Finite Group Databasebeta

← #32962

Subgroups — #32962

Order 256.

711 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#518C211yesmaximal, characteristic
2C2×#476C211yesmaximal, characteristic, complemented
3C2×#474C211yesmaximal, characteristic, complemented
4#2628C211yesmaximal, characteristic, complemented
5#1864C212yesmaximal, complemented
6#1862C212yesmaximal, complemented
7#1854C212yesmaximal, complemented
8#1804C211yesmaximal, characteristic, complemented
9#1804C211yesmaximal, characteristic, complemented
10#1798C211yesmaximal, characteristic, complemented
11#1786C212yesmaximal
12C23×D4C2211yescharacteristic
13C2×C4⋊Q8C2211yescharacteristic, complemented
14#518—24no—
15#518—24no—
16C2×C4×D4—21no—
17C2×C4×D4—21no—
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×#117C2211yescharacteristic
23C2×#117C2211yescharacteristic
24C22×C22⋊C4—21no—
25C22×C22⋊C4—21no—
26#476—24no—
27#474—24no—
28C2×#107C2212yescomplemented
29C2×#106C2212yescomplemented
30C2×C4⋊C8C2211yescharacteristic, complemented
31C2×C4⋊C8C2211yescharacteristic, complemented
32C2×#102C2212yes—
33C2×#102C2212yes—
34C2×#102C2212yes—
35C2×#102C2212yes—
36C2×C22⋊C8C2212yes—
37#400—21no—
38#399—21no—
39#395—21no—
40#393—21no—
41#389C2212yescomplemented
42#382C2211yescharacteristic, complemented
43#380C2212yes—
44#340C2212yescomplemented
45#340C2212yescomplemented
46#336C2212yescomplemented
47C25—21no—
48C22×Q8—21no—
49C22×D4—22no—
50C22×D4—22no—
51C22×D4—22no—
52C22×D4—22no—
53C22×D4—21no—
54C22×D4—22no—
55C22×D4Q814yescomplemented
56C22×D4C2311yescharacteristic
57C22×D4C2311yescharacteristic
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4—21no—
62C23×C4C2311yescharacteristic
63C22×C8C2312yes—
64C22×C8C2312yes—
65C4⋊Q8—22no—
66C4⋊Q8—22no—
67C4×D4—28no—
68C4×D4—28no—
69C4×D4—22no—
70C4×D4—22no—
71C4×D4—22no—
72C4×D4—22no—
73C4×D4—22no—
74C4×D4—22no—
75C4×D4D414yes—
76C4×D4D414yes—
77#117—22no—
78#117—22no—
79#117—22no—
80#117—22no—
81C2×C4⋊C4—21no—
82C2×C4⋊C4—22no—
83C2×C4⋊C4D411yescharacteristic
84C2×C4⋊C4D411yescharacteristic
85C2×C4⋊C4C2312yes—
86C2×C4⋊C4C2312yes—
87C2×C4⋊C4C2311yescharacteristic
88C2×C4⋊C4C2311yescharacteristic
89C2×C22⋊C4—28no—
90C2×C22⋊C4—28no—
91C2×C22⋊C4—21no—
92C2×C22⋊C4—21no—
93C2×C22⋊C4—21no—
94C2×C22⋊C4—21no—
95C2×C22⋊C4—21no—
96C2×C22⋊C4—22no—
97C2×C22⋊C4D411yescharacteristic
98C2×C22⋊C4D411yescharacteristic
99C2×C42—21no—
100C2×C42C2311yescharacteristic
101C2×C42C2311yescharacteristic
102#107—24no—
103#106—24no—
104C4⋊C8—22no—
105C4⋊C8—22no—
106#102—24no—
107#102—24no—
108#102—24no—
109#102—24no—
110C22⋊C8—24no—
111#95—22no—
112#95—21no—
113#95—21no—
114#95—22no—
115C24—24no—
116C24—24no—
117C24—21no—
118C24—21no—
119C24—24no—
120C24—82no—
121C24—82no—
122C24C2×D411yescharacteristic
123C2×Q8—42no—
124C2×Q8—82no—
125C2×D4—24no—
126C2×D4—24no—
127C2×D4—28no—
128C2×D4—28no—
129C2×D4—28no—
130C2×D4—28no—
131C2×D4—24no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—21no—
137C2×D4—21no—
138C2×D4C2×D412yes—
139C2×D4C2×D412yes—
140C2×D4C2×Q812yes—
141C2×D4C2×Q812yes—
142C2×D4#4112yes—
