Finite Group Databasebeta

← #356848

Subgroups — #356848

Order 512.

1778 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2360C211yesmaximal, characteristic
2C2×#1996C212yesmaximal
3C22⋊C4×C4⋊C4C212yesmaximal
4#10912C214yesmaximal
5#10846C211yesmaximal, characteristic
6#10801C212yesmaximal
7C4×#342C211yesmaximal, characteristic
8C4×#342C212yesmaximal
9C22×C22⋊Q8C2211yescharacteristic
10C22×C4×Q8C411yescharacteristic, complemented
11C23×C4⋊C4C2211yescharacteristic
12#2360—22no—
13#2360—22no—
14#2360—24no—
15C2×#391C411yescharacteristic, complemented
16C2×#389C411yescharacteristic, complemented
17C2×#389C412yescomplemented
18C2×#387C412yescomplemented
19C2×#385C2212yes—
20C2×#380C2212yes—
21C2×C4×C4⋊C4—22no—
22C2×C4×C4⋊C4C411yescharacteristic, complemented
23C2×C4×C22⋊C4C2214yes—
24C2×C4×C22⋊C4C2212yes—
25C2×C4×C22⋊C4C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1996C4116yescomplemented
28C2×#342C2211yescharacteristic
29C2×#342C2212yes—
30C2×#342C2212yes—
31C2×#342C2212yes—
32C2×#342C2214yes—
33C2×#342C2214yes—
34C2×#342C2211yescharacteristic
35#1436—24no—
36C4⋊#95—22no—
37C22⋊#95C2214yes—
38C23×Q8C2×C411yescharacteristic
39C24×C4C2312yes—
40C24×C4C2311yescharacteristic
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8C2×C418yes—
46C2×C4×Q8—21no—
47C2×C4×Q8—22no—
48C2×C4×Q8—21no—
49C2×C4×Q8—44no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—44no—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4D412yes—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4D414yes—
62C22×C22⋊C4C2311yescharacteristic
63C22×C22⋊C4C2312yes—
64C22×C22⋊C4C2314yes—
65C22×C22⋊C4C2312yes—
66C22×C22⋊C4C2311yescharacteristic
67C22×C22⋊C4C2312yes—
68C22×C42—22no—
69C22×C42—24no—
70C22×C42C2×C412yes—
71C22×C42C2×C411yescharacteristic
72C22×C42C2×C411yescharacteristic
73C22×C42C2×C412yes—
74#391—22no—
75#391—22no—
76#389—22no—
77#389—22no—
78#389—24no—
79#389—24no—
80#387—24no—
81#387—24no—
82#385—24no—
83#385—24no—
84#380—24no—
85#380C2×C418yes—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—48no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—24no—
93C4×C22⋊C4—24no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—28no—
96C4×C22⋊C4—44no—
97C4×C22⋊C4C2×C418yes—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—24no—
101C2×#95—24no—
102C2×#95—24no—
103C2×#95C2×C411yescharacteristic
104C2×#95C2×C412yes—
105C2×#95C2×C411yescharacteristic
106C2×#95C2×C412yes—
107C2×#95C2×C412yes—
108C2×#95C2×C412yes—
109#342—22no—
110#342—24no—
111#342—24no—
112#342—42no—
113#342C2×C4116yes—
114#342C2×C4116yes—
115#342Q814yescomplemented
116C25C2411yescharacteristic
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8C4214yescomplemented
123C23×C4—22no—
124C23×C4—24no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—44no—
132C23×C4—82no—
133C23×C4C22×C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C412yes—
138C23×C4C22×C412yes—
139C23×C4C22×C412yes—
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C414yes—
144C23×C4C2×D414yes—
145C23×C4C2×D412yes—
146C23×C4C2×D414yes—
147C23×C4C2×D412yes—
148C23×C4#4112yes—
149C23×C4#4112yes—
150C22⋊Q8—28no—
151C22⋊Q8—28no—
152C22⋊Q8—28no—
153C22⋊Q8—28no—
154C4×Q8—44no—
155C4×Q8—48no—
156C4×Q8—44no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4C4214yescomplemented
177C2×C4⋊C4C4218yescomplemented
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C414yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C414yes—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C411yescharacteristic
185C2×C22⋊C4—24no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4C22×C414yes—
208C2×C22⋊C4C22×C418yes—
209C2×C22⋊C4C22×C418yes—
210C2×C22⋊C4C22×C414yes—
211C2×C22⋊C4C2×Q812yes—
212C2×C22⋊C4C2×Q814yes—
213C2×C22⋊C4#4112yes—
214C2×C22⋊C4#4114yes—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—28no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—44no—
225C2×C42—44no—
226C2×C42—42no—
227C2×C42—44no—
228C2×C42—42no—
229C2×C42—48no—
230C2×C42—48no—
231C2×C42—84no—
232C2×C42—82no—
233#95—22no—
234#95—24no—
235#95—22no—
236#95—24no—
237#95—22no—
238#95—22no—
239#95—24no—
240#95—28no—
241#95—28no—
242#95—48no—
243#95—48no—
244#95—84no—
245C24—41no—
246C24—41no—
247C24—41no—
248C24—41no—
249C24C23×C411yescharacteristic
250C24C23×C412yes—
251C24C22×D412yes—
252C24C22×Q811yescharacteristic
253C24C2×#4111yescharacteristic
254C24C2×#4111yescharacteristic
255C24C2×#4112yes—
256C24C2×#4111yescharacteristic
257C24H(5,𝔽2)11yescharacteristic
258C242−1+411yescharacteristic
259C242−1+412yes—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—48no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—48no—
266C2×Q8—42no—
267C2×Q8—42no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—28no—
