Finite Group Databasebeta

← #584852

Subgroups — #584852

Order 512.

2514 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2842C211yesmaximal, characteristic
2#14653C216yesmaximal, complemented
3#14636C211yesmaximal, characteristic, complemented
4#14562C212yesmaximal, complemented
5#14551C213yesmaximal, complemented
6#14366C213yesmaximal
7#14331C212yesmaximal, complemented
8#14240C213yesmaximal, complemented
9#14200C216yesmaximal, complemented
10#13991C213yesmaximal
11C4×#399C211yesmaximal, characteristic, complemented
12C23×C22⋊C4C2211yescharacteristic
13#2842C4116yescomplemented
14C2×#399C2211yescharacteristic
15C2×#399C2213yes—
16C2×#396C2216yes—
17C2×#396C2212yes—
18C2×#389C2216yescomplemented
19C2×#388C2216yes—
20C2×#388C2216yes—
21C2×#385C2213yes—
22C2×#379C2213yes—
23C2×#379C2213yes—
24C2×C4×C4⋊C4C2213yescomplemented
25C2×C4×C4⋊C4C2213yescomplemented
26C2×C4×C22⋊C4C2213yes—
27#1445C2213yescomplemented
28#1445C2213yescomplemented
29#1445C2216yescomplemented
30#1444C2211yescharacteristic, complemented
31#1443C2212yescomplemented
32#1442C2213yescomplemented
33#1441C2216yescomplemented
34#1441C2216yescomplemented
35#1440C2216yes—
36#1440C2216yes—
37#1439C2213yes—
38#1439C2213yes—
39#1439C2216yes—
40#1439C2216yes—
41#1437C2216yes—
42#1437C2216yes—
43#1437C2216yes—
44#1436C2216yescomplemented
45#1436C2216yescomplemented
46#1436C2216yescomplemented
47#1434C2213yescomplemented
48C22⋊#95C2213yes—
49C22⋊#95C2213yes—
50C4×#95C2211yescharacteristic, complemented
51C26C2311yescharacteristic
52C24×C4C2311yescharacteristic
53C24×C4C2311yescharacteristic
54C22×C4⋊C4C2313yes—
55C22×C4⋊C4C2316yes—
56C22×C4⋊C4C2313yes—
57C22×C4⋊C4C2316yes—
58C22×C4⋊C4C2313yes—
59C22×C4⋊C4C2313yes—
60C22×C22⋊C4—46no—
61C22×C22⋊C4C2312yes—
62C22×C22⋊C4C2313yes—
63C22×C22⋊C4C2313yes—
64C22×C22⋊C4C2316yes—
65C22×C22⋊C4C2312yes—
66C22×C22⋊C4C2316yes—
67C22×C22⋊C4C2316yes—
68C22×C22⋊C4C2316yes—
69C22×C22⋊C4C2313yes—
70C22×C22⋊C4C2313yes—
71C22×C42C2313yes—
72C22×C42C2313yes—
73C22×C42C2316yes—
74#399C2×C418yes—
75#399C2×C4124yes—
76#396C2×C4148yescomplemented
77#396C2×C4116yescomplemented
78#389—412no—
79#388—412no—
80#388—412no—
81#385—46no—
82#379—46no—
83#379C2×C4124yes—
84C4×C4⋊C4—46no—
85C4×C4⋊C4—46no—
86C4×C22⋊C4—46no—
87C2×#95C2316yes—
88C2×#95C2316yes—
89C2×#95C2313yes—
90C2×#95C2316yes—
91C2×#95C2316yes—
92C2×#95C2316yes—
93C2×#95C2313yes—
94C2×#95C2316yes—
95C2×#95C2313yes—
96C2×#95C2316yes—
97C2×#95C2316yes—
98C2×#95C2316yes—
99C2×#95C2316yes—
100C2×#95C2313yes—
101C2×#95C2311yescharacteristic
102C2×#95C2313yes—
103C25—22no—
104C25—22no—
105C25—43no—
106C25—43no—
107C25—43no—
108C25—43no—
109C25C22×C414yes—
110C25C2412yes—
111C25C2411yescharacteristic
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—26no—
115C23×C4—43no—
116C23×C4—43no—
117C23×C4—43no—
118C23×C4C2416yes—
