Finite Group Databasebeta

← #4607016

Subgroups — #4607016

Order 512.

2326 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2734C211yesmaximal, characteristic, complemented
2C2×#2547C211yesmaximal, characteristic
3#17106C214yesmaximal, complemented
4#16651C214yesmaximal, complemented
5#16629C212yesmaximal, complemented
6#16510C212yesmaximal
7#16303C214yesmaximal, complemented
8#15995C214yesmaximal, complemented
9#15762C212yesmaximal, complemented
10#15761C212yesmaximal
11#15108C212yesmaximal, complemented
12#14824C212yesmaximal
13#14668C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4⋊Q8C2212yes—
16C22×#124C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19#2734—28no—
20#2547—28no—
21C2×#399C2214yes—
22C2×#398C2212yes—
23C2×#397C2214yes—
24C2×#397C2212yes—
25C2×#395C2214yes—
26C2×#395C2214yes—
27C2×#395C2212yes—
28C2×#395C2212yes—
29C2×#394C2214yes—
30C2×#393C2214yes—
31C2×#391C2211yescharacteristic
32C2×#391C2211yescharacteristic
33C2×#389C2212yes—
34C2×#387C2214yes—
35C2×#380C2214yes—
36C2×#380C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38#1446C2212yes—
39#1444C2214yes—
40#1444C2214yes—
41#1444C2214yes—
42#1443C2214yes—
43#1443C2214yes—
44#1443C2214yes—
45#1442C2212yes—
46#1442C2214yes—
47#1442C2212yes—
48#1442C2214yes—
49#1442C2214yes—
50#1441C2212yes—
51#1441C2212yes—
52#1441C2214yes—
53#1441C2212yes—
54#1440C2214yes—
55#1439C2212yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1437C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64#1436C2212yes—
65#1436C2212yes—
66#1434C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2311yescharacteristic
71C23×Q8C2312yes—
72C23×D4C2311yescharacteristic
73C24×C4C2312yes—
74C2×C4⋊Q8—28no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8—42no—
77C2×C4⋊Q8—42no—
78C2×C4⋊Q8—48no—
79C2×#124—24no—
80C2×#124—24no—
81C2×#124—44no—
82C2×#124—44no—
83C2×C4×D4—24no—
84C2×C4×D4—42no—
85C2×C4×D4—42no—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2312yes—
108C22×C42C2311yescharacteristic
109#399—48no—
110#398—28no—
111#397—216no—
112#397—44no—
113#395—216no—
114#395—44no—
115#395—44no—
116#395—48no—
117#394—48no—
118#393—48no—
119#391—24no—
120#391D418yes—
121#389—28no—
122#387—216no—
123#380—216no—
124#380—44no—
125C4×C4⋊C4—24no—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C25C2412yes—
152C22×Q8—22no—
153C22×Q8—24no—
154C22×Q8—24no—
155C22×Q8—24no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8—41no—
161C22×Q8—41no—
162C22×Q8—44no—
163C22×Q8—44no—
164C22×Q8—42no—
165C22×Q8—42no—
166C22×Q8C2×D414yes—
167C22×D4—24no—
168C22×D4—22no—
169C22×D4—24no—
170C22×D4—41no—
171C22×D4—41no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—28no—
175C23×C4—42no—
176C23×C4—44no—
177C23×C4C2412yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2412yes—
184C23×C4C2412yes—
185C23×C4C2411yescharacteristic
186C23×C4C2414yes—
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C4⋊Q8—416no—
190C4⋊Q8—432no—
191#124—416no—
192#124—416no—
193C4×D4—416no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—28no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—44no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4C2×D414yes—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—24no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—44no—
256C2×C42C2×D414yes—
257#95—24no—
258#95—28no—
259#95—24no—
