Finite Group Databasebeta

← #4462645

Subgroups — #4462645

Order 512.

2945 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2614C211yesmaximal, characteristic
2C22×#395C211yesmaximal, characteristic, complemented
3#17223C212yesmaximal, complemented
4#17219C216yesmaximal, complemented
5#17218C212yesmaximal, complemented
6#17217C216yesmaximal, complemented
7#15073C213yesmaximal, complemented
8#15069C213yesmaximal
9#14919C213yesmaximal, complemented
10#14912C213yesmaximal
11#14336C211yesmaximal, characteristic, complemented
12C24×Q8C2211yescharacteristic
13C22×#123C2213yes—
14C22×#123C2211yescharacteristic
15C23×C4⋊C4C2213yes—
16#2614—28no—
17C2×#395—28no—
18C2×#395C2211yescharacteristic
19C2×#395C2213yes—
20C2×#395C2216yes—
21C2×#395C2212yes—
22C2×#393C2213yes—
23C2×#393C2211yescharacteristic
24C2×#386C2213yes—
25C2×#385C2213yes—
26C2×#381C2213yes—
27C2×#381C2213yes—
28C2×#381C2213yes—
29C2×#381C2213yes—
30C22×#95C2213yes—
31#1446C2216yes—
32#1446C2212yes—
33#1445C2216yes—
34#1444C2216yes—
35#1444C2216yes—
36#1443C2216yes—
37#1443C2216yes—
38#1442C2216yes—
39#1442C2216yes—
40#1442C2216yes—
41#1442C2216yes—
42#1441C2216yes—
43#1441C2216yes—
44#1441C2216yes—
45#1440C2216yes—
46#1439C2216yes—
47#1439C2216yes—
48#1439C2216yes—
49C22⋊#95C2213yes—
50C22⋊#95C2213yes—
51C23×Q8—46no—
52C23×Q8—46no—
53C23×Q8D418yes—
54C23×Q8C2313yes—
55C23×Q8C2311yescharacteristic
56C23×D4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2313yes—
59C24×C4C2313yes—
60C2×#123—212no—
61C2×#123—46no—
62C2×#123—46no—
63C2×#123—46no—
64C22×C4⋊C4—212no—
65C22×C4⋊C4—46no—
66C22×C4⋊C4—46no—
67C22×C4⋊C4C2316yes—
68C22×C4⋊C4C2316yes—
69C22×C4⋊C4C2313yes—
70C22×C4⋊C4C2313yes—
71C22×C4⋊C4C2316yes—
72C22×C4⋊C4C2312yes—
73C22×C22⋊C4C2311yescharacteristic
74C22×C22⋊C4C2313yes—
75C22×C22⋊C4C2313yes—
76C22×C22⋊C4C2313yes—
77C22×C22⋊C4C2313yes—
78C22×C22⋊C4C2316yes—
79C22×C22⋊C4C2312yes—
80#395—24no—
81#395—212no—
82#395—224no—
83#395—28no—
84#393—212no—
85#393—24no—
86#386—46no—
87#385—46no—
88#381—212no—
89#381—46no—
90#381—46no—
91#381—46no—
92C2×#95—212no—
93C2×#95C2313yes—
94C2×#95C2313yes—
95C2×#95C2313yes—
96C2×#95C2313yes—
97C2×#95C2316yes—
98C2×#95C2316yes—
99C2×#95C2316yes—
100C2×#95C2316yes—
101C2×#95C2316yes—
102C2×#95C2316yes—
103C2×#95C2316yes—
104C2×#95C2316yes—
105C2×#95C2316yes—
106C2×#95C2316yes—
107C2×#95C2316yes—
108C2×#95C2316yes—
109C2×#95C2316yes—
110C2×#95C2316yes—
111C25C2411yescharacteristic
112C25C2411yescharacteristic
113C22×Q8—448no—
114C22×Q8—43no—
115C22×Q8—43no—
116C22×Q8—43no—
117C22×Q8—43no—
118C22×Q8—43no—
119C22×Q8—43no—
120C22×Q8—43no—
121C22×Q8—43no—
