Finite Group Databasebeta

← #21014

Subgroups — #21014

Order 256.

874 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3449C211yesmaximal, characteristic
2#3428C211yesmaximal, characteristic, complemented
3C2×#410C211yesmaximal, characteristic, complemented
4C2×#410C212yesmaximal, complemented
5#1913C218yesmaximal, complemented
6#1740C212yesmaximal, complemented
7C22×#41C2211yescharacteristic
8C22×#41C2211yescharacteristic
9#537—22no—
10#536—21no—
11#536C412yescomplemented
12#535—21no—
13#535C412yescomplemented
14#534C2214yes—
15#532—22no—
16#525—22no—
17#525C414yescomplemented
18#520—24no—
19#519—21no—
20#518—21no—
21#518C2214yes—
22C4×#41—24no—
23C2×#117—22no—
24C2×#117C2212yes—
25C2×#117C2211yescharacteristic
26C2×#117C2212yes—
27#410—22no—
28#410—22no—
29#410—24no—
30#410—24no—
31C2×#99C2214yes—
32C2×#99C2218yes—
33#342C2218yes—
34C2×#41—22no—
35C2×#41—22no—
36C2×#41—21no—
37C2×#41—21no—
38C2×#41—21no—
39C2×#41—21no—
40C2×#41—22no—
41C2×#41—22no—
42C2×#41—22no—
43C2×#41—22no—
44C2×#41—22no—
45C2×#41—22no—
46C2×#41C2×C412yes—
47C2×#41C2×C412yes—
48C2×#41D412yes—
49C2×#41D412yes—
50C22×Q8C2×C411yescharacteristic, complemented
51C22×Q8D411yescharacteristic
52C22×D4C2×C411yescharacteristic, complemented
53C22×D4D411yescharacteristic
54C22×D4C2312yes—
55C22×D4C2312yes—
56C23×C4C2×C412yescomplemented
57C23×C4D412yes—
58C23×C4C2311yescharacteristic
59C4⋊Q8—22no—
60#127—22no—
61#124—22no—
62#124—22no—
63#123—24no—
64#123C2×C418yescomplemented
65C22⋊Q8—24no—
66C22⋊Q8—24no—
67#121—24no—
68#121—28no—
69#121—24no—
70#120—24no—
71#120C2×C418yescomplemented
72C4×Q8—28no—
73C4×D4—28no—
74C4×D4—28no—
75C4×D4—216no—
76#117—24no—
77#117—24no—
78#117—22no—
79#117—24no—
80#117—22no—
81#117—21no—
82#117—21no—
83#117—22no—
84#117—22no—
85#117—22no—
86#117—22no—
87#117C2×C412yescomplemented
88#117C2×C412yescomplemented
89#117D414yescomplemented
90#117D414yescomplemented
91C2×C4⋊C4—24no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—22no—
95C2×C22⋊C4—24no—
96C2×C22⋊C4C2314yes—
97C2×C22⋊C4C2312yes—
98C2×C22⋊C4C2314yes—
99C2×C42—24no—
100C2×C42—22no—
101C2×C42—21no—
102C2×C42—22no—
103#99—28no—
104#99—216no—
105C24C22×C412yes—
106C24C2×D412yes—
107C24C2411yescharacteristic
108#41—24no—
109#41—24no—
110#41—24no—
111#41—24no—
112#41—24no—
113#41—24no—
114#41—24no—
115#41—24no—
116#41—42no—
117#41—41no—
118#41—41no—
119#41—42no—
120#41—42no—
121#41—42no—
122#41—42no—
123#41—41no—
124#41—42no—
125#41—41no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—22no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8—22no—
132C2×Q8C22⋊C412yes—
133C2×Q8C22⋊C412yes—
134C2×D4—24no—
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—24no—
140C2×D4—28no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4C22⋊C412yes—
148C2×D4C22⋊C412yes—
149C2×D4C22×C414yes—
150C2×D4C2×D414yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—28no—
156C22×C4—24no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—24no—
163C22×C4C22⋊C414yes—
164C22×C4C22⋊C414yes—
165C22×C4C22×C412yes—
166C22×C4C22×C411yescharacteristic
