Finite Group Databasebeta

← #5084468

Subgroups — #5084468

Order 512.

2733 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2811C211yesmaximal, characteristic, complemented
2C2×#2614C211yesmaximal, characteristic
3#17223C212yesmaximal, complemented
4#17085C213yesmaximal, complemented
5#16375C216yesmaximal, complemented
6#15854C213yesmaximal, complemented
7#15668C213yesmaximal
8#15371C216yesmaximal, complemented
9#15262C213yesmaximal
10#14874C212yesmaximal, complemented
11#14655C211yesmaximal, characteristic, complemented
12C24×Q8C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×#123C2213yes—
15#2811—28no—
16#2614—28no—
17C2×#395C2212yes—
18C2×#395C2213yes—
19C2×#395C2213yes—
20C2×#395C2216yes—
21C2×#395C2213yes—
22C2×#395C2211yescharacteristic
23C2×#393C2211yescharacteristic
24C2×#393C2213yes—
25C2×#393C2213yes—
26C2×#393C2213yes—
27C2×#391C2216yes—
28C2×#391C2213yes—
29C2×#385C2216yes—
30C2×#381C2213yes—
31#1446C2216yes—
32#1446C2212yes—
33#1446C2216yes—
34#1445C2216yes—
35#1445C2213yes—
36#1443C2216yes—
37#1443C2216yes—
38#1442C2216yes—
39#1442C2216yes—
40#1442C2216yes—
41#1440C2216yes—
42#1440C2216yes—
43#1439C2216yes—
44#1439C2216yes—
45#1438C2213yes—
46#1437C2213yes—
47#1437C2213yes—
48#1436C2213yes—
49#1436C2213yes—
50#1434C2216yes—
51C22⋊#95C2216yes—
52C23×Q8—24no—
53C23×Q8—46no—
54C23×Q8—46no—
55C23×Q8C2313yes—
56C23×Q8C2313yes—
57C23×Q8C2311yescharacteristic
58C23×Q8C2311yescharacteristic
59C23×D4C2311yescharacteristic
60C24×C4C2313yes—
61C2×#124—46no—
62C2×#123—212no—
63C2×#123—46no—
64C2×#123—46no—
65C22×C4⋊C4C2313yes—
66C22×C4⋊C4C2316yes—
67C22×C4⋊C4C2316yes—
68C22×C4⋊C4C2313yes—
69C22×C22⋊C4C2312yes—
70C22×C22⋊C4C2313yes—
71C22×C22⋊C4C2313yes—
72C22×C22⋊C4C2316yes—
73C22×C22⋊C4C2313yes—
74C22×C22⋊C4C2316yes—
75C22×C22⋊C4C2312yes—
76C22×C42C2313yes—
77C22×C42C2311yescharacteristic
78#395—28no—
79#395—212no—
80#395—212no—
81#395—412no—
82#395—46no—
83#395D418yes—
84#393—24no—
85#393—212no—
86#393—46no—
87#393—46no—
88#391—212no—
89#391—224no—
90#385—412no—
91#381—212no—
92C2×#95C2316yes—
93C2×#95C2316yes—
94C2×#95C2313yes—
95C2×#95C2316yes—
96C2×#95C2313yes—
97C2×#95C2316yes—
98C2×#95C2316yes—
99C2×#95C2316yes—
100C2×#95C2316yes—
101C2×#95C2313yes—
102C2×#95C2313yes—
103C2×#95C2316yes—
104C2×#95C2316yes—
105C2×#95C2313yes—
106C2×#95C2316yes—
107C2×#95C2316yes—
108C2×#95C2316yes—
109C2×#95C2316yes—
110C2×#95C2313yes—
111C25C2411yescharacteristic
112C25C2411yescharacteristic
113C22×Q8—26no—
114C22×Q8—26no—