122C22×Q8—448no—
123C22×Q8—416no—
124C22×Q8C2×D4112yes—
125C22×Q8C2×D414yes—
126C22×D4—22no—
127C22×D4—26no—
128C22×D4—26no—
129C23×C4—26no—
130C23×C4—26no—
131C23×C4—26no—
132C23×C4—26no—
133C23×C4—26no—
134C23×C4—26no—
135C23×C4—26no—
136C23×C4—212no—
137C23×C4—22no—
138C23×C4—26no—
139C23×C4—26no—
140C23×C4—26no—
141C23×C4—26no—
142C23×C4—46no—
143C23×C4C2×D4112yes—
144C23×C4C2413yes—
145C23×C4C2413yes—
146C23×C4C2416yes—
147C23×C4C2412yes—
148C23×C4C2412yes—
149C23×C4C2411yescharacteristic
150C23×C4C2413yes—
151C23×C4C2413yes—
152C23×C4C2416yes—
153#123—448no—
154#123—416no—
155C2×C4⋊C4—212no—
156C2×C4⋊C4—212no—
157C2×C4⋊C4—26no—
158C2×C4⋊C4—26no—
159C2×C4⋊C4—212no—
160C2×C4⋊C4—46no—
161C2×C4⋊C4—46no—
162C2×C4⋊C4—46no—
163C2×C4⋊C4—46no—
164C2×C4⋊C4—43no—
165C2×C4⋊C4—43no—
166C2×C4⋊C4—43no—
167C2×C4⋊C4—43no—
168C2×C4⋊C4—46no—
169C2×C4⋊C4—448no—
170C2×C4⋊C4—46no—
171C2×C4⋊C4—46no—
172C2×C22⋊C4—26no—
173C2×C22⋊C4—26no—
174C2×C22⋊C4—26no—
175C2×C22⋊C4—26no—
176C2×C22⋊C4—26no—
177C2×C22⋊C4—26no—
178C2×C22⋊C4—26no—
179C2×C22⋊C4—26no—
180C2×C22⋊C4—26no—
181C2×C22⋊C4—212no—
182C2×C22⋊C4—43no—
183C2×C22⋊C4—43no—
184C2×C22⋊C4—43no—
185C2×C22⋊C4—43no—
186C2×C22⋊C4—46no—
187C2×C22⋊C4—46no—
188C2×C22⋊C4—46no—
189#95—212no—
190#95—212no—
191#95—26no—
192#95—26no—
193#95—26no—
194#95—26no—
195#95—212no—
196#95—212no—
197#95—46no—
198#95—46no—
199#95—46no—
200#95—46no—
201#95—46no—
202#95—46no—
203#95—46no—
204#95—46no—
205#95—46no—
206#95—46no—
207C24—26no—
208C24—22no—
209C24—21no—
210C24—23no—
211C24—23no—
212C24—26no—
213C24—22no—
214C24C22×D412yes—
215C24H(5,𝔽2)16yes—
216C24H(5,𝔽2)16yes—
217C24C2511yescharacteristic
218C2×Q8—412no—
219C2×Q8—412no—
220C2×Q8—412no—
221C2×Q8—412no—
222C2×Q8—412no—
223C2×Q8—412no—
224C2×Q8—412no—
225C2×Q8—412no—
226C2×Q8—412no—
227C2×Q8—44no—
228C2×Q8—4128no—
229C2×D4—412no—
230C2×D4—412no—
231C2×D4—44no—
232C22×C4—224no—
233C22×C4—224no—
234C22×C4—224no—
235C22×C4—224no—
236C22×C4—26no—
237C22×C4—23no—
238C22×C4—23no—
239C22×C4—23no—
240C22×C4—23no—
241C22×C4—26no—
242C22×C4—26no—
243C22×C4—23no—
244C22×C4—23no—
245C22×C4—23no—
246C22×C4—23no—
247C22×C4—23no—
248C22×C4—23no—
249C22×C4—23no—
250C22×C4—23no—
251C22×C4—26no—
252C22×C4—26no—
253C22×C4—26no—
254C22×C4—23no—
255C22×C4—23no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—26no—
259C22×C4—26no—
260C22×C4—26no—
261C22×C4—23no—
262C22×C4—23no—
263C22×C4—23no—
264C22×C4—26no—
265C22×C4—26no—
266C22×C4—23no—
267C22×C4—23no—
268C22×C4—26no—
269C22×C4—23no—
270C22×C4—26no—
271C22×C4—26no—
272C22×C4—26no—
273C22×C4—26no—
274C22×C4—26no—
275C22×C4—26no—
276C22×C4—26no—
277C22×C4—22no—
