Finite Group Databasebeta

← #32907

Subgroups — #32907

Order 256.

734 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#41C211yesmaximal, characteristic
2C2×#462C211yesmaximal, characteristic, complemented
3C2×#461C211yesmaximal, characteristic, complemented
4C2×#452C212yesmaximal
5#2857C211yesmaximal, characteristic, complemented
6#1878C212yesmaximal, complemented
7#1877C212yesmaximal, complemented
8#1873C212yesmaximal, complemented
9#1803C211yesmaximal, characteristic, complemented
10#1802C211yesmaximal, characteristic, complemented
11#1797C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×Q16C2212yes—
14C22×SD16C2212yes—
15C2×C4⋊Q8C2211yescharacteristic
16C2×C22⋊Q8C2212yes—
17C4×#41—24no—
18C4×#41—24no—
19C2×C4×Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×C4×D4—21no—
22C2×C4×D4—21no—
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×#117—21no—
26C2×#117C2211yescharacteristic
27C22×C42—21no—
28C22×C42C2211yescharacteristic
29#462—24no—
30#461—24no—
31#452—28no—
32C2×C4⋊C8C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
34C2×#103C2212yes—
35C2×#103C2212yes—
36C2×#103C2212yes—
37C2×#102C2212yes—
38C2×C22⋊C8C2212yes—
39#391C2212yes—
40#389—21no—
41#389—21no—
42#384C2211yescharacteristic
43#380—21no—
44#380—21no—
45#340C2212yes—
46#340C2212yes—
47#336C2212yes—
48C2×#41—22no—
49C2×#41—22no—
50C2×#41—22no—
51C2×#41—22no—
52C2×#41—22no—
53C2×#41D414yes—
54C22×Q8C2312yes—
55C22×Q8C2311yescharacteristic
56C22×D4—21no—
57C22×D4C2311yescharacteristic
58C23×C4—21no—
59C23×C4—21no—
60C23×C4—21no—
61C23×C4D411yescharacteristic
62C23×C4D411yescharacteristic
63C23×C4C2311yescharacteristic
64C2×Q16—24no—
65C2×Q16—24no—
66C2×Q16—24no—
67C2×SD16—24no—
68C2×SD16—24no—
69C2×SD16—24no—
70C22×C8C2312yes—
71C22×C8C2312yes—
72C4⋊Q8—22no—
73C4⋊Q8—22no—
74C22⋊Q8—24no—
75C22⋊Q8—24no—
76C4×Q8—22no—
77C4×Q8—22no—
78C4×Q8—22no—
79C4×Q8D414yes—
80C4×D4—28no—
81C4×D4—28no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4—22no—
85C4×D4D414yes—
86#117—28no—
87#117—22no—
88#117—22no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—22no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4C2312yes—
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—22no—
103C2×C42—28no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—21no—
107C2×C42—22no—
108C2×C42—22no—
109C2×C42D411yescharacteristic
110C2×C42D411yescharacteristic
111C2×C42C2311yescharacteristic
112C2×C42C2311yescharacteristic
113C4⋊C8—22no—
114C4⋊C8—22no—
115#103—24no—
116#103—24no—
117#103—24no—
118#102—24no—
119C22⋊C8—24no—
120#95—22no—
121#95—22no—
122C24—21no—
123C24C2×D411yescharacteristic
124#41—28no—
125#41—28no—
126#41—28no—
127#41—28no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—22no—
132C2×Q8—22no—
133C2×Q8—22no—
134C2×Q8—22no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8C2×D412yes—
138C2×Q8C2×D412yes—
139C2×Q8#4112yes—
140C2×D4—24no—
141C2×D4—24no—
142C2×D4—24no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4C2×D412yes—
147C2×D4C2×D412yes—
148C2×D4#4112yes—
149C22×C4—24no—
150C22×C4—24no—
151C22×C4—21no—
152C22×C4—24no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—24no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—41no—
169C22×C4—42no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D412yes—
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4#4112yes—
180C22×C4C2411yescharacteristic
181Q16—48no—
182SD16—48no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—22no—
189C4⋊C4—24no—
190C4⋊C4—24no—
191C4⋊C4—21no—
192C4⋊C4—21no—
193C4⋊C4—22no—
194C4⋊C4—22no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—44no—
199C4⋊C4—42no—
200C4⋊C4C2×D412yes—
201C4⋊C4C2×D412yes—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—24no—
205C22⋊C4—44no—
206C42—24no—
207C42—24no—
208C42—22no—
209C42—24no—
210C42—22no—
211C42—21no—
212C42—21no—
213C42C2×D412yes—
214C42C2×D412yes—
215C23—21no—
216C23—22no—
217C23—22no—
218C23—21no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23#12112yes—
229C23C22×D411yescharacteristic
230Q8—22no—
231Q8—22no—
232Q8—22no—
233Q8—42no—
234Q8—42no—
235Q8—42no—
236Q8—42no—
237Q8#12114yes—
238D4—216no—
239D4—22no—
240D4—22no—
241D4—22no—
242D4#12114yes—
243C2×C4—216no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—44no—
263C2×C4—48no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—44no—
269C2×C4—48no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4#12012yes—
277C2×C4#12012yes—
278C2×C4#12112yes—
279C2×C4#13512yes—
280C2×C4#13712yes—
281C2×C4#13712yes—
282C2×C4C22×D411yescharacteristic
283C2×C4C22×D411yescharacteristic
284C2×C4C22×D411yescharacteristic
285C2×C4C2×#4111yescharacteristic
286C2×C4C2×#4111yescharacteristic
287C2×C4C2×#4111yescharacteristic
288C8—42no—
289C8—42no—
290C22—22no—
291C22—24no—
292C22—22no—
293C22—22no—
294C22—41no—
295C22—42no—
296C22—41no—
297C22—42no—
298C22—41no—
299C22—42no—
300C22—41no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22C2×#12011yescharacteristic
305C22C2×#12111yescharacteristic
306C22#52511yescharacteristic
307C22C2×#13511yescharacteristic
308C22C2×#13711yescharacteristic
309C22C2×#13711yescharacteristic
310C22#57511yescharacteristic
311C4—22no—
312C4—24no—
313C4—22no—
314C4—22no—
315C4—41no—
316C4—42no—
317C4—42no—
318C4—82no—
319C4—82no—
320C4#45214yescomplemented
321C4C2×#12111yescharacteristic
322C4C2×#12111yescharacteristic
323C4#52511yescharacteristic
324C4#52511yescharacteristic
325C2—42no—
326C2—44no—
327C2#284711yescharacteristic
328C2C2×#45211yescharacteristic
329C2#300711yescharacteristic
330C2#303511yescharacteristic
331C2#303911yescharacteristic
332C2#310711yescharacteristic
333C2#311611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
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
2. C2×#462C2semidirect
3. C2×#461C2semidirect
5. #2857C2semidirect
6. #1878C2semidirect
7. #1877C2semidirect
8. #1873C2semidirect
9. #1803C2semidirect
10. #1802C2semidirect
11. #1797C2semidirect
320. C4#452semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#412
C2×#4622
C2×#4612
C2×#4522
#28572
#18782
#18772
#18732
#18032
#18022
#17972