Finite Group Databasebeta

← #382122

Subgroups — #382122

Order 512.

1214 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2025C211yesmaximal, characteristic
2#14283C211yesmaximal, characteristic, complemented
3#13785C211yesmaximal, characteristic
4#11385C211yesmaximal, characteristic, complemented
5#11324C211yesmaximal, characteristic, complemented
6#11316C212yesmaximal, complemented
7#11123C212yesmaximal
8#11049C211yesmaximal, characteristic
9#10907C212yesmaximal, complemented
10#10796C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×#125C2211yescharacteristic
13C22×C4×D4C2211yescharacteristic
14C22×#107C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C2×#390C2211yescharacteristic
18C2×#390C2211yescharacteristic
19C2×#384C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×#376C2211yescharacteristic
22#2025C418yescomplemented
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2212yes—
31C2×#336C2211yescharacteristic
32#1441—22no—
33#1441—22no—
34#1437—22no—
35C4⋊#95C2212yes—
36C4⋊#95C2211yescharacteristic
37C4⋊#95C2212yes—
38C4⋊#95C2211yescharacteristic
39C22⋊#95—22no—
40C23×D4C2311yescharacteristic
41C24×C4—21no—
42C23×C8C2312yes—
43C23×C8C2312yes—
44C2×#125—22no—
45C2×#125—42no—
46C2×#125—42no—
47C2×#125C2×C414yescomplemented
48C2×C4×D4—22no—
49C2×C4×D4—42no—
50C2×C4×D4—42no—
51C2×C4×D4Q814yes—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C22⋊C4—21no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C42—21no—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×#107—24no—
67C2×#107—42no—
68C2×#107—42no—
69C2×#102—24no—
70C2×#102—42no—
71C2×#102—42no—
72C2×#102—42no—
73C2×#102—42no—
74C2×#102C2×C418yes—
75#390—22no—
76#390—22no—
77#390—22no—
78#390C2×C414yes—
79#384—42no—
80C4×C4⋊C4—42no—
81#376—42no—
82C2×#95—21no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95D412yes—
90C2×#95D412yes—
91#340—42no—
92#340—42no—
93#340—42no—
94#340—42no—
95#340C2×C418yescomplemented
96#336—22no—
97#336—24no—
98#336—42no—
99#336C2×C414yescomplemented
100C25—21no—
101C22×D4—21no—
102C22×D4—21no—
103C22×D4—21no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4—22no—
107C22×D4—22no—
108C22×D4C22×C412yes—
109C22×D4C2×Q812yes—
110C22×D4#4112yes—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—24no—
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×C8C22×C414yes—
140#125—48no—
141#125—48no—
142C4×D4—48no—
143C4×D4—48no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
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—28no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
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—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C22×C414yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C2×Q812yes—
173C2×C4⋊C4#4112yes—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—82no—
180C2×C22⋊C4—82no—
181C2×C22⋊C4—82no—
182C2×C22⋊C4—82no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—42no—
192C2×C42—82no—
193C2×C42C22×C412yes—
194C2×C42C2×Q812yes—
195C2×C42#4112yes—
196C2×C42#4112yes—
197#107—48no—
198#102—48no—
199#102—48no—
200#95—42no—
201#95—42no—
202#95—42no—
203#95—82no—
204#95—82no—
205#95—82no—
206#95—82no—
207#95—82no—
208#95—82no—
209C24—22no—
210C24—22no—
211C24—22no—
212C24—41no—
213C24—41no—
214C24—41no—
215C24—41no—
216C24—42no—
217C24—42no—
218C24—42no—
219C24—42no—
220C24C22×D411yescharacteristic
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—24no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4C4×Q814yes—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
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—24no—
