Finite Group Databasebeta

← #369416

Subgroups — #369416

Order 512.

1744 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic
2C2×#2344C211yesmaximal, characteristic
3C2×#1996C211yesmaximal, characteristic
4C2×#1996C211yesmaximal, characteristic
5#13904C211yesmaximal, characteristic
6#11197C214yesmaximal
7#10846C211yesmaximal, characteristic
8#10736C214yesmaximal
9C4×#342C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C22×#117C2212yes—
12C23×C4⋊C4C2211yescharacteristic
13#2503—22no—
14#2503—22no—
15#2503—21no—
16#2503—22no—
17#2503—21no—
18C2×#400C414yescomplemented
19#2344—24no—
20#2344—24no—
21C2×#391C412yescomplemented
22C2×#389—21no—
23C2×#387—22no—
24C2×#382C412yescomplemented
25C2×#381C2212yes—
26C2×#381C2214yes—
27C2×#381C2214yes—
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C2×C4×C22⋊C4C2211yescharacteristic
34#1996—24no—
35#1996—24no—
36C2×#342C2212yes—
37C2×#342C2214yes—
38C2×#342C2212yes—
39C2×#342C2212yes—
40C2×#342C2212yes—
41C2×#342C2214yes—
42#1445—22no—
43C4⋊#95—22no—
44C23×Q8C2×C411yescharacteristic
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⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—28no—
58C2×#117—44no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2×C412yes—
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C4⋊C4C2×C412yes—
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4D412yes—
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4D412yes—
74C22×C4⋊C4D411yescharacteristic
75C22×C22⋊C4C2312yes—
76C22×C22⋊C4C2312yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2312yes—
79C22×C22⋊C4C2311yescharacteristic
80C22×C22⋊C4C2312yes—
81C22×C22⋊C4C2311yescharacteristic
82C22×C42—22no—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42C2×C412yes—
86#400—28no—
87#400—24no—
88#400—24no—
89#391—24no—
90#391—22no—
91#391—22no—
92#389—28no—
93#387—216no—
94#382—24no—
95#382—22no—
96#382—22no—
97#381—24no—
98#381—28no—
99#381—24no—
100#381—24no—
101#381—24no—
102#381—28no—
103#381—44no—
104#380—22no—
105#380—22no—
106#380—22no—
107#380D414yes—
108C4×C4⋊C4—28no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—21no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4D414yes—
117C2×#95—21no—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—28no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95C2×C412yes—
125C2×#95C2×C418yes—
126C2×#95C2×C418yes—
127#342—28no—
128#342—24no—
129#342—24no—
130#342—24no—
131#342—24no—
132#342—44no—
133C25C2411yescharacteristic
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—44no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—24no—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—44no—
153C23×C4C22×C414yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C412yes—
156C23×C4C22×C412yes—
157C23×C4C22×C414yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D414yes—
161C23×C4C2×D412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D412yes—
167C23×C4C2×D412yes—
168C23×C4#4112yes—
169C23×C4#4112yes—
170C22⋊Q8—42no—
171C22⋊Q8—42no—
172C22⋊Q8—42no—
173C22⋊Q8—42no—
174C22⋊Q8—42no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178#117—416no—
179#117—48no—
180#117—48no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4C22⋊C414yes—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C414yes—
213C2×C4⋊C4C22⋊C412yes—
214C2×C4⋊C4C22⋊C412yes—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4C2×D414yes—
245C2×C22⋊C4C2×D412yes—
246C2×C22⋊C4C2×D412yes—
247C2×C22⋊C4#4114yes—
248C2×C22⋊C4#4114yes—
249C2×C42—28no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—48no—
254C2×C42—44no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42—42no—
258#95—28no—
259#95—28no—
260#95—22no—
261#95—22no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—24no—
267#95—24no—
268#95—416no—
269#95—42no—
270#95—42no—
271C24—42no—
272C24—42no—
273C24C23×C411yescharacteristic
274C24C22×D411yescharacteristic
275C24C22×D411yescharacteristic
276C24C22×D411yescharacteristic
