Finite Group Databasebeta

← #2037539

Subgroups — #2037539

Order 512.

2216 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2734C211yesmaximal, characteristic, complemented
2C2×#2861C211yesmaximal, characteristic
3#17069C212yesmaximal
4#16480C214yesmaximal, complemented
5#16093C212yesmaximal, complemented
6#15881C212yesmaximal, complemented
7#15818C214yesmaximal, complemented
8#15814C212yesmaximal
9#15772C214yesmaximal, complemented
10#15761C212yesmaximal
11#14699C212yesmaximal, complemented
12#14530C214yesmaximal, complemented
13#14471C211yesmaximal, characteristic, complemented
14C22×C4⋊Q8C2212yes—
15C22×#125C2211yescharacteristic
16C22×#124C2211yescharacteristic
17#2861—28no—
18#2734—28no—
19C2×#400C2212yes—
20C2×#398C2212yes—
21C2×#397C2214yes—
22C2×#395C2214yes—
23C2×#394C2214yes—
24C2×#393C2212yes—
25C2×#393C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#387C2214yes—
35C2×#387C2214yes—
36C2×#382C2212yes—
37C2×C43C2211yescharacteristic
38#1446C2214yes—
39#1446C2212yes—
40#1445C2212yes—
41#1442C2214yes—
42#1442C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1441C2214yes—
46#1441C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1437C2212yes—
56#1437C2212yes—
57#1437C2214yes—
58#1436C2214yes—
59#1436C2212yes—
60#1436C2212yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68#1434C2214yes—
69C23×Q8C2311yescharacteristic
70C23×Q8C2312yes—
71C23×D4C2311yescharacteristic
72C2×C4⋊Q8—28no—
73C2×C4⋊Q8—48no—
74C2×#125—24no—
75C2×#125—42no—
76C2×#125—42no—
77C2×#124—24no—
78C2×#124—42no—
79C2×#124—42no—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C42C2311yescharacteristic
101C22×C42C2312yes—
102C22×C42C2311yescharacteristic
103C22×C42C2312yes—
104C22×C42C2312yes—
105C22×C42C2312yes—
106C22×C42C2311yescharacteristic
107#400—44no—
108#398—28no—
109#397—216no—
110#395—216no—
111#394—48no—
112#393—44no—
113#393—44no—
114#391—24no—
115#391D418yes—
116#390—24no—
117#390—24no—
118#389—28no—
119#389—28no—
120#388—216no—
121#388—216no—
122#387—216no—
123#387—48no—
124#382—44no—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C43—24no—
149C25C2412yes—
150C22×Q8—24no—
151C22×Q8—24no—
152C22×Q8—41no—
153C22×Q8—41no—
154C22×Q8—44no—
155C22×Q8—42no—
156C22×Q8—44no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8C2×D414yes—
160C22×D4—22no—
161C22×D4—24no—
162C22×D4—41no—
163C22×D4—41no—
164C22×D4—42no—
165C23×C4C2414yes—
166C23×C4C2412yes—
167C23×C4C2414yes—
168C23×C4C2414yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2414yes—
172C23×C4C2411yescharacteristic
173C23×C4C2411yescharacteristic
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177C4⋊Q8—432no—
178#125—416no—
179#124—416no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C2×D414yes—
215C2×C4⋊C4#4118yes—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—24no—
246C2×C42—22no—
247C2×C42—24no—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—24no—
251C2×C42—44no—
252C2×C42—42no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—44no—
256C2×C42C2×D414yes—
257#95—24no—
258#95—28no—
259#95—28no—
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—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—42no—
276#95—42no—
277#95—42no—
278#95—44no—
279#95C2×D4116yes—
280C24—22no—
281C24—24no—
282C24—22no—
283C24—22no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24C2511yescharacteristic
290C2×Q8—28no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—48no—
294C2×Q8—84no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×Q8—84no—
298C2×Q8—88no—
299C2×Q8—88no—
300C2×D4—44no—
301C2×D4—44no—
302C2×D4—44no—
303C2×D4—84no—
304C2×D4—84no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
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—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4C22×D414yes—
367C22×C4C22×D414yes—
368C22×C4C22×D412yes—
369C22×C4C22×D414yes—
370C22×C4C2×#4118yes—
371C22×C4C2×#4114yes—
372C22×C4H(5,𝔽2)12yes—
373C22×C4H(5,𝔽2)12yes—
374C22×C4H(5,𝔽2)14yes—
375C22×C42−1+414yes—
376C22×C42−1+414yes—
377C4⋊C4—216no—
378C4⋊C4—44no—
379C4⋊C4—416no—
380C4⋊C4—48no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—44no—
384C4⋊C4—416no—
385C4⋊C4—416no—
386C4⋊C4—84no—
387C4⋊C4—84no—
388C4⋊C4—84no—
389C22⋊C4—416no—
390C22⋊C4—48no—
391C22⋊C4—48no—
392C22⋊C4—416no—
393C22⋊C4—88no—
394C22⋊C4—88no—
395C22⋊C4—88no—
396C22⋊C4—88no—
397C42—28no—
398C42—44no—
399C42—48no—
400C42—48no—
401C42—48no—
402C42—44no—
403C42—48no—
404C23—42no—
405C23—44no—
406C23—44no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—42no—
416C23—44no—
417C23—42no—
418C23—42no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23C23×D411yescharacteristic
426C23C22×#4112yes—
427C23C2×H(5,𝔽2)12yes—
428C23C2×H(5,𝔽2)11yescharacteristic
429C23C2×H(5,𝔽2)11yescharacteristic
430C23C2×2−1+412yes—
431C23C2×2−1+412yes—
432C23#58512yes—
433C23#58512yes—
434Q8—88no—
435Q8—816no—
436D4—88no—
437C2×C4—24no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—28no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—24no—
458C2×C4—28no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
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—42no—
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#53514yes—
508C2×C4#54818yes—
509C22—82no—
510C22—84no—
511C22—84no—
512C22—84no—
513C22—84no—
514C22—82no—
515C22—84no—
516C22—82no—
517C22—84no—
518C22C2×#53511yescharacteristic
519C22#346112yes—
520C22C2×#54812yes—
521C22#348311yescharacteristic
522C22#348411yescharacteristic
523C22#348511yescharacteristic
524C22#348912yes—
525C22#349312yes—
526C22#351412yes—
527C22#352012yes—
528C22#352212yes—
529C22#352911yescharacteristic
530C22#354712yes—
531C22#355011yescharacteristic
532C22#355011yescharacteristic
533C22#355212yes—
534C22#355812yes—
535C22#355812yes—
536C22#355912yes—
537C22#356312yes—
538C22#356412yes—
539C4—24no—
540C4—28no—
541C4—44no—
542C4—42no—
543C4—44no—
544C4—42no—
545C4—48no—
546C4—44no—
547C4—84no—
548C4—84no—
549C4—84no—
550C4—82no—
551C2—162no—
552C2#3696711yescharacteristic
553C2#3725612yes—
554C2#3775512yes—
555C2#3882912yes—
556C2#3912412yes—
557C2#4595812yes—
558C2#4622811yescharacteristic
559C2#5335612yes—
560C2#5908411yescharacteristic
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
4. #16480C2semidirect
5. #16093C2semidirect
6. #15881C2semidirect
7. #15818C2semidirect
9. #15772C2semidirect
11. #14699C2semidirect
12. #14530C2semidirect
13. #14471C2semidirect

Maximal subgroups

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

subgroupindex
C2×#27342
C2×#28612
#170692
#164802
#160932
#158812
#158182
#158142
#157722
#157612
#146992
#145302
#144712