Finite Group Databasebeta

← #369250

Subgroups — #369250

Order 512.

1666 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic
2#34212C211yesmaximal, characteristic
3C2×#1996C211yesmaximal, characteristic
4#13904C211yesmaximal, characteristic
5#13904C211yesmaximal, characteristic
6#11070C214yesmaximal
7#10936C214yesmaximal
8#10854C211yesmaximal, characteristic
9C4×#342C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12#2503—24no—
13#2503C418yescomplemented
14C2×#400—22no—
15#2368—22no—
16#2360—22no—
17#2345—22no—
18#2345—21no—
19Q8×C22⋊C4—21no—
20#2329—22no—
21#2315C2212yes—
22C4×C22⋊Q8—22no—
23C2×#391—21no—
24C2×#382—21no—
25C2×#381C2212yes—
26C2×#381C2214yes—
27C2×#381C2214yes—
28C2×#381C2212yes—
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31#2273C2212yes—
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35#1996—24no—
36C2×#342C2212yes—
37C2×#342C2214yes—
38C2×#342C2212yes—
39C2×#342C2214yes—
40#1445—22no—
41#1445—22no—
42C4⋊#95—22no—
43C4⋊#95—22no—
44C23×Q8D411yescharacteristic
45C24×C4C2312yes—
46C24×C4C2311yescharacteristic
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8C2×C414yes—
54C2×C22⋊Q8D414yes—
55C2×C4×Q8—22no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2×C414yes—
67C22×C4⋊C4D412yes—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D412yes—
70C22×C22⋊C4C2312yes—
71C22×C22⋊C4C2311yescharacteristic
72C22×C22⋊C4C2312yes—
73C22×C22⋊C4C2312yes—
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2312yes—
76C22×C22⋊C4C2311yescharacteristic
77C22×C42—21no—
78C22×C42—21no—
79C22×C42D412yes—
80#400—216no—
81#391—24no—
82#391—22no—
83#391—28no—
84#389—24no—
85#389—22no—
86#387—24no—
87#387—24no—
88#387—24no—
89#385—24no—
90#385—24no—
91#382—28no—
92#381—28no—
93#381—28no—
94#381—28no—
95#381—44no—
96#381—44no—
97#381C2×C4116yes—
98#380—22no—
99#380—22no—
100#380—21no—
101#380—21no—
102#380—21no—
103#380—21no—
104#380—22no—
105#380—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—24no—
108C4×C22⋊C4—24no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—24no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4C2×C414yes—
120C2×#95—21no—
121C2×#95—24no—
122C2×#95—24no—
123C2×#95—28no—
124C2×#95—24no—
125C2×#95—24no—
126C2×#95—21no—
127C2×#95—21no—
128C2×#95D412yes—
129C2×#95D412yes—
130#342—28no—
131#342—24no—
132#342—24no—
133#342—44no—
134C25C2411yescharacteristic
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—22no—
138C22×Q8—44no—
139C22×Q8C22⋊C414yes—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—24no—
149C23×C4—24no—
150C23×C4—24no—
151C23×C4—24no—
152C23×C4—22no—
153C23×C4—44no—
154C23×C4C22×C414yes—
155C23×C4C22×C412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D414yes—
160C23×C4C2×D412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4#4112yes—
169C23×C4#4112yes—
170C22⋊Q8—24no—
171C22⋊Q8—24no—
172C22⋊Q8—24no—
173C22⋊Q8—24no—
174C22⋊Q8—24no—
175C22⋊Q8—24no—
176C22⋊Q8—24no—
177C22⋊Q8—24no—
178C4×Q8—84no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4C22⋊C414yes—
209C2×C4⋊C4C22⋊C418yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4C22×C414yes—
240C2×C22⋊C4C22×C414yes—
241C2×C22⋊C4C2×D414yes—
242C2×C22⋊C4#4114yes—
243C2×C22⋊C4#4114yes—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—44no—
250C2×C42—44no—
251C2×C42—44no—
252C2×C42—42no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—24no—
258#95—216no—
259#95—216no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—416no—
265#95—48no—
266#95—42no—
267#95—42no—
268#95—48no—
269C24—42no—
270C24—42no—
271C24C23×C411yescharacteristic
272C24C22×D411yescharacteristic
273C24C22×D411yescharacteristic
274C24C22×D411yescharacteristic
275C24C22×D411yescharacteristic
276C24C2×#4111yescharacteristic
