Finite Group Databasebeta

← #380084

Subgroups — #380084

Order 512.

1342 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#479C211yesmaximal, characteristic
2C2×#2320C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5#14283C211yesmaximal, characteristic
6#11220C212yesmaximal
7#11219C212yesmaximal
8#11190C212yesmaximal
9#11190C212yesmaximal
10#10849C211yesmaximal, characteristic
11#10833C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#479—24no—
15C2×#479—24no—
16C2×#479C418yescomplemented
17C22×#107C2211yescharacteristic
18C22×#106C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C22×C4⋊C8C2211yescharacteristic
21C22×#103C2211yescharacteristic
22C22×#103C2211yescharacteristic
23C22×#103C2212yes—
24#2320—24no—
25#2320—24no—
26C2×#391C2212yes—
27C2×#389C2212yes—
28C2×#387—22no—
29C2×#384C2211yescharacteristic
30C2×#383—21no—
31C2×#382—21no—
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1989—24no—
35#1803—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1441—21no—
44#1441—21no—
45#1441—22no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×Q8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125C2×C414yes—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8C2×C414yes—
62C2×C4×Q8D414yes—
63C2×C4×Q8D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79#479—232no—
80C2×#107—22no—
81C2×#107—22no—
82C2×#107C2×C414yes—
83C2×#106—22no—
84C2×#106—22no—
85C2×#106C2×C414yes—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8C2×C414yes—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—24no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103C2×C414yes—
100C2×#103C2×C414yes—
101#391—24no—
102#391—24no—
103#389—24no—
104#389—24no—
105#387—216no—
106#384—22no—
107#384—22no—
108#383—28no—
109#382—28no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4D414yes—
112#376—22no—
113#376—22no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95D411yescharacteristic
123C2×#95D411yescharacteristic
124C2×#95D412yes—
125#340—24no—
126#340—24no—
127#340—24no—
128#340—42no—
129#340—42no—
130#336—42no—
131#336—42no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8C22×C412yes—
140C22×Q8C2×D412yes—
141C22×Q8C2×D412yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8C22×C412yes—
169C22×C8C22×C412yes—
170#125—28no—
171#125—28no—
172#125—44no—
173#125—44no—
174C4×Q8—28no—
175C4×Q8—28no—
176C4×Q8—28no—
177C4×Q8C22⋊C4116yes—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—81no—
215C2×C4⋊C4—82no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C4⋊C4C22×C412yes—
222C2×C4⋊C4C2×D414yes—
223C2×C4⋊C4C2×D412yes—
224C2×C4⋊C4C2×D412yes—
225C2×C42—24no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—81no—
236C2×C42—81no—
237C2×C42C22×C412yes—
238C2×C42C2×D412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42#4112yes—
242C2×C42#4112yes—
243#107—28no—
244#106—28no—
245C4⋊C8—28no—
246C4⋊C8—44no—
247#103—28no—
248#103—28no—
249#103—48no—
250#95—28no—
251#95—28no—
252#95—28no—
253#95—24no—
254#95—22no—
255#95—22no—
256#95—24no—
257#95—82no—
258#95—82no—
259#95—82no—
260#95—82no—
261C24C22×D411yescharacteristic
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—24no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8C2×C22⋊C414yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
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—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4C4×D412yes—
339C22×C4C4×D412yes—
340C22×C4#12012yes—
341C22×C4#12012yes—
342C22×C4#12112yes—
343C22×C4#12112yes—
344C22×C4C23×C411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D411yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4H(5,𝔽2)12yes—
351C22×C42−1+412yes—
352C22×C42−1+412yes—
353C22×C42−1+412yes—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—216no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C4⋊C4—88no—
396C4⋊C4C2×C22⋊C414yes—
397C42—216no—
398C42—42no—
399C42—42no—
400C42—42no—
401C42—42no—
402C42—84no—
403C42C2×C22⋊C414yes—
404C42C4×D414yes—
405C42#12114yes—
406C42#12314yes—
407C23C2×C4×D411yescharacteristic
408C23C2×#12011yescharacteristic
409C23C2×#12111yescharacteristic
410C23#53612yes—
411C23#53712yes—
412C23#57611yescharacteristic
413C23#57712yes—
414C23#57711yescharacteristic
415C23#57711yescharacteristic
416C23#57812yes—
417C23#57811yescharacteristic
418Q8—24no—
419Q8—24no—
420Q8—44no—
421Q8—42no—
422Q8—42no—
423Q8—44no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—44no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—81no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4#38614yes—
487C2×C4#38614yes—
488C2×C4C22×C22⋊C411yescharacteristic
489C2×C4C2×C4×D411yescharacteristic
490C2×C4#51912yes—
491C2×C4#52012yes—
492C2×C4C2×#12111yescharacteristic
493C2×C4C2×#12311yescharacteristic
494C2×C4#53511yescharacteristic
495C2×C4#53611yescharacteristic
496C2×C4#53611yescharacteristic
497C2×C4#53612yes—
498C2×C4#53611yescharacteristic
499C2×C4#53612yes—
500C2×C4#53712yes—
501C2×C4#53712yes—
502C2×C4#54112yes—
503C2×C4#54112yes—
504C2×C4#54212yes—
505C2×C4#54212yes—
506C8—42no—
507C8—42no—
508C8—82no—
509C22C2×#38611yescharacteristic
510C22#232312yes—
511C22#261612yes—
512C22#262011yescharacteristic
513C22#262111yescharacteristic
514C22#294712yes—
515C22#295111yescharacteristic
516C22#295211yescharacteristic
517C22#301211yescharacteristic
518C22#301311yescharacteristic
519C22#301312yes—
520C22#304311yescharacteristic
521C22#304411yescharacteristic
522C22#304511yescharacteristic
523C22#304611yescharacteristic
524C22#324212yes—
525C22#324212yes—
526C22#324312yes—
527C22#324412yes—
528C22#325712yes—
529C22#325812yes—
530C22#326012yes—
531C22#326112yes—
532C4—24no—
533C4—24no—
534C4—44no—
535C4—44no—
536C4—82no—
537C4—82no—
538C4—84no—
539C4—82no—
540C4—82no—
541C4—162no—
542C4#231011yescharacteristic
543C4#231111yescharacteristic
544C4#232312yes—
545C4#262011yescharacteristic
546C4#262111yescharacteristic
547C4#262512yes—
548C2#1425811yescharacteristic
549C2#2093011yescharacteristic
550C2#2093711yescharacteristic
551C2#2212412yes—
552C2#2212812yes—
553C2#2970112yes—
554C2#2970212yes—
555C2#2983511yescharacteristic
556C2#2983711yescharacteristic
557C2#2987311yescharacteristic
558C2#2987511yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×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
16. C2×#479C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4792
C2×#23202
C2×#19892
C2×#18032
#142832
#112202
#112192
#111902
#108492
#108332