Finite Group Databasebeta

← #355534

Subgroups — #355534

Order 512.

1646 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2326C211yesmaximal, characteristic
2C2×#1959C211yesmaximal, characteristic
3C2×#1958C211yesmaximal, characteristic
4C2×#1746C211yesmaximal, characteristic, complemented
5#13779C211yesmaximal, characteristic, complemented
6#10932C212yesmaximal, complemented
7#10931C212yesmaximal
8#10843C211yesmaximal, characteristic, complemented
9#10841C211yesmaximal, characteristic, complemented
10#10789C212yesmaximal, complemented
11#10788C212yesmaximal
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#102C2212yes—
18C22×#102C2212yes—
19C22×#97C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21#2326—24no—
22#2326C418yescomplemented
23C2×#391C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#388—22no—
26C2×#385—22no—
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4C2212yes—
31C2×#376C2211yescharacteristic
32#1959C418yescomplemented
33#1958C418yescomplemented
34#1746—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1436—22no—
42#1436—22no—
43C4⋊#95C2212yes—
44C23×Q8C2311yescharacteristic
45C23×D4C2311yescharacteristic
46C24×C4—21no—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×#124—22no—
50C2×#124—22no—
51C2×#124—22no—
52C2×#124C2×C414yes—
53C2×#124C2×C414yes—
54C2×#124C2×C414yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—42no—
57C2×C4×Q8—42no—
58C2×C4×Q8C2×C414yes—
59C2×C4×D4—22no—
60C2×C4×D4—42no—
61C2×C4×D4—42no—
62C2×C4×D4C2×C414yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C22⋊C4—22no—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—22no—
73C22×C42—21no—
74C22×C42—22no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103C2×C418yescomplemented
83C2×#103C2×C418yescomplemented
84C2×#102—42no—
85C2×#102—42no—
86C2×#102—42no—
87C2×#102—42no—
88C2×#102C2×C418yes—
89C2×#102C2×C418yes—
90C2×#97—22no—
91C2×#97—22no—
92C2×#97C2×C414yescomplemented
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8C2×C414yescomplemented
96#391—22no—
97#391C2×C414yes—
98#390—22no—
99#390C2×C414yes—
100#388—216no—
101#385—216no—
102#384—22no—
103#384—22no—
104#384C2×C414yes—
105#384C2×C414yescomplemented
106#382—22no—
107#382C2×C414yescomplemented
108C4×C4⋊C4—42no—
109C4×C4⋊C4—42no—
110#376C2×C414yes—
111#376D414yes—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95D412yes—
118C2×#95D412yes—
119#340—42no—
120#340—42no—
121#340—42no—
122#340—42no—
123#336—24no—
124#336—24no—
125C25—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8C22×C412yes—
134C22×Q8C22×C412yes—
135C22×Q8C22×C412yes—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4—22no—
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4C22×C412yes—
144C22×D4C22×C412yes—
145C22×D4C22×C412yes—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—82no—
155C23×C4—82no—
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2411yescharacteristic
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8C22×C414yes—
174C22×C8C22×C414yes—
175#124—28no—
176#124—28no—
177#124—28no—
178#124C42116yescomplemented
179C4×Q8—48no—
180C4×Q8—48no—
181C4×D4—48no—
182C4×D4—48no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C414yes—
211C2×C4⋊C4C22×C414yes—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—82no—
220C2×C22⋊C4—82no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—24no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42—44no—
229C2×C42—82no—
230C2×C42—82no—
231C2×C42C22×C412yes—
232C2×C42C22×C412yes—
233C2×C42C22×C412yes—
234C2×C42C22×C412yes—
235C2×C42C22×C412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D412yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241#103—48no—
242#103—48no—
243#102—48no—
244#102—48no—
245#97—28no—
246C4×C8—28no—
247#95—28no—
248#95—28no—
249#95—44no—
250#95—82no—
251#95—82no—
252#95C22⋊C418yes—
253#95C22⋊C418yes—
254C24—22no—
255C24—22no—
256C24—22no—
257C24—41no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—42no—
265C24C22×D411yescharacteristic
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8C2×C4214yes—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—24no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—24no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—42no—
