Finite Group Databasebeta

← #134403

Subgroups — #134403

Order 512.

398 conjugacy classes of subgroups.

The subgroup lattice has 301 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21275C211yesmaximal, characteristic
2C4×C4⋊C16C211yesmaximal, characteristic, complemented
3C2×#1331C212yesmaximal
4#8806C211yesmaximal, characteristic, complemented
5#7171C212yesmaximal, complemented
6C2×C4×D8C2211yescharacteristic
7#2293—21no—
8C2×C4⋊C16—21no—
9C2×C4×C16C2212yes—
10#1970C411yescharacteristic
11#1970C411yescharacteristic
12#1942C411yescharacteristic
13#1942C411yescharacteristic
14#1934—21no—
15#1934—21no—
16#1842—21no—
17C4×#107C2211yescharacteristic
18C4×#102—21no—
19C42×C8C2211yescharacteristic
20C2×#335—21no—
21C2×#335C2212yes—
22#1374—21no—
23#1331—24no—
24C22×D8C2×C411yescharacteristic
25C2×C4×D4—21no—
26C2×#117—41no—
27C22×C16—22no—
28C4×D8—21no—
29C4×D8—21no—
30C4×D8D412yes—
31C4×D8Q812yes—
32C2×#107C2×C411yescharacteristic
33C2×#107C2×C411yescharacteristic
34C2×#107C2×C411yescharacteristic
35C2×C4⋊C8—22no—
36C2×C4⋊C8—41no—
37C2×C4⋊C8—41no—
38C2×#102—21no—
39C2×#102—21no—
40C2×#102—21no—
41C2×C4×C8C2×C411yescharacteristic
42C2×C4×C8C2×C411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8D411yescharacteristic
45C2×C4×C8D411yescharacteristic
46C2×C4×C8C2311yescharacteristic
47#390—21no—
48#390—21no—
49#389—21no—
50#389—21no—
51#388—41no—
52#388—41no—
53C4×C4⋊C4—21no—
54C4×C22⋊C4—41no—
55C43D411yescharacteristic
56C4⋊C16—24no—
57C4×C16—22no—
58C4×C16—22no—
59#340—21no—
60#340—21no—
61#340—21no—
62#335—22no—
63#335—22no—
64#335—42no—
65C22×D4—21no—
66C23×C4—41no—
67C2×D8—21no—
68C2×D8—22no—
69C2×D8—22no—
70C2×D8C4⋊C412yes—
71C22×C8—42no—
72C22×C8C22⋊C411yescharacteristic
73C22×C8C22⋊C411yescharacteristic
74C22×C8C22×C411yescharacteristic
75C22×C8C2×D411yescharacteristic
76C4×D4—22no—
77C4×D4—22no—
78C4×D4—41no—
79C4×D4—41no—
80#117—82no—
81#117—82no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—21no—
85C2×C4⋊C4—41no—
86C2×C4⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C22⋊C4—41no—
92C2×C42—41no—
93C2×C42C22⋊C411yescharacteristic
94C2×C42C22⋊C411yescharacteristic
95C2×C42D811yescharacteristic
96C2×C42D811yescharacteristic
97C2×C42SD1611yescharacteristic
98C2×C42SD1611yescharacteristic
99C2×C42C2×D411yescharacteristic
100C2×C16—24no—
101C2×C16—42no—
102C2×C16—42no—
103#107—21no—
104#107—22no—
105#107—22no—
106#107C4⋊C412yes—
107C4⋊C8—42no—
108C4⋊C8—42no—
109C4⋊C8—82no—
110C4⋊C8—82no—
111#102—22no—
112#102—22no—
113#102—42no—
114#102—42no—
115C4×C8—22no—
116C4×C8—22no—
117C4×C8—22no—
118C4×C8—21no—
119C4×C8—21no—
120C4×C8—41no—
121C4×C8C2×D411yescharacteristic
122C4×C8C2×Q811yescharacteristic
123C4×C8#4111yescharacteristic
124C4×C8#4111yescharacteristic
125#95—41no—
126#95—41no—
127#95—41no—
128C24—41no—
129C2×D4—22no—
130C2×D4—41no—
131C2×D4—41no—
132C2×D4—41no—
133C2×D4—41no—
134C2×D4—41no—
135C22×C4—41no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—41no—
139C22×C4—41no—
140C22×C4—81no—
141C22×C4—81no—
142C22×C4—81no—
143C22×C4—81no—
144C22×C4#10211yescharacteristic
145C22×C4#10211yescharacteristic
146C22×C4#10211yescharacteristic
147C22×C4#10211yescharacteristic
