Finite Group Databasebeta

← C2×#3118 #60547

Subgroups — C2×#3118 #60547

Order 256.

1137 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#542C211yesmaximal, characteristic, complemented
2C2×#518C211yesmaximal, characteristic, complemented
3#3118C2116yesmaximal, direct product factor
4C2×#485C211yesmaximal, characteristic, complemented
5C2×#482C211yesmaximal, characteristic, complemented
6C2×#476C211yesmaximal, characteristic, complemented
7C2×#474C211yesmaximal, characteristic, complemented
8C2×#463C211yesmaximal, characteristic, complemented
9C2×#463C211yesmaximal, characteristic, complemented
10C2×#450C211yesmaximal, characteristic, complemented
11C2×#447C211yesmaximal, characteristic
12C2×#442C211yesmaximal, characteristic, complemented
13C2×#440C211yesmaximal, characteristic, complemented
14C2×C4×SD16C211yesmaximal, characteristic, complemented
15C2×C4×D8C211yesmaximal, characteristic, complemented
16C2×#433C211yesmaximal, characteristic, complemented
17C23×D4C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×D8C2211yescharacteristic
21C22×D8C2211yescharacteristic
22#542—24no—
23#542C2218yescomplemented
24C2×C4⋊Q8C2211yescharacteristic, complemented
25C2×#126—21no—
26C2×#124—21no—
27C2×#124C2211yescharacteristic, complemented
28C2×#123—21no—
29C2×C22⋊Q8—21no—
30C2×C22⋊Q8C2211yescharacteristic
31C2×#121C2211yescharacteristic
32#518—24no—
33#518C2218yescomplemented
34C2×C4×Q8C2211yescharacteristic, complemented
35C2×C4×D4—21no—
36C2×C4×D4—21no—
37C2×C4×D4C2211yescharacteristic
38C2×C4×D4C2211yescharacteristic
39C2×C4×D4C2211yescharacteristic
40C2×C4×D4C2211yescharacteristic
41C2×C4×D4C2211yescharacteristic, complemented
42C2×#117C2211yescharacteristic
43C2×#117C2211yescharacteristic
44C22×C22⋊C4—21no—
45C22×C22⋊C4—21no—
46#485C2218yescomplemented
47#482C2218yescomplemented
48#476C2218yescomplemented
49#474C2218yescomplemented
50#463C2218yescomplemented
51#463C2218yescomplemented
52#450C2218yescomplemented
53#447C2218yes—
54#442C2218yescomplemented
55#440C2218yescomplemented
56C4×SD16C2218yescomplemented
57C4×D8C2218yescomplemented
58#433C2218yescomplemented
59C2×#107C2211yescharacteristic, complemented
60C2×#107C2211yescharacteristic, complemented
61C2×#106C2211yescharacteristic, complemented
62C2×#106C2211yescharacteristic, complemented
63C2×C4⋊C8C2211yescharacteristic, complemented
64C2×C4⋊C8C2211yescharacteristic, complemented
65C2×#103C2211yescharacteristic, complemented
66C2×#103C2211yescharacteristic, complemented
67C2×#102C2211yescharacteristic
68C2×#102C2211yescharacteristic
69C2×#102C2211yescharacteristic
70C2×#102C2211yescharacteristic
71C2×#102C2211yescharacteristic, complemented
72C2×#102C2211yescharacteristic, complemented
73C2×C22⋊C8C2211yescharacteristic
74C2×C22⋊C8C2211yescharacteristic
75C2×#97C2211yescharacteristic, complemented
76C2×C4×C8C2211yescharacteristic, complemented
77C25—21no—
78C22×Q8—21no—
79C22×Q8C2311yescharacteristic
80C22×D4—24no—
81C22×D4—24no—
82C22×D4—21no—
83C22×D4—21no—
84C22×D4—22no—
85C22×D4C2311yescharacteristic
86C22×D4C2314yes—
87C22×D4C2311yescharacteristic