119C23×C4C2413yes—
120C23×C4C2412yes—
121C23×C4C2411yescharacteristic
122C23×C4C2416yes—
123C23×C4C2413yes—
124C23×C4C2413yes—
125C23×C4C2413yes—
126C23×C4C2411yescharacteristic
127C2×C4⋊C4—212no—
128C2×C4⋊C4—212no—
129C2×C4⋊C4—43no—
130C2×C4⋊C4—46no—
131C2×C4⋊C4—43no—
132C2×C4⋊C4—46no—
133C2×C4⋊C4—43no—
134C2×C4⋊C4—43no—
135C2×C4⋊C4—43no—
136C2×C4⋊C4—43no—
137C2×C4⋊C4—46no—
138C2×C4⋊C4—46no—
139C2×C4⋊C4—46no—
140C2×C4⋊C4C22×C4124yes—
141C2×C4⋊C4C22×C4112yes—
142C2×C4⋊C4C22×C4112yes—
143C2×C22⋊C4—26no—
144C2×C22⋊C4—212no—
145C2×C22⋊C4—26no—
146C2×C22⋊C4—26no—
147C2×C22⋊C4—212no—
148C2×C22⋊C4—46no—
149C2×C22⋊C4—43no—
150C2×C22⋊C4—43no—
151C2×C22⋊C4—46no—
152C2×C22⋊C4—46no—
153C2×C22⋊C4—43no—
154C2×C22⋊C4—43no—
155C2×C22⋊C4—43no—
156C2×C22⋊C4—46no—
157C2×C22⋊C4—46no—
158C2×C22⋊C4—416no—
159C2×C22⋊C4—43no—
160C2×C22⋊C4—46no—
161C2×C22⋊C4—46no—
162C2×C22⋊C4—43no—
163C2×C22⋊C4—46no—
164C2×C22⋊C4—46no—
165C2×C22⋊C4—46no—
166C2×C22⋊C4C22×C4124yes—
167C2×C22⋊C4C22×C4124yes—
168C2×C22⋊C4C22×C4112yes—
169C2×C22⋊C4C22×C4112yes—
170C2×C42—26no—
171C2×C42—26no—
172C2×C42—212no—
173C2×C42—46no—
174C2×C42—46no—
175C2×C42—46no—
176C2×C42—46no—
177#95—212no—
178#95—212no—
179#95—43no—
180#95—46no—
181#95—46no—
182#95—46no—
183#95—43no—
184#95—43no—
185#95—43no—
186#95—46no—
187#95—46no—
188#95—46no—
189#95—46no—
190#95—46no—
191#95C22×C414yes—
192#95C22×C4112yes—
193C24—23no—
194C24—23no—
195C24—23no—
196C24—23no—
197C24—26no—
198C24—26no—
199C24—22no—
200C24—22no—
201C24—26no—
202C24—26no—
203C24—28no—
204C24—412no—
205C24—412no—
206C24—44no—
207C24—412no—
208C24—86no—
209C24—86no—
210C24—86no—
211C24—86no—
212C24—86no—
213C24—86no—
214C24—812no—
215C24C23×C414yes—
216C24C23×C412yes—
217C242−1+412yes—
218C242−1+412yes—
219C24C2511yescharacteristic
220C22×C4—21no—
221C22×C4—23no—
222C22×C4—26no—
223C22×C4—26no—
224C22×C4—26no—
225C22×C4—26no—
226C22×C4—23no—
227C22×C4—23no—
228C22×C4—23no—
229C22×C4—26no—
230C22×C4—26no—
231C22×C4—26no—
232C22×C4—23no—
233C22×C4—23no—
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—23no—
243C22×C4—23no—
244C22×C4—23no—
245C22×C4—26no—
246C22×C4—26no—
247C22×C4—26no—
248C22×C4—26no—
249C22×C4—23no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—26no—
253C22×C4—26no—
254C22×C4—23no—
255C22×C4—26no—
256C22×C4—26no—
257C22×C4—26no—
258C22×C4—412no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—86no—
262C22×C4—86no—
263C22×C4—86no—
264C22×C4C23×C416yes—
265C22×C4C23×C4112yes—
266C22×C4C23×C416yes—
267C22×C4H(5,𝔽2)16yes—
268C22×C4H(5,𝔽2)16yes—
269C4⋊C4—412no—
270C4⋊C4—412no—
271C4⋊C4—424no—
272C4⋊C4—412no—
273C4⋊C4—412no—
274C4⋊C4—812no—
275C22⋊C4—412no—
276C22⋊C4—412no—
277C22⋊C4—424no—
278C22⋊C4—424no—
279C22⋊C4—412no—
280C22⋊C4—48no—
281C22⋊C4—48no—
282C22⋊C4—812no—
283C22⋊C4—86no—
284C22⋊C4—812no—
285C42—412no—
286C42—412no—
287C42—812no—
