Finite Group Databasebeta

← #4935917

Subgroups — #4935917

Order 512.

2808 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2732C213yesmaximal, complemented
2C2×#2421C213yesmaximal, complemented
3#16964C213yesmaximal, complemented
4#16692C213yesmaximal, complemented
5#15765C216yesmaximal, complemented
6#15526C216yesmaximal, complemented
7#14977C213yesmaximal
8#14969C213yesmaximal, complemented
9#13834C211yesmaximal, characteristic, complemented
10C22×C22⋊Q8C2213yes—
11C22×C4×Q8C2211yescharacteristic
12#2732—224no—
13#2421—224no—
14C2×#400C2216yes—
15C2×#399C2216yes—
16C2×#399C2213yes—
17C2×#398C2216yes—
18C2×#397C2216yes—
19C2×#396C2216yes—
20C2×#392C2211yescharacteristic
21C2×#391C2213yes—
22C2×#390C2213yes—
23C2×#389C2213yes—
24C2×#389C2213yes—
25C2×#388C2216yes—
26C2×#387C2216yes—
27C2×#386C2216yes—
28C2×#385C2216yes—
29C2×#384C2212yes—
30C2×#382C2213yes—
31C2×#380C2213yes—
32C2×C4×C22⋊C4C2213yes—
33#1445C2213yes—
34#1445C2216yes—
35#1445C2216yes—
36#1444C2216yes—
37#1444C2211yescharacteristic
38#1444C2213yes—
39#1439C2213yes—
40#1439C2216yes—
41#1439C2216yes—
42#1439C2213yes—
43#1438C2216yes—
44#1438C2213yes—
45#1437C2216yes—
46#1436C2216yes—
47#1434C2213yes—
48#1434C2213yes—
49C23×Q8C2311yescharacteristic
50C23×D4C2313yes—
51C24×C4C2313yes—
52C2×C22⋊Q8—212no—
53C2×C22⋊Q8—224no—
54C2×C4×Q8—212no—
55C22×C4⋊C4C2316yes—
56C22×C4⋊C4C2313yes—
57C22×C4⋊C4C2313yes—
58C22×C4⋊C4C2316yes—
59C22×C4⋊C4C2316yes—
60C22×C4⋊C4C2313yes—
61C22×C4⋊C4C2313yes—
62C22×C4⋊C4C2313yes—
63C22×C4⋊C4C2313yes—
64C22×C22⋊C4C2316yes—
65C22×C22⋊C4C2313yes—
66C22×C22⋊C4C2313yes—
67C22×C22⋊C4C2316yes—
68C22×C22⋊C4C2316yes—
69C22×C22⋊C4C2316yes—
70C22×C22⋊C4C2316yes—
71C22×C42C2313yes—
72C22×C42C2312yes—
73C22×C42C2313yes—
74#400—224no—
75#399—224no—
76#399—46no—
77#398—224no—
78#397—224no—
79#396—224no—
80#392—42no—
81#391D4124yescomplemented
82#390—212no—
83#389—212no—
84#389—212no—
85#388—224no—
86#387—224no—
87#386—224no—
88#385—224no—
89#384—44no—
90#382—212no—
91#380—212no—
92C4×C22⋊C4—212no—
93C2×#95C2316yes—
94C2×#95C2316yes—
95C2×#95C2316yes—
96C2×#95C2316yes—
97C2×#95C2313yes—
98C2×#95C2316yes—
99C2×#95C2316yes—
100C2×#95C2313yes—
101C2×#95C2311yescharacteristic
102C2×#95C2313yes—
103C2×#95C2311yescharacteristic
104C2×#95C2316yes—
105C2×#95C2313yes—
106C2×#95C2316yes—
107C2×#95C2316yes—
108C25C2413yes—
109C22×Q8—43no—
110C22×Q8—41no—
111C22×Q8C2×D4112yes—
112C22×D4—212no—
113C22×D4—26no—
114C22×D4—46no—
115C22×D4—43no—
116C22×D4—43no—
117C23×C4—26no—
118C23×C4—212no—
