Finite Group Databasebeta

← C2×#3301 #61119

Subgroups — C2×#3301 #61119

Order 256.

1105 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#553C211yesmaximal, characteristic, complemented
2C2×#548C211yesmaximal, characteristic, complemented
3#3301C2116yesmaximal, direct product factor
4C2×#488C211yesmaximal, characteristic, complemented
5C2×#481C211yesmaximal, characteristic, complemented
6C2×#485C211yesmaximal, characteristic, complemented
7C2×#479C211yesmaximal, characteristic, complemented
8C2×#466C211yesmaximal, characteristic, complemented
9C2×#465C211yesmaximal, characteristic, complemented
10C2×#463C211yesmaximal, characteristic, complemented
11C2×#449C211yesmaximal, characteristic, complemented
12C2×#448C211yesmaximal, characteristic, complemented
13C2×C4×SD16C211yesmaximal, characteristic, complemented
14C2×C8×D4C211yesmaximal, characteristic, complemented
15C2×#416C211yesmaximal, characteristic, complemented
16C22×#103C211yesmaximal, characteristic, complemented
17C22×#41C2211yescharacteristic
18C23×Q8C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×D8C2211yescharacteristic
22C23×C8C2211yescharacteristic
23C23×C8C2211yescharacteristic
24#553—24no—
25#553C2218yescomplemented
26#548—24no—
27#548C2218yescomplemented
28C2×#126—21no—
29C2×#125C2211yescharacteristic
30C2×#124—21no—
31C2×#124C2211yescharacteristic
32C2×#123—21no—
33C2×C22⋊Q8—21no—
34C2×C22⋊Q8C2211yescharacteristic
35C2×#121—21no—
36C2×#121—21no—
37C2×#121C2211yescharacteristic
38C2×#121C2211yescharacteristic
39C2×#121C2211yescharacteristic
40C2×C4×Q8C2211yescharacteristic
41C2×C4×D4—21no—
42C2×C4×D4—21no—
43C2×C4×D4C2211yescharacteristic
44C2×C4×D4C2211yescharacteristic
45C2×#117C2211yescharacteristic
46C22×C4⋊C4C2211yescharacteristic
47#488C2218yescomplemented
48#485C2218yescomplemented
49#481C2218yescomplemented
50#479C2218yescomplemented
51#466C2218yescomplemented
52#465C2218yescomplemented
53#463C2218yescomplemented
54#449C2218yescomplemented
55#448C2218yescomplemented
56C4×SD16C2218yescomplemented
57C8×D4C2218yescomplemented
58C2×#107C2211yescharacteristic
59C2×#106C2211yescharacteristic
60C2×#106C2211yescharacteristic
61C2×C4⋊C8C2211yescharacteristic
62#416C2218yescomplemented
63C2×#103C2211yescharacteristic
64C2×#103C2211yescharacteristic
65C2×#103C2211yescharacteristic
66C2×#103C2211yescharacteristic
67C2×#103C2211yescharacteristic
68C2×#103C2211yescharacteristic
69C2×#103C2218yescomplemented
70C2×#102C2211yescharacteristic
71C2×#102C2211yescharacteristic
72C2×#102C2211yescharacteristic
73C2×#102C2211yescharacteristic
74C2×C22⋊C8C2211yescharacteristic
75C2×C22⋊C8C2211yescharacteristic
76C2×C4×C8C2211yescharacteristic
77C2×#41—22no—
78C2×#41—24no—
79C2×#41—42no—
80C2×#41C2314yes—
81C22×Q8—21no—
82C22×Q8—22no—
83C22×Q8—24no—
84C22×Q8—42no—
85C22×Q8C2311yescharacteristic
86C22×Q8C2314yes—
87C22×Q8C2311yescharacteristic
88C22×D4—21no—
89C22×D4—21no—
90C22×D4—21no—
91C22×D4—21no—
92C22×D4D411yescharacteristic
93C22×D4C2311yescharacteristic
94C22×D4C2311yescharacteristic
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4C2311yescharacteristic
100C23×C4C2311yescharacteristic
101C2×SD16—24no—
102C2×SD16—42no—
103C2×SD16C2314yes—
104C2×SD16C2314yes—
105C2×D8—42no—
106C2×D8C2314yes—
107C22×C8—42no—
108C22×C8—42no—
109C22×C8C2311yescharacteristic
110C22×C8C2311yescharacteristic
111C22×C8C2311yescharacteristic
112C22×C8C2311yescharacteristic
113C22×C8C2314yes—
114C22×C8C2314yes—
115#126—28no—
116#125—22no—
117#125C2314yes—
118#124—28no—
119#124—22no—
120#124C2314yes—
121#123—28no—
122C22⋊Q8—22no—
123C22⋊Q8—28no—
124C22⋊Q8C2314yes—
125#121—28no—
126#121—28no—
127#121—22no—
128#121—22no—
129#121—22no—
130#121C2314yes—
131#121C2314yes—
132#121C2314yes—
133C4×Q8—22no—
134C4×Q8C2314yes—
135C4×D4—28no—
136C4×D4—28no—
137C4×D4—22no—
138C4×D4D414yes—
139C4×D4C2314yes—
140C4×D4C2314yes—
141#117—22no—
142#117C2314yes—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4D411yescharacteristic
150C2×C4⋊C4C2311yescharacteristic
151C2×C4⋊C4C2311yescharacteristic
152C2×C4⋊C4C2314yes—
153C2×C4⋊C4C2311yescharacteristic
154C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4D411yescharacteristic
164C2×C22⋊C4D411yescharacteristic