288C23—26no—
289C23—26no—
290C23—26no—
291C23—26no—
292C23—26no—
293C23—26no—
294C23—24no—
295C23—212no—
296C23—212no—
297C23—41no—
298C23—43no—
299C23—43no—
300C23—43no—
301C23—46no—
302C23—416no—
303C23—42no—
304C23—46no—
305C23—46no—
306C23—46no—
307C23—46no—
308C23—46no—
309C23—46no—
310C23—46no—
311C23—46no—
312C23—46no—
313C23—824no—
314C23—824no—
315C23—88no—
316C23—824no—
317C23—824no—
318C23—88no—
319C23#51914yes—
320C23C24×C411yescharacteristic
321C23C2×H(5,𝔽2)13yes—
322C23C2×H(5,𝔽2)13yes—
323C23C2×2−1+411yescharacteristic
324C23C2×2−1+411yescharacteristic
325C23#58513yes—
326C23#58513yes—
327C2×C4—26no—
328C2×C4—26no—
329C2×C4—212no—
330C2×C4—26no—
331C2×C4—212no—
332C2×C4—26no—
333C2×C4—212no—
334C2×C4—212no—
335C2×C4—212no—
336C2×C4—26no—
337C2×C4—212no—
338C2×C4—43no—
339C2×C4—43no—
340C2×C4—43no—
341C2×C4—46no—
342C2×C4—43no—
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—41no—
352C2×C4—42no—
353C2×C4—46no—
354C2×C4—43no—
355C2×C4—43no—
356C2×C4—43no—
357C2×C4—46no—
358C2×C4—43no—
359C2×C4—43no—
360C2×C4—46no—
361C2×C4—46no—
362C2×C4—46no—
363C2×C4—46no—
364C2×C4—46no—
365C2×C4—46no—
366C2×C4—46no—
367C2×C4—46no—
368C2×C4—46no—
369C2×C4—88no—
370C2×C4—88no—
371C2×C4#56014yes—
372C22—212no—
373C22—42no—
374C22—46no—
375C22—42no—
376C22—46no—
377C22—46no—
378C22—42no—
379C22—44no—
380C22—412no—
381C22—412no—
382C22—832no—
383C22—82no—
384C22—86no—
385C22—86no—
386C22—82no—
387C22C2×#51911yescharacteristic
388C22C4×H(5,𝔽2)13yes—
389C22C4×H(5,𝔽2)13yes—
390C22C2×#56011yescharacteristic
391C22#352913yes—
392C22#353313yes—
393C22#353513yes—
394C22#353613yes—
395C22#353713yes—
396C22#356113yes—
397C22#356216yes—
398C22#356413yes—
399C4—24no—
400C4—46no—
401C4—46no—
402C4—412no—
403C4—46no—
404C4—81no—
405C4—83no—
406C4—86no—
407C4—82no—
408C2—44no—
409C2—84no—
410C2C4×#56011yescharacteristic
411C2#3443313yes—
412C2#3451813yes—
413C2#5274211yescharacteristic
414C2#5275113yes—
415C2#5289113yes—
416C2#5298811yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC26C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14653C2semidirect
3. #14636C2semidirect
4. #14562C2semidirect
5. #14551C2semidirect
7. #14331C2semidirect
8. #14240C2semidirect
9. #14200C2semidirect
11. C4×#399C2semidirect
13. #2842C4semidirect
18. C2×#389C22semidirect
24. C2×C4×C4⋊C4C22semidirect
25. C2×C4×C4⋊C4C22semidirect
27. #1445C22semidirect
28. #1445C22semidirect
29. #1445C22semidirect
30. #1444C22semidirect
31. #1443C22semidirect
32. #1442C22semidirect
33. #1441C22semidirect
34. #1441C22semidirect
44. #1436C22semidirect
45. #1436C22semidirect
46. #1436C22semidirect
47. #1434C22semidirect
50. C4×#95C22semidirect
76. #396C2×C4semidirect
77. #396C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28422
#146532
#146362
#145622
#145512
#143662
#143312
#142402
#142002
#139912
C4×#3992