Finite Group Databasebeta

← #468288

Subgroups — #468288

Order 512.

1580 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2057C211yesmaximal, characteristic, complemented
3C2×#1862C211yesmaximal, characteristic
4#15149C211yesmaximal, characteristic, complemented
5#11384C211yesmaximal, characteristic, complemented
6#11346C211yesmaximal, characteristic, complemented
7#11324C212yesmaximal, complemented
8#11207C212yesmaximal, complemented
9#11123C212yesmaximal
10#11066C212yesmaximal
11#10788C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—21no—
14C22×C22⋊Q8—21no—
15C22×C4×D4C2211yescharacteristic
16C22×#107C2211yescharacteristic
17C22×#107C2212yes—
18C22×#102C2211yescharacteristic
19C22×#102C2211yescharacteristic
20C22×#102C2212yes—
21#2542—24no—
22#2542—24no—
23C2×#399—22no—
24C2×#398—21no—
25C2×#390C2212yes—
26C2×#389C2211yescharacteristic
27C2×#389C2212yes—
28C2×#384C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30#2057—24no—
31#1862—24no—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41#1442—22no—
42#1441—21no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—42no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8D414yes—
55C2×C22⋊Q8—28no—
56C2×C22⋊Q8—28no—
57C2×C22⋊Q8—44no—
58C2×C22⋊Q8—48no—
59C2×C22⋊Q8—44no—
60C2×C4×D4—22no—
61C2×C4×D4—42no—
62C2×C4×D4—42no—
63C2×C4×D4D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C22⋊C4—21no—
77C22×C22⋊C4—22no—
78C22×C22⋊C4—22no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C2×#107—24no—
82C2×#107—22no—
83C2×#107—42no—
84C2×#107—42no—
85C2×#107—42no—
86C2×#102—22no—
87C2×#102—22no—
88C2×#102—22no—
89C2×#102—22no—
90C2×#102—42no—
91C2×#102—42no—
92C2×#102—42no—
93C2×#102Q818yescomplemented
94#399—216no—
95#398—44no—
96#390—24no—
97#390—24no—
98#389—24no—
99#389—22no—
100#389—22no—
101#389—42no—
102#384—24no—
103#384—24no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95D412yes—
113#340—24no—
114#340—22no—
115#340—42no—
116#340—42no—
117#340—42no—
118#336—24no—
119#336—22no—
120#336—22no—
121#336—42no—
122C25—21no—
123C22×Q8—24no—
124C22×Q8—44no—
125C22×Q8—44no—
126C22×Q8—42no—
127C22×Q8—42no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—22no—
136C22×D4C2×D412yes—
137C22×D4C2×Q814yes—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—24no—
143C23×C4—21no—
144C23×C4—44no—
145C23×C4—44no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—21no—
168C22×C8C2×Q814yes—
169C22×C8#4112yes—
170C4⋊Q8—48no—
171C4⋊Q8—44no—
172C4⋊Q8—44no—
173C22⋊Q8—432no—
174C22⋊Q8—432no—
175C4×D4—44no—
176C4×D4—44no—
177C4×D4—48no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
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—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4D814yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D414yes—
221C2×C4⋊C4C2×Q814yes—
222C2×C4⋊C4#4112yes—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241#107—44no—
242#107—48no—
243#102—28no—
244#102—48no—
245#102—44no—
246#95—24no—
247#95—44no—
248#95—42no—
249#95—42no—
250#95—44no—
251#95—44no—
252#95—81no—
253C24—22no—
254C24—24no—
255C24—42no—
256C24—41no—
257C24—41no—
258C24—41no—
259C24—42no—
260C24—41no—
261C24—41no—
262C24—42no—
263C24—41no—
264C24C22×D411yescharacteristic
265C2×Q8—48no—
266C2×Q8—48no—
267C2×Q8—48no—
268C2×Q8—88no—
269C2×Q8—88no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—24no—
278C2×D4—24no—
279C2×D4—24no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4C4⋊Q814yes—
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—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—48no—
311C22×C4—48no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—48no—
335C22×C4—41no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—88no—
341C22×C4—88no—
342C22×C4#12012yes—
343C22×C4#12012yes—
344C22×C4C2×D814yes—
