Finite Group Databasebeta

← #364131

Subgroups — #364131

Order 512.

1868 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2C2×#1899C212yesmaximal
3#14685C211yesmaximal, characteristic, complemented
4#13756C211yesmaximal, characteristic, complemented
5#11203C212yesmaximal, complemented
6#11043C212yesmaximal
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×#380C2212yes—
26C2×C4×C4⋊C4C412yescomplemented
27C2×C4×C22⋊C4C412yescomplemented
28C2×C4×C22⋊C4C2211yescharacteristic
29#1899—28no—
30C2×#342C2214yes—
31C2×#342C2214yes—
32C2×#342C2214yes—
33C4⋊#95—22no—
34C4⋊#95—22no—
35#1434—22no—
36C22⋊#95C2212yes—
37C22⋊#95C2212yes—
38C22⋊#95C2212yes—
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—22no—
49C2×#123—22no—
50C2×#123D414yes—
51C2×#123Q814yes—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4—44no—
55C2×#117—28no—
56C2×#117—24no—
57C2×#117—24no—
58C2×#117—44no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C412yes—
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C411yescharacteristic
66C22×C22⋊C4C2×C412yes—
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2311yescharacteristic
69C22×C22⋊C4C2311yescharacteristic
70C22×C22⋊C4C2314yes—
71C22×C22⋊C4C2311yescharacteristic
72C22×C42—22no—
73C22×C42—44no—
74C22×C42C2×C411yescharacteristic
75C22×C42C2×C412yes—
76C22×C42C2×C412yes—
77C22×C42C2×C412yes—
78C22×C42D412yes—
79C22×C42D412yes—
80C22×C42D412yes—
81C22×C42D412yes—
82C2×#99—28no—
83C2×#99—28no—
84C2×#99—44no—
85#388—22no—
86#388—22no—
87#388—22no—
88#388—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#386—21no—
94#386—22no—
95#386—21no—
96#385—21no—
97#385—22no—
98#385—21no—
99#381—21no—
100#381—22no—
101#381—21no—
102#381—24no—
103#381—24no—
104#380—44no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C2×#95—24no—
116C2×#95—24no—
117C2×#95—24no—
118C2×#95—22no—
119C2×#95—24no—
120C2×#95C2×C412yes—
121C2×#95C2×C412yes—
122C2×#95C2×C412yes—
123C2×#95C2×C412yes—
124C2×#95C2×C412yes—
125#342—28no—
126#342—28no—
127#342—44no—
128C25C22×C411yescharacteristic
129C25C2411yescharacteristic
130C22×D4—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—24no—
135C22×D4—24no—
136C22×D4C2×D412yes—
137C22×D4C2×Q812yes—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—44no—
146C23×C4—44no—
147C23×C4—42no—
148C23×C4—44no—
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C412yes—
152C23×C4C22×C412yes—
153C23×C4C22×C411yescharacteristic
154C23×C4C22×C412yes—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C412yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×Q812yes—
168C23×C4#4112yes—
169C23×C4#4112yes—
170#123—28no—
171#123—24no—
172#123—24no—
173#123—28no—
174#123—24no—
175#123—24no—
176C4×D4—48no—
177C4×D4—48no—
178#117—416no—
179#117—416no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—48no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4C4⋊C418yescomplemented
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4C4⋊C418yescomplemented
224C2×C22⋊C4C2×D412yes—
225C2×C22⋊C4C2×Q812yes—
226C2×C22⋊C4#4118yes—
227C2×C42—28no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—42no—
232C2×C42—44no—
233C2×C42—44no—
234C2×C42—44no—
235C2×C42—44no—
236C2×C42—44no—
237C2×C42—44no—
238C2×C42—44no—
239C2×C42—44no—
240C2×C42—44no—
241C2×C42C22⋊C412yes—
242C2×C42C22⋊C412yes—
243C2×C42C22⋊C412yes—
244C2×C42C22⋊C412yes—
245C2×C42C22⋊C412yes—
246C2×C42C22⋊C412yes—
247C2×C42C22⋊C412yes—
248C2×C42C22⋊C412yes—
249#99—416no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—48no—
261#95—48no—
262#95—48no—
263#95—44no—
264#95—48no—
265C24—21no—
266C24—21no—
267C24—22no—
268C24—22no—
269C24—42no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24C2×C4⋊C412yes—
274C24C23×C411yescharacteristic
275C24C22×D411yescharacteristic
276C24C22×D411yescharacteristic
277C24C22×D411yescharacteristic
278C24C22×Q811yescharacteristic
279C24C2×#4111yescharacteristic
280C24C2×#4111yescharacteristic
