Finite Group Databasebeta

← #355465

Subgroups — #355465

Order 512.

1928 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2C2×#1899C212yesmaximal
3C4×#381C211yesmaximal, characteristic, complemented
4#13756C211yesmaximal, characteristic, complemented
5#10960C212yesmaximal
6#10912C212yesmaximal, complemented
7#10801C212yesmaximal, complemented
8#10761C212yesmaximal
9#10736C212yesmaximal, complemented
10C22×#123C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C22×#117C2212yes—
13C23×C42C2211yescharacteristic
14C22×#99C2214yes—
15C4×#123—22no—
16C4×#123—22no—
17C4×#123—22no—
18C4×#123—22no—
19C2×#388C412yescomplemented
20C2×#387C412yescomplemented
21C2×#386C2211yescharacteristic
22C2×#385C2211yescharacteristic
23C2×#381C2212yes—
24C2×#381C2211yescharacteristic
25C2×C4×C4⋊C4C412yescomplemented
26C2×C4×C22⋊C4C412yescomplemented
27C2×C4×C22⋊C4C2212yes—
28C2×C4×C22⋊C4C2211yescharacteristic
29#1899C4116yescomplemented
30C2×#342C2214yes—
31C2×#342C2214yes—
32C2×#342C2214yes—
33C4⋊#95—22no—
34#1434—22no—
35C22⋊#95C2212yes—
36C22⋊#95C2212yes—
37C22⋊#95C2212yes—
38C4×#95—22no—
39C4×#95—22no—
40C23×D4C2311yescharacteristic
41C24×C4C2×C411yescharacteristic
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C2×#123—21no—
46C2×#123—22no—
47C2×#123—21no—
48C2×#123—24no—
49C2×#123C2×C418yes—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4—44no—
53C2×#117—24no—
54C2×#117—44no—
55C2×#117—44no—
56C2×#117C2×C418yescomplemented
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C412yes—
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C411yescharacteristic
63C22×C22⋊C4C2×C412yes—
64C22×C22⋊C4C2311yescharacteristic
65C22×C22⋊C4C2311yescharacteristic
66C22×C22⋊C4C2314yes—
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2311yescharacteristic
69C22×C42—22no—
70C22×C42—22no—
71C22×C42—44no—
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C412yes—
74C22×C42C2×C412yes—
75C22×C42C2×C412yes—
76C22×C42D412yes—
77C22×C42D412yes—
78C22×C42D412yes—
79C22×C42D412yes—
80C2×#99—44no—
81C2×#99—44no—
82C2×#99C2×C4116yes—
83#388—22no—
84#388—22no—
85#388—22no—
86#388—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#386—21no—
92#386—22no—
93#386—21no—
94#385—21no—
95#385—22no—
96#385—21no—
97#381—21no—
98#381—22no—
99#381—21no—
100#381—24no—
101#381C2×C418yescomplemented
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—44no—
113C2×#95—24no—
114C2×#95—24no—
115C2×#95—24no—
116C2×#95—24no—
117C2×#95—22no—
118C2×#95C2×C412yes—
119C2×#95C2×C412yes—
120C2×#95C2×C412yes—
121C2×#95C2×C412yes—
122C2×#95C2×C412yes—
123#342—44no—
124#342—44no—
125#342C2×C4116yescomplemented
126C25C22×C411yescharacteristic
127C25C2411yescharacteristic
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—22no—
131C22×D4—24no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4C22×C414yes—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—44no—
144C23×C4—44no—
145C23×C4—42no—
146C23×C4—44no—
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C412yes—
150C23×C4C22×C412yes—
151C23×C4C22×C414yes—
152C23×C4C22×C411yescharacteristic
153C23×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C412yes—
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D412yes—
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4#4114yes—
166#123—216no—
167#123—28no—
168#123—28no—
169C4×D4—48no—
170C4×D4—48no—
171#117—416no—
172#117—416no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4C4218yescomplemented
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4C4218yescomplemented
214C2×C22⋊C4C22×C418yes—
215C2×C22⋊C4C22×C418yes—
216C2×C22⋊C4C22×C414yes—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—28no—
221C2×C42—42no—
222C2×C42—44no—
223C2×C42—44no—
224C2×C42—44no—
225C2×C42—44no—
226C2×C42—44no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42—44no—
231C2×C42—44no—
232C2×C42—48no—
233C2×C42C22⋊C412yes—
234C2×C42C22⋊C412yes—
235C2×C42C22⋊C412yes—
236C2×C42C22⋊C412yes—
237C2×C42C22⋊C412yes—
238C2×C42C22⋊C412yes—
239C2×C42C22⋊C412yes—
240C2×C42C22⋊C412yes—
241#99—416no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—216no—
253#95—48no—
254#95—48no—
255#95—48no—
