Finite Group Databasebeta

← #137640

Subgroups — #137640

Order 512.

478 conjugacy classes of subgroups.

The subgroup lattice has 333 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2162C211yesmaximal, characteristic, complemented
2C8×#121C211yesmaximal, characteristic
3#12783C211yesmaximal, characteristic, complemented
4C2×D4⋊C16C212yesmaximal, complemented
5#7167C212yesmaximal, complemented
6C2×C8×D4—21no—
7C2×C8×D4C2211yescharacteristic
8C22×C4×C8C2211yescharacteristic
9C4×#121C411yescharacteristic
10#2162—24no—
11C2×C4⋊C16—21no—
12C2×C4⋊C16C2212yes—
13C2×C22⋊C16—21no—
14C2×C22⋊C16—21no—
15C2×C4×C16C2212yes—
16#1954C411yescharacteristic
17#1939C411yescharacteristic
18#1923—21no—
19#1869—21no—
20#1797C411yescharacteristic
21C8×C4⋊C4C2211yescharacteristic
22C8×C22⋊C4—21no—
23#1374—21no—
24D4⋊C16—24no—
25C23×C8—21no—
26C23×C8D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×#121C811yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4C2×C411yescharacteristic
31C22×C42C2×C411yescharacteristic
32C22×C16—22no—
33C22×C16—22no—
34C8×D4—22no—
35C8×D4—22no—
36C8×D4—44no—
37C2×C4⋊C8—21no—
38C2×C4⋊C8C2×C411yescharacteristic
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C22⋊C8—21no—
41C2×C22⋊C8—21no—
42C2×C22⋊C8—21no—
43C2×C22⋊C8—21no—
44C2×C4×C8—22no—
45C2×C4×C8—22no—
46C2×C4×C8—21no—
47C2×C4×C8C2×C411yescharacteristic
48C2×C4×C8D411yescharacteristic
49C2×C4×C8D411yescharacteristic
50C2×C4×C8C2311yescharacteristic
51#390C811yescharacteristic
52#389C811yescharacteristic
53#388—21no—
54#386—21no—
55#380C811yescharacteristic
56C4×C4⋊C4C2×C411yescharacteristic
57C4×C22⋊C4—21no—
58C4⋊C16—24no—
59C4⋊C16—22no—
60C4⋊C16—22no—
61C22⋊C16—24no—
62C22⋊C16—42no—
63C4×C16—22no—
64C4×C16D414yescomplemented
65#336—21no—
66#336—21no—
67#336—21no—
68C22×D4—21no—
69C22×D4C2×C811yescharacteristic
70C23×C4—21no—
71C23×C4C22⋊C411yescharacteristic
72C23×C4C2×C811yescharacteristic
73C23×C4M1611yescharacteristic
74C22×C8—21no—
75C22×C8—22no—
76C22×C8—21no—
77C22×C8—21no—
78C22×C8—22no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8—42no—
85C22×C8—41no—
86C22×C8—42no—
87C22×C8—41no—
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90C22×C8C2×D411yescharacteristic
91C22×C8C2×D411yescharacteristic
92#121—22no—
93#121—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—42no—
97C4×D4—42no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4C2×C811yescharacteristic
100C2×C4⋊C4C2×C811yescharacteristic
101C2×C4⋊C4C2×C811yescharacteristic
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42—21no—
110C2×C42C22⋊C411yescharacteristic
111C2×C42C2×C811yescharacteristic
112C2×C42M1611yescharacteristic
113C2×C42C22×C411yescharacteristic
114C2×C16—24no—
115C2×C16—42no—
116C2×C16—42no—
117C2×C16—42no—
118C2×C16—42no—
119C2×C16—42no—
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—42no—
123C22⋊C8—42no—
124C22⋊C8—42no—
125C22⋊C8—42no—
126C22⋊C8—41no—
127C22⋊C8—41no—
128C4×C8—21no—
129C4×C8—22no—
130C4×C8—21no—
131C4×C8—22no—
132C4×C8—42no—
133C4×C8C2×D412yes—
134C4×C8#4112yes—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—21no—
139C24C22⋊C811yescharacteristic
140C2×D4—24no—
141C2×D4—22no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4—42no—
145C2×D4—42no—
146C2×D4M3212yes—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—22no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4C22⋊C811yescharacteristic
170C22×C4C22⋊C811yescharacteristic
171C22×C4C22⋊C811yescharacteristic
172C22×C4C2×C22⋊C411yescharacteristic
173C22×C4C22×C811yescharacteristic
174C22×C4C2×M1611yescharacteristic
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—42no—
184C2×C8—42no—
185C2×C8—42no—
186C2×C8—42no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—82no—
194C2×C8—82no—
195C2×C8#12011yescharacteristic
196C2×C8#12112yes—
197C2×C8#12311yescharacteristic
198C2×C8AGL(1,ℤ/8)12yes—
199C2×C8AGL(1,ℤ/8)11yescharacteristic
200C2×C8#13711yescharacteristic
201C4⋊C4—21no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—42no—
205C4⋊C4M3212yes—
206C22⋊C4—24no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—42no—
211C42—21no—
212C42—21no—
213C42—22no—
214C42—22no—
215C42—41no—
216C42—41no—
217C42C4×D412yes—
218C42#13112yes—
219C16—44no—
220C16—82no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23—41no—
234C23#35012yes—
235C23C2×C22⋊C811yescharacteristic
236D4—24no—
237D4—48no—
238D4—42no—
239C2×C4—28no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—81no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4#35012yes—
270C2×C4#38611yescharacteristic
271C2×C4#40811yescharacteristic
272C2×C4#41711yescharacteristic
273C2×C4#41811yescharacteristic
274C2×C4#42111yescharacteristic
275C2×C4#42111yescharacteristic
276C2×C4C8×D412yes—
277C2×C4#43512yes—
278C2×C4C2×M3211yescharacteristic
279C2×C4C2×M3211yescharacteristic
280C2×C4#50412yes—
281C8—21no—
282C8—21no—
283C8—41no—
284C8—41no—
285C8—81no—
286C8—81no—
287C8#46812yes—
288C8#46912yes—
289C22—22no—
290C22—22no—
291C22—22no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22—81no—
303C22#147711yescharacteristic
304C22#147811yescharacteristic
305C22#186911yescharacteristic
306C22#211211yescharacteristic
307C22#211311yescharacteristic
308C22C2×#35011yescharacteristic
309C22C2×#35011yescharacteristic
310C4—22no—
311C4—22no—
312C4—42no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—81no—
317C4#149511yescharacteristic
318C4#149611yescharacteristic
319C4#189511yescharacteristic
320C4#189711yescharacteristic
321C4#216812yes—
322C4#216912yes—
323C2—42no—
324C2—82no—
325C2#907411yescharacteristic
326C2#1162211yescharacteristic
327C2#1162311yescharacteristic
328C2#1163011yescharacteristic
329C2#1163111yescharacteristic
330C2#1297711yescharacteristic
331C2#1298311yescharacteristic
subgroupG/H
centerC22×C8C2×D4
commutatorC22×C4C22×C8
Frattini, cosocleC2×C4×C8C23
socleC23C2×C22⋊C8
prime radicalC2×#121C8
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C8 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2162C2semidirect
3. #12783C2semidirect
4. C2×D4⋊C16C2semidirect
5. #7167C2semidirect
64. C4×C16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#21622
C8×#1212
#127832
C2×D4⋊C162
#71672