Finite Group Databasebeta

← #4414343

Subgroups — #4414343

Order 512.

2694 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2843C211yesmaximal, characteristic, complemented
2C22×#384C211yesmaximal, characteristic
3#16570C213yesmaximal, complemented
4#16523C216yesmaximal, complemented
5#16522C213yesmaximal, complemented
6#14822C213yesmaximal
7#14728C216yesmaximal
8#14720C213yesmaximal
9#14683C212yesmaximal
10#14678C212yesmaximal, complemented
11#14676C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2211yescharacteristic
13C23×C4⋊C4C2213yes—
14C23×C4⋊C4C2213yes—
15C23×C42C2211yescharacteristic
16#2843—28no—
17C2×#400C2213yes—
18C2×#400C2213yes—
19C2×#398C2216yes—
20C2×#398C2216yes—
21C2×#395C2213yes—
22C2×#395C2213yes—
23C2×#385C2216yes—
24C2×#384—28no—
25C2×#384C2211yescharacteristic
26C2×#384C2213yes—
27C2×#384C2213yes—
28C2×#384C2211yescharacteristic
29C2×#383C2212yes—
30C2×#383C2212yes—
31C2×#380C2213yes—
32#1444C2216yes—
33#1444C2216yes—
34#1444C2216yes—
35#1444C2216yes—
36#1442C2216yes—
37#1442C2216yes—
38#1442C2216yes—
39#1442C2216yes—
40#1441C2213yes—
41#1441C2216yes—
42#1441C2213yes—
43#1441C2213yes—
44#1441C2216yes—
45#1441C2213yes—
46#1434C2216yes—
47#1434C2216yes—
48C22⋊#95C2216yes—
49C22⋊#95C2216yes—
50C22⋊#95C2213yes—
51C22⋊#95C2213yes—
52C23×Q8C2311yescharacteristic
53C24×C4C2313yes—
54C24×C4C2311yescharacteristic
55C24×C4C2313yes—
56C2×C22⋊Q8—46no—
57C22×C4⋊C4—212no—
58C22×C4⋊C4—212no—
59C22×C4⋊C4—46no—
60C22×C4⋊C4—46no—
61C22×C4⋊C4—46no—
62C22×C4⋊C4—46no—
63C22×C4⋊C4C2313yes—
64C22×C4⋊C4C2316yes—
65C22×C4⋊C4C2313yes—
66C22×C4⋊C4C2316yes—
67C22×C4⋊C4C2313yes—
68C22×C4⋊C4C2313yes—
69C22×C4⋊C4C2313yes—
70C22×C4⋊C4C2316yes—
71C22×C4⋊C4C2313yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2313yes—
75C22×C4⋊C4C2313yes—
76C22×C22⋊C4C2312yes—
77C22×C22⋊C4C2316yes—
78C22×C42—24no—
79C22×C42—24no—
80C22×C42Q818yescomplemented
81C22×C42C2313yes—
82C22×C42C2311yescharacteristic
83#400—212no—
84#400—46no—
85#398—224no—
86#398—412no—
87#395—212no—
88#395—46no—
89#385—412no—
90#384—212no—
91#384—212no—
92#384—24no—
93#384D418yes—
94#383—28no—
95#383—28no—
96#380—46no—
97C2×#95C2316yes—
98C2×#95C2316yes—
99C2×#95C2313yes—
100C2×#95C2313yes—
101C2×#95C2316yes—
102C2×#95C2316yes—
103C2×#95C2316yes—
104C2×#95C2316yes—
105C2×#95C2316yes—
106C2×#95C2316yes—
107C2×#95C2313yes—
108C2×#95C2316yes—
109C2×#95C2313yes—
110C2×#95C2316yes—
111C2×#95C2316yes—
112C2×#95C2316yes—
113C2×#95C2316yes—
114C25C2411yescharacteristic
115C22×Q8—22no—
116C22×Q8—43no—
117C22×Q8—43no—
118C23×C4—26no—
119C23×C4—26no—
120C23×C4—26no—
121C23×C4—26no—
122C23×C4—26no—
123C23×C4—26no—
124C23×C4—26no—
125C23×C4—26no—
126C23×C4—212no—
