Finite Group Databasebeta

← #30013

Subgroups — #30013

Order 256.

370 conjugacy classes of subgroups.

The subgroup lattice has 319 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2855C211yesmaximal, characteristic
2C4×C22⋊Q8C211yesmaximal, characteristic
3#2088C211yesmaximal, characteristic, complemented
4#2075C211yesmaximal, characteristic
5#2068C211yesmaximal, characteristic, complemented
6#2066C211yesmaximal, characteristic, complemented
7#2056C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic
9#2035C211yesmaximal, characteristic
10#2001C211yesmaximal, characteristic
11#1928C211yesmaximal, characteristic, complemented
12#1919C211yesmaximal, characteristic, complemented
13#1878C211yesmaximal, characteristic, complemented
14#1865C211yesmaximal, characteristic, complemented
15#1797C211yesmaximal, characteristic
16C22×Q16C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×C4×Q8C2211yescharacteristic
20C22×C42C2211yescharacteristic
21C2×#107C2211yescharacteristic
22C2×#106C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34#391C2211yescharacteristic
35#391C2211yescharacteristic
36#391C2211yescharacteristic
37#389C2211yescharacteristic
38#389C2211yescharacteristic
39#389C2211yescharacteristic
40#387—21no—
41#387—21no—
42#387—21no—
43#385—21no—
44#385—21no—
45#381—21no—
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48C4×C4⋊C4—21no—
49C4×C4⋊C4C2211yescharacteristic
50C4×C22⋊C4—21no—
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C22×Q8C2311yescharacteristic
60C22×Q8C2311yescharacteristic
61C23×C4D411yescharacteristic
62C23×C4D411yescharacteristic
63C23×C4C2311yescharacteristic
64C2×Q16—22no—
65C2×Q16—41no—
66C2×Q16—41no—
67C22×C8C2311yescharacteristic
68C22×C8C2311yescharacteristic
69C22×C8C2311yescharacteristic
70C22×C8C2311yescharacteristic
71C22⋊Q8—22no—
72C22⋊Q8—22no—
73C22⋊Q8—41no—
74C22⋊Q8—41no—
75C4×Q8—41no—
76C4×Q8—41no—
77C2×C4⋊C4—21no—
78C2×C4⋊C4—21no—
79C2×C4⋊C4—21no—
80C2×C4⋊C4—21no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—21no—
83C2×C4⋊C4C2311yescharacteristic
84C2×C4⋊C4C2311yescharacteristic
85C2×C4⋊C4C2311yescharacteristic
86C2×C4⋊C4C2311yescharacteristic
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—21no—
90C2×C22⋊C4—21no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C42—21no—
94C2×C42—21no—
95C2×C42—41no—
96C2×C42—41no—
97C2×C42D411yescharacteristic
98C2×C42D411yescharacteristic
99C2×C42C2311yescharacteristic
100C2×C42C2311yescharacteristic
101#107—41no—
102#106—22no—
103#106—41no—
104C4⋊C8—41no—
105C4⋊C8—41no—
106#103—22no—
107#103—22no—
108#103—41no—
109#103—41no—
110#103—41no—
111#103—41no—
112C22⋊C8—41no—
113C22⋊C8—41no—
114#95—21no—
115#95—21no—
116#95—21no—
117#95—21no—
118#95—21no—
119#95—21no—
120#95—21no—
121#95—21no—
122#95—21no—
123#95—21no—
124#95—21no—
125#95—21no—
126C24C2×D411yescharacteristic
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8#4112yes—
139C22×C4—22no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—41no—
159C22×C4C2×D411yescharacteristic
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2411yescharacteristic
165Q16—44no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—21no—
178C4⋊C4—24no—
179C4⋊C4—21no—
180C4⋊C4—21no—
181C4⋊C4—21no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—21no—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—42no—
195C4⋊C4#4112yes—
196C22⋊C4—24no—
197C22⋊C4—42no—
198C22⋊C4—42no—
199C22⋊C4—42no—
200C42—21no—
201C42—21no—
202C42—21no—
203C42—21no—
204C42—41no—
205C42—41no—
206C42—42no—
207C42—42no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—21no—
212C23—21no—
213C23—21no—
214C23#13512yes—
215C23C22×D411yescharacteristic
216Q8—22no—
217Q8—22no—
218Q8—41no—
219Q8—41no—
220Q8—41no—
221Q8—41no—
222Q8—41no—
223Q8—41no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—44no—
240C2×C4—44no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
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—41no—
266C2×C4—41no—
267C2×C4#13712yes—
268C2×C4#13712yes—
269C2×C4C2×#4111yescharacteristic
270C2×C4C2×#4111yescharacteristic
271C2×C4H(5,𝔽2)11yescharacteristic
272C2×C4H(5,𝔽2)11yescharacteristic
273C2×C4H(5,𝔽2)11yescharacteristic
274C2×C42−1+411yescharacteristic
275C8—41no—
276C8—41no—
277C8—41no—
278C8—41no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22#52511yescharacteristic
287C22#53411yescharacteristic
288C22#53711yescharacteristic
289C22C2×#13511yescharacteristic
290C22C2×#13711yescharacteristic
291C22C2×#13711yescharacteristic
292C22#57511yescharacteristic
293C4—22no—
294C4—22no—
295C4—42no—
296C4—41no—
297C4—42no—
298C4—41no—
299C4—41no—
300C4—81no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4—81no—
305C4—81no—
306C4#53811yescharacteristic
307C4#53911yescharacteristic
308C4#54011yescharacteristic
309C4#54211yescharacteristic
310C2—42no—
311C2#263011yescharacteristic
312C2#310811yescharacteristic
313C2#311611yescharacteristic
314C2#319011yescharacteristic
315C2#319611yescharacteristic
316C2#325011yescharacteristic
317C2#325511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2088C2semidirect
5. #2068C2semidirect
6. #2066C2semidirect
7. #2056C2semidirect
11. #1928C2semidirect
12. #1919C2semidirect
13. #1878C2semidirect
14. #1865C2semidirect

Maximal subgroups

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

subgroupindex
#28552
C4×C22⋊Q82
#20882
#20752
#20682
#20662
#20562
#20352
#20012
#19282
#19192
#18782
#18652
#17972