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—48no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
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—42no—
288C22×C4—88no—
289C22×C4—88no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4C4×D414yes—
293C22×C4C22⋊Q812yes—
294C22×C4C22⋊Q812yes—
295C22×C4C23×C411yescharacteristic
296C22×C4C22×Q811yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C4H(5,𝔽2)12yes—
302C22×C4H(5,𝔽2)12yes—
303C22×C4H(5,𝔽2)12yes—
304C22×C42−1+412yes—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
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—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—416no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—88no—
335C22⋊C4—48no—
336C22⋊C4—88no—
337C22⋊C4—88no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42—24no—
342C42—42no—
343C42—42no—
344C42—84no—
345C23—24no—
346C23—44no—
347C23—44no—
348C23—42no—
349C23—42no—
350C23—44no—
351C23—44no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—44no—
357C23—44no—
358C23—82no—
359C23—82no—
360C23—81no—
361C23—81no—
362C23—82no—
363C23—81no—
364C23—81no—
365C23—82no—
366C23—82no—
367C23—82no—
368C23C2×C4×D411yescharacteristic
369C23C2×C22⋊Q811yescharacteristic
370C23#52511yescharacteristic
371C23#53411yescharacteristic
372C23#53511yescharacteristic
373C23#53511yescharacteristic
374C23#53611yescharacteristic
375C23#57612yes—
376C23#57612yes—
377C23#57712yes—
378C23#57812yes—
379D4—28no—
380D4—44no—
381D4—44no—
382D4—44no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—816no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
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×C4C2×C4×Q811yescharacteristic
438C2×C4C4×#4111yescharacteristic
439C2×C4#51812yes—
440C2×C4#52012yes—
441C2×C4#52911yescharacteristic
442C2×C4#53311yescharacteristic
443C2×C4#53812yes—
444C2×C4#53812yes—
445C2×C4#53812yes—
446C2×C4#53812yes—
447C2×C4#53912yes—
448C2×C4#54212yes—
449C2×C4#54212yes—
450C2×C4#54212yes—
451C2×C4#54312yes—
452C2×C4#54412yes—
453C8—44no—
454C8—82no—
455C22—48no—
456C22—44no—
457C22—82no—
458C22—82no—
459C22—84no—
460C22—84no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—84no—
466C22C4×C22⋊Q811yescharacteristic
467C22#232111yescharacteristic
468C22#232611yescharacteristic
469C22#262611yescharacteristic
470C22#262811yescharacteristic
471C22#263111yescharacteristic
472C22#263811yescharacteristic
473C22#294612yes—
474C22#295112yes—
475C22#308112yes—
476C22#308312yes—
477C22#312012yes—
478C22#312212yes—
479C22#319812yes—
480C22#320312yes—
481C22#322011yescharacteristic
482C22#322011yescharacteristic
483C22#322012yes—
484C22#322211yescharacteristic
485C22#322311yescharacteristic
486C22#322512yes—
487C22#324112yes—
488C22#324312yes—
489C4—42no—
490C4—42no—
491C4—44no—
492C4—82no—
493C4—84no—
494C4—82no—
495C4—82no—
496C4—82no—
497C4—162no—
498C4#232912yes—
499C4#233712yes—
500C4#265312yes—
501C4#266512yes—
502C2—84no—
503C2#1431911yescharacteristic
504C2#2162012yes—
505C2#2205512yes—
506C2#2222811yescharacteristic
507C2#2223211yescharacteristic
508C2#2996211yescharacteristic
509C2#2996411yescharacteristic
510C2#2999112yes—
511C2#3004811yescharacteristic
512C2#3005411yescharacteristic
513C2#3017812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×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
2. #14283C2semidirect
4. #11385C2semidirect
5. #11324C2semidirect
6. #11316C2semidirect
9. #10907C2semidirect
10. #10796C2semidirect
22. #2025C4semidirect
47. C2×#125C2×C4semidirect
95. #340C2×C4semidirect
99. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20252
#142832
#137852
#113852
#113242
#113162
#111232
#110492
#109072
#107962
#107882