256#95—44no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24C2×C4212yes—
266C24C23×C411yescharacteristic
267C24C23×C412yes—
268C24C22×D411yescharacteristic
269C24C22×D411yescharacteristic
270C24C2×#4112yes—
271C24H(5,𝔽2)14yes—
272C24H(5,𝔽2)12yes—
273C242−1+412yes—
274C2×D4—24no—
275C2×D4—24no—
276C2×D4—28no—
277C2×D4—28no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—44no—
284C2×D4—44no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—28no—
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—24no—
299C22×C4—28no—
300C22×C4—28no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—416no—
334C22×C4—416no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—48no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—48no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—88no—
353C22×C4C2×C4212yes—
354C22×C4C2×C4214yes—
355C22×C4C2×C4214yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4#11714yes—
363C22×C4C4×D418yes—
364C22×C4C4×D414yes—
365C22×C4C4×D414yes—
366C4⋊C4—28no—
367C4⋊C4—28no—
368C4⋊C4—44no—
369C4⋊C4—416no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C22⋊C4—24no—
373C22⋊C4—24no—
374C22⋊C4—28no—
375C22⋊C4—28no—
376C22⋊C4—28no—
377C22⋊C4—28no—
378C22⋊C4—28no—
379C22⋊C4—28no—
380C22⋊C4—44no—
381C22⋊C4—44no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—42no—
387C22⋊C4—44no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C42—42no—
393C42—42no—
394C42—42no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42—42no—
400C42—44no—
401C42—416no—
402C42—44no—
403C42—44no—
404C42—88no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—24no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—41no—
422C23—41no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—82no—
427C23—84no—
428C23—82no—
429C23C22×C4211yescharacteristic
430C23C22×C22⋊C411yescharacteristic
431C23C2×#11711yescharacteristic
432C23C2×C4×D412yes—
433C23C2×C4×D412yes—
434C23#51812yes—
435C23#51814yes—
436C23#52012yes—
437C23#52014yes—
438C23#53412yes—
439C23#53412yes—
440C23#53511yescharacteristic
441C23#53611yescharacteristic
442C23#53712yes—
443C23#53814yes—
444C23#53914yes—
445D4—44no—
446D4—44no—
447D4—48no—
448C2×C4—28no—
449C2×C4—28no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—216no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—28no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—48no—
477C2×C4—48no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—416no—
488C2×C4—416no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—48no—
493C2×C4—48no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—88no—
499C2×C4—84no—
500C2×C4—88no—
501C2×C4—88no—
502C2×C4—84no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4—84no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4C4×C22⋊C414yes—
509C2×C4C4×C22⋊C414yes—
510C2×C4#41014yes—
511C2×C4#41014yes—
512C22—22no—
513C22—24no—
514C22—24no—
515C22—22no—
516C22—24no—
517C22—24no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—42no—
532C22—84no—
533C22—84no—
534C22—82no—
535C22—81no—
536C22—81no—
537C22C2×C4×C22⋊C411yescharacteristic
538C22#227512yes—
539C22#230811yescharacteristic
540C22#231111yescharacteristic
541C22#231812yes—
542C22#232112yes—
543C22#232212yes—
544C22#232514yes—
545C22C2×#41012yes—
546C22C2×#41012yes—
547C4—24no—
548C4—24no—
549C4—24no—
550C4—24no—
551C4—44no—
552C4—42no—
553C4—42no—
554C4—42no—
555C4—88no—
556C4—88no—
557C4—82no—
558C4—81no—
559C4—81no—
560C2—22no—
561C2—24no—
562C2—22no—
563C2—22no—
564C2—22no—
565C2—82no—
566C2#1378411yescharacteristic
567C2C4×#41012yes—
568C2#2174711yescharacteristic
569C2#2174811yescharacteristic
570C2#2187512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4×#381C2semidirect
4. #13756C2semidirect
6. #10912C2semidirect
7. #10801C2semidirect
9. #10736C2semidirect
19. C2×#388C4semidirect
20. C2×#387C4semidirect
25. C2×C4×C4⋊C4C4semidirect
26. C2×C4×C22⋊C4C4semidirect
29. #1899C4semidirect
56. C2×#117C2×C4semidirect
101. #381C2×C4semidirect
125. #342C2×C4semidirect
184. C2×C4⋊C4C42semidirect
213. C2×C22⋊C4C42semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1232
C2×#18992
C4×#3812
#137562
#109602
#109122
#108012
#107612
#107362