Finite Group Databasebeta

← #4606927

Subgroups — #4606927

Order 512.

2130 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2811C211yesmaximal, characteristic, complemented
2C2×#2503C211yesmaximal, characteristic
3#17112C214yesmaximal, complemented
4#16694C212yesmaximal, complemented
5#16686C212yesmaximal, complemented
6#16389C214yesmaximal, complemented
7#16262C214yesmaximal
8#16110C212yesmaximal, complemented
9#15465C212yesmaximal
10#15371C214yesmaximal
11#15050C212yesmaximal
12#14720C212yesmaximal
13#14668C211yesmaximal, characteristic, complemented
14C22×C22⋊Q8C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C23×C4⋊C4C2211yescharacteristic
17#2811—28no—
18#2503—28no—
19C2×#400C2214yes—
20C2×#400C2212yes—
21C2×#397C2212yes—
22C2×#397C2212yes—
23C2×#395C2212yes—
24C2×#395C2214yes—
25C2×#395C2212yes—
26C2×#393C2212yes—
27C2×#391C2214yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#385C2214yes—
33C2×#382C2211yescharacteristic
34C2×#382C2211yescharacteristic
35C2×#381C2214yes—
36C2×C4×C22⋊C4C2211yescharacteristic
37#1446C2212yes—
38#1446C2214yes—
39#1446C2214yes—
40#1446C2212yes—
41#1445C2214yes—
42#1445C2212yes—
43#1444C2214yes—
44#1444C2214yes—
45#1444C2214yes—
46#1444C2214yes—
47#1443C2214yes—
48#1443C2214yes—
49#1443C2214yes—
50#1442C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2212yes—
58#1441C2214yes—
59#1441C2212yes—
60#1440C2214yes—
61#1440C2214yes—
62#1437C2212yes—
63#1436C2212yes—
64#1434C2214yes—
65#1434C2214yes—
66#1434C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2214yes—
69C23×Q8C2312yes—
70C23×Q8C2311yescharacteristic
71C23×Q8C2312yes—
72C24×C4C2311yescharacteristic
73C24×C4C2312yes—
74C2×C22⋊Q8—24no—
75C2×C22⋊Q8—48no—
76C2×C22⋊Q8—42no—
77C2×C22⋊Q8—44no—
78C2×C22⋊Q8—42no—
79C22×C4⋊C4—24no—
80C22×C4⋊C4—42no—
81C22×C4⋊C4—42no—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2314yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C42C2311yescharacteristic
98C22×C42C2312yes—
99C22×C42C2311yescharacteristic
100#400—216no—
101#400—44no—
102#397—44no—
103#397—44no—
104#395—28no—
105#395—216no—
106#395—44no—
107#393—44no—
108#391—24no—
109#391—28no—
110#391—28no—
111#391—216no—
112#391D418yes—
113#385—48no—
114#382—24no—
115#382—24no—
116#381—216no—
117C4×C22⋊C4—24no—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2312yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C25C2411yescharacteristic
149C22×Q8—24no—
150C22×Q8—24no—
151C22×Q8—28no—
152C22×Q8—24no—
153C22×Q8—41no—
154C22×Q8—41no—
155C22×Q8—44no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8—44no—
159C22×Q8—42no—
160C22×Q8—44no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8C2×D414yes—
164C23×C4—22no—
165C23×C4—22no—
166C23×C4—22no—
167C23×C4—24no—
168C23×C4—28no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—44no—
174C23×C4C2414yes—
175C23×C4C2412yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179C23×C4C2412yes—
180C23×C4C2412yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C23×C4C2414yes—
185C23×C4C2411yescharacteristic
186C22⋊Q8—416no—
187C22⋊Q8—432no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—416no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4C2×D414yes—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C42—22no—
234C2×C42—24no—
235C2×C42—24no—
236C2×C42—42no—
237C2×C42—44no—
238C2×C42C2×D414yes—
239#95—24no—
240#95—28no—
241#95—28no—
242#95—28no—
243#95—24no—
244#95—28no—
245#95—28no—
246#95—28no—
