Finite Group Databasebeta

← C2×#3345 #61237

Subgroups — C2×#3345 #61237

Order 256.

1079 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#553C211yesmaximal, characteristic, complemented
2C2×#547C211yesmaximal, characteristic, complemented
3#3345C2116yesmaximal, direct product factor
4C2×#501C211yesmaximal, characteristic, complemented
5C2×#481C211yesmaximal, characteristic, complemented
6C2×#481C211yesmaximal, characteristic, complemented
7C2×#482C211yesmaximal, characteristic, complemented
8C2×#480C211yesmaximal, characteristic, complemented
9C2×#478C211yesmaximal, characteristic, complemented
10C2×#476C211yesmaximal, characteristic, complemented
11C2×#469C211yesmaximal, characteristic, complemented
12C2×#466C211yesmaximal, characteristic, complemented
13C2×#442C211yesmaximal, characteristic, complemented
14C2×#435C211yesmaximal, characteristic, complemented
15C2×#428C211yesmaximal, characteristic, complemented
16C22×#106C211yesmaximal, characteristic, complemented
17C22×#41—21no—
18C22×D8C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C23×C8C2211yescharacteristic
21#553—24no—
22#553C2218yescomplemented
23#547—24no—
24#547C2218yescomplemented
25C2×C4⋊Q8C2211yescharacteristic
26C2×#126—21no—
27C2×#125C2211yescharacteristic
28C2×#123—21no—
29C2×#123—21no—
30C2×#123—21no—
31C2×C22⋊Q8—21no—
32C2×#121—21no—
33C2×#121C2211yescharacteristic
34C2×#121C2211yescharacteristic
35C2×#121C2211yescharacteristic
36C2×#121C2211yescharacteristic
37C2×C4×D4—21no—
38C2×C4×D4C2211yescharacteristic
39C2×C4×D4C2211yescharacteristic
40C2×C4×D4C2211yescharacteristic
41C2×#117C2211yescharacteristic
42C22×C4⋊C4C2211yescharacteristic
43C22×C4⋊C4C2211yescharacteristic
44C22×C4⋊C4C2211yescharacteristic
45#501C2218yescomplemented
46#482C2218yescomplemented
47#481C2218yescomplemented
48#481C2218yescomplemented
49#480C2218yescomplemented
50#478C2218yescomplemented
51#476C2218yescomplemented
52#469C2218yescomplemented
53#466C2218yescomplemented
54#442C2218yescomplemented
55#435C2218yescomplemented
56#428C2218yescomplemented
57C2×#107C2211yescharacteristic
58C2×#107C2211yescharacteristic
59C2×#107C2211yescharacteristic
60C2×#106C2211yescharacteristic
61C2×#106C2211yescharacteristic
62C2×#106C2211yescharacteristic
63C2×#106C2211yescharacteristic
64C2×#106C2211yescharacteristic
65C2×#106C2211yescharacteristic
66C2×#106C2218yescomplemented
67C2×C4⋊C8C2211yescharacteristic
68C2×#102C2211yescharacteristic
69C2×#102C2211yescharacteristic
70C2×#102C2211yescharacteristic
71C2×#102C2211yescharacteristic
72C2×#102C2211yescharacteristic
73C2×#102C2211yescharacteristic
74C2×C22⋊C8C2211yescharacteristic
75C2×C22⋊C8C2211yescharacteristic
76C2×#97C2211yescharacteristic
77C2×#41—28no—
78C2×#41—48no—
79C22×Q8—21no—
80C22×D4—21no—
81C22×D4—21no—
82C22×D4—21no—
83C22×D4—21no—
84C22×D4D411yescharacteristic
85C22×D4C2311yescharacteristic
86C22×D4C2311yescharacteristic
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4C2311yescharacteristic
93C23×C4C2311yescharacteristic
94C2×D8—24no—
95C2×D8C2314yes—
96C2×M16—24no—
97C2×M16C2314yes—
98C22×C8—24no—
99C22×C8C2311yescharacteristic
100C22×C8C2314yes—
101C22×C8C2311yescharacteristic
102C22×C8C2311yescharacteristic
103C22×C8C2311yescharacteristic
104C4⋊Q8—22no—
105C4⋊Q8C2314yes—
106#126—28no—
107#125—22no—
108#125C2314yes—
109#123—28no—
110#123—28no—
111#123—28no—
112C22⋊Q8—28no—
113#121—28no—
114#121—22no—
115#121—22no—
116#121—22no—
117#121—22no—
118#121C2314yes—
119#121C2314yes—
120#121C2314yes—
121#121C2314yes—
122C4×D4—28no—
123C4×D4—22no—
124C4×D4—22no—
125C4×D4D414yes—
126C4×D4C2314yes—
127C4×D4C2314yes—
128C4×D4C2314yes—
129#117—22no—
130#117C2314yes—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4D411yescharacteristic
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2311yescharacteristic
145C2×C4⋊C4C2311yescharacteristic
146C2×C4⋊C4C2314yes—
147C2×C4⋊C4C2314yes—
148C2×C4⋊C4C2314yes—
