Finite Group Databasebeta

← C2×#3201 #60745

Subgroups — C2×#3201 #60745

Order 256.

1093 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#537C211yesmaximal, characteristic
2C2×#536C211yesmaximal, characteristic, complemented
3#3201C2116yesmaximal, direct product factor
4C2×#484C211yesmaximal, characteristic, complemented
5C2×#485C211yesmaximal, characteristic, complemented
6C2×#482C211yesmaximal, characteristic, complemented
7C2×#483C211yesmaximal, characteristic, complemented
8C2×#470C211yesmaximal, characteristic, complemented
9C2×#468C211yesmaximal, characteristic, complemented
10C2×#467C211yesmaximal, characteristic, complemented
11C2×#465C211yesmaximal, characteristic, complemented
12C2×#452C211yesmaximal, characteristic, complemented
13C2×#452C211yesmaximal, characteristic, complemented
14C2×#451C211yesmaximal, characteristic, complemented
15C2×#448C211yesmaximal, characteristic, complemented
16C2×#408C211yesmaximal, characteristic, complemented
17C22×#41C2211yescharacteristic
18C22×#41C2211yescharacteristic
19C23×Q8C2211yescharacteristic
20C22×Q16C2211yescharacteristic
21C22×Q16C2211yescharacteristic
22C22×SD16C2211yescharacteristic
23C22×SD16C2211yescharacteristic
24C22×M16C2211yescharacteristic
25C23×C8C2211yescharacteristic
26#537—24no—
27#537C2218yes—
28#536—24no—
29#536C2218yescomplemented
30C2×C4⋊Q8—21no—
31C2×#124—21no—
32C2×#123—21no—
33C2×#123—21no—
34C2×C22⋊Q8—21no—
35C2×C22⋊Q8C2211yescharacteristic
36C2×C22⋊Q8C2211yescharacteristic
37C2×C22⋊Q8C2211yescharacteristic
38C2×C22⋊Q8C2211yescharacteristic
39C2×C22⋊Q8C2211yescharacteristic
40C2×C22⋊Q8C2211yescharacteristic
41C2×#121—21no—
42C2×#121—21no—
43C2×#121—21no—
44C2×#121C2211yescharacteristic
45C2×#121C2211yescharacteristic
46C2×#117C2211yescharacteristic
47C22×C4⋊C4C2211yescharacteristic
48#485C2218yescomplemented
49#484C2218yescomplemented
50#483C2218yescomplemented
51#482C2218yescomplemented
52#470C2218yescomplemented
53#468C2218yescomplemented
54#467C2218yescomplemented
55#465C2218yescomplemented
56#452C2218yescomplemented
57#452C2218yescomplemented
58#451C2218yescomplemented
59#448C2218yescomplemented
60C2×#107C2211yescharacteristic
61C2×#107C2211yescharacteristic
62C2×#106C2211yescharacteristic
63C2×#106C2211yescharacteristic
64C2×#103C2211yescharacteristic
65C2×#103C2211yescharacteristic
66C2×#103C2211yescharacteristic
67C2×#103C2211yescharacteristic
68C2×#103C2211yescharacteristic
69C2×#103C2211yescharacteristic
70C2×#102C2211yescharacteristic
71C2×#102C2211yescharacteristic
72#408C2218yescomplemented
73C2×C22⋊C8C2211yescharacteristic
74C2×C22⋊C8C2211yescharacteristic
75C2×C22⋊C8C2211yescharacteristic
76C2×C22⋊C8C2211yescharacteristic
77C2×#41—22no—
78C2×#41—42no—
79C2×#41—42no—
80C2×#41—42no—
81C2×#41—42no—
82C2×#41D414yes—
83C2×#41C2314yes—
84C2×#41C2314yes—
85C22×Q8—21no—
86C22×Q8—22no—
87C22×Q8—42no—
88C22×Q8—42no—
89C22×Q8D411yescharacteristic
90C22×Q8C2311yescharacteristic
91C22×Q8C2311yescharacteristic
92C22×Q8C2311yescharacteristic
93C22×Q8C2314yes—
94C22×D4—21no—
95C22×D4—21no—
96C22×D4—21no—
97C22×D4D411yescharacteristic
98C22×D4D411yescharacteristic
99C22×D4D411yescharacteristic
100C22×D4C2311yescharacteristic
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4C2311yescharacteristic
105C23×C4C2311yescharacteristic
106C23×C4C2311yescharacteristic
107C2×Q16—42no—
108C2×Q16—42no—
109C2×Q16C2314yes—
110C2×Q16C2314yes—
111C2×SD16—42no—
112C2×SD16—42no—
113C2×SD16C2314yes—
114C2×SD16C2314yes—
115C2×M16—42no—
116C2×M16C2314yes—
117C22×C8—42no—
118C22×C8C2311yescharacteristic
119C22×C8C2311yescharacteristic
120C22×C8C2311yescharacteristic
121C22×C8C2311yescharacteristic
122C22×C8C2314yes—
123C4⋊Q8—28no—
124#124—28no—
125#123—28no—
126#123—28no—
127C22⋊Q8—28no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8C2314yes—
135C22⋊Q8C2314yes—
136C22⋊Q8C2314yes—
137C22⋊Q8C2314yes—
138C22⋊Q8C2314yes—
139C22⋊Q8C2314yes—
140#121—28no—
141#121—28no—
142#121—28no—
143#121—22no—
144#121—22no—
145#121C2314yes—
146#121C2314yes—
147#117—22no—
148#117C2314yes—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2311yescharacteristic
158C2×C4⋊C4C2311yescharacteristic
159C2×C4⋊C4C2314yes—
