Finite Group Databasebeta

← C2×#3124 #60553

Subgroups — C2×#3124 #60553

Order 256.

1067 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#542C211yesmaximal, characteristic, complemented
2C2×#520C211yesmaximal, characteristic, complemented
3#3124C2116yesmaximal, direct product factor
4C2×#484C211yesmaximal, characteristic, complemented
5C2×#480C211yesmaximal, characteristic, complemented
6C2×#477C211yesmaximal, characteristic, complemented
7C2×#474C211yesmaximal, characteristic, complemented
8C2×#464C211yesmaximal, characteristic, complemented
9C2×#463C211yesmaximal, characteristic, complemented
10C2×#452C211yesmaximal, characteristic, complemented
11C2×#449C211yesmaximal, characteristic
12C2×#440C211yesmaximal, characteristic, complemented
13C2×#440C211yesmaximal, characteristic, complemented
14C2×C4×Q16C211yesmaximal, characteristic, complemented
15C2×C4×D8C211yesmaximal, characteristic, complemented
16C2×#433C211yesmaximal, characteristic, complemented
17C22×#41C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×D8C2211yescharacteristic
22#542—24no—
23#542C2218yescomplemented
24C2×C4⋊Q8C2211yescharacteristic
25C2×#126—21no—
26C2×#124—21no—
27C2×#124C2211yescharacteristic
28C2×#123—21no—
29C2×C22⋊Q8—21no—
30C2×C22⋊Q8C2211yescharacteristic
31C2×#121C2211yescharacteristic
32#520—24no—
33#520C2218yescomplemented
34C2×C4×Q8C2211yescharacteristic
35C2×C4×Q8C2211yescharacteristic
36C2×C4×Q8C2211yescharacteristic
37C2×C4×D4—21no—
38C2×C4×D4—21no—
39C2×C4×D4C2211yescharacteristic
40C2×C4×D4C2211yescharacteristic
41C2×C4×D4C2211yescharacteristic
42C2×#117—21no—
43C2×#117C2211yescharacteristic
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2211yescharacteristic
46#484C2218yescomplemented
47#480C2218yescomplemented
48#477C2218yescomplemented
49#474C2218yescomplemented
50#464C2218yescomplemented
51#463C2218yescomplemented
52#452C2218yescomplemented
53#449C2218yes—
54#440C2218yescomplemented
55#440C2218yescomplemented
56C4×Q16C2218yescomplemented
57C4×D8C2218yescomplemented
58#433C2218yescomplemented
59C2×#107C2211yescharacteristic
60C2×#107C2211yescharacteristic
61C2×#107C2211yescharacteristic
62C2×#107C2211yescharacteristic
63C2×C4⋊C8C2211yescharacteristic
64C2×C4⋊C8C2211yescharacteristic
65C2×#103C2211yescharacteristic
66C2×#103C2211yescharacteristic
67C2×#103C2211yescharacteristic
68C2×#103C2211yescharacteristic
69C2×#102C2211yescharacteristic
70C2×#102C2211yescharacteristic
71C2×#102C2211yescharacteristic
72C2×#102C2211yescharacteristic
73C2×C22⋊C8C2211yescharacteristic
74C2×C22⋊C8C2211yescharacteristic
75C2×#97C2211yescharacteristic
76C2×C4×C8C2211yescharacteristic
77C2×#41—24no—
78C2×#41—24no—
79C2×#41—22no—
80C2×#41C2314yes—
81C22×Q8—21no—
82C22×Q8C2311yescharacteristic
83C22×Q8C2311yescharacteristic
84C22×D4—21no—
85C22×D4—21no—
86C22×D4C2311yescharacteristic
87C22×D4C2311yescharacteristic
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4C2311yescharacteristic
94C2×Q16—24no—
95C2×Q16C2314yes—
96C2×SD16—24no—
97C2×SD16—24no—
98C2×SD16C2314yes—
99C2×SD16C2314yes—
100C2×D8—24no—
101C2×D8C2314yes—
102C22×C8C2311yescharacteristic
103C22×C8C2311yescharacteristic
104C22×C8C2311yescharacteristic
105C22×C8C2311yescharacteristic
106C4⋊Q8—22no—
107C4⋊Q8C2314yes—
108#126—28no—
109#124—28no—
110#124—22no—
111#124C2314yes—
112#123—28no—
113C22⋊Q8—22no—
114C22⋊Q8—28no—
115C22⋊Q8C2314yes—
116#121—22no—
117#121C2314yes—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×Q8C2314yes—
122C4×Q8C2314yes—
123C4×Q8C2314yes—
124C4×D4—28no—
125C4×D4—28no—
126C4×D4—22no—
127C4×D4—22no—
128C4×D4—22no—
129C4×D4C2314yes—
130C4×D4C2314yes—
131C4×D4C2314yes—
132#117—28no—
133#117D414yes—
134#117C2314yes—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4D411yescharacteristic
144C2×C4⋊C4D411yescharacteristic
145C2×C4⋊C4C2311yescharacteristic
146C2×C4⋊C4C2311yescharacteristic
147C2×C4⋊C4C2314yes—
148C2×C4⋊C4C2311yescharacteristic
149C2×C4⋊C4C2311yescharacteristic
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4D411yescharacteristic
