Finite Group Databasebeta

← C2×#2334 #34077

Subgroups — C2×#2334 #34077

Order 256.

1307 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#126C211yesmaximal, characteristic
2#2334C2116yesmaximal, direct product factor
3C2×#389C211yesmaximal, characteristic, complemented
4C2×#389C211yesmaximal, characteristic, complemented
5C2×#388C211yesmaximal, characteristic
6C2×#388C211yesmaximal, characteristic
7C2×#388C211yesmaximal, characteristic
8C2×#387C211yesmaximal, characteristic, complemented
9C2×#387C211yesmaximal, characteristic, complemented
10C2×#387C211yesmaximal, characteristic, complemented
11C2×#385C211yesmaximal, characteristic
12C2×#385C211yesmaximal, characteristic
13C2×#382C211yesmaximal, characteristic, complemented
14C2×C4×C4⋊C4C211yesmaximal, characteristic, complemented
15C2×C4×C4⋊C4C211yesmaximal, characteristic, complemented
16C2×C4×C22⋊C4C211yesmaximal, characteristic
17C24×C4C2211yescharacteristic
18C2×#126C4116yescomplemented
19C2×#126C2218yes—
20C22×C4⋊C4C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23C22×C4⋊C4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28C22×C4⋊C4C2211yescharacteristic
29C22×C4⋊C4C2211yescharacteristic
30C22×C22⋊C4C2211yescharacteristic
31C22×C22⋊C4C2211yescharacteristic
32C22×C22⋊C4C2211yescharacteristic
33C22×C22⋊C4C2211yescharacteristic
34C22×C22⋊C4C2211yescharacteristic
35C22×C22⋊C4C2211yescharacteristic
36C22×C42C2211yescharacteristic
37C22×C42C2211yescharacteristic
38C22×C42C2211yescharacteristic
39C22×C42C2211yescharacteristic
40C22×C42C2211yescharacteristic
41C22×C42C2211yescharacteristic
42#389C2218yescomplemented
43#389C2218yescomplemented
44#388C2218yes—
45#388C2218yes—
46#388C2218yes—
47#387C2218yescomplemented
48#387C2218yescomplemented
49#387C2218yescomplemented
50#385C2218yes—
51#385C2218yes—
52#382C2218yescomplemented
53C4×C4⋊C4C2218yescomplemented
54C4×C4⋊C4C2218yescomplemented
55C4×C22⋊C4C2218yes—
56C2×#95C2211yescharacteristic
57C2×#95C2211yescharacteristic
58C2×#95C2211yescharacteristic
59C2×#95C2211yescharacteristic
60C2×#95C2211yescharacteristic
61C2×#95C2211yescharacteristic
62C2×#95C2211yescharacteristic
63C2×#95C2211yescharacteristic
64C2×#95C2211yescharacteristic
65C2×#95C2211yescharacteristic
66C2×#95C2211yescharacteristic
67C2×#95C2211yescharacteristic
68C25C2311yescharacteristic
69C23×C4—24no—
70C23×C4—42no—
71C23×C4—42no—
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C23×C4C2311yescharacteristic
79C23×C4C2311yescharacteristic
80C23×C4C2311yescharacteristic
81C23×C4C2311yescharacteristic
82C23×C4C2311yescharacteristic
83C23×C4C2311yescharacteristic
84C23×C4C2311yescharacteristic
85C23×C4C2311yescharacteristic
86C23×C4C2314yes—
87#126C2×C4164yescomplemented
88C2×C4⋊C4—24no—
89C2×C4⋊C4—24no—
90C2×C4⋊C4—24no—
91C2×C4⋊C4—42no—
92C2×C4⋊C4—42no—
93C2×C4⋊C4—42no—
94C2×C4⋊C4—42no—
95C2×C4⋊C4C2×C418yescomplemented
96C2×C4⋊C4C2×C418yescomplemented
97C2×C4⋊C4C2×C418yescomplemented
98C2×C4⋊C4C2314yes—
99C2×C4⋊C4C2314yes—
100C2×C4⋊C4C2314yes—
101C2×C4⋊C4C2314yes—
102C2×C4⋊C4C2314yes—
103C2×C4⋊C4C2314yes—
104C2×C4⋊C4C2314yes—
105C2×C4⋊C4C2314yes—
106C2×C4⋊C4C2314yes—
107C2×C4⋊C4C2314yes—
108C2×C22⋊C4—24no—
109C2×C22⋊C4—42no—
110C2×C22⋊C4—42no—
111C2×C22⋊C4C2×C418yes—
112C2×C22⋊C4C2×C418yes—
113C2×C22⋊C4C2×C418yes—
114C2×C22⋊C4C2314yes—
115C2×C22⋊C4C2314yes—
116C2×C22⋊C4C2314yes—
117C2×C22⋊C4C2314yes—
118C2×C22⋊C4C2314yes—
119C2×C22⋊C4C2314yes—
120C2×C42—24no—
121C2×C42—42no—
122C2×C42—42no—
123C2×C42—42no—
124C2×C42—42no—
125C2×C42C2×C418yescomplemented
126C2×C42C2314yes—
127C2×C42C2314yes—
128C2×C42C2314yes—
129C2×C42C2314yes—
130C2×C42C2314yes—
131C2×C42C2314yes—
132#95C2314yes—
133#95C2314yes—
134#95C2314yes—
135#95C2314yes—
136#95C2314yes—
137#95C2314yes—
138#95C2314yes—
139#95C2314yes—
140#95C2314yes—
141#95C2314yes—
142#95C2314yes—
143#95C2314yes—
