Finite Group Databasebeta

← #369340

Subgroups — #369340

Order 512.

1570 conjugacy classes of subgroups.

The subgroup lattice has 575 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#491C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4C2×#1981C211yesmaximal, characteristic
5#14160C211yesmaximal, characteristic
6#11190C212yesmaximal
7#11190C212yesmaximal
8#10835C211yesmaximal, characteristic
9#10789C212yesmaximal
10#10789C212yesmaximal
11#10703C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C4×Q8C2212yes—
15C2×#491—24no—
16C2×#491—24no—
17C2×#491C418yescomplemented
18C22×C4⋊C8C2211yescharacteristic
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×#97C2211yescharacteristic
24#2365—24no—
25#2365—24no—
26C2×#391C2212yes—
27C2×#389—21no—
28C2×#387—22no—
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1989—24no—
34#1981—24no—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1441—21no—
44#1441—21no—
45#1436—22no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×Q8C2312yes—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C414yes—
56C2×C4⋊Q8D414yes—
57C2×C4⋊Q8D414yes—
58C2×#125—22no—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125—22no—
62C2×#125—22no—
63C2×#125C2×C414yes—
64C2×C4×Q8—24no—
65C2×C4×Q8—24no—
66C2×C4×Q8—24no—
67C2×C4×Q8—24no—
68C2×C4×Q8—42no—
69C2×C4×Q8—42no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D411yescharacteristic
76C22×C4⋊C4D411yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C42—22no—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42C2312yes—
86C22×C42C2311yescharacteristic
87#491—232no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—24no—
98C2×#103—24no—
99C2×#103—42no—
100C2×#103—42no—
101C2×#103C2×C418yes—
102C2×#103C2×C418yes—
103C2×#97—22no—
104C2×#97—22no—
105C2×#97C2×C414yes—
106#391—24no—
107#391—24no—
108#389—28no—
109#387—216no—
110C4×C4⋊C4—28no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4D414yes—
117C2×#95—22no—
118C2×#95—21no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—21no—
123C2×#95—22no—
124C2×#95—22no—
125C2×#95D412yes—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#340—24no—
131#340—24no—
132#336—42no—
133#336—42no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8C22×C414yes—
146C22×Q8C2×D414yes—
147C22×Q8C2×D414yes—
148C23×C4—22no—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2411yescharacteristic
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8C22×C414yes—
176C4⋊Q8—28no—
177C4⋊Q8—28no—
178C4⋊Q8—28no—
179C4⋊Q8C22⋊C4116yes—
180#125—28no—
181#125—28no—
182#125—44no—
183#125—44no—
184C4×Q8—48no—
185C4×Q8—48no—
186C4×Q8—48no—
187C4×Q8—48no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—28no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4C22×C412yes—
228C2×C4⋊C4C22×C412yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4#4114yes—
232C2×C4⋊C4#4114yes—
233C2×C42—28no—
234C2×C42—24no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42—44no—
244C2×C42—42no—
245C2×C42—81no—
246C2×C42—81no—
247C2×C42C22×C412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42#4112yes—
252C2×C42#4112yes—
253C4⋊C8—44no—
254#103—216no—
255#103—216no—
256#103—48no—
257#103—48no—
258#97—28no—
259#95—28no—
260#95—28no—
261#95—24no—
262#95—24no—
263#95—44no—
264#95—42no—
265#95—44no—
266#95—82no—
267#95—82no—
268C24C22×D411yescharacteristic
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—24no—
282C2×Q8—24no—
283C2×Q8—24no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8C2×C22⋊C418yes—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4C4×D412yes—
353C22×C4C4×D412yes—
354C22×C4#12012yes—
355C22×C4#12012yes—
356C22×C4#12112yes—
357C22×C4#12112yes—
358C22×C4#13714yes—
359C22×C4C23×C411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4112yes—
365C22×C4C2×#4112yes—
366C22×C4C2×#4111yescharacteristic
367C22×C4H(5,𝔽2)12yes—
368C22×C42−1+412yes—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C4⋊C4—28no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—216no—
390C4⋊C4—216no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—44no—
400C4⋊C4—416no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4#12414yes—
411C42—48no—
412C42—416no—
413C42—42no—
414C42—42no—
415C42—42no—
416C42—42no—
417C42—84no—
418C42C2×C22⋊C414yes—
419C42C4×D414yes—
420C42#12114yes—
421C42#12314yes—
422C23C2×C4×D411yescharacteristic
423C23C2×#12011yescharacteristic
424C23C2×#12111yescharacteristic
425C23#54612yes—
426C23#54812yes—
427C23C2×#13712yes—
428C23#57512yes—
429C23#57711yescharacteristic
430C23#57711yescharacteristic
431C23#57811yescharacteristic
432C23#57811yescharacteristic
433Q8—28no—
434Q8—28no—
435Q8—44no—
436Q8—44no—
437Q8—44no—
438Q8—44no—
439Q8—44no—
440Q8—44no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—24no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—28no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4#38614yes—
502C2×C4#38614yes—
503C2×C4C22×C22⋊C411yescharacteristic
504C2×C4C2×C4×D411yescharacteristic
505C2×C4C4×#4112yes—
506C2×C4#52012yes—
507C2×C4C2×#12111yescharacteristic
508C2×C4C2×#12311yescharacteristic
509C2×C4C2×#12411yescharacteristic
510C2×C4#52912yes—
511C2×C4#53211yescharacteristic
512C2×C4#53511yescharacteristic
513C2×C4#53611yescharacteristic
514C2×C4#54212yes—
515C2×C4#54612yes—
516C2×C4#54612yes—
517C2×C4#54712yes—
518C2×C4#54712yes—
519C2×C4#55612yes—
520C2×C4#55912yes—
521C8—44no—
522C8—81no—
523C8—81no—
524C22C2×#38611yescharacteristic
525C22#236712yes—
526C22#242712yes—
527C22#243211yescharacteristic
528C22#243511yescharacteristic
529C22C4×#13712yes—
530C22#294711yescharacteristic
531C22#294711yescharacteristic
532C22#300411yescharacteristic
533C22#300511yescharacteristic
534C22#301311yescharacteristic
535C22#301311yescharacteristic
536C22#303712yes—
537C22#304311yescharacteristic
538C22#304411yescharacteristic
539C22#330712yes—
540C22#331212yes—
541C22#331512yes—
542C22#332012yes—
543C22#337212yes—
544C22#337412yes—
545C22#337612yes—
546C22#337712yes—
547C4—24no—
548C4—24no—
549C4—44no—
550C4—44no—
551C4—48no—
552C4—82no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4—84no—
557C4C4×#12411yescharacteristic
558C4#232611yescharacteristic
559C4#233912yes—
560C4#244712yes—
561C4#254411yescharacteristic
562C4#254711yescharacteristic
563C2#1402711yescharacteristic
564C2#2092611yescharacteristic
565C2#2093011yescharacteristic
566C2#2322712yes—
567C2#2323312yes—
568C2#2533112yes—
569C2#2533312yes—
570C2#2565611yescharacteristic
571C2#2565711yescharacteristic
572C2#2589211yescharacteristic
573C2#2589411yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
17. C2×#491C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4912
C2×#23652
C2×#19892
C2×#19812
#141602
#111902
#108352
#107892
#107032