Finite Group Databasebeta

← #413114

Subgroups — #413114

Order 512.

1344 conjugacy classes of subgroups; only the 298 normal subgroups are shown.

normal subgroups only

This group has 1344 conjugacy classes of subgroups — too many to draw. Only its 298 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1968C213yesmaximal
2C2×#1941C213yesmaximal, complemented
3#15189C211yesmaximal, characteristic, complemented
4#15141C211yesmaximal, characteristic
5#11067C213yesmaximal, complemented
6#11066C213yesmaximal
7C8×#95C211yesmaximal, characteristic, complemented
8C23×SD16C2211yescharacteristic
9C22×#106C2213yes—
10C22×#103C2213yes—
11C22×#102C2213yes—
12C22×C4×C8C2213yes—
13C2×#396—21no—
14C2×#391C2213yes—
15C2×#390C2213yes—
16C2×#389C2213yes—
17C2×#389C2213yes—
18#1968—212no—
19#1941—212no—
20C2×#340C2213yes—
21C2×#340C2213yes—
22C2×#336C2213yes—
23#1443—21no—
24#1441—23no—
25#1439—23no—
26C4×#95C2211yescharacteristic
27C23×Q8C2311yescharacteristic
28C23×D4C2311yescharacteristic
29C22×SD16—26no—
30C22×SD16—26no—
31C22×SD16Q814yes—
32C23×C8C2313yes—
33C23×C8C2311yescharacteristic
34C22×C4⋊C4—23no—
35C22×C4⋊C4—23no—
36C22×C4⋊C4C2313yes—
37C22×C4⋊C4C2313yes—
38C22×C22⋊C4—23no—
39C22×C22⋊C4—23no—
40C22×C42C2313yes—
41C2×#106—26no—
42C2×#106—26no—
43C2×#106D4112yes—
44C2×#103—26no—
45C2×#103—26no—
46C2×#103—43no—
47C2×#102—26no—
48C2×#102—26no—
49C2×#102—43no—
50C2×C4×C8—26no—
51C2×C4×C8—26no—
52C2×C4×C8—26no—
53#396—44no—
54#391—26no—
55#391—43no—
56#390—26no—
57#390—43no—
58#389—26no—
59#389—26no—
60#389—43no—
61#389—43no—
62C2×#95—23no—
63C2×#95—23no—
64C2×#95—23no—
65C2×#95—23no—
66C2×#95D413yes—
67C2×#95D411yescharacteristic
68#340—26no—
69#340—26no—
70#336—26no—
71C25—21no—
72C22×Q8—21no—
73C22×Q8—23no—
74C22×Q8—23no—
75C22×Q8—23no—
76C22×Q8—23no—
77C22×Q8C2×Q812yes—
78C22×D4—23no—
79C22×D4—23no—
80C22×D4—23no—
81C22×D4—23no—
82C22×D4—21no—
83C22×D4C2×Q812yes—
84C23×C4—23no—
85C23×C4—23no—
86C23×C4—21no—
87C23×C4C2×D413yes—
88C23×C4C2×D413yes—
89C23×C4C2411yescharacteristic
90C2×SD16—224no—
91C2×SD16—412no—
92C2×SD16—44no—
93C22×C8—23no—
94C22×C8—23no—
95C22×C8—23no—
96C22×C8—23no—
97C22×C8—23no—
98C22×C8—23no—
99C22×C8C2×D416yes—
100C22×C8C2×Q812yes—
101C22×C8#4116yes—
102C22×C8#4116yes—
103C2×C4⋊C4—23no—
104C2×C4⋊C4—23no—
105C2×C4⋊C4—23no—
106C2×C4⋊C4—23no—
107C2×C4⋊C4—23no—
108C2×C4⋊C4—23no—
109C2×C4⋊C4—23no—
110C2×C4⋊C4—23no—
111C2×C4⋊C4—23no—
112C2×C4⋊C4—23no—
113C2×C4⋊C4—46no—
114C2×C4⋊C4—46no—
115C2×C4⋊C4—46no—
116C2×C4⋊C4—46no—
117C2×C4⋊C4—83no—
118C2×C4⋊C4—83no—
119C2×C4⋊C4C2×D416yes—
120C2×C4⋊C4C2×D416yes—
121C2×C22⋊C4—46no—
122C2×C22⋊C4—46no—
123C2×C22⋊C4—46no—
124C2×C22⋊C4—46no—
125C2×C22⋊C4—83no—
126C2×C22⋊C4—83no—
127C2×C42—23no—
128C2×C42—23no—
129C2×C42—23no—
130C2×C42—23no—
131C2×C42—23no—
132C2×C42—23no—
133#106—224no—
134#103—412no—
135#102—412no—
136C4×C8—412no—
137#95—26no—
138#95—46no—
139#95—46no—
140#95—46no—
141#95—46no—
142#95SD1614yescomplemented
143C24—22no—
144C24—43no—
145C24—43no—
146C24—41no—
147C24—43no—
148C24—43no—
149C24—41no—
150C24C22×D411yescharacteristic
151C2×Q8—26no—
152C2×Q8—26no—
153C2×Q8—43no—
154C2×Q8—43no—
155C2×Q8—43no—
156C2×Q8—41no—
157C2×Q8—43no—
158C2×Q8—43no—
159C2×Q8—43no—
160C2×Q8—43no—
161C2×D4—26no—
162C2×D4—26no—
163C2×D4—43no—
164C2×D4—41no—
165C2×D4—43no—
166C2×D4—43no—