144C24—22no—
145C24—22no—
146C24—22no—
147C24—22no—
148C24—22no—
149C24—22no—
150C24C22×C414yes—
151C24C2412yes—
152C24C2411yescharacteristic
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×C4—22no—
166C22×C4—22no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—28no—
188C22×C4—44no—
189C22×C4—44no—
190C22×C4—416no—
191C22×C4C22×C414yes—
192C22×C4C22×C414yes—
193C22×C4C22×C414yes—
194C22×C4C22×C414yes—
195C22×C4C22×C414yes—
196C22×C4C22×C414yes—
197C22×C4#4114yes—
198C22×C4#4114yes—
199C22×C4C2412yes—
200C22×C4C2412yes—
201C22×C4C2412yes—
202C22×C4C2412yes—
203C22×C4C2412yes—
204C22×C4C2412yes—
205C22×C4C2412yes—
206C22×C4C2412yes—
207C22×C4C2412yes—
208C22×C4C2412yes—
209C22×C4C2412yes—
210C22×C4C2412yes—
211C22×C4C2412yes—
212C22×C4C2412yes—
213C4⋊C4—28no—
214C4⋊C4—28no—
215C4⋊C4—28no—
216C4⋊C4—44no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—44no—
220C4⋊C4C22×C4116yescomplemented
221C4⋊C4C22×C4116yescomplemented
222C4⋊C4C22×C4116yescomplemented
223C22⋊C4—28no—
224C22⋊C4—44no—
225C22⋊C4—44no—
226C22⋊C4C22×C4116yes—
227C22⋊C4C22×C4116yes—
228C22⋊C4C22×C4116yes—
229C42—28no—
230C42—44no—
231C42—44no—
232C42—44no—
233C42—44no—
234C42C22×C4116yescomplemented
235C23—28no—
236C23—28no—
237C23—28no—
238C23—22no—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—22no—
243C23—22no—
244C23—44no—
245C23—44no—
246C23—44no—
247C23—44no—
248C23C23×C414yes—
249C23C23×C412yes—
250C23C2×#4112yes—
251C23C2×#4112yes—
252C23H(5,𝔽2)12yes—
253C23H(5,𝔽2)12yes—
254C232−1+412yes—
255C232−1+412yes—
256C23C2511yescharacteristic
257C2×C4—28no—
258C2×C4—28no—
259C2×C4—28no—
260C2×C4—28no—
261C2×C4—28no—
262C2×C4—28no—
263C2×C4—28no—
264C2×C4—28no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—416no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—44no—
306C2×C4C23×C414yes—
307C2×C4C23×C414yes—
308C2×C4C23×C414yes—
309C2×C4C23×C414yes—
310C2×C4C23×C414yes—
311C2×C4C23×C414yes—
312C2×C4C2×#4114yes—
313C2×C4C2×#4114yes—
314C22—24no—
315C22—24no—
316C22—24no—
317C22—416no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22C4×#4114yes—
323C22#52014yes—
324C22#52014yes—
325C22#53814yes—
326C22#53914yes—
327C22#54114yes—
328C22#54214yes—
329C22C24×C411yescharacteristic
330C22C22×#4111yescharacteristic
331C22C22×#4111yescharacteristic
332C22C2×H(5,𝔽2)11yescharacteristic
333C22C2×H(5,𝔽2)11yescharacteristic
334C22C2×2−1+411yescharacteristic
335C22C2×2−1+411yescharacteristic
336C4—24no—
337C4—24no—
338C4—24no—
339C4—24no—
340C4—24no—
341C4—24no—
342C4—24no—
343C4—24no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—42no—
350C2—44no—
351C2#233418yesdirect product factor
352C2C2×C4×#4111yescharacteristic
353C2C2×#52011yescharacteristic
354C2C2×#52011yescharacteristic
355C2C2×#53811yescharacteristic
356C2C2×#53911yescharacteristic
357C2C2×#54111yescharacteristic
358C2C2×#54211yescharacteristic
subgroupG/H
center, socleC24C24
commutatorC22C24×C4
Frattini, cosocleC23C25
prime radicalC25C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2334C2direct (N × H)
3. C2×#389C2semidirect
4. C2×#389C2semidirect
8. C2×#387C2semidirect
9. C2×#387C2semidirect
10. C2×#387C2semidirect
13. C2×#382C2semidirect
14. C2×C4×C4⋊C4C2semidirect
15. C2×C4×C4⋊C4C2semidirect
18. C2×#126C4semidirect
42. #389C22semidirect
43. #389C22semidirect
47. #387C22semidirect
48. #387C22semidirect
49. #387C22semidirect
52. #382C22semidirect
53. C4×C4⋊C4C22semidirect
54. C4×C4⋊C4C22semidirect
87. #126C2×C4semidirect
95. C2×C4⋊C4C2×C4semidirect
96. C2×C4⋊C4C2×C4semidirect
97. C2×C4⋊C4C2×C4semidirect
125. C2×C42C2×C4semidirect
220. C4⋊C4C22×C4semidirect
221. C4⋊C4C22×C4semidirect
222. C4⋊C4C22×C4semidirect
234. C42C22×C4semidirect
351. C2#2334direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#1262
#23342
C2×#3892
C2×#3882
C2×#3872
C2×#3852
C2×#3822
C2×C4×C4⋊C42
C2×C4×C22⋊C42