Finite Group Databasebeta

← #360382

Subgroups — #360382

Order 512.

1142 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1835C211yesmaximal, characteristic
2C2×#1805C211yesmaximal, characteristic
3C2×C4×#103C211yesmaximal, characteristic
4#13942C211yesmaximal, characteristic
5#13804C211yesmaximal, characteristic
6#11315C212yesmaximal
7#11190C212yesmaximal
8#10932C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10834C211yesmaximal, characteristic
11#10807C212yesmaximal
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2211yescharacteristic
16C22×#103C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#391C2212yes—
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2212yes—
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1835—24no—
25#1805—24no—
26C4×#103C418yescomplemented
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1441—22no—
37#1436—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40#1434—21no—
41#1434—21no—
42#1434—21no—
43C4×#95—21no—
44C23×Q8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8C2×C414yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C42—21no—
62C22×C42—21no—
63C22×C42—22no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—42no—
76C2×#103C2×C414yes—
77C2×#103C2×C414yes—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80C2×C4×C8C2×C414yes—
81#391—24no—
82#391—42no—
83#382—22no—
84#382—22no—
85#382—22no—
86#382D414yes—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—42no—
92C4×C4⋊C4C2×C414yes—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95D411yescharacteristic
106C2×#95D411yescharacteristic
107#340—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—42no—
112#340C2×C414yes—
113#340C2×C414yes—
114#336—42no—
115#336—42no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8C22×C412yes—
128C22×Q8#4112yes—
129C22×Q8#4112yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4C2×D412yes—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8C22×C412yes—
161C22×C8C22×C412yes—
162C4×Q8—28no—
163C4×Q8—28no—
164C4×Q8—44no—
165C4×Q8—44no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
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—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C2×D414yes—
200C2×C4⋊C4#4112yes—
201C2×C4⋊C4#4112yes—
202C2×C42—24no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42—44no—
213C2×C42—44no—
214C2×C42—82no—
215C2×C42—82no—
216C2×C42C22×C412yes—
217C2×C42C2×D412yes—
218C2×C42#4112yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C4⋊C8—44no—
223#103—28no—
224#103—28no—
225#103—48no—
226C4×C8—28no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—22no—
231#95—22no—
232#95—44no—
233#95—44no—
234#95—44no—
235#95—44no—
236#95—42no—
237#95—42no—
238#95—44no—
239#95—44no—
240#95—82no—
241C24C22×D411yescharacteristic
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8#11714yes—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
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×C4C4×D412yes—
324C22×C4C4×D412yes—
325C22×C4#12112yes—
326C22×C4#12112yes—
327C22×C4#12312yes—
328C22×C4#12312yes—
329C22×C4#13514yes—
330C22×C4C23×C411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C42−1+411yescharacteristic
343C22×C42−1+411yescharacteristic
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—48no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—48no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4#11714yes—
387C42—42no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42—48no—
392C42—84no—
393C42—88no—
394C42#11714yes—
395C42C4×D414yes—
396C42#12414yes—
397C42#12614yes—
398C23C2×C4×D411yescharacteristic
399C23C2×#12111yescharacteristic
400C23C2×#12311yescharacteristic
401C23#54611yescharacteristic
402C23#54611yescharacteristic
403C23#54811yescharacteristic
404C23#54811yescharacteristic
405C23C2×#13512yes—
406C23#57512yes—
407C23#57712yes—
408C23#57812yes—
409Q8—24no—
410Q8—24no—
411Q8—42no—
412Q8—42no—
413Q8—42no—
414Q8—42no—
415Q8—42no—
416Q8—42no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4#38814yes—
476C2×C4#38814yes—
477C2×C4C2×#11711yescharacteristic
478C2×C4C2×#11711yescharacteristic
479C2×C4C2×#11711yescharacteristic
480C2×C4C2×C4×D411yescharacteristic
481C2×C4#52011yescharacteristic
482C2×C4#52011yescharacteristic
483C2×C4C2×#12411yescharacteristic
484C2×C4C2×#12611yescharacteristic
485C2×C4#52911yescharacteristic
486C2×C4#52911yescharacteristic
487C2×C4#52911yescharacteristic
488C2×C4#52911yescharacteristic
489C2×C4#54011yescharacteristic
490C2×C4#54111yescharacteristic
491C2×C4#54211yescharacteristic
492C2×C4#54211yescharacteristic
493C2×C4#54611yescharacteristic
494C2×C4#54711yescharacteristic
495C2×C4#54811yescharacteristic
496C2×C4#55011yescharacteristic
497C2×C4#55211yescharacteristic
498C2×C4#55211yescharacteristic
499C2×C4#55211yescharacteristic
500C2×C4#55311yescharacteristic
501C2×C4#55311yescharacteristic
502C2×C4#55311yescharacteristic
503C2×C4#55511yescharacteristic
504C2×C4#55611yescharacteristic
505C8—42no—
506C8—42no—
507C8—82no—
508C22C2×#38811yescharacteristic
509C22#235211yescharacteristic
510C22#235211yescharacteristic
511C22#242011yescharacteristic
512C22#242111yescharacteristic
513C22#242611yescharacteristic
514C22#242711yescharacteristic
515C22C4×#13511yescharacteristic
516C22#294311yescharacteristic
517C22#294911yescharacteristic
518C22#295011yescharacteristic
519C22#303512yes—
520C22#304612yes—
521C22#309112yes—
522C22#309712yes—
523C22#330012yes—
524C22#330112yes—
525C22#332312yes—
526C22#332712yes—
527C22#336412yes—
528C22#336512yes—
529C22#337612yes—
530C22#337712yes—
531C4—24no—
532C4—24no—
533C4—44no—
534C4—44no—
535C4—44no—
536C4—82no—
537C4—82no—
538C4—84no—
539C4—82no—
540C4—162no—
541C4#228311yescharacteristic
542C4#231411yescharacteristic
543C4#235211yescharacteristic
544C4#235411yescharacteristic
545C4#248911yescharacteristic
546C4#249011yescharacteristic
547C4#249411yescharacteristic
548C4#249511yescharacteristic
549C2#1387511yescharacteristic
550C2#2105811yescharacteristic
551C2#2106611yescharacteristic
552C2#2276011yescharacteristic
553C2#2276111yescharacteristic
554C2#2276911yescharacteristic
555C2#2277311yescharacteristic
556C2#2497512yes—
557C2#2500012yes—
558C2#2528212yes—
559C2#2532812yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
26. C4×#103C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18352
C2×#18052
C2×C4×#1032
#139422
#138042
#113152
#111902
#109322
#108502
#108342
#108072