Finite Group Databasebeta

← #409261

Subgroups — #409261

Order 512.

1288 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1854C212yesmaximal
2#14340C211yesmaximal, characteristic
3#13795C211yesmaximal, characteristic
4#11358C211yesmaximal, characteristic, complemented
5#11358C211yesmaximal, characteristic, complemented
6#11324C211yesmaximal, characteristic, complemented
7#11323C211yesmaximal, characteristic, complemented
8#11257C212yesmaximal, complemented
9#11256C212yesmaximal, complemented
10#10730C212yesmaximal
11#10713C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C22×C22⋊C8C2212yes—
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#380C2212yes—
21#1854—28no—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2211yescharacteristic
29C2×#336C2211yescharacteristic
30C2×#336C2212yes—
31#1441—22no—
32#1439—22no—
33C4⋊#95—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2211yescharacteristic
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C22⋊#95—22no—
40C24×C4—21no—
41C24×C4C2311yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×#117—22no—
45C2×#117—24no—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—42no—
49C2×#117—42no—
50C2×#117—42no—
51C2×#117D414yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—42no—
58C22×C4⋊C4D412yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4D412yes—
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C2×#107—24no—
72C2×#107—42no—
73C2×#107—42no—
74C2×#106—24no—
75C2×#106—42no—
76C2×#106—42no—
77C2×C22⋊C8—24no—
78C2×C22⋊C8—24no—
79C2×C22⋊C8Q818yes—
80#384—42no—
81#382—22no—
82#382—22no—
83#380—24no—
84#380—24no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—42no—
98#340—42no—
99#336—22no—
100#336—22no—
101#336—42no—
102C25C2×D411yescharacteristic
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—82no—
119C23×C4—82no—
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×Q814yes—
125C23×C4#4112yes—
126C23×C4#4114yes—
127C23×C4C2411yescharacteristic
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8C2×Q814yes—
141C22×C8C2×Q814yes—
142#117—48no—
143#117—48no—
144#117—48no—
145#117—48no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—48no—
174C2×C4⋊C4—48no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4#4112yes—
184C2×C4⋊C4#4112yes—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4—82no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—24no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42#4112yes—
204C2×C42#4112yes—
205#107—48no—
206#106—48no—
207C22⋊C8—216no—
208#95—44no—
209#95—44no—
210#95—82no—
211#95—82no—
212#95—82no—
213#95—82no—
214#95—82no—
215#95—82no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24#12112yes—
225C24#12312yes—
226C24C4⋊Q812yes—
227C24C22×D411yescharacteristic
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—48no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—88no—
295C22×C4—88no—
296C22×C4—84no—
297C22×C4—84no—
298C22×C4#12112yes—
299C22×C4C22⋊Q818yes—
300C22×C4#12314yes—
301C22×C4#12312yes—
302C22×C4#12518yes—
303C22×C4C4⋊Q812yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×Q812yes—
306C22×C4C2×#4112yes—
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4H(5,𝔽2)12yes—
313C22×C4H(5,𝔽2)12yes—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—48no—
333C4⋊C4—48no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—88no—
347C22⋊C4—48no—
348C22⋊C4—48no—
349C22⋊C4—88no—
350C42—24no—
351C42—24no—
352C42—48no—
353C42—42no—
354C42—42no—
355C23—22no—
356C23—24no—
357C23—24no—
358C23—22no—
359C23—22no—
360C23—24no—
361C23—24no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—24no—
366C23—24no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23#39814yes—
378C23C2×#12111yescharacteristic
379C23C2×#12311yescharacteristic
380C23C2×#12311yescharacteristic
381C23#52511yescharacteristic
382C23C2×C4⋊Q811yescharacteristic
383C23#53411yescharacteristic
384C23#53511yescharacteristic
385C23#57612yes—
386C23#57712yes—
387C23#57712yes—
388C23#57812yes—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—28no—
396C2×C4—28no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—816no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4#39814yes—
449C2×C4C2×C22⋊Q812yes—
450C2×C4C2×#12512yes—
451C2×C4#52911yescharacteristic
452C2×C4#52911yescharacteristic
453C2×C4#53211yescharacteristic
454C2×C4#53211yescharacteristic
455C2×C4#53412yes—
456C2×C4#53812yes—
457C2×C4#55112yes—
458C2×C4#55112yes—
459C2×C4#55112yes—
460C2×C4#55112yes—
461C2×C4#55412yes—
462C2×C4#55412yes—
463C2×C4#55412yes—
464C2×C4#55412yes—
465C8—44no—
466C8—44no—
467C22—28no—
468C22—24no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—44no—
474C22—44no—
475C22—42no—
476C22—42no—
477C22—44no—
478C22C2×#39811yescharacteristic
479C22#248211yescharacteristic
480C22#248311yescharacteristic
481C22#250211yescharacteristic
482C22#258311yescharacteristic
483C22#261711yescharacteristic
484C22#285411yescharacteristic
485C22#304512yes—
486C22#304612yes—
487C22#309612yes—
488C22#309712yes—
489C22#309812yes—
490C22#309912yes—
491C22#311812yes—
492C22#312112yes—
493C22#316312yes—
494C22#316412yes—
495C22#319812yes—
496C22#320112yes—
497C22#322011yescharacteristic
498C22#322111yescharacteristic
499C22#322211yescharacteristic
500C22#322311yescharacteristic
501C4—24no—
502C4—44no—
503C4—82no—
504C4—82no—
505C4—82no—
506C4—82no—
507C4—82no—
508C4—162no—
509C4—162no—
510C4#250512yes—
511C4#250512yes—
512C4#256412yes—
513C4#256412yes—
514C2—44no—
515C2#1501411yescharacteristic
516C2#2366212yes—
517C2#2759411yescharacteristic
518C2#2759611yescharacteristic
519C2#2760612yes—
520C2#2795711yescharacteristic
521C2#2795811yescharacteristic
522C2#2930812yes—
523C2#2972511yescharacteristic
524C2#2972611yescharacteristic
525C2#3311512yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11358C2semidirect
5. #11358C2semidirect
6. #11324C2semidirect
7. #11323C2semidirect
8. #11257C2semidirect
9. #11256C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18542
#143402
#137952
#113582
#113242
#113232
#112572
#112562
#107302
#107132