Finite Group Databasebeta

← #392244

Subgroups — #392244

Order 512.

3770 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#520C211yesmaximal, characteristic
2C2×#1824C211yesmaximal, characteristic, complemented
3C2×#1825C211yesmaximal, characteristic, complemented
4C2×#1825C211yesmaximal, characteristic, complemented
5C2×#1823C211yesmaximal, characteristic, complemented
6C2×#1780C211yesmaximal, characteristic
7#14337C211yesmaximal, characteristic
8#10731C214yesmaximal, complemented
9#10730C214yesmaximal, complemented
10C23×#41C2211yescharacteristic
11C2×#520—24no—
12C2×#520—24no—
13C2×#520—24no—
14C2×#520—24no—
15C2×#520—24no—
16C2×#520C418yescomplemented
17C22×C4×Q8—21no—
18C22×C4×D4—21no—
19C22×C4×D4—22no—
20C22×#117C2212yes—
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23C23×C4⋊C4C2211yescharacteristic
24C22×C22⋊C8C2211yescharacteristic
25C22×C22⋊C8C2211yescharacteristic
26C22×C22⋊C8C2214yes—
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×#380C2214yes—
30#1825—24no—
31#1825—24no—
32#1824—24no—
33#1823—24no—
34#1780—24no—
35C2×#340C2214yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2214yes—
41#1446—21no—
42#1438—22no—
43#1438—21no—
44C22×#41—22no—
45C22×#41—22no—
46C22×#41—22no—
47C22×#41—22no—
48C22×#41—216no—
49C22×#41C2×C414yes—
50C22×#41D414yes—
51C23×Q8D411yescharacteristic
52C23×D4D411yescharacteristic
53C23×D4D412yes—
54C24×C4—22no—
55C24×C4—21no—
56C24×C4C2311yescharacteristic
57C24×C4C2312yes—
58C23×C8C2312yes—
59C23×C8C2312yes—
60#520—232no—
61#520—232no—
62#520—232no—
63#520—232no—
64C2×C4×Q8—28no—
65C2×C4×Q8—28no—
66C2×C4×Q8—28no—
67C2×C4×D4—216no—
68C2×C4×D4—216no—
69C2×C4×D4—28no—
70C2×C4×D4—28no—
71C2×C4×D4—216no—
72C2×C4×D4—28no—
73C2×#117—24no—
74C2×#117—22no—
75C2×#117—22no—
76C2×#117—22no—
77C2×#117—24no—
78C2×#117—24no—
79C2×#117—24no—
80C2×#117—24no—
81C2×#117C2×C418yes—
82C2×#117C2×C414yes—
83C2×#117D414yes—
84C2×#117D414yes—
85C22×C4⋊C4—22no—
86C22×C4⋊C4—21no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—22no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4C2×C418yes—
96C22×C4⋊C4C2×C414yes—
97C22×C4⋊C4D414yes—
98C22×C4⋊C4D414yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4—22no—
105C22×C22⋊C4—21no—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—24no—
109C22×C42—22no—
110C22×C42—21no—
111C2×C22⋊C8—22no—
112C2×C22⋊C8—22no—
113C2×C22⋊C8—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8—22no—
117C2×C22⋊C8—28no—
118C2×C22⋊C8—216no—
119#380—22no—
120#380—22no—
121#380—22no—
122#380—22no—
123#380—216no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—24no—
127#340—28no—
128#340—28no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—24no—
133C25C2×D411yescharacteristic
134C25C2×D412yes—
135C2×#41—232no—
136C2×#41—232no—
137C2×#41—232no—
138C2×#41—232no—
139C2×#41—28no—
140C2×#41—28no—
141C2×#41—232no—
142C2×#41—28no—
143C2×#41—28no—
144C2×#41—232no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8—28no—
148C22×Q8C22⋊C414yes—
149C22×D4—24no—
150C22×D4—24no—
151C22×D4—28no—
152C22×D4—28no—
153C22×D4—22no—
154C22×D4—22no—
155C22×D4—28no—
156C22×D4C22⋊C414yes—
157C22×D4C22⋊C418yes—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—28no—
163C23×C4—28no—
164C23×C4—21no—
165C23×C4—22no—
166C23×C4—21no—
167C23×C4—28no—
168C23×C4—28no—
169C23×C4—28no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—82no—
176C23×C4—84no—
177C23×C4—82no—
178C23×C4—84no—
179C23×C4C22×C412yes—
180C23×C4C22×C414yes—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D412yes—
183C23×C4C2×D412yes—
184C23×C4C2×D412yes—
185C23×C4C2×D412yes—
186C23×C4C2×D412yes—
187C23×C4C2×D412yes—
188C23×C4C2×D414yes—
189C23×C4#41116yes—
190C23×C4C2411yescharacteristic
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—28no—
198C22×C8—28no—
199C4×Q8—264no—
200C4×D4—2128no—
201C4×D4—264no—
202#117—216no—
203#117—216no—
204#117—28no—
205#117—28no—
206#117—28no—
207#117—216no—
208#117—216no—
209#117C22⋊C4116yescomplemented
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—216no—
213C2×C4⋊C4—216no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—216no—
231C2×C4⋊C4—216no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4C22⋊C4116yescomplemented
239C2×C4⋊C4C22×C412yes—
240C2×C4⋊C4C22×C414yes—
241C2×C4⋊C4C22×C412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D414yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D412yes—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—48no—
257C2×C22⋊C4—84no—
258C2×C22⋊C4—82no—
259C2×C22⋊C4—84no—
260C2×C22⋊C4—82no—
261C2×C42—28no—
262C2×C42—28no—
263C2×C42—28no—
264C2×C42—24no—
265C2×C42—24no—
266C2×C42—24no—
