432—22no—
85#432C2×C414yes—
86#432C2×C414yescomplemented
87#431—22no—
88#431—22no—
89#431—22no—
90#431—22no—
91#431C2×C414yes—
92#431C2×C414yes—
93#431C2×C414yescomplemented
94#431C2×C414yescomplemented
95#431C2×C414yescomplemented
96#431C2314yes—
97#423—22no—
98#423—22no—
99#423—22no—
100#423—22no—
101#423C2×C414yescomplemented
102#423C2314yes—
103C2×C4⋊C8—22no—
104C2×C4⋊C8—22no—
105C2×C4⋊C8C2×C414yescomplemented
106#408—28no—
107#407—22no—
108#407—22no—
109#407—22no—
110#407—22no—
111C2×C22⋊C8—22no—
112#405—28no—
113C4×M16—22no—
114C4×M16—22no—
115C4×M16C2×C414yes—
116C4×M16C2×C414yescomplemented
117C2×C4×C8C2×C414yescomplemented
118#391—22no—
119#390—22no—
120#388—22no—
121#388—22no—
122#388—22no—
123#388—22no—
124#387—22no—
125#387—22no—
126#387—22no—
127#387—22no—
128#386—22no—
129#386—22no—
130#384—21no—
131#383—21no—
132#383—21no—
133#383C2×C412yes—
134#383C2×C412yes—
135#382—21no—
136#382—21no—
137#382C2×C412yes—
138#382C2311yescharacteristic
139#382C2311yescharacteristic
140#382C2311yescharacteristic
141#382C2311yescharacteristic
142#381—21no—
143#381—21no—
144#381C2×C412yes—
145#381C2×C412yes—
146#380—21no—
147#380—21no—
148#380—21no—
149#380—21no—
150#380—21no—
151#380C2×C412yes—
152C4×C4⋊C4—22no—
153C4×C4⋊C4—22no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4C2×C412yes—
160C4×C22⋊C4—21no—
161C4×C22⋊C4—21no—
162C4×C22⋊C4—22no—
163C4×C22⋊C4—22no—
164C4×C22⋊C4C2×C412yes—
165C4×C22⋊C4C2×C412yes—
166#376—21no—
167#376—21no—
168#376C2×C412yes—
169#376C2×C412yes—
170C2×#95—21no—
171C2×#95C2×C412yes—
172C2×#41—42no—
173C2×#41—42no—
174C2×#41C22⋊C414yes—
175C2×#41C4⋊C414yes—
176C22×Q8C22⋊C411yescharacteristic
177C22×D4C22⋊C411yescharacteristic
178C22×D4C22⋊C411yescharacteristic
179C22×D4C22⋊C411yescharacteristic
180C23×C4—22no—
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C22×C411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C2×M16—24no—
191C2×M16—24no—
192C2×M16—24no—
193C2×M16—24no—
194C2×M16—44no—
195C2×M16—44no—
196C22×C8—24no—
197C22×C8—24no—
198C4×Q8C4214yes—
199C4×Q8C22⋊C414yes—
200C4×Q8C4⋊C414yes—
201C4×Q8C4⋊C414yes—
202C4×D4C4214yes—
203C4×D4C4214yes—
204C4×D4C4214yes—
205C4×D4C22⋊C414yes—
206C4×D4C22⋊C414yes—
207C4×D4C22⋊C414yes—
208C4×D4C4⋊C414yes—
209C4×D4C4⋊C414yes—
210C4×D4C4⋊C414yes—
211C4×D4C4⋊C414yes—
212C4×D4C4⋊C414yes—
213C4×D4C4⋊C414yes—
214#117—44no—
215#117C22⋊C414yes—
216#117C22⋊C414yes—
217#117C22⋊C414yes—
218#117C4⋊C414yes—
219#117C4⋊C414yes—
220#117C4⋊C414yes—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4C22⋊C411yescharacteristic
241C2×C22⋊C4C22⋊C411yescharacteristic
242C2×C22⋊C4C22⋊C411yescharacteristic
243C2×C22⋊C4C22⋊C411yescharacteristic
244C2×C22⋊C4C22⋊C411yescharacteristic
245C2×C22⋊C4C22⋊C411yescharacteristic
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C412yes—
255C2×C42C22×C412yes—
256C2×C42C22×C412yes—
257C2×C42C22×C412yes—
258C2×C42C22×C412yes—
259C2×C42C22×C412yes—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D412yes—
267C2×C42C2×D412yes—
268C2×C42C2×D412yes—
269C2×C42C2×Q812yes—
270C2×C42C2×Q812yes—
271C2×C42C2×Q812yes—
272C2×C42C2411yescharacteristic
273C4⋊C8—24no—
274C4⋊C8—24no—
275C4⋊C8—24no—
276C4⋊C8—24no—
277C4⋊C8—24no—
278C4⋊C8—24no—
279C4⋊C8—24no—
280C4⋊C8C22×C418yes—
281C22⋊C8—24no—
282C22⋊C8—24no—
283C22⋊C8—24no—
284C22⋊C8—24no—
285C22⋊C8—24no—
286C22⋊C8—24no—
287C22⋊C8—24no—
288C22⋊C8—24no—
289#97—24no—
290#97—24no—
291#97—24no—
292#97—24no—
293C4×C8—24no—
294C4×C8—24no—
295C4×C8—24no—
296C4×C8C22×C418yes—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—22no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95C22×C412yes—
312#95C22×C412yes—
313#95C22×C412yes—
314#95C22×C412yes—
315C24C2×C22⋊C411yescharacteristic
316C24C2×C22⋊C411yescharacteristic
317C24C2×C22⋊C411yescharacteristic
318#41—44no—
319#41—44no—
