Finite Group Databasebeta

← #14325

Subgroups — #14325

Order 256.

825 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#394C211yesmaximal, characteristic
2C2×#390C211yesmaximal, characteristic
3C2×#387C211yesmaximal, characteristic, complemented
4#1442C211yesmaximal, characteristic, complemented
5#1441C211yesmaximal, characteristic, complemented
6#1440C212yesmaximal, complemented
7#1439C212yesmaximal, complemented
8#1438C211yesmaximal, characteristic
9#1437C212yesmaximal, complemented
10#1437C212yesmaximal, complemented
11C4×#95C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2211yescharacteristic
16C22×C22⋊C4C2212yes—
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C42C2212yes—
23C22×C42C2211yescharacteristic
24#394C418yescomplemented
25#390—24no—
26#387—24no—
27C2×#95C2211yescharacteristic
28C2×#95C2212yes—
29C2×#95C2211yescharacteristic
30C2×#95C2211yescharacteristic
31C2×#95C2211yescharacteristic
32C2×#95C2211yescharacteristic
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2211yescharacteristic
38C25C2312yes—
39C22×D4—22no—
40C22×D4—22no—
41C22×D4—42no—
42C22×D4—42no—
43C22×D4C2×C414yes—
44C23×C4C2311yescharacteristic
45C23×C4C2312yes—
46C23×C4C2311yescharacteristic
47C23×C4C2311yescharacteristic
48C23×C4C2311yescharacteristic
49C23×C4C2312yes—
50C23×C4C2311yescharacteristic
51C23×C4C2312yes—
52C23×C4C2311yescharacteristic
53C23×C4C2311yescharacteristic
54C2×C4⋊C4—22no—
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—42no—
59C2×C4⋊C4—42no—
60C2×C4⋊C4C2×C414yes—
61C2×C22⋊C4—24no—
62C2×C22⋊C4—24no—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—42no—
68C2×C22⋊C4—42no—
69C2×C22⋊C4—42no—
70C2×C22⋊C4—42no—
71C2×C22⋊C4—42no—
72C2×C22⋊C4—42no—
73C2×C22⋊C4—42no—
74C2×C22⋊C4—42no—
75C2×C22⋊C4—42no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4C2×C418yes—
78C2×C22⋊C4C2×C418yes—
79C2×C42—24no—
80C2×C42—42no—
81C2×C42—42no—
82C2×C42—42no—
83C2×C42D414yes—
84#95—22no—
85#95—22no—
86#95—22no—
87#95—22no—
88#95—24no—
89#95—22no—
90#95—42no—
91#95—42no—
92#95—42no—
93#95—42no—
94#95C2×C414yes—
95C24—22no—
96C24—22no—
97C24—22no—
98C24—22no—
99C24—22no—
100C24—22no—
101C24—22no—
102C24—22no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24C22×C414yes—
110C24C2411yescharacteristic
111C2×D4—28no—
112C2×D4—48no—
113C2×D4—44no—
114C2×D4—44no—
115C2×D4—48no—
116C22×C4—22no—
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—22no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—22no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—22no—
135C22×C4—22no—
136C22×C4—22no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—22no—
163C22×C4—21no—
164C22×C4C22×C414yes—
165C22×C4C22×C412yes—
166C22×C4C22×C412yes—
167C22×C4C22×C412yes—
168C22×C4C2×D412yes—
169C22×C4C2×D412yes—
170C22×C4C2×D412yes—
171C22×C4#4114yes—
172C22×C4#4112yes—
173C22×C4#4114yes—
174C4⋊C4—28no—
175C4⋊C4—44no—
176C4⋊C4—44no—
177C22⋊C4—48no—
178C22⋊C4—48no—
179C22⋊C4—48no—
180C22⋊C4—48no—
181C22⋊C4—48no—
182C22⋊C4—48no—
183C42—44no—
184C42—48no—
185C23—24no—
186C23—24no—
187C23—24no—
188C23—24no—
189C23—24no—
190C23—24no—
191C23—24no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23—42no—
198C23—42no—
199C23—42no—
200C23—42no—
201C23—42no—
202C23—42no—
203C23—42no—
204C23—42no—
205C23—42no—
206C23—42no—
207C23—42no—
208C23—42no—
209C23—42no—
210C23—42no—
211C23—42no—
212C23—42no—
213C23—42no—
214C23—42no—
215C23—42no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23C23×C411yescharacteristic
221C23C22×D411yescharacteristic
222C23C2×#4111yescharacteristic
223C23C2×#4112yes—
224C23C2×#4112yes—
225C23H(5,𝔽2)12yes—
226C23H(5,𝔽2)12yes—
227C23H(5,𝔽2)11yescharacteristic
228C23H(5,𝔽2)11yescharacteristic
229C23H(5,𝔽2)11yescharacteristic
230C232−1+411yescharacteristic
231D4—416no—
232C2×C4—24no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—22no—
248C2×C4—24no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4C4×D414yes—
289C2×C4#12414yes—
290C22—44no—
291C22—44no—
292C22—44no—
293C22—44no—
294C22—44no—
295C22—44no—
296C22—44no—
297C22—82no—
298C22—82no—
299C22—82no—
300C22—82no—
301C22—82no—
302C22—82no—
303C22—82no—
304C22—82no—
305C22C2×C4×D411yescharacteristic
306C22C4×#4112yes—
307C22#51812yes—
308C22#51811yescharacteristic
309C22#52011yescharacteristic
310C22C2×#12411yescharacteristic
311C22#52912yes—
312C22#53211yescharacteristic
313C22#53411yescharacteristic
314C22#53511yescharacteristic
315C22#53511yescharacteristic
316C22#53711yescharacteristic
317C22#53812yes—
318C22#53812yes—
319C22#53812yes—
320C22#53812yes—
321C22#53912yes—
322C22#54011yescharacteristic
323C22#54012yes—
324C22#54012yes—
325C22#54012yes—
326C22#54211yescharacteristic
327C22#54212yes—
328C4—24no—
329C4—24no—
330C4—42no—
331C4—42no—
332C4—44no—
333C4—44no—
334C4—42no—
335C4—82no—
336C4—81no—
337C4—81no—
338C2—84no—
339C2C4×#12411yescharacteristic
340C2#232111yescharacteristic
341C2#232411yescharacteristic
342C2#233312yes—
343C2#233512yes—
344C2#263911yescharacteristic
345C2#264111yescharacteristic
346C2#264812yes—
347C2#265112yes—
348C2#266111yescharacteristic
349C2#266211yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C24
commutatorC23C23×C4
prime radicalC23×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#387C2semidirect
4. #1442C2semidirect
5. #1441C2semidirect
6. #1440C2semidirect
7. #1439C2semidirect
9. #1437C2semidirect
10. #1437C2semidirect
11. C4×#95C2semidirect
24. #394C4semidirect

Maximal subgroups

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

subgroupindex
C2×#3942
C2×#3902
C2×#3872
#14422
#14412
#14402
#14392
#14382
#14372
C4×#952