Finite Group Databasebeta

← #415047

Subgroups — #415047

Order 512.

1334 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#1992C211yesmaximal, characteristic, complemented
3C2×#1991C211yesmaximal, characteristic, complemented
4C2×#1847C211yesmaximal, characteristic, complemented
5C2×#1750C211yesmaximal, characteristic
6#14340C211yesmaximal, characteristic
7#11220C212yesmaximal, complemented
8#11219C212yesmaximal, complemented
9#11096C212yesmaximal
10#11095C212yesmaximal
11#10841C211yesmaximal, characteristic, complemented
12C22×#125C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×C22⋊C8C2212yes—
17C22×#97C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2629—24no—
20#2629—24no—
21C2×#398—22no—
22C2×#396—22no—
23C2×#389C2212yes—
24C2×#384C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29C2×#380C2211yescharacteristic
30#1992—24no—
31#1991—24no—
32#1847—24no—
33#1750—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41#1441—22no—
42#1439—22no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C24×C4C2311yescharacteristic
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×#125—22no—
49C2×#125—22no—
50C2×#125—22no—
51C2×#125—22no—
52C2×#125—22no—
53C2×#125D414yes—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—22no—
59C2×#117—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4D412yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C22⋊C4D412yes—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C2×#107—24no—
78C2×#107—24no—
79C2×#107—42no—
80C2×#106—24no—
81C2×#106—24no—
82C2×#106—42no—
83C2×C22⋊C8—24no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8Q818yes—
86C2×#97—22no—
87C2×#97—22no—
88C2×#97D414yes—
89C2×C4×C8—22no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92#398—216no—
93#396—216no—
94#389—24no—
95#389—24no—
96#384—22no—
97#384—22no—
98#384—22no—
99#384D414yes—
100#382—22no—
101#382—22no—
102#380—24no—
103#380—24no—
104#380—42no—
105#380—42no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113#340—24no—
114#340—24no—
115#340—24no—
116#340—24no—
117#340—42no—
118#340—42no—
119#336—24no—
120C25C2×D411yescharacteristic
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×Q814yes—
140C23×C4C2411yescharacteristic
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8C2×D414yes—
152C22×C8C2×Q814yes—
153C22×C8C2×Q814yes—
154C22×C8#4114yes—
155#125—28no—
156#125—28no—
157#125—44no—
158#125—44no—
159#117—44no—
160#117—44no—
161#117—44no—
162#117—44no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
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—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×D414yes—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—82no—
213C2×C22⋊C4—82no—
214C2×C22⋊C4—82no—
215C2×C22⋊C4—82no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42#4112yes—
226C2×C42#4112yes—
227C2×C42#4112yes—
228#107—48no—
229#106—48no—
230C22⋊C8—216no—
231#97—28no—
232C4×C8—44no—
233#95—28no—
234#95—28no—
235#95—42no—
236#95—42no—
237#95—44no—
238#95—82no—
239#95—82no—
240C24—21no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—21no—
246C24—22no—
247C24—22no—
248C24—22no—
249C24—22no—
250C24C4⋊Q812yes—
251C24C22×D411yescharacteristic
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
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#12014yes—
320C22×C4#12114yes—
321C22×C4#12314yes—
322C22×C4C4⋊Q812yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×Q812yes—
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4112yes—
330C22×C4C2×#4111yescharacteristic
331C22×C4H(5,𝔽2)12yes—
332C22×C42−1+412yes—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8C22⋊Q818yes—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—48no—
356C4⋊C4—416no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—88no—
366C4⋊C4—88no—
367C4⋊C4—88no—
368C22⋊C4—48no—
369C22⋊C4—88no—
370C22⋊C4—84no—
371C22⋊C4—84no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42#12114yes—
377C42#12114yes—
378C42#12314yes—
379C42#12414yes—
380C23—22no—
381C23—24no—
382C23—24no—
383C23—22no—
384C23—22no—
385C23—42no—
386C23—42no—
387C23—41no—
388C23—41no—
389C23—42no—
390C23—42no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23C2×#12011yescharacteristic
404C23C2×#12111yescharacteristic
405C23C2×#12311yescharacteristic
406C23C2×C4⋊Q811yescharacteristic
407C23#53211yescharacteristic
408C23#53511yescharacteristic
409C23#53611yescharacteristic
410C23#57612yes—
411C23#57712yes—
412C23#57712yes—
413C23#57812yes—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4#39418yes—
473C2×C4C2×#12111yescharacteristic
474C2×C4C2×#12111yescharacteristic
475C2×C4C2×C22⋊Q812yes—
476C2×C4C2×#12311yescharacteristic
477C2×C4C2×#12411yescharacteristic
478C2×C4#53312yes—
479C2×C4#53712yes—
480C2×C4#54212yes—
481C2×C4#54612yes—
482C2×C4D4×Q812yes—
483C2×C4D4×Q812yes—
484C2×C4#55012yes—
485C2×C4#55212yes—
486C2×C4#55412yes—
487C2×C4#55412yes—
488C2×C4#55612yes—
489C8—28no—
490C8—44no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—44no—
495C22—44no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—82no—
501C22—82no—
502C22C2×#39411yescharacteristic
503C22#243111yescharacteristic
504C22#243411yescharacteristic
505C22#249911yescharacteristic
506C22#250211yescharacteristic
507C22#262011yescharacteristic
508C22#285711yescharacteristic
509C22#301212yes—
510C22#301312yes—
511C22#306212yes—
512C22#306312yes—
513C22#309812yes—
514C22#309912yes—
515C22#316312yes—
516C22#316412yes—
517C22#317812yes—
518C22#317912yes—
519C22#322011yescharacteristic
520C22#322111yescharacteristic
521C22#322211yescharacteristic
522C22#322311yescharacteristic
523C22#324112yes—
524C22#324212yes—
525C4—28no—
526C4—42no—
527C4—84no—
528C4—82no—
529C4—84no—
530C4—161no—
531C4—161no—
532C4—161no—
533C4—161no—
534C4#241712yes—
535C4#242312yes—
536C4#249212yes—
537C4#253812yes—
538C2—82no—
539C2#1518211yescharacteristic
540C2#2348512yes—
541C2#2561111yescharacteristic
542C2#2561211yescharacteristic
543C2#2582212yes—
544C2#2788812yes—
545C2#2795711yescharacteristic
546C2#2795811yescharacteristic
547C2#2985511yescharacteristic
548C2#2985611yescharacteristic
549C2#3321912yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1992C2semidirect
3. C2×#1991C2semidirect
4. C2×#1847C2semidirect
7. #11220C2semidirect
8. #11219C2semidirect
11. #10841C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#19922
C2×#19912
C2×#18472
C2×#17502
#143402
#112202
#112192
#110962
#110952
#108412