320#41—48no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8#9514yes—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4#9514yes—
331C2×D4#9514yes—
332C2×D4#9514yes—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—44no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4—81no—
347C22×C4—81no—
348C22×C4—81no—
349C22×C4—81no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4C2×C4212yes—
353C22×C4C2×C4212yes—
354C22×C4C2×C4212yes—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C4⋊C412yes—
371C22×C4C2×C4⋊C412yes—
372C22×C4C2×C4⋊C412yes—
373C22×C4C2×C4⋊C412yes—
374C22×C4C2×C4⋊C412yes—
375C22×C4C2×C4⋊C412yes—
376C22×C4C2×C4⋊C412yes—
377C22×C4C2×C4⋊C412yes—
378C22×C4C2×C4⋊C412yes—
379C22×C4C23×C411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382M16—48no—
383M16—44no—
384M16—44no—
385M16—84no—
386M16—84no—
387C2×C8—28no—
388C2×C8—28no—
389C2×C8—48no—
390C2×C8—84no—
391C4⋊C4—44no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4#9514yes—
399C4⋊C4#9514yes—
400C4⋊C4#9514yes—
401C4⋊C4#9514yes—
402C4⋊C4#9514yes—
403C4⋊C4#9514yes—
404C22⋊C4—44no—
405C22⋊C4—82no—
406C22⋊C4—82no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C22⋊C4#9514yes—
410C22⋊C4#9514yes—
411C22⋊C4#9514yes—
412C22⋊C4#9514yes—
413C22⋊C4#9514yes—
414C22⋊C4#9514yes—
415C42—44no—
416C42—82no—
417C42—82no—
418C42—82no—
419C42—82no—
420C42C2×C4212yes—
421C42C2×C4212yes—
422C42C2×C4212yes—
423C42C2×C22⋊C412yes—
424C42C2×C22⋊C412yes—
425C42C2×C22⋊C412yes—
426C42C2×C22⋊C412yes—
427C42C2×C22⋊C412yes—
428C42C2×C22⋊C412yes—
429C42C2×C4⋊C412yes—
430C42C2×C4⋊C412yes—
431C42C2×C4⋊C412yes—
432C42C2×C4⋊C412yes—
433C42C2×C4⋊C412yes—
434C42C2×C4⋊C412yes—
435C42C2×C4⋊C412yes—
436C42C2×C4⋊C412yes—
437C42C2×C4⋊C412yes—
438C42C23×C411yescharacteristic
439C42C23×C411yescharacteristic
440C42C22×D411yescharacteristic
441C42C22×Q811yescharacteristic
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23C2×#9512yes—
455C23C2×#9512yes—
456C23C2×#9512yes—
457C23C22×C22⋊C411yescharacteristic
458Q8—44no—
459D4—44no—
460D4—44no—
461D4—44no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
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—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—164no—
508C2×C4C2×#9512yes—
509C2×C4C2×#9512yes—
510C2×C4C2×#9512yes—
511C2×C4C2×#9512yes—
512C2×C4C2×#9512yes—
513C2×C4C2×#9512yes—
514C2×C4C2×#9512yes—
515C2×C4C2×#9512yes—
516C2×C4C2×#9512yes—
517C2×C4C22×C4211yescharacteristic
518C2×C4C22×C22⋊C411yescharacteristic
519C2×C4C22×C22⋊C411yescharacteristic
520C2×C4C22×C4⋊C411yescharacteristic
521C2×C4C22×C4⋊C411yescharacteristic
522C2×C4C22×C4⋊C411yescharacteristic
523C2×C4#57512yes—
524C2×C4#57512yes—
525C8—48no—
526C8—48no—
527C22—21no—
528C22—21no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—42no—
539C22—42no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22C22×#9511yescharacteristic
545C22#289611yescharacteristic
546C22#289611yescharacteristic
547C4—21no—
548C4—21no—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—22no—
553C4—22no—
554C4—22no—
555C4—21no—
556C4—21no—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—82no—
561C4—82no—
562C4—82no—
563C4—82no—
564C4#289612yes—
565C4#291112yes—
566C2—21no—
567C2—21no—
568C2—42no—
569C2—42no—
570C2—42no—
571C2#2033811yescharacteristic
572C2#2034711yescharacteristic
573C2#2036011yescharacteristic
subgroupG/H
center, socleC22C22×#95
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7962C2semidirect
5. #7961C2semidirect
6. #7960C2semidirect
7. #7959C2semidirect
8. #7958C2semidirect
9. #7956C2semidirect
16. #2965C4semidirect
18. #2925C4semidirect
86. #432C2×C4semidirect
93. #431C2×C4semidirect
94. #431C2×C4semidirect
95. #431C2×C4semidirect
101. #423C2×C4semidirect
105. C2×C4⋊C8C2×C4semidirect
116. C4×M16C2×C4semidirect
117. C2×C4×C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602772
#601252
#341892
#79622
#79612
#79602
#79592
#79582
#79562