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×C42C2311yescharacteristic
161#107C2314yes—
162#107C2314yes—
163#107C2314yes—
164#106C2314yes—
165#106C2314yes—
166#106C2314yes—
167#106C2314yes—
168#106C2314yes—
169#106C2314yes—
170C4⋊C8C2314yes—
171#102C2314yes—
172#102C2314yes—
173#102C2314yes—
174#102C2314yes—
175#102C2314yes—
176#102C2314yes—
177C22⋊C8C2314yes—
178C22⋊C8C2314yes—
179#97C2314yes—
180C24—21no—
181C24—21no—
182C24C2×D411yescharacteristic
183C24C2×D411yescharacteristic
184#41—432no—
185C2×Q8—24no—
186C2×Q8—44no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—24no—
190C2×D4—24no—
191C2×D4—24no—
192C2×D4—21no—
193C2×D4—21no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—44no—
199C2×D4—44no—
200C2×D4—44no—
201C2×D4—82no—
202C2×D4C2×D414yes—
203C2×D4C2412yes—
204C2×D4C2412yes—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4C2×D412yes—
226C22×C4C2×D412yes—
227C22×C4C2×D411yescharacteristic
228C22×C4C2×D411yescharacteristic
229C22×C4C2×D411yescharacteristic
230C22×C4C2×D411yescharacteristic
231C22×C4C2412yes—
232C22×C4C2412yes—
233C22×C4C2411yescharacteristic
234D8—28no—
235M16—28no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—28no—
240C2×C8#4114yescomplemented
241C2×C8C2412yes—
242C2×C8C2412yes—
243C2×C8C2412yes—
244C2×C8C2412yes—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—21no—
257C4⋊C4C2×D414yes—
258C4⋊C4C2412yes—
259C4⋊C4C2412yes—
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—
274C42C2×D412yes—
275C42C2412yes—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23AGL(1,ℤ/8)14yes—
290C23C22×D412yes—
291C23C22×D412yes—
292C23C22×D411yescharacteristic
293Q8—48no—
294D4—28no—
295D4—22no—
296D4—22no—
297D4—22no—
298D4—22no—
299D4—48no—
300D4—48no—
301D4—48no—
302D4—42no—
303D4—42no—
304D4—84no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—48no—
322C2×C4—48no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—48no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—84no—
331C2×C4—84no—
332C2×C4C22×D412yes—
333C2×C4C22×D412yes—
334C2×C4C22×D412yes—
335C2×C4C22×D412yes—
336C2×C4C22×D411yescharacteristic
337C2×C4C2×#4112yes—
338C2×C42−1+412yes—
339C2×C4C2511yescharacteristic
340C8—22no—
341C8—22no—
342C8—22no—
343C8C2×#4114yescomplemented
344C22—28no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—44no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—84no—
358C22—84no—
359C22#54712yes—
360C22C2×AGL(1,ℤ/8)14yes—
361C22C2×AGL(1,ℤ/8)12yes—
362C22#57712yes—
363C22C23×D411yescharacteristic
364C4—22no—
365C4—22no—
366C4—22no—
367C4—22no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—42no—
372C4—42no—
373C4—42no—
374C4#54712yes—
375C4C22×#4111yescharacteristic
376C4C2×2−1+411yescharacteristic
377C2—24no—
378C2—42no—
379C2—82no—
380C2—82no—
381C2#334514yesdirect product factor
382C2C2×#54711yescharacteristic
383C2C22×AGL(1,ℤ/8)11yescharacteristic
384C2C2×#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. C2×#553C2semidirect
2. C2×#547C2semidirect
3. #3345C2direct (N × H)
4. C2×#501C2semidirect
5. C2×#481C2semidirect
6. C2×#481C2semidirect
7. C2×#482C2semidirect
8. C2×#480C2semidirect
9. C2×#478C2semidirect
10. C2×#476C2semidirect
11. C2×#469C2semidirect
12. C2×#466C2semidirect
13. C2×#442C2semidirect
14. C2×#435C2semidirect
15. C2×#428C2semidirect
16. C22×#106C2semidirect
22. #553C22semidirect
24. #547C22semidirect
45. #501C22semidirect
46. #482C22semidirect
47. #481C22semidirect
48. #481C22semidirect
49. #480C22semidirect
50. #478C22semidirect
51. #476C22semidirect
52. #469C22semidirect
53. #466C22semidirect
54. #442C22semidirect
55. #435C22semidirect
56. #428C22semidirect
66. C2×#106C22semidirect
240. C2×C8#41semidirect
343. C8C2×#41semidirect
381. C2#3345direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5532
C2×#5472
#33452
C2×#5012
C2×#4812
C2×#4822
C2×#4802
C2×#4782
C2×#4762
C2×#4692
C2×#4662
C2×#4422
C2×#4352
C2×#4282
C22×#1062