Finite Group Databasebeta

← C2×#2438 #34731

Subgroups — C2×#2438 #34731

Order 256.

1503 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C22⋊Q8C211yesmaximal, characteristic
2C22×#117C211yesmaximal, characteristic
3#2438C2116yesmaximal, direct product factor
4C2×#400C211yesmaximal, characteristic, complemented
5C2×#398C211yesmaximal, characteristic, complemented
6C2×#398C211yesmaximal, characteristic, complemented
7C2×#395C211yesmaximal, characteristic, complemented
8C2×#391C211yesmaximal, characteristic, complemented
9C2×#389C211yesmaximal, characteristic, complemented
10C2×#387C211yesmaximal, characteristic, complemented
11C2×#387C211yesmaximal, characteristic, complemented
12C2×#385C211yesmaximal, characteristic
13C2×#385C211yesmaximal, characteristic
14C2×#381C211yesmaximal, characteristic
15C2×#380C211yesmaximal, characteristic
16C22×#95C211yesmaximal, characteristic
17C23×Q8C2211yescharacteristic
18C24×C4C2211yescharacteristic
19C24×C4C2211yescharacteristic
20C24×C4C2211yescharacteristic
21C2×C22⋊Q8—28no—
22C2×C22⋊Q8C2218yes—
23C2×#117—28no—
24C2×#117C2218yes—
25C22×C4⋊C4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28C22×C4⋊C4C2211yescharacteristic
29C22×C4⋊C4C2211yescharacteristic
30C22×C4⋊C4C2211yescharacteristic
31C22×C4⋊C4C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33C22×C4⋊C4C2211yescharacteristic
34C22×C4⋊C4C2211yescharacteristic
35C22×C4⋊C4C2211yescharacteristic
36C22×C22⋊C4C2211yescharacteristic
37C22×C22⋊C4C2211yescharacteristic
38C22×C22⋊C4C2211yescharacteristic
39C22×C22⋊C4C2211yescharacteristic
40C22×C42C2211yescharacteristic
41C22×C42C2211yescharacteristic
42#400C2218yescomplemented
43#398C2218yescomplemented
44#398C2218yescomplemented
45#395C2218yescomplemented
46#391C2218yescomplemented
47#389C2218yescomplemented
48#387C2218yescomplemented
49#387C2218yescomplemented
50#385C2218yes—
51#385C2218yes—
52#381C2218yes—
53#380C2218yes—
54C2×#95C2211yescharacteristic
55C2×#95C2211yescharacteristic
56C2×#95C2211yescharacteristic
57C2×#95C2211yescharacteristic
58C2×#95C2211yescharacteristic
59C2×#95C2211yescharacteristic
60C2×#95C2211yescharacteristic
61C2×#95C2211yescharacteristic
62C2×#95C2211yescharacteristic
63C2×#95C2211yescharacteristic
64C2×#95C2218yes—
65C2×#95C2211yescharacteristic
66C2×#95C2211yescharacteristic
67C2×#95C2211yescharacteristic
68C2×#95C2211yescharacteristic
69C25C2311yescharacteristic
70C22×Q8—24no—
71C22×Q8—24no—
72C22×Q8—42no—
73C22×Q8C2314yes—
74C23×C4—24no—
75C23×C4—24no—
76C23×C4—24no—
77C23×C4—24no—
78C23×C4—24no—
79C23×C4—24no—
80C23×C4—42no—
81C23×C4—42no—
82C23×C4Q818yescomplemented
83C23×C4C2311yescharacteristic
84C23×C4C2311yescharacteristic
85C23×C4C2311yescharacteristic
86C23×C4C2311yescharacteristic
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C23×C4C2311yescharacteristic
91C23×C4C2314yes—
92C23×C4C2314yes—
93C23×C4C2311yescharacteristic
94C23×C4C2311yescharacteristic
95C23×C4C2311yescharacteristic
96C23×C4C2311yescharacteristic
97C23×C4C2314yes—
98C23×C4C2311yescharacteristic
99C23×C4C2311yescharacteristic
100C22⋊Q8—232no—
101#117—232no—
102C2×C4⋊C4—24no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—24no—
106C2×C4⋊C4—24no—
107C2×C4⋊C4—24no—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—42no—
111C2×C4⋊C4—42no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4C2314yes—
114C2×C4⋊C4C2314yes—
115C2×C4⋊C4C2314yes—
116C2×C4⋊C4C2314yes—
117C2×C4⋊C4C2314yes—
118C2×C4⋊C4C2314yes—
119C2×C4⋊C4C2314yes—
120C2×C4⋊C4C2314yes—
121C2×C4⋊C4C2314yes—
122C2×C4⋊C4C2314yes—
123C2×C4⋊C4C2314yes—
124C2×C22⋊C4—24no—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—24no—
127C2×C22⋊C4D418yes—
128C2×C22⋊C4C2314yes—
129C2×C22⋊C4C2314yes—
130C2×C22⋊C4C2314yes—
131C2×C22⋊C4C2314yes—
132C2×C42—24no—
133C2×C42—24no—
134C2×C42C2314yes—
135C2×C42C2314yes—
136#95C2314yes—
137#95C2314yes—
138#95C2314yes—
139#95C2314yes—
140#95C2314yes—
141#95C2314yes—
