Finite Group Databasebeta

← #1045731

Subgroups — #1045731

Order 512.

2142 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17204C212yesmaximal, complemented
2#16591C216yesmaximal, complemented
3#15446C216yesmaximal, complemented
4#15219C213yesmaximal
5#15198C211yesmaximal, characteristic, complemented
6#15155C211yesmaximal, characteristic, complemented
7#14618C212yesmaximal, complemented
8#14022C213yesmaximal, complemented
9#13909C213yesmaximal, complemented
10#13876C213yesmaximal, complemented
11C4×#396C211yesmaximal, characteristic, complemented
12C22×#121C2211yescharacteristic
13C2×#399C2212yes—
14C2×#399C2216yes—
15C2×#396C2211yescharacteristic
16C2×#390C2213yes—
17C2×#390C2213yes—
18C2×#388C2216yes—
19C2×#388C2216yes—
20C2×#384C2213yes—
21C2×#382C2213yes—
22C2×#380C2213yes—
23C2×C4×C4⋊C4C2213yes—
24C2×C4×C4⋊C4C2213yes—
25C2×C4×C22⋊C4C2213yes—
26#1443C2211yescharacteristic
27#1443C2212yes—
28#1443C2216yes—
29#1442C2216yes—
30#1441C2213yes—
31#1441C2216yes—
32#1441C2216yes—
33#1440C2216yes—
34#1440C2216yes—
35#1440C2216yes—
36#1439C2213yes—
37#1439C2216yes—
38#1439C2216yes—
39#1438C2216yes—
40#1438C2216yes—
41#1437C2216yes—
42#1436C2216yes—
43#1436C2216yes—
44#1436C2216yes—
45C4⋊#95C2213yes—
46C4⋊#95C2213yes—
47C4⋊#95C2213yes—
48C4×#95C2211yescharacteristic
49C23×D4C2311yescharacteristic
50C23×D4C2312yes—
51C24×C4C2311yescharacteristic
52C2×#121—46no—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2313yes—
55C22×C4⋊C4C2313yes—
56C22×C4⋊C4C2313yes—
57C22×C4⋊C4C2313yes—
58C22×C4⋊C4C2316yes—
59C22×C4⋊C4C2316yes—
60C22×C22⋊C4C2313yes—
61C22×C22⋊C4C2313yes—
62C22×C22⋊C4C2316yes—
63C22×C22⋊C4C2316yes—
64C22×C22⋊C4C2316yes—
65C22×C22⋊C4C2316yes—
66C22×C22⋊C4C2312yes—
67C22×C22⋊C4C2316yes—
68C22×C42C2313yes—
69C22×C42C2313yes—
70C22×C42C2316yes—
71#399—28no—
72#399—412no—
73#396—24no—
74#390—212no—
75#390—46no—
76#388—412no—
77#388—412no—
78#384—46no—
79#382—46no—
80#380—46no—
81C4×C4⋊C4—212no—
82C4×C4⋊C4—46no—
83C4×C22⋊C4—212no—
84C2×#95C2313yes—
85C2×#95C2316yes—
86C2×#95C2313yes—
87C2×#95C2313yes—
88C2×#95C2313yes—
89C2×#95C2313yes—
90C2×#95C2316yes—
91C2×#95C2316yes—
92C2×#95C2316yes—
93C2×#95C2316yes—
94C2×#95C2316yes—
95C2×#95C2316yes—
96C2×#95C2316yes—
97C2×#95C2316yes—
98C2×#95C2316yes—
99C2×#95C2311yescharacteristic
100C25C2412yes—
101C25C2411yescharacteristic
102C22×D4—24no—
103C22×D4—26no—
104C22×D4—26no—
105C22×D4—22no—
106C22×D4—46no—
107C22×D4—46no—
108C23×C4—26no—
109C23×C4—26no—
110C23×C4—22no—
111C23×C4C2416yes—
112C23×C4C2416yes—
113C23×C4C2411yescharacteristic
114C23×C4C2412yes—
115C23×C4C2413yes—
116C23×C4C2413yes—
