Finite Group Databasebeta

← #948195

Subgroups — #948195

Order 512.

2390 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2564C212yesmaximal
2#17228C211yesmaximal, characteristic, complemented
3#17144C212yesmaximal, complemented
4#17085C211yesmaximal, characteristic, complemented
5#16300C214yesmaximal, complemented
6#16299C214yesmaximal, complemented
7#15952C212yesmaximal
8#15952C212yesmaximal
9#15122C212yesmaximal, complemented
10#14763C214yesmaximal, complemented
11#14314C212yesmaximal, complemented
12#13972C214yesmaximal, complemented
13#13847C211yesmaximal, characteristic
14C22×C22⋊Q8C2212yes—
15C22×C4×D4C2211yescharacteristic
16#2564—216no—
17C2×#400C2212yes—
18C2×#399C2212yes—
19C2×#398C2212yes—
20C2×#395C2212yes—
21C2×#395C2214yes—
22C2×#393C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#387C2214yes—
26C2×#386C2214yes—
27C2×#385C2214yes—
28C2×#381C2214yes—
29C2×#381C2214yes—
30C2×#380C2214yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2214yes—
33C2×#376C2211yescharacteristic
34C22×#95C2212yes—
35#1446C2214yes—
36#1445C2212yes—
37#1445C2212yes—
38#1445C2211yescharacteristic
39#1445C2211yescharacteristic
40#1445C2212yes—
41#1444C2212yes—
42#1444C2214yes—
43#1443C2214yes—
44#1443C2214yes—
45#1443C2214yes—
46#1443C2214yes—
47#1442C2212yes—
48#1442C2214yes—
49#1442C2212yes—
50#1442C2214yes—
51#1442C2214yes—
52#1441C2214yes—
53#1441C2212yes—
54#1441C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2211yescharacteristic
61#1438C2211yescharacteristic
62#1436C2214yes—
63#1436C2214yes—
64#1434C2212yes—
65#1434C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C23×Q8C2312yes—
69C23×D4C2311yescharacteristic
70C24×C4C2314yes—
71C24×C4C2312yes—
72C2×C22⋊Q8—28no—
73C2×C22⋊Q8—44no—
74C2×C22⋊Q8—44no—
75C2×C4×D4—42no—
76C2×C4×D4—42no—
77C2×C4×D4D418yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2311yescharacteristic
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C42C2312yes—
95C22×C42C2311yescharacteristic
96C22×C42C2312yes—
97#400—28no—
98#399—28no—
99#398—28no—
100#395—28no—
101#395—48no—
102#393—28no—
103#389—44no—
104#389—44no—
105#387—216no—
106#386—216no—
107#385—216no—
108#381—216no—
109#381—48no—
110#380—216no—
111C4×C4⋊C4—42no—
112C4×C22⋊C4—48no—
113#376—42no—
114C2×#95—28no—
115C2×#95C2312yes—
116C2×#95C2314yes—
117C2×#95C2312yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138C2×#95C2311yescharacteristic
139C2×#95C2311yescharacteristic
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2312yes—
145C25C2412yes—
146C22×Q8—24no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—44no—
150C22×Q8—44no—
151C22×D4—28no—
152C22×D4—41no—
153C22×D4—41no—
154C22×D4C2×D414yes—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—28no—
158C23×C4—28no—
159C23×C4—28no—
160C23×C4—28no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4C2×D418yes—
167C23×C4C2414yes—
168C23×C4C2414yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2414yes—
172C23×C4C2411yescharacteristic
173C23×C4C2412yes—
174C23×C4C2411yescharacteristic
175C23×C4C2412yes—
176C23×C4C2414yes—
177C23×C4C2412yes—
178C23×C4C2411yescharacteristic
179C22⋊Q8—432no—
180C4×D4—416no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4C2×D414yes—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4C2×D418yes—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—42no—
233C2×C42—44no—
234C2×C42C2×D414yes—
235#95—28no—
236#95—24no—
237#95—24no—