127C23×C4—22no—
128C23×C4—26no—
129C23×C4—43no—
130C23×C4—43no—
131C23×C4—46no—
132C23×C4C2×Q8112yes—
133C23×C4C2412yes—
134C23×C4C2416yes—
135C23×C4C2416yes—
136C23×C4C2412yes—
137C23×C4C2411yescharacteristic
138C23×C4C2413yes—
139C23×C4C2413yes—
140C23×C4C2411yescharacteristic
141C23×C4C2413yes—
142C23×C4C2413yes—
143C22⋊Q8—416no—
144C2×C4⋊C4—26no—
145C2×C4⋊C4—212no—
146C2×C4⋊C4—26no—
147C2×C4⋊C4—26no—
148C2×C4⋊C4—26no—
149C2×C4⋊C4—26no—
150C2×C4⋊C4—212no—
151C2×C4⋊C4—26no—
152C2×C4⋊C4—212no—
153C2×C4⋊C4—26no—
154C2×C4⋊C4—43no—
155C2×C4⋊C4—43no—
156C2×C4⋊C4—43no—
157C2×C4⋊C4—43no—
158C2×C4⋊C4—46no—
159C2×C4⋊C4—43no—
160C2×C4⋊C4—43no—
161C2×C4⋊C4—43no—
162C2×C4⋊C4—43no—
163C2×C4⋊C4—43no—
164C2×C4⋊C4—43no—
165C2×C4⋊C4—43no—
166C2×C4⋊C4—46no—
167C2×C4⋊C4—448no—
168C2×C4⋊C4—448no—
169C2×C4⋊C4—43no—
170C2×C4⋊C4—43no—
171C2×C4⋊C4—46no—
172C2×C4⋊C4—46no—
173C2×C4⋊C4—46no—
174C2×C4⋊C4—46no—
175C2×C4⋊C4—46no—
176C2×C4⋊C4—46no—
177C2×C4⋊C4—43no—
178C2×C4⋊C4C2×D4112yes—
179C2×C4⋊C4C2×D4112yes—
180C2×C22⋊C4—212no—
181C2×C22⋊C4—212no—
182C2×C22⋊C4—212no—
183C2×C22⋊C4—212no—
184C2×C42—26no—
185C2×C42—26no—
186C2×C42—232no—
187C2×C42C2×D414yes—
188C2×C42C2×Q8112yes—
189C2×C42C2×Q814yes—
190C2×C42#4114yes—
191#95—212no—
192#95—212no—
193#95—26no—
194#95—26no—
195#95—212no—
196#95—212no—
197#95—212no—
198#95—212no—
199#95—212no—
200#95—212no—
201#95—46no—
202#95—46no—
203#95—46no—
204#95—43no—
205#95—46no—
206#95—43no—
207#95—46no—
208C24—21no—
209C24—21no—
210C24—23no—
211C24—23no—
212C24C22×Q812yes—
213C24H(5,𝔽2)16yes—
214C242−1+416yes—
215C24C2511yescharacteristic
216C2×Q8—44no—
217C2×Q8—86no—
218C2×Q8—86no—
219C22×C4—224no—
220C22×C4—224no—
221C22×C4—224no—
222C22×C4—224no—
223C22×C4—224no—
224C22×C4—23no—
225C22×C4—23no—
226C22×C4—23no—
227C22×C4—23no—
228C22×C4—23no—
229C22×C4—23no—
230C22×C4—23no—
231C22×C4—23no—
232C22×C4—224no—
233C22×C4—224no—
234C22×C4—23no—
235C22×C4—23no—
236C22×C4—26no—
237C22×C4—26no—
238C22×C4—23no—
239C22×C4—23no—
240C22×C4—26no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—23no—
244C22×C4—26no—
245C22×C4—22no—
246C22×C4—26no—
247C22×C4—26no—
248C22×C4—26no—
249C22×C4—26no—
250C22×C4—26no—
251C22×C4—26no—
252C22×C4—26no—
253C22×C4—23no—
254C22×C4—23no—
255C22×C4—26no—
256C22×C4—26no—
257C22×C4—26no—
258C22×C4—23no—
259C22×C4—23no—
260C22×C4—26no—
261C22×C4—26no—
262C22×C4—26no—
263C22×C4—26no—
264C22×C4—26no—
265C22×C4—26no—
266C22×C4—26no—
267C22×C4—26no—
268C22×C4—412no—
269C22×C4—424no—
270C22×C4—412no—
271C22×C4—44no—
272C22×C4—412no—
273C22×C4—86no—
274C22×C4—812no—
275C22×C4—86no—
276C22×C4C22×D412yes—
277C22×C4C22×D416yes—
278C22×C4C22×D416yes—
279C22×C4C22×Q816yes—
