Finite Group Databasebeta

← #772309

Subgroups — #772309

Order 512.

2856 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15871C214yesmaximal, complemented
2#15412C214yesmaximal, complemented
3#15334C214yesmaximal, complemented
4#15270C214yesmaximal, complemented
5#15249C211yesmaximal, characteristic, complemented
6#15172C211yesmaximal, characteristic, complemented
7#14886C212yesmaximal, complemented
8#14207C214yesmaximal, complemented
9#14095C214yesmaximal, complemented
10#13852C211yesmaximal, characteristic, complemented
11#13791C212yesmaximal, complemented
12C22×#123C2214yes—
13C22×#117C2211yescharacteristic
14C2×#399C2214yes—
15C2×#394C2212yes—
16C2×#392C2214yes—
17C2×#392C2212yes—
18C2×#392C2214yes—
19C2×#390C2212yes—
20C2×#390C2214yes—
21C2×#390C2214yes—
22C2×#387C2214yes—
23C2×#386C2214yes—
24C2×#386C2214yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C22⋊C4C2214yes—
29C2×C4×C22⋊C4C2212yes—
30#1445C2214yes—
31#1442C2214yes—
32#1441C2214yes—
33#1441C2214yes—
34#1441C2212yes—
35#1440C2214yes—
36#1439C2214yes—
37#1439C2214yes—
38#1439C2214yes—
39#1439C2214yes—
40#1439C2212yes—
41#1439C2214yes—
42#1439C2214yes—
43#1438C2214yes—
44#1438C2214yes—
45#1437C2214yes—
46#1437C2214yes—
47#1437C2214yes—
48#1437C2214yes—
49#1436C2214yes—
50#1436C2214yes—
51#1436C2214yes—
52#1434C2212yes—
53C22⋊#95C2214yes—
54C22⋊#95C2214yes—
55C4×#95C2211yescharacteristic
56C4×#95C2211yescharacteristic
57C23×D4C2312yes—
58C23×D4C2314yes—
59C23×D4C2314yes—
60C24×C4C2311yescharacteristic
61C24×C4C2314yes—
62C2×#123—216no—
63C2×#123—48no—
64C2×#123—48no—
65C2×#117—24no—
66C2×#117—44no—
67C22×C4⋊C4C2314yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2314yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2312yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C42C2311yescharacteristic
86C22×C42C2312yes—
87C22×C42C2312yes—
88C22×C42C2314yes—
89C22×C42C2311yescharacteristic
90#399—48no—
91#394—44no—
92#392—216no—
93#392—48no—
94#392—44no—
95#390—28no—
96#390—48no—
97#390—48no—
98#387—48no—
99#386—216no—
100#386—48no—
101#386—48no—
102#386—48no—
103C4×C4⋊C4—44no—
104C4×C22⋊C4—44no—
105C4×C22⋊C4—48no—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2312yes—
114C2×#95C2312yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C25C2414yes—
124C25C2411yescharacteristic
125C22×D4—28no—
126C22×D4—28no—
127C22×D4—28no—
128C22×D4—28no—
129C22×D4—28no—
130C22×D4—24no—
131C22×D4—28no—
132C22×D4—28no—
133C22×D4—44no—
134C22×D4—44no—
135C22×D4—44no—
136C22×D4—44no—
137C22×D4—44no—
138C22×D4—44no—
139C22×D4—44no—
140C22×D4—44no—
141C22×D4—42no—
142C22×D4—44no—
143C22×D4—42no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—28no—
148C23×C4—24no—
149C23×C4—28no—
150C23×C4—42no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4C2414yes—
157C23×C4C2414yes—
158C23×C4C2414yes—
159C23×C4C2412yes—
160C23×C4C2414yes—
161C23×C4C2412yes—
162C23×C4C2411yescharacteristic
163C23×C4C2414yes—
164C23×C4C2411yescharacteristic
165#123—464no—
166#117—416no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4#41116yes—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—44no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42—44no—
231C2×C42—44no—
232C2×C42#4114yes—
233#95—28no—
234#95—28no—
235#95—28no—
236#95—28no—
237#95—28no—
238#95—28no—
239#95—28no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95—42no—
249#95—42no—
250C24—21no—
251C24—24no—
252C24—24no—
253C24—24no—
254C24—24no—
255C24—24no—
256C24—24no—
257C24—24no—
258C24—24no—
259C24—24no—
260C24—24no—
261C24—24no—
262C24—22no—
263C24—24no—
264C24—24no—
265C24—21no—
266C24—24no—
267C24—22no—
268C24—24no—
269C24—24no—
270C24C2×#4118yes—
271C242−1+412yes—
272C24C2511yescharacteristic
273C2×D4—216no—
274C2×D4—416no—
275C2×D4—416no—
276C2×D4—416no—
277C2×D4—416no—
278C2×D4—416no—
