Finite Group Databasebeta

← #134405

Subgroups — #134405

Order 512.

388 conjugacy classes of subgroups.

The subgroup lattice has 287 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#8807C211yesmaximal, 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#1831—21no—
17C4×#107C2211yescharacteristic
18C4×#102—21no—
19C42×C8C2211yescharacteristic
20C2×#335C2212yes—
21C2×#334—21no—
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×#102—21no—
37C2×#102—21no—
38C2×#102—21no—
39C2×C4×C8C2×C411yescharacteristic
40C2×C4×C8C2×C411yescharacteristic
41C2×C4×C8D411yescharacteristic
42C2×C4×C8D411yescharacteristic
43C2×C4×C8D411yescharacteristic
44C2×C4×C8C2311yescharacteristic
45#390—21no—
46#390—21no—
47#389—21no—
48#389—21no—
49#388—41no—
50#388—41no—
51C4×C4⋊C4—21no—
52C4×C22⋊C4—41no—
53C43D411yescharacteristic
54C4⋊C16—24no—
55C4×C16—22no—
56C4×C16—22no—
57#340—21no—
58#340—21no—
59#340—21no—
60#336—41no—
61#336—41no—
62#335—22no—
63#335—22no—
64#334—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—84no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4—21no—
84C2×C4⋊C4—41no—
85C2×C4⋊C4—41no—
86C2×C22⋊C4—41no—
87C2×C22⋊C4—41no—
88C2×C22⋊C4—41no—
89C2×C22⋊C4—41no—
90C2×C22⋊C4—41no—
91C2×C42—41no—
92C2×C42C22⋊C411yescharacteristic
93C2×C42C22⋊C411yescharacteristic
94C2×C42D811yescharacteristic
95C2×C42D811yescharacteristic
96C2×C42SD1611yescharacteristic
97C2×C42SD1611yescharacteristic
98C2×C42C2×D411yescharacteristic
99C2×C16—24no—
100C2×C16—42no—
101C2×C16—42no—
102#107—21no—
103#107—22no—
104#107—22no—
105#107C4⋊C412yes—
106C4⋊C8—42no—
107C4⋊C8—42no—
108#102—22no—
109#102—22no—
110#102—42no—
111#102—42no—
112C4×C8—22no—
113C4×C8—22no—
114C4×C8—22no—
115C4×C8—21no—
116C4×C8—21no—
117C4×C8—41no—
118C4×C8C2×D411yescharacteristic
119C4×C8C2×Q811yescharacteristic
120C4×C8#4111yescharacteristic
121C4×C8#4111yescharacteristic
122#95—41no—
123#95—41no—
124#95—41no—
125C24—41no—
126C2×D4—22no—
127C2×D4—41no—
128C2×D4—41no—
129C2×D4—41no—
130C2×D4—41no—
131C2×D4—41no—
132C22×C4—41no—
133C22×C4—42no—
134C22×C4—42no—
135C22×C4—41no—
136C22×C4—41no—
137C22×C4—82no—
138C22×C4—82no—
139C22×C4#10211yescharacteristic
140C22×C4#10211yescharacteristic
141C22×C4#10211yescharacteristic
142C22×C4#10211yescharacteristic
143C22×C4C2×C22⋊C411yescharacteristic
144C22×C4C2×D811yescharacteristic
145C22×C4C2×SD1611yescharacteristic
146D8—24no—
147D8—42no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—82no—
156C2×C8—82no—
157C2×C8—82no—
158C2×C8#10412yes—
159C2×C8C2×C4⋊C411yescharacteristic
160C2×C8#11711yescharacteristic
161C2×C8#12111yescharacteristic
162C2×C8C22⋊Q811yescharacteristic
163C2×C8#13112yes—
164C2×C8#13112yes—
165C4⋊C4—22no—
166C4⋊C4—41no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—82no—
172C4⋊C4—82no—
173C22⋊C4—42no—
174C22⋊C4—42no—
175C22⋊C4—82no—
176C22⋊C4—82no—
177C22⋊C4—82no—
178C22⋊C4—82no—
179C42—22no—
180C42—21no—
181C42—21no—
182C42—21no—
183C42—21no—
184C42—22no—
185C42—22no—
186C42—41no—
187C42—82no—
188C42#12111yescharacteristic
189C42C22⋊Q811yescharacteristic
190C42AGL(1,ℤ/8)11yescharacteristic
191C42#13711yescharacteristic
192C16—44no—
193C23—42no—
194C23—81no—
195C23—81no—
196C23—81no—
197C23—81no—
198C23—81no—
199C23—81no—
200C23C2×#10211yescharacteristic
201D4—42no—
202D4—42no—
203D4—81no—
204D4—81no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—81no—
220C2×C4—81no—
221C2×C4—84no—
222C2×C4—84no—
223C2×C4—81no—
224C2×C4—81no—
225C2×C4—81no—
226C2×C4—81no—
227C2×C4—81no—
228C2×C4—81no—
229C2×C4#35912yes—
230C2×C4#38011yescharacteristic
231C2×C4#40811yescharacteristic
232C2×C4#40811yescharacteristic
233C2×C4#41711yescharacteristic
234C2×C4C2×#10411yescharacteristic
235C2×C4#42111yescharacteristic
236C2×C4#45911yescharacteristic
237C2×C4#46111yescharacteristic
238C2×C4#47411yescharacteristic
239C2×C4#47611yescharacteristic
240C2×C4#50812yes—
241C2×C4#50912yes—
242C8—21no—
243C8—21no—
244C8—22no—
245C8—41no—
246C8—162no—
247C8#42412yes—
248C8#43312yes—
249C22—82no—
250C22—82no—
251C22—82no—
252C22—161no—
253C22—161no—
254C22—161no—
255C22—161no—
256C22#148011yescharacteristic
257C22#149211yescharacteristic
258C22#180211yescharacteristic
259C22#213811yescharacteristic
260C22#214011yescharacteristic
261C22C2×#35911yescharacteristic
262C22#214311yescharacteristic
263C4—21no—
264C4—21no—
265C4—22no—
266C4—41no—
267C4—41no—
268C4—41no—
269C4—82no—
270C4—161no—
271C4—161no—
272C4#148311yescharacteristic
273C4#149511yescharacteristic
274C4#179811yescharacteristic
275C4#180611yescharacteristic
276C4#221012yes—
277C4#222812yes—
278C2—162no—
279C2#856711yescharacteristic
280C2#1164711yescharacteristic
281C2#1164911yescharacteristic
282C2#1167311yescharacteristic
283C2#1167511yescharacteristic
284C2#1281311yescharacteristic
285C2#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. #8807C2semidirect
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
#88072
#71712