280C22×C4C22×Q816yes—
281C22×C4C2×#4116yes—
282C22×C4H(5,𝔽2)16yes—
283C22×C42−1+416yes—
284C4⋊C4—412no—
285C4⋊C4—412no—
286C4⋊C4—424no—
287C4⋊C4—412no—
288C4⋊C4—412no—
289C4⋊C4—412no—
290C4⋊C4—412no—
291C4⋊C4—44no—
292C4⋊C4—48no—
293C4⋊C4—86no—
294C4⋊C4—812no—
295C4⋊C4—86no—
296C4⋊C4—812no—
297C22⋊C4—48no—
298C22⋊C4—424no—
299C42—224no—
300C42C22⋊Q8116yescomplemented
301C23—22no—
302C23—26no—
303C23—26no—
304C23—26no—
305C23—26no—
306C23—26no—
307C23—26no—
308C23—26no—
309C23—212no—
310C23#544112yes—
311C23#56014yes—
312C23#563112yes—
313C23C23×D411yescharacteristic
314C23C23×Q811yescharacteristic
315C23C22×#4111yescharacteristic
316C23C2×H(5,𝔽2)13yes—
317C23C2×H(5,𝔽2)13yes—
318C23C2×2−1+413yes—
319C23C2×2−1+413yes—
320Q8—88no—
321C2×C4—296no—
322C2×C4—26no—
323C2×C4—26no—
324C2×C4—26no—
325C2×C4—26no—
326C2×C4—26no—
327C2×C4—26no—
328C2×C4—26no—
329C2×C4—26no—
330C2×C4—26no—
331C2×C4—26no—
332C2×C4—26no—
333C2×C4—26no—
334C2×C4—26no—
335C2×C4—212no—
336C2×C4—26no—
337C2×C4—41no—
338C2×C4—43no—
339C2×C4—43no—
340C2×C4—46no—
341C2×C4—43no—
342C2×C4—43no—
343C2×C4—43no—
344C2×C4—46no—
345C2×C4—46no—
346C2×C4—46no—
347C2×C4—46no—
348C2×C4—46no—
349C2×C4—46no—
350C2×C4—46no—
351C2×C4—46no—
352C2×C4—46no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—43no—
357C2×C4—46no—
358C2×C4—46no—
359C2×C4—46no—
360C2×C4—46no—
361C2×C4—46no—
362C2×C4—43no—
363C2×C4—43no—
364C2×C4—46no—
365C2×C4—46no—
366C2×C4—46no—
367C2×C4—46no—
368C2×C4—46no—
369C2×C4—46no—
370C2×C4—824no—
371C2×C4—88no—
372C2×C4C2×C22⋊Q8112yes—
373C2×C4#544112yes—
374C22—212no—
375C22—24no—
376C22—212no—
377C22—212no—
378C22—212no—
379C22—24no—
380C22#284318yescomplemented
381C22C22×C22⋊Q811yescharacteristic
382C22C2×#54413yes—
383C22C2×#54413yes—
384C22#348413yes—
385C22#348613yes—
386C22#349213yes—
387C22#349213yes—
388C22C2×#56011yescharacteristic
389C22C2×#56313yes—
390C22#354313yes—
391C22#354413yes—
392C22#355216yes—
393C4—212no—
394C4—212no—
395C4—46no—
396C4—42no—
397C4—81no—
398C4—83no—
399C4—82no—
400C4—86no—
401C4—86no—
402C4—82no—
403C2—28no—
404C2#3522413yes—
405C2C2×#284311yescharacteristic
406C2#3762313yes—
407C2#4632411yescharacteristic
408C2#4632513yes—
409C2#4632613yes—
410C2#4632711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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. #16570C2semidirect
4. #16523C2semidirect
5. #16522C2semidirect
10. #14678C2semidirect
11. #14676C2semidirect
80. C22×C42Q8semidirect
300. C42C22⋊Q8semidirect
380. C22#2843semidirect

Maximal subgroups

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

subgroupindex
C2×#28432
C22×#3842
#165702
#165232
#165222
#148222
#147282
#147202
#146832
#146782
#146762