Finite Group Databasebeta

← C4×#388 #13763

Subgroups — C4×#388 #13763

Order 256.

1067 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#388C212yesmaximal
2C2×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C4C211yesmaximal, characteristic
4C2×C4×C22⋊C4C211yesmaximal, characteristic
5C2×C4×C22⋊C4C211yesmaximal, characteristic
6C2×C43C211yesmaximal, characteristic, complemented
7#1437C212yesmaximal
8#1436C212yesmaximal, complemented
9#1434C212yesmaximal, complemented
10C4×#95C211yesmaximal, characteristic, complemented
11C4×#95C211yesmaximal, characteristic, complemented
12C24×C4C2211yescharacteristic
13C22×C4⋊C4C2212yes—
14C22×C22⋊C4C2211yescharacteristic
15C22×C22⋊C4C2212yes—
16C22×C22⋊C4C2212yes—
17C22×C22⋊C4C2211yescharacteristic
18C22×C42C2212yes—
19C22×C42C2211yescharacteristic
20C22×C42C2211yescharacteristic
21C22×C42C2211yescharacteristic
22C22×C42C2211yescharacteristic
23C22×C42C2212yes—
24C22×C42C2211yescharacteristic
25C22×C42C2211yescharacteristic
26#388C4116yesdirect product factor
27C4×C4⋊C4—24no—
28C4×C22⋊C4—24no—
29C4×C22⋊C4—24no—
30C4×C22⋊C4C418yescomplemented
31C2×#95C2212yes—
32C2×#95C2212yes—
33C2×#95C2212yes—
34C2×#95C2212yes—
35C2×#95C2212yes—
36C2×#95C2211yescharacteristic
37C2×#95C2211yescharacteristic
38C2×#95C2212yes—
39C2×#95C2212yes—
40C43—24no—
41C25C2311yescharacteristic
42C23×C4—22no—
43C23×C4—22no—
44C23×C4—22no—
45C23×C4—22no—
46C23×C4—22no—
47C23×C4—22no—
48C23×C4C2×C414yes—
49C23×C4C2312yes—
50C23×C4C2311yescharacteristic
51C23×C4C2312yes—
52C23×C4C2311yescharacteristic
53C23×C4C2311yescharacteristic
54C23×C4C2311yescharacteristic
55C23×C4C2312yes—
56C23×C4C2312yes—
57C23×C4C2311yescharacteristic
58C23×C4C2311yescharacteristic
59C2×C4⋊C4—24no—
60C2×C4⋊C4—24no—
61C2×C4⋊C4C2×C418yescomplemented
62C2×C22⋊C4—24no—
63C2×C22⋊C4—24no—
64C2×C22⋊C4—24no—
65C2×C22⋊C4—24no—
66C2×C22⋊C4—24no—
67C2×C22⋊C4C2×C414yes—
68C2×C22⋊C4C2×C418yes—
69C2×C22⋊C4C2×C418yes—
70C2×C22⋊C4C2×C414yes—
71C2×C22⋊C4C2×C418yes—
72C2×C42—24no—
73C2×C42—22no—
74C2×C42—24no—
75C2×C42—22no—
76C2×C42—22no—
77C2×C42—24no—
78C2×C42—24no—
79C2×C42—22no—
80C2×C42—22no—
81C2×C42—24no—
82C2×C42—22no—
83C2×C42—22no—
84C2×C42—22no—
85C2×C42—22no—
86C2×C42—22no—
87C2×C42—22no—
88C2×C42—22no—
89C2×C42—22no—
90C2×C42—22no—
91C2×C42C2×C414yes—
92C2×C42C2×C414yes—
93C2×C42C2×C418yescomplemented
94C2×C42D414yes—
95C2×C42D414yes—
96#95—24no—
97#95—24no—
98#95—24no—
99#95—24no—
100#95—24no—
101#95C2×C414yes—
102#95C2×C414yes—
103#95C2×C418yescomplemented
104#95C2×C418yescomplemented
105C24—21no—
106C24—21no—
107C24—22no—
108C24—22no—
109C24—22no—
110C24—21no—
111C24—21no—
112C24—21no—
113C24—21no—
114C24C22×C414yes—
115C24C22×C412yes—
116C24C2411yescharacteristic
117C22×C4—21no—
118C22×C4—22no—
119C22×C4—21no—
120C22×C4—21no—
121C22×C4—21no—
122C22×C4—21no—
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—22no—
126C22×C4—22no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—22no—
139C22×C4—22no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—28no—
157C22×C4—28no—
158C22×C4—28no—
159C22×C4—44no—
160C22×C4—44no—
161C22×C4—44no—
162C22×C4—44no—
163C22×C4C22×C412yes—
164C22×C4C22×C412yes—
165C22×C4C22×C414yes—
166C22×C4C22×C414yes—
167C22×C4C22×C412yes—
168C22×C4C22×C412yes—
169C22×C4C22×C414yes—
170C22×C4C22×C414yes—
171C22×C4C22×C412yes—
172C22×C4C22×C414yes—
173C22×C4C22×C414yes—
