Finite Group Databasebeta

← #4029000

Subgroups — #4029000

Order 512.

2366 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17072C211yesmaximal, characteristic, complemented
2#16684C213yesmaximal, complemented
3#15772C216yesmaximal, complemented
4#15476C216yesmaximal, complemented
5#14886C213yesmaximal
6#14476C213yesmaximal, complemented
7#14224C213yesmaximal, complemented
8#13957C213yesmaximal, complemented
9#13879C213yesmaximal, complemented
10C22×#117C2213yes—
11C2×#395C2216yes—
12C2×#394C2213yes—
13C2×#393C2216yes—
14C2×#392C2211yescharacteristic
15C2×#390C2213yes—
16C2×#388C2216yes—
17C2×#386C2216yes—
18C2×#384C2211yescharacteristic
19C2×#382C2213yes—
20C2×#382C2213yes—
21C2×#380C2213yes—
22C2×C4×C22⋊C4C2213yes—
23C2×#376C2211yescharacteristic
24#1446C2212yes—
25#1442C2216yes—
26#1442C2216yes—
27#1441C2213yes—
28#1441C2213yes—
29#1441C2213yes—
30#1439C2216yes—
31#1439C2213yes—
32#1439C2213yes—
33#1439C2216yes—
34#1439C2216yes—
35#1439C2213yes—
36#1439C2216yes—
37#1438C2216yes—
38#1437C2216yes—
39#1436C2216yes—
40#1436C2216yes—
41C4⋊#95C2213yes—
42C4⋊#95C2213yes—
43#1434C2213yes—
44#1434C2216yes—
45#1434C2216yes—
46C22⋊#95C2216yes—
47C23×Q8C2312yes—
48C23×D4C2313yes—
49C24×C4C2313yes—
50C2×#117—212no—
51C2×#117—46no—
52C2×#117—46no—
53C22×C4⋊C4C2313yes—
54C22×C4⋊C4C2313yes—
55C22×C4⋊C4C2313yes—
56C22×C4⋊C4C2313yes—
57C22×C4⋊C4C2313yes—
58C22×C4⋊C4C2313yes—
59C22×C4⋊C4C2313yes—
60C22×C22⋊C4C2316yes—
61C22×C22⋊C4C2313yes—
62C22×C22⋊C4C2316yes—
63C22×C22⋊C4C2316yes—
64C22×C22⋊C4C2313yes—
65C22×C22⋊C4C2316yes—
66C22×C22⋊C4C2316yes—
67C22×C42C2313yes—
68C22×C42C2313yes—
69C22×C42C2313yes—
70C22×C42C2311yescharacteristic
71#395—412no—
72#394—212no—
73#393—412no—
74#392—42no—
75#390—46no—
76#388—412no—
77#386—412no—
78#384—42no—
79#382—212no—
80#382—46no—
81#380—46no—
82C4×C22⋊C4—212no—
83#376—42no—
84C2×#95C2316yes—
85C2×#95C2316yes—
86C2×#95C2313yes—
87C2×#95C2316yes—
88C2×#95C2316yes—
89C2×#95C2316yes—
90C2×#95C2316yes—
91C2×#95C2316yes—
92C2×#95C2311yescharacteristic
93C2×#95C2316yes—
94C2×#95C2316yes—
95C2×#95C2313yes—
96C2×#95C2316yes—
97C2×#95C2316yes—
98C2×#95C2316yes—
99C2×#95C2311yescharacteristic
100C25C2413yes—
101C22×Q8—42no—
102C22×Q8—46no—
103C22×Q8—46no—
104C22×D4—212no—
105C22×D4—26no—
106C22×D4—46no—
107C22×D4—43no—
108C22×D4—43no—
109C23×C4—26no—
110C23×C4—26no—
111C23×C4—26no—
112C23×C4—26no—
113C23×C4—26no—
114C23×C4—43no—
115C23×C4—43no—
116C23×C4C2416yes—
117C23×C4C2413yes—
118C23×C4C2416yes—
119C23×C4C2413yes—
120C23×C4C2413yes—
121C23×C4C2411yescharacteristic
122C23×C4C2413yes—
123C23×C4C2413yes—
124#117—448no—
125C2×C4⋊C4—26no—
126C2×C4⋊C4—26no—
127C2×C4⋊C4—26no—
128C2×C4⋊C4—26no—
129C2×C4⋊C4—43no—
130C2×C4⋊C4—43no—
131C2×C4⋊C4—43no—
132C2×C4⋊C4—43no—
133C2×C4⋊C4—43no—
134C2×C4⋊C4—43no—
135C2×C4⋊C4—43no—
136C2×C4⋊C4—43no—
137C2×C4⋊C4—43no—
138C2×C4⋊C4—46no—
139C2×C4⋊C4—43no—
140C2×C4⋊C4—46no—
141C2×C4⋊C4—46no—
142C2×C4⋊C4#41112yes—
143C2×C22⋊C4—212no—
144C2×C22⋊C4—26no—
145C2×C22⋊C4—26no—
146C2×C22⋊C4—212no—
147C2×C22⋊C4—212no—
148C2×C22⋊C4—212no—
149C2×C22⋊C4—26no—
150C2×C22⋊C4—212no—
151C2×C22⋊C4—212no—
152C2×C22⋊C4—43no—
153C2×C22⋊C4—43no—
154C2×C22⋊C4—43no—
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×C22⋊C4—46no—
163C2×C22⋊C4—46no—
164C2×C42—26no—
165C2×C42—26no—
166C2×C42—26no—
167C2×C42—26no—
168C2×C42—26no—
169C2×C42—26no—
170C2×C42—26no—
171C2×C42—26no—
172C2×C42—43no—
173C2×C42—41no—
174C2×C42—41no—
175C2×C42#4114yes—
176#95—212no—
177#95—212no—
178#95—212no—
179#95—26no—
180#95—212no—
181#95—212no—
182#95—212no—
183#95—212no—
184#95—43no—