238#95—28no—
239#95—28no—
240#95—28no—
241#95—28no—
242#95—28no—
243#95—28no—
244#95—28no—
245#95—28no—
246#95—28no—
247#95—24no—
248#95—28no—
249#95—42no—
250#95—44no—
251#95—42no—
252#95—44no—
253#95—42no—
254#95—41no—
255#95—41no—
256#95—44no—
257#95—41no—
258#95—41no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95#4118yes—
265C24—24no—
266C24—24no—
267C24—22no—
268C24—22no—
269C24—24no—
270C24—24no—
271C24—24no—
272C24C22×D414yes—
273C242−1+418yes—
274C24C2511yescharacteristic
275C2×Q8—48no—
276C2×Q8—88no—
277C2×Q8—88no—
278C2×Q8—84no—
279C2×Q8—84no—
280C2×D4—216no—
281C2×D4—44no—
282C2×D4—48no—
283C2×D4—48no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—24no—
311C22×C4—216no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—416no—
344C22×C4—416no—
345C22×C4—416no—
346C22×C4—416no—
347C22×C4—416no—
348C22×C4—88no—
349C22×C4—84no—
350C22×C4—84no—
351C22×C4—88no—
352C22×C4—88no—
353C22×C4—88no—
354C22×C4C22×D412yes—
355C22×C4C22×D412yes—
356C22×C4C22×D414yes—
357C22×C4C22×D412yes—
358C22×C4C2×#4118yes—
359C22×C4C2×#4114yes—
360C22×C4H(5,𝔽2)14yes—
361C22×C4H(5,𝔽2)12yes—
362C22×C4H(5,𝔽2)12yes—
363C4⋊C4—44no—
364C4⋊C4—416no—
365C4⋊C4—48no—
366C4⋊C4—48no—
367C4⋊C4—48no—
368C4⋊C4—48no—
369C4⋊C4—416no—
370C4⋊C4—88no—
371C4⋊C4—84no—
372C4⋊C4—84no—
373C22⋊C4—48no—
374C22⋊C4—416no—
375C22⋊C4—416no—
376C22⋊C4—416no—
377C22⋊C4—416no—
378C22⋊C4—88no—
379C42—28no—
380C42—48no—
381C42—84no—
382C23—24no—
383C23—24no—
384C23—28no—
385C23—28no—
386C23—28no—
387C23—28no—
388C23—28no—
389C23—28no—
390C23—28no—
391C23—28no—
392C23—44no—
393C23—42no—
394C23—44no—
395C23—44no—
396C23—42no—
397C23—44no—
398C23—44no—
399C23—42no—
400C23—42no—
401C23—44no—
402C23#53618yes—
403C23C23×D411yescharacteristic
404C23C22×#4112yes—
405C23C2×H(5,𝔽2)11yescharacteristic
406C23C2×H(5,𝔽2)11yescharacteristic
407C23C2×H(5,𝔽2)12yes—
408C23C2×2−1+414yes—
409C23#58512yes—
410C23#58512yes—
411Q8—816no—
412D4—416no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—28no—
423C2×C4—24no—
424C2×C4—28no—
425C2×C4—28no—
426C2×C4—28no—
427C2×C4—28no—
428C2×C4—24no—
429C2×C4—28no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—816no—
480C2×C4—832no—
481C2×C4#53414yes—
482C22—216no—
483C22—28no—
484C22—44no—
485C22—48no—
486C22—48no—
487C22—48no—
488C22—48no—
489C22—44no—
490C22—48no—
491C22C2×#53411yescharacteristic
492C22C2×#53612yes—
493C22#346012yes—
494C22D4×#4112yes—
495C22#348311yescharacteristic
496C22#349514yes—
497C22#349612yes—
498C22#352014yes—
499C22#352911yescharacteristic
500C22#353412yes—
501C22#353514yes—
502C22#353812yes—
503C22#353812yes—
504C22#354311yescharacteristic
505C22#354511yescharacteristic
506C22#356414yes—
507C4—24no—
508C4—44no—
509C4—48no—
510C4—42no—
511C4—44no—
512C4—44no—
513C4—42no—
514C4—84no—
515C4—84no—
516C4—82no—
517C4—84no—
518C4—82no—
519C2—48no—
520C2#3712911yescharacteristic
521C2#3745311yescharacteristic
522C2#3842214yes—
523C2#4136512yes—
524C2#4240412yes—
525C2#4348812yes—
526C2#5216711yescharacteristic
527C2#5267812yes—
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
2. #17228C2semidirect
3. #17144C2semidirect
4. #17085C2semidirect
5. #16300C2semidirect
6. #16299C2semidirect
9. #15122C2semidirect
10. #14763C2semidirect
11. #14314C2semidirect
12. #13972C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25642
#172282
#171442
#170852
#163002
#162992
#159522
#151222
#147632
#143142
#139722
#138472