Finite Group Databasebeta

← #6854475

Subgroups — #6854475

Order 512.

1738 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36559C211yesmaximal, characteristic, complemented
2#36559C211yesmaximal, characteristic, complemented
3#33289C211yesmaximal, characteristic, complemented
4#31523C218yesmaximal, complemented
5#30307C212yesmaximal, complemented
6#29190C212yesmaximal, complemented
7#29143C212yesmaximal, complemented
8#29139C211yesmaximal, characteristic, complemented
9#28509C211yesmaximal, characteristic, complemented
10#24812C214yesmaximal, complemented
11#24327C214yesmaximal, complemented
12#22269C212yesmaximal
13C4×#493C211yesmaximal, characteristic
14#21339C211yesmaximal, characteristic, complemented
15C2×#554—21no—
16C2×#554C2212yes—
17C2×#493C2211yescharacteristic
18C2×#476C2214yes—
19C2×C4×D8C2214yes—
20#2868C2211yescharacteristic
21#2868C2211yescharacteristic
22#2859C2212yes—
23#2859C2212yes—
24#2566—22no—
25#2566C2214yes—
26#2563—21no—
27#2563C2212yes—
28#2538—21no—
29#2538—21no—
30#2538—21no—
31#2538C2212yes—
32#2452—22no—
33#2452—22no—
34#2452—22no—
35#2452C2214yes—
36D4×C4⋊C4—21no—
37D4×C4⋊C4—21no—
38D4×C4⋊C4—21no—
39D4×C4⋊C4C2212yes—
40C4×#127C2212yes—
41C4×#127C2211yescharacteristic
42C4×#127C2211yescharacteristic
43C4×#127C2212yes—
44C42×Q8C2211yescharacteristic
45#2094C2214yes—
46#2085C2218yes—
47#2065C2218yes—
48#2059C2218yes—
49#2025C2218yes—
50#2023C2214yes—
51#1990C2214yes—
52#1990C2214yes—
53#1984C2212yes—
54#1970C2214yes—
55#1957C2211yescharacteristic
56#1957C2212yes—
57#1957C2212yes—
58#1957C2211yescharacteristic
59#1939C2212yes—
60#1916C2214yes—
61#1879C2218yes—
62#1879C2218yes—
63#1864C2218yes—
64#1845C2211yescharacteristic
65#1844C2212yes—
66#1834C2212yes—
67#1834C2212yes—
68#1834C2212yes—
69#1804C2218yes—
70#1802C2214yes—
71C4×C4⋊C8C2212yes—
72C4×C4⋊C8C2211yescharacteristic
73#1744C2212yes—
74C42×C8C2211yescharacteristic
75C22×D8C2314yes—
76#554—48no—
77#554—48no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8—21no—
80C2×C4⋊Q8—21no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84C2×#127C2311yescharacteristic
85C2×#127C2311yescharacteristic
86C2×#125—22no—
87C2×#125C2314yes—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C4×Q8D411yescharacteristic
93C2×C4×Q8D411yescharacteristic
94C2×C4×Q8C2311yescharacteristic
95C2×C4×D4—21no—
96C2×C4×D4—22no—
97C2×C4×D4—21no—
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×C4×D4—22no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4C2312yes—
104C2×C4×D4C2312yes—
105C2×C4×D4C2314yes—
106C2×C4×D4C2314yes—
107C22×C4⋊C4—22no—
108C22×C4⋊C4—22no—
109#493Q818yescomplemented
110#476—48no—
111C4×D8—48no—
112C2×#107C2314yes—
113C2×#106C2318yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2314yes—
116C2×#102C2318yes—
117C2×#102C2314yes—
118C2×#102C2318yes—
119C2×#102C2314yes—
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2312yes—
122C2×C4×C8C2312yes—
123C2×C4×C8C2312yes—
124C2×C4×C8C2311yescharacteristic
125#396—24no—
126#396—24no—
127#396—24no—
128#396—24no—
129#393—24no—
130#393—24no—
131#390—22no—
132#390—21no—
133#390—21no—
