Finite Group Databasebeta

← #380078

Subgroups — #380078

Order 512.

1408 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#479C211yesmaximal, characteristic
2C2×#2295C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4C2×#1805C211yesmaximal, characteristic
5#14709C211yesmaximal, characteristic
6#11220C212yesmaximal
7#11219C212yesmaximal
8#11190C212yesmaximal
9#11190C212yesmaximal
10#10850C211yesmaximal, characteristic
11#10832C211yesmaximal, 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#2295—24no—
25#2295—24no—
26C2×#391C2212yes—
27C2×#389C2212yes—
28C2×#387—22no—
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376—21no—
33C2×C43C2211yescharacteristic
34#1989—24no—
35#1805—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1436—21no—
44#1436—22no—
45#1436—21no—
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⋊C4C2312yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C42—21no—
74C22×C42—21no—
75C22×C42D411yescharacteristic
76C22×C42D412yes—
77C22×C42D411yescharacteristic
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80#479—232no—
81C2×#107—22no—
82C2×#107—22no—
83C2×#107C2×C414yes—
84C2×#106—22no—
85C2×#106—22no—
86C2×#106C2×C414yes—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×C4⋊C8C2×C414yes—
93C2×#103—22no—
94C2×#103—22no—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—24no—
98C2×#103—42no—
99C2×#103—42no—
100C2×#103C2×C414yes—
101C2×#103C2×C414yes—
102#391—24no—
103#391—24no—
104#389—24no—
105#389—24no—
106#387—216no—
107#382—22no—
108#382—22no—
109C4×C4⋊C4—28no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4D414yes—
112#376—28no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—21no—
123C43—22no—
124C43—22no—
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—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C412yes—
216C2×C4⋊C4C22×C412yes—
217C2×C4⋊C4C2×D414yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4C2×D412yes—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—22no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—44no—
238C2×C42C22×C412yes—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42#4112yes—
243C2×C42#4112yes—
244#107—28no—
245#106—28no—
246C4⋊C8—28no—
247C4⋊C8—44no—
248#103—28no—
249#103—28no—
250#103—48no—
251#95—28no—
252#95—28no—
253#95—28no—
254#95—24no—
255#95—44no—
256#95—44no—
257#95—42no—
258#95—42no—
259#95—44no—
260#95—44no—
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—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
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—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
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—42no—
330C22×C4C4×D412yes—
331C22×C4C4×D412yes—
332C22×C4#12012yes—
333C22×C4#12012yes—
334C22×C4#12112yes—
335C22×C4#12112yes—
336C22×C4#13514yes—
337C22×C4#13714yes—
338C22×C4C23×C411yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C22×D411yescharacteristic
342C22×C4C2×#4111yescharacteristic
343C22×C4C2×#4111yescharacteristic
344C22×C4H(5,𝔽2)12yes—
345C22×C42−1+412yes—
346C22×C42−1+412yes—
347C22×C42−1+412yes—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—42no—
357C2×C8—42no—
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—216no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—416no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—48no—
383C4⋊C4—48no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4C2×C22⋊C414yes—
389C42—28no—
390C42—216no—
391C42—48no—
392C42—44no—
393C42—48no—
394C42—42no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42C2×C22⋊C414yes—
399C42C4×D414yes—
400C42#12114yes—
401C42#12314yes—
402C23C2×C4×D411yescharacteristic
403C23C2×#12011yescharacteristic
404C23C2×#12111yescharacteristic
405C23#53612yes—
406C23#53712yes—
407C23C2×#13512yes—
408C23C2×#13712yes—
409C23#57512yes—
410C23#57512yes—
411Q8—24no—
412Q8—24no—
413Q8—44no—
414Q8—42no—
415Q8—42no—
416Q8—44no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
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—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
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—44no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—81no—
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#38614yes—
474C2×C4#38614yes—
475C2×C4C22×C22⋊C411yescharacteristic
476C2×C4C2×C4×D411yescharacteristic
477C2×C4#51912yes—
478C2×C4#52012yes—
479C2×C4C2×#12111yescharacteristic
480C2×C4C2×#12311yescharacteristic
481C2×C4#53511yescharacteristic
482C2×C4#53611yescharacteristic
483C2×C4#53611yescharacteristic
484C2×C4#53612yes—
485C2×C4#53611yescharacteristic
486C2×C4#53612yes—
487C2×C4#53712yes—
488C2×C4#53712yes—
489C2×C4#54112yes—
490C2×C4#54112yes—
491C2×C4#54212yes—
492C2×C4#54212yes—
493C8—42no—
494C8—42no—
495C8—82no—
496C22C2×#38611yescharacteristic
497C22#232312yes—
498C22#261612yes—
499C22#262011yescharacteristic
500C22#262111yescharacteristic
501C22C4×#13511yescharacteristic
502C22#294311yescharacteristic
503C22C4×#13712yes—
504C22#300212yes—
505C22#300411yescharacteristic
506C22#300511yescharacteristic
507C22#303512yes—
508C22#303712yes—
509C22#323112yes—
510C22#323212yes—
511C22#323612yes—
512C22#324012yes—
513C22#324812yes—
514C22#325012yes—
515C22#325212yes—
516C22#325612yes—
517C4—24no—
518C4—24no—
519C4—44no—
520C4—44no—
521C4—44no—
522C4—44no—
523C4—82no—
524C4—82no—
525C4—84no—
526C4—162no—
527C4#231011yescharacteristic
528C4#231111yescharacteristic
529C4#232312yes—
530C4#262011yescharacteristic
531C4#262111yescharacteristic
532C4#262512yes—
533C2#1425811yescharacteristic
534C2#2092411yescharacteristic
535C2#2092611yescharacteristic
536C2#2211512yes—
537C2#2212112yes—
538C2#2969312yes—
539C2#2969512yes—
540C2#2983312yes—
541C2#2987112yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ 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×#22952
C2×#19892
C2×#18052
#147092
#112202
#112192
#111902
#108502
#108322