Finite Group Databasebeta

← #214580

Subgroups — #214580

Order 512.

1025 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#21853C211yesmaximal, characteristic
3#20629C211yesmaximal, characteristic
4#9140C212yesmaximal, complemented
5#9139C212yesmaximal, complemented
6#9137C212yesmaximal, complemented
7#9129C212yesmaximal, complemented
8#8726C212yesmaximal, complemented
9#8340C212yesmaximal, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C2×#408—21no—
17#2857C2212yes—
18#2857C2212yes—
19#2628—21no—
20#2628—21no—
21#2568—21no—
22#2567C412yes—
23#2563C412yes—
24#2542—21no—
25#2487C412yes—
26#2487C412yes—
27#2409—21no—
28#2409—21no—
29#2317—21no—
30#2311—21no—
31#2285C2211yescharacteristic
32#1985—21no—
33#1955—21no—
34#1870—21no—
35#1870—21no—
36#1798—21no—
37#1798—21no—
38#1797C2212yes—
39#1797C2212yes—
40#1797C2212yes—
41#1797C2212yes—
42#1729—21no—
43C2×#332C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1296C2212yes—
51C22×#41D411yescharacteristic
52C23×Q8—21no—
53C22×M16—21no—
54C22×M16—21no—
55C23×C8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—41no—
59C2×C4⋊Q8—41no—
60C2×C4⋊Q8—41no—
61C2×C4⋊Q8—41no—
62C2×C4⋊Q8C2×C412yes—
63C2×C4⋊Q8C2312yes—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C4×#41—41no—
67C4×#41—41no—
68C4×#41—41no—
69C4×#41—41no—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C2×C4×D4D411yescharacteristic
78C2×#117—21no—
79C2×#117D411yescharacteristic
80C2×#117D411yescharacteristic
81C2×#117D411yescharacteristic
82C22×C4⋊C4—21no—
83C22×C4⋊C4C2×C412yes—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86C22×C42C2311yescharacteristic
87#432—41no—
88#432—41no—
89#423—41no—
90#423—41no—
91#423—41no—
92#423—41no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—41no—
97C2×C4⋊C8—41no—
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2312yes—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2312yes—
102#408—44no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×#97—21no—
109#400—22no—
110#400—22no—
111#399—22no—
112#399—22no—
113#395—22no—
114#395—22no—
115#393—22no—
116#393—22no—
117#391—22no—
118#391—22no—
119#391—22no—
120#390—22no—
121#389—22no—
122#389—22no—
123#389C2×C414yes—
124#389C2×C414yes—
125#386—22no—
126#386—22no—
127#385—22no—
128#385—22no—
129#384—22no—
130#384C2312yes—
131#382—22no—
132#382—21no—
133#380—22no—
134#380—22no—
135#380—21no—
136#380C2×C414yes—
137#380C2×C414yes—
138#380C2×C412yes—
139C4×C4⋊C4—21no—
140C4×C4⋊C4C2×C412yes—
141#336—22no—
142#336—22no—
143#336—22no—
144#336—22no—
145#332—42no—
146#332—42no—
147#332—42no—
148#332—42no—
149C2×#41—22no—
150C2×#41—22no—
151C2×#41—22no—
152C2×#41—22no—
153C2×#41—41no—
154C2×#41—41no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8—82no—
160C22×Q8—82no—
161C22×Q8C22⋊C411yescharacteristic
162C22×D4C22⋊C411yescharacteristic
163C22×D4C22⋊C411yescharacteristic
164C22×D4C22⋊C411yescharacteristic
165C23×C4—22no—
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C2×M16—42no—
176C2×M16—42no—
177C2×M16—42no—
178C2×M16—42no—
179C2×M16—82no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—42no—
185C22×C8—42no—
186C22×C8—42no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191C4⋊Q8—42no—
192C4⋊Q8—42no—
193C4⋊Q8—42no—
194C4⋊Q8—42no—
195C4⋊Q8—42no—
196C4⋊Q8—42no—
197C22⋊Q8—82no—
198C22⋊Q8—82no—
199C22⋊Q8—82no—
200C22⋊Q8—82no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—41no—
208C4×D4—41no—
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—41no—
212C4×D4—41no—
213C4×D4—41no—
214C4×D4—41no—
215C4×D4—41no—
216C4×D4—41no—
217#117—41no—
218#117—41no—
219#117—41no—
220#117—41no—
221#117—41no—
222#117—41no—
223#117—42no—
224#117—42no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22⋊C411yescharacteristic
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C22⋊C411yescharacteristic
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22×C414yes—