148C22×C4C2×C22⋊C411yescharacteristic
149C22×C4C2×D811yescharacteristic
150C22×C4C2×SD1611yescharacteristic
151D8—24no—
152D8—42no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8—82no—
161C2×C8—82no—
162C2×C8—82no—
163C2×C8#10412yes—
164C2×C8C2×C4⋊C411yescharacteristic
165C2×C8#11711yescharacteristic
166C2×C8#12111yescharacteristic
167C2×C8C22⋊Q811yescharacteristic
168C2×C8#13112yes—
169C2×C8#13112yes—
170C4⋊C4—22no—
171C4⋊C4—41no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—82no—
177C4⋊C4—82no—
178C22⋊C4—42no—
179C22⋊C4—42no—
180C22⋊C4—82no—
181C22⋊C4—82no—
182C22⋊C4—82no—
183C22⋊C4—82no—
184C42—22no—
185C42—21no—
186C42—21no—
187C42—21no—
188C42—21no—
189C42—21no—
190C42—21no—
191C42—41no—
192C42—81no—
193C42—81no—
194C42D1612yes—
195C42SD3212yes—
196C42#12111yescharacteristic
197C42C22⋊Q811yescharacteristic
198C42AGL(1,ℤ/8)11yescharacteristic
199C42#13711yescharacteristic
200C16—44no—
201C23—42no—
202C23—81no—
203C23—81no—
204C23—81no—
205C23—81no—
206C23—81no—
207C23—81no—
208C23C2×#10211yescharacteristic
209D4—42no—
210D4—42no—
211D4—81no—
212D4—81no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—82no—
228C2×C4—82no—
229C2×C4—82no—
230C2×C4—82no—
231C2×C4—81no—
232C2×C4—81no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—81no—
236C2×C4—81no—
237C2×C4#35712yes—
238C2×C4#35712yes—
239C2×C4#35912yes—
240C2×C4#38011yescharacteristic
241C2×C4#40811yescharacteristic
242C2×C4#40811yescharacteristic
243C2×C4#41711yescharacteristic
244C2×C4C2×#10411yescharacteristic
245C2×C4#42111yescharacteristic
246C2×C4#45911yescharacteristic
247C2×C4#46111yescharacteristic
248C2×C4#47411yescharacteristic
249C2×C4#47611yescharacteristic
250C2×C4C2×D1611yescharacteristic
251C2×C4C2×SD3211yescharacteristic
252C2×C4#50911yescharacteristic
253C2×C4#51011yescharacteristic
254C8—21no—
255C8—21no—
256C8—22no—
257C8—41no—
258C8—162no—
259C8#42412yes—
260C8#43312yes—
261C22—82no—
262C22—82no—
263C22—82no—
264C22—161no—
265C22—161no—
266C22—161no—
267C22—161no—
268C22#148011yescharacteristic
269C22#149211yescharacteristic
270C22#180211yescharacteristic
271C22C2×#35711yescharacteristic
272C22#213911yescharacteristic
273C22C2×#35911yescharacteristic
274C22#214311yescharacteristic
275C4—21no—
276C4—21no—
277C4—22no—
278C4—22no—
279C4—22no—
280C4—41no—
281C4—82no—
282C4—161no—
283C4—161no—
284C4#148311yescharacteristic
285C4#149511yescharacteristic
286C4#179811yescharacteristic
287C4#180611yescharacteristic
288C4#220611yescharacteristic
289C4#220811yescharacteristic
290C4#222411yescharacteristic
291C4#222611yescharacteristic
292C2—162no—
293C2#856711yescharacteristic
294C2#1164511yescharacteristic
295C2#1164811yescharacteristic
296C2#1167111yescharacteristic
297C2#1167411yescharacteristic
298C2#1281111yescharacteristic
299C2#1281511yescharacteristic
subgroupG/H
centerC22×C4C2×D8
commutatorC22×C8C22×C4
Frattini, cosocleC2×C4×C8C23
socleC23C2×#102
prime radicalC22×D8C2×C4
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×C4⋊C16C2semidirect
4. #8806C2semidirect
5. #7171C2semidirect

Maximal subgroups

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

subgroupindex
#212752
C4×C4⋊C162
C2×#13312
#88062
#71712