277C24C22×D411yescharacteristic
278C24C2×#4111yescharacteristic
279C24C2×#4111yescharacteristic
280C24C2×#4112yes—
281C24C2×#4112yes—
282C24H(5,𝔽2)12yes—
283C242−1+412yes—
284C2×Q8—44no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—44no—
288C2×Q8—84no—
289C2×Q8—84no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—28no—
296C22×C4—28no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—28no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—28no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—44no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—44no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—42no—
359C22×C4—44no—
360C22×C4C2×C22⋊C414yes—
361C22×C4C2×C22⋊C414yes—
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C4×D412yes—
367C22×C4C4×D412yes—
368C22×C4C4×D412yes—
369C22×C4C4×D412yes—
370C22×C4#12012yes—
371C22×C4#12012yes—
372C22×C4#12112yes—
373C22×C4#12112yes—
374C22×C4#12112yes—
375C22×C4#12112yes—
376C22×C4#12312yes—
377C22×C4#12312yes—
378C4⋊C4—24no—
379C4⋊C4—28no—
380C4⋊C4—24no—
381C4⋊C4—28no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—48no—
385C4⋊C4—416no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—44no—
389C4⋊C4—44no—
390C4⋊C4—84no—
391C22⋊C4—22no—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—22no—
395C22⋊C4—22no—
396C22⋊C4—28no—
397C22⋊C4—24no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—44no—
414C22⋊C4#12414yes—
415C42—44no—
416C42—48no—
417C42—416no—
418C42—44no—
419C42—84no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—44no—
433C23—44no—
434C23—41no—
435C23—41no—
436C23—44no—
437C23—44no—
438C23C22×C22⋊C411yescharacteristic
439C23C2×C4×D411yescharacteristic
440C23C2×C4×D411yescharacteristic
441C23C4×#4112yes—
442C23#52012yes—
443C23C2×#12011yescharacteristic
444C23C2×#12111yescharacteristic
445C23C2×#12111yescharacteristic
446C23C2×#12311yescharacteristic
447C23C2×#12411yescharacteristic
448C23#52912yes—
449C23#53211yescharacteristic
450C23#53511yescharacteristic
451C23#53611yescharacteristic
452C23#54212yes—
453C23#54612yes—
454C23#54612yes—
455C23#54612yes—
456C23#54712yes—
457C23#54712yes—
458C23#54812yes—
459C23#55612yes—
460C23#55912yes—
461Q8—84no—
462Q8—84no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—48no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—48no—
488C2×C4—48no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—48no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—82no—
514C2×C4—88no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—82no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—88no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4#38612yes—
529C2×C4#38612yes—
530C2×C4#38612yes—
531C2×C4#38612yes—
532C2×C4#41014yes—
533C22—24no—
534C22—24no—
535C22—24no—
536C22—24no—
537C22—24no—
538C22—24no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—41no—
552C22—42no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22C2×#38611yescharacteristic
562C22C4×#12411yescharacteristic
563C22#232611yescharacteristic
564C22#233912yes—
565C22#236712yes—
566C22#242712yes—
567C22#243211yescharacteristic
568C22#243511yescharacteristic
569C22#244712yes—
570C22#254411yescharacteristic
571C22#254711yescharacteristic
572C22C2×#41012yes—
573C22#288412yes—
574C4—22no—
575C4—22no—
576C4—22no—
577C4—44no—
578C4—44no—
579C4—44no—
580C4—84no—
581C4—84no—
582C4—82no—
583C4—84no—
584C4—82no—
585C4—82no—
586C4—162no—
587C4#189914yes—
588C2—21no—
589C2—22no—
590C2—21no—
591C2—21no—
592C2—21no—
593C2—22no—
594C2—22no—
595C2—21no—
596C2—21no—
597C2#1402711yescharacteristic
598C2C2×#189911yescharacteristic
599C2#2098611yescharacteristic
600C2#2247814yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, prime radical, cosocleC25C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
18. C2×#400C4semidirect
21. C2×#391C4semidirect
24. C2×#382C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#23442
C2×#19962
#139042
#111972
#108462
#107362
C4×#3422