Finite Group Databasebeta

← #370185

Subgroups — #370185

Order 512.

1234 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1855C211yesmaximal, characteristic
2#20846C211yesmaximal, characteristic
3C2×#1852C211yesmaximal, characteristic
4#20379C211yesmaximal, characteristic
5C2×#1736C211yesmaximal, characteristic
6#11270C212yesmaximal, complemented
7#11249C212yesmaximal, complemented
8#10896C212yesmaximal, complemented
9#10870C212yesmaximal, complemented
10#10739C211yesmaximal, characteristic
11#10727C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#108C2212yes—
16C2×#428C2211yescharacteristic
17C2×#428C2211yescharacteristic
18C22×#107C2212yes—
19C2×#407C2211yescharacteristic
20C2×#407C2211yescharacteristic
21C22×C22⋊C8C2211yescharacteristic
22C2×#405C2212yes—
23C2×#380C2211yescharacteristic
24#1855C418yescomplemented
25#1854—21no—
26#1854—21no—
27#1853—21no—
28#1852—21no—
29#1852—21no—
30#1852—22no—
31#1852—21no—
32#1750—22no—
33#1749—21no—
34C4×C22⋊C8—21no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#341C2212yes—
40#1736—21no—
41#1736—21no—
42#1736C2211yescharacteristic
43#1736C2212yes—
44#1736C2211yescharacteristic
45#1736C2212yes—
46C2×#340—21no—
47C2×#340—21no—
48C2×#340C2212yes—
49#1729C2212yes—
50C24×C4—21no—
51C24×C4C2311yescharacteristic
52C22×M16—21no—
53C22×M16—22no—
54C22×M16—42no—
55C22×M16—42no—
56C22×M16C2×C414yes—
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2312yes—
60C22×M16C2312yes—
61C23×C8—21no—
62C23×C8C2312yes—
63C2×#117—21no—
64C2×#117—21no—
65C2×#117—24no—
66C2×#117—42no—
67C2×#117D411yescharacteristic
68C2×#117D411yescharacteristic
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—24no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—42no—
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C4⋊C4C2311yescharacteristic
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—21no—
84C22×C22⋊C4—21no—
85C22×C42—21no—
86C2×#108—42no—
87C2×#108—42no—
88C2×#108C2×C418yescomplemented
89#428—24no—
90#428—42no—
91C2×#107—22no—
92C2×#107—22no—
93C2×#107—21no—
94C2×#107—21no—
95C2×#107—24no—
96C2×#107—42no—
97C2×#106—21no—
98C2×#106—21no—
99#407—22no—
100#407—22no—
101#407C2×C414yes—
102#407Q814yes—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8C2×C414yes—
108C2×C22⋊C8D411yescharacteristic
109C2×C22⋊C8D411yescharacteristic
110C2×C22⋊C8D412yes—
111C2×C22⋊C8Q814yes—
112#405—24no—
113#405—42no—
114C2×#97—22no—
115C2×C4×C8—22no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380—21no—
120C2×#95—21no—
121#343—24no—
122#343—42no—
123#343—42no—
124#343—42no—
125#341C2×C418yescomplemented
126#340—22no—
127#340—22no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#336—22no—
137#336—22no—
138C25#4111yescharacteristic
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—44no—
149C23×C4—41no—
150C23×C4—41no—
151C23×C4—82no—
152C23×C4—82no—
153C23×C4C22×C412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×Q812yes—
159C23×C4C2×Q812yes—
160C23×C4#4112yes—
161C23×C4#4111yescharacteristic
162C23×C4C2411yescharacteristic
163C2×M16—24no—
164C2×M16—22no—
165C2×M16—21no—
166C2×M16—21no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—22no—
180C2×M16—48no—
181C2×M16—48no—
182C2×M16—42no—
183C2×M16—42no—
184C2×M16C22×C414yes—
185C2×M16C22×C412yes—
186C2×M16C22×C412yes—
187C2×M16C2×Q814yes—
188C22×C8—22no—
189C22×C8—22no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—24no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—42no—
198C22×C8—42no—
199C22×C8C22×C414yes—
200C22×C8C2×D412yes—
201C22×C8C2×D412yes—
202C22×C8C2×Q814yes—
203#117—22no—
204#117—22no—
205#117—22no—
206#117—22no—
207#117—22no—
208#117—44no—
209#117—44no—
210#117—41no—
211#117—41no—
212#117#4114yes—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4#4114yes—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—82no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—44no—
259#108—48no—
260#107—44no—
261#107—44no—
262#107—42no—
263#107—42no—
264#106—42no—
265#106—42no—
266C22⋊C8—24no—
267C22⋊C8—24no—
268C22⋊C8—24no—
269C22⋊C8—24no—
270C22⋊C8C4⋊C414yes—
271C22⋊C8C4⋊C414yes—
272C22⋊C8C4⋊C418yes—
273#97—44no—
274C4×C8—44no—
275#95—41no—
276#95—41no—
277C24—22no—
278C24—22no—
279C24—22no—
