Finite Group Databasebeta

← #359988

Subgroups — #359988

Order 512.

1610 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
1C22×#433C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3C2×#1824C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5#14343C211yesmaximal, characteristic
6#11228C212yesmaximal, complemented
7#11227C212yesmaximal, complemented
8#10881C212yesmaximal, complemented
9#10880C212yesmaximal, complemented
10#10731C211yesmaximal, characteristic
11#10731C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#433—24no—
16C2×#433—24no—
17C2×#433—24no—
18C22×C4⋊C8C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C22×#97C2211yescharacteristic
23C22×C4×C8C2211yescharacteristic
24#2317—24no—
25#2317C418yescomplemented
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#385—21no—
29C2×#385—21no—
30C2×#384—21no—
31C2×#382—21no—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34#1824—24no—
35#1823—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44#1445—21no—
45#1445—21no—
46#1440—21no—
47#1440—21no—
48C4⋊#95C2212yes—
49C4⋊#95C2212yes—
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C2×#117—22no—
61C2×#117—24no—
62C2×#117—24no—
63C2×#117C2×C414yes—
64C2×#117C2×C414yes—
65C2×#117D414yes—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4C2×C414yes—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C22⋊C4—21no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89#433—232no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8D414yes—
101C2×C22⋊C8D414yes—
102C2×#97—22no—
103C2×#97—22no—
104C2×#97—22no—
105C2×C4×C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108#389—24no—
109#389—24no—
110#389C2×C418yes—
111#389C2×C418yes—
112#385—28no—
113#385—28no—
114#384—28no—
115#382—28no—
116#380—22no—
117#380—22no—
118#380—22no—
119#380—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—22no—
125C2×#95—22no—
126C2×#95—22no—
127C2×#95—22no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—24no—
133#340—24no—
134#340—24no—
135#340—24no—
136C25C2×D411yescharacteristic
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—82no—
149C23×C4—82no—
150C23×C4C22×C412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4#4112yes—
160C23×C4#4112yes—
161C23×C4C2411yescharacteristic
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8C2×D412yes—
179C22×C8C2×D412yes—
180C22×C8C2×D412yes—
181C22×C8C2×D412yes—
182#117—28no—
183#117—28no—
184#117—28no—
185#117—28no—
186#117—28no—
187#117—28no—
188#117—216no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—82no—
223C2×C4⋊C4—82no—
224C2×C4⋊C4—82no—
225C2×C4⋊C4—82no—
226C2×C4⋊C4C22⋊C414yes—
227C2×C4⋊C4C22⋊C414yes—
228C2×C4⋊C4C22×C414yes—
229C2×C4⋊C4C22×C414yes—
230C2×C4⋊C4#4114yes—
231C2×C4⋊C4#4114yes—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—82no—
241C2×C22⋊C4—82no—
242C2×C22⋊C4C22⋊C414yes—
243C2×C22⋊C4C22⋊C414yes—
244C2×C42—28no—
245C2×C42—24no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42C22×C412yes—
255C2×C42C22×C412yes—
256C2×C42C2×D412yes—
257C2×C42C2×D412yes—
258C4⋊C8—28no—
259C4⋊C8—28no—
260C22⋊C8—28no—
261C22⋊C8—28no—
262#97—28no—
263C4×C8—28no—
264#95—28no—
265#95—28no—
266#95—42no—
267#95—42no—
268#95—82no—
269#95—82no—
270#95—82no—
271#95—82no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24C2×C22⋊C412yes—
279C24#12112yes—
280C24#12112yes—
281C24C22×D411yescharacteristic
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—48no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—88no—
344C22×C4—84no—
345C22×C4—84no—
346C22×C4—88no—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C4×D414yes—
353C22×C4C4×D414yes—
354C22×C4#12014yes—
355C22×C4#12112yes—
356C22×C4#12112yes—
357C22×C4#12314yes—
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C2×#4112yes—
363C22×C4C2×#4112yes—
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C22×C42−1+412yes—
367C22×C42−1+412yes—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—28no—
392C4⋊C4—28no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—48no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—88no—
407C4⋊C4—88no—
408C4⋊C4#11718yes—
409C22⋊C4—216no—
410C22⋊C4—28no—
411C22⋊C4—28no—
412C22⋊C4—84no—
413C22⋊C4—84no—
414C42—216no—
415C42—22no—
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C23—22no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23#38614yes—
444C23C22×C22⋊C411yescharacteristic
445C23C2×#12111yescharacteristic
446C23C2×#12111yescharacteristic
447C23#52512yes—
448C23#53612yes—
449C23#57612yes—
450C23#57712yes—
451C23#57712yes—
452C23#57812yes—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—28no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—816no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4#38614yes—
513C2×C4C2×#11712yes—
514C2×C4C2×C4×D411yescharacteristic
515C2×C4C2×C4×D411yescharacteristic
516C2×C4#51912yes—
517C2×C4C2×#12011yescharacteristic
518C2×C4C2×#12311yescharacteristic
519C2×C4#52511yescharacteristic
520C2×C4#52511yescharacteristic
521C2×C4#53611yescharacteristic
522C2×C4#53611yescharacteristic
523C2×C4#54712yes—
524C2×C4#54712yes—
525C2×C4#54712yes—
526C2×C4#54712yes—
527C2×C4#55212yes—
528C2×C4#55212yes—
529C2×C4#55212yes—
530C2×C4#55212yes—
531C8—42no—
532C8—42no—
533C8—42no—
534C8—42no—
535C22—24no—
536C22—24no—
537C22—24no—
538C22—44no—
539C22—44no—
540C22—44no—
541C22—44no—
542C22—44no—
543C22—44no—
544C22#228011yescharacteristic
545C22C2×#38611yescharacteristic
546C22#231011yescharacteristic
547C22#241312yes—
548C22#241312yes—
549C22#289712yes—
550C22#289812yes—
551C22#305711yescharacteristic
552C22#305711yescharacteristic
553C22#305811yescharacteristic
554C22#305911yescharacteristic
555C22#306212yes—
556C22#306312yes—
557C22#311912yes—
558C22#312012yes—
559C22#312412yes—
560C22#312512yes—
561C22#324112yes—
562C22#324212yes—
563C22#324312yes—
564C22#324412yes—
565C4—24no—
566C4—48no—
567C4—42no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—84no—
572C4—162no—
573C4—162no—
574C4#234412yes—
575C4#234412yes—
576C4#240611yescharacteristic
577C4#240611yescharacteristic
578C4#248411yescharacteristic
579C4#248411yescharacteristic
580C2—44no—
581C2#1387011yescharacteristic
582C2#2074612yes—
583C2#2097411yescharacteristic
584C2#2097511yescharacteristic
585C2#2184312yes—
586C2#2461212yes—
587C2#2461412yes—
588C2#2461612yes—
589C2#2461812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11228C2semidirect
7. #11227C2semidirect
8. #10881C2semidirect
9. #10880C2semidirect
25. #2317C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4332
C2×#23172
C2×#18242
C2×#18232
#143432
#112282
#112272
#108812
#108802
#107312