88C22×D4C2311yescharacteristic
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4C2311yescharacteristic
95C2×SD16—24no—
96C2×SD16—24no—
97C2×SD16C2314yes—
98C2×SD16C2314yes—
99C2×D8—24no—
100C2×D8—24no—
101C2×D8C2314yes—
102C2×D8C2314yes—
103C22×C8C2311yescharacteristic
104C22×C8C2311yescharacteristic
105C22×C8C2311yescharacteristic
106C22×C8C2311yescharacteristic
107C4⋊Q8—22no—
108C4⋊Q8C2314yescomplemented
109#126—28no—
110#124—28no—
111#124—22no—
112#124C2314yescomplemented
113#123—28no—
114C22⋊Q8—22no—
115C22⋊Q8—28no—
116C22⋊Q8C2314yes—
117#121—22no—
118#121C2314yes—
119C4×Q8—22no—
120C4×Q8C2314yescomplemented
121C4×D4—28no—
122C4×D4—28no—
123C4×D4—22no—
124C4×D4—22no—
125C4×D4—22no—
126C4×D4—22no—
127C4×D4—22no—
128C4×D4C2314yes—
129C4×D4C2314yes—
130C4×D4C2314yes—
131C4×D4C2314yes—
132C4×D4C2314yescomplemented
133#117—22no—
134#117D414yes—
135#117C2314yes—
136#117C2314yes—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4D411yescharacteristic
141C2×C4⋊C4D411yescharacteristic
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2311yescharacteristic
145C2×C4⋊C4C2311yescharacteristic
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
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—28no—
157C2×C22⋊C4—28no—
158C2×C22⋊C4D411yescharacteristic
159C2×C22⋊C4D411yescharacteristic
160C2×C42—21no—
161C2×C42—21no—
162C2×C42C2311yescharacteristic
163C2×C42C2311yescharacteristic
164#107C2314yescomplemented
165#107C2314yescomplemented
166#106C2314yescomplemented
167#106C2314yescomplemented
168C4⋊C8C2314yescomplemented
169C4⋊C8C2314yescomplemented
170#103C2314yescomplemented
171#103C2314yescomplemented
172#102C2314yes—
173#102C2314yes—
174#102C2314yes—
175#102C2314yes—
176#102C2314yescomplemented
177#102C2314yescomplemented
178C22⋊C8C2314yes—
179C22⋊C8C2314yes—
180#97C2314yescomplemented
181C4×C8C2314yescomplemented
182C24—28no—
183C24—21no—
184C24—21no—
185C24—24no—
186C24—21no—
187C24—82no—
188C24—82no—
189C24C2×D411yescharacteristic
190C2×Q8—21no—
191C2×Q8—24no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—82no—
195C2×Q8C2412yes—
196C2×D4—28no—
197C2×D4—216no—
198C2×D4—28no—
199C2×D4—28no—
200C2×D4—24no—
201C2×D4—24no—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—82no—
210C2×D4#4114yes—
211C2×D4#4114yes—
212C2×D4C2412yes—
213C2×D4C2412yes—
214C2×D4C2412yes—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4C2×D412yes—
233C22×C4C2×D411yescharacteristic
234C22×C4C2×D411yescharacteristic
235C22×C4C2×D411yescharacteristic
236C22×C4C2×D411yescharacteristic
237C22×C4C2×D411yescharacteristic
238C22×C4C2412yes—
239C22×C4C2411yescharacteristic
240SD16—28no—
241SD16—28no—
242D8—28no—
243D8—28no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8C2412yes—
249C2×C8C2412yes—
250C2×C8C2412yes—
251C2×C8C2412yes—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4C2×D414yes—
260C4⋊C4C2×D414yes—
261C4⋊C4C2412yes—
262C4⋊C4C2412yes—
263C4⋊C4C2412yes—
264C4⋊C4C2412yes—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4C2×D414yes—
276C22⋊C4C2×D414yes—
277C42—24no—
278C42—24no—
279C42C2×D412yes—
280C42C2×D412yes—
