Finite Group Databasebeta

← #508322

Subgroups — #508322

Order 512.

638 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2235C212yesmaximal
2#33172C211yesmaximal, characteristic
3C2×#1942C211yesmaximal, characteristic
4#13506C212yesmaximal, complemented
5#13477C214yesmaximal, complemented
6#13409C212yesmaximal, complemented
7#12993C212yesmaximal
8#12783C211yesmaximal, characteristic
9C2×#494C2211yescharacteristic
10C2×#467—21no—
11C2×#467C2212yes—
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#2857—21no—
16C2×#389—21no—
17#2235—28no—
18C2×#367C2214yes—
19C2×C4⋊C16C2212yes—
20C2×#358C2214yes—
21C2×#358C2214yes—
22C2×C22⋊C16C2212yes—
23#1969—21no—
24#1969C2212yes—
25#1956—21no—
26#1942—21no—
27#1942—21no—
28#1942C2212yes—
29#1942C2212yes—
30#1845C2211yescharacteristic
31C2×#340—21no—
32#1380C2214yes—
33#1374C2212yes—
34C22×Q16—21no—
35C22×Q16C2312yes—
36C23×C8D411yescharacteristic
37C23×C8C2311yescharacteristic
38C2×C4⋊Q8—21no—
39C2×C22⋊Q8—22no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4—41no—
43C22×C42D411yescharacteristic
44C22×C16C2314yes—
45#494—42no—
46#467—24no—
47#467—24no—
48#467—44no—
49C2×#107—21no—
50C2×#107—22no—
51C2×#107—21no—
52C2×#107—21no—
53C2×#107—42no—
54C2×#107D414yes—
55C2×#107C2312yes—
56C2×#107C2312yes—
57C2×#107C2312yes—
58C2×#103—22no—
59C2×#103—22no—
60C2×C4×C8—42no—
61C2×C4×C8D411yescharacteristic
62C2×C4×C8D411yescharacteristic
63C2×C4×C8C2311yescharacteristic
64C2×C4×C8C2311yescharacteristic
65#391—22no—
66#389—22no—
67#389—22no—
68#389—41no—
69#389—41no—
70#389—41no—
71#389—41no—
72#384—21no—
73#380—41no—
74#380—41no—
75C2×#95—41no—
76#367—28no—
77C4⋊C16—42no—
78#358—28no—
79#358—44no—
80C22⋊C16—24no—
81#340—22no—
82#340—22no—
83C22×Q8—22no—
84C23×C4—41no—
85C23×C4—41no—
86C23×C4D811yescharacteristic
87C23×C4D811yescharacteristic
88C23×C4C2×D411yescharacteristic
89C2×Q16—22no—
90C2×Q16—22no—
91C2×Q16—24no—
92C2×Q16—24no—
93C2×Q16—42no—
94C2×Q16—42no—
95C2×Q16—42no—
96C22×C8—22no—
97C22×C8—22no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—41no—
101C22×C8C2×D411yescharacteristic
102C22×C8C2×D411yescharacteristic
103C22×C8C2×D412yes—
104C22×C8C2×D411yescharacteristic
105C22×C8#4112yes—
106C22×C8C2411yescharacteristic
107C4⋊Q8—42no—
108C4⋊Q8—42no—
109C22⋊Q8—44no—
110C22⋊Q8—44no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4—42no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4—42no—
121C2×C4⋊C4—42no—
122C2×C4⋊C4—42no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—84no—
128C2×C22⋊C4—42no—
129C2×C42—41no—
130C2×C42—41no—
131C2×C42D811yescharacteristic
132C2×C42D811yescharacteristic
133C2×C42C2×D411yescharacteristic
134C2×C42C2×D411yescharacteristic
135C2×C16—24no—
136C2×C16—24no—
137C2×C16—24no—
138#107—24no—
139#107—22no—
140#107—22no—
141#107—22no—
142#107—24no—
143#107—42no—
144#107—42no—
145#107C2×D414yes—
146#103—44no—
147#103—44no—
148C4×C8—22no—
149C4×C8—22no—
150C4×C8—41no—
151C4×C8—41no—
152#95—42no—
153#95—42no—
154#95—42no—
155#95—42no—
156C24C2×D811yescharacteristic
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—42no—
162C2×Q8—42no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4—42no—
173C22×C4—44no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—82no—
178C22×C4—82no—
179C22×C4—82no—
180C22×C4—82no—
181C22×C4—82no—
182C22×C4SD3214yes—
183C22×C4#12112yes—
184C22×C4C2×D811yescharacteristic
185C22×C4C2×D811yescharacteristic
186C22×C4C2×D811yescharacteristic
187C22×C4C2×D811yescharacteristic
188C22×C4C2×D811yescharacteristic
189C22×C4AGL(1,ℤ/8)12yes—
190C22×C4C22×D411yescharacteristic
191Q16—44no—
192Q16—44no—
193Q16—84no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—24no—
199C2×C8—24no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—44no—
204C2×C8#12112yes—
205C2×C8#13712yes—
206C2×C8C22×D411yescharacteristic
207C2×C8C2×#4111yescharacteristic
208C2×C8H(5,𝔽2)12yes—
209C2×C82−1+412yes—
210C4⋊C4—24no—
211C4⋊C4—42no—
212C4⋊C4—44no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—82no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C4⋊C4—82no—
221C4⋊C4—84no—
222C4⋊C4—84no—
223C4⋊C4—84no—
224C4⋊C4—82no—
225C22⋊C4—84no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—41no—
231C42—41no—
232C16—44no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23#45912yes—
238C23C2×SD3212yes—
239C23#51012yes—
240C23C22×D811yescharacteristic
241Q8—82no—
242Q8—82no—
243Q8—82no—
244Q8—82no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—44no—
261C2×C4—44no—
262C2×C4—44no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—88no—
266C2×C4—82no—
267C2×C4—82no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—82no—
275C2×C4—164no—
276C2×C4#45912yes—
277C2×C4C2×SD3212yes—
278C2×C4C2×SD3212yes—
279C2×C4#51012yes—
280C2×C4C2×#12111yescharacteristic
281C2×C4#53411yescharacteristic
282C2×C4#53611yescharacteristic
283C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
284C2×C4C2×#13711yescharacteristic
285C2×C4#57511yescharacteristic
286C8—22no—
287C8—22no—
288C8—41no—
289C8#53712yes—
290C8#53912yes—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22—22no—
297C22#220814yes—
298C22C2×#45911yescharacteristic
299C22#318611yescharacteristic
300C22#322611yescharacteristic
301C22C22×SD3211yescharacteristic
302C22C2×#51011yescharacteristic
303C22C2×#51011yescharacteristic
304C22#341411yescharacteristic
305C4—22no—
306C4—22no—
307C4—22no—
308C4—22no—
309C4—41no—
310C4—84no—
311C4—162no—
312C4—162no—
313C4—162no—
314C4#261811yescharacteristic
315C4#303711yescharacteristic
316C4#319311yescharacteristic
317C4#323411yescharacteristic
318C2—24no—
319C2#2975411yescharacteristic
320C2C2×#220811yescharacteristic
321C2#3358111yescharacteristic
322C2#3375311yescharacteristic
323C2#3376111yescharacteristic
324C2#3379511yescharacteristic
325C2#3380211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13506C2semidirect
5. #13477C2semidirect
6. #13409C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22352
#331722
C2×#19422
#135062
#134772
#134092
#129932
#127832