Finite Group Databasebeta

← #17005

Subgroups — #17005

Order 256.

681 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#385C211yesmaximal, characteristic
2C2×#380C211yesmaximal, characteristic
3#1443C212yesmaximal, complemented
4#1442C211yesmaximal, characteristic, complemented
5#1442C211yesmaximal, characteristic, complemented
6#1442C212yesmaximal, complemented
7#1441C211yesmaximal, characteristic, complemented
8#1441C211yesmaximal, characteristic, complemented
9#1440C212yesmaximal
10#1440C212yesmaximal
11C22⋊#95C211yesmaximal, characteristic
12C24×C4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C4⋊C4C2211yescharacteristic
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2211yescharacteristic
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2211yescharacteristic
21#385—24no—
22#380—24no—
23C2×#95C2211yescharacteristic
24C2×#95C2211yescharacteristic
25C2×#95C2212yes—
26C2×#95C2212yes—
27C2×#95C2211yescharacteristic
28C2×#95C2212yes—
29C2×#95C2211yescharacteristic
30C2×#95C2212yes—
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C25C2311yescharacteristic
38C23×C4—22no—
39C23×C4—22no—
40C23×C4—42no—
41C23×C4—42no—
42C23×C4D414yes—
43C23×C4C2311yescharacteristic
44C23×C4C2312yes—
45C23×C4C2312yes—
46C23×C4C2311yescharacteristic
47C23×C4C2312yes—
48C23×C4C2312yes—
49C23×C4C2311yescharacteristic
50C23×C4C2311yescharacteristic
51C23×C4C2311yescharacteristic
52C23×C4C2311yescharacteristic
53C2×C4⋊C4—22no—
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4—42no—
61C2×C4⋊C4—42no—
62C2×C22⋊C4—22no—
63C2×C22⋊C4—22no—
64C2×C22⋊C4—22no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—22no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—42no—
71C2×C22⋊C4—42no—
72C2×C22⋊C4—42no—
73#95—24no—
74#95—22no—
75#95—22no—
76#95—24no—
77#95—24no—
78#95—22no—
79#95—22no—
80#95—42no—
81#95—42no—
82#95—42no—
83#95—42no—
84#95—42no—
85#95—42no—
86#95—42no—
87C24—21no—
88C24—21no—
89C24—21no—
90C24—21no—
91C24—21no—
92C24—22no—
93C24—22no—
94C24—22no—
95C24—21no—
96C24C2×D412yes—
97C24#4114yes—
98C24C2411yescharacteristic
99C22×C4—22no—
100C22×C4—22no—
101C22×C4—21no—
102C22×C4—21no—
103C22×C4—21no—
104C22×C4—21no—
105C22×C4—21no—
106C22×C4—21no—
107C22×C4—21no—
108C22×C4—21no—
109C22×C4—21no—
110C22×C4—22no—
111C22×C4—22no—
112C22×C4—22no—
113C22×C4—22no—
114C22×C4—21no—
115C22×C4—28no—
116C22×C4—21no—
117C22×C4—22no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—22no—
121C22×C4—22no—
122C22×C4—22no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—21no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—44no—
155C22×C4—44no—
156C22×C4—48no—
157C22×C4—48no—
158C22×C4C2×D412yes—
159C22×C4C2×D412yes—
160C22×C4#4112yes—
161C22×C4#4112yes—
162C22×C4#4112yes—
163C22×C4#4112yes—
164C4⋊C4—44no—
165C4⋊C4—44no—
166C4⋊C4—44no—
167C4⋊C4—44no—
168C22⋊C4—28no—
169C22⋊C4—44no—
170C22⋊C4—48no—
171C22⋊C4—48no—
172C23—22no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—22no—
178C23—24no—
179C23—24no—
180C23—24no—
181C23—24no—
182C23—24no—
183C23—24no—
184C23—42no—
185C23—42no—
186C23—42no—
187C23—42no—
188C23—41no—
189C23—41no—
190C23—41no—
191C23—41no—
192C23—42no—
193C23—42no—
194C23#12414yes—
195C23C22×D411yescharacteristic
196C23C2×#4111yescharacteristic
197C23C2×#4111yescharacteristic
198C23C2×#4111yescharacteristic
199C23C2×#4111yescharacteristic
200C23C2×#4112yes—
201C23H(5,𝔽2)12yes—
202C23H(5,𝔽2)12yes—
203C23H(5,𝔽2)11yescharacteristic
204C232−1+412yes—
205C232−1+411yescharacteristic
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—24no—
209C2×C4—24no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—416no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—42no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—42no—
241C2×C4—42no—
242C2×C4—42no—
243C2×C4—42no—
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—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4#12314yes—
267C22—24no—
268C22—24no—
269C22—24no—
270C22—42no—
271C22—42no—
272C22—42no—
273C22—44no—
274C22—44no—
275C22—44no—
276C22—44no—
277C22—42no—
278C22C2×#12311yescharacteristic
279C22#52511yescharacteristic
280C22C2×#12411yescharacteristic
281C22#53411yescharacteristic
282C22#53411yescharacteristic
283C22#53611yescharacteristic
284C22#53711yescharacteristic
285C22#53812yes—
286C22#53812yes—
287C22#53812yes—
288C22#53912yes—
289C22#53912yes—
290C22#54212yes—
291C22#55211yescharacteristic
292C22#55212yes—
293C22#55212yes—
294C22#55312yes—
295C22#55312yes—
296C22#55311yescharacteristic
297C22#55511yescharacteristic
298C22#55611yescharacteristic
299C22#56212yes—
300C22#56312yes—
301C4—24no—
302C4—42no—
303C4—42no—
304C4—42no—
305C4—82no—
306C4—81no—
307C4—82no—
308C4—82no—
309C4—82no—
310C4—81no—
311C2—44no—
312C2#248511yescharacteristic
313C2#250311yescharacteristic
314C2#259111yescharacteristic
315C2#262211yescharacteristic
316C2#263811yescharacteristic
317C2#263911yescharacteristic
318C2#277012yes—
319C2#277912yes—
320C2#278112yes—
321C2#278712yes—
322C2#281411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C24
prime radicalC25C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #1443C2semidirect
4. #1442C2semidirect
5. #1442C2semidirect
6. #1442C2semidirect
7. #1441C2semidirect
8. #1441C2semidirect

Maximal subgroups

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

subgroupindex
C2×#3852
C2×#3802
#14432
#14422
#14412
#14402
C22⋊#952