Finite Group Databasebeta

← #96918

Subgroups — #96918

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2#10712C211yesmaximal, characteristic
3#10712C211yesmaximal, characteristic
4C2×#1272C212yesmaximal, complemented
5C2×#1271C212yesmaximal, complemented
6C22×C22⋊Q8C411yescharacteristic
7C22×C4×Q8C411yescharacteristic
8C23×C42C2211yescharacteristic
9C22×C22⋊C8—21no—
10C22×C22⋊C8—21no—
11C4×C22⋊Q8—24no—
12C4×C22⋊Q8—24no—
13C2×#391C2211yescharacteristic
14C2×#389C2211yescharacteristic
15C2×#387—21no—
16C2×#385—21no—
17C2×#380C411yescharacteristic
18C2×C4×C4⋊C4—21no—
19C2×C4×C4⋊C4C411yescharacteristic
20C2×C4×C22⋊C4—21no—
21C2×#336—21no—
22C2×#336—21no—
23C2×#336C2212yes—
24C2×#336C2212yes—
25#1272—24no—
26#1271—24no—
27C23×Q8C2×C411yescharacteristic
28C24×C4C2×C411yescharacteristic
29C24×C4D411yescharacteristic
30C24×C4D411yescharacteristic
31C23×C8—22no—
32C23×C8—22no—
33C2×C22⋊Q8—22no—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8C814yescomplemented
39C2×C4×Q8—22no—
40C2×C4×Q8—22no—
41C2×C4×Q8—42no—
42C2×C4×Q8—42no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4C2×C411yescharacteristic
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C22⋊C4—21no—
50C22×C22⋊C4—21no—
51C22×C42—21no—
52C22×C42—21no—
53C22×C42—22no—
54C22×C42—22no—
55C22×C42—42no—
56C22×C42—42no—
57C22×C42C2×C411yescharacteristic
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42C2311yescharacteristic
61C2×C22⋊C8—24no—
62C2×C22⋊C8—24no—
63C2×C22⋊C8—42no—
64C2×C22⋊C8—42no—
65C2×C22⋊C8—42no—
66C2×C22⋊C8—42no—
67#391—22no—
68#391D414yes—
69#389—22no—
70#389D414yes—
71#387—28no—
72#385—28no—
73#380—22no—
74#380—22no—
75C4×C4⋊C4—28no—
76C4×C4⋊C4—22no—
77C4×C4⋊C4—22no—
78C4×C22⋊C4—28no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—21no—
83C2×#95—21no—
84#336—22no—
85#336—22no—
86#336—22no—
87#336—22no—
88#336—42no—
89#336—42no—
90C25C22⋊C411yescharacteristic
91C22×Q8—21no—
92C22×Q8—21no—
93C22×Q8—21no—
94C22×Q8—41no—
95C22×Q8—41no—
96C22×Q8—41no—
97C22×Q8—41no—
98C22×Q8C22⋊C412yes—
99C22×Q8C2×C812yes—
100C22×Q8M1612yes—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—41no—
120C23×C4—41no—
121C23×C4—41no—
122C23×C4—41no—
123C23×C4—41no—
124C23×C4—41no—
125C23×C4—41no—
126C23×C4—41no—
127C23×C4C22⋊C411yescharacteristic
128C23×C4C22⋊C411yescharacteristic
129C23×C4C22⋊C411yescharacteristic
130C23×C4C2×C812yes—
131C23×C4M1612yes—
132C23×C4SD1612yes—
133C23×C4Q1612yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C22×C8—24no—
138C22×C8—24no—
139C22×C8—42no—
140C22×C8—42no—
141C22×C8—42no—
142C22×C8—42no—
143C22×C8—42no—
144C22×C8—42no—
145C22×C8—42no—
146C22×C8—42no—
147C22×C8—42no—
148C22×C8—42no—
149C22×C8—42no—
150C22×C8—42no—
151C22⋊Q8—28no—
152C22⋊Q8—28no—
153C22⋊Q8—28no—
154C22⋊Q8—28no—
155C4×Q8—48no—
156C4×Q8—48no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C2×C812yes—
180C2×C4⋊C4M1612yes—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—48no—
198C2×C42—48no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42—41no—
202C2×C42—41no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C22⋊C8—48no—
210C22⋊C8—48no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—21no—
225C24—21no—
226C24—21no—
227C24—21no—
228C24C22⋊C812yes—
229C24#10312yes—
230C24#10412yes—
231C24C2×C22⋊C411yescharacteristic
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8C22⋊C814yescomplemented
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
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—22no—
293C22×C4—22no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4C22⋊C812yes—
341C22×C4#9912yes—
342C22×C4#9912yes—
343C22×C4#10312yes—
344C22×C4#10412yes—
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4#12012yes—
348C22×C4#12012yes—
349C22×C4C22×C811yescharacteristic
350C22×C4C2×M1611yescharacteristic
351C22×C4C2×M1611yescharacteristic
352C22×C4C2×M1611yescharacteristic
