Finite Group Databasebeta

← #355522

Subgroups — #355522

Order 512.

1638 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2326C211yesmaximal, characteristic
2C2×#1983C212yesmaximal
3C2×#1768C211yesmaximal, characteristic, complemented
4#13797C211yesmaximal, characteristic, complemented
5#10928C211yesmaximal, characteristic, complemented
6#10927C211yesmaximal, characteristic
7#10926C211yesmaximal, characteristic, complemented
8#10925C211yesmaximal, characteristic
9#10849C212yesmaximal, complemented
10#10789C212yesmaximal, complemented
11#10788C212yesmaximal
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×#97C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22#2326—24no—
23#2326C418yescomplemented
24C2×#391C2211yescharacteristic
25C2×#390C2211yescharacteristic
26C2×#388—22no—
27C2×#385—22no—
28C2×#384C2211yescharacteristic
29C2×C4×C4⋊C4C2212yes—
30C2×#376C2211yescharacteristic
31#1983C4116yescomplemented
32#1768—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41#1434—22no—
42#1434—22no—
43C23×Q8C2311yescharacteristic
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×#124—22no—
49C2×#124—24no—
50C2×#124C2×C414yes—
51C2×#124C2×C418yes—
52C2×C4×Q8—22no—
53C2×C4×Q8—42no—
54C2×C4×Q8—42no—
55C2×C4×Q8C2×C414yes—
56C2×C4×D4—22no—
57C2×C4×D4—42no—
58C2×C4×D4—42no—
59C2×C4×D4C2×C414yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C22⋊C4—22no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C42—21no—
70C22×C42—22no—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×C4⋊C8—24no—
75C2×C4⋊C8—24no—
76C2×C4⋊C8C2×C418yescomplemented
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103C2×C418yescomplemented
82C2×#103C2×C418yescomplemented
83C2×#102—42no—
84C2×#102—42no—
85C2×#102—42no—
86C2×#102—42no—
87C2×#102C2×C418yes—
88C2×#102C2×C418yes—
89C2×#97—22no—
90C2×#97—42no—
91C2×C4×C8—22no—
92C2×C4×C8—42no—
93#391—22no—
94#391C2×C414yes—
95#390—22no—
96#390C2×C414yes—
97#388—216no—
98#385—216no—
99#384—22no—
100#384C2×C414yes—
101C4×C4⋊C4—24no—
102C4×C4⋊C4C2×C418yescomplemented
103#376C2×C414yes—
104#376D414yes—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95D412yes—
112C2×#95D412yes—
113#340—42no—
114#340—42no—
115#340—42no—
116#340—42no—
117#336—24no—
118C25—21no—
119C22×Q8—21no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8C22×C412yes—
126C22×Q8C22×C414yes—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4C22×C412yes—
134C22×D4C22×C414yes—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—24no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—82no—
144C23×C4—82no—
145C23×C4C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2411yescharacteristic
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×C8—22no—
162C22×C8C22×C414yes—
163C22×C8C22×C414yes—
164#124—28no—
165#124—216no—
166#124C42116yes—
167C4×Q8—48no—
168C4×Q8—48no—
169C4×D4—48no—
170C4×D4—48no—
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—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4C22×C412yes—
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C414yes—
196C2×C4⋊C4C22×C414yes—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4—82no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—24no—
211C2×C42—28no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—82no—
215C2×C42—82no—
216C2×C42C22×C412yes—
217C2×C42C22×C412yes—
218C2×C42C22×C412yes—
219C2×C42C22×C414yes—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42#4114yes—
224C4⋊C8—216no—
225#103—48no—
226#103—48no—
227#102—48no—
228#102—48no—
229#97—44no—
230C4×C8—44no—
231#95—28no—
232#95—28no—
233#95—82no—
234#95—82no—
235#95—82no—
236#95—82no—
237#95C22⋊C418yes—
238#95C22⋊C418yes—
239C24—22no—
240C24—24no—
241C24—41no—
242C24—41no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24C22×D411yescharacteristic