119C23×C4—46no—
120C23×C4—43no—
121C23×C4—43no—
122C23×C4C2416yes—
123C23×C4C2413yes—
124C23×C4C2413yes—
125C23×C4C2416yes—
126C23×C4C2413yes—
127C23×C4C2413yes—
128C23×C4C2411yescharacteristic
129C23×C4C2413yes—
130C22⋊Q8—448no—
131C4×Q8—416no—
132C2×C4⋊C4—26no—
133C2×C4⋊C4—212no—
134C2×C4⋊C4—26no—
135C2×C4⋊C4—212no—
136C2×C4⋊C4—212no—
137C2×C4⋊C4—212no—
138C2×C4⋊C4—212no—
139C2×C4⋊C4—212no—
140C2×C4⋊C4—26no—
141C2×C4⋊C4—212no—
142C2×C4⋊C4—26no—
143C2×C4⋊C4—26no—
144C2×C4⋊C4—212no—
145C2×C4⋊C4—43no—
146C2×C4⋊C4—43no—
147C2×C4⋊C4—46no—
148C2×C4⋊C4—46no—
149C2×C4⋊C4—46no—
150C2×C4⋊C4—46no—
151C2×C4⋊C4—43no—
152C2×C4⋊C4—46no—
153C2×C4⋊C4—43no—
154C2×C4⋊C4C2×D4112yes—
155C2×C22⋊C4—26no—
156C2×C22⋊C4—212no—
157C2×C22⋊C4—212no—
158C2×C22⋊C4—212no—
159C2×C22⋊C4—212no—
160C2×C22⋊C4—26no—
161C2×C22⋊C4—26no—
162C2×C22⋊C4—212no—
163C2×C22⋊C4—212no—
164C2×C22⋊C4—26no—
165C2×C22⋊C4—212no—
166C2×C22⋊C4—212no—
167C2×C22⋊C4—43no—
168C2×C22⋊C4—43no—
169C2×C22⋊C4—46no—
170C2×C22⋊C4—46no—
171C2×C22⋊C4—46no—
172C2×C22⋊C4—46no—
173C2×C22⋊C4—46no—
174C2×C22⋊C4—46no—
175C2×C22⋊C4—46no—
176C2×C42—26no—
177C2×C42—212no—
178C2×C42—42no—
179C2×C42—46no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42C2×D4112yes—
183#95—26no—
184#95—26no—
185#95—212no—
186#95—212no—
187#95—212no—
188#95—212no—
189#95—212no—
190#95—212no—
191#95—41no—
192#95—43no—
193#95—46no—
194#95—41no—
195#95—43no—
196#95C2×D4124yes—
197#95C2×D4124yes—
198C24—23no—
199C24—26no—
200C24—23no—
201C24—26no—
202C24—23no—
203C24—23no—
204C24—23no—
205C24—26no—
206C24—26no—
207C24—23no—
208C242−1+416yes—
209C24C2511yescharacteristic
210C2×Q8—412no—
211C2×Q8—412no—
212C2×Q8#127116yes—
213C2×D4—224no—
214C2×D4—86no—
215C2×D4—86no—
216C2×D4—812no—
217C2×D4—812no—
218C22×C4—26no—
219C22×C4—23no—
220C22×C4—23no—
221C22×C4—23no—
222C22×C4—23no—
223C22×C4—26no—
224C22×C4—26no—
225C22×C4—23no—
226C22×C4—26no—
227C22×C4—23no—
228C22×C4—26no—
229C22×C4—26no—
230C22×C4—26no—
231C22×C4—26no—
232C22×C4—26no—
233C22×C4—26no—
234C22×C4—23no—
235C22×C4—26no—
236C22×C4—23no—
237C22×C4—23no—
238C22×C4—23no—
239C22×C4—26no—
240C22×C4—26no—
241C22×C4—26no—
242C22×C4—26no—
243C22×C4—26no—
244C22×C4—26no—
245C22×C4—23no—
246C22×C4—26no—
247C22×C4—23no—
248C22×C4—26no—
249C22×C4—23no—
250C22×C4—26no—
251C22×C4—26no—
252C22×C4—26no—
253C22×C4—23no—
254C22×C4—412no—
255C22×C4—412no—
256C22×C4—412no—
257C22×C4—812no—
258C22×C4—812no—
259C22×C4C22×D416yes—
260C22×C4C22×D416yes—
261C22×C4C22×D4112yes—
262C22×C4C22×D4112yes—
263C22×C4C22×D416yes—
