Finite Group Databasebeta

← #4414372

Subgroups — #4414372

Order 512.

2874 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2843C211yesmaximal, characteristic, complemented
2C2×C4×#127C211yesmaximal, characteristic
3#16599C213yesmaximal, complemented
4#16556C213yesmaximal, complemented
5#16555C216yesmaximal, complemented
6#14825C213yesmaximal, complemented
7#14733C216yesmaximal, complemented
8#14723C213yesmaximal, complemented
9#14710C212yesmaximal, complemented
10#14676C211yesmaximal, characteristic, complemented
11#14661C212yesmaximal
12C22×#127C2211yescharacteristic
13C22×C22⋊Q8C2211yescharacteristic
14C22×C4×D4C2213yes—
15C22×C4×D4C2213yes—
16#2843—28no—
17C2×#400C2213yes—
18C2×#399C2213yes—
19C2×#398C2216yes—
20C2×#396C2216yes—
21C2×#395C2213yes—
22C2×#393C2213yes—
23C4×#127—28no—
24C2×#390C2213yes—
25C2×#390C2213yes—
26C2×#385C2216yes—
27C2×#384C2211yescharacteristic
28C2×#383C2212yes—
29C2×#383C2212yes—
30C2×#380C2213yes—
31C2×C43C2211yescharacteristic
32#1441C2213yes—
33#1441C2216yes—
34#1441C2213yes—
35#1439C2216yes—
36#1439C2213yes—
37#1439C2216yes—
38#1439C2216yes—
39#1439C2216yes—
40#1439C2213yes—
41#1439C2216yes—
42#1438C2216yes—
43#1438C2216yes—
44#1438C2216yes—
45#1438C2216yes—
46#1436C2216yes—
47#1436C2216yes—
48#1436C2216yes—
49#1436C2216yes—
50C22⋊#95C2213yes—
51C22⋊#95C2213yes—
52C23×Q8C2311yescharacteristic
53C23×D4C2313yes—
54C23×D4C2313yes—
55C24×C4C2311yescharacteristic
56C24×C4C2313yes—
57C2×#127—24no—
58C2×#127—24no—
59C2×#127Q818yescomplemented
60C2×C22⋊Q8—46no—
61C2×C4×D4—212no—
62C2×C4×D4—212no—
63C2×C4×D4—46no—
64C2×C4×D4—46no—
65C2×C4×D4—46no—
66C2×C4×D4—46no—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2316yes—
69C22×C4⋊C4C2316yes—
70C22×C4⋊C4C2313yes—
71C22×C4⋊C4C2313yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2313yes—
74C22×C4⋊C4C2313yes—
75C22×C22⋊C4C2313yes—
76C22×C22⋊C4C2316yes—
77C22×C22⋊C4C2313yes—
78C22×C22⋊C4C2316yes—
79C22×C22⋊C4C2316yes—
80C22×C22⋊C4C2316yes—
81C22×C22⋊C4C2312yes—
82C22×C22⋊C4C2316yes—
83C22×C22⋊C4C2316yes—
84C22×C42C2313yes—
85C22×C42C2313yes—
86C22×C42C2311yescharacteristic
87#400—212no—
88#399—46no—
89#398—224no—
90#396—412no—
91#395—212no—
92#393—46no—
93#390—212no—
94#390—212no—
95#385—412no—
96#384D418yes—
97#383—28no—
98#383—28no—
99#380—46no—
100C2×#95C2316yes—
101C2×#95C2316yes—
102C2×#95C2313yes—
103C2×#95C2313yes—
104C2×#95C2316yes—
105C2×#95C2316yes—
106C2×#95C2316yes—
107C2×#95C2316yes—
108C2×#95C2313yes—
109C2×#95C2313yes—
110C2×#95C2316yes—
111C2×#95C2316yes—
112C2×#95C2316yes—
113C43—24no—
114C25C2413yes—
115C25C2411yescharacteristic
116C22×Q8—22no—
117C22×Q8—43no—
118C22×Q8—43no—
119C22×D4—26no—
120C22×D4—26no—
121C22×D4—26no—
122C22×D4—26no—
123C22×D4—43no—
124C22×D4—43no—
125C22×D4—43no—
126C22×D4—43no—
127C22×D4—43no—
128C22×D4—43no—
129C22×D4—43no—
130C22×D4—43no—
131C22×D4C2×Q8112yes—
132C22×D4C2×Q8112yes—
133C23×C4—22no—
134C23×C4—26no—
135C23×C4—26no—