278C22×C4—26no—
279C22×C4—26no—
280C22×C4—424no—
281C22×C4—424no—
282C22×C4—412no—
283C22×C4—412no—
284C22×C4—412no—
285C22×C4—412no—
286C22×C4—412no—
287C22×C4—412no—
288C22×C4—412no—
289C22×C4—412no—
290C22×C4—44no—
291C22×C4C22×D416yes—
292C22×C4C22×D416yes—
293C22×C42−1+4112yes—
294C4⋊C4—424no—
295C4⋊C4—424no—
296C4⋊C4—412no—
297C4⋊C4—412no—
298C4⋊C4—48no—
299C4⋊C4—812no—
300C22⋊C4—412no—
301C22⋊C4—412no—
302C22⋊C4—48no—
303C22⋊C4—412no—
304C22⋊C4—412no—
305C22⋊C4—44no—
306C22⋊C4—812no—
307C23—26no—
308C23—22no—
309C23—212no—
310C23—212no—
311C23—212no—
312C23—212no—
313C23—43no—
314C23—43no—
315C23—41no—
316C23—43no—
317C23—41no—
318C23—46no—
319C23—46no—
320C23—46no—
321C23—46no—
322C23#534112yes—
323C23#560112yes—
324C23H(3,𝔽4)14yes—
325C23C23×D411yescharacteristic
326C23C2×H(5,𝔽2)13yes—
327C23C2×H(5,𝔽2)13yes—
328C23C2×2−1+416yes—
329C23#58512yes—
330Q8—448no—
331Q8—416no—
332D4—88no—
333C2×C4—26no—
334C2×C4—26no—
335C2×C4—26no—
336C2×C4—26no—
337C2×C4—26no—
338C2×C4—26no—
339C2×C4—212no—
340C2×C4—212no—
341C2×C4—212no—
342C2×C4—448no—
343C2×C4—46no—
344C2×C4—46no—
345C2×C4—43no—
346C2×C4—43no—
347C2×C4—43no—
348C2×C4—43no—
349C2×C4—43no—
350C2×C4—43no—
351C2×C4—43no—
352C2×C4—43no—
353C2×C4—46no—
354C2×C4—46no—
355C2×C4—448no—
356C2×C4—43no—
357C2×C4—43no—
358C2×C4—46no—
359C2×C4—43no—
360C2×C4—43no—
361C2×C4—43no—
362C2×C4—43no—
363C2×C4—43no—
364C2×C4—46no—
365C2×C4—46no—
366C2×C4—43no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—46no—
371C2×C4—46no—
372C2×C4—46no—
373C2×C4—46no—
374C2×C4—46no—
375C2×C4—46no—
376C2×C4—46no—
377C2×C4—46no—
378C2×C4—46no—
379C2×C4—46no—
380C2×C4—46no—
381C2×C4—46no—
382C2×C4—88no—
383C22—212no—
384C22—212no—
385C22—224no—
386C22—28no—
387C22—81no—
388C22—83no—
389C22—83no—
390C22—86no—
391C22—82no—
392C22#281218yes—
393C22C2×#53413yes—
394C22#348613yes—
395C22#348711yescharacteristic
396C22C2×#56013yes—
397C22C2×H(3,𝔽4)11yescharacteristic
398C22#353516yes—
399C22#353516yes—
400C22#354416yes—
401C22#354416yes—
402C4—46no—
403C4—46no—
404C4—412no—
405C4—82no—
406C4—81no—
407C4—83no—
408C4—83no—
409C4—86no—
410C4—82no—
411C2—28no—
412C2—161no—
413C2C2×#281211yescharacteristic
414C2#3718613yes—
415C2#3725813yes—
416C2#5291316yes—
417C2#5291412yes—
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
2. C22×#395C2semidirect
3. #17223C2semidirect
4. #17219C2semidirect
5. #17218C2semidirect
6. #17217C2semidirect
7. #15073C2semidirect
9. #14919C2semidirect
11. #14336C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26142
C22×#3952
#172232
#172192
#172182
#172172
#150732
#150692
#149192
#149122
#143362