Finite Group Databasebeta

← #188144

Subgroups — #188144

Order 512.

2388 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#539C211yesmaximal, characteristic
2#21830C212yesmaximal, complemented
3#17398C212yesmaximal, complemented
4#17364C212yesmaximal, complemented
5#8556C212yesmaximal, complemented
6#8555C212yesmaximal, complemented
7#8553C212yesmaximal, complemented
8#8553C212yesmaximal, complemented
9C2×#539—28no—
10C2×#539—28no—
11C2×#539—24no—
12C2×#539—24no—
13C22×#126C2211yescharacteristic
14C22×#126C2211yescharacteristic
15C22×#125C411yescharacteristic
16C22×#125C2211yescharacteristic
17C22×#123C411yescharacteristic
18C22×#123C411yescharacteristic
19C22×#123C2211yescharacteristic
20C22×#123C2211yescharacteristic
21C22×C22⋊Q8C411yescharacteristic
22C22×C22⋊Q8C411yescharacteristic
23C22×C22⋊Q8C2211yescharacteristic
24C22×#121C411yescharacteristic
25C22×C4×D4C411yescharacteristic
26C22×C4×D4C2211yescharacteristic
27C23×C4⋊C4C411yescharacteristic
28#2920C2214yes—
29#1998—28no—
30#1821C2214yes—
31#1821C2214yes—
32#1473—28no—
33#1466—24no—
34#1463C2214yes—
35#1462C2214yes—
36#1462C2214yes—
37#1462C2214yes—
38#1460—24no—
39C23×Q8C2×C411yescharacteristic
40C23×D4C2×C411yescharacteristic
41C23×D4C2×C411yescharacteristic
42C23×D4C2×C411yescharacteristic
43C24×C4C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2×C411yescharacteristic
47C24×C4C2×C411yescharacteristic
48#539—232no—
49#539—232no—
50#539—232no—
51#539—232no—
52C2×#126—24no—
53C2×#126—24no—
54C2×#126—24no—
55C2×#126—24no—
56C2×#126—22no—
57C2×#126—22no—
58C2×#126D418yes—
59C2×#125—24no—
60C2×#125—24no—
61C2×#125—24no—
62C2×#125—24no—
63C2×#125—22no—
64C2×#125—22no—
65C2×#125D418yes—
66C2×#123—24no—
67C2×#123—22no—
68C2×#123—22no—
69C2×#123—24no—
70C2×#123—24no—
71C2×#123—24no—
72C2×#123—24no—
73C2×#123—24no—
74C2×#123—24no—
75C2×#123—24no—
76C2×#123—24no—
77C2×#123—24no—
78C2×#123—24no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×#121—24no—
90C2×#121—24no—
91C2×#121—24no—
92C2×C4×D4—22no—
93C2×C4×D4—22no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4D414yes—
99C2×C4×D4D414yes—
100C22×C4⋊C4—22no—
101C22×C4⋊C4—22no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4C2×C411yescharacteristic
105C22×C4⋊C4C2×C411yescharacteristic
106C22×C4⋊C4C2×C411yescharacteristic
107C22×C4⋊C4C2×C411yescharacteristic
108C22×C4⋊C4C2×C411yescharacteristic
109C22×C4⋊C4C2×C411yescharacteristic
110C22×C4⋊C4C2×C411yescharacteristic
111C22×C4⋊C4C2×C411yescharacteristic
112C22×C4⋊C4C2×C411yescharacteristic
113C22×C4⋊C4C2×C411yescharacteristic
114C22×C4⋊C4C2×C411yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4C2×C411yescharacteristic
119C22×C22⋊C4C2×C411yescharacteristic
120C22×C22⋊C4C2×C411yescharacteristic
121C22×C22⋊C4C2×C411yescharacteristic
122C22×C22⋊C4C2×C411yescharacteristic
123C22×C22⋊C4C2×C411yescharacteristic
124C22×C22⋊C4C2×C411yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C42C2×C411yescharacteristic
129C22×C42C2311yescharacteristic
130#432—24no—
131#432—24no—
132#431—24no—
133#431—24no—
134#423—24no—
135#407C2318yes—
136C4×M16—24no—
137#343—28no—
138#343—28no—
139#324—28no—
140#323—28no—
141#323—28no—
142#323—28no—
143C25C22×C411yescharacteristic
144C25C22×C411yescharacteristic
145C25C22×C411yescharacteristic
146C22×Q8—22no—
147C22×Q8—22no—
148C22×Q8—22no—
149C22×Q8—82no—
150C22×D4—22no—
151C22×D4—22no—
152C22×D4—22no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—22no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—44no—
159C22×D4—44no—
160C22×D4—82no—
161C22×D4C22⋊C414yes—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—22no—
171C23×C4—22no—
172C23×C4—22no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—44no—
176C23×C4—82no—
177C23×C4—82no—
178C23×C4—82no—
179C23×C4—82no—
180C23×C4C22⋊C412yes—
181C23×C4C22⋊C412yes—
182C23×C4C22⋊C412yes—
183C23×C4C22⋊C412yes—
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C22×C411yescharacteristic
191C23×C4C22×C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C22×C411yescharacteristic
195C23×C4C2411yescharacteristic
196C2×M16—28no—
197#126—232no—
198#126—28no—
199#126—28no—
200#126—28no—
201#126—28no—
202#125—232no—
203#125—28no—
204#125—28no—
205#125—28no—
206#125—28no—
207#123—232no—