243C2×C4⋊C4C22×C414yes—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4C22⋊C411yescharacteristic
248C2×C22⋊C4C22⋊C411yescharacteristic
249C2×C22⋊C4C22⋊C411yescharacteristic
250C2×C22⋊C4C22⋊C411yescharacteristic
251C2×C22⋊C4C22⋊C411yescharacteristic
252C2×C22⋊C4C22⋊C411yescharacteristic
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—22no—
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2411yescharacteristic
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—42no—
285C4⋊C8—42no—
286C4⋊C8—42no—
287C22⋊C8—44no—
288C22⋊C8—42no—
289C22⋊C8—44no—
290C22⋊C8—82no—
291C22⋊C8—82no—
292#97—42no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299C24C2×C22⋊C411yescharacteristic
300C24C2×C22⋊C411yescharacteristic
301C24C2×C22⋊C411yescharacteristic
302#41—44no—
303#41—44no—
304#41—44no—
305#41—44no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—41no—
309C2×Q8—44no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—42no—
314C2×Q8—42no—
315C2×Q8—42no—
316C2×Q8—84no—
317C2×Q8—88no—
318C2×Q8—84no—
319C2×Q8—84no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
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—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—84no—
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×C4C2×C22⋊C411yescharacteristic
377C22×C4AGL(1,ℤ/8)12yes—
378C22×C4#13712yes—
379C22×C4#13712yes—
380C22×C4#13712yes—
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384M16—84no—
385M16—84no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—42no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—84no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—84no—
417C4⋊C4—84no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—44no—
425C22⋊C4—84no—
426C22⋊C4—84no—
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—21no—
444C42—21no—
445C42—44no—
446C42H(5,𝔽2)11yescharacteristic
447C42H(5,𝔽2)11yescharacteristic
448C42H(5,𝔽2)11yescharacteristic
449C42H(5,𝔽2)11yescharacteristic
450C23—21no—
451C23—21no—
452C23—22no—
453C23—22no—
454C23—21no—
455C23—21no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23—22no—
460C23—21no—
461C23—21no—
462C23—22no—
463C23#41712yes—
464C23#41912yes—
465C23C22×C22⋊C411yescharacteristic
466Q8—44no—
467Q8—88no—
468Q8—82no—
469Q8—82no—
470Q8—82no—
471Q8—82no—
472D4—44no—
473D4—44no—
474D4—44no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—88no—
529C2×C4#41712yes—
530C2×C4#41912yes—
531C2×C4#51811yescharacteristic
532C2×C4#51811yescharacteristic
533C2×C4#53411yescharacteristic
534C2×C4#53411yescharacteristic
535C2×C4#53411yescharacteristic
536C2×C4#53411yescharacteristic
537C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
538C2×C4C2×#13711yescharacteristic
539C2×C4C2×#13711yescharacteristic
540C2×C4C2×#13711yescharacteristic
541C2×C4#57612yes—
542C2×C4#57712yes—
543C8—82no—
544C8—82no—
545C8—82no—
546C8—82no—
547C22—22no—
548C22—22no—
549C22—24no—
550C22—24no—
551C22—41no—
552C22—41no—
553C22—42no—
554C22—42no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—41no—
559C22—42no—
560C22—42no—
561C22—41no—
562C22#230811yescharacteristic
563C22#288912yes—
564C22C2×#41911yescharacteristic
565C22C2×#41711yescharacteristic
566C22#289712yes—
567C4—22no—
568C4—22no—
569C4—21no—
570C4—21no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—84no—
579C4—84no—
580C4#319311yescharacteristic
581C4#319311yescharacteristic
582C4#319511yescharacteristic
583C4#319511yescharacteristic
584C4#319812yes—
585C4#319911yescharacteristic
586C4#320011yescharacteristic
587C2—42no—
588C2—42no—
589C2—81no—
590C2#1776112yes—
591C2#1776512yes—
592C2#2175611yescharacteristic
593C2#2176511yescharacteristic
594C2#2177011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9140C2semidirect
5. #9139C2semidirect
6. #9137C2semidirect
7. #9129C2semidirect
8. #8726C2semidirect
9. #8340C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#218532
#206292
#91402
#91392
#91372
#91292
#87262
#83402