Finite Group Databasebeta

← #4462764

Subgroups — #4462764

Order 512.

1940 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2645C211yesmaximal, characteristic, complemented
2C2×#2401C211yesmaximal, characteristic
3#17230C212yesmaximal, complemented
4#17213C212yesmaximal
5#17209C216yesmaximal, complemented
6#17124C216yesmaximal
7#15236C213yesmaximal
8#15221C213yesmaximal, complemented
9#15184C213yesmaximal, complemented
10#15131C213yesmaximal
11#14354C211yesmaximal, characteristic, complemented
12C22×C22⋊Q8C2211yescharacteristic
13#2645—28no—
14#2401—28no—
15C2×#401C2212yes—
16C2×#401C2211yescharacteristic
17C2×#400C2213yes—
18C2×#400C2216yes—
19C2×#397C2213yes—
20C2×#396C2211yescharacteristic
21C2×#396C2216yes—
22C2×#391C2213yes—
23C2×#391C2213yes—
24C2×#389C2213yes—
25C2×#389C2213yes—
26C2×#384C2213yes—
27C2×#382C2213yes—
28C2×#380C2213yes—
29C2×C4×C4⋊C4C2213yes—
30C2×C4×C22⋊C4C2213yes—
31C2×#376C2211yescharacteristic
32#1445C2216yes—
33#1444C2216yes—
34#1444C2212yes—
35#1444C2216yes—
36#1442C2216yes—
37#1442C2216yes—
38#1442C2216yes—
39#1442C2216yes—
40#1441C2216yes—
41#1441C2216yes—
42#1441C2216yes—
43#1441C2216yes—
44#1441C2216yes—
45#1441C2216yes—
46#1440C2216yes—
47#1440C2216yes—
48#1439C2216yes—
49C4⋊#95C2213yes—
50C4⋊#95C2213yes—
51C23×Q8C2311yescharacteristic
52C24×C4C2311yescharacteristic
53C2×C22⋊Q8—46no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2313yes—
56C22×C4⋊C4C2313yes—
57C22×C4⋊C4C2313yes—
58C22×C4⋊C4C2313yes—
59C22×C4⋊C4C2316yes—
60C22×C4⋊C4C2316yes—
61C22×C4⋊C4C2316yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2316yes—
64C22×C22⋊C4C2313yes—
65C22×C22⋊C4C2313yes—
66C22×C22⋊C4C2316yes—
67C22×C22⋊C4C2312yes—
68C22×C42C2313yes—
69C22×C42C2313yes—
70#401—28no—
71#401—24no—
72#400—224no—
73#400—212no—
74#397—212no—
75#396—24no—
76#396—412no—
77#391—212no—
78#391—46no—
79#389—212no—
80#389—46no—
81#384—46no—
82#382—46no—
83#380—46no—
84C4×C4⋊C4—212no—
85C4×C22⋊C4—212no—
86#376D418yes—
87C2×#95C2313yes—
88C2×#95C2313yes—
89C2×#95C2316yes—
90C2×#95C2313yes—
91C2×#95C2316yes—
92C2×#95C2316yes—
93C2×#95C2313yes—
94C2×#95C2316yes—
95C2×#95C2316yes—
96C2×#95C2316yes—
97C2×#95C2316yes—
98C2×#95C2316yes—
99C2×#95C2316yes—
100C2×#95C2316yes—
101C2×#95C2316yes—
102C2×#95C2316yes—
103C2×#95C2316yes—
104C2×#95C2311yescharacteristic
105C2×#95C2313yes—
106C25C2411yescharacteristic
107C22×Q8—22no—
108C22×Q8—26no—
109C22×Q8—26no—
110C23×C4—22no—
111C23×C4—26no—
112C23×C4—26no—
113C23×C4C2416yes—
114C23×C4C2416yes—
115C23×C4C2413yes—
116C23×C4C2412yes—
117C23×C4C2412yes—
118C23×C4C2411yescharacteristic
119C23×C4C2413yes—
120C23×C4C2413yes—
121C23×C4C2413yes—
122C23×C4C2411yescharacteristic
123C22⋊Q8—416no—
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—26no—
133C2×C4⋊C4—26no—
134C2×C4⋊C4—26no—
135C2×C4⋊C4—212no—
136C2×C4⋊C4—43no—
137C2×C4⋊C4—43no—
138C2×C4⋊C4—46no—
139C2×C4⋊C4—46no—
140C2×C4⋊C4—46no—
141C2×C4⋊C4—46no—
142C2×C4⋊C4—46no—
143C2×C4⋊C4—46no—
144C2×C4⋊C4—46no—
145C2×C4⋊C4—46no—
146C2×C4⋊C4—46no—
147C2×C4⋊C4—46no—
148C2×C4⋊C4—46no—
149C2×C4⋊C4—46no—
150C2×C22⋊C4—26no—
151C2×C22⋊C4—26no—
152C2×C22⋊C4—26no—
153C2×C22⋊C4—26no—
154C2×C22⋊C4—212no—
155C2×C22⋊C4—43no—
156C2×C22⋊C4—43no—
157C2×C22⋊C4—46no—
158C2×C22⋊C4—46no—
159C2×C22⋊C4—46no—
160C2×C42—26no—
161C2×C42—26no—
162C2×C42—26no—
163C2×C42—26no—
164C2×C42—26no—
165C2×C42C2×D4112yes—
166#95—26no—
167#95—26no—
168#95—26no—
169#95—26no—
170#95—212no—
171#95—212no—
172#95—212no—
173#95—212no—
174#95—46no—
175#95—46no—
176#95—46no—
177#95—46no—
178#95—46no—
179#95—46no—
180#95—46no—
181#95—46no—
182#95—46no—
183#95C2×D414yes—
