Finite Group Databasebeta

← #4386590

Subgroups — #4386590

Order 512.

2562 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic, complemented
2C2×#2317C211yesmaximal, characteristic
3#16686C212yesmaximal, complemented
4#16522C212yesmaximal, complemented
5#16059C214yesmaximal, complemented
6#15462C214yesmaximal
7#15318C214yesmaximal, complemented
8#15265C214yesmaximal
9#14887C212yesmaximal
10#14720C212yesmaximal
11#14676C211yesmaximal, characteristic, complemented
12#14298C212yesmaximal, complemented
13#13938C212yesmaximal
14C22×C4⋊Q8C2214yes—
15C22×C22⋊Q8C2212yes—
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19#2551—28no—
20C2×#398C2214yes—
21C2×#395C2212yes—
22C2×#395C2212yes—
23C2×#395C2212yes—
24C2×#395C2214yes—
25C2×#393C2212yes—
26#2317—28no—
27C2×#391C2212yes—
28C2×#389C2214yes—
29C2×#389C2214yes—
30C2×#385C2214yes—
31C2×#385C2214yes—
32C2×#384C2211yescharacteristic
33C2×#384C2211yescharacteristic
34C2×#383C2212yes—
35C2×#382C2211yescharacteristic
36C2×#382C2211yescharacteristic
37C2×C4×C4⋊C4C2212yes—
38#1446C2214yes—
39#1444C2214yes—
40#1444C2214yes—
41#1444C2214yes—
42#1444C2214yes—
43#1443C2214yes—
44#1442C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1441C2212yes—
49#1441C2212yes—
50#1441C2214yes—
51#1441C2214yes—
52#1441C2214yes—
53#1441C2212yes—
54#1441C2212yes—
55#1441C2212yes—
56#1439C2212yes—
57#1439C2214yes—
58#1439C2212yes—
59#1439C2212yes—
60#1437C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1434C2212yes—
66#1434C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C23×Q8C2312yes—
70C23×Q8C2312yes—
71C24×C4C2311yescharacteristic
72C24×C4C2312yes—
73C2×C4⋊Q8—216no—
74C2×C4⋊Q8—48no—
75C2×C4⋊Q8—48no—
76C2×C22⋊Q8—44no—
77C2×C22⋊Q8—48no—
78C2×#117—24no—
79C2×#117—24no—
80C2×#117—24no—
81C2×#117—42no—
82C2×#117—42no—
83C2×#117Q818yescomplemented
84C22×C4⋊C4—24no—
85C22×C4⋊C4—42no—
86C22×C4⋊C4—42no—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109C22×C42C2314yes—
110C22×C42C2311yescharacteristic
111#398—48no—
112#395—28no—
113#395—28no—
114#395—44no—
115#395—48no—
116#393—44no—
117#391—44no—
118#389—216no—
119#389—216no—
120#385—216no—
121#385—48no—
122#384—24no—
123#384D418yes—
124#383—28no—
125#382—24no—
126#382—24no—
127C4×C4⋊C4—28no—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C25C2411yescharacteristic
154C22×Q8—24no—
155C22×Q8—24no—
156C22×Q8—28no—
157C22×Q8—28no—
158C22×Q8—42no—
159C22×Q8—44no—
160C22×Q8—44no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—42no—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—42no—
173C23×C4—44no—
174C23×C4—44no—
175C23×C4C2×Q814yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C4⋊Q8—464no—
189C22⋊Q8—432no—
190#117—232no—
191#117—416no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—416no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4C2×D414yes—
234C2×C4⋊C4C2×D414yes—
235C2×C4⋊C4C2×D4116yes—
236C2×C4⋊C4C2×Q818yes—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—42no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4C2×Q818yes—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—24no—
252C2×C42—28no—
253C2×C42—24no—
254C2×C42—42no—
255C2×C42—44no—
256C2×C42—44no—
257C2×C42C2×D414yes—
258C2×C42C2×Q814yes—
259C2×C42C2×Q814yes—
260C2×C42#4114yes—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—24no—
265#95—24no—
266#95—28no—
267#95—24no—
268#95—24no—
269#95—28no—
270#95—28no—
271#95—28no—
