Finite Group Databasebeta

← C2×#3271 #61029

Subgroups — C2×#3271 #61029

Order 256.

1413 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#137C211yesmaximal, characteristic, complemented
2C22×#135C211yesmaximal, characteristic, complemented
3C2×#550C211yesmaximal, characteristic, complemented
4C2×#548C211yesmaximal, characteristic, complemented
5#3271C2116yesmaximal, direct product factor
6C2×#496C211yesmaximal, characteristic, complemented
7C2×#469C211yesmaximal, characteristic, complemented
8C2×#465C211yesmaximal, characteristic, complemented
9C2×#464C211yesmaximal, characteristic, complemented
10C2×#460C211yesmaximal, characteristic, complemented
11C2×#452C211yesmaximal, characteristic, complemented
12C2×#449C211yesmaximal, characteristic, complemented
13C2×#449C211yesmaximal, characteristic, complemented
14C2×#448C211yesmaximal, characteristic, complemented
15C2×#441C211yesmaximal, characteristic, complemented
16C2×#435C211yesmaximal, characteristic, complemented
17C22×#41C2211yescharacteristic
18C22×#41C2211yescharacteristic
19C22×#41C2211yescharacteristic
20C23×Q8C2211yescharacteristic
21C2×#137—28no—
22C2×#137C2218yescomplemented
23C2×#135—28no—
24C2×#135C2218yescomplemented
25C22×Q16C2211yescharacteristic
26C22×Q16C2211yescharacteristic
27C22×SD16C2211yescharacteristic
28C22×SD16C2211yescharacteristic
29C22×SD16C2211yescharacteristic
30C22×SD16C2211yescharacteristic
31C22×SD16C2211yescharacteristic
32C22×D8C2211yescharacteristic
33C22×D8C2211yescharacteristic
34C22×M16C2211yescharacteristic
35C23×C8C2211yescharacteristic
36#550—24no—
37#550C2218yescomplemented
38#548—24no—
39#548C2218yescomplemented
40C2×#127—21no—
41C2×#127C2211yescharacteristic
42C2×#124—21no—
43C2×#124C2211yescharacteristic
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8C2211yescharacteristic
46C2×#121—21no—
47C2×#121—21no—
48C2×#121—21no—
49C2×#121C2211yescharacteristic
50C2×#121C2211yescharacteristic
51C2×#121C2211yescharacteristic
52C2×C4×Q8C2211yescharacteristic
53C2×C4×Q8C2211yescharacteristic
54C2×C4×D4—21no—
55C2×C4×D4—21no—
56C2×C4×D4C2211yescharacteristic
57#496C2218yescomplemented
58#469C2218yescomplemented
59#465C2218yescomplemented
60#464C2218yescomplemented
61#460C2218yescomplemented
62#452C2218yescomplemented
63#449C2218yescomplemented
64#449C2218yescomplemented
65#448C2218yescomplemented
66#441C2218yescomplemented
67#435C2218yescomplemented
68C2×#106C2211yescharacteristic
69C2×C4⋊C8C2211yescharacteristic
70C2×#103C2211yescharacteristic
71C2×#103C2211yescharacteristic
72C2×#103C2211yescharacteristic
73C2×#102C2211yescharacteristic
74C2×#102C2211yescharacteristic
75C2×#102C2211yescharacteristic
76C2×C22⋊C8C2211yescharacteristic
77C2×C22⋊C8C2211yescharacteristic
78C2×#97C2211yescharacteristic
79C2×#41—22no—
80C2×#41—22no—
81C2×#41—22no—
82C2×#41—24no—
83C2×#41—24no—
84C2×#41—24no—
85C2×#41—42no—
86C2×#41—42no—
87C2×#41—42no—
88C2×#41C2314yes—
89C2×#41C2314yes—
90C2×#41C2314yes—
91C22×Q8—21no—
92C22×Q8—22no—
93C22×Q8—24no—
94C22×Q8—42no—
95C22×Q8C2311yescharacteristic
96C22×Q8C2314yes—