184#95C2×D4112yes—
185C24—21no—
186C24—23no—
187C24—23no—
188C24—23no—
189C24—21no—
190C24—23no—
191C24H(5,𝔽2)12yes—
192C24C2511yescharacteristic
193C2×Q8—412no—
194C2×Q8—44no—
195C2×Q8—412no—
196C22×C4—23no—
197C22×C4—23no—
198C22×C4—23no—
199C22×C4—23no—
200C22×C4—23no—
201C22×C4—23no—
202C22×C4—23no—
203C22×C4—23no—
204C22×C4—23no—
205C22×C4—23no—
206C22×C4—23no—
207C22×C4—23no—
208C22×C4—23no—
209C22×C4—23no—
210C22×C4—23no—
211C22×C4—23no—
212C22×C4—23no—
213C22×C4—23no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—23no—
217C22×C4—26no—
218C22×C4—26no—
219C22×C4—23no—
220C22×C4—23no—
221C22×C4—26no—
222C22×C4—26no—
223C22×C4—26no—
224C22×C4—26no—
225C22×C4—23no—
226C22×C4—26no—
227C22×C4—26no—
228C22×C4—26no—
229C22×C4—26no—
230C22×C4—26no—
231C22×C4—23no—
232C22×C4—26no—
233C22×C4—23no—
234C22×C4—23no—
235C22×C4—26no—
236C22×C4—22no—
237C22×C4—23no—
238C22×C4—26no—
239C22×C4—26no—
240C22×C4—26no—
241C22×C4—26no—
242C22×C4—23no—
243C22×C4—26no—
244C22×C4—412no—
245C22×C4—44no—
246C22×C4—412no—
247C22×C4C22×D416yes—
248C22×C4C22×D416yes—
249C22×C4C22×D412yes—
250C22×C4H(5,𝔽2)16yes—
251C22×C42−1+416yes—
252C22×C42−1+416yes—
253C4⋊C4—412no—
254C4⋊C4—412no—
255C4⋊C4—48no—
256C4⋊C4—412no—
257C4⋊C4—412no—
258C4⋊C4—44no—
259C4⋊C4—812no—
260C4⋊C4—812no—
261C4⋊C4—812no—
262C4⋊C4—812no—
263C22⋊C4—412no—
264C22⋊C4—412no—
265C22⋊C4—48no—
266C22⋊C4—812no—
267C42—224no—
268C42—412no—
269C23—22no—
270C23—26no—
271C23—26no—
272C23—43no—
273C23—43no—
274C23—41no—
275C23—43no—
276C23—43no—
277C23—43no—
278C23—43no—
279C23—43no—
280C23—43no—
281C23—43no—
282C23C23×D411yescharacteristic
283C23C2×H(5,𝔽2)13yes—
284C23C2×H(5,𝔽2)11yescharacteristic
285C23C2×2−1+413yes—
286C23C2×2−1+413yes—
287C23#58511yescharacteristic
288C23#58513yes—
289Q8—88no—
290C2×C4—26no—
291C2×C4—26no—
292C2×C4—26no—
293C2×C4—26no—
294C2×C4—26no—
295C2×C4—26no—
296C2×C4—26no—
297C2×C4—26no—
298C2×C4—26no—
299C2×C4—43no—
300C2×C4—43no—
301C2×C4—43no—
302C2×C4—43no—
303C2×C4—43no—
304C2×C4—43no—
305C2×C4—43no—
306C2×C4—43no—
307C2×C4—43no—
308C2×C4—43no—
309C2×C4—43no—
310C2×C4—43no—
311C2×C4—43no—
312C2×C4—46no—
313C2×C4—43no—
314C2×C4—46no—
315C2×C4—43no—
316C2×C4—46no—
317C2×C4—43no—
318C2×C4—46no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—46no—
323C2×C4—43no—
324C2×C4—46no—
325C2×C4—43no—
326C2×C4—46no—
327C2×C4—46no—
328C2×C4—46no—
329C2×C4—43no—
330C2×C4—46no—
331C2×C4—46no—
332C2×C4—46no—
333C2×C4—43no—
334C2×C4—46no—
335C2×C4—46no—
336C2×C4—46no—
337C2×C4—46no—
338C2×C4—46no—
339C2×C4—43no—
340C2×C4—46no—
341C2×C4—88no—
342C2×C4#537112yes—
343C2×C4#56014yes—
344C2×C4#563112yes—
345C22—42no—
346C22—46no—
347C22—46no—
348C22—83no—
349C22—81no—
350C22—81no—
351C22—83no—
352C22C2×#53713yes—
353C22#348811yescharacteristic
354C22#348913yes—
355C22C2×#56011yescharacteristic
356C22C2×#56313yes—
357C22#354413yes—
358C22#355213yes—
359C22#355413yes—
360C22#355513yes—
361C22#355813yes—
362C22#356213yes—
363C22#356313yes—
364C22#356513yes—
365C4—46no—
366C4—46no—
367C4—46no—
368C4—86no—
369C4—81no—
370C4—82no—
371C4—82no—
372C4—83no—
373C4—86no—
374C4#282318yes—
375C2—82no—
376C2C2×#282311yescharacteristic
377C2#3728013yes—
378C2#3737813yes—
379C2#5300411yescharacteristic
380C2#5303813yes—
381C2#5308511yescharacteristic
382C2#5309213yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2645C2semidirect
3. #17230C2semidirect
5. #17209C2semidirect
8. #15221C2semidirect
9. #15184C2semidirect
11. #14354C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26452
C2×#24012
#172302
#172132
#172092
#171242
#152362
#152212
#151842
#151312
#143542