280C24—22no—
281C24—41no—
282C24—41no—
283C24C4×Q812yes—
284C24#12111yescharacteristic
285C24#12111yescharacteristic
286C24#12512yes—
287C24C2×#4111yescharacteristic
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4—82no—
330C22×C4—82no—
331C22×C4—82no—
332C22×C4—816no—
333C22×C4—82no—
334C22×C4—82no—
335C22×C4—82no—
336C22×C4—81no—
337C22×C4—81no—
338C22×C4—81no—
339C22×C4—81no—
340C22×C4C2×C4⋊C412yes—
341C22×C4C2×C4⋊C412yes—
342C22×C4C4×D412yes—
343C22×C4C4×D412yes—
344C22×C4C4×Q812yes—
345C22×C4#12111yescharacteristic
346C22×C4#12111yescharacteristic
347C22×C4C22⋊Q812yes—
348C22×C4C22⋊Q812yes—
349C22×C4#12512yes—
350C22×C4C4⋊Q812yes—
351C22×C4C4⋊Q812yes—
352C22×C4C23×C411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×Q811yescharacteristic
356C22×C4C22×Q811yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4112yes—
359C22×C4C2×#4112yes—
360C22×C4H(5,𝔽2)12yes—
361C22×C4H(5,𝔽2)12yes—
362M16—24no—
363M16—24no—
364M16—24no—
365M16—24no—
366M16—24no—
367M16—24no—
368M16—48no—
369M16—42no—
370M16—42no—
371M16—42no—
372M16—42no—
373M16—42no—
374M16—42no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—22no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—48no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—41no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8C2×C4⋊C414yes—
397C2×C8C2×C4⋊C414yes—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—24no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—24no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4#12312yes—
419C4⋊C4#12312yes—
420C22⋊C4—24no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—82no—
425C22⋊C4—82no—
426C42—24no—
427C42—42no—
428C42—42no—
429C42—42no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—24no—
439C23—21no—
440C23—21no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—42no—
449C23—42no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23#38912yes—
455C23#38912yes—
456C23#47212yes—
457C23#47312yes—
458C23C2×C4×Q811yescharacteristic
459C23C2×#12111yescharacteristic
460C23C2×#12511yescharacteristic
461C23#55112yes—
462C23#55112yes—
463C2×C4—24no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—44no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—88no—
498C2×C4—88no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—84no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—162no—
509C2×C4—162no—
510C2×C4#38912yes—
511C2×C4#38912yes—
512C2×C4C22×C4⋊C411yescharacteristic
513C2×C4C2×C4×D411yescharacteristic
514C2×C4C4×#4112yes—
515C2×C4#51812yes—
516C2×C4C2×C22⋊Q811yescharacteristic
517C2×C4C2×#12311yescharacteristic
518C2×C4#52511yescharacteristic
519C2×C4#52912yes—
520C2×C4C2×C4⋊Q811yescharacteristic
521C2×C4#53411yescharacteristic
522C2×C4#53511yescharacteristic
523C2×C4#53812yes—
524C2×C4#54612yes—
525C2×C4#54612yes—
526C2×C4#55412yes—
527C2×C4#55412yes—
528C2×C4#55412yes—
529C2×C4#55412yes—
530C8—24no—
531C8—24no—
532C8—44no—
533C8—42no—
534C8—44no—
535C8—42no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—48no—
554C22—48no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22#195214yes—
560C22C2×#38911yescharacteristic
561C22#236812yes—
562C22#241111yescharacteristic
563C22#241211yescharacteristic
564C22#250512yes—
565C22C2×#47211yescharacteristic
566C22C2×#47311yescharacteristic
567C22#306811yescharacteristic
568C22#306911yescharacteristic
569C4—22no—
570C4—22no—
571C4—42no—
572C4—42no—
573C4—84no—
574C4—81no—
575C4—81no—
576C4—161no—
577C4—161no—
578C4C4×#12311yescharacteristic
579C4#232111yescharacteristic
580C4#234912yes—
581C4#245012yes—
582C4#250211yescharacteristic
583C4#258311yescharacteristic
584C2—22no—
585C2—24no—
586C2—22no—
587C2—42no—
588C2#1404411yescharacteristic
589C2C2×#195211yescharacteristic
590C2#2118211yescharacteristic
591C2#2452011yescharacteristic
592C2#2452211yescharacteristic
593C2#2457011yescharacteristic
594C2#2457211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11270C2semidirect
7. #11249C2semidirect
8. #10896C2semidirect
9. #10870C2semidirect
24. #1855C4semidirect
88. C2×#108C2×C4semidirect
125. #341C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18552
#208462
C2×#18522
#203792
C2×#17362
#112702
#112492
#108962
#108702
#107392
#107272