Finite Group Databasebeta

← C2×#3121 #60550

Subgroups — C2×#3121 #60550

Order 256.

1029 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#542C211yesmaximal, characteristic
2C2×#519C211yesmaximal, characteristic, complemented
3#3121C2116yesmaximal, direct product factor
4C2×#485C211yesmaximal, characteristic, complemented
5C2×#482C211yesmaximal, characteristic
6C2×#477C211yesmaximal, characteristic, complemented
7C2×#475C211yesmaximal, characteristic, complemented
8C2×#464C211yesmaximal, characteristic, complemented
9C2×#464C211yesmaximal, characteristic, complemented
10C2×#451C211yesmaximal, characteristic, complemented
11C2×#448C211yesmaximal, characteristic
12C2×#441C211yesmaximal, characteristic, complemented
13C2×#440C211yesmaximal, characteristic, complemented
14C2×C4×Q16C211yesmaximal, characteristic, complemented
15C2×C4×SD16C211yesmaximal, characteristic, complemented
16C2×#433C211yesmaximal, characteristic, complemented
17C23×Q8C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×Q16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×SD16C2211yescharacteristic
22#542—24no—
23#542C2218yes—
24C2×C4⋊Q8C2211yescharacteristic
25C2×#126—21no—
26C2×#124—21no—
27C2×#124C2211yescharacteristic
28C2×#123—21no—
29C2×C22⋊Q8—21no—
30C2×C22⋊Q8C2211yescharacteristic
31C2×#121C2211yescharacteristic
32#519—24no—
33#519C2218yescomplemented
34C2×C4×Q8—21no—
35C2×C4×Q8—21no—
36C2×C4×Q8C2211yescharacteristic
37C2×C4×Q8C2211yescharacteristic
38C2×C4×Q8C2211yescharacteristic
39C2×C4×Q8C2211yescharacteristic
40C2×C4×Q8C2211yescharacteristic
41C2×C4×D4C2211yescharacteristic
42C2×#117—21no—
43C2×#117—21no—
44C2×#117C2211yescharacteristic
45C2×#117C2211yescharacteristic
46#485C2218yescomplemented
47#482C2218yes—
48#477C2218yescomplemented
49#475C2218yescomplemented
50#464C2218yescomplemented
51#464C2218yescomplemented
52#451C2218yescomplemented
53#448C2218yes—
54#441C2218yescomplemented
55#440C2218yescomplemented
56C4×Q16C2218yescomplemented
57C4×SD16C2218yescomplemented
58#433C2218yescomplemented
59C2×#107C2211yescharacteristic
60C2×#107C2211yescharacteristic
61C2×#106C2211yescharacteristic
62C2×#106C2211yescharacteristic
63C2×C4⋊C8C2211yescharacteristic
64C2×C4⋊C8C2211yescharacteristic
65C2×#103C2211yescharacteristic
66C2×#103C2211yescharacteristic
67C2×#103C2211yescharacteristic
68C2×#103C2211yescharacteristic
69C2×#103C2211yescharacteristic
70C2×#103C2211yescharacteristic
71C2×#102C2211yescharacteristic
72C2×#102C2211yescharacteristic
73C2×C22⋊C8C2211yescharacteristic
74C2×C22⋊C8C2211yescharacteristic
75C2×#97C2211yescharacteristic
76C2×C4×C8C2211yescharacteristic
77C22×Q8—24no—
78C22×Q8—24no—
79C22×Q8—21no—
80C22×Q8—22no—
81C22×Q8—21no—
82C22×Q8C2311yescharacteristic
83C22×Q8C2314yes—
84C22×Q8C2311yescharacteristic
85C22×Q8C2311yescharacteristic
86C22×D4—21no—
87C22×D4C2311yescharacteristic
88C23×C4—21no—
89C23×C4—21no—
90C23×C4C2311yescharacteristic
91C2×Q16—24no—
92C2×Q16—24no—
93C2×Q16C2314yes—
94C2×Q16C2314yes—
95C2×SD16—24no—
96C2×SD16—24no—
97C2×SD16C2314yes—
98C2×SD16C2314yes—
99C22×C8C2311yescharacteristic
100C22×C8C2311yescharacteristic
101C22×C8C2311yescharacteristic
102C22×C8C2311yescharacteristic
103C4⋊Q8—22no—
104C4⋊Q8C2314yes—
105#126—28no—
106#124—28no—
107#124—22no—
108#124C2314yes—
109#123—28no—
110C22⋊Q8—22no—
111C22⋊Q8—28no—
112C22⋊Q8C2314yes—
113#121—22no—
114#121C2314yes—
115C4×Q8—28no—
116C4×Q8—28no—
117C4×Q8—22no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8C2314yes—
123C4×Q8C2314yes—
124C4×Q8C2314yes—
125C4×Q8C2314yes—
126C4×Q8C2314yes—
127C4×D4—22no—
128C4×D4C2314yes—
129#117—22no—
130#117—28no—
131#117—28no—
132#117D414yes—
133#117C2314yes—
134#117C2314yes—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4D411yescharacteristic
143C2×C4⋊C4D411yescharacteristic
144C2×C4⋊C4C2311yescharacteristic
145C2×C4⋊C4C2311yescharacteristic
146C2×C4⋊C4C2311yescharacteristic
147C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
155C2×C22⋊C4D411yescharacteristic