247#95—28no—
248#95—28no—
249#95—28no—
250#95—28no—
251#95—28no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—44no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95—44no—
265#95—44no—
266#95—42no—
267#95—44no—
268#95C2×D4116yes—
269C24—21no—
270C24—21no—
271C24—24no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24H(5,𝔽2)14yes—
276C24H(5,𝔽2)12yes—
277C24C2511yescharacteristic
278C2×Q8—216no—
279C2×Q8—28no—
280C2×Q8—44no—
281C2×Q8—44no—
282C2×Q8—48no—
283C2×Q8—84no—
284C2×Q8—84no—
285C2×Q8—84no—
286C2×Q8—84no—
287C2×Q8—84no—
288C2×Q8—88no—
289C2×Q8—88no—
290C2×Q8—88no—
291C2×Q8—84no—
292C2×Q8—88no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—28no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—24no—
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—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—24no—
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—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—48no—
356C22×C4—48no—
357C22×C4—44no—
358C22×C4—48no—
359C22×C4—416no—
360C22×C4—44no—
361C22×C4—48no—
362C22×C4—84no—
363C22×C4—88no—
364C22×C4C22×D414yes—
365C22×C4C22×D414yes—
366C22×C4C22×D414yes—
367C22×C4C22×D412yes—
368C22×C4H(5,𝔽2)12yes—
369C22×C42−1+418yes—
370C22×C42−1+414yes—
371C4⋊C4—44no—
372C4⋊C4—48no—
373C4⋊C4—48no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—48no—
377C4⋊C4—416no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C4⋊C4—88no—
381C4⋊C4—88no—
382C22⋊C4—48no—
383C22⋊C4—416no—
384C22⋊C4—48no—
385C22⋊C4—416no—
386C42—44no—
387C42—44no—
388C42—84no—
389C23—22no—
390C23—24no—
391C23—28no—
392C23—24no—
393C23—28no—
394C23—24no—
395C23—22no—
396C23—24no—
397C23—41no—
398C23—41no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—44no—
403C23—44no—
404C23H(3,𝔽4)14yes—
405C23C23×D411yescharacteristic
406C23C2×H(5,𝔽2)11yescharacteristic
407C23C2×H(5,𝔽2)11yescharacteristic
408C23C2×H(5,𝔽2)12yes—
409C23C2×2−1+412yes—
410C23C2×2−1+414yes—
411C23#58512yes—
412C23#58512yes—
413Q8—88no—
414Q8—816no—
415Q8—816no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—28no—
428C2×C4—24no—
429C2×C4—28no—
430C2×C4—28no—
431C2×C4—24no—
432C2×C4—28no—
433C2×C4—416no—
434C2×C4—41no—
435C2×C4—44no—
436C2×C4—41no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—816no—
485C2×C4#53614yes—
486C22—24no—
487C22—28no—
488C22—48no—
489C22—44no—
490C22—42no—
491C22—42no—
492C22—44no—
493C22—44no—
494C22C2×#53611yescharacteristic
495C22#348411yescharacteristic
496C22#348511yescharacteristic
497C22#348612yes—
498C22#348812yes—
499C22C2×H(3,𝔽4)11yescharacteristic
500C22#353512yes—
501C22#353514yes—
502C22#353512yes—
503C22#354414yes—
504C22#354412yes—
505C22#354511yescharacteristic
506C22#355012yes—
507C22#355514yes—
508C22#355812yes—
509C22#356414yes—
510C4—24no—
511C4—42no—
512C4—44no—
513C4—44no—
514C4—48no—
515C4—44no—
516C4—84no—
517C4—82no—
518C4—82no—
519C4—82no—
520C4—84no—
521C4—84no—
522C2—44no—
523C2#3672011yescharacteristic
524C2#4281312yes—
525C2#4597711yescharacteristic
526C2#4877912yes—
527C2#5207211yescharacteristic
528C2#5291412yes—
529C2#5397512yes—
530C2#5532614yes—
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×#2811C2semidirect
3. #17112C2semidirect
4. #16694C2semidirect
5. #16686C2semidirect
6. #16389C2semidirect
8. #16110C2semidirect
13. #14668C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28112
C2×#25032
#171122
#166942
#166862
#163892
#162622
#161102
#154652
#153712
#150502
#147202
#146682