143C2×D4#4112yes—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—24no—
151C22×C4—24no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—82no—
157C22×C4—82no—
158C22×C4—82no—
159C22×C4—82no—
160C22×C4C2×D411yescharacteristic
161C22×C4C2×D411yescharacteristic
162C22×C4C2×D411yescharacteristic
163C22×C4C2×D411yescharacteristic
164C22×C4C2×D411yescharacteristic
165C22×C4C2×Q812yes—
166C22×C4#4112yes—
167C22×C4C2411yescharacteristic
168C2×C8—22no—
169C2×C8—22no—
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—22no—
173C2×C8—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—22no—
179C4⋊C4—22no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—42no—
183C4⋊C4—82no—
184C4⋊C4C2×D412yes—
185C4⋊C4C2×D412yes—
186C22⋊C4—24no—
187C22⋊C4—24no—
188C22⋊C4—24no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—24no—
192C22⋊C4—24no—
193C22⋊C4—82no—
194C42—24no—
195C42—21no—
196C42—21no—
197C42C2×D412yes—
198C42C2×D412yes—
199C23—216no—
200C23—21no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—21no—
206C23—22no—
207C23—22no—
208C23—22no—
209C23—22no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—88no—
215C23—88no—
216C23—84no—
217C23—84no—
218C23C22⋊Q812yes—
219C23C22×D411yescharacteristic
220Q8—84no—
221D4—216no—
222D4—22no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—22no—
228D4C22⋊Q814yes—
229D4C22⋊Q814yes—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—84no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4—84no—
256C2×C4#12012yes—
257C2×C4#12012yes—
258C2×C4C22⋊Q812yes—
259C2×C4C22×D411yescharacteristic
260C2×C4C22×D411yescharacteristic
261C2×C4C22×Q811yescharacteristic
262C2×C4C2×#4111yescharacteristic
263C2×C4C2×#4111yescharacteristic
264C2×C4C2×#4111yescharacteristic
265C8—42no—
266C8—42no—
267C22—24no—
268C22—22no—
269C22—24no—
270C22—24no—
271C22—44no—
272C22—44no—
273C22—42no—
274C22—44no—
275C22—44no—
276C22—42no—
277C22—42no—
278C22—41no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—816no—
283C22C2×#12011yescharacteristic
284C22C2×C22⋊Q811yescharacteristic
285C22#52511yescharacteristic
286C22#57612yes—
287C22#57712yes—
288C4—22no—
289C4—22no—
290C4—22no—
291C4—41no—
292C4—42no—
293C4—41no—
294C4—42no—
295C4—82no—
296C4—82no—
297C4C2×C22⋊Q811yescharacteristic
298C4C2×C22⋊Q811yescharacteristic
299C4#52511yescharacteristic
300C4#52511yescharacteristic
301C2—42no—
302C2—44no—
303C2—44no—
304C2#284811yescharacteristic
305C2#300811yescharacteristic
306C2#300911yescharacteristic
307C2#308112yes—
308C2#311812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#476C2semidirect
3. C2×#474C2semidirect
4. #2628C2semidirect
5. #1864C2semidirect
6. #1862C2semidirect
7. #1854C2semidirect
8. #1804C2semidirect
9. #1804C2semidirect
10. #1798C2semidirect
13. C2×C4⋊Q8C22semidirect
28. C2×#107C22semidirect
29. C2×#106C22semidirect
30. C2×C4⋊C8C22semidirect
31. C2×C4⋊C8C22semidirect
41. #389C22semidirect
42. #382C22semidirect
44. #340C22semidirect
45. #340C22semidirect
46. #336C22semidirect
55. C22×D4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#5182
C2×#4762
C2×#4742
#26282
#18642
#18622
#18542
#18042
#17982
#17862