249C2×Q8—22no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—42no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8C2×C4214yes—
265C2×D4—22no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4C2×C4214yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
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—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—48no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
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—42no—
328C22×C4—42no—
329C22×C4—88no—
330C22×C4—84no—
331C22×C4—84no—
332C22×C4—88no—
333C22×C4C2×C22⋊C414yes—
334C22×C4C2×C22⋊C412yes—
335C22×C4C2×C22⋊C412yes—
336C22×C4C2×C22⋊C414yes—
337C22×C4C4×D414yes—
338C22×C4C4×D414yes—
339C22×C4C23×C411yescharacteristic
340C22×C4C23×C412yes—
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4112yes—
343C22×C4H(5,𝔽2)12yes—
344C22×C4H(5,𝔽2)12yes—
345C22×C4H(5,𝔽2)12yes—
346C22×C42−1+412yes—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—48no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—88no—
375C22⋊C4—232no—
376C22⋊C4—48no—
377C22⋊C4—88no—
378C42—24no—
379C42—24no—
380C42—22no—
381C42—24no—
382C42—22no—
383C42—48no—
384C42—88no—
385C42C2×C4214yes—
386C42#11714yes—
387C42C4×D418yes—
388C23—24no—
389C23—42no—
390C23—42no—
391C23—44no—
392C23—44no—
393C23—44no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—81no—
400C23—81no—
401C23—81no—
402C23—81no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23C22×C22⋊C411yescharacteristic
410C23C2×C4×D412yes—
411C23#53511yescharacteristic
412C23#53511yescharacteristic
413C23#53712yes—
414C23#57612yes—
415C23#57712yes—
416C23#57712yes—
417C23#57812yes—
418Q8—28no—
419Q8—44no—
420Q8—44no—
421Q8—44no—
422D4—28no—
423D4—44no—
424D4—44no—
425D4—44no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—816no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—81no—
468C2×C4—82no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4C4×C22⋊C418yes—
480C2×C4C22×C4211yescharacteristic
481C2×C4C2×#11711yescharacteristic
482C2×C4C2×C4×D412yes—
483C2×C4#51812yes—
484C2×C4#51812yes—
485C2×C4#51812yes—
486C2×C4#52012yes—
487C2×C4#52012yes—
488C2×C4#52012yes—
489C2×C4#53512yes—
490C2×C4#53511yescharacteristic
491C2×C4#53611yescharacteristic
492C2×C4#54012yes—
493C2×C4#54012yes—
494C2×C4#54212yes—
495C2×C4#54212yes—
496C8—44no—
497C8—44no—
498C22—48no—
499C22—44no—
500C22—82no—
501C22—82no—
502C22—84no—
503C22—84no—
504C22—84no—
505C22—84no—
506C22—84no—
507C22C2×C4×C22⋊C411yescharacteristic
508C22#230911yescharacteristic
509C22#231111yescharacteristic
510C22#232412yes—
511C22#232412yes—
512C22#289712yes—
513C22#289812yes—
514C22#294612yes—
515C22#294712yes—
516C22#295112yes—
517C22#295212yes—
518C22#322011yescharacteristic
519C22#322111yescharacteristic
520C22#322211yescharacteristic
521C22#322311yescharacteristic
522C22#322412yes—
523C22#322512yes—
524C22#325712yes—
525C22#325812yes—
526C22#326212yes—
527C22#326312yes—
528C4—28no—
529C4—42no—
530C4—42no—
531C4—44no—
532C4—44no—
533C4—82no—
534C4—82no—
535C4—84no—
536C4—84no—
537C4#227512yes—
538C4#231912yes—
539C4#232612yes—
540C4#232812yes—
541C2—84no—
542C2#1378511yescharacteristic
543C2#2041512yes—
544C2#2181311yescharacteristic
545C2#2181411yescharacteristic
546C2#2189612yes—
547C2#2215712yes—
548C2#2215912yes—
549C2#2216112yes—
550C2#2216312yes—
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
3. C2×#1768C2semidirect
4. #13797C2semidirect
5. #10928C2semidirect
7. #10926C2semidirect
9. #10849C2semidirect
10. #10789C2semidirect
23. #2326C4semidirect
31. #1983C4semidirect
76. C2×C4⋊C8C2×C4semidirect
81. C2×#103C2×C4semidirect
82. C2×#103C2×C4semidirect
102. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23262
C2×#19832
C2×#17682
#137972
#109282
#109272
#109262
#109252
#108492
#107892
#107882