Finite Group Databasebeta

← #360108

Subgroups — #360108

Order 512.

2094 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2C22×#382C211yesmaximal, characteristic
3C2×#1872C212yesmaximal
4C2×#1805C212yesmaximal, complemented
5C2×#1797C211yesmaximal, characteristic
6#11208C212yesmaximal, complemented
7#11052C212yesmaximal, complemented
8#10850C212yesmaximal, complemented
9#10730C212yesmaximal
10C22×C22⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C23×C4⋊C4C2212yes—
13C23×C42C2211yescharacteristic
14C22×C4⋊C8C2212yes—
15C22×#103C2214yes—
16C22×C22⋊C8C2212yes—
17C4×C22⋊Q8—24no—
18C4×C22⋊Q8C418yescomplemented
19C2×#391C2211yescharacteristic
20C2×#389C2211yescharacteristic
21C2×#387—21no—
22C2×#385—21no—
23C2×#382—21no—
24C2×#382—21no—
25C2×#382—28no—
26C2×#382C2212yes—
27C2×#382C2212yes—
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4—21no—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C22⋊C4—21no—
32C22×#95—21no—
33C22×#95—21no—
34#1872—28no—
35#1805—28no—
36#1797—24no—
37C2×#340C2214yes—
38C2×#340C2214yes—
39C2×#340C2214yes—
40C2×#336C2212yes—
41C23×Q8C2311yescharacteristic
42C24×C4—22no—
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8C2314yes—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8C2×C414yes—
51C2×C22⋊Q8D414yes—
52C2×C22⋊Q8Q814yes—
53C2×C4×Q8—22no—
54C2×C4×Q8—24no—
55C2×C4×Q8—24no—
56C2×C4×Q8C2×C414yescomplemented
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—28no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—44no—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C22⋊C4—21no—
70C22×C22⋊C4—21no—
71C22×C42—24no—
72C22×C42—24no—
73C22×C42—21no—
74C22×C42—21no—
75C22×C42C2×C414yes—
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D414yes—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×C4⋊C8—24no—
82C2×C4⋊C8—24no—
83C2×C4⋊C8—24no—
84C2×#103—28no—
85C2×#103—28no—
86C2×#103—44no—
87C2×C22⋊C8—24no—
88C2×C22⋊C8—24no—
89C2×C22⋊C8—24no—
90#391—22no—
91#391C2×C414yescomplemented
92#389—22no—
93#389C2×C414yescomplemented
94#387—28no—
95#385—28no—
96#382—28no—
97#382—28no—
98#382—28no—
99#382—24no—
100#382D418yes—
101#380—22no—
102#380C2×C414yes—
103C4×C4⋊C4—28no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4C2×C414yescomplemented
106C4×C22⋊C4—28no—
107C2×#95—28no—
108C2×#95—28no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118#340—28no—
119#340—28no—
120#340—44no—
121#336—24no—
122C25C2×D411yescharacteristic
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8C22×C412yes—
131C22×Q8C2×D412yes—
132C22×Q8C2×Q812yes—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—28no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—24no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4C22⋊C414yes—
158C23×C4C22⋊C414yes—
159C23×C4C22×C412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×D412yes—
167C23×C4C2×Q812yes—
168C23×C4#4112yes—
169C23×C4#4112yes—
170C23×C4C2411yescharacteristic
171C22×C8—24no—
172C22×C8—24no—
173C22×C8—24no—
174C22×C8—24no—
175C22×C8—24no—
176C22×C8—24no—
177C22×C8—24no—
178C22⋊Q8—28no—
179C22⋊Q8—28no—
180C22⋊Q8—28no—
181C22⋊Q8C4⋊C4116yes—
182C4×Q8—216no—
183C4×Q8—48no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—416no—
205C2×C4⋊C4—416no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C22×C412yes—
214C2×C4⋊C4C2×D418yes—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×Q812yes—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—44no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—216no—
229C2×C42—216no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—44no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42C22⋊C414yes—
236C2×C42C22⋊C414yes—
237C2×C42C22×C412yes—
238C2×C42C22×C412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D414yes—
241C2×C42C2×D412yes—
242C2×C42#4114yes—
243C4⋊C8—48no—
244#103—416no—
245C22⋊C8—48no—
246#95—28no—
247#95—28no—
248#95—24no—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—28no—
254#95—28no—
255C24—21no—
256C24—21no—
257C24—21no—
258C24—21no—
259C24—22no—
260C24—22no—
261C24C2×C22⋊C412yes—
262C24#12112yes—
263C24C22⋊Q812yes—
264C24#13514yes—
265C24#13714yes—
266C24C22×D411yescharacteristic
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8C2×C4⋊C414yes—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—28no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—28no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—48no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—416no—