97C22×Q8C2311yescharacteristic
98C22×Q8C2311yescharacteristic
99C22×D4—21no—
100C22×D4—21no—
101C22×D4—21no—
102C22×D4—21no—
103C22×D4—21no—
104C22×D4—21no—
105C22×D4—21no—
106C22×D4—21no—
107C22×D4D411yescharacteristic
108C22×D4C2311yescharacteristic
109C22×D4C2311yescharacteristic
110C22×D4C2311yescharacteristic
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4C2311yescharacteristic
116C23×C4C2311yescharacteristic
117#137—232no—
118#135—232no—
119C2×Q16—24no—
120C2×Q16D418yescomplemented
121C2×Q16C2314yes—
122C2×Q16C2314yes—
123C2×SD16—24no—
124C2×SD16—24no—
125C2×SD16—24no—
126C2×SD16—24no—
127C2×SD16—42no—
128C2×SD16C2314yes—
129C2×SD16C2314yes—
130C2×SD16C2314yes—
131C2×SD16C2314yes—
132C2×SD16C2314yes—
133C2×D8—24no—
134C2×D8—42no—
135C2×D8C2314yes—
136C2×D8C2314yes—
137C2×M16—24no—
138C2×M16C2314yes—
139C22×C8—24no—
140C22×C8C2311yescharacteristic
141C22×C8C2314yes—
142C22×C8C2311yescharacteristic
143C22×C8C2311yescharacteristic
144C22×C8C2311yescharacteristic
145#127—28no—
146#127—22no—
147#127C2314yes—
148#124—28no—
149#124—22no—
150#124C2314yes—
151C22⋊Q8—22no—
152C22⋊Q8—28no—
153C22⋊Q8C2314yes—
154#121—28no—
155#121—22no—
156#121—22no—
157#121—22no—
158#121—28no—
159#121—28no—
160#121C2314yes—
161#121C2314yes—
162#121C2314yes—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8C2314yes—
166C4×Q8C2314yes—
167C4×D4—28no—
168C4×D4—28no—
169C4×D4D414yes—
170C4×D4C2314yes—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4D411yescharacteristic
175C2×C4⋊C4C2311yescharacteristic
176C2×C4⋊C4C2311yescharacteristic
177C2×C22⋊C4—21no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—21no—
181C2×C22⋊C4—21no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4D411yescharacteristic
184C2×C22⋊C4D411yescharacteristic
185C2×C42—21no—
186C2×C42—21no—
187C2×C42C2311yescharacteristic
188#106C2314yes—
189C4⋊C8C2314yes—
190#103C2314yes—
191#103C2314yes—
192#103C2314yes—
193#102C2314yes—
194#102C2314yes—
195#102C2314yes—
196C22⋊C8C2314yes—
197C22⋊C8C2314yes—
198#97C2314yes—
199C24—21no—
200C24—21no—
201C24—21no—
202C24C2×D411yescharacteristic
203C24C2×D411yescharacteristic
204#41—28no—
205#41—28no—
206#41—28no—
207#41—44no—
208#41—44no—
209#41—44no—
210#41—48no—
211#41—48no—
212#41—48no—
213C2×Q8—21no—
214C2×Q8—21no—
215C2×Q8—24no—
216C2×Q8—21no—
217C2×Q8—22no—
218C2×Q8—22no—
219C2×Q8—28no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—48no—
225C2×Q8C2×D414yes—
226C2×Q8C2×D414yes—
227C2×Q8C2412yes—
228C2×Q8C2412yes—
229C2×Q8C2412yes—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—24no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—24no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—82no—
254C2×D4—82no—
255C2×D4—82no—
256C2×D4C2×D414yes—
257C2×D4C2412yes—
258C2×D4C2412yes—
259C2×D4C2412yes—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4C2×D412yes—
278C22×C4C2×D412yes—
279C22×C4C2×D411yescharacteristic
280C22×C4C2×D411yescharacteristic
281C22×C4C2×D411yescharacteristic
282C22×C4C2×D411yescharacteristic
283C22×C4C2412yes—
284C22×C4C2412yes—
285C22×C4C2411yescharacteristic
