Finite Group Databasebeta

← #476371

Subgroups — #476371

Order 512.

1310 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
1C2×#2043C212yesmaximal, complemented
2C2×#2035C212yesmaximal
3#15178C211yesmaximal, characteristic
4#15135C211yesmaximal, characteristic
5#11220C212yesmaximal, complemented
6#11147C211yesmaximal, characteristic
7#11122C212yesmaximal, complemented
8#11050C212yesmaximal, complemented
9#10970C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×C22⋊Q8C2212yes—
12C22×#106C2212yes—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×C22⋊C8C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#399—21no—
20C2×#391C2212yes—
21C2×#389C2212yes—
22C2×#384C2211yescharacteristic
23C2×#380C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#2043—28no—
26#2035—28no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C2×#336C2212yes—
32#1445—21no—
33#1444—21no—
34#1444—21no—
35#1444—21no—
36#1439—21no—
37#1439—21no—
38#1439—21no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41C23×Q8C2312yes—
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4⋊Q8—24no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—42no—
49C2×C22⋊Q8—24no—
50C2×C22⋊Q8—24no—
51C2×C22⋊Q8—42no—
52C2×C22⋊Q8—42no—
53C2×C22⋊Q8—42no—
54C2×C22⋊Q8—42no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4—22no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4D411yescharacteristic
67C22×C22⋊C4D411yescharacteristic
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×#106—24no—
71C2×#106—24no—
72C2×#106—42no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103D418yes—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—41no—
87C2×C22⋊C8—42no—
88C2×C22⋊C8—41no—
89#399—44no—
90#391—24no—
91#391—42no—
92#389—24no—
93#389—42no—
94#384—41no—
95#384—41no—
96#380—24no—
97#380—42no—
98C4×C22⋊C4—41no—
99C4×C22⋊C4—41no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95D411yescharacteristic
109C2×#95D411yescharacteristic
110#340—24no—
111#340—42no—
112#340—42no—
113#336—24no—
114#336—42no—
115C25C2×D411yescharacteristic
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8C2×D414yes—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8C2×D414yes—
162C4⋊Q8—44no—
163C4⋊Q8—48no—
164C4⋊Q8—44no—
165C22⋊Q8—48no—
166C22⋊Q8—48no—
167C22⋊Q8—48no—
168C22⋊Q8—48no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4#4114yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4SD1614yes—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—21no—
221C2×C42—21no—
222#106—48no—
223#103—48no—
224#103—48no—
225#103—48no—
226#103—48no—
227C22⋊C8—44no—
228C22⋊C8—44no—
229#95—41no—
230#95—41no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—42no—
236#95—42no—
237#95—42no—
238#95—42no—
239C24—21no—
240C24—21no—
241C24—21no—
242C24—22no—
243C24—22no—
244C24—21no—
245C24—21no—
246C24—21no—
247C24—22no—
248C24—22no—
249C24C2×SD1612yes—
250C24C22×D411yescharacteristic
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—42no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4C2×SD1612yes—
345C22×C4C2×SD1612yes—
346C22×C4#13712yes—
347C22×C4#13712yes—
348C22×C4C22×D412yes—
349C22×C4C2×#4112yes—
350C22×C4H(5,𝔽2)12yes—
351C22×C4H(5,𝔽2)12yes—
352C22×C4H(5,𝔽2)11yescharacteristic
353C22×C4H(5,𝔽2)12yes—
354C22×C4H(5,𝔽2)11yescharacteristic
355C22×C42−1+412yes—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—416no—
375C4⋊C4—48no—
376C4⋊C4—48no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—88no—
386C4⋊C4—84no—
387C4⋊C4—84no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—416no—
391C22⋊C4—88no—
392C42—42no—
393C42—42no—
394C42—41no—
395C42—41no—
396C42—41no—
397C42—41no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—24no—
402C23—24no—
403C23—41no—
404C23—42no—
405C23—41no—
406C23—42no—
407C23—42no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23—42no—
414C23—41no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23#53411yescharacteristic
422C23#53511yescharacteristic
423C23#53611yescharacteristic
424C23D4212yes—
425C23#54612yes—
426C23C22×SD1611yescharacteristic
427C23C2×#13711yescharacteristic
428C23C2×#13711yescharacteristic
429C23#57511yescharacteristic
430C23#57712yes—
431C23#57812yes—
432Q8—44no—
433Q8—44no—
434Q8—44no—
435Q8—44no—
436Q8—82no—
437Q8—82no—
438Q8—82no—
439Q8—82no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—41no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4#53412yes—
498C2×C4#53512yes—
499C2×C4#53712yes—
500C2×C4#53812yes—
501C2×C4#54012yes—
502C2×C4#54212yes—
503C2×C4D4211yescharacteristic
504C2×C4#54612yes—
505C2×C4#54712yes—
506C2×C4#55911yescharacteristic
507C2×C4#56011yescharacteristic
508C2×C4#56012yes—
509C2×C4#56011yescharacteristic
510C2×C4#56212yes—
511C2×C4#56212yes—
512C2×C4#56212yes—
513C8—44no—
514C8—82no—
515C22—42no—
516C22—44no—
517C22—42no—
518C22—42no—
519C22—44no—
520C22—81no—
521C22—82no—
522C22—81no—
523C22—81no—
524C22—82no—
525C22—81no—
526C22#266911yescharacteristic
527C22#267212yes—
528C22#267812yes—
529C22#270911yescharacteristic
530C22#281611yescharacteristic
531C22#318711yescharacteristic
532C22#319511yescharacteristic
533C22#320011yescharacteristic
534C22#320211yescharacteristic
535C22#320611yescharacteristic
536C22#321611yescharacteristic
537C22#322512yes—
538C22#322911yescharacteristic
539C22#323711yescharacteristic
540C22#324212yes—
541C22#327012yes—
542C22#327112yes—
543C22#327612yes—
544C22D4×SD1612yes—
545C22#329812yes—
546C22#331512yes—
547C22#332012yes—
548C22#333112yes—
549C4—42no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—84no—
554C4—84no—
555C4—81no—
556C4—84no—
557C4—162no—
558C4#267112yes—
559C4#271012yes—
560C4#281312yes—
561C4#282912yes—
562C2—82no—
563C2#1708811yescharacteristic
564C2#3050611yescharacteristic
565C2#3051711yescharacteristic
566C2#3063312yes—
567C2#3065012yes—
568C2#3098612yes—
569C2#3101312yes—
570C2#3202711yescharacteristic
571C2#3203711yescharacteristic
572C2#3277111yescharacteristic
573C2#3278011yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2043C2semidirect
5. #11220C2semidirect
7. #11122C2semidirect
8. #11050C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20432
C2×#20352
#151782
#151352
#112202
#111472
#111222
#110502
#109702