272#95—28no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—44no—
277#95—44no—
278#95—44no—
279#95—42no—
280#95—44no—
281#95—44no—
282#95—42no—
283#95—44no—
284#95—44no—
285#95—44no—
286C24—21no—
287C24—21no—
288C24—24no—
289C24—21no—
290C24—24no—
291C24—21no—
292C24C22×Q812yes—
293C24H(5,𝔽2)12yes—
294C242−1+412yes—
295C24C2511yescharacteristic
296C2×Q8—48no—
297C2×Q8—416no—
298C2×Q8—416no—
299C2×Q8—48no—
300C2×Q8—84no—
301C2×Q8—84no—
302C2×Q8—88no—
303C2×Q8—88no—
304C2×Q8—84no—
305C2×Q8—84no—
306C22×C4—28no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—28no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—24no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—48no—
372C22×C4—48no—
373C22×C4—88no—
374C22×C4—84no—
375C22×C4—88no—
376C22×C4C22×D414yes—
377C22×C4C22×D414yes—
378C22×C4C22×D412yes—
379C22×C4C22×D414yes—
380C22×C4C22×Q814yes—
381C22×C4C22×Q812yes—
382C22×C4C22×Q814yes—
383C22×C4C22×Q812yes—
384C22×C4C2×#4112yes—
385C22×C4C2×#4114yes—
386C22×C4H(5,𝔽2)18yes—
387C22×C4H(5,𝔽2)12yes—
388C22×C42−1+418yes—
389C22×C42−1+412yes—
390C4⋊C4—216no—
391C4⋊C4—216no—
392C4⋊C4—216no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—416no—
396C4⋊C4—48no—
397C4⋊C4—416no—
398C4⋊C4—416no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—416no—
402C4⋊C4—48no—
403C4⋊C4—88no—
404C4⋊C4—88no—
405C4⋊C4—84no—
406C22⋊C4—216no—
407C22⋊C4—416no—
408C22⋊C4—48no—
409C22⋊C4—416no—
410C42—28no—
411C42—48no—
412C42—416no—
413C42C22⋊Q8116yescomplemented
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—28no—
421C23—24no—
422C23—28no—
423C23—24no—
424C23—42no—
425C23—42no—
426C23—44no—
427C23—42no—
428C23—42no—
429C23—44no—
430C23#54414yes—
431C23C23×D411yescharacteristic
432C23C23×Q811yescharacteristic
433C23C22×#4111yescharacteristic
434C23C2×H(5,𝔽2)11yescharacteristic
435C23C2×H(5,𝔽2)11yescharacteristic
436C23C2×H(5,𝔽2)14yes—
437C23C2×2−1+411yescharacteristic
438C23C2×2−1+411yescharacteristic
439C23C2×2−1+414yes—
440Q8—816no—
441Q8—816no—
442C2×C4—232no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—28no—
458C2×C4—28no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—816no—
513C2×C4C2×C22⋊Q814yes—
514C2×C4C2×C22⋊Q818yes—
515C2×C4#54414yes—
516C22—28no—
517C22—24no—
518C22—42no—
519C22—44no—
520C22—42no—
521C22—44no—
522C22—44no—
523C22—44no—
524C22—44no—
525C22C22×C22⋊Q811yescharacteristic
526C22C2×#54411yescharacteristic
527C22C2×#54411yescharacteristic
528C22#348412yes—
529C22#348411yescharacteristic
530C22#348611yescharacteristic
531C22#348612yes—
532C22#349211yescharacteristic
533C22#349211yescharacteristic
534C22#350314yes—
535C22#351114yes—
536C22#353212yes—
537C22#353712yes—
538C22#353814yes—
539C22#354412yes—
540C22#355012yes—
541C22#355214yes—
542C4—24no—
543C4—24no—
544C4—28no—
545C4—44no—
546C4—44no—
547C4—48no—
548C4—48no—
549C4—44no—
550C4—82no—
551C4—82no—
552C4—84no—
553C4—84no—
554C2—44no—
555C2#3522411yescharacteristic
556C2#3557412yes—
557C2#4199912yes—
558C2#4293312yes—
559C2#4775012yes—
560C2#4775212yes—
561C2#5200411yescharacteristic
562C2#5200712yes—
563C2#5207211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2551C2semidirect
3. #16686C2semidirect
4. #16522C2semidirect
5. #16059C2semidirect
7. #15318C2semidirect
11. #14676C2semidirect
12. #14298C2semidirect
83. C2×#117Q8semidirect
413. C42C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#23172
#166862
#165222
#160592
#154622
#153182
#152652
#148872
#147202
#146762
#142982
#139382