286Q16—28no—
287Q16C2×D4116yescomplemented
288SD16—28no—
289SD16—28no—
290SD16—28no—
291SD16—28no—
292SD16—44no—
293D8—28no—
294D8—44no—
295M16—28no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—28no—
300C2×C8C2×D414yes—
301C2×C8C2412yes—
302C2×C8C2412yes—
303C2×C8C2412yes—
304C2×C8C2412yes—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4C2×D414yes—
311C4⋊C4C2412yes—
312C4⋊C4C2412yes—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—24no—
316C22⋊C4—24no—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4C2×D414yes—
320C22⋊C4C2×D414yes—
321C42—24no—
322C42—24no—
323C42C2×D412yes—
324C42C2412yes—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23C22×D412yes—
344C23C22×D412yes—
345C23C22×D411yescharacteristic
346Q8—24no—
347Q8—24no—
348Q8—22no—
349Q8—22no—
350Q8—22no—
351Q8—22no—
352Q8—48no—
353Q8—42no—
354Q8C22×D414yes—
355Q8C22×D414yes—
356D4—22no—
357D4—22no—
358D4—22no—
359D4—22no—
360D4—22no—
361D4—22no—
362D4—44no—
363D4—48no—
364D4—48no—
365D4—48no—
366D4—48no—
367D4—48no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—84no—
372D4—84no—
373D4—84no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—48no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4C22×D412yes—
399C2×C4C22×D412yes—
400C2×C4C22×D412yes—
401C2×C4C22×D412yes—
402C2×C4C22×D411yescharacteristic
403C2×C4C22×D412yes—
404C2×C4H(5,𝔽2)12yes—
405C2×C4C2511yescharacteristic
406C8—22no—
407C8—22no—
408C8—22no—
409C8C22×D414yes—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—44no—
417C22—44no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—84no—
428C22—84no—
429C22—84no—
430C22D4212yes—
431C22#57512yes—
432C22#57712yes—
433C22C23×D411yescharacteristic
434C4—22no—
435C4—22no—
436C4—24no—
437C4—24no—
438C4—22no—
439C4—22no—
440C4—42no—
441C4—42no—
442C4—42no—
443C4D4212yes—
444C4C23×D411yescharacteristic
445C4C2×H(5,𝔽2)11yescharacteristic
446C2—42no—
447C2—42no—
448C2—82no—
449C2—82no—
450C2—82no—
451C2#327114yesdirect product factor
452C2C2×D4211yescharacteristic
453C2C2×#57511yescharacteristic
454C2C2×#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. C22×#137C2semidirect
2. C22×#135C2semidirect
3. C2×#550C2semidirect
4. C2×#548C2semidirect
5. #3271C2direct (N × H)
6. C2×#496C2semidirect
7. C2×#469C2semidirect
8. C2×#465C2semidirect
9. C2×#464C2semidirect
10. C2×#460C2semidirect
11. C2×#452C2semidirect
12. C2×#449C2semidirect
13. C2×#449C2semidirect
14. C2×#448C2semidirect
15. C2×#441C2semidirect
16. C2×#435C2semidirect
22. C2×#137C22semidirect
24. C2×#135C22semidirect
37. #550C22semidirect
39. #548C22semidirect
57. #496C22semidirect
58. #469C22semidirect
59. #465C22semidirect
60. #464C22semidirect
61. #460C22semidirect
62. #452C22semidirect
63. #449C22semidirect
64. #449C22semidirect
65. #448C22semidirect
66. #441C22semidirect
67. #435C22semidirect
120. C2×Q16D4semidirect
287. Q16C2×D4semidirect
451. C2#3271direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#1372
C22×#1352
C2×#5502
C2×#5482
#32712
C2×#4962
C2×#4692
C2×#4652
C2×#4642
C2×#4602
C2×#4522
C2×#4492
C2×#4482
C2×#4412
C2×#4352