Finite Group Databasebeta

← #464874

Subgroups — #464874

Order 512.

1434 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
1C2×#2551C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic
3C2×#2058C212yesmaximal
4C2×#2024C212yesmaximal
5C2×#1927C211yesmaximal, characteristic
6#15193C211yesmaximal, characteristic
7#11357C211yesmaximal, characteristic
8#11346C211yesmaximal, characteristic
9#11327C212yesmaximal
10#11067C212yesmaximal
11#10806C211yesmaximal, characteristic
12C22×C4⋊Q8—21no—
13C22×C4⋊Q8C2212yes—
14C22×#106C2211yescharacteristic
15C22×#106C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19#2551—24no—
20#2551—24no—
21C2×#395—22no—
22C2×#391C2211yescharacteristic
23C2×#391C2212yes—
24C2×#389—21no—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#384C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2096—24no—
31#2058—28no—
32#2024—28no—
33#1927—24no—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42C2×#336C2211yescharacteristic
43#1445—22no—
44#1445—21no—
45#1444—21no—
46C4⋊#95C2211yescharacteristic
47C23×Q8—22no—
48C23×Q8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C2×C4⋊Q8—28no—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8—44no—
60C2×C4⋊Q8—44no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C42—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C2×#106—24no—
77C2×#106—24no—
78C2×#106—24no—
79C2×#106—22no—
80C2×#106—42no—
81C2×#103—22no—
82C2×#103—22no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—22no—
86C2×#103—24no—
87C2×#103—42no—
88C2×#103Q814yescomplemented
89#395—216no—
90#391—24no—
91#391—22no—
92#391—22no—
93#391—42no—
94#389—28no—
95#389—24no—
96#389—24no—
97#389—24no—
98#389—42no—
99#384—22no—
100#384D414yes—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4D414yes—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95D412yes—
115#340—24no—
116#340—22no—
117#340—24no—
118#340—24no—
119#340D418yes—
120#336—24no—
121#336—24no—
122#336—22no—
123#336—22no—
124C22×Q8—28no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—21no—
134C22×Q8—44no—
135C22×Q8—44no—
136C22×Q8—44no—
137C22×Q8—44no—
138C22×Q8—82no—
139C22×Q8—82no—
140C22×Q8C2×Q812yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D411yescharacteristic
151C23×C4C2411yescharacteristic
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8C2×D414yes—
168C22×C8C2×Q812yes—
169C4⋊Q8—48no—
170C4⋊Q8—48no—
171C4⋊Q8—48no—
172C4⋊Q8—432no—
173C4⋊Q8—48no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
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—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4SD1614yes—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D414yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×Q812yes—
218C2×C4⋊C4#4112yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232#106—48no—
233#106—44no—
234#103—28no—
235#103—48no—
236#103—44no—
237#95—28no—
238#95—24no—
239#95—24no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—42no—
244#95—42no—
245#95—42no—
246#95—81no—
247C24C22×D411yescharacteristic
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—22no—
251C2×Q8—24no—
252C2×Q8—416no—
253C2×Q8—416no—
254C2×Q8—41no—
255C2×Q8—42no—
256C2×Q8—41no—
257C2×Q8—42no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—88no—
269C2×Q8—88no—
270C2×Q8—88no—
271C2×Q8—88no—
272C2×Q8—88no—
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—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—22no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
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—41no—
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—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4#12012yes—
328C22×C4#12012yes—
329C22×C4C2×SD1614yes—
330C22×C4C2×SD1612yes—
331C22×C4AGL(1,ℤ/8)12yes—
332C22×C4#13714yes—
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D412yes—
335C22×C4C22×D411yescharacteristic
336C22×C4C22×Q811yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4H(5,𝔽2)12yes—
339C22×C42−1+412yes—
340C22×C42−1+411yescharacteristic
341C22×C42−1+411yescharacteristic
342C22×C42−1+412yes—
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—41no—
349C2×C8—42no—
350C2×C8—41no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—42no—
359C2×C8—42no—
360C4⋊C4—28no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—28no—
367C4⋊C4—24no—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—416no—
372C4⋊C4—416no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—48no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—84no—
382C4⋊C4—88no—
383C4⋊C4C22⋊Q814yes—
384C4⋊C4#12714yes—
385C42—22no—
386C42—22no—
387C42—24no—
388C42—42no—
389C42—42no—
390C42—42no—
391C42—42no—
392C42—48no—
393C23C2×#12011yescharacteristic
394C23#53612yes—
395C23D4212yes—
396C23#54811yescharacteristic
397C23D4×Q811yescharacteristic
398C23C22×SD1611yescharacteristic
399C23C2×AGL(1,ℤ/8)11yescharacteristic
400C23C2×#13712yes—
401C23#57712yes—
402C23#57711yescharacteristic
403C23#57811yescharacteristic
404Q8—42no—
405Q8—44no—
406Q8—42no—
407Q8—832no—
408Q8—81no—
409Q8—81no—
410Q8—82no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
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—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—82no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4#45014yes—
471C2×C4C2×C22⋊Q811yescharacteristic
472C2×C4C2×#12711yescharacteristic
473C2×C4#53511yescharacteristic
474C2×C4#53611yescharacteristic
475C2×C4#53611yescharacteristic
476C2×C4#53611yescharacteristic
477C2×C4#53612yes—
478C2×C4#53612yes—
479C2×C4#53612yes—
480C2×C4#54212yes—
481C2×C4#54412yes—
482C2×C4D4212yes—
483C2×C4#54712yes—
484C2×C4#54811yescharacteristic
485C2×C4D4×Q811yescharacteristic
486C2×C4D4×Q812yes—
487C2×C4#56312yes—
488C2×C4#56312yes—
489C8—44no—
490C8—42no—
491C8—81no—
492C22#240911yescharacteristic
493C22#243311yescharacteristic
494C22#261411yescharacteristic
495C22#267412yes—
496C22#267712yes—
497C22C2×#45011yescharacteristic
498C22#300411yescharacteristic
499C22#300911yescharacteristic
500C22#301311yescharacteristic
501C22#322912yes—
502C22#323812yes—
503C22#324212yes—
504C22#324412yes—
505C22#326612yes—
506C22#327012yes—
507C22#327212yes—
508C22D4×SD1612yes—
509C22#335911yescharacteristic
510C22#336912yes—
511C22#337311yescharacteristic
512C22Q8×SD1611yescharacteristic
513C22#338611yescharacteristic
514C22#338712yes—
515C4—24no—
516C4—44no—
517C4—44no—
518C4—44no—
519C4—44no—
520C4—42no—
521C4—84no—
522C4—82no—
523C4—84no—
524C4—161no—
525C4#254211yescharacteristic
526C4#262811yescharacteristic
527C4#265811yescharacteristic
528C4#267412yes—
529C4#267712yes—
530C4#282111yescharacteristic
531C2#1651111yescharacteristic
532C2#2433011yescharacteristic
533C2#2434511yescharacteristic
534C2#2575311yescharacteristic
535C2#2576611yescharacteristic
536C2#2964111yescharacteristic
537C2#2964811yescharacteristic
538C2#3080812yes—
539C2#3082912yes—
540C2#3094912yes—
541C2#3097312yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×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
88. C2×#103Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#20962
C2×#20582
C2×#20242
C2×#19272
#151932
#113572
#113462
#113272
#110672
#108062