Finite Group Databasebeta

← #14312

Subgroups — #14312

Order 256.

883 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#394C211yesmaximal, characteristic
2C2×#390C211yesmaximal, characteristic
3C2×#388C212yesmaximal, complemented
4C2×C4×C4⋊C4C211yesmaximal, characteristic, complemented
5#1444C211yesmaximal, characteristic, complemented
6#1441C211yesmaximal, characteristic, complemented
7#1439C212yesmaximal, complemented
8#1439C212yesmaximal, complemented
9#1438C211yesmaximal, characteristic
10#1437C212yesmaximal, complemented
11#1434C211yesmaximal, characteristic, complemented
12C23×D4C2211yescharacteristic
13C22×C4⋊C4C2211yescharacteristic
14C22×C4⋊C4C2211yescharacteristic
15C22×C4⋊C4C2212yes—
16C22×C4⋊C4C2211yescharacteristic
17C22×C22⋊C4C2212yes—
18C22×C22⋊C4C2212yes—
19C22×C22⋊C4C2212yes—
20C22×C22⋊C4C2212yes—
21C22×C22⋊C4C2212yes—
22C22×C22⋊C4C2212yes—
23C22×C42C2211yescharacteristic
24C22×C42C2211yescharacteristic
25C22×C42C2211yescharacteristic
26#394C418yescomplemented
27#390—24no—
28#388—28no—
29C4×C4⋊C4—24no—
30C2×#95C2212yes—
31C2×#95C2211yescharacteristic
32C2×#95C2211yescharacteristic
33C2×#95C2211yescharacteristic
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2212yes—
37C2×#95C2212yes—
38C2×#95C2211yescharacteristic
39C25C2312yes—
40C22×D4—22no—
41C22×D4—22no—
42C22×D4—42no—
43C22×D4—42no—
44C22×D4C2×C414yes—
45C23×C4C2312yes—
46C23×C4C2312yes—
47C23×C4C2312yes—
48C23×C4C2311yescharacteristic
49C23×C4C2311yescharacteristic
50C23×C4C2311yescharacteristic
51C23×C4C2311yescharacteristic
52C23×C4C2311yescharacteristic
53C23×C4C2311yescharacteristic
54C23×C4C2311yescharacteristic
55C2×C4⋊C4—22no—
56C2×C4⋊C4—22no—
57C2×C4⋊C4—22no—
58C2×C4⋊C4—22no—
59C2×C4⋊C4—22no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4—22no—
62C2×C4⋊C4—42no—
63C2×C4⋊C4—42no—
64C2×C4⋊C4—42no—
65C2×C4⋊C4C2×C414yes—
66C2×C22⋊C4—24no—
67C2×C22⋊C4—24no—
68C2×C22⋊C4—24no—
69C2×C22⋊C4—24no—
70C2×C22⋊C4—24no—
71C2×C22⋊C4—24no—
72C2×C22⋊C4—24no—
73C2×C22⋊C4—24no—
74C2×C22⋊C4—42no—
75C2×C22⋊C4—42no—
76C2×C22⋊C4—42no—
77C2×C22⋊C4—42no—
78C2×C22⋊C4—42no—
79C2×C22⋊C4—42no—
80C2×C22⋊C4—42no—
81C2×C22⋊C4—42no—
82C2×C22⋊C4C2×C418yes—
83C2×C22⋊C4C2×C418yes—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42—42no—
88C2×C42—42no—
89C2×C42D414yes—
90C2×C42D414yescomplemented
91#95—22no—
92#95—22no—
93#95—24no—
94#95—24no—
95#95—22no—
96#95—42no—
97#95—42no—
98#95—42no—
99#95C2×C414yes—
100C24—22no—
101C24—22no—
102C24—22no—
103C24—22no—
104C24—22no—
105C24—22no—
106C24—22no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—22no—
111C24—22no—
112C24—22no—
113C24—22no—
114C24C22×C414yes—
115C24C2411yescharacteristic
116C2×D4—28no—
117C2×D4—48no—
118C2×D4—44no—
119C2×D4—44no—
120C2×D4—48no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—21no—
138C22×C4—21no—
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—22no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4C22×C414yes—
