Finite Group Databasebeta

← C2×#3313 #61139

Subgroups — C2×#3313 #61139

Order 256.

1057 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#556C211yesmaximal, characteristic, complemented
2C2×#548C211yesmaximal, characteristic
3#3313C2116yesmaximal, direct product factor
4C2×#489C211yesmaximal, characteristic, complemented
5C2×#485C211yesmaximal, characteristic, complemented
6C2×#482C211yesmaximal, characteristic, complemented
7C2×#477C211yesmaximal, characteristic, complemented
8C2×#468C211yesmaximal, characteristic, complemented
9C2×#467C211yesmaximal, characteristic, complemented
10C2×#464C211yesmaximal, characteristic, complemented
11C2×#452C211yesmaximal, characteristic, complemented
12C2×#451C211yesmaximal, characteristic, complemented
13C2×#441C211yesmaximal, characteristic, complemented
14C2×#435C211yesmaximal, characteristic, complemented
15C2×#419C211yesmaximal, characteristic, complemented
16C2×#416C211yesmaximal, characteristic, complemented
17C22×#41C2211yescharacteristic
18C23×Q8C2211yescharacteristic
19C22×Q16C2211yescharacteristic
20C22×Q16C2211yescharacteristic
21C22×SD16C2211yescharacteristic
22C22×M16C2211yescharacteristic
23C23×C8C2211yescharacteristic
24#556—24no—
25#556C2218yescomplemented
26#548—24no—
27#548C2218yes—
28C2×C4⋊Q8C2211yescharacteristic
29C2×#126—21no—
30C2×#126—21no—
31C2×#124—21no—
32C2×#124—21no—
33C2×#124C2211yescharacteristic
34C2×C22⋊Q8—21no—
35C2×C22⋊Q8C2211yescharacteristic
36C2×C22⋊Q8C2211yescharacteristic
37C2×C22⋊Q8C2211yescharacteristic
38C2×#121—21no—
39C2×#121C2211yescharacteristic
40C2×C4×Q8—21no—
41C2×C4×Q8C2211yescharacteristic
42C2×C4×Q8C2211yescharacteristic
43C2×C4×D4—21no—
44C2×C4×D4C2211yescharacteristic
45C2×#117C2211yescharacteristic
46C2×#117C2211yescharacteristic
47#489C2218yescomplemented
48#485C2218yescomplemented
49#482C2218yescomplemented
50#477C2218yescomplemented
51#468C2218yescomplemented
52#467C2218yescomplemented
53#464C2218yescomplemented
54#452C2218yescomplemented
55#451C2218yescomplemented
56#441C2218yescomplemented
57#435C2218yescomplemented
58C2×#107C2211yescharacteristic
59C2×#107C2211yescharacteristic
60C2×#106C2211yescharacteristic
61C2×C4⋊C8C2211yescharacteristic
62#419C2218yescomplemented
63#416C2218yescomplemented
64C2×#103C2211yescharacteristic
65C2×#103C2211yescharacteristic
66C2×#103C2211yescharacteristic
67C2×#103C2211yescharacteristic
68C2×#103C2211yescharacteristic
69C2×#103C2211yescharacteristic
70C2×#103C2211yescharacteristic
71C2×#102C2211yescharacteristic
72C2×#102C2211yescharacteristic
73C2×#102C2211yescharacteristic
74C2×C22⋊C8C2211yescharacteristic
75C2×C22⋊C8C2211yescharacteristic
76C2×#97C2211yescharacteristic
77C2×#41—22no—
78C2×#41—24no—
79C2×#41—42no—
80C2×#41C2314yes—
81C22×Q8—21no—
82C22×Q8—21no—
83C22×Q8—22no—
84C22×Q8—24no—
85C22×Q8—42no—
86C22×Q8C2311yescharacteristic
87C22×Q8C2311yescharacteristic
88C22×Q8C2314yes—
89C22×Q8C2311yescharacteristic
90C22×D4—21no—
91C22×D4—21no—
92C22×D4D411yescharacteristic
93C22×D4C2311yescharacteristic
94C23×C4—21no—
95C23×C4—21no—
96C23×C4C2311yescharacteristic
97C23×C4C2311yescharacteristic
98C2×Q16—24no—
99C2×Q16—42no—
100C2×Q16C2314yes—
101C2×Q16C2314yes—
102C2×SD16—42no—
103C2×SD16C2314yes—
104C2×M16—42no—
105C2×M16C2314yes—
106C22×C8—42no—
107C22×C8C2311yescharacteristic
108C22×C8C2311yescharacteristic
109C22×C8C2311yescharacteristic
110C22×C8C2311yescharacteristic
111C22×C8C2314yes—
112C4⋊Q8—22no—
113C4⋊Q8C2314yes—
114#126—28no—
115#126—28no—
116#124—28no—
117#124—28no—
118#124—22no—
119#124C2314yes—
120C22⋊Q8—22no—
121C22⋊Q8—22no—
122C22⋊Q8—22no—
123C22⋊Q8—28no—
124C22⋊Q8C2314yes—
125C22⋊Q8C2314yes—
126C22⋊Q8C2314yes—
127#121—28no—
128#121—22no—
129#121C2314yes—
130C4×Q8—28no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8C2314yes—
134C4×Q8C2314yes—
135C4×D4—28no—
136C4×D4D414yes—
137C4×D4C2314yes—
138#117—22no—
139#117—22no—
140#117C2314yes—
141#117C2314yes—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4D411yescharacteristic
149C2×C4⋊C4C2311yescharacteristic
150C2×C4⋊C4C2311yescharacteristic
151C2×C4⋊C4C2311yescharacteristic
152C2×C4⋊C4C2311yescharacteristic