115C22×Q8—22no—
116C22×Q8—212no—
117C22×Q8—212no—
118C22×Q8—43no—
119C22×Q8—46no—
120C22×Q8—43no—
121C22×Q8—43no—
122C22×Q8—43no—
123C22×Q8—43no—
124C22×Q8—46no—
125C22×Q8—43no—
126C22×Q8—43no—
127C22×Q8—43no—
128C22×Q8—448no—
129C22×Q8—448no—
130C22×Q8—416no—
131C22×Q8C2×D414yes—
132C22×D4—26no—
133C22×D4—22no—
134C22×D4—43no—
135C23×C4—26no—
136C23×C4—26no—
137C23×C4—26no—
138C23×C4—212no—
139C23×C4—46no—
140C23×C4C2412yes—
141C23×C4C2413yes—
142C23×C4C2416yes—
143C23×C4C2412yes—
144C23×C4C2413yes—
145C23×C4C2413yes—
146C23×C4C2411yescharacteristic
147C23×C4C2413yes—
148C23×C4C2416yes—
149#124—416no—
150#123—448no—
151C2×C4⋊C4—26no—
152C2×C4⋊C4—26no—
153C2×C4⋊C4—212no—
154C2×C4⋊C4—212no—
155C2×C4⋊C4—43no—
156C2×C4⋊C4—43no—
157C2×C4⋊C4—43no—
158C2×C4⋊C4—46no—
159C2×C4⋊C4—46no—
160C2×C4⋊C4—46no—
161C2×C4⋊C4—46no—
162C2×C4⋊C4C2×D4112yes—
163C2×C22⋊C4—26no—
164C2×C22⋊C4—212no—
165C2×C22⋊C4—212no—
166C2×C22⋊C4—26no—
167C2×C22⋊C4—26no—
168C2×C22⋊C4—26no—
169C2×C22⋊C4—26no—
170C2×C22⋊C4—212no—
171C2×C22⋊C4—212no—
172C2×C22⋊C4—46no—
173C2×C22⋊C4—43no—
174C2×C22⋊C4—43no—
175C2×C22⋊C4—43no—
176C2×C22⋊C4—43no—
177C2×C22⋊C4—46no—
178C2×C22⋊C4—46no—
179C2×C22⋊C4—46no—
180C2×C42—26no—
181C2×C42—43no—
182C2×C42—46no—
183#95—212no—
184#95—26no—
185#95—212no—
186#95—26no—
187#95—212no—
188#95—212no—
189#95—212no—
190#95—212no—
191#95—46no—
192#95—46no—
193#95—46no—
194#95—46no—
195#95—43no—
196#95—43no—
197#95—46no—
198#95—43no—
199#95—46no—
200#95—46no—
201#95C2×D4112yes—
202C24—21no—
203C24—23no—
204C24—26no—
205C24—22no—
206C24—21no—
207C24—23no—
208C24—23no—
209C24—26no—
210C24—22no—
211C24H(5,𝔽2)16yes—
212C24C2511yescharacteristic
213C2×Q8—224no—
214C2×Q8—412no—
215C2×Q8—424no—
216C2×Q8—412no—
217C2×Q8—412no—
218C2×Q8—424no—
219C2×Q8—4128no—
220C2×Q8—44no—
221C2×Q8—412no—
222C2×Q8—412no—
223C2×Q8—44no—
224C2×Q8—86no—
225C2×Q8—86no—
226C2×Q8—86no—
227C2×Q8—812no—
228C2×Q8—812no—
229C2×Q8—86no—
230C2×D4—412no—
231C2×D4—44no—
232C2×D4—86no—
233C22×C4—224no—
234C22×C4—26no—
235C22×C4—23no—
236C22×C4—26no—
237C22×C4—26no—
238C22×C4—23no—
239C22×C4—23no—
240C22×C4—23no—
241C22×C4—26no—
242C22×C4—26no—
243C22×C4—26no—
244C22×C4—26no—
245C22×C4—23no—
246C22×C4—23no—
247C22×C4—23no—
248C22×C4—23no—
249C22×C4—26no—
250C22×C4—26no—
251C22×C4—26no—
252C22×C4—26no—
253C22×C4—26no—
254C22×C4—23no—
255C22×C4—22no—
256C22×C4—26no—
257C22×C4—26no—
258C22×C4—26no—
259C22×C4—26no—
260C22×C4—23no—
261C22×C4—23no—
262C22×C4—26no—
263C22×C4—26no—