136C23×C4—26no—
137C23×C4—212no—
138C23×C4—43no—
139C23×C4—43no—
140C23×C4—46no—
141C23×C4C2412yes—
142C23×C4C2416yes—
143C23×C4C2416yes—
144C23×C4C2412yes—
145C23×C4C2413yes—
146C23×C4C2411yescharacteristic
147C23×C4C2413yes—
148C23×C4C2413yes—
149C23×C4C2411yescharacteristic
150#127—232no—
151C22⋊Q8—416no—
152C4×D4—448no—
153C4×D4—448no—
154C2×C4⋊C4—212no—
155C2×C4⋊C4—212no—
156C2×C4⋊C4—26no—
157C2×C4⋊C4—212no—
158C2×C4⋊C4—43no—
159C2×C4⋊C4—43no—
160C2×C4⋊C4—43no—
161C2×C4⋊C4—43no—
162C2×C4⋊C4—43no—
163C2×C4⋊C4—46no—
164C2×C4⋊C4—46no—
165C2×C4⋊C4—43no—
166C2×C4⋊C4—43no—
167C2×C4⋊C4—46no—
168C2×C4⋊C4—46no—
169C2×C4⋊C4—46no—
170C2×C4⋊C4—43no—
171C2×C4⋊C4C2×D4112yes—
172C2×C4⋊C4C2×D4112yes—
173C2×C22⋊C4—26no—
174C2×C22⋊C4—212no—
175C2×C22⋊C4—26no—
176C2×C22⋊C4—26no—
177C2×C22⋊C4—212no—
178C2×C22⋊C4—26no—
179C2×C22⋊C4—46no—
180C2×C22⋊C4—46no—
181C2×C22⋊C4—46no—
182C2×C22⋊C4—43no—
183C2×C22⋊C4—43no—
184C2×C22⋊C4—46no—
185C2×C22⋊C4—46no—
186C2×C22⋊C4—46no—
187C2×C22⋊C4—46no—
188C2×C22⋊C4—46no—
189C2×C22⋊C4—46no—
190C2×C22⋊C4—46no—
191C2×C22⋊C4—46no—
192C2×C22⋊C4—46no—
193C2×C22⋊C4—46no—
194C2×C22⋊C4—46no—
195C2×C22⋊C4—46no—
196C2×C22⋊C4—46no—
197C2×C22⋊C4—46no—
198C2×C42—26no—
199C2×C42—26no—
200C2×C42—26no—
201C2×C42—26no—
202C2×C42—43no—
203C2×C42—43no—
204C2×C42C2×D414yes—
205C2×C42C2×Q814yes—
206C2×C42#4114yes—
207#95—212no—
208#95—212no—
209#95—26no—
210#95—26no—
211#95—212no—
212#95—212no—
213#95—212no—
214#95—212no—
215#95—46no—
216#95—46no—
217#95—46no—
218#95—43no—
219#95—43no—
220C24—23no—
221C24—21no—
222C24—23no—
223C24—21no—
224C24—23no—
225C24—26no—
226C24—23no—
227C24—23no—
228C24—26no—
229C24—23no—
230C24—23no—
231C24—26no—
232C24—23no—
233C24—26no—
234C24—23no—
235C24—23no—
236C24C22×Q816yes—
237C24C22×Q812yes—
238C24C2511yescharacteristic
239C2×Q8—44no—
240C2×Q8—86no—
241C2×Q8—86no—
242C2×D4—224no—
243C2×D4—224no—
244C2×D4—412no—
245C2×D4—412no—
246C2×D4—412no—
247C2×D4—412no—
248C2×D4—412no—
249C2×D4—412no—
250C2×D4—412no—
251C2×D4—412no—
252C2×D4—412no—
253C2×D4—412no—
254C2×D4—412no—
255C2×D4—412no—
256C22×C4—21no—
257C22×C4—23no—
258C22×C4—23no—
259C22×C4—23no—
260C22×C4—23no—
261C22×C4—23no—
262C22×C4—23no—
263C22×C4—23no—
264C22×C4—23no—
265C22×C4—26no—
266C22×C4—23no—
267C22×C4—26no—
268C22×C4—23no—
269C22×C4—23no—
270C22×C4—26no—
271C22×C4—26no—
272C22×C4—26no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—26no—
276C22×C4—26no—
277C22×C4—26no—
278C22×C4—26no—
279C22×C4—26no—
280C22×C4—26no—
281C22×C4—26no—
282C22×C4—26no—
283C22×C4—26no—
284C22×C4—26no—
285C22×C4—26no—
286C22×C4—26no—
287C22×C4—26no—
288C22×C4—26no—
289C22×C4—26no—
290C22×C4—26no—
291C22×C4—26no—
292C22×C4—412no—
293C22×C4—424no—
294C22×C4—412no—
295C22×C4—44no—
296C22×C4—412no—
297C22×C4—86no—
298C22×C4—812no—
299C22×C4—86no—
300C22×C4C22×D412yes—
301C22×C4C22×D416yes—
302C22×C4C22×D416yes—
303C22×C4C22×Q816yes—
304C22×C4C2×#4116yes—
305C22×C4H(5,𝔽2)16yes—