277C24C2×#4111yescharacteristic
278C24C2×#4112yes—
279C24C2×#4112yes—
280C24H(5,𝔽2)12yes—
281C242−1+412yes—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—48no—
285C2×Q8—44no—
286C2×Q8—82no—
287C2×Q8—84no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—28no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—44no—
320C22×C4—42no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—42no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C414yes—
357C22×C4C2×C22⋊C414yes—
358C22×C4C2×C22⋊C412yes—
359C22×C4C4×D412yes—
360C22×C4C4×D412yes—
361C22×C4C4×D412yes—
362C22×C4C4×D412yes—
363C22×C4#12011yescharacteristic
364C22×C4#12011yescharacteristic
365C22×C4#12011yescharacteristic
366C22×C4#12011yescharacteristic
367C22×C4#12112yes—
368C22×C4#12112yes—
369C22×C4#12112yes—
370C22×C4#12112yes—
371C22×C4#12312yes—
372C22×C4#12312yes—
373C22×C4#12414yes—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—28no—
377C4⋊C4—28no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—82no—
384C4⋊C4—84no—
385C4⋊C4—84no—
386C4⋊C4—84no—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—24no—
396C22⋊C4—24no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—44no—
408C42—48no—
409C42—44no—
410C42—82no—
411C42—84no—
412C42—84no—
413C42—84no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—44no—
427C23—41no—
428C23—41no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23C22×C22⋊C411yescharacteristic
433C23C2×C4×D411yescharacteristic
434C23C2×C4×D411yescharacteristic
435C23C4×#4112yes—
436C23#52012yes—
437C23C2×#12011yescharacteristic
438C23C2×#12111yescharacteristic
439C23C2×#12111yescharacteristic
440C23C2×#12311yescharacteristic
441C23C2×#12411yescharacteristic
442C23#52912yes—
443C23#53211yescharacteristic
444C23#53511yescharacteristic
445C23#53611yescharacteristic
446C23#54212yes—
447C23#54612yes—
448C23#54812yes—
449C23#54812yes—
450C23#54812yes—
451C23#55012yes—
452C23#55012yes—
453C23#55612yes—
454C23#55912yes—
455Q8—48no—
456Q8—84no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
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—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—48no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—48no—
489C2×C4—48no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4—82no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—84no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—88no—
515C2×C4—82no—
516C2×C4—84no—
517C2×C4—82no—
518C2×C4—84no—
519C2×C4—82no—
520C2×C4#38612yes—
521C2×C4#38612yes—
522C2×C4#38612yes—
523C2×C4#38612yes—
524C2×C4#45814yes—
525C22—22no—
526C22—22no—
527C22—24no—
528C22—24no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—22no—
542C22—24no—
543C22—22no—
544C22—24no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22C2×#38611yescharacteristic
554C22C4×#12411yescharacteristic
555C22#232611yescharacteristic
556C22#234112yes—
557C22#236712yes—
558C22#242612yes—
559C22#243111yescharacteristic
560C22#243611yescharacteristic
561C22#244812yes—
562C22#254511yescharacteristic
563C22#254811yescharacteristic
564C22C2×#45812yes—
565C22#302812yes—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—44no—
570C4—44no—
571C4—44no—
572C4—84no—
573C4—84no—
574C4—84no—
575C4—82no—
576C4—84no—
577C4—162no—
578C4#189014yes—
579C2—22no—
580C2—22no—
581C2—21no—
582C2—21no—
583C2—22no—
584C2—22no—
585C2—21no—
586C2—21no—
587C2#1402611yescharacteristic
588C2C2×#189011yescharacteristic
589C2#2096211yescharacteristic
590C2#2558612yes—
591C2#2594812yes—
subgroupG/H
center, socleC23C2×#120
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2503C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
#342122
C2×#19962
#139042
#110702
#109362
#108542
C4×#3422