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42C2311yescharacteristic
163C2×C42C2311yescharacteristic
164#107C2314yes—
165#107C2314yes—
166#106C2314yes—
167#106C2314yes—
168C4⋊C8C2314yes—
169C4⋊C8C2314yes—
170#103C2314yes—
171#103C2314yes—
172#103C2314yes—
173#103C2314yes—
174#103C2314yes—
175#103C2314yes—
176#102C2314yes—
177#102C2314yes—
178C22⋊C8C2314yes—
179C22⋊C8C2314yes—
180#97C2314yes—
181C4×C8C2314yes—
182C24—21no—
183C24C2×D411yescharacteristic
184C2×Q8—216no—
185C2×Q8—28no—
186C2×Q8—28no—
187C2×Q8—28no—
188C2×Q8—24no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—24no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—82no—
198C2×Q8#4114yes—
199C2×Q8#4114yes—
200C2×Q8C2412yes—
201C2×Q8C2412yes—
202C2×Q8C2412yes—
203C2×D4—24no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—82no—
208C2×D4C2412yes—
209C22×C4—28no—
210C22×C4—24no—
211C22×C4—21no—
212C22×C4—21no—
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—82no—
222C22×C4C2×D412yes—
223C22×C4C2×D411yescharacteristic
224C22×C4C2×D411yescharacteristic
225C22×C4C2×D411yescharacteristic
226C22×C4C2×D411yescharacteristic
227C22×C4C2×D411yescharacteristic
228C22×C4C2412yes—
229C22×C4C2411yescharacteristic
230Q16—28no—
231Q16—28no—
232SD16—28no—
233SD16—28no—
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8C2412yes—
239C2×C8C2412yes—
240C2×C8C2412yes—
241C2×C8C2412yes—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4C2×D414yes—
254C4⋊C4C2×D414yes—
255C4⋊C4C2412yes—
256C4⋊C4C2412yes—
257C4⋊C4C2412yes—
258C4⋊C4C2412yes—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4C2×D414yes—
266C22⋊C4C2×D414yes—
267C42—24no—
268C42—24no—
269C42—24no—
270C42—24no—
271C42—24no—
272C42—24no—
273C42C2×D412yes—
274C42C2×D412yes—
275C42C2412yes—
276C42C2412yes—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23C22×D412yes—
285C23C22×D411yescharacteristic
286Q8—216no—
287Q8—24no—
288Q8—24no—
289Q8—22no—
290Q8—22no—
291Q8—22no—
292Q8—22no—
293Q8—42no—
294Q8—84no—
295Q8C2×#4114yes—
296Q8C2×#4114yes—
297D4—22no—
298D4—22no—
299D4—42no—
300D4—84no—
301C2×C4—216no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—24no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—84no—
325C2×C4C22×D412yes—
326C2×C4C22×D412yes—
327C2×C4C22×D412yes—
328C2×C4C22×D412yes—
329C2×C4C22×D412yes—
330C2×C4C22×D411yescharacteristic
331C2×C4C2×#4112yes—
332C2×C4C2×#4112yes—
333C2×C4C2511yescharacteristic
334C8—22no—
335C8—22no—
336C8—22no—
337C8—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—44no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—84no—
346C22#52512yes—
347C22#57712yes—
348C22#57812yes—
349C22C23×D411yescharacteristic
350C4—24no—
351C4—24no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—22no—
356C4—22no—
357C4—42no—
358C4—42no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4#52512yes—
363C4C22×#4111yescharacteristic
364C4C22×#4111yescharacteristic
365C2—42no—
366C2—82no—
367C2#312114yesdirect product factor
368C2C2×#52511yescharacteristic
369C2C2×#57711yescharacteristic
370C2C2×#57811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC2×#121C22
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
2. C2×#519C2semidirect
3. #3121C2direct (N × H)
4. C2×#485C2semidirect
6. C2×#477C2semidirect
7. C2×#475C2semidirect
8. C2×#464C2semidirect
9. C2×#464C2semidirect
10. C2×#451C2semidirect
12. C2×#441C2semidirect
13. C2×#440C2semidirect
14. C2×C4×Q16C2semidirect
15. C2×C4×SD16C2semidirect
16. C2×#433C2semidirect
33. #519C22semidirect
46. #485C22semidirect
48. #477C22semidirect
49. #475C22semidirect
50. #464C22semidirect
51. #464C22semidirect
52. #451C22semidirect
54. #441C22semidirect
55. #440C22semidirect
56. C4×Q16C22semidirect
57. C4×SD16C22semidirect
58. #433C22semidirect
367. C2#3121direct (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×#5192
#31212
C2×#4852
C2×#4822
C2×#4772
C2×#4752
C2×#4642
C2×#4512
C2×#4482
C2×#4412
C2×#4402
C2×C4×Q162
C2×C4×SD162
C2×#4332