142#95C2314yes—
143#95C2314yes—
144#95C2314yes—
145#95C2314yes—
146#95C2314yes—
147#95C2314yes—
148#95C2314yes—
149#95C2314yes—
150C24—22no—
151C24—22no—
152C24—22no—
153C24—22no—
154C24C2×D414yes—
155C24C2×Q814yes—
156C24#4114yes—
157C24C2412yes—
158C24C2411yescharacteristic
159C2×Q8—28no—
160C2×Q8—28no—
161C2×Q8—44no—
162C2×Q8—416no—
163C22×C4—232no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—28no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—28no—
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—28no—
184C22×C4—22no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—28no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—28no—
205C22×C4—28no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—416no—
209C22×C4—416no—
210C22×C4C2×D414yes—
211C22×C4C2×D414yes—
212C22×C4C2×Q814yes—
213C22×C4C2×Q814yes—
214C22×C4C2×Q8116yescomplemented
215C22×C4#4114yes—
216C22×C4#4114yes—
217C22×C4C2412yes—
218C22×C4C2412yes—
219C22×C4C2412yes—
220C22×C4C2412yes—
221C22×C4C2412yes—
222C22×C4C2412yes—
223C22×C4C2412yes—
224C22×C4C2412yes—
225C22×C4C2412yes—
226C22×C4C2412yes—
227C22×C4C2412yes—
228C22×C4C2412yes—
229C22×C4C2412yes—
230C22×C4C2412yes—
231C4⋊C4—28no—
232C4⋊C4—28no—
233C4⋊C4—28no—
234C4⋊C4—28no—
235C4⋊C4—28no—
236C4⋊C4—28no—
237C4⋊C4—28no—
238C4⋊C4—28no—
239C4⋊C4—44no—
240C4⋊C4—44no—
241C4⋊C4—44no—
242C22⋊C4—28no—
243C22⋊C4—28no—
244C22⋊C4—28no—
245C22⋊C4C2×D4116yes—
246C42—28no—
247C42—28no—
248C23—28no—
249C23—28no—
250C23—28no—
251C23—28no—
252C23—28no—
253C23—28no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23C22⋊Q8116yes—
259C23C22×D414yes—
260C23C22×D412yes—
261C23C22×Q814yes—
262C23C22×Q812yes—
263C23C2×#4114yes—
264C23C2×#4112yes—
265C23C2×#4112yes—
266C23C2×#4112yes—
267C23H(5,𝔽2)12yes—
268C232−1+412yes—
269C23C2511yescharacteristic
270Q8—416no—
271C2×C4—28no—
272C2×C4—28no—
273C2×C4—232no—
274C2×C4—28no—
275C2×C4—28no—
276C2×C4—28no—
277C2×C4—28no—
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—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—416no—
315C2×C4—416no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4C22×D414yes—
325C2×C4C22×D414yes—
326C2×C4C22×Q814yes—
327C2×C4C22×Q814yes—
328C2×C4C2×#4114yes—
329C2×C4C2×#4114yes—
330C22—24no—
331C22—24no—
332C22—232no—
333C22—24no—
334C22—24no—
335C22—24no—
336C22—24no—
337C22C2×C22⋊Q818yes—
338C22C2×C22⋊Q814yes—
339C22#53314yes—
340C22#54414yes—
341C22#54614yes—
342C22#54714yes—
343C22#55114yes—
344C22#55214yes—
345C22C23×D411yescharacteristic
346C22C23×Q811yescharacteristic
347C22C22×#4111yescharacteristic
348C22C22×#4111yescharacteristic
349C22C22×#4111yescharacteristic
350C22C2×H(5,𝔽2)11yescharacteristic
351C22C2×2−1+411yescharacteristic
352C4—24no—
353C4—24no—
354C4—24no—
355C4—24no—
356C4—24no—
357C4—24no—
358C4—42no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4—42no—
363C4—42no—
364C4—42no—
365C4—42no—
366C2—28no—
367C2#243818yesdirect product factor
368C2C22×C22⋊Q811yescharacteristic
369C2C2×#53311yescharacteristic
370C2C2×#54411yescharacteristic
371C2C2×#54611yescharacteristic
372C2C2×#54711yescharacteristic
373C2C2×#55111yescharacteristic
374C2C2×#55211yescharacteristic
subgroupG/H
center, socleC24C24
commutator, Frattini, cosocleC23C25
prime radicalC25C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2438C2direct (N × H)
4. C2×#400C2semidirect
5. C2×#398C2semidirect
6. C2×#398C2semidirect
7. C2×#395C2semidirect
8. C2×#391C2semidirect
9. C2×#389C2semidirect
10. C2×#387C2semidirect
11. C2×#387C2semidirect
42. #400C22semidirect
43. #398C22semidirect
44. #398C22semidirect
45. #395C22semidirect
46. #391C22semidirect
47. #389C22semidirect
48. #387C22semidirect
49. #387C22semidirect
82. C23×C4Q8semidirect
214. C22×C4C2×Q8semidirect
367. C2#2438direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C22⋊Q82
C22×#1172
#24382
C2×#4002
C2×#3982
C2×#3952
C2×#3912
C2×#3892
C2×#3872
C2×#3852
C2×#3812
C2×#3802
C22×#952