281C42C2412yes—
282C42C2412yes—
283C23—216no—
284C23—22no—
285C23—24no—
286C23—24no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—84no—
300C23—84no—
301C23—816no—
302C23C22×D412yes—
303C23C22×D411yescharacteristic
304Q8—22no—
305Q8—22no—
306Q8—42no—
307Q8—84no—
308D4—216no—
309D4—24no—
310D4—24no—
311D4—22no—
312D4—22no—
313D4—22no—
314D4—22no—
315D4—42no—
316D4—84no—
317D4C2×#4114yes—
318D4C2×#4114yes—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—84no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4C22×D412yes—
343C2×C4C22×D412yes—
344C2×C4C22×D412yes—
345C2×C4C22×D412yes—
346C2×C4C22×D412yes—
347C2×C4C22×D411yescharacteristic
348C2×C4C2×#4112yes—
349C2×C4C2×#4112yes—
350C2×C4C2511yescharacteristic
351C8—22no—
352C8—22no—
353C8—22no—
354C8—22no—
355C22—22no—
356C22—24no—
357C22—24no—
358C22—22no—
359C22—22no—
360C22—44no—
361C22—48no—
362C22—48no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—816no—
371C22—84no—
372C22#52512yes—
373C22#57612yes—
374C22#57712yes—
375C22C23×D411yescharacteristic
376C4—22no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4#52512yes—
387C4C22×#4111yescharacteristic
388C4C22×#4111yescharacteristic
389C2—42no—
390C2—44no—
391C2—44no—
392C2—82no—
393C2#311814yesdirect product factor
394C2C2×#52511yescharacteristic
395C2C2×#57611yescharacteristic
396C2C2×#57711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC2×#447C2
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×#518C2semidirect
3. #3118C2direct (N × H)
4. C2×#485C2semidirect
5. C2×#482C2semidirect
6. C2×#476C2semidirect
7. C2×#474C2semidirect
8. C2×#463C2semidirect
9. C2×#463C2semidirect
10. C2×#450C2semidirect
12. C2×#442C2semidirect
13. C2×#440C2semidirect
14. C2×C4×SD16C2semidirect
15. C2×C4×D8C2semidirect
16. C2×#433C2semidirect
23. #542C22semidirect
24. C2×C4⋊Q8C22semidirect
27. C2×#124C22semidirect
33. #518C22semidirect
34. C2×C4×Q8C22semidirect
41. C2×C4×D4C22semidirect
46. #485C22semidirect
47. #482C22semidirect
48. #476C22semidirect
49. #474C22semidirect
50. #463C22semidirect
51. #463C22semidirect
52. #450C22semidirect
54. #442C22semidirect
55. #440C22semidirect
56. C4×SD16C22semidirect
57. C4×D8C22semidirect
58. #433C22semidirect
59. C2×#107C22semidirect
60. C2×#107C22semidirect
61. C2×#106C22semidirect
62. C2×#106C22semidirect
63. C2×C4⋊C8C22semidirect
64. C2×C4⋊C8C22semidirect
65. C2×#103C22semidirect
66. C2×#103C22semidirect
71. C2×#102C22semidirect
72. C2×#102C22semidirect
75. C2×#97C22semidirect
76. C2×C4×C8C22semidirect
108. C4⋊Q8C23semidirect
112. #124C23semidirect
120. C4×Q8C23semidirect
132. C4×D4C23semidirect
164. #107C23semidirect
165. #107C23semidirect
166. #106C23semidirect
167. #106C23semidirect
168. C4⋊C8C23semidirect
169. C4⋊C8C23semidirect
170. #103C23semidirect
171. #103C23semidirect
176. #102C23semidirect
177. #102C23semidirect
180. #97C23semidirect
181. C4×C8C23semidirect
393. C2#3118direct (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×#5182
#31182
C2×#4852
C2×#4822
C2×#4762
C2×#4742
C2×#4632
C2×#4502
C2×#4472
C2×#4422
C2×#4402
C2×C4×SD162
C2×C4×D82
C2×#4332