117C23×C4C2413yes—
118C23×C4C2413yes—
119C23×C4C2411yescharacteristic
120#121—416no—
121C2×C4⋊C4—26no—
122C2×C4⋊C4—26no—
123C2×C4⋊C4—26no—
124C2×C4⋊C4—26no—
125C2×C4⋊C4—26no—
126C2×C4⋊C4—26no—
127C2×C4⋊C4—26no—
128C2×C4⋊C4—26no—
129C2×C4⋊C4—26no—
130C2×C4⋊C4—26no—
131C2×C4⋊C4—26no—
132C2×C4⋊C4—212no—
133C2×C4⋊C4—212no—
134C2×C4⋊C4—43no—
135C2×C4⋊C4—43no—
136C2×C4⋊C4—46no—
137C2×C4⋊C4—46no—
138C2×C4⋊C4—46no—
139C2×C4⋊C4—46no—
140C2×C22⋊C4—26no—
141C2×C22⋊C4—26no—
142C2×C22⋊C4—26no—
143C2×C22⋊C4—26no—
144C2×C22⋊C4—212no—
145C2×C22⋊C4—26no—
146C2×C22⋊C4—212no—
147C2×C22⋊C4—212no—
148C2×C22⋊C4—212no—
149C2×C22⋊C4—43no—
150C2×C22⋊C4—46no—
151C2×C22⋊C4—46no—
152C2×C22⋊C4—46no—
153C2×C22⋊C4—46no—
154C2×C22⋊C4—46no—
155C2×C22⋊C4—46no—
156C2×C22⋊C4—46no—
157C2×C22⋊C4—46no—
158C2×C22⋊C4—46no—
159C2×C22⋊C4—46no—
160C2×C22⋊C4—46no—
161C2×C22⋊C4—46no—
162C2×C42—26no—
163C2×C42—26no—
164C2×C42—26no—
165C2×C42—26no—
166C2×C42—26no—
167C2×C42—46no—
168C2×C42—46no—
169C2×C42—46no—
170C2×C42#41112yes—
171#95—26no—
172#95—26no—
173#95—26no—
174#95—26no—
175#95—212no—
176#95—26no—
177#95—46no—
178#95—46no—
179#95—46no—
180#95—46no—
181#95—46no—
182#95—46no—
183#95—46no—
184#95—46no—
185#95—46no—
186#95#4114yes—
187C24—21no—
188C24—23no—
189C24—23no—
190C24—23no—
191C24—21no—
192C24—23no—
193C24—22no—
194C24—26no—
195C24—26no—
196C24—26no—
197C24—22no—
198C24—22no—
199C24—26no—
200C242−1+412yes—
201C24C2511yescharacteristic
202C2×D4—48no—
203C2×D4—412no—
204C2×D4—412no—
205C2×D4—44no—
206C2×D4—812no—
207C2×D4—812no—
208C22×C4—23no—
209C22×C4—23no—
210C22×C4—23no—
211C22×C4—23no—
212C22×C4—23no—
213C22×C4—23no—
214C22×C4—23no—
215C22×C4—23no—
216C22×C4—23no—
217C22×C4—23no—
218C22×C4—23no—
219C22×C4—23no—
220C22×C4—23no—
221C22×C4—23no—
222C22×C4—23no—
223C22×C4—23no—
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—23no—
233C22×C4—26no—
234C22×C4—26no—
235C22×C4—26no—
236C22×C4—26no—
237C22×C4—26no—
238C22×C4—23no—
239C22×C4—26no—
240C22×C4—26no—
241C22×C4—26no—
242C22×C4—26no—
243C22×C4—26no—
244C22×C4—26no—
245C22×C4—26no—
246C22×C4—26no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—26no—
250C22×C4—26no—
251C22×C4—26no—
252C22×C4—412no—
253C22×C4—412no—
254C22×C4—44no—
255C22×C4C2×#4116yes—
256C22×C4C2×#4116yes—
257C22×C4C2×#4116yes—
258C22×C4C2×#4116yes—
259C22×C4H(5,𝔽2)16yes—
260C22×C4H(5,𝔽2)16yes—
261C22×C42−1+412yes—
262C4⋊C4—412no—
263C4⋊C4—412no—
264C4⋊C4—412no—
265C4⋊C4—412no—
266C4⋊C4—44no—
267C4⋊C4—812no—
268C4⋊C4—812no—
269C22⋊C4—224no—
270C22⋊C4—412no—
271C22⋊C4—424no—
272C22⋊C4—48no—
273C22⋊C4—812no—
274C22⋊C4—812no—
275C22⋊C4—812no—
276C22⋊C4—812no—
