Finite Group Databasebeta

← #409291

Subgroups — #409291

Order 512.

1216 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14340C211yesmaximal, characteristic
2#13795C211yesmaximal, characteristic
3#11373C211yesmaximal, characteristic, complemented
4#11371C211yesmaximal, characteristic, complemented
5#11316C212yesmaximal, complemented
6#11315C212yesmaximal, complemented
7#11268C211yesmaximal, characteristic, complemented
8#11268C211yesmaximal, characteristic, complemented
9#10731C212yesmaximal
10#10730C212yesmaximal
11#10713C211yesmaximal, characteristic
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C2×#384C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#380C2212yes—
19C2×#340C2212yes—
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#336C2211yescharacteristic
28C2×#336C2211yescharacteristic
29C2×#336C2212yes—
30#1441—22no—
31#1439—22no—
32C4⋊#95—22no—
33C4⋊#95C2211yescharacteristic
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2211yescharacteristic
38C22⋊#95—22no—
39C24×C4—21no—
40C24×C4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×#117—24no—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—24no—
49C2×#117—42no—
50C2×#117—42no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—42no—
55C22×C4⋊C4—42no—
56C22×C4⋊C4D412yes—
57C22×C4⋊C4Q814yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C22⋊C4—21no—
65C22×C22⋊C4—22no—
66C22×C22⋊C4D412yes—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70C2×C22⋊C8—24no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—24no—
73#384—22no—
74#384—22no—
75#382—42no—
76#380—24no—
77#380—24no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86#340—24no—
87#340—24no—
88#340—42no—
89#340—42no—
90#340—42no—
91#340—42no—
92#340—42no—
93#340—42no—
94#336—22no—
95#336—22no—
96#336—42no—
97C25C2×D411yescharacteristic
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—24no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—82no—
114C23×C4—82no—
115C23×C4C2×D412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×Q812yes—
119C23×C4#4112yes—
120C23×C4#4114yes—
121C23×C4#4114yes—
122C23×C4C2411yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
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—
137#117—48no—
138#117—48no—
139#117—48no—
140#117—48no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
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—48no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—48no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4C2×Q812yes—
177C2×C4⋊C4C2×Q812yes—
178C2×C4⋊C4#4112yes—
179C2×C4⋊C4#4112yes—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—82no—
188C2×C22⋊C4—82no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—22no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42#4112yes—
199C2×C42#4112yes—
200C22⋊C8—48no—
201#95—44no—
202#95—44no—
203#95—82no—
204#95—82no—
205#95—82no—
206#95—82no—
207#95—82no—
208#95—82no—
209C24—21no—
210C24—21no—
211C24—21no—
212C24—21no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24—22no—
217C24C22⋊Q812yes—
218C24#12312yes—
219C24#12412yes—
220C24C22×D411yescharacteristic
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
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—48no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
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—88no—
288C22×C4—88no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4C22⋊Q812yes—
292C22×C4#12314yes—
293C22×C4#12312yes—
294C22×C4#12412yes—
295C22×C4#12518yes—
296C22×C4#12618yes—
297C22×C4C22×Q811yescharacteristic
298C22×C4C2×#4112yes—
299C22×C4C2×#4112yes—
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4C2×#4111yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C4H(5,𝔽2)12yes—
306C22×C4H(5,𝔽2)12yes—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—48no—
326C4⋊C4—48no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—88no—
340C22⋊C4—48no—
341C22⋊C4—48no—
342C22⋊C4—88no—
343C42—24no—
344C42—24no—
345C42—48no—
346C42—42no—
347C42—42no—
348C23—24no—
349C23—24no—
350C23—24no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—24no—
355C23—24no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23#40014yes—
371C23C2×C22⋊Q811yescharacteristic
372C23C2×#12311yescharacteristic
373C23C2×#12311yescharacteristic
374C23#52511yescharacteristic
375C23C2×#12411yescharacteristic
376C23#53411yescharacteristic
377C23#53511yescharacteristic
378C23#57612yes—
379C23#57712yes—
380C23#57712yes—
381C23#57812yes—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—28no—
389C2×C4—28no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—816no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4#40014yes—
442C2×C4C2×#12512yes—
443C2×C4C2×#12612yes—
444C2×C4#52911yescharacteristic
445C2×C4#52911yescharacteristic
446C2×C4#53311yescharacteristic
447C2×C4#53311yescharacteristic
448C2×C4#53812yes—
449C2×C4#54312yes—
450C2×C4#55112yes—
451C2×C4#55112yes—
452C2×C4#55112yes—
453C2×C4#55112yes—
454C2×C4#55312yes—
455C2×C4#55312yes—
456C2×C4#55312yes—
457C2×C4#55312yes—
458C8—82no—
459C8—82no—
460C22—24no—
461C22—28no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—44no—
467C22—44no—
468C22—42no—
469C22—42no—
470C22—44no—
471C22C2×#40011yescharacteristic
472C22#248211yescharacteristic
473C22#248311yescharacteristic
474C22#250411yescharacteristic
475C22#259111yescharacteristic
476C22#262611yescharacteristic
477C22#285511yescharacteristic
478C22#308112yes—
479C22#308212yes—
480C22#309612yes—
481C22#309712yes—
482C22#309812yes—
483C22#309912yes—
484C22#312412yes—
485C22#312512yes—
486C22#313612yes—
487C22#313712yes—
488C22#319812yes—
489C22#320112yes—
490C22#322011yescharacteristic
491C22#322111yescharacteristic
492C22#322211yescharacteristic
493C22#322311yescharacteristic
494C4—24no—
495C4—44no—
496C4—82no—
497C4—82no—
498C4—82no—
499C4—82no—
500C4—82no—
501C4—162no—
502C4—162no—
503C4#251012yes—
504C4#251012yes—
505C4#256912yes—
506C4#256912yes—
507C2—44no—
508C2#1501511yescharacteristic
509C2#2374912yes—
510C2#2759511yescharacteristic
511C2#2759711yescharacteristic
512C2#2760612yes—
513C2#2798511yescharacteristic
514C2#2798611yescharacteristic
515C2#2934512yes—
516C2#2996211yescharacteristic
517C2#2996311yescharacteristic
518C2#3315912yes—
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
3. #11373C2semidirect
4. #11371C2semidirect
5. #11316C2semidirect
6. #11315C2semidirect
7. #11268C2semidirect
8. #11268C2semidirect

Maximal subgroups

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

subgroupindex
#143402
#137952
#113732
#113712
#113162
#113152
#112682
#107312
#107302
#107132