Finite Group Databasebeta

← #454523

Subgroups — #454523

Order 512.

1454 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2629C211yesmaximal, characteristic
2C2×#2628C211yesmaximal, characteristic
3C2×#2059C212yesmaximal, complemented
4C2×#2058C211yesmaximal, characteristic, complemented
5C2×#2057C211yesmaximal, characteristic, complemented
6#11208C211yesmaximal, characteristic, complemented
7#11208C211yesmaximal, characteristic, complemented
8#11207C212yesmaximal, complemented
9#11096C212yesmaximal
10#11095C212yesmaximal
11#10713C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C22⋊C8C2212yes—
18#2629—24no—
19#2629—24no—
20#2628—28no—
21C2×#400—21no—
22C2×#399—21no—
23C2×#398—21no—
24C2×#398—21no—
25C2×#396—21no—
26C2×#396—21no—
27C2×#395—21no—
28C2×#393—21no—
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#384C2211yescharacteristic
33C2×#382C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37#2059—28no—
38#2058—24no—
39#2057—24no—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#340C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46C2×#336C2212yes—
47C2×#336C2211yescharacteristic
48C2×#336C2211yescharacteristic
49C23×Q8—21no—
50C24×C4C2311yescharacteristic
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×C4⋊Q8—24no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—42no—
57C2×#125—22no—
58C2×#125—22no—
59C2×#125—24no—
60C2×#125—42no—
61C2×#117—24no—
62C2×#117—22no—
63C2×#117—22no—
64C2×#117—42no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D411yescharacteristic
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—22no—
82C22×C22⋊C4—21no—
83C22×C22⋊C4—21no—
84C22×C22⋊C4D411yescharacteristic
85C22×C22⋊C4D411yescharacteristic
86C22×C42C2311yescharacteristic
87C22×C42C2311yescharacteristic
88C2×#107—24no—
89C2×#107—24no—
90C2×#107—42no—
91C2×#106—24no—
92C2×#106—24no—
93C2×#106—42no—
94C2×C22⋊C8—42no—
95C2×C22⋊C8—42no—
96C2×C22⋊C8Q818yescomplemented
97#400—28no—
98#399—28no—
99#398—28no—
100#398—28no—
101#396—28no—
102#396—28no—
103#395—28no—
104#393—28no—
105#389—24no—
106#389—22no—
107#389—22no—
108#389D414yes—
109#389D414yes—
110#389D418yes—
111#384—22no—
112#384—22no—
113#382—24no—
114#380—24no—
115#380—24no—
116#380—22no—
117#380—22no—
118#380—22no—
119#380—22no—
120C2×#95—21no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—21no—
127C2×#95—21no—
128#340—24no—
129#340—24no—
130#340—24no—
131#340—24no—
132#340—42no—
133#340—42no—
134#336—22no—
135#336—22no—
136#336—24no—
137C25C2×D411yescharacteristic
138C22×Q8—24no—
139C22×Q8—42no—
140C22×Q8—42no—
141C22×Q8—82no—
142C22×Q8—82no—
143C23×C4—22no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×Q814yes—
164C23×C4C2411yescharacteristic
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8C2×Q814yes—
178C22×C8C2×Q814yes—
179C4⋊Q8—48no—
180C4⋊Q8—48no—
181#125—48no—
182#125—48no—
183#117—48no—
184#117—48no—
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—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—82no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4—82no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4—82no—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4C2×D414yes—
241C2×C4⋊C4C2×D412yes—
242C2×C4⋊C4C2×D412yes—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—82no—
253C2×C22⋊C4—82no—
254C2×C22⋊C4—82no—
255C2×C22⋊C4—82no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—22no—
259C2×C42—22no—
260C2×C42—22no—
261C2×C42C2×D414yes—
262C2×C42C2×D412yes—
263C2×C42C2×D412yes—
264#107—48no—
265#106—48no—
266C22⋊C8—48no—
267#95—28no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—28no—