306C22×C4H(5,𝔽2)16yes—
307C4⋊C4—412no—
308C4⋊C4—412no—
309C4⋊C4—44no—
310C4⋊C4—48no—
311C4⋊C4—86no—
312C4⋊C4—812no—
313C4⋊C4—812no—
314C4⋊C4—86no—
315C22⋊C4—424no—
316C22⋊C4—412no—
317C22⋊C4—412no—
318C22⋊C4—424no—
319C22⋊C4—48no—
320C22⋊C4—812no—
321C22⋊C4—812no—
322C22⋊C4—812no—
323C22⋊C4—812no—
324C42—224no—
325C42—412no—
326C42C22⋊Q8116yescomplemented
327C23—26no—
328C23—22no—
329C23—26no—
330C23—212no—
331C23—26no—
332C23—212no—
333C23—26no—
334C23—26no—
335C23—43no—
336C23—46no—
337C23—43no—
338C23—43no—
339C23—46no—
340C23—43no—
341C23—46no—
342C23—43no—
343C23—43no—
344C23—43no—
345C23—46no—
346C23—43no—
347C23—41no—
348C23—43no—
349C23—46no—
350C23—43no—
351C23—43no—
352C23—46no—
353C23—43no—
354C23—43no—
355C23—43no—
356C23—43no—
357C23—46no—
358C23—46no—
359C23—46no—
360C23—46no—
361C23—46no—
362C23C23×D411yescharacteristic
363C23C23×Q811yescharacteristic
364C23C22×#4111yescharacteristic
365C23C2×H(5,𝔽2)13yes—
366C23C2×H(5,𝔽2)13yes—
367C23#58513yes—
368C23#58513yes—
369Q8—88no—
370D4—448no—
371D4—448no—
372C2×C4—26no—
373C2×C4—26no—
374C2×C4—26no—
375C2×C4—26no—
376C2×C4—26no—
377C2×C4—26no—
378C2×C4—26no—
379C2×C4—26no—
380C2×C4—26no—
381C2×C4—26no—
382C2×C4—212no—
383C2×C4—26no—
384C2×C4—212no—
385C2×C4—46no—
386C2×C4—41no—
387C2×C4—43no—
388C2×C4—43no—
389C2×C4—43no—
390C2×C4—46no—
391C2×C4—43no—
392C2×C4—43no—
393C2×C4—46no—
394C2×C4—46no—
395C2×C4—46no—
396C2×C4—46no—
397C2×C4—46no—
398C2×C4—46no—
399C2×C4—46no—
400C2×C4—46no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—46no—
404C2×C4—46no—
405C2×C4—46no—
406C2×C4—46no—
407C2×C4—46no—
408C2×C4—46no—
409C2×C4—46no—
410C2×C4—46no—
411C2×C4—46no—
412C2×C4—46no—
413C2×C4—824no—
414C2×C4—88no—
415C2×C4C2×C22⋊Q8112yes—
416C2×C4#56014yes—
417C22—46no—
418C22—412no—
419C22—46no—
420C22—412no—
421C22—46no—
422C22—46no—
423C22—42no—
424C22—46no—
425C22—81no—
426C22—83no—
427C22—86no—
428C22—83no—
429C22—83no—
430C22—86no—
431C22—83no—
432C22—83no—
433C22—81no—
434C22—83no—
435C22C22×C22⋊Q811yescharacteristic
436C22#346013yes—
437C22#346013yes—
438C22#348413yes—
439C22#348713yes—
440C22#350113yes—
441C22#350113yes—
442C22C2×#56011yescharacteristic
443C22#352913yes—
444C22#354313yes—
445C22#355113yes—
446C22#356016yes—
447C4—212no—
448C4—24no—
449C4—46no—
450C4—46no—
451C4—42no—
452C4—82no—
453C4—86no—
454C4—86no—
455C4—82no—
456C2—86no—
457C2—82no—
458C2#3517413yes—
459C2#3669611yescharacteristic
460C2#3760513yes—
461C2#4628213yes—
462C2#4631013yes—
463C2#4632411yescharacteristic
464C2#4633011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2843C2semidirect
3. #16599C2semidirect
4. #16556C2semidirect
5. #16555C2semidirect
6. #14825C2semidirect
7. #14733C2semidirect
8. #14723C2semidirect
9. #14710C2semidirect
10. #14676C2semidirect
59. C2×#127Q8semidirect
326. C42C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28432
C2×C4×#1272
#165992
#165562
#165552
#148252
#147332
#147232
#147102
#146762
#146612