134#390—21no—
135#390—21no—
136#390—22no—
137#390—22no—
138#390—22no—
139#390C2314yes—
140#390C2312yes—
141#390C2312yes—
142#390C2314yes—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—22no—
147#389—22no—
148#389—22no—
149#389C2314yes—
150#389C2314yes—
151#385—24no—
152#385—24no—
153#385—24no—
154#385—24no—
155#384—21no—
156#384—21no—
157#384—21no—
158#384—21no—
159#384C2312yes—
160#384C2311yescharacteristic
161#384C2312yes—
162#384C2311yescharacteristic
163#384C2312yes—
164#384C2312yes—
165#382—22no—
166#382C2314yes—
167#380—22no—
168#380—22no—
169#380—22no—
170#380—22no—
171C4×C4⋊C4D411yescharacteristic
172C4×C4⋊C4D412yes—
173C4×C4⋊C4D411yescharacteristic
174C4×C4⋊C4D412yes—
175C4×C4⋊C4C2311yescharacteristic
176C4×C4⋊C4C2312yes—
177C43C2312yes—
178C43C2311yescharacteristic
179#340C2318yes—
180#340C2318yes—
181#340C2318yes—
182#336C2314yes—
183#336C2314yes—
184#336C2312yes—
185#336C2314yes—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8C2×D411yescharacteristic
189C22×D4—21no—
190C22×D4—21no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4C2412yes—
194C22×D4C2412yes—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C2×D8—44no—
200C2×D8—44no—
201C2×D8C2×Q8116yes—
202C22×C8C2414yes—
203C22×C8C2412yes—
204C22×C8C2412yes—
205C4⋊Q8—28no—
206C4⋊Q8—44no—
207C4⋊Q8—44no—
208C4⋊Q8—82no—
209C4⋊Q8—82no—
210C4⋊Q8—82no—
211#127—22no—
212#127—22no—
213#127C2×Q814yes—
214#127C2×Q814yes—
215#125—48no—
216#125—48no—
217C22⋊Q8—416no—
218C22⋊Q8—416no—
219C22⋊Q8—48no—
220C22⋊Q8—48no—
221C4×Q8—24no—
222C4×Q8—24no—
223C4×Q8—24no—
224C4×D4—28no—
225C4×D4—28no—
226C4×D4—48no—
227C4×D4—48no—
228C4×D4—48no—
229C4×D4—44no—
230C4×D4—44no—
231C4×D4—44no—
232C4×D4—44no—
233C4×D4—84no—
234C4×D4—84no—
235C4×D4—84no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—48no—
255C2×C4⋊C4—48no—
256C2×C4⋊C4C2×D411yescharacteristic
257C2×C4⋊C4C2×D411yescharacteristic
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2414yes—
265C2×C4⋊C4C2412yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C42C2×D412yes—
279C2×C42C2×D412yes—
280C2×C42C2×D412yes—
281C2×C42C2×D412yes—
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2412yes—
287C2×C42C2412yes—
288C2×C42C2411yescharacteristic
289C2×C42C2411yescharacteristic
290C2×C42C2411yescharacteristic
291#107—44no—
292#106—48no—
293C4⋊C8—24no—
294C4⋊C8—44no—
295#102—48no—
296#102—44no—
297#102—44no—
298#102—48no—
299C4×C8—24no—
300C4×C8—22no—
301C4×C8—24no—
302C4×C8—42no—
303C4×C8C2×Q814yes—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95C2×D412yes—
309#95C2×D412yes—
310#95C2×D412yes—
311C24—22no—
312C24—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—44no—
317C2×Q8—44no—
318C2×Q8—82no—
319C2×Q8—82no—
320C2×Q8—82no—
321C2×Q8—82no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4C22×Q814yes—
341C2×D4C22×Q814yes—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—48no—
347C22×C4—48no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—84no—
355C22×C4—84no—
356C22×C4—84no—
357C22×C4—84no—
358C22×C4C22×D412yes—
359C22×C4C22×D412yes—
360C22×C4C22×D412yes—
361C22×C4C22×D412yes—