345C22×C4C2×D812yes—
346C22×C4AGL(1,ℤ/8)14yes—
347C22×C4AGL(1,ℤ/8)12yes—
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×Q812yes—
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4H(5,𝔽2)12yes—
354C22×C4H(5,𝔽2)12yes—
355C22×C4H(5,𝔽2)11yescharacteristic
356C22×C42−1+412yes—
357C22×C42−1+411yescharacteristic
358C2×C8—22no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—42no—
374C2×C8—42no—
375C4⋊C4—28no—
376C4⋊C4—28no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—24no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—416no—
386C4⋊C4—416no—
387C4⋊C4—48no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—88no—
399C4⋊C4—84no—
400C22⋊C4—48no—
401C22⋊C4—416no—
402C22⋊C4—416no—
403C42—22no—
404C42—22no—
405C42—24no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42—42no—
410C23—24no—
411C23—44no—
412C23—42no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—42no—
417C23—44no—
418C23—44no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—81no—
424C23—81no—
425C23—82no—
426C23—81no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23—82no—
431C23—81no—
432C23—82no—
433C23C2×#12011yescharacteristic
434C23#53512yes—
435C23#54611yescharacteristic
436C23#54711yescharacteristic
437C23D4×Q812yes—
438C23C22×D811yescharacteristic
439C23C2×AGL(1,ℤ/8)11yescharacteristic
440C23C2×AGL(1,ℤ/8)12yes—
441C23#57711yescharacteristic
442C23#57811yescharacteristic
443C23#57812yes—
444Q8—816no—
445D4—24no—
446D4—24no—
447D4—42no—
448D4—44no—
449D4—44no—
450D4—42no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—816no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—82no—
503C2×C4—82no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4#44714yes—
507C2×C4#52511yescharacteristic
508C2×C4C2×C4⋊Q811yescharacteristic
509C2×C4#53411yescharacteristic
510C2×C4#53411yescharacteristic
511C2×C4#53412yes—
512C2×C4#53611yescharacteristic
513C2×C4#53711yescharacteristic
514C2×C4#53912yes—
515C2×C4#54212yes—
516C2×C4#54312yes—
517C2×C4#54412yes—
518C2×C4#54811yescharacteristic
519C2×C4D4×Q812yes—
520C2×C4#55011yescharacteristic
521C2×C4#55412yes—
522C2×C4#55412yes—
523C2×C4#56212yes—
524C2×C4#56312yes—
525C8—42no—
526C8—44no—
527C8—81no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—84no—
532C22—82no—
533C22—84no—
534C22—82no—
535C22—84no—
536C22—84no—
537C22—82no—
538C22—82no—
539C22#240711yescharacteristic
540C22#243411yescharacteristic
541C22#261311yescharacteristic
542C22#269212yes—
543C22#272012yes—
544C22C2×#44711yescharacteristic
545C22#300311yescharacteristic
546C22#301111yescharacteristic
547C22#301311yescharacteristic
548C22#320512yes—
549C22#321712yes—
550C22#322312yes—
551C22#322512yes—
552C22#329411yescharacteristic
553C22#330812yes—
554C22#333011yescharacteristic
555C22#333411yescharacteristic
556C22#334312yes—
557C22#335411yescharacteristic
558C22Q8×D812yes—
559C22#338412yes—
560C22#338612yes—
561C22#338612yes—
562C4—24no—
563C4—44no—
564C4—44no—
565C4—42no—
566C4—84no—
567C4—82no—
568C4—82no—
569C4—84no—
570C4—161no—
571C4#258311yescharacteristic
572C4#263811yescharacteristic
573C4#265911yescharacteristic
574C4#272912yes—
575C4#273212yes—
576C4#281411yescharacteristic
577C2—84no—
578C2#1660611yescharacteristic
579C2#2411611yescharacteristic
580C2#2413211yescharacteristic
581C2#2583811yescharacteristic
582C2#2584411yescharacteristic
583C2#2962211yescharacteristic
584C2#2963311yescharacteristic
585C2#3151812yes—
586C2#3154012yes—
587C2#3234212yes—
588C2#3236412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
2. C2×#2057C2semidirect
4. #15149C2semidirect
5. #11384C2semidirect
6. #11346C2semidirect
7. #11324C2semidirect
8. #11207C2semidirect
93. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#20572
C2×#18622
#151492
#113842
#113462
#113242
#112072
#111232
#110662
#107882