273#95—24no—
274#95—24no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—21no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24C4⋊Q812yes—
286C24C22×D411yescharacteristic
287C2×Q8—48no—
288C2×Q8—84no—
289C2×Q8—84no—
290C2×Q8—88no—
291C2×Q8—88no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—42no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4#12012yes—
365C22×C4#12012yes—
366C22×C4#12012yes—
367C22×C4#12012yes—
368C22×C4C4⋊Q812yes—
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C22×D412yes—
372C22×C4C22×Q812yes—
373C22×C4H(5,𝔽2)11yescharacteristic
374C22×C4H(5,𝔽2)12yes—
375C22×C42−1+411yescharacteristic
376C22×C42−1+411yescharacteristic
377C22×C42−1+411yescharacteristic
378C22×C42−1+412yes—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C2×C8—42no—
391C2×C8—42no—
392C2×C8—42no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—48no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—42no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4—88no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—84no—
426C22⋊C4—84no—
427C22⋊C4—88no—
428C42—24no—
429C42—24no—
430C42—24no—
431C42—24no—
432C23—24no—
433C23—24no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—42no—
454C23—42no—
455C23C2×#12011yescharacteristic
456C23C2×#12011yescharacteristic
457C23C2×C4⋊Q811yescharacteristic
458C23#53411yescharacteristic
459C23#53611yescharacteristic
460C23#53611yescharacteristic
461C23#53711yescharacteristic
462C23#57612yes—
463C23#57712yes—
464C23#57712yes—
465C23#57812yes—
466Q8—816no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—82no—
531C2×C4—82no—
532C2×C4#53611yescharacteristic
533C2×C4#53611yescharacteristic
534C2×C4#53612yes—
535C2×C4#53711yescharacteristic
536C2×C4#53711yescharacteristic
537C2×C4#53712yes—
538C2×C4#54312yes—
539C2×C4#54412yes—
540C2×C4D4212yes—
541C2×C4D4212yes—
542C2×C4D4×Q812yes—
543C2×C4D4×Q812yes—
544C2×C4D4×Q812yes—
545C2×C4D4×Q812yes—
546C2×C4#56312yes—
547C2×C4#56312yes—
548C8—44no—
549C8—44no—
550C22—44no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—44no—
555C22—81no—
556C22—81no—
557C22—82no—
558C22—82no—
559C22—81no—
560C22—81no—
561C22#240311yescharacteristic
562C22#243411yescharacteristic
563C22#243411yescharacteristic
564C22#261611yescharacteristic
565C22#261611yescharacteristic
566C22#265911yescharacteristic
567C22#281711yescharacteristic
568C22#300811yescharacteristic
569C22#300911yescharacteristic
570C22#301011yescharacteristic
571C22#301111yescharacteristic
572C22#301212yes—
573C22#301312yes—
574C22#316312yes—
575C22#316412yes—
576C22#319812yes—
577C22#320112yes—
578C22#324112yes—
579C22#324212yes—
580C22#324312yes—
581C22#324412yes—
582C22#325712yes—
583C22#325812yes—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—42no—
588C4—42no—
589C4—82no—
590C4—82no—
591C4—84no—
592C4—82no—
593C4—84no—
594C4—82no—
595C4#267712yes—
596C4#267712yes—
597C4#268312yes—
598C4#268312yes—
599C2—82no—
600C2#1625411yescharacteristic
601C2#2392411yescharacteristic
602C2#2392711yescharacteristic
603C2#2582011yescharacteristic
604C2#2582111yescharacteristic
605C2#2582212yes—
606C2#2969811yescharacteristic
607C2#2969911yescharacteristic
608C2#2970212yes—
609C2#3029912yes—
610C2#3279312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2059C2semidirect
4. C2×#2058C2semidirect
5. C2×#2057C2semidirect
6. #11208C2semidirect
7. #11208C2semidirect
8. #11207C2semidirect
96. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26292
C2×#26282
C2×#20592
C2×#20582
C2×#20572
#112082
#112072
#110962
#110952
#107132