167C22×C4C22×C411yescharacteristic
168C22×C4C2×D412yes—
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D412yes—
174C22×C4C2×D412yes—
175C22×C4#4112yes—
176C22×C4#4112yes—
177C4⋊C4—28no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—28no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C22⋊C4—28no—
186C22⋊C4—28no—
187C22⋊C4—24no—
188C22⋊C4—22no—
189C22⋊C4—28no—
190C22⋊C4—24no—
191C22⋊C4—24no—
192C22⋊C4C22×C414yes—
193C22⋊C4C2×D418yes—
194C42—28no—
195C42—22no—
196C42—22no—
197C42—24no—
198C42—24no—
199C42—42no—
200C42—42no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—24no—
205C23—81no—
206C23—82no—
207C23—82no—
208C23C2×C22⋊C414yes—
209C23C23×C411yescharacteristic
210C23C22×D411yescharacteristic
211C23C22×D411yescharacteristic
212C23C22×D412yes—
213C23C2×#4112yes—
214Q8—24no—
215Q8—24no—
216Q8—41no—
217Q8—41no—
218Q8—42no—
219D4—28no—
220D4—28no—
221D4—24no—
222D4—24no—
223D4—42no—
224D4—44no—
225D4—42no—
226D4—41no—
227D4—41no—
228D4—42no—
229D4—88no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—24no—
233C2×C4—24no—
234C2×C4—24no—
235C2×C4—24no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—28no—
240C2×C4—42no—
241C2×C4—44no—
242C2×C4—42no—
243C2×C4—44no—
244C2×C4—84no—
245C2×C4—88no—
246C2×C4—82no—
247C2×C4—84no—
248C2×C4C2×C22⋊C414yes—
249C2×C4C2×C22⋊C411yescharacteristic
250C2×C4C2×C22⋊C411yescharacteristic
251C2×C4C2×C22⋊C411yescharacteristic
252C2×C4C2×C22⋊C411yescharacteristic
253C2×C4C4×D412yes—
254C2×C4C4×D412yes—
255C2×C4C4×D412yes—
256C2×C4C4×D412yes—
257C2×C4#12012yes—
258C2×C4#12011yescharacteristic
259C2×C4#12011yescharacteristic
260C2×C4#12112yes—
261C2×C4#12112yes—
262C2×C4#12112yes—
263C2×C4#12112yes—
264C2×C4#12311yescharacteristic
265C2×C4#12312yes—
266C2×C4#12311yescharacteristic
267C22—24no—
268C22—42no—
269C22—41no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—44no—
274C22—44no—
275C22—44no—
276C22—42no—
277C22—82no—
278C22—84no—
279C22C22×C22⋊C411yescharacteristic
280C22C2×C4×D412yes—
281C22C2×#12011yescharacteristic
282C22C2×#12112yes—
283C22C2×#12311yescharacteristic
284C4—24no—
285C4—48no—
286C4—44no—
287C4—42no—
288C4—84no—
289C4#38611yescharacteristic
290C4#38611yescharacteristic
291C4#38611yescharacteristic
292C4#38612yes—
293C4#38612yes—
294C4#38611yescharacteristic
295C2—21no—
296C2—21no—
297C2—22no—
298C2—22no—
299C2—21no—
300C2—44no—
301C2—82no—
302C2C2×#38611yescharacteristic
subgroupG/H
centerC4#386
commutatorC23C23×C4
Frattini, cosocleC24C24
socleC2C2×#386
prime radical#3449C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #3428C2semidirect
3. C2×#410C2semidirect
4. C2×#410C2semidirect
5. #1913C2semidirect
6. #1740C2semidirect
11. #536C4semidirect
13. #535C4semidirect
17. #525C4semidirect
50. C22×Q8C2×C4semidirect
52. C22×D4C2×C4semidirect
56. C23×C4C2×C4semidirect
64. #123C2×C4semidirect
71. #120C2×C4semidirect
87. #117C2×C4semidirect
88. #117C2×C4semidirect
89. #117D4semidirect
90. #117D4semidirect

Maximal subgroups

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

subgroupindex
#34492
#34282
C2×#4102
#19132
#17402