337C22×C4—416no—
338C22×C4—416no—
339C22×C4—41no—
340C22×C4—44no—
341C22×C4—41no—
342C22×C4—44no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—88no—
354C22×C4—88no—
355C22×C4—88no—
356C22×C4—88no—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C412yes—
360C22×C4C2×C22⋊C412yes—
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C4⋊C414yes—
363C22×C4#11714yes—
364C22×C4#12012yes—
365C22×C4#12012yes—
366C22×C4#12114yes—
367C22×C4#12112yes—
368C22×C4C22⋊Q814yes—
369C22×C4C22⋊Q812yes—
370C22×C4#12312yes—
371C22×C4#12312yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D412yes—
375C22×C4C22×D411yescharacteristic
376C22×C4C22×Q811yescharacteristic
377C22×C4C2×#4111yescharacteristic
378C22×C4C2×#4111yescharacteristic
379C22×C4C2×#4112yes—
380C22×C4H(5,𝔽2)12yes—
381C22×C42−1+412yes—
382C2×C8—44no—
383C2×C8—44no—
384C2×C8—44no—
385C2×C8—44no—
386C2×C8—44no—
387C2×C8—44no—
388C2×C8—44no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—216no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—28no—
396C4⋊C4—28no—
397C4⋊C4—48no—
398C4⋊C4—416no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—48no—
404C4⋊C4C2×C4⋊C414yes—
405C22⋊C4—216no—
406C22⋊C4—48no—
407C42—24no—
408C42—24no—
409C42—28no—
410C42—28no—
411C42—48no—
412C42—48no—
413C42C4×D418yes—
414C42#12118yes—
415C23—22no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—22no—
428C23—22no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—22no—
433C23#38014yes—
434C23#41614yes—
435C23#41914yes—
436C23#46418yes—
437C23#47918yes—
438C23C22×C22⋊C411yescharacteristic
439C23C2×#12011yescharacteristic
440C23C2×#12111yescharacteristic
441C23C2×C22⋊Q811yescharacteristic
442C23C2×#12311yescharacteristic
443C23#53511yescharacteristic
444C23#53611yescharacteristic
445C23C2×#13512yes—
446C23C2×#13712yes—
447C23#57512yes—
448C23#57512yes—
449Q8—28no—
450Q8—44no—
451Q8—44no—
452Q8—44no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—28no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—416no—
480C2×C4—416no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—48no—
501C2×C4—48no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—832no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—84no—
508C2×C4—84no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4#38014yes—
514C2×C4#38614yes—
515C2×C4#38614yes—
516C2×C4C22×C4⋊C411yescharacteristic
517C2×C4C2×#11711yescharacteristic
518C2×C4C2×C4×D412yes—
519C2×C4#52012yes—
520C2×C4C2×#12111yescharacteristic
521C2×C4C2×#12112yes—
522C2×C4C2×C22⋊Q811yescharacteristic
523C2×C4#53712yes—
524C2×C4D4212yes—
525C2×C4#54612yes—
526C2×C4#54712yes—
527C2×C4#54812yes—
528C2×C4D4×Q812yes—
529C2×C4#55212yes—
530C2×C4#55312yes—
531C2×C4#55412yes—
532C8—84no—
533C22—28no—
534C22—28no—
535C22—24no—
536C22—24no—
537C22—24no—
538C22—28no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—28no—
543C22#180518yescomplemented
544C22C2×#38011yescharacteristic
545C22C2×#38611yescharacteristic
546C22#231111yescharacteristic
547C22#240811yescharacteristic
548C22#240911yescharacteristic
549C22#248611yescharacteristic
550C22#248711yescharacteristic
551C22C2×#41611yescharacteristic
552C22C2×#41911yescharacteristic
553C22#289611yescharacteristic
554C22#289611yescharacteristic
555C22#300212yes—
556C22#300712yes—
557C22C2×#46412yes—
558C22#304012yes—
559C22C2×#47912yes—
560C22#308012yes—
561C22#309112yes—
562C22#309512yes—
563C22#321112yes—
564C22#321512yes—
565C22#323212yes—
566C22#323612yes—
567C4—24no—
568C4—24no—
569C4—24no—
570C4—44no—
571C4—44no—
572C4—42no—
573C4—42no—
574C4—82no—
575C4—88no—
576C4—82no—
577C4D4×C4⋊C412yes—
578C4#235212yes—
579C4#241612yes—
580C4#241712yes—
581C2—28no—
582C2#1387211yescharacteristic
583C2C2×#180511yescharacteristic
584C2#2064911yescharacteristic
585C2#2090811yescharacteristic
586C2#2091111yescharacteristic
587C2#2186111yescharacteristic
588C2#2186611yescharacteristic
589C2#2421712yes—
590C2#2430712yes—
591C2#2765312yes—
592C2#2768612yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1805C2semidirect
6. #11208C2semidirect
7. #11052C2semidirect
8. #10850C2semidirect
18. C4×C22⋊Q8C4semidirect
56. C2×C4×Q8C2×C4semidirect
91. #391C2×C4semidirect
93. #389C2×C4semidirect
105. C4×C4⋊C4C2×C4semidirect
543. C22#1805semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C22×#3822
C2×#18722
C2×#18052
C2×#17972
#112082
#110522
#108502
#107302