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4D411yescharacteristic
162C2×C22⋊C4D411yescharacteristic
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42C2311yescharacteristic
168#107C2314yes—
169#107C2314yes—
170#106C2314yes—
171C4⋊C8C2314yes—
172#103C2314yes—
173#103C2314yes—
174#103C2314yes—
175#103C2314yes—
176#103C2314yes—
177#103C2314yes—
178#103C2314yes—
179#102C2314yes—
180#102C2314yes—
181#102C2314yes—
182C22⋊C8C2314yes—
183C22⋊C8C2314yes—
184#97C2314yes—
185C24—21no—
186C24C2×D411yescharacteristic
187C24C2×D411yescharacteristic
188#41—28no—
189#41—44no—
190#41—48no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—24no—
194C2×Q8—24no—
195C2×Q8—21no—
196C2×Q8—22no—
197C2×Q8—28no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—48no—
205C2×Q8—44no—
206C2×Q8#4114yes—
207C2×Q8C2412yes—
208C2×Q8C2412yes—
209C2×Q8C2412yes—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—21no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—42no—
216C2×D4—44no—
217C2×D4—44no—
218C2×D4C2×D414yes—
219C2×D4C2412yes—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4C2×D412yes—
236C22×C4C2×D412yes—
237C22×C4C2×D411yescharacteristic
238C22×C4C2×D411yescharacteristic
239C22×C4C2×D411yescharacteristic
240C22×C4C2×D411yescharacteristic
241C22×C4C2412yes—
242C22×C4C2412yes—
243C22×C4C2411yescharacteristic
244Q16—28no—
245Q16—44no—
246SD16—44no—
247M16—44no—
248C2×C8—22no—
249C2×C8—22no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—44no—
253C2×C8C2412yes—
254C2×C8C2412yes—
255C2×C8C2412yes—
256C2×C8C2412yes—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4C2×D414yes—
268C4⋊C4C2412yes—
269C4⋊C4C2412yes—
270C4⋊C4C2412yes—
271C4⋊C4C2412yes—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4C2×D414yes—
281C22⋊C4C2×D414yes—
282C42—24no—
283C42—24no—
284C42—24no—
285C42—24no—
286C42C2×D412yes—
287C42C2412yes—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23C22×D412yes—
299C23C22×D412yes—
300C23C22×D411yescharacteristic
301Q8—24no—
302Q8—22no—
303Q8—22no—
304Q8—22no—
305Q8—22no—
306Q8—22no—
307Q8—48no—
308Q8—48no—
309Q8—42no—
310Q8—42no—
311Q8C2×#4114yes—
312D4—22no—
313D4—22no—
314D4—44no—
315D4—48no—
316D4—48no—
317D4—42no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—48no—
337C2×C4—48no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4C22×D412yes—
345C2×C4C22×D412yes—
346C2×C4C22×D412yes—
347C2×C4C22×D412yes—
348C2×C4C22×D411yescharacteristic
349C2×C4C2×#4112yes—
350C2×C4H(5,𝔽2)12yes—
351C2×C4C2511yescharacteristic
352C8—22no—
353C8—22no—
354C8—22no—
355C8—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—44no—
363C22—44no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—84no—
368C22#54612yes—
369C22#57512yes—
370C22#57812yes—
371C22C23×D411yescharacteristic
372C4—22no—
373C4—22no—
374C4—24no—
375C4—22no—
376C4—22no—
377C4—42no—
378C4—42no—
379C4—42no—
380C4—42no—
381C4—42no—
382C4—42no—
383C4#54612yes—
384C4C22×#4111yescharacteristic
385C4C2×H(5,𝔽2)11yescharacteristic
386C2—42no—
387C2—42no—
388C2—82no—
389C2#331314yesdirect product factor
390C2C2×#54611yescharacteristic
391C2C2×#57511yescharacteristic
392C2C2×#57811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC2×#548C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#556C2semidirect
3. #3313C2direct (N × H)
4. C2×#489C2semidirect
5. C2×#485C2semidirect
6. C2×#482C2semidirect
7. C2×#477C2semidirect
8. C2×#468C2semidirect
9. C2×#467C2semidirect
10. C2×#464C2semidirect
11. C2×#452C2semidirect
12. C2×#451C2semidirect
13. C2×#441C2semidirect
14. C2×#435C2semidirect
15. C2×#419C2semidirect
16. C2×#416C2semidirect
25. #556C22semidirect
47. #489C22semidirect
48. #485C22semidirect
49. #482C22semidirect
50. #477C22semidirect
51. #468C22semidirect
52. #467C22semidirect
53. #464C22semidirect
54. #452C22semidirect
55. #451C22semidirect
56. #441C22semidirect
57. #435C22semidirect
62. #419C22semidirect
63. #416C22semidirect
389. C2#3313direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#5482
#33132
C2×#4892
C2×#4852
C2×#4822
C2×#4772
C2×#4682
C2×#4672
C2×#4642
C2×#4522
C2×#4512
C2×#4412
C2×#4352
C2×#4192
C2×#4162