281C24C2×#4114yes—
282C24H(5,𝔽2)14yes—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—28no—
288C2×D4—28no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—44no—
294C2×D4—44no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—28no—
306C22×C4—28no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—28no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—416no—
341C22×C4—416no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—88no—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C412yes—
363C22×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C412yes—
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C412yes—
367C22×C4C2×C4⋊C412yes—
368C22×C4C2×C4⋊C414yes—
369C22×C4C2×C4⋊C414yes—
370C22×C4#11714yes—
371C22×C4#12114yes—
372C22×C4#12112yes—
373C22×C4#12112yes—
374C22×C4C22⋊Q814yes—
375C22×C4C22⋊Q812yes—
376C22×C4C22⋊Q812yes—
377C4⋊C4—28no—
378C4⋊C4—28no—
379C4⋊C4—44no—
380C4⋊C4—416no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—88no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—28no—
389C22⋊C4—28no—
390C22⋊C4—28no—
391C22⋊C4—28no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—44no—
399C22⋊C4—44no—
400C22⋊C4—44no—
401C22⋊C4—44no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—42no—
413C42—44no—
414C42—416no—
415C42—44no—
416C42—44no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—24no—
425C23—44no—
426C23—44no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—48no—
432C23—44no—
433C23—44no—
434C23—41no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—41no—
439C23—82no—
440C23—84no—
441C23—82no—
442C23C22×C22⋊C411yescharacteristic
443C23C22×C4⋊C411yescharacteristic
444C23C2×#11711yescharacteristic
445C23C4×#4112yes—
446C23#51812yes—
447C23C2×#12111yescharacteristic
448C23C2×#12111yescharacteristic
449C23C2×C22⋊Q811yescharacteristic
450C23C2×C22⋊Q811yescharacteristic
451C23#52512yes—
452C23#53212yes—
453C23#53412yes—
454C23#53412yes—
455C23#54614yes—
456C23#55114yes—
457C23#55314yes—
458C23#55414yes—
459D4—44no—
460D4—44no—
461D4—44no—
462D4—44no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—28no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—48no—
493C2×C4—48no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—44no—
503C2×C4—416no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—88no—
513C2×C4—88no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—84no—
520C2×C4—88no—
521C2×C4—88no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4#38014yes—
525C2×C4#38014yes—
526C2×C4#41014yes—
527C2×C4#41014yes—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—24no—
535C22—24no—
536C22—24no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—44no—
544C22—44no—
545C22—44no—
546C22—44no—
547C22—42no—
548C22—44no—
549C22—84no—
550C22—84no—
551C22—82no—
552C22—81no—
553C22—81no—
554C22C2×#38011yescharacteristic
555C22#229711yescharacteristic
556C22#230811yescharacteristic
557C22#234912yes—
558C22#235312yes—
559C22#253412yes—
560C22#254312yes—
561C22#256412yes—
562C22#257912yes—
563C22C2×#41012yes—
564C22C2×#41012yes—
565C4—24no—
566C4—24no—
567C4—24no—
568C4—24no—
569C4—42no—
570C4—44no—
571C4—42no—
572C4—42no—
573C4—88no—
574C4—82no—
575C4—81no—
576C4—81no—
577C4—164no—
578C2—22no—
579C2—22no—
580C2—22no—
581C2—22no—
582C2—22no—
583C2—22no—
584C2—82no—
585C2#1396011yescharacteristic
586C2#2063212yes—
587C2#2142112yes—
588C2#2174711yescharacteristic
589C2#2174811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ 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. #14685C2semidirect
4. #13756C2semidirect
5. #11203C2semidirect
7. #10801C2semidirect
9. #10736C2semidirect
19. C2×#388C4semidirect
20. C2×#387C4semidirect
26. C2×C4×C4⋊C4C4semidirect
27. C2×C4×C22⋊C4C4semidirect
193. C2×C4⋊C4C4⋊C4semidirect
223. C2×C22⋊C4C4⋊C4semidirect

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
#146852
#137562
#112032
#110432
#108012
#107612
#107362