260#95—28no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—28no—
268#95—42no—
269#95—42no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—42no—
278#95—42no—
279#95—42no—
280#95—44no—
281#95C2×D4116yes—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24H(5,𝔽2)14yes—
293C24C2511yescharacteristic
294C2×Q8—28no—
295C2×Q8—44no—
296C2×Q8—44no—
297C2×Q8—44no—
298C2×Q8—44no—
299C2×Q8—48no—
300C2×Q8—44no—
301C2×Q8—84no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×Q8—84no—
305C2×Q8—84no—
306C2×Q8—84no—
307C2×Q8—88no—
308C2×Q8—88no—
309C2×D4—28no—
310C2×D4—48no—
311C2×D4—48no—
312C2×D4—44no—
313C2×D4—44no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—21no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—22no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—48no—
373C22×C4—416no—
374C22×C4—48no—
375C22×C4—84no—
376C22×C4—88no—
377C22×C4C22×D414yes—
378C22×C4C22×D414yes—
379C22×C4C22×D414yes—
380C22×C4C22×D412yes—
381C22×C4H(5,𝔽2)12yes—
382C22×C4H(5,𝔽2)14yes—
383C22×C4H(5,𝔽2)12yes—
384C22×C42−1+418yes—
385C22×C42−1+414yes—
386C4⋊C4—216no—
387C4⋊C4—44no—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—416no—
392C4⋊C4—416no—
393C4⋊C4—84no—
394C4⋊C4—84no—
395C4⋊C4—84no—
396C4⋊C4—88no—
397C4⋊C4—88no—
398C22⋊C4—48no—
399C22⋊C4—416no—
400C22⋊C4—416no—
401C22⋊C4—88no—
402C22⋊C4—88no—
403C22⋊C4—88no—
404C22⋊C4—88no—
405C42—28no—
406C42—44no—
407C42—44no—
408C42—48no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—28no—
413C23—28no—
414C23—42no—
415C23—42no—
416C23—44no—
417C23—44no—
418C23—42no—
419C23—42no—
420C23—42no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23C23×D411yescharacteristic
433C23C2×H(5,𝔽2)12yes—
434C23C2×H(5,𝔽2)11yescharacteristic
435C23C2×H(5,𝔽2)12yes—
436C23C2×H(5,𝔽2)11yescharacteristic
437C23C2×2−1+414yes—
438C23C2×2−1+412yes—
439C23#58512yes—
440Q8—88no—
441Q8—88no—
442Q8—816no—
443D4—416no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—24no—
461C2×C4—28no—
462C2×C4—24no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
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—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—816no—
510C2×C4#53514yes—
511C2×C4#56314yes—
512C22—48no—
513C22—44no—
514C22—44no—
515C22—48no—
516C22—44no—
517C22—84no—
518C22—82no—
519C22—84no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22C2×#53511yescharacteristic
524C22#348411yescharacteristic
525C22#348411yescharacteristic
526C22#348612yes—
527C22#348912yes—
528C22C2×#56311yescharacteristic
529C22#353712yes—
530C22#353712yes—
531C22#354412yes—
532C22#354414yes—
533C22#354511yescharacteristic
534C22#355012yes—
535C22#355214yes—
536C22#355914yes—
537C22#356112yes—
538C22#356114yes—
539C4—24no—
540C4—24no—
541C4—44no—
542C4—48no—
543C4—42no—
544C4—44no—
545C4—44no—
546C4—84no—
547C4—84no—
548C4—82no—
549C4—84no—
550C4—82no—
551C2—84no—
552C2#3671711yescharacteristic
553C2#4281212yes—
554C2#4613611yescharacteristic
555C2#4954812yes—
556C2#5207211yescharacteristic
557C2#5296812yes—
558C2#5351014yes—
559C2#5364212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2734C2semidirect
3. #17106C2semidirect
4. #16651C2semidirect
5. #16629C2semidirect
7. #16303C2semidirect
8. #15995C2semidirect
9. #15762C2semidirect
11. #15108C2semidirect
13. #14668C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27342
C2×#25472
#171062
#166512
#166292
#165102
#163032
#159952
#157622
#157612
#151082
#148242
#146682