Finite Group Databasebeta

← #214579

Subgroups — #214579

Order 512.

1025 conjugacy classes of subgroups.

The subgroup lattice has 588 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#9136C212yesmaximal, complemented
5#9135C212yesmaximal, complemented
6#9135C212yesmaximal, complemented
7#9127C212yesmaximal, complemented
8#8726C212yesmaximal, complemented
9#8342C212yesmaximal, 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#2568C412yes—
22#2567—21no—
23#2563—21no—
24#2542C412yes—
25#2487—21no—
26#2487—21no—
27#2409C412yes—
28#2409C412yes—
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×#333C2212yes—
44C2×#333C2212yes—
45C2×#333C2212yes—
46C2×#333C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1296C2212yes—
51C22×#41D411yescharacteristic
52C23×Q8C2×C412yes—
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⋊Q8C2×C414yes—
65C2×C22⋊Q8C2×C414yes—
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⋊C4—21no—
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#391C2×C414yes—
119#391C2×C414yes—
120#390—22no—
121#389—22no—
122#389—22no—
123#389—22no—
124#389—22no—
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—22no—
136#380—22no—
137#380—21no—
138#380C2×C412yes—
139C4×C4⋊C4—21no—
140C4×C4⋊C4C2×C412yes—
141#336—22no—
142#336—22no—
143#336—22no—
144#336—22no—
145#333—42no—
146#333—42no—
147#333—42no—
148#333—42no—
149C2×#41—22no—
150C2×#41—22no—
151C2×#41—22no—
152C2×#41—22no—
153C2×#41—41no—
154C2×#41—41no—
155C22×Q8—22no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8—84no—
159C22×Q8C22⋊C411yescharacteristic
160C22×Q8C22×C414yes—
161C22×D4C22⋊C411yescharacteristic
162C22×D4C22⋊C411yescharacteristic
163C22×D4C22⋊C411yescharacteristic
164C23×C4—22no—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C2×M16—42no—
175C2×M16—42no—
176C2×M16—42no—
177C2×M16—42no—
178C2×M16—82no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—42no—
184C22×C8—42no—
185C22×C8—42no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—42no—
191C4⋊Q8—42no—
192C4⋊Q8—42no—
193C4⋊Q8—42no—
194C4⋊Q8—42no—
195C4⋊Q8—42no—
196C22⋊Q8—84no—
197C22⋊Q8—84no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×Q8—41no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—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—
214#117—41no—
215#117—41no—
216#117—41no—
217#117—41no—
218#117—41no—
219#117—41no—
220#117—42no—
221#117—42no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22⋊C411yescharacteristic
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C22×C414yes—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4C22⋊C411yescharacteristic
245C2×C22⋊C4C22⋊C411yescharacteristic
246C2×C22⋊C4C22⋊C411yescharacteristic
247C2×C22⋊C4C22⋊C411yescharacteristic
248C2×C22⋊C4C22⋊C411yescharacteristic
249C2×C22⋊C4C22⋊C411yescharacteristic
250C2×C42—21no—
251C2×C42—21no—
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—22no—
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2411yescharacteristic
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—42no—
282C4⋊C8—42no—
283C4⋊C8—42no—
284C22⋊C8—44no—
285C22⋊C8—42no—
286C22⋊C8—44no—
287C22⋊C8—82no—
288C22⋊C8—82no—
289#97—42no—
290#95—22no—
291#95—22no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296C24C2×C22⋊C411yescharacteristic
297C24C2×C22⋊C411yescharacteristic
298C24C2×C22⋊C411yescharacteristic
299#41—44no—
300#41—44no—
301#41—44no—
302#41—44no—
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—24no—
306C2×Q8—24no—
307C2×Q8—41no—
308C2×Q8—44no—
309C2×Q8—44no—
310C2×Q8—44no—
311C2×Q8—84no—
312C2×Q8—88no—
313C2×Q8—88no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
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—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—84no—
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
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×C4AGL(1,ℤ/8)12yes—
372C22×C4#13712yes—
373C22×C4#13712yes—
374C22×C4#13712yes—
375C22×C4C23×C411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C22×D411yescharacteristic
378M16—84no—
379M16—84no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—84no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—84no—
411C4⋊C4—84no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—44no—
419C22⋊C4—84no—
420C22⋊C4—84no—
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—44no—
440C42H(5,𝔽2)11yescharacteristic
441C42H(5,𝔽2)11yescharacteristic
442C42H(5,𝔽2)11yescharacteristic
443C42H(5,𝔽2)11yescharacteristic
444C23—21no—
445C23—21no—
446C23—22no—
447C23—22no—
448C23—21no—
449C23—21no—
450C23—22no—
451C23—22no—
452C23—21no—
453C23—22no—
454C23—21no—
455C23—21no—
456C23—22no—
457C23#41712yes—
458C23#41912yes—
459C23C22×C22⋊C411yescharacteristic
460Q8—44no—
461Q8—84no—
462Q8—84no—
463Q8—88no—
464D4—44no—
465D4—44no—
466D4—44no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—88no—
521C2×C4#41712yes—
522C2×C4#41912yes—
523C2×C4#51811yescharacteristic
524C2×C4#51811yescharacteristic
525C2×C4#53411yescharacteristic
526C2×C4#53411yescharacteristic
527C2×C4#53411yescharacteristic
528C2×C4#53411yescharacteristic
529C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
530C2×C4C2×#13711yescharacteristic
531C2×C4C2×#13711yescharacteristic
532C2×C4C2×#13711yescharacteristic
533C2×C4#57712yes—
534C2×C4#57812yes—
535C8—82no—
536C8—82no—
537C8—82no—
538C8—82no—
539C22—22no—
540C22—22no—
541C22—24no—
542C22—24no—
543C22—41no—
544C22—41no—
545C22—42no—
546C22—42no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22#230811yescharacteristic
555C22#288812yes—
556C22C2×#41911yescharacteristic
557C22C2×#41711yescharacteristic
558C22#289812yes—
559C4—22no—
560C4—22no—
561C4—21no—
562C4—21no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—84no—
571C4—84no—
572C4#319311yescharacteristic
573C4#319311yescharacteristic
574C4#319511yescharacteristic
575C4#319511yescharacteristic
576C4#320011yescharacteristic
577C4#320112yes—
578C4#320211yescharacteristic
579C2—42no—
580C2—42no—
581C2—81no—
582C2#1776012yes—
583C2#1776412yes—
584C2#2175511yescharacteristic
585C2#2176511yescharacteristic
586C2#2177111yescharacteristic
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. #9136C2semidirect
5. #9135C2semidirect
6. #9135C2semidirect
7. #9127C2semidirect
8. #8726C2semidirect
9. #8342C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#218532
#206292
#91362
#91352
#91272
#87262
#83422