Finite Group Databasebeta

← #409305

Subgroups — #409305

Order 512.

1160 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2094C211yesmaximal, characteristic
2C2×#2073C211yesmaximal, characteristic
3#14348C211yesmaximal, characteristic, complemented
4#13785C211yesmaximal, characteristic
5#11373C211yesmaximal, characteristic, complemented
6#11358C211yesmaximal, characteristic, complemented
7#11257C212yesmaximal, complemented
8#11189C212yesmaximal
9#10908C212yesmaximal, complemented
10#10796C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4×D4C2211yescharacteristic
13C22×#106C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16C2×#390C2211yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×#376C2211yescharacteristic
22#2094—24no—
23#2073—24no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2212yes—
33#1442—22no—
34#1442—22no—
35#1437—22no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C22⋊#95—22no—
41C23×D4C2311yescharacteristic
42C24×C4—21no—
43C23×C8C2312yes—
44C23×C8C2312yes—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—24no—
48C2×C4×D4—42no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×#106—24no—
63C2×#106—24no—
64C2×#106—42no—
65C2×#102—24no—
66C2×#102—24no—
67C2×#102—24no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—42no—
71#390—22no—
72#390—22no—
73#390—22no—
74#390D414yes—
75#384—22no—
76#384—22no—
77#382—22no—
78#382—22no—
79C4×C4⋊C4—42no—
80#376—24no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95D412yes—
90C2×#95D412yes—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—42no—
95#340—42no—
96#336—24no—
97#336—22no—
98#336—22no—
99#336—24no—
100C25—21no—
101C22×D4—22no—
102C22×D4—22no—
103C22×D4—21no—
104C22×D4—21no—
105C22×D4—21no—
106C22×D4—22no—
107C22×D4—22no—
108C22×D4C2×D412yes—
109C22×D4#4112yes—
110C22×D4#4112yes—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—24no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—42no—
118C23×C4—42no—
119C23×C4—82no—
120C23×C4—82no—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2411yescharacteristic
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—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8#4114yes—
140C4×D4—48no—
141C4×D4—48no—
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—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
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—44no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4#4112yes—
168C2×C4⋊C4#4112yes—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—82no—
175C2×C22⋊C4—82no—
176C2×C22⋊C4—82no—
177C2×C22⋊C4—82no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—42no—
188C2×C42—82no—
189C2×C42C2×D412yes—
190C2×C42#4112yes—
191C2×C42#4114yes—
192#106—48no—
193#102—48no—
194#102—48no—
195#95—24no—
196#95—24no—
197#95—42no—
198#95—44no—
199#95—82no—
200#95—82no—
201#95—82no—
202#95—82no—
203#95—82no—
204#95—82no—
205C24—22no—
206C24—22no—
207C24—22no—
208C24—41no—
209C24—41no—
210C24—41no—
211C24—41no—
212C24—42no—
213C24—42no—
214C24—42no—
215C24—42no—
216C24C22×D411yescharacteristic
217C2×D4—24no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—24no—
225C2×D4—24no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4#12414yes—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
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—22no—
257C22×C4—48no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
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—88no—
285C22×C4—88no—
286C22×C4—84no—
287C22×C4—84no—
288C22×C4#12114yes—
289C22×C4#12312yes—
290C22×C4#12312yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4112yes—
296C22×C4C2×#4112yes—
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4H(5,𝔽2)12yes—
300C22×C42−1+412yes—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—88no—
330C22⋊C4—48no—
331C22⋊C4—88no—
332C22⋊C4—88no—
333C42—24no—
334C42—24no—
335C42—24no—
336C42—24no—
337C42—42no—
338C42—42no—
339C42—84no—
340C23—24no—
341C23—44no—
342C23—44no—
343C23—42no—
344C23—44no—
345C23—44no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—44no—
351C23—44no—
352C23—42no—
353C23—82no—
354C23—82no—
355C23—82no—
356C23—82no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—82no—
362C23—82no—
363C23C2×#12111yescharacteristic
364C23C2×#12311yescharacteristic
365C23#52511yescharacteristic
366C23#52511yescharacteristic
367C23#53211yescharacteristic
368C23#53511yescharacteristic
369C23#53611yescharacteristic
370C23#57612yes—
371C23#57712yes—
372C23#57712yes—
373C23#57712yes—
374D4—28no—
375D4—44no—
376D4—44no—
377D4—44no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—816no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4C2×#12411yescharacteristic
433C2×C4#52911yescharacteristic
434C2×C4#52912yes—
435C2×C4#52912yes—
436C2×C4#52911yescharacteristic
437C2×C4#53211yescharacteristic
438C2×C4#53912yes—
439C2×C4#53912yes—
440C2×C4#54612yes—
441C2×C4#54712yes—
442C2×C4#55112yes—
443C2×C4#55112yes—
444C2×C4#55212yes—
445C2×C4#55212yes—
446C2×C4#55212yes—
447C2×C4#55912yes—
448C8—44no—
449C8—82no—
450C22—48no—
451C22—44no—
452C22—84no—
453C22—82no—
454C22—82no—
455C22—84no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22—84no—
460C22—82no—
461C22#239511yescharacteristic
462C22#248211yescharacteristic
463C22#248611yescharacteristic
464C22#254411yescharacteristic
465C22#261111yescharacteristic
466C22#263311yescharacteristic
467C22#285011yescharacteristic
468C22#304212yes—
469C22#304512yes—
470C22#309612yes—
471C22#309912yes—
472C22#311812yes—
473C22#311812yes—
474C22#312512yes—
475C22#312512yes—
476C22#317412yes—
477C22#317912yes—
478C22#322011yescharacteristic
479C22#322111yescharacteristic
480C22#322211yescharacteristic
481C22#322211yescharacteristic
482C22#324112yes—
483C22#324412yes—
484C4—44no—
485C4—42no—
486C4—42no—
487C4—82no—
488C4—82no—
489C4—82no—
490C4—82no—
491C4—162no—
492C4—162no—
493C4#248812yes—
494C4#251712yes—
495C4#255312yes—
496C4#257512yes—
497C2—84no—
498C2#1501611yescharacteristic
499C2#2376612yes—
500C2#2759411yescharacteristic
501C2#2759711yescharacteristic
502C2#2765412yes—
503C2#2862211yescharacteristic
504C2#2863011yescharacteristic
505C2#2958812yes—
506C2#3008411yescharacteristic
507C2#3009011yescharacteristic
508C2#3303412yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #14348C2semidirect
5. #11373C2semidirect
6. #11358C2semidirect
7. #11257C2semidirect
9. #10908C2semidirect
10. #10796C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20942
C2×#20732
#143482
#137852
#113732
#113582
#112572
#111892
#109082
#107962
#107882