279C2×D4—416no—
280C2×D4—84no—
281C2×D4—88no—
282C2×D4—84no—
283C2×D4—88no—
284C2×D4—88no—
285C2×D4—88no—
286C2×D4—88no—
287C2×D4—88no—
288C2×D4—88no—
289C2×D4—88no—
290C2×D4—88no—
291C2×D4—88no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—28no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—21no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—21no—
325C22×C4—24no—
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—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—48no—
341C22×C4—48no—
342C22×C4—44no—
343C22×C4—416no—
344C22×C4—416no—
345C22×C4—44no—
346C22×C4—88no—
347C22×C4—88no—
348C22×C4—88no—
349C22×C4—88no—
350C22×C4—88no—
351C22×C4C2×#4118yes—
352C22×C4C2×#4118yes—
353C22×C4C2×#4114yes—
354C22×C4C2×#4112yes—
355C22×C4H(5,𝔽2)12yes—
356C22×C4H(5,𝔽2)18yes—
357C22×C4H(5,𝔽2)14yes—
358C22×C4H(5,𝔽2)14yes—
359C4⋊C4—232no—
360C4⋊C4—48no—
361C4⋊C4—416no—
362C4⋊C4—84no—
363C4⋊C4—88no—
364C22⋊C4—416no—
365C22⋊C4—416no—
366C22⋊C4—416no—
367C22⋊C4—48no—
368C22⋊C4—416no—
369C22⋊C4—416no—
370C22⋊C4—416no—
371C22⋊C4—88no—
372C22⋊C4—88no—
373C22⋊C4—88no—
374C22⋊C4—88no—
375C22⋊C4—88no—
376C22⋊C4—88no—
377C42—28no—
378C42—44no—
379C42—84no—
380C42—84no—
381C42—88no—
382C23—28no—
383C23—28no—
384C23—28no—
385C23—28no—
386C23—22no—
387C23—28no—
388C23—28no—
389C23—28no—
390C23—28no—
391C23—24no—
392C23—42no—
393C23—44no—
394C23—44no—
395C23—41no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—41no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—42no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—42no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23C22×#4111yescharacteristic
431C23C22×#4114yes—
432C23C2×H(5,𝔽2)12yes—
433C23C2×H(5,𝔽2)12yes—
434C23C2×H(5,𝔽2)14yes—
435C23C2×H(5,𝔽2)11yescharacteristic
436C23C2×2−1+411yescharacteristic
437D4—816no—
438D4—832no—
439D4—832no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—28no—
446C2×C4—28no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—28no—
450C2×C4—28no—
451C2×C4—28no—
452C2×C4—28no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—28no—
456C2×C4—28no—
457C2×C4—28no—
458C2×C4—24no—
459C2×C4—28no—
460C2×C4—416no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—44no—
464C2×C4—41no—
465C2×C4—44no—
466C2×C4—42no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
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—44no—
486C2×C4—44no—
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—832no—
496C2×C4#54014yes—
497C22—48no—
498C22—44no—
499C22—48no—
500C22—42no—
501C22—48no—
502C22—48no—
503C22—48no—
504C22—48no—
505C22—48no—
506C22—48no—
507C22—84no—
508C22—82no—
509C22—84no—
510C22—84no—
511C22—81no—
512C22—84no—
513C22—81no—
514C22—84no—
515C22—84no—
516C22—84no—
517C22—84no—
518C22—84no—
519C22#344412yes—
520C22C2×#54011yescharacteristic
521C22#348114yes—
522C22#349011yescharacteristic
523C22#349014yes—
524C22#349211yescharacteristic
525C22#350812yes—
526C22#350814yes—
527C22#351914yes—
528C22#352014yes—
529C22#353612yes—
530C22#353712yes—
531C22#354314yes—
532C4—24no—
533C4—44no—
534C4—44no—
535C4—42no—
536C4—48no—
537C4—48no—
538C4—48no—
539C4—84no—
540C4—84no—
541C2—82no—
542C2—88no—
543C2#3496711yescharacteristic
544C2#3541912yes—
545C2#3543812yes—
546C2#3672711yescharacteristic
547C2#3731911yescharacteristic
548C2#4347614yes—
549C2#4424414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #15871C2semidirect
2. #15412C2semidirect
3. #15334C2semidirect
4. #15270C2semidirect
5. #15249C2semidirect
6. #15172C2semidirect
7. #14886C2semidirect
8. #14207C2semidirect
9. #14095C2semidirect
10. #13852C2semidirect
11. #13791C2semidirect

Maximal subgroups

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

subgroupindex
#158712
#154122
#153342
#152702
#152492
#151722
#148862
#142072
#140952
#138522
#137912