160C2×C4⋊C4C2311yescharacteristic
161C2×C4⋊C4C2311yescharacteristic
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C42—21no—
172#107C2314yes—
173#107C2314yes—
174#106C2314yes—
175#106C2314yes—
176#103C2314yes—
177#103C2314yes—
178#103C2314yes—
179#103C2314yes—
180#103C2314yes—
181#103C2314yes—
182#102C2314yes—
183#102C2314yes—
184C22⋊C8C2314yes—
185C22⋊C8C2314yes—
186C22⋊C8C2314yes—
187C22⋊C8C2314yes—
188C24—21no—
189C24C2×D411yescharacteristic
190C24C2×D411yescharacteristic
191C24C2×D411yescharacteristic
192#41—48no—
193#41—44no—
194#41—44no—
195#41—44no—
196#41—44no—
197#41—48no—
198C2×Q8—21no—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—24no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—42no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—44no—
212C2×Q8—48no—
213C2×Q8C2×D414yes—
214C2×Q8C2412yes—
215C2×Q8C2412yes—
216C2×Q8C2412yes—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—21no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—44no—
227C2×D4—82no—
228C2×D4—82no—
229C2×D4C2×D414yes—
230C2×D4C2×D414yes—
231C2×D4C2×D414yes—
232C2×D4C2412yes—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—82no—
252C22×C4C2×D412yes—
253C22×C4C2×D412yes—
254C22×C4C2×D412yes—
255C22×C4C2×D411yescharacteristic
256C22×C4C2×D411yescharacteristic
257C22×C4C2×D411yescharacteristic
258C22×C4C2412yes—
259C22×C4C2412yes—
260C22×C4C2412yes—
261C22×C4C2411yescharacteristic
262Q16—44no—
263Q16—44no—
264SD16—44no—
265SD16—44no—
266M16—44no—
267C2×C8—22no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—44no—
272C2×C8C2412yes—
273C2×C8C2412yes—
274C2×C8C2412yes—
275C2×C8C2412yes—
276C4⋊C4—24no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4C2412yes—
288C4⋊C4C2412yes—
289C4⋊C4C2412yes—
290C4⋊C4C2412yes—
291C22⋊C4—24no—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C42—24no—
301C23—22no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23C22×D412yes—
315C23C22×D412yes—
316C23C22×D412yes—
317C23C22×D411yescharacteristic
318Q8—22no—
319Q8—22no—
320Q8—22no—
321Q8—22no—
322Q8—22no—
323Q8—22no—
324Q8—44no—
325Q8—48no—
326Q8—42no—
327Q8—42no—
328Q8—42no—
329D4—22no—
330D4—22no—
331D4—44no—
332D4—44no—
333D4—44no—
334D4—48no—
335D4—42no—
336D4—84no—
337D4—84no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—48no—
358C2×C4—48no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—84no—
367C2×C4C22×D412yes—
368C2×C4C22×D412yes—
369C2×C4C22×D412yes—
370C2×C4C22×D411yescharacteristic
371C2×C4H(5,𝔽2)12yes—
372C2×C4H(5,𝔽2)12yes—
373C2×C4C2511yescharacteristic
374C8—22no—
375C8—22no—
376C8—22no—
377C8—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—44no—
388C22—44no—
389C22—44no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—84no—
394C22#53412yes—
395C22#57712yes—
396C22#57812yes—
397C22C23×D411yescharacteristic
398C4—22no—
399C4—22no—
400C4—22no—
401C4—42no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—42no—
406C4—42no—
407C4—42no—
408C4#53412yes—
409C4C2×H(5,𝔽2)11yescharacteristic
410C4C2×H(5,𝔽2)11yescharacteristic
411C2—42no—
412C2—42no—
413C2—42no—
414C2—82no—
415C2#320114yesdirect product factor
416C2C2×#53411yescharacteristic
417C2C2×#57711yescharacteristic
418C2C2×#57811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC2×#537C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#536C2semidirect
3. #3201C2direct (N × H)
4. C2×#484C2semidirect
5. C2×#485C2semidirect
6. C2×#482C2semidirect
7. C2×#483C2semidirect
8. C2×#470C2semidirect
9. C2×#468C2semidirect
10. C2×#467C2semidirect
11. C2×#465C2semidirect
12. C2×#452C2semidirect
13. C2×#452C2semidirect
14. C2×#451C2semidirect
15. C2×#448C2semidirect
16. C2×#408C2semidirect
29. #536C22semidirect
48. #485C22semidirect
49. #484C22semidirect
50. #483C22semidirect
51. #482C22semidirect
52. #470C22semidirect
53. #468C22semidirect
54. #467C22semidirect
55. #465C22semidirect
56. #452C22semidirect
57. #452C22semidirect
58. #451C22semidirect
59. #448C22semidirect
72. #408C22semidirect
415. C2#3201direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5372
C2×#5362
#32012
C2×#4842
C2×#4852
C2×#4822
C2×#4832
C2×#4702
C2×#4682
C2×#4672
C2×#4652
C2×#4522
C2×#4512
C2×#4482
C2×#4082