158C2×C22⋊C4D411yescharacteristic
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42C2311yescharacteristic
163C2×C42C2311yescharacteristic
164#107C2314yes—
165#107C2314yes—
166#107C2314yes—
167#107C2314yes—
168C4⋊C8C2314yes—
169C4⋊C8C2314yes—
170#103C2314yes—
171#103C2314yes—
172#103C2314yes—
173#103C2314yes—
174#102C2314yes—
175#102C2314yes—
176#102C2314yes—
177#102C2314yes—
178C22⋊C8C2314yes—
179C22⋊C8C2314yes—
180#97C2314yes—
181C4×C8C2314yes—
182C24—21no—
183C24—21no—
184C24C2×D411yescharacteristic
185#41—216no—
186#41—28no—
187#41—28no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—24no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—82no—
195C2×Q8#4114yes—
196C2×Q8C2412yes—
197C2×Q8C2412yes—
198C2×D4—28no—
199C2×D4—24no—
200C2×D4—24no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—82no—
207C2×D4#4114yes—
208C2×D4C2412yes—
209C2×D4C2412yes—
210C22×C4—28no—
211C22×C4—21no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4C2×D412yes—
229C22×C4C2×D411yescharacteristic
230C22×C4C2×D411yescharacteristic
231C22×C4C2×D411yescharacteristic
232C22×C4C2×D411yescharacteristic
233C22×C4C2×D411yescharacteristic
234C22×C4C2412yes—
235C22×C4C2411yescharacteristic
236Q16—28no—
237SD16—28no—
238SD16—28no—
239D8—28no—
240C2×C8—22no—
241C2×C8—22no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8C2412yes—
245C2×C8C2412yes—
246C2×C8C2412yes—
247C2×C8C2412yes—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4C2×D414yes—
259C4⋊C4C2×D414yes—
260C4⋊C4C2412yes—
261C4⋊C4C2412yes—
262C4⋊C4C2412yes—
263C4⋊C4C2412yes—
264C22⋊C4—24no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4C2×D414yes—
272C22⋊C4C2×D414yes—
273C42—24no—
274C42—24no—
275C42—24no—
276C42C2×D412yes—
277C42C2×D412yes—
278C42C2412yes—
279C42C2412yes—
280C23—22no—
281C23—24no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23C22×D412yes—
292C23C22×D411yescharacteristic
293Q8—24no—
294Q8—22no—
295Q8—22no—
296Q8—22no—
297Q8—42no—
298Q8—84no—
299Q8C2×#4114yes—
300D4—216no—
301D4—24no—
302D4—22no—
303D4—22no—
304D4—22no—
305D4—42no—
306D4—84no—
307D4C2×#4114yes—
308C2×C4—216no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4—84no—
332C2×C4—84no—
333C2×C4C22×D412yes—
334C2×C4C22×D412yes—
335C2×C4C22×D412yes—
336C2×C4C22×D412yes—
337C2×C4C22×D412yes—
338C2×C4C22×D411yescharacteristic
339C2×C4C2×#4112yes—
340C2×C4C2×#4112yes—
341C2×C4C2511yescharacteristic
342C8—22no—
343C8—22no—
344C8—22no—
345C8—22no—
346C22—22no—
347C22—24no—
348C22—22no—
349C22—22no—
350C22—44no—
351C22—48no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—84no—
358C22#52512yes—
359C22#57612yes—
360C22#57812yes—
361C22C23×D411yescharacteristic
362C4—22no—
363C4—24no—
364C4—22no—
365C4—22no—
366C4—22no—
367C4—22no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—42no—
372C4—42no—
373C4#52512yes—
374C4C22×#4111yescharacteristic
375C4C22×#4111yescharacteristic
376C2—42no—
377C2—44no—
378C2—82no—
379C2#312414yesdirect product factor
380C2C2×#52511yescharacteristic
381C2C2×#57611yescharacteristic
382C2C2×#57811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC2×#449C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#542C2semidirect
2. C2×#520C2semidirect
3. #3124C2direct (N × H)
4. C2×#484C2semidirect
5. C2×#480C2semidirect
6. C2×#477C2semidirect
7. C2×#474C2semidirect
8. C2×#464C2semidirect
9. C2×#463C2semidirect
10. C2×#452C2semidirect
12. C2×#440C2semidirect
13. C2×#440C2semidirect
14. C2×C4×Q16C2semidirect
15. C2×C4×D8C2semidirect
16. C2×#433C2semidirect
23. #542C22semidirect
33. #520C22semidirect
46. #484C22semidirect
47. #480C22semidirect
48. #477C22semidirect
49. #474C22semidirect
50. #464C22semidirect
51. #463C22semidirect
52. #452C22semidirect
54. #440C22semidirect
55. #440C22semidirect
56. C4×Q16C22semidirect
57. C4×D8C22semidirect
58. #433C22semidirect
379. C2#3124direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#5202
#31242
C2×#4842
C2×#4802
C2×#4772
C2×#4742
C2×#4642
C2×#4632
C2×#4522
C2×#4492
C2×#4402
C2×C4×Q162
C2×C4×D82
C2×#4332