185#95—41no—
186#95—46no—
187#95—46no—
188#95—46no—
189#95—41no—
190#95—46no—
191#95—46no—
192C24—23no—
193C24—23no—
194C24—26no—
195C24—26no—
196C24—26no—
197C24—23no—
198C24—23no—
199C24—23no—
200C24—26no—
201C24—23no—
202C242−1+416yes—
203C24C2511yescharacteristic
204C2×Q8—812no—
205C2×Q8—84no—
206C2×Q8—812no—
207C2×D4—424no—
208C2×D4—412no—
209C2×D4—86no—
210C2×D4—812no—
211C2×D4—86no—
212C22×C4—23no—
213C22×C4—23no—
214C22×C4—23no—
215C22×C4—23no—
216C22×C4—224no—
217C22×C4—23no—
218C22×C4—23no—
219C22×C4—23no—
220C22×C4—26no—
221C22×C4—26no—
222C22×C4—23no—
223C22×C4—23no—
224C22×C4—23no—
225C22×C4—23no—
226C22×C4—23no—
227C22×C4—23no—
228C22×C4—23no—
229C22×C4—26no—
230C22×C4—23no—
231C22×C4—26no—
232C22×C4—23no—
233C22×C4—26no—
234C22×C4—26no—
235C22×C4—26no—
236C22×C4—26no—
237C22×C4—23no—
238C22×C4—26no—
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—26no—
248C22×C4—23no—
249C22×C4—23no—
250C22×C4—26no—
251C22×C4—26no—
252C22×C4—412no—
253C22×C4—412no—
254C22×C4—412no—
255C22×C4—412no—
256C22×C4—412no—
257C22×C4—412no—
258C22×C4C2×#4116yes—
259C22×C4C2×#4116yes—
260C22×C4C2×#4116yes—
261C22×C4C2×#4116yes—
262C22×C4H(5,𝔽2)16yes—
263C22×C4H(5,𝔽2)12yes—
264C22×C4H(5,𝔽2)16yes—
265C4⋊C4—412no—
266C4⋊C4—412no—
267C4⋊C4—86no—
268C4⋊C4—86no—
269C4⋊C4—86no—
270C4⋊C4—86no—
271C4⋊C4—86no—
272C22⋊C4—412no—
273C22⋊C4—424no—
274C22⋊C4—412no—
275C22⋊C4—424no—
276C22⋊C4—812no—
277C22⋊C4—812no—
278C22⋊C4—812no—
279C42—224no—
280C42—44no—
281C42—412no—
282C42—412no—
283C23—26no—
284C23—26no—
285C23—212no—
286C23—26no—
287C23—212no—
288C23—43no—
289C23—46no—
290C23—43no—
291C23—43no—
292C23—46no—
293C23—43no—
294C23—46no—
295C23—46no—
296C23—46no—
297C23—46no—
298C23—46no—
299C23—46no—
300C23—43no—
301C23—46no—
302C23—43no—
303C23—43no—
304C23—46no—
305C23—43no—
306C23C22×#4111yescharacteristic
307C23C22×#4113yes—
308C23C2×H(5,𝔽2)11yescharacteristic
309C23C2×H(5,𝔽2)13yes—
310C23C2×H(5,𝔽2)13yes—
311C23C2×2−1+413yes—
312C23#58511yescharacteristic
313Q8—816no—
314D4—824no—
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—212no—
327C2×C4—212no—
328C2×C4—26no—
329C2×C4—448no—
330C2×C4—43no—
331C2×C4—43no—
332C2×C4—43no—
333C2×C4—43no—
334C2×C4—46no—
335C2×C4—43no—
336C2×C4—46no—
337C2×C4—43no—
338C2×C4—43no—
339C2×C4—43no—
340C2×C4—43no—
341C2×C4—41no—
342C2×C4—46no—
343C2×C4—46no—
344C2×C4—46no—
345C2×C4—43no—
346C2×C4—46no—
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—46no—
358C2×C4—46no—
359C2×C4—46no—
360C2×C4—46no—
361C2×C4#555112yes—
362C22—46no—
363C22—46no—
364C22—46no—
365C22—412no—
366C22—412no—
367C22—86no—
368C22—86no—
369C22—83no—
370C22—83no—
371C22—83no—
372C22—83no—
373C22#345911yescharacteristic
374C22C2×#55513yes—
375C22#348313yes—
376C22#349013yes—
377C22#349213yes—
378C22#349513yes—
379C22#352016yes—
380C22#353413yes—
381C22#353613yes—
382C22#353813yes—
383C22#354913yes—
384C22#354911yescharacteristic
385C4—212no—
386C4—46no—
387C4—42no—
388C4—46no—
389C4—46no—
390C4—86no—
391C4—86no—
392C4—83no—
393C2—86no—
394C2#3794813yes—
395C2#4348813yes—
396C2#4780411yescharacteristic
397C2#4966013yes—
398C2#5898911yescharacteristic
399C2#5923313yes—
400C2#6004811yescharacteristic
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. #17072C2semidirect
2. #16684C2semidirect
3. #15772C2semidirect
4. #15476C2semidirect
6. #14476C2semidirect
7. #14224C2semidirect
8. #13957C2semidirect
9. #13879C2semidirect

Maximal subgroups

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

subgroupindex
#170722
#166842
#157722
#154762
#148862
#144762
#142242
#139572
#138792