Finite Group Databasebeta

← #482385

Subgroups — #482385

Order 512.

1009 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic, complemented
3#17365C211yesmaximal, characteristic, complemented
4#11883C212yesmaximal, complemented
5#11882C212yesmaximal, complemented
6#11805C214yesmaximal, complemented
7#11778C214yesmaximal, complemented
8C2×#536—21no—
9C2×#532C411yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11#2920—21no—
12#2920—21no—
13#2854—21no—
14#2854—21no—
15#2618C412yes—
16#2618C2212yes—
17#2548C411yescharacteristic
18#2548C2211yescharacteristic
19#2546C411yescharacteristic
20#2546C2211yescharacteristic
21#2545—21no—
22#2545—21no—
23#2538—21no—
24#2538—21no—
25#2536C412yes—
26#2536C2212yes—
27#2311—21no—
28#2285C411yescharacteristic
29#1821C2212yes—
30#1801—21no—
31#1801C2212yes—
32#1784—21no—
33C4×#101—21no—
34C4×#101C2212yes—
35#1467—21no—
36#1467C2212yes—
37#1463—21no—
38#1463C2212yes—
39#1461—21no—
40#1461C2212yes—
41#1346C2218yes—
42#1345C2218yes—
43C22×#41—21no—
44C23×Q8D411yescharacteristic
45#536—48no—
46#532—41no—
47#532—41no—
48#532—41no—
49#532—41no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#127C2×C411yescharacteristic
54C2×#124—21no—
55C2×#124—21no—
56C2×#124—21no—
57C2×#123—22no—
58C2×#123—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×#121—21no—
62C2×#121C2×C412yes—
63C2×C4×Q8—41no—
64C2×C4×Q8—41no—
65C2×C4×Q8—41no—
66C2×C4×Q8—41no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4C2×C412yes—
77C2×#117—21no—
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84#432—22no—
85#432—41no—
86#432—41no—
87#432—41no—
88#431—22no—
89#431—41no—
90#431—41no—
91#431—41no—
92#423—22no—
93#423—41no—
94C2×#101—22no—
95C2×#101—22no—
96#407—22no—
97#407—22no—
98C4×M16—22no—
99C4×M16—41no—
100#398C2×C412yes—
101#398C2×C412yes—
102#398C2×C412yes—
103#398C2312yes—
104#396—22no—
105#396—22no—
106#396—22no—
107#396—22no—
108#394C2×C412yes—
109#394C2×C412yes—
110#394C2×C412yes—
111#394C2312yes—
112#391—21no—
113#391—21no—
114#390—21no—
115#390—21no—
116#390—21no—
117#390—21no—
118#390—21no—
119#390—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#388—22no—
125#388—22no—
126#388—22no—
127#388—22no—
128#386—22no—
129#386—22no—
130#384—21no—
131#382C2×C412yes—
132#380—21no—
133#380—22no—
134#380—22no—
135#380C2×C412yes—
136C4×C4⋊C4—21no—
137C4×C4⋊C4C2×C411yescharacteristic
138C4×C4⋊C4C2×C411yescharacteristic
139#343—22no—
140#343C2314yes—
141#324—22no—
142#324—22no—
143#324—22no—
144#324C2314yes—
145C2×#41—42no—
146C2×#41—42no—
147C2×#41—44no—
148C2×#41—44no—
149C22×Q8—24no—
150C22×Q8—21no—
151C22×Q8—41no—
152C22×Q8—41no—
153C22×Q8—42no—
154C22×Q8C22⋊C411yescharacteristic
155C22×Q8C22⋊C411yescharacteristic
156C22×Q8C2×D411yescharacteristic
157C22×Q8C2×D411yescharacteristic
158C22×D4—22no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4C22×C412yes—
162C23×C4—22no—
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C2×M16—24no—
171C2×M16—42no—
172C4⋊Q8—41no—
173C4⋊Q8—41no—
174C4⋊Q8—44no—
175C4⋊Q8—44no—
176#127—41no—
177#127—41no—
178#124—42no—
179#124—42no—
180#124—44no—
181#124—44no—
182#123—48no—
183#123—48no—
184C22⋊Q8—48no—
185C22⋊Q8—48no—
186#121—42no—
187#121—42no—
188#121—41no—
189#121—41no—
190#121—41no—
191#121—41no—
192C4×Q8—41no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×Q8—41no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212#117—44no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22×C412yes—
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2411yescharacteristic
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C22×C411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C4⋊C8—42no—
279C4⋊C8—42no—
280C4⋊C8—42no—
281C4⋊C8—42no—
282#101—44no—
283#101—44no—
284C22⋊C8—24no—
285C22⋊C8—42no—
286C22⋊C8—42no—
287C22⋊C8—42no—
288#97—42no—
289#97—42no—
290C4×C8—42no—
291C4×C8—42no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95C22×C412yes—
298#95C22×C412yes—
299C24—22no—
300C24C2×C22⋊C411yescharacteristic
301#41—88no—
302#41—88no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—41no—
314C2×Q8—48no—
315C2×Q8—48no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×Q8AGL(1,ℤ/8)14yes—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—44no—
326C2×D4—44no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—42no—
332C2×D4—82no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—42no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—44no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4#9911yescharacteristic
362C22×C4#9911yescharacteristic
363C22×C4#9911yescharacteristic
364C22×C4#9911yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C23×C411yescharacteristic
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379M16—44no—
380M16—44no—
381M16—84no—
382C2×C8—44no—
383C2×C8—44no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—44no—
406C4⋊C4—82no—
407C22⋊C4—44no—
408C22⋊C4—44no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—44no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—42no—
444C42—42no—
445C23—21no—
446C23—21no—
447C23—41no—
448C23—41no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—42no—
456C23C2×#9911yescharacteristic
457C23C2×#9911yescharacteristic
458C23C22×C22⋊C411yescharacteristic
459Q8—44no—
460Q8—44no—
461Q8—44no—
462Q8—44no—
463Q8—84no—
464D4—82no—
465D4—84no—
466D4—84no—
467D4—84no—
468D4—84no—
469D4—82no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—88no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4C2×#9911yescharacteristic
525C2×C4C2×#9911yescharacteristic
526C2×C4C2×#9911yescharacteristic
527C2×C4C2×#9911yescharacteristic
528C2×C4#41812yes—
529C2×C4#41812yes—
530C2×C4#42112yes—
531C2×C4C2×AGL(1,ℤ/8)12yes—
532C8—84no—
533C8—84no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—42no—
543C22—42no—
544C22—82no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22C22×#9911yescharacteristic
552C22C2×#41811yescharacteristic
553C22C2×#42111yescharacteristic
554C4—21no—
555C4—21no—
556C4—21no—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—82no—
570C4—82no—
571C4—82no—
572C4#213312yes—
573C2—21no—
574C2—21no—
575C2—41no—
576C2—41no—
577C2—162no—
578C2#1757711yescharacteristic
579C2#3334711yescharacteristic
580C2C2×#213311yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#532C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20727C2semidirect
3. #17365C2semidirect
4. #11883C2semidirect
5. #11882C2semidirect
6. #11805C2semidirect
7. #11778C2semidirect

Maximal subgroups

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

subgroupindex
#351782
#207272
#173652
#118832
#118822
#118052
#117782