264C22×C4H(5,𝔽2)12yes—
265C4⋊C4—412no—
266C4⋊C4—424no—
267C4⋊C4—412no—
268C4⋊C4—412no—
269C4⋊C4—412no—
270C4⋊C4—424no—
271C4⋊C4—86no—
272C4⋊C4—86no—
273C4⋊C4—812no—
274C22⋊C4—412no—
275C22⋊C4—412no—
276C22⋊C4—424no—
277C22⋊C4—424no—
278C22⋊C4—424no—
279C22⋊C4—812no—
280C22⋊C4—812no—
281C42—412no—
282C42—48no—
283C42—412no—
284C23—212no—
285C23—26no—
286C23—26no—
287C23—26no—
288C23—212no—
289C23—43no—
290C23—46no—
291C23—43no—
292C23—43no—
293C23—46no—
294C23—43no—
295C23—46no—
296C23—43no—
297C23—43no—
298C23—46no—
299C23—46no—
300C23—43no—
301C23—46no—
302C23—46no—
303C23—46no—
304C23—46no—
305C23—46no—
306C23—43no—
307C23C23×D413yes—
308C23C2×H(5,𝔽2)11yescharacteristic
309C23C2×2−1+413yes—
310C23#58511yescharacteristic
311C23#58513yes—
312C23#58513yes—
313C23#58511yescharacteristic
314Q8—416no—
315D4—824no—
316C2×C4—26no—
317C2×C4—26no—
318C2×C4—212no—
319C2×C4—212no—
320C2×C4—26no—
321C2×C4—26no—
322C2×C4—212no—
323C2×C4—26no—
324C2×C4—26no—
325C2×C4—26no—
326C2×C4—26no—
327C2×C4—212no—
328C2×C4—26no—
329C2×C4—212no—
330C2×C4—41no—
331C2×C4—43no—
332C2×C4—46no—
333C2×C4—46no—
334C2×C4—46no—
335C2×C4—46no—
336C2×C4—46no—
337C2×C4—46no—
338C2×C4—46no—
339C2×C4—46no—
340C2×C4—43no—
341C2×C4—46no—
342C2×C4—43no—
343C2×C4—46no—
344C2×C4—46no—
345C2×C4—43no—
346C2×C4—43no—
347C2×C4—43no—
348C2×C4—46no—
349C2×C4—46no—
350C2×C4—46no—
351C2×C4—46no—
352C2×C4—46no—
353C2×C4—46no—
354C2×C4—46no—
355C2×C4—43no—
356C2×C4—46no—
357C2×C4—824no—
358C2×C4C2×#127112yes—
359C22—46no—
360C22—412no—
361C22—412no—
362C22—46no—
363C22—46no—
364C22—83no—
365C22—83no—
366C22—83no—
367C22—86no—
368C22—86no—
369C22—83no—
370C22C22×#12711yescharacteristic
371C22#348513yes—
372C22#348513yes—
373C22#348613yes—
374C22#348713yes—
375C22#348813yes—
376C22#348913yes—
377C22#352911yescharacteristic
378C22#355313yes—
379C22#356113yes—
380C22#356416yes—
381C22#356713yes—
382C4—212no—
383C4—46no—
384C4—46no—
385C4—412no—
386C4—42no—
387C4—83no—
388C4—83no—
389C4—86no—
390C2—86no—
391C2#3546311yescharacteristic
392C2#3592711yescharacteristic
393C2#3606611yescharacteristic
394C2#4597213yes—
395C2#4877813yes—
396C2#4878313yes—
397C2#5207313yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2732C2semidirect
2. C2×#2421C2semidirect
3. #16964C2semidirect
4. #16692C2semidirect
5. #15765C2semidirect
6. #15526C2semidirect
8. #14969C2semidirect
9. #13834C2semidirect
81. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27322
C2×#24212
#169642
#166922
#157652
#155262
#149772
#149692
#138342