174C22×C4C22×C412yes—
175C22×C4C2×D414yes—
176C22×C4C2×D414yes—
177C22×C4C2×D412yes—
178C22×C4C2×D412yes—
179C22×C4#4114yes—
180C22×C4#4114yes—
181C22×C4#4114yes—
182C22×C4#4112yes—
183C22×C4#4112yes—
184C22×C4#4112yes—
185C22×C4#4112yes—
186C4⋊C4—216no—
187C22⋊C4—28no—
188C22⋊C4—216no—
189C22⋊C4—216no—
190C22⋊C4C42116yescomplemented
191C42—216no—
192C42—28no—
193C42—28no—
194C42—216no—
195C42—28no—
196C42—28no—
197C42—44no—
198C42—44no—
199C23—22no—
200C23—22no—
201C23—24no—
202C23—24no—
203C23—22no—
204C23—22no—
205C23—24no—
206C23—24no—
207C23—24no—
208C23—24no—
209C23—22no—
210C23—22no—
211C23—41no—
212C23—41no—
213C23—42no—
214C23—41no—
215C23—41no—
216C23—42no—
217C23—42no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23C2×C4214yes—
222C23C23×C411yescharacteristic
223C23C23×C412yes—
224C23C22×D411yescharacteristic
225C23C22×D411yescharacteristic
226C23C2×#4111yescharacteristic
227C23C2×#4111yescharacteristic
228C23C2×#4112yes—
229C23C2×#4111yescharacteristic
230C23C2×#4112yes—
231C23C2×#4111yescharacteristic
232C23C2×#4112yes—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—24no—
238C2×C4—24no—
239C2×C4—24no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—24no—
247C2×C4—24no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—24no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—416no—
284C2×C4C2×C4214yes—
285C2×C4C2×C4214yes—
286C2×C4#11714yes—
287C2×C4#11714yes—
288C2×C4C4×D418yes—
289C2×C4C4×D418yes—
290C2×C4C4×D414yes—
291C2×C4C4×D414yes—
292C2×C4#12114yes—
293C2×C4#12314yes—
294C2×C4#12414yes—
295C2×C4#12614yes—
296C22—24no—
297C22—24no—
298C22—24no—
299C22—44no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—44no—
305C22—44no—
306C22—44no—
307C22C22×C4211yescharacteristic
308C22C2×#11711yescharacteristic
309C22C2×#11711yescharacteristic
310C22C2×C4×D411yescharacteristic
311C22C2×C4×D411yescharacteristic
312C22C2×C4×D412yes—
313C22C2×C4×D412yes—
314C22C4×#4112yes—
315C22C4×#4112yes—
316C22C4×#4112yes—
317C22C4×#4112yes—
318C22C4×#4111yescharacteristic
319C22C4×#4111yescharacteristic
320C22C2×#12111yescharacteristic
321C22C2×#12311yescharacteristic
322C22#52512yes—
323C22C2×#12411yescharacteristic
324C22C2×#12611yescharacteristic
325C22#52912yes—
326C22#52912yes—
327C22#52912yes—
328C22#52912yes—
329C22#53212yes—
330C4—24no—
331C4—24no—
332C4—24no—
333C4—28no—
334C4—28no—
335C4—42no—
336C4—42no—
337C4—44no—
338C4—44no—
339C4#38818yesdirect product factor
340C2—44no—
341C2C4×#11711yescharacteristic
342C2C42×D411yescharacteristic
343C2C42×D411yescharacteristic
344C2C2×#38811yescharacteristic
345C2#229011yescharacteristic
346C2#229311yescharacteristic
347C2#229811yescharacteristic
348C2C4×#12112yes—
349C2C4×#12312yes—
350C2C4×#12412yes—
351C2C4×#12612yes—
subgroupG/H
centerC23×C4C23
commutatorC22C22×C42
Frattini, socle, cosocleC24C24
prime radicalC25C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×C4×C4⋊C4C2semidirect
6. C2×C43C2semidirect
8. #1436C2semidirect
9. #1434C2semidirect
10. C4×#95C2semidirect
11. C4×#95C2semidirect
26. #388C4direct (N × H)
30. C4×C22⋊C4C4semidirect
61. C2×C4⋊C4C2×C4semidirect
93. C2×C42C2×C4semidirect
103. #95C2×C4semidirect
104. #95C2×C4semidirect
190. C22⋊C4C42semidirect
339. C4#388direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#3882
C2×C4×C4⋊C42
C2×C4×C22⋊C42
C2×C432
#14372
#14362
#14342
C4×#952