167C2×D4—43no—
168C2×D4—43no—
169C2×D4—43no—
170C2×D4—43no—
171C22×C4—23no—
172C22×C4—23no—
173C22×C4—23no—
174C22×C4—23no—
175C22×C4—23no—
176C22×C4—23no—
177C22×C4—23no—
178C22×C4—23no—
179C22×C4—23no—
180C22×C4—23no—
181C22×C4—23no—
182C22×C4—23no—
183C22×C4—26no—
184C22×C4—26no—
185C22×C4—22no—
186C22×C4—43no—
187C22×C4—43no—
188C22×C4—43no—
189C22×C4—43no—
190C22×C4—43no—
191C22×C4—43no—
192C22×C4—43no—
193C22×C4—43no—
194C22×C4—43no—
195C22×C4—43no—
196C22×C4—43no—
197C22×C4—43no—
198C22×C4—43no—
199C22×C4—43no—
200C22×C4C2×SD1616yes—
201C22×C4#13516yes—
202C22×C4C22×D413yes—
203C22×C4C22×Q811yescharacteristic
204C22×C4C2×#4113yes—
205C22×C4C2×#4113yes—
206C22×C4H(5,𝔽2)13yes—
207C22×C42−1+411yescharacteristic
208SD16—416no—
209C2×C8—26no—
210C2×C8—26no—
211C2×C8—26no—
212C2×C8—43no—
213C2×C8—43no—
214C2×C8—43no—
215C2×C8—43no—
216C2×C8#121112yes—
217C2×C8C22⋊Q8112yes—
218C2×C8#12614yes—
219C4⋊C4—26no—
220C4⋊C4—26no—
221C4⋊C4—26no—
222C4⋊C4—26no—
223C4⋊C4—43no—
224C4⋊C4—43no—
225C4⋊C4—43no—
226C4⋊C4—43no—
227C4⋊C4—812no—
228C4⋊C4—812no—
229C22⋊C4—812no—
230C22⋊C4—812no—
231C42—43no—
232C42—43no—
233C42—43no—
234C42—43no—
235C23—42no—
236C23—46no—
237C23—46no—
238C23—81no—
239C23—83no—
240C23—83no—
241C23—83no—
242C23—83no—
243C23—83no—
244C23—83no—
245C23—83no—
246C23—83no—
247C23—83no—
248C23#53213yes—
249C23D4213yes—
250C23D4×Q811yescharacteristic
251C23C22×SD1611yescharacteristic
252C23C2×#13513yes—
253C23#57711yescharacteristic
254C23#57713yes—
255Q8—42no—
256Q8—46no—
257Q8—81no—
258Q8—83no—
259D4—42no—
260D4—46no—
261D4—81no—
262D4—83no—
263C2×C4—26no—
264C2×C4—26no—
265C2×C4—26no—
266C2×C4—26no—
267C2×C4—26no—
268C2×C4—26no—
269C2×C4—43no—
270C2×C4—43no—
271C2×C4—43no—
272C2×C4—43no—
273C2×C4—43no—
274C2×C4—43no—
275C2×C4—43no—
276C2×C4—43no—
277C2×C4—46no—
278C2×C4—46no—
279C2×C4—46no—
280C2×C4—46no—
281C2×C4—46no—
282C2×C4—46no—
283C2×C4—46no—
284C2×C4—81no—
285C2×C4—83no—
286C2×C4—83no—
287C2×C4—83no—
288C2×C4—83no—
289C2×C4—83no—
290C2×C4—83no—
291C2×C4—83no—
292C2×C4—83no—
293C2×C4—83no—
294C2×C4C2×#12113yes—
295C2×C4C2×C22⋊Q813yes—
296C2×C4C2×#12611yescharacteristic
297C2×C4#52913yes—
298C2×C4#54013yes—
299C2×C4#54213yes—
300C2×C4#54613yes—
301C2×C4#54813yes—
302C2×C4#55013yes—
303C2×C4#55413yes—
304C8—46no—
305C8#39618yescomplemented
306C22—82no—
307C22—86no—
308C22—86no—
309C22—163no—
310C22—163no—
311C22—161no—
312C22—161no—
313C22#242813yes—
314C22#245213yes—
315C22#286011yescharacteristic
316C22#316713yes—
317C22#316813yes—
318C22#317313yes—
319C22#317413yes—
320C22D4×SD1613yes—
321C22#328213yes—
322C22#328213yes—
323C22#328413yes—
324C22Q8×SD1611yescharacteristic
325C22#338113yes—
326C4—46no—
327C4—46no—
328C4—82no—
329C4—86no—
330C4—86no—
331C4C2×#39611yescharacteristic
332C4#242613yes—
333C4#245413yes—
334C4#264411yescharacteristic
335C2—162no—
336C2#1514111yescharacteristic
337C2#2541013yes—
338C2#2541113yes—
339C2#2648013yes—
340C2#2648113yes—
341C2#3326511yescharacteristic
342C2#3327111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1941C2semidirect
3. #15189C2semidirect
5. #11067C2semidirect
7. C8×#95C2semidirect
142. #95SD16semidirect
305. C8#396semidirect

Maximal subgroups

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

subgroupindex
C2×#19682
C2×#19412
#151892
#151412
#110672
#110662
C8×#952