Finite Group Databasebeta

← #32916

Subgroups — #32916

Order 256.

703 conjugacy classes of subgroups.

The subgroup lattice has 309 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#520C211yesmaximal, characteristic
2C2×#462C211yesmaximal, characteristic, complemented
3C2×#461C211yesmaximal, characteristic, complemented
4C2×#452C212yesmaximal
5#2621C211yesmaximal, characteristic, complemented
6#1878C212yesmaximal, complemented
7#1877C212yesmaximal, complemented
8#1872C212yesmaximal, complemented
9#1805C211yesmaximal, characteristic, complemented
10#1804C211yesmaximal, characteristic, complemented
11#1798C211yesmaximal, characteristic, complemented
12C22×#41C2211yescharacteristic
13C22×Q16C2212yes—
14C22×SD16C2212yes—
15C2×C4⋊Q8C2211yescharacteristic
16C2×C22⋊Q8C2212yes—
17#520—24no—
18#520—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×C4⋊C4—21no—
28C22×C4⋊C4C2211yescharacteristic
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#398—21no—
40#398—21no—
41#397—21no—
42#397—21no—
43#391C2212yes—
44#382C2211yescharacteristic
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×C4—21no—
62C23×C4C2311yescharacteristic
63C2×Q16—24no—
64C2×Q16—24no—
65C2×Q16—24no—
66C2×SD16—24no—
67C2×SD16—24no—
68C2×SD16—24no—
69C22×C8C2312yes—
70C22×C8C2312yes—
71C4⋊Q8—22no—
72C4⋊Q8—22no—
73C22⋊Q8—24no—
74C22⋊Q8—24no—
75C4×Q8—22no—
76C4×Q8—22no—
77C4×Q8—22no—
78C4×Q8D414yes—
79C4×D4—28no—
80C4×D4—28no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—22no—
84C4×D4D414yes—
85#117—28no—
86#117—22no—
87#117—22no—
88C2×C4⋊C4—28no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—22no—
93C2×C4⋊C4—22no—
94C2×C4⋊C4—22no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4D411yescharacteristic
98C2×C4⋊C4D411yescharacteristic
99C2×C4⋊C4C2312yes—
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—22no—
105C2×C22⋊C4D411yescharacteristic
106C2×C22⋊C4D411yescharacteristic
107C2×C42—21no—
108C2×C42—21no—
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111C4⋊C8—22no—
112C4⋊C8—22no—
113#103—24no—
114#103—24no—
115#103—24no—
116#102—24no—
117C22⋊C8—24no—
118#95—21no—
119#95—21no—
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—21no—
158C22×C4—21no—
159C22×C4—24no—
160C22×C4—24no—
161C22×C4—24no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—82no—
165C22×C4—82no—
166C22×C4—82no—
167C22×C4C2×D411yescharacteristic
168C22×C4C2×D411yescharacteristic
169C22×C4C2×D411yescharacteristic
170C22×C4C2×D411yescharacteristic
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D412yes—
173C22×C4#4112yes—
174C22×C4C2411yescharacteristic
175Q16—48no—
176SD16—48no—
177C2×C8—22no—
178C2×C8—22no—
179C2×C8—22no—
180C2×C8—22no—
181C2×C8—22no—
182C2×C8—22no—
183C4⋊C4—24no—
184C4⋊C4—24no—
185C4⋊C4—22no—
186C4⋊C4—24no—
187C4⋊C4—22no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—44no—
195C4⋊C4C2×D412yes—
196C4⋊C4C2×D412yes—
197C22⋊C4—24no—
198C22⋊C4—24no—
199C22⋊C4—22no—
200C22⋊C4—22no—
201C22⋊C4—44no—
202C42—24no—
203C42—24no—
204C42—21no—
205C42—21no—
206C42C2×D412yes—
207C42C2×D412yes—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—21no—
212C23—22no—
213C23—22no—
214C23—22no—
215C23—42no—
216C23—42no—
217C23#12112yes—
218C23C22×D411yescharacteristic
219Q8—22no—
220Q8—22no—
221Q8—22no—
222Q8—42no—
223Q8—42no—
224Q8—42no—
225Q8—42no—
226Q8#12114yes—
227D4—216no—
228D4—22no—
229D4—22no—
230D4—22no—
231D4#12114yes—
232C2×C4—216no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—84no—
259C2×C4—84no—
260C2×C4—84no—
261C2×C4#12012yes—
262C2×C4#12012yes—
263C2×C4#12112yes—
264C2×C4C22×D411yescharacteristic
265C2×C4C22×D411yescharacteristic
266C2×C4C22×D411yescharacteristic
267C2×C4C2×#4111yescharacteristic
268C2×C4C2×#4111yescharacteristic
269C2×C4C2×#4111yescharacteristic
270C8—42no—
271C8—42no—
272C22—22no—
273C22—24no—
274C22—24no—
275C22—42no—
276C22—44no—
277C22—41no—
278C22—42no—
279C22—41no—
280C22—42no—
281C22—44no—
282C22C2×#12011yescharacteristic
283C22C2×#12111yescharacteristic
284C22#52511yescharacteristic
285C22#57712yes—
286C22#57812yes—
287C4—22no—
288C4—24no—
289C4—22no—
290C4—22no—
291C4—41no—
292C4—42no—
293C4—42no—
294C4—41no—
295C4—82no—
296C4—82no—
297C4C2×#12111yescharacteristic
298C4C2×#12111yescharacteristic
299C4#52511yescharacteristic
300C4#52511yescharacteristic
301C2—42no—
302C2—44no—
303C2#284711yescharacteristic
304C2#300911yescharacteristic
305C2#301111yescharacteristic
306C2#304612yes—
307C2#312012yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
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×#462C2semidirect
3. C2×#461C2semidirect
5. #2621C2semidirect
6. #1878C2semidirect
7. #1877C2semidirect
8. #1872C2semidirect
9. #1805C2semidirect
10. #1804C2semidirect
11. #1798C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5202
C2×#4622
C2×#4612
C2×#4522
#26212
#18782
#18772
#18722
#18052
#18042
#17982