208#123—216no—
209#123—216no—
210#123—232no—
211#123—232no—
212C22⋊Q8—232no—
213C22⋊Q8—216no—
214C22⋊Q8—216no—
215C22⋊Q8—232no—
216#121—232no—
217C4×D4—216no—
218C4×D4—216no—
219C4×D4—28no—
220C4×D4—28no—
221C4×D4—28no—
222C4×D4—28no—
223C2×C4⋊C4—216no—
224C2×C4⋊C4—216no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4C22⋊C414yes—
261C2×C4⋊C4C22⋊C414yes—
262C2×C4⋊C4C22⋊C414yes—
263C2×C4⋊C4C22⋊C414yes—
264C2×C4⋊C4C22⋊C414yes—
265C2×C4⋊C4C22⋊C414yes—
266C2×C4⋊C4C22⋊C414yes—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4C22⋊C414yes—
297C2×C22⋊C4C22⋊C414yes—
298C2×C22⋊C4C2×D414yes—
299C2×C22⋊C4C2×D414yes—
300C2×C22⋊C4C2×D414yes—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C4⋊C8—44no—
309C4⋊C8—44no—
310C22⋊C8—28no—
311C22⋊C8—28no—
312#97—44no—
313C4×C8—44no—
314C24—21no—
315C24—21no—
316C24—24no—
317C24—21no—
318C24—21no—
319C24—21no—
320C24—21no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24—42no—
325C24—42no—
326C24—42no—
327C24—42no—
328C24—42no—
329C24C2×C22⋊C412yes—
330C24C2×C22⋊C412yes—
331C24C2×C22⋊C412yes—
332C24C23×C411yescharacteristic
333C2×Q8—28no—
334C2×Q8—84no—
335C2×Q8—84no—
336C2×Q8—84no—
337C2×D4—28no—
338C2×D4—28no—
339C2×D4—28no—
340C2×D4—28no—
341C2×D4—44no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—44no—
346C2×D4—48no—
347C2×D4—44no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—84no—
351C22×C4—28no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—28no—
371C22×C4—28no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—84no—
393C22×C4—84no—
394C22×C4—84no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4—84no—
398C22×C4—84no—
399C22×C4—84no—
400C22×C4—84no—
401C22×C4—84no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4C2×C22⋊C412yes—
407C22×C4C2×C22⋊C412yes—
408C22×C4C2×C22⋊C412yes—
409C22×C4C2×C22⋊C412yes—
410C22×C4C2×C22⋊C412yes—
411C22×C4C2×C22⋊C412yes—
412C22×C4C2×C22⋊C412yes—
413C22×C4C2×C22⋊C412yes—
414C22×C4C2×C22⋊C412yes—
415C22×C4C2×C22⋊C412yes—
416C22×C4C2×C22⋊C412yes—
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C412yes—
420C22×C4C2×C22⋊C412yes—
421C22×C4C2×C22⋊C412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425M16—416no—
426C2×C8—48no—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—28no—
430C4⋊C4—28no—
431C4⋊C4—28no—
432C4⋊C4—28no—
433C4⋊C4—28no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—28no—
444C22⋊C4—28no—
445C22⋊C4—28no—
446C22⋊C4—28no—
447C22⋊C4—28no—
448C22⋊C4—28no—
449C22⋊C4—28no—
450C22⋊C4—28no—
451C22⋊C4—28no—
452C22⋊C4—24no—
453C22⋊C4—24no—
454C22⋊C4—28no—
455C42—28no—
456C42#12014yes—
457C42#12014yes—
458C42#12014yes—
459C42#12014yes—
460C23—22no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—416no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—88no—
495C23—88no—
496C23#41314yes—
497C23C22×C22⋊C411yescharacteristic
498C23C22×C22⋊C411yescharacteristic
499C23C22×C22⋊C411yescharacteristic
500Q8—88no—
501D4—416no—
502D4—416no—
503D4—88no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—416no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—84no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—88no—
556C2×C4—88no—
557C2×C4#37914yes—
558C2×C4#37912yes—
559C2×C4#37912yes—
560C2×C4C2×#12014yes—
561C8—88no—
562C22—28no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—24no—
570C22—44no—
571C22—48no—
572C22—44no—
573C22—48no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—84no—
583C22—84no—
584C22—84no—
585C22—84no—
586C22—84no—
587C22—84no—
588C22C2×#37911yescharacteristic
589C22#288412yes—
590C22C2×#41312yes—
591C22#288812yes—
592C4—24no—
593C4—24no—
594C4—22no—
595C4—22no—
596C4—42no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—42no—
605C2—22no—
606C2—22no—
607C2—22no—
608C2—44no—
609C2—82no—
610C2—82no—
611C2#2058711yescharacteristic
612C2#2059111yescharacteristic
613C2#2059211yescharacteristic
subgroupG/H
center, socleC22C2×#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#121C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21830C2semidirect
3. #17398C2semidirect
4. #17364C2semidirect
5. #8556C2semidirect
6. #8555C2semidirect
7. #8553C2semidirect
8. #8553C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5392
#218302
#173982
#173642
#85562
#85552
#85532