264C22×C4—26no—
265C22×C4—26no—
266C22×C4—26no—
267C22×C4—26no—
268C22×C4—26no—
269C22×C4—26no—
270C22×C4—22no—
271C22×C4—26no—
272C22×C4—424no—
273C22×C4—424no—
274C22×C4—412no—
275C22×C4—412no—
276C22×C4C22×D416yes—
277C22×C4C22×D416yes—
278C22×C4C22×D412yes—
279C22×C4H(5,𝔽2)16yes—
280C22×C42−1+4112yes—
281C4⋊C4—412no—
282C4⋊C4—412no—
283C4⋊C4—812no—
284C4⋊C4—812no—
285C22⋊C4—412no—
286C22⋊C4—424no—
287C22⋊C4—412no—
288C22⋊C4—412no—
289C22⋊C4—48no—
290C22⋊C4—48no—
291C22⋊C4—812no—
292C42—44no—
293C42—86no—
294C23—26no—
295C23—26no—
296C23—212no—
297C23—212no—
298C23—43no—
299C23—41no—
300C23—43no—
301C23—43no—
302C23—46no—
303C23—43no—
304C23—43no—
305C23—41no—
306C23—41no—
307C23—43no—
308C23—46no—
309C23—46no—
310C23—43no—
311C23—43no—
312C23—46no—
313C23H(3,𝔽4)14yes—
314C23C23×D411yescharacteristic
315C23C2×H(5,𝔽2)13yes—
316C23C2×H(5,𝔽2)13yes—
317C23C2×2−1+416yes—
318C23#58512yes—
319Q8—416no—
320Q8—448no—
321Q8—824no—
322Q8—88no—
323D4—88no—
324C2×C4—26no—
325C2×C4—26no—
326C2×C4—26no—
327C2×C4—26no—
328C2×C4—212no—
329C2×C4—26no—
330C2×C4—212no—
331C2×C4—26no—
332C2×C4—212no—
333C2×C4—212no—
334C2×C4—212no—
335C2×C4—448no—
336C2×C4—46no—
337C2×C4—46no—
338C2×C4—46no—
339C2×C4—41no—
340C2×C4—43no—
341C2×C4—43no—
342C2×C4—46no—
343C2×C4—43no—
344C2×C4—43no—
345C2×C4—43no—
346C2×C4—46no—
347C2×C4—46no—
348C2×C4—46no—
349C2×C4—46no—
350C2×C4—43no—
351C2×C4—46no—
352C2×C4—46no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—46no—
356C2×C4—46no—
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—
368C22—24no—
369C22—28no—
370C22—46no—
371C22—46no—
372C22—412no—
373C22—83no—
374C22—83no—
375C22—81no—
376C22—86no—
377C22—82no—
378C22#348413yes—
379C22#348613yes—
380C22#350011yescharacteristic
381C22C2×H(3,𝔽4)11yescharacteristic
382C22#353213yes—
383C22#353516yes—
384C22#354416yes—
385C22#355013yes—
386C22#355013yes—
387C22#356116yes—
388C4—46no—
389C4—412no—
390C4—46no—
391C4—46no—
392C4—42no—
393C4—83no—
394C4—82no—
395C4—86no—
396C4—82no—
397C2—44no—
398C2—161no—
399C2#4543911yescharacteristic
400C2#4635713yes—
401C2#5207213yes—
402C2#5291412yes—
403C2#5491316yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2811C2semidirect
3. #17223C2semidirect
4. #17085C2semidirect
5. #16375C2semidirect
6. #15854C2semidirect
8. #15371C2semidirect
10. #14874C2semidirect
11. #14655C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28112
C2×#26142
#172232
#170852
#163752
#158542
#156682
#153712
#152622
#148742
#146552