277C42—224no—
278C42—412no—
279C42—812no—
280C23—26no—
281C23—22no—
282C23—26no—
283C23—43no—
284C23—43no—
285C23—43no—
286C23—43no—
287C23—43no—
288C23—43no—
289C23—41no—
290C23—43no—
291C23—43no—
292C23—43no—
293C23—46no—
294C23—46no—
295C23—46no—
296C23—46no—
297C23—46no—
298C23—46no—
299C23—46no—
300C23—42no—
301C23—42no—
302C23—46no—
303C23—46no—
304C23—46no—
305C23—46no—
306C23C22×#4113yes—
307C23C22×#4111yescharacteristic
308C23C2×H(5,𝔽2)13yes—
309C23C2×H(5,𝔽2)13yes—
310C23C2×2−1+411yescharacteristic
311C23C2×2−1+411yescharacteristic
312C23#58513yes—
313D4—816no—
314D4—88no—
315C2×C4—26no—
316C2×C4—26no—
317C2×C4—26no—
318C2×C4—26no—
319C2×C4—26no—
320C2×C4—26no—
321C2×C4—26no—
322C2×C4—26no—
323C2×C4—26no—
324C2×C4—26no—
325C2×C4—26no—
326C2×C4—26no—
327C2×C4—43no—
328C2×C4—43no—
329C2×C4—43no—
330C2×C4—43no—
331C2×C4—43no—
332C2×C4—43no—
333C2×C4—43no—
334C2×C4—43no—
335C2×C4—43no—
336C2×C4—43no—
337C2×C4—43no—
338C2×C4—43no—
339C2×C4—43no—
340C2×C4—43no—
341C2×C4—43no—
342C2×C4—43no—
343C2×C4—46no—
344C2×C4—43no—
345C2×C4—43no—
346C2×C4—46no—
347C2×C4—46no—
348C2×C4—46no—
349C2×C4—46no—
350C2×C4—46no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—46no—
354C2×C4—46no—
355C2×C4—46no—
356C2×C4—46no—
357C2×C4—46no—
358C2×C4—46no—
359C2×C4—46no—
360C2×C4—46no—
361C2×C4—46no—
362C2×C4—46no—
363C2×C4—88no—
364C2×C4#542112yes—
365C2×C4#553112yes—
366C2×C4#56014yes—
367C22—42no—
368C22—46no—
369C22—46no—
370C22—81no—
371C22—83no—
372C22—81no—
373C22—83no—
374C22—82no—
375C22—82no—
376C22—86no—
377C22—86no—
378C22—82no—
379C22—86no—
380C22—86no—
381C22C2×#54213yes—
382C22C2×#55313yes—
383C22#347513yes—
384C22#349213yes—
385C22#349413yes—
386C22#350613yes—
387C22#350813yes—
388C22#352311yescharacteristic
389C22C2×#56011yescharacteristic
390C22#354313yes—
391C22#355413yes—
392C22#356113yes—
393C22#356213yes—
394C4—46no—
395C4—46no—
396C4—46no—
397C4—46no—
398C4—81no—
399C4—82no—
400C4—86no—
401C4—86no—
402C4#280118yes—
403C2—82no—
404C2—162no—
405C2#3990313yes—
406C2#4247213yes—
407C2#4492213yes—
408C2#4601011yescharacteristic
409C2C2×#280111yescharacteristic
410C2#5304311yescharacteristic
411C2#5930713yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17204C2semidirect
2. #16591C2semidirect
3. #15446C2semidirect
5. #15198C2semidirect
6. #15155C2semidirect
7. #14618C2semidirect
8. #14022C2semidirect
9. #13909C2semidirect
10. #13876C2semidirect
11. C4×#396C2semidirect

Maximal subgroups

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

subgroupindex
#172042
#165912
#154462
#152192
#151982
#151552
#146182
#140222
#139092
#138762
C4×#3962