Finite Group Databasebeta

← #471905

Subgroups — #471905

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2093C211yesmaximal, characteristic
2C2×#2043C211yesmaximal, characteristic, complemented
3C2×#1956C211yesmaximal, characteristic, complemented
4C2×#1934C211yesmaximal, characteristic
5#15243C211yesmaximal, characteristic
6#15153C211yesmaximal, characteristic, complemented
7#11124C212yesmaximal, complemented
8#11124C212yesmaximal, complemented
9#11009C212yesmaximal
10#11009C212yesmaximal
11#10797C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#124C2212yes—
14C22×#107C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#102C2211yescharacteristic
20C22×#102C2211yescharacteristic
21C22×#102C2212yes—
22C22×C4×C8C2211yescharacteristic
23C2×#400—22no—
24C2×#391C2212yes—
25C2×#390C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#384C2211yescharacteristic
28#2093—24no—
29#2043—24no—
30#1956—24no—
31#1934—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37#1445—22no—
38#1439—22no—
39#1439—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4×#95C2211yescharacteristic
43C23×Q8C2312yes—
44C23×D4C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—42no—
52C2×#124—24no—
53C2×#124—24no—
54C2×#124—42no—
55C2×#124—42no—
56C2×#124—42no—
57C2×#124—42no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C22⋊C4—24no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—21no—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72C2×#107—24no—
73C2×#107—42no—
74C2×#107—42no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#102—22no—
88C2×#102—22no—
89C2×#102—22no—
90C2×#102—24no—
91C2×#102—22no—
92C2×#102—42no—
93C2×#102—42no—
94C2×#102—42no—
95C2×C4×C8—42no—
96C2×C4×C8D414yes—
97#400—48no—
98#391—42no—
99#391—42no—
100#390—22no—
101#390—22no—
102#389—22no—
103#389—22no—
104#384—22no—
105#384—22no—
106C2×#95—24no—
107C2×#95—24no—
108C2×#95—24no—
109C2×#95—21no—
110C2×#95D412yes—
111C2×#95D412yes—
112#340—22no—
113#340—22no—
114#336—42no—
115#336—42no—
116#336D414yes—
117C25—21no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8#4114yes—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—22no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4#4112yes—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—21no—
147C23×C4C2×D414yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2411yescharacteristic
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8C2×D414yes—
166C22×C8C2×D412yes—
167C22×C8C2×D412yes—
168C4⋊Q8—28no—
169C4⋊Q8—48no—
170C4⋊Q8—44no—
171#124—48no—
172#124—48no—
173#124—48no—
174#124—48no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
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—22no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4#4112yes—
205C2×C22⋊C4—48no—
206C2×C22⋊C4—48no—
207C2×C22⋊C4—48no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222#107—48no—
223#103—48no—
224#103—48no—
225#103—48no—
226#103—48no—
227#102—28no—
228#102—44no—
229#102—48no—
230C4×C8—44no—
231#95—24no—
232#95—24no—
233#95—48no—
234#95—48no—
235#95—48no—
236#95—42no—
237C24—22no—
238C24—41no—
239C24—42no—
240C24—42no—
241C24—41no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24C22×D411yescharacteristic
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×D4—22no—
276C2×D4—24no—
277C2×D4—22no—
278C2×D4—24no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—41no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—44no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4#12712yes—
330C22×C4#12712yes—
331C22×C4#13514yes—
332C22×C4AGL(1,ℤ/8)14yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4112yes—
337C22×C4C2×#4111yescharacteristic
338C22×C4H(5,𝔽2)12yes—
339C22×C4H(5,𝔽2)12yes—
340C22×C4H(5,𝔽2)12yes—
341C22×C4H(5,𝔽2)11yescharacteristic
342C22×C42−1+411yescharacteristic
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—24no—
347C2×C8—22no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—84no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C22⋊C4—432no—
380C22⋊C4—84no—
381C22⋊C4—84no—
382C42—22no—
383C42—22no—
384C42—24no—
385C42—42no—
386C42—42no—
387C42—42no—
388C42—42no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—42no—
393C23—82no—
394C23—82no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23—81no—
400C23—84no—
401C23—82no—
402C23—84no—
403C23—84no—
404C23—81no—
405C23—84no—
406C23C2×#12711yescharacteristic
407C23#53512yes—
408C23#54611yescharacteristic
409C23#54612yes—
410C23#54811yescharacteristic
411C23C2×#13512yes—
412C23C2×AGL(1,ℤ/8)12yes—
413C23#57812yes—
414C23#57812yes—
415Q8—44no—
416Q8—44no—
417Q8—44no—
418Q8—44no—
419Q8—82no—
420Q8—82no—
421Q8—82no—
422Q8—82no—
423D4—42no—
424D4—42no—
425D4—44no—
426D4—81no—
427D4—82no—
428D4—81no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—28no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—28no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—81no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4—84no—
465C2×C4—81no—
466C2×C4—84no—
467C2×C4#49514yes—
468C2×C4#49614yes—
469C2×C4#52511yescharacteristic
470C2×C4#52511yescharacteristic
471C2×C4#53411yescharacteristic
472C2×C4#53511yescharacteristic
473C2×C4#53511yescharacteristic
474C2×C4#53512yes—
475C2×C4#53711yescharacteristic
476C2×C4#53812yes—
477C2×C4#53812yes—
478C2×C4#53912yes—
479C2×C4#54012yes—
480C2×C4#54612yes—
481C2×C4#54611yescharacteristic
482C2×C4#54811yescharacteristic
483C2×C4#55112yes—
484C2×C4#55312yes—
485C2×C4#56012yes—
486C2×C4#56212yes—
487C8—42no—
488C8—44no—
489C8—42no—
490C22—88no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—161no—
495C22—161no—
496C22—164no—
497C22—161no—
498C22—161no—
499C22#253911yescharacteristic
500C22#254011yescharacteristic
501C22#265711yescharacteristic
502C22#268512yes—
503C22#271112yes—
504C22C2×#49511yescharacteristic
505C22C2×#49611yescharacteristic
506C22#315611yescharacteristic
507C22#315611yescharacteristic
508C22#320714yes—
509C22#322314yes—
510C22#330014yes—
511C22#330212yes—
512C22#330814yes—
513C22#333012yes—
514C22#336412yes—
515C22#336712yes—
516C4—24no—
517C4—24no—
518C4—84no—
519C4—84no—
520C4—82no—
521C4—161no—
522C4—162no—
523C4—161no—
524C4#259811yescharacteristic
525C4#263511yescharacteristic
526C4#263611yescharacteristic
527C4#269612yes—
528C4#270312yes—
529C4#281311yescharacteristic
530C2—162no—
531C2#1680911yescharacteristic
532C2#2833511yescharacteristic
533C2#2834611yescharacteristic
534C2#2840311yescharacteristic
535C2#2841411yescharacteristic
536C2#3026011yescharacteristic
537C2#3027211yescharacteristic
538C2#3128714yes—
539C2#3208514yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2043C2semidirect
3. C2×#1956C2semidirect
6. #15153C2semidirect
7. #11124C2semidirect
8. #11124C2semidirect
11. #10797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20932
C2×#20432
C2×#19562
C2×#19342
#152432
#151532
#111242
#110092
#107972