165C2×C42—21no—
166C2×C42—21no—
167C2×C42C2311yescharacteristic
168#107C2314yes—
169#106C2314yes—
170#106C2314yes—
171C4⋊C8C2314yes—
172#103C2314yes—
173#103C2314yes—
174#103C2314yes—
175#103C2314yes—
176#103C2314yes—
177#103C2314yes—
178#102C2314yes—
179#102C2314yes—
180#102C2314yes—
181#102C2314yes—
182C22⋊C8C2314yes—
183C22⋊C8C2314yes—
184C4×C8C2314yes—
185C24—21no—
186C24—21no—
187C24C2×D411yescharacteristic
188C24C2×D411yescharacteristic
189#41—28no—
190#41—44no—
191#41—48no—
192C2×Q8—21no—
193C2×Q8—24no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—28no—
197C2×Q8—22no—
198C2×Q8—22no—
199C2×Q8—44no—
200C2×Q8—44no—
201C2×Q8—48no—
202C2×Q8#4114yes—
203C2×Q8C2412yes—
204C2×Q8C2412yes—
205C2×D4—24no—
206C2×D4—24no—
207C2×D4—24no—
208C2×D4—24no—
209C2×D4—24no—
210C2×D4—21no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—44no—
217C2×D4—44no—
218C2×D4—44no—
219C2×D4—82no—
220C2×D4C2×D414yes—
221C2×D4C2412yes—
222C2×D4C2412yes—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—82no—
240C22×C4—82no—
241C22×C4C2×D412yes—
242C22×C4C2×D412yes—
243C22×C4C2×D411yescharacteristic
244C22×C4C2×D411yescharacteristic
245C22×C4C2×D411yescharacteristic
246C22×C4C2×D411yescharacteristic
247C22×C4C2412yes—
248C22×C4C2412yes—
249C22×C4C2411yescharacteristic
250SD16—28no—
251SD16—44no—
252D8—44no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—44no—
258C2×C8—44no—
259C2×C8C2412yes—
260C2×C8C2412yes—
261C2×C8C2412yes—
262C2×C8C2412yes—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4C2×D414yes—
273C4⋊C4C2412yes—
274C4⋊C4C2412yes—
275C4⋊C4C2412yes—
276C4⋊C4C2412yes—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4C2×D414yes—
286C22⋊C4C2×D414yes—
287C42—24no—
288C42—24no—
289C42C2×D412yes—
290C42C2412yes—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23#13514yes—
305C23C22×D412yes—
306C23C22×D412yes—
307C23C22×D411yescharacteristic
308Q8—24no—
309Q8—22no—
310Q8—22no—
311Q8—22no—
312Q8—48no—
313Q8—42no—
314Q8C2×#4114yes—
315D4—28no—
316D4—22no—
317D4—22no—
318D4—22no—
319D4—22no—
320D4—48no—
321D4—48no—
322D4—48no—
323D4—42no—
324D4—42no—
325D4—84no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—48no—
344C2×C4—48no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—84no—
351C2×C4—84no—
352C2×C4C22×D412yes—
353C2×C4C22×D412yes—
354C2×C4C22×D412yes—
355C2×C4C22×D412yes—
356C2×C4C22×D411yescharacteristic
357C2×C4C2×#4112yes—
358C2×C4H(5,𝔽2)12yes—
359C2×C4C2511yescharacteristic
360C8—22no—
361C8—22no—
362C8—22no—
363C8—22no—
364C22—22no—
365C22—28no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—44no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—84no—
378C22—84no—
379C22#54612yes—
380C22C2×#13514yes—
381C22C2×#13512yes—
382C22#57712yes—
383C22C23×D411yescharacteristic
384C4—22no—
385C4—22no—
386C4—24no—
387C4—22no—
388C4—22no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—42no—
393C4—42no—
394C4#54612yes—
395C4C22×#4111yescharacteristic
396C4C2×H(5,𝔽2)11yescharacteristic
397C2—24no—
398C2—42no—
399C2—82no—
400C2—82no—
401C2#330114yesdirect product factor
402C2C2×#54611yescharacteristic
403C2C22×#13511yescharacteristic
404C2C2×#57711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
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×#553C2semidirect
2. C2×#548C2semidirect
3. #3301C2direct (N × H)
4. C2×#488C2semidirect
5. C2×#481C2semidirect
6. C2×#485C2semidirect
7. C2×#479C2semidirect
8. C2×#466C2semidirect
9. C2×#465C2semidirect
10. C2×#463C2semidirect
11. C2×#449C2semidirect
12. C2×#448C2semidirect
13. C2×C4×SD16C2semidirect
14. C2×C8×D4C2semidirect
15. C2×#416C2semidirect
16. C22×#103C2semidirect
25. #553C22semidirect
27. #548C22semidirect
47. #488C22semidirect
48. #485C22semidirect
49. #481C22semidirect
50. #479C22semidirect
51. #466C22semidirect
52. #465C22semidirect
53. #463C22semidirect
54. #449C22semidirect
55. #448C22semidirect
56. C4×SD16C22semidirect
57. C8×D4C22semidirect
62. #416C22semidirect
69. C2×#103C22semidirect
401. C2#3301direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5532
C2×#5482
#33012
C2×#4882
C2×#4812
C2×#4852
C2×#4792
C2×#4662
C2×#4652
C2×#4632
C2×#4492
C2×#4482
C2×C4×SD162
C2×C8×D42
C2×#4162
C22×#1032