166C22×C4C22×C412yes—
167C22×C4C22×C412yes—
168C22×C4C22×C412yes—
169C22×C4C2×D412yes—
170C22×C4C2×D412yes—
171C22×C4C2×D412yes—
172C22×C4C2×D412yes—
173C22×C4C2×D412yes—
174C22×C4C2×D412yes—
175C22×C4#4112yes—
176C22×C4#4112yes—
177C22×C4#4112yes—
178C22×C4#4112yes—
179C4⋊C4—28no—
180C4⋊C4—28no—
181C4⋊C4—44no—
182C4⋊C4—48no—
183C22⋊C4—216no—
184C22⋊C4—48no—
185C22⋊C4—48no—
186C22⋊C4—48no—
187C22⋊C4—48no—
188C22⋊C4—48no—
189C42—28no—
190C42—44no—
191C42—44no—
192C23—24no—
193C23—24no—
194C23—24no—
195C23—24no—
196C23—24no—
197C23—24no—
198C23—24no—
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—
220C23—42no—
221C23—42no—
222C23—42no—
223C23—42no—
224C23—42no—
225C23—42no—
226C23—42no—
227C23C23×C411yescharacteristic
228C23C22×D411yescharacteristic
229C23C22×D411yescharacteristic
230C23C2×#4111yescharacteristic
231C23C2×#4111yescharacteristic
232C23C2×#4111yescharacteristic
233C23C2×#4111yescharacteristic
234C23H(5,𝔽2)12yes—
235C23H(5,𝔽2)12yes—
236C23H(5,𝔽2)12yes—
237C232−1+412yes—
238D4—416no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—24no—
264C2×C4—24no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
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×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4C4×D414yes—
295C2×C4C4×D414yes—
296C2×C4#12114yes—
297C22—44no—
298C22—44no—
299C22—44no—
300C22—44no—
301C22—44no—
302C22—44no—
303C22—44no—
304C22—82no—
305C22—82no—
306C22—82no—
307C22—82no—
308C22—82no—
309C22—82no—
310C22—82no—
311C22—82no—
312C22C2×C4×D411yescharacteristic
313C22C2×C4×D411yescharacteristic
314C22C4×#4111yescharacteristic
315C22#51812yes—
316C22#52012yes—
317C22C2×#12111yescharacteristic
318C22#52511yescharacteristic
319C22#52911yescharacteristic
320C22#53211yescharacteristic
321C22#53411yescharacteristic
322C22#53511yescharacteristic
323C22#53512yes—
324C22#53511yescharacteristic
325C22#53611yescharacteristic
326C22#53712yes—
327C22#53812yes—
328C22#53812yes—
329C22#53912yes—
330C22#54012yes—
331C22#54012yes—
332C22#54212yes—
333C22#54212yes—
334C22#54212yes—
335C4—24no—
336C4—24no—
337C4—24no—
338C4—42no—
339C4—44no—
340C4—42no—
341C4—42no—
342C4—42no—
343C4—82no—
344C4—82no—
345C2—84no—
346C2C4×#12111yescharacteristic
347C2#232111yescharacteristic
348C2#232412yes—
349C2#232611yescharacteristic
350C2#233512yes—
351C2#261811yescharacteristic
352C2#261911yescharacteristic
353C2#263211yescharacteristic
354C2#263511yescharacteristic
355C2#265112yes—
356C2#266212yes—
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×#388C2semidirect
4. C2×C4×C4⋊C4C2semidirect
5. #1444C2semidirect
6. #1441C2semidirect
7. #1439C2semidirect
8. #1439C2semidirect
10. #1437C2semidirect
11. #1434C2semidirect
26. #394C4semidirect
90. C2×C42D4semidirect

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×#3882
C2×C4×C4⋊C42
#14442
#14412
#14392
#14382
#14372
#14342