353C22×C4C2×SD1611yescharacteristic
354C22×C4C2×Q1611yescharacteristic
355C22×C4#13512yes—
356C22×C4AGL(1,ℤ/8)11yescharacteristic
357C22×C4#13712yes—
358C22×C4#13711yescharacteristic
359C2×C8—44no—
360C2×C8—44no—
361C2×C8—44no—
362C2×C8—44no—
363C2×C8—44no—
364C2×C8—44no—
365C2×C8—82no—
366C2×C8—82no—
367C2×C8—82no—
368C2×C8—82no—
369C2×C8—82no—
370C2×C8—82no—
371C2×C8—82no—
372C2×C8—82no—
373C4⋊C4—216no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—48no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4C22⋊C814yescomplemented
385C22⋊C4—216no—
386C22⋊C4—48no—
387C42—24no—
388C42—24no—
389C42—42no—
390C42—42no—
391C42—44no—
392C42—44no—
393C42—48no—
394C42—48no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—22no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23Q8⋊C814yes—
430C23#37911yescharacteristic
431C23C2×C22⋊C811yescharacteristic
432C23#40711yescharacteristic
433C23C2×#9911yescharacteristic
434C23C2×#9911yescharacteristic
435C23#41311yescharacteristic
436C23#41311yescharacteristic
437C23C2×#10311yescharacteristic
438C23#41611yescharacteristic
439C23#41711yescharacteristic
440C23#41911yescharacteristic
441C23C2×#10411yescharacteristic
442C23C2×#10411yescharacteristic
443C23#42111yescharacteristic
444C23#42111yescharacteristic
445Q8—44no—
446Q8—44no—
447Q8—44no—
448Q8—44no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—22no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—48no—
492C2×C4—48no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—48no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4#32814yes—
534C2×C4C2×C22⋊C811yescharacteristic
535C2×C4C2×C22⋊C811yescharacteristic
536C2×C4#40711yescharacteristic
537C2×C4#40711yescharacteristic
538C2×C4#44812yes—
539C2×C4#44912yes—
540C2×C4#45112yes—
541C2×C4#45212yes—
542C8—84no—
543C8—84no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22#145911yescharacteristic
560C22#145911yescharacteristic
561C22#146511yescharacteristic
562C22#146511yescharacteristic
563C22C2×Q8⋊C811yescharacteristic
564C22#147611yescharacteristic
565C22#147711yescharacteristic
566C22#147911yescharacteristic
567C22#148711yescharacteristic
568C22#148811yescharacteristic
569C22#148911yescharacteristic
570C22#149111yescharacteristic
571C22C2×#32811yescharacteristic
572C22#150111yescharacteristic
573C22#150411yescharacteristic
574C22#150811yescharacteristic
575C22#151111yescharacteristic
576C22#151311yescharacteristic
577C22#151611yescharacteristic
578C22#152011yescharacteristic
579C22#152411yescharacteristic
580C22#152911yescharacteristic
581C22#153211yescharacteristic
582C22#153511yescharacteristic
583C22#154211yescharacteristic
584C22#155011yescharacteristic
585C22#155411yescharacteristic
586C22#155911yescharacteristic
587C22#177911yescharacteristic
588C22#178111yescharacteristic
589C22#178511yescharacteristic
590C22#178711yescharacteristic
591C22#178811yescharacteristic
592C22#178911yescharacteristic
593C22#178911yescharacteristic
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—22no—
599C4—24no—
600C4—22no—
601C4—22no—
602C4—22no—
603C4—42no—
604C4—44no—
605C4—44no—
606C4—42no—
607C4—82no—
608C4—82no—
609C2—44no—
610C2#756611yescharacteristic
611C2#758311yescharacteristic
612C2#758911yescharacteristic
613C2#760011yescharacteristic
614C2#761711yescharacteristic
615C2#762311yescharacteristic
616C2#763411yescharacteristic
617C2#764811yescharacteristic
618C2#830111yescharacteristic
619C2#831111yescharacteristic
620C2#831211yescharacteristic
621C2#832911yescharacteristic
622C2#833111yescharacteristic
623C2#833611yescharacteristic
624C2#834311yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1272C2semidirect
5. C2×#1271C2semidirect
38. C2×C22⋊Q8C8semidirect
247. C2×Q8C22⋊C8semidirect
384. C4⋊C4C22⋊C8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
#107122
C2×#12722
C2×#12712