292C22×C4—24no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—28no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—48no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—44no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—44no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—84no—
343C22×C4—88no—
344C22×C4—84no—
345C22×C4C2×C4212yes—
346C22×C4C2×C4214yes—
347C22×C4C2×C4214yes—
348C22×C4C2×C4212yes—
349C22×C4C2×C22⋊C414yes—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4#11712yes—
356C22×C4#11712yes—
357C22×C4C4×D414yes—
358C22×C4C4×D414yes—
359C22×C4C4×D414yes—
360C22×C4C4×D414yes—
361C22×C4C4×Q812yes—
362C22×C4C4×Q812yes—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—216no—
366C4⋊C4—28no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—42no—
371C4⋊C4—44no—
372C4⋊C4—416no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—84no—
376C4⋊C4#9914yes—
377C22⋊C4—28no—
378C22⋊C4—28no—
379C22⋊C4—24no—
380C22⋊C4—24no—
381C22⋊C4—24no—
382C22⋊C4—24no—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—28no—
386C22⋊C4—24no—
387C22⋊C4—28no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4C4×Q818yescomplemented
395C42—42no—
396C42—416no—
397C42—42no—
398C42—44no—
399C42—44no—
400C42—42no—
401C42—42no—
402C42—82no—
403C42—82no—
404C42—816no—
405C23—21no—
406C23—22no—
407C23—21no—
408C23—22no—
409C23—21no—
410C23—22no—
411C23—21no—
412C23—22no—
413C23—42no—
414C23—41no—
415C23—41no—
416C23—42no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—42no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—42no—
430C23—42no—
431C23C22×C4211yescharacteristic
432C23C22×C22⋊C411yescharacteristic
433C23C2×#11711yescharacteristic
434C23C2×C4×D412yes—
435C23C2×C4×D412yes—
436C23C2×C4×Q811yescharacteristic
437C23C2×C4×Q812yes—
438C23C4×#4111yescharacteristic
439C23C4×#4112yes—
440C23#51911yescharacteristic
441C23#51912yes—
442C23#52011yescharacteristic
443C23#52012yes—
444C23#54612yes—
445C23#54712yes—
446C23#54812yes—
447C23D4×Q812yes—
448C23#55212yes—
449C23#55411yescharacteristic
450C23#55611yescharacteristic
451C23#55612yes—
452C23#55711yescharacteristic
453C23#55911yescharacteristic
454Q8—48no—
455Q8—48no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—28no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—48no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—48no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—48no—
509C2×C4—44no—
510C2×C4—48no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—44no—
514C2×C4—48no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—84no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—84no—
523C2×C4—84no—
524C2×C4—82no—
525C2×C4—84no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—84no—
529C2×C4—82no—
530C2×C4—84no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4C4×C22⋊C414yes—
536C2×C4C4×C22⋊C412yes—
537C2×C4C4×C22⋊C412yes—
538C2×C4C2×#9914yes—
539C2×C4C2×#9912yes—
540C2×C4#41012yes—
541C22—21no—
542C22—22no—
543C22—22no—
544C22—21no—
545C22—21no—
546C22—22no—
547C22—22no—
548C22—21no—
549C22—22no—
550C22—21no—
551C22—21no—
552C22—22no—
553C22—22no—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—21no—
558C22—21no—
559C22—22no—
560C22—22no—
561C22—21no—
562C22—21no—
563C22—22no—
564C22—22no—
565C22—42no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—41no—
571C22—42no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×C4×C22⋊C411yescharacteristic
576C22C42×Q811yescharacteristic
577C22#227611yescharacteristic
578C22#233911yescharacteristic
579C22Q8×C22⋊C411yescharacteristic
580C22#235511yescharacteristic
581C22#235611yescharacteristic
582C22#236012yes—
583C22#236212yes—
584C22#236512yes—
585C22#236712yes—
586C22C22×#9911yescharacteristic
587C22C2×#41011yescharacteristic
588C22#288311yescharacteristic
589C22#288411yescharacteristic
590C4—24no—
591C4—22no—
592C4—24no—
593C4—48no—
594C4—44no—
595C4—44no—
596C4—42no—
597C4—44no—
598C4—42no—
599C4—42no—
600C4—88no—
601C4—84no—
602C4—81no—
603C4—81no—
604C4C4×#9914yes—
605C2—22no—
606C2—22no—
607C2—22no—
608C2—21no—
609C2—21no—
610C2—22no—
611C2—21no—
612C2—21no—
613C2#1381711yescharacteristic
614C2C2×C4×#9911yescharacteristic
615C2#2039311yescharacteristic
616C2#2247611yescharacteristic
617C2#2247811yescharacteristic
618C2Q8×#9911yescharacteristic
619C2#2251311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC23C22×C42
Frattini, prime radical, cosocleC25C24
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4⋊C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
10. C22×C4×Q8C4semidirect
15. C2×#391C4semidirect
16. C2×#389C4semidirect
17. C2×#389C4semidirect
18. C2×#387C4semidirect
22. C2×C4×C4⋊C4C4semidirect
27. #1996C4semidirect
115. #342Q8semidirect
122. C22×Q8C42semidirect
176. C2×C4⋊C4C42semidirect
177. C2×C4⋊C4C42semidirect
394. C22⋊C4C4×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23602
C2×#19962
C22⋊C4×C4⋊C42
#109122
#108462
#108012
C4×#3422