267C22⋊C8—44no—
268C22⋊C8—44no—
269C22⋊C8—416no—
270#95—42no—
271#95—42no—
272#95—48no—
273C24—22no—
274C24—22no—
275C24—24no—
276C24—24no—
277C24—24no—
278C24—22no—
279C24—24no—
280C24—22no—
281C24—24no—
282C24—22no—
283C24—24no—
284C24—22no—
285C24C2×C22⋊C412yes—
286C24C2×C22⋊C414yes—
287C24#12012yes—
288C24#12012yes—
289C24#12318yes—
290C24C22×D411yescharacteristic
291#41—264no—
292#41—264no—
293#41—264no—
294#41—264no—
295C2×Q8—216no—
296C2×Q8—216no—
297C2×Q8—28no—
298C2×Q8—216no—
299C2×D4—216no—
300C2×D4—216no—
301C2×D4—232no—
302C2×D4—232no—
303C2×D4—216no—
304C2×D4—216no—
305C2×D4—216no—
306C2×D4—28no—
307C2×D4—216no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—28no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—216no—
325C22×C4—216no—
326C22×C4—216no—
327C22×C4—216no—
328C22×C4—22no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—232no—
338C22×C4—216no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—44no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—88no—
356C22×C4—816no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4—816no—
363C22×C4C2×C22⋊C414yes—
364C22×C4C2×C22⋊C414yes—
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C414yes—
367C22×C4#117116yes—
368C22×C4#12014yes—
369C22×C4#12014yes—
370C22×C4#12012yes—
371C22×C4#12012yes—
372C22×C4#123116yes—
373C22×C4#12318yes—
374C22×C4C23×C411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C2×#4118yes—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—44no—
384C2×C8—44no—
385C2×C8—44no—
386C2×C8—44no—
387C2×C8—44no—
388C2×C8—44no—
389C4⋊C4—232no—
390C4⋊C4—24no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—216no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4C2×C22⋊C414yes—
405C4⋊C4C2×C22⋊C418yes—
406C4⋊C4C2×C22⋊C414yes—
407C22⋊C4—232no—
408C22⋊C4—216no—
409C22⋊C4—816no—
410C22⋊C4—84no—
411C22⋊C4—84no—
412C42—232no—
413C42—216no—
414C23—28no—
415C23—28no—
416C23—28no—
417C23—28no—
418C23—24no—
419C23—24no—
420C23—28no—
421C23—28no—
422C23—24no—
423C23—28no—
424C23—24no—
425C23—28no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—24no—
431C23—24no—
432C23—24no—
433C23—24no—
434C23—24no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—44no—
448C23—44no—
449C23#37914yes—
450C23#38118yes—
451C23C22×C22⋊C411yescharacteristic
452C23C2×#12011yescharacteristic
453C23C2×#12011yescharacteristic
454C23C2×#12314yes—
455C23#57612yes—
456C23#57712yes—
457C23#57712yes—
458C23#57812yes—
459Q8—232no—
460D4—232no—
461D4—264no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—216no—
472C2×C4—216no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—216no—
476C2×C4—216no—
477C2×C4—24no—
478C2×C4—28no—
479C2×C4—24no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—48no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—816no—
493C2×C4—832no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—84no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4#37914yes—
505C2×C4#38118yes—
506C2×C4C22×C22⋊C411yescharacteristic
507C2×C4C22×C22⋊C411yescharacteristic
508C2×C4C2×#11714yes—
509C2×C4C2×#12011yescharacteristic
510C2×C4C2×#12011yescharacteristic
511C2×C4C2×#12314yes—
512C2×C4#52514yes—
513C2×C4#52514yes—
514C2×C4#52514yes—
515C2×C4#52514yes—
516C8—82no—
517C8—82no—
518C22—24no—
519C22—28no—
520C22—24no—
521C22—24no—
522C22—216no—
523C22—44no—
524C22—44no—
525C22—48no—
526C22—44no—
527C22—48no—
528C22—44no—
529C22—48no—
530C22—44no—
531C22—48no—
532C22—44no—
533C22—48no—
534C22—48no—
535C22C2×#37911yescharacteristic
536C22C2×#38112yes—
537C22#284912yes—
538C22#284912yes—
539C22#289712yes—
540C22#289812yes—
541C22#300811yescharacteristic
542C22#300911yescharacteristic
543C22#301011yescharacteristic
544C22#301111yescharacteristic
545C22#301212yes—
546C22#301312yes—
547C22#309614yes—
548C22#309714yes—
549C22#309814yes—
550C22#309914yes—
551C4—24no—
552C4—28no—
553C4—48no—
554C4—44no—
555C4—44no—
556C4—162no—
557C4—162no—
558C4#228012yes—
559C4#228012yes—
560C4#284912yes—
561C4#284912yes—
562C2—44no—
563C2—48no—
564C2#1468111yescharacteristic
565C2#2060211yescharacteristic
566C2#2060311yescharacteristic
567C2#2070314yes—
568C2#3298612yes—
569C2#3298712yes—
570C2#3298812yes—
571C2#3298912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×#41 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1824C2semidirect
3. C2×#1825C2semidirect
4. C2×#1825C2semidirect
5. C2×#1823C2semidirect
8. #10731C2semidirect
9. #10730C2semidirect
16. C2×#520C4semidirect
209. #117C22⋊C4semidirect
238. C2×C4⋊C4C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#5202
C2×#18242
C2×#18252
C2×#18232
C2×#17802
#143372
#107312
#107302