297C2×D4C2×C4214yes—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
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—22no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—48no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
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×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—88no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4—88no—
352C22×C4C2×C22⋊C414yes—
353C22×C4C2×C22⋊C412yes—
354C22×C4C2×C22⋊C412yes—
355C22×C4C2×C22⋊C414yes—
356C22×C4C4×D414yes—
357C22×C4C4×D414yes—
358C22×C4C23×C411yescharacteristic
359C22×C4C23×C411yescharacteristic
360C22×C4C23×C411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4H(5,𝔽2)12yes—
365C22×C4H(5,𝔽2)12yes—
366C22×C4H(5,𝔽2)12yes—
367C22×C42−1+412yes—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—24no—
373C2×C8—24no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—24no—
388C4⋊C4—48no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—88no—
396C4⋊C4—88no—
397C22⋊C4—232no—
398C22⋊C4—48no—
399C22⋊C4—88no—
400C42—24no—
401C42—24no—
402C42—24no—
403C42—24no—
404C42—48no—
405C42—88no—
406C42C2×C4214yes—
407C42#11714yes—
408C42C4×D414yes—
409C42C4×D414yes—
410C23—24no—
411C23—42no—
412C23—42no—
413C23—42no—
414C23—42no—
415C23—42no—
416C23—42no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—81no—
424C23—81no—
425C23—81no—
426C23—81no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23C22×C22⋊C411yescharacteristic
434C23C2×C4×D411yescharacteristic
435C23C2×C4×D411yescharacteristic
436C23#53511yescharacteristic
437C23#53511yescharacteristic
438C23#53511yescharacteristic
439C23#53611yescharacteristic
440C23#57612yes—
441C23#57712yes—
442C23#57712yes—
443C23#57812yes—
444Q8—28no—
445Q8—44no—
446Q8—44no—
447Q8—44no—
448D4—28no—
449D4—44no—
450D4—44no—
451D4—44no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
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—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—816no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4C4×C22⋊C418yes—
508C2×C4C22×C4211yescharacteristic
509C2×C4C2×#11711yescharacteristic
510C2×C4C2×C4×D411yescharacteristic
511C2×C4C2×C4×D411yescharacteristic
512C2×C4#51812yes—
513C2×C4#51812yes—
514C2×C4#51812yes—
515C2×C4#52012yes—
516C2×C4#52012yes—
517C2×C4#52012yes—
518C2×C4#53512yes—
519C2×C4#53712yes—
520C2×C4#54012yes—
521C2×C4#54012yes—
522C2×C4#54212yes—
523C2×C4#54212yes—
524C8—44no—
525C8—44no—
526C22—48no—
527C22—44no—
528C22—84no—
529C22—82no—
530C22—82no—
531C22—82no—
532C22—82no—
533C22—84no—
534C22—82no—
535C22—82no—
536C22—84no—
537C22C2×C4×C22⋊C411yescharacteristic
538C22#230911yescharacteristic
539C22#231111yescharacteristic
540C22#232611yescharacteristic
541C22#232611yescharacteristic
542C22#232811yescharacteristic
543C22#232811yescharacteristic
544C22#289712yes—
545C22#289812yes—
546C22#294612yes—
547C22#294712yes—
548C22#295112yes—
549C22#295212yes—
550C22#322011yescharacteristic
551C22#322111yescharacteristic
552C22#322211yescharacteristic
553C22#322311yescharacteristic
554C22#322412yes—
555C22#322412yes—
556C22#322512yes—
557C22#322512yes—
558C22#324312yes—
559C22#324412yes—
560C4—28no—
561C4—42no—
562C4—42no—
563C4—44no—
564C4—44no—
565C4—84no—
566C4—84no—
567C4—82no—
568C4—82no—
569C4#227512yes—
570C4#231912yes—
571C4#232412yes—
572C4#232412yes—
573C2—84no—
574C2#1378511yescharacteristic
575C2#2041512yes—
576C2#2181311yescharacteristic
577C2#2181411yescharacteristic
578C2#2187112yes—
579C2#2222711yescharacteristic
580C2#2222811yescharacteristic
581C2#2223312yes—
582C2#2227412yes—
583C2#2227811yescharacteristic
584C2#2227911yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1746C2semidirect
5. #13779C2semidirect
6. #10932C2semidirect
8. #10843C2semidirect
9. #10841C2semidirect
10. #10789C2semidirect
22. #2326C4semidirect
32. #1959C4semidirect
33. #1958C4semidirect
82. C2×#103C2×C4semidirect
83. C2×#103C2×C4semidirect
92. C2×#97C2×C4semidirect
95. C2×C4×C8C2×C4semidirect
105. #384C2×C4semidirect
107. #382C2×C4semidirect
178. #124C42semidirect

Maximal subgroups

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

subgroupindex
C2×#23262
C2×#19592
C2×#19582
C2×#17462
#137792
#109322
#109312
#108432
#108412
#107892
#107882