Finite Group Databasebeta

← #374537

Subgroups — #374537

Order 512.

1500 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#433C211yesmaximal, characteristic
2C2×#2628C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic, complemented
4C2×#1987C211yesmaximal, characteristic, complemented
5#13777C211yesmaximal, characteristic
6#11200C212yesmaximal
7#11199C212yesmaximal
8#10881C212yesmaximal, complemented
9#10880C212yesmaximal, complemented
10#10850C211yesmaximal, characteristic, complemented
11#10850C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C2×#433—24no—
15C2×#433—24no—
16C2×#433—24no—
17C22×C4⋊C8C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20C22×C22⋊C8C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2628—24no—
24#2628C418yescomplemented
25C2×#400—21no—
26C2×#399—21no—
27C2×#395—21no—
28C2×#393—21no—
29C2×#389C2212yes—
30C2×#382C2211yescharacteristic
31C2×#380C2212yes—
32C2×#380C2212yes—
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1988—24no—
36#1987—24no—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45#1437—21no—
46#1437—21no—
47#1436—21no—
48#1436—21no—
49C4⋊#95C2212yes—
50C23×Q8—21no—
51C24×C4C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8C2×C414yes—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117C2×C414yes—
67C2×#117D414yes—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C22⋊C4—22no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4D411yescharacteristic
79C22×C22⋊C4D411yescharacteristic
80C22×C42—21no—
81C22×C42—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#433—232no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8D414yes—
90C2×C4⋊C8D414yes—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×#97—22no—
98C2×#97—22no—
99C2×#97—22no—
100C2×C4×C8—22no—
101C2×C4×C8—22no—
102C2×C4×C8—22no—
103#400—28no—
104#399—28no—
105#395—28no—
106#393—28no—
107#389—24no—
108#389C2×C418yes—
109#382—22no—
110#382C2×C414yes—
111#380—24no—
112#380—24no—
113#380—24no—
114#380C2×C418yes—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95—21no—
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
137C22×Q8—24no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8—82no—
141C22×Q8—82no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4C22×C412yes—
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
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—
182C4⋊Q8—28no—
183C4⋊Q8—28no—
184C4⋊Q8—44no—
185C4⋊Q8—44no—
186#117—28no—
187#117—28no—
188#117—28no—
189#117—28no—
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—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
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—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4C22⋊C414yes—
221C2×C4⋊C4C22⋊C414yes—
222C2×C4⋊C4C22×C414yes—
223C2×C4⋊C4C22×C414yes—
224C2×C4⋊C4#4114yes—
225C2×C4⋊C4#4114yes—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4C22⋊C414yes—
237C2×C22⋊C4C22⋊C414yes—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—42no—
243C2×C42—42no—
244C2×C42—82no—
245C2×C42—82no—
246C2×C42C22×C412yes—
247C2×C42C22×C412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C2×C42#4112yes—
253C2×C42#4112yes—
254C4⋊C8—28no—
255C4⋊C8—28no—
256C22⋊C8—28no—
257C22⋊C8—28no—
258#97—28no—
259C4×C8—28no—
260#95—28no—
261#95—28no—
262#95—24no—
263#95—24no—
264#95—28no—
265#95—42no—
266#95—42no—
267#95—82no—
268#95—82no—
269C24—21no—
270C24—21no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24C2×C22⋊C412yes—
278C24C22×D411yescharacteristic
279C2×Q8—48no—
280C2×Q8—88no—
281C2×Q8—84no—
282C2×Q8—84no—
283C2×Q8—88no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
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—42no—
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—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C412yes—
351C22×C4#12112yes—
352C22×C4#12112yes—
353C22×C4#12112yes—
354C22×C4#12112yes—
355C22×C4C23×C411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4112yes—
362C22×C4C2×#4112yes—
363C22×C42−1+412yes—
364C22×C42−1+412yes—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C4⋊C4—28no—
386C4⋊C4—28no—
387C4⋊C4—24no—
388C4⋊C4—24no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—48no—
404C4⋊C4—88no—
405C22⋊C4—28no—
406C22⋊C4—28no—
407C22⋊C4—88no—
408C22⋊C4—88no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—84no—
414C42—84no—
415C42C4×D414yes—
416C42C4×D414yes—
417C42#12014yes—
418C42#12314yes—
419C23—22no—
420C23—24no—
421C23—24no—
422C23—22no—
423C23—22no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—42no—
439C23—42no—
440C23C22×C22⋊C411yescharacteristic
441C23C2×#12111yescharacteristic
442C23C2×#12111yescharacteristic
443C23#52512yes—
444C23#53612yes—
445C23#57612yes—
446C23#57712yes—
447C23#57712yes—
448C23#57812yes—
449Q8—816no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
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#38614yes—
508C2×C4#38614yes—
509C2×C4C2×C4×D411yescharacteristic
510C2×C4C2×C4×D411yescharacteristic
511C2×C4C4×#4112yes—
512C2×C4#51912yes—
513C2×C4C2×#12011yescharacteristic
514C2×C4C2×#12311yescharacteristic
515C2×C4#53211yescharacteristic
516C2×C4#53211yescharacteristic
517C2×C4#53611yescharacteristic
518C2×C4#53611yescharacteristic
519C2×C4#54712yes—
520C2×C4#54712yes—
521C2×C4#54812yes—
522C2×C4#54812yes—
523C2×C4#55212yes—
524C2×C4#55212yes—
525C2×C4#55612yes—
526C2×C4#55612yes—
527C8—42no—
528C8—42no—
529C8—42no—
530C8—42no—
531C22—42no—
532C22—44no—
533C22—44no—
534C22—42no—
535C22—42no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22C2×#38611yescharacteristic
541C22#229711yescharacteristic
542C22#231011yescharacteristic
543C22#241512yes—
544C22#241512yes—
545C22#289712yes—
546C22#289812yes—
547C22#305711yescharacteristic
548C22#305711yescharacteristic
549C22#305811yescharacteristic
550C22#305911yescharacteristic
551C22#306212yes—
552C22#306312yes—
553C22#311812yes—
554C22#312112yes—
555C22#312412yes—
556C22#312512yes—
557C22#324112yes—
558C22#324212yes—
559C22#324312yes—
560C22#324412yes—
561C4—24no—
562C4—24no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—84no—
567C4—84no—
568C4—84no—
569C4—84no—
570C4#236212yes—
571C4#236212yes—
572C4#243211yescharacteristic
573C4#243211yescharacteristic
574C4#249911yescharacteristic
575C4#249911yescharacteristic
576C2—82no—
577C2#1413811yescharacteristic
578C2#2097411yescharacteristic
579C2#2097511yescharacteristic
580C2#2139712yes—
581C2#2184312yes—
582C2#2471212yes—
583C2#2471412yes—
584C2#2471612yes—
585C2#2471812yes—
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
3. C2×#1988C2semidirect
4. C2×#1987C2semidirect
8. #10881C2semidirect
9. #10880C2semidirect
10. #10850C2semidirect
11. #10850C2semidirect
24. #2628C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4332
C2×#26282
C2×#19882
C2×#19872
#137772
#112002
#111992
#108812
#108802
#108502