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C2511yescharacteristic
369D8—416no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—24no—
374C2×C8—24no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—24no—
378C2×C8C22×Q814yes—
379C4⋊C4—22no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—28no—
387C4⋊C4—24no—
388C4⋊C4—28no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—48no—
392C4⋊C4—48no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C22⋊C4—44no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—84no—
416C22⋊C4—84no—
417C22⋊C4—84no—
418C22⋊C4—84no—
419C42—24no—
420C42—24no—
421C42—24no—
422C42—22no—
423C42—24no—
424C42—22no—
425C42—24no—
426C42—22no—
427C42—24no—
428C42#13514yes—
429C42AGL(1,ℤ/8)14yes—
430C42C22×Q812yes—
431C42H(5,𝔽2)14yes—
432C422−1+412yes—
433C23—24no—
434C23—24no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23C23×D411yescharacteristic
448Q8—44no—
449Q8—164no—
450Q8—164no—
451D4—28no—
452D4—28no—
453D4—48no—
454D4—48no—
455D4—48no—
456D4—48no—
457D4—44no—
458D4—44no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
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—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—88no—
490C2×C4—88no—
491C2×C4—168no—
492C2×C4—168no—
493C2×C4#53512yes—
494C2×C4#53512yes—
495C2×C4D4×Q812yes—
496C2×C4D4×Q812yes—
497C2×C4C2×#13512yes—
498C2×C4C2×#13512yes—
499C2×C4C2×#13512yes—
500C2×C4C2×AGL(1,ℤ/8)12yes—
501C2×C4C2×AGL(1,ℤ/8)12yes—
502C2×C4C2×AGL(1,ℤ/8)12yes—
503C2×C4#57612yes—
504C2×C4#57712yes—
505C2×C4C23×Q811yescharacteristic
506C2×C4C2×H(5,𝔽2)12yes—
507C2×C4C2×2−1+411yescharacteristic
508C2×C4#58512yes—
509C8—24no—
510C8—42no—
511C8—44no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—44no—
516C22—44no—
517C22—44no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22C2×#53511yescharacteristic
527C22C2×D4×Q811yescharacteristic
528C22#350011yescharacteristic
529C22C22×#13511yescharacteristic
530C22C22×AGL(1,ℤ/8)11yescharacteristic
531C22C2×#57611yescharacteristic
532C22C2×#57711yescharacteristic
533C4—22no—
534C4—22no—
535C4—22no—
536C4—42no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—84no—
541C4—82no—
542C4—82no—
543C4—84no—
544C4#320714yes—
545C4#338014yes—
546C4#338414yes—
547C4#350112yes—
548C4#354712yes—
549C2—84no—
550C2—84no—
551C2#3961811yescharacteristic
552C2C2×#320711yescharacteristic
553C2#6082211yescharacteristic
554C2C2×#338011yescharacteristic
555C2C2×#338411yescharacteristic
556C2#6253211yescharacteristic
557C2#6253511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#493C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36559C2semidirect
2. #36559C2semidirect
3. #33289C2semidirect
4. #31523C2semidirect
5. #30307C2semidirect
6. #29190C2semidirect
7. #29143C2semidirect
8. #29139C2semidirect
9. #28509C2semidirect
10. #24812C2semidirect
11. #24327C2semidirect
14. #21339C2semidirect
109. #493Q8semidirect

Maximal subgroups

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

subgroupindex
#365592
#332892
#315232
#303072
#291902
#291432
#291392
#285092
#248122
#243272
#222692
C4×#4932
#213392