Finite Group Databasebeta

← C2×#3401 #61398

Subgroups — C2×#3401 #61398

Order 256.

1001 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#559C211yesmaximal, characteristic, complemented
2C2×#556C211yesmaximal, characteristic, complemented
3#3401C2116yesmaximal, direct product factor
4C2×#497C211yesmaximal, characteristic, complemented
5C2×#496C211yesmaximal, characteristic
6C2×#495C211yesmaximal, characteristic, complemented
7C2×#464C211yesmaximal, characteristic, complemented
8C2×#464C211yesmaximal, characteristic, complemented
9C2×#460C211yesmaximal, characteristic, complemented
10C2×#463C211yesmaximal, characteristic, complemented
11C2×#463C211yesmaximal, characteristic, complemented
12C2×#461C211yesmaximal, characteristic, complemented
13C2×#446C211yesmaximal, characteristic, complemented
14C2×#442C211yesmaximal, characteristic, complemented
15C2×#441C211yesmaximal, characteristic, complemented
16C2×C4×SD16C211yesmaximal, characteristic, complemented
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×SD16C2211yescharacteristic
22C22×SD16C2211yescharacteristic
23C22×SD16C2211yescharacteristic
24C22×D8C2211yescharacteristic
25C22×D8C2211yescharacteristic
26#559—24no—
27#559C2218yescomplemented
28#556—24no—
29#556C2218yescomplemented
30C2×C4⋊Q8C2211yescharacteristic
31C2×#127C2211yescharacteristic
32C2×#126—21no—
33C2×#126—21no—
34C2×#124—21no—
35C2×#124C2211yescharacteristic
36C2×#124C2211yescharacteristic
37C2×#124C2211yescharacteristic
38C2×#124C2211yescharacteristic
39C2×#123—21no—
40C2×#123—21no—
41C2×C22⋊Q8—21no—
42C2×C4×Q8C2211yescharacteristic
43C2×C4×Q8C2211yescharacteristic
44C2×C4×Q8C2211yescharacteristic
45C2×C4×Q8C2211yescharacteristic
46C2×C4×D4—21no—
47C2×C4×D4C2211yescharacteristic
48C2×C4×D4C2211yescharacteristic
49C2×C4×D4C2211yescharacteristic
50C2×#117—21no—
51#497C2218yescomplemented
52#496C2218yes—
53#495C2218yescomplemented
54#464C2218yescomplemented
55#464C2218yescomplemented
56#463C2218yescomplemented
57#463C2218yescomplemented
58#461C2218yescomplemented
59#460C2218yescomplemented
60#446C2218yescomplemented
61#442C2218yescomplemented
62#441C2218yescomplemented
63C4×SD16C2218yescomplemented
64C2×#106C2211yescharacteristic
65C2×C4⋊C8C2211yescharacteristic
66C2×C4⋊C8C2211yescharacteristic
67C2×C4⋊C8C2211yescharacteristic
68C2×#103C2211yescharacteristic
69C2×#103C2211yescharacteristic
70C2×#103C2211yescharacteristic
71C2×#102C2211yescharacteristic
72C2×#102C2211yescharacteristic
73C2×#102C2211yescharacteristic
74C2×#97C2211yescharacteristic
75C2×#97C2211yescharacteristic
76C2×C4×C8C2211yescharacteristic
77C22×Q8D411yescharacteristic
78C22×Q8C2311yescharacteristic
79C22×Q8C2311yescharacteristic
80C22×Q8C2311yescharacteristic
81C22×D4—21no—
82C22×D4—21no—
83C22×D4C2311yescharacteristic
84C22×D4C2311yescharacteristic
85C22×D4C2311yescharacteristic
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C2×Q16—24no—
90C2×Q16—42no—
91C2×Q16C2314yes—
92C2×Q16C2314yes—
93C2×SD16—24no—
94C2×SD16—42no—
95C2×SD16—42no—
96C2×SD16—42no—
97C2×SD16—42no—
98C2×SD16C2314yes—
99C2×SD16C2314yes—
100C2×SD16C2314yes—
101C2×SD16C2314yes—
102C2×SD16C2314yes—
103C2×D8—24no—
104C2×D8—42no—
105C2×D8C2314yes—
106C2×D8C2314yes—
107C22×C8C2311yescharacteristic
108C22×C8C2311yescharacteristic
109C22×C8C2311yescharacteristic
110C22×C8C2311yescharacteristic
111C4⋊Q8—22no—
112C4⋊Q8C2314yes—
113#127—22no—
114#127C2314yes—
115#126—28no—
116#126—28no—
117#124—28no—
118#124—22no—
119#124—22no—
120#124—22no—
121#124—22no—
122#124C2314yes—
123#124C2314yes—
124#124C2314yes—
125#124C2314yes—
126#123—28no—
127#123—28no—
128C22⋊Q8—28no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8D414yes—
133C4×Q8C2314yes—
134C4×Q8C2314yes—
135C4×Q8C2314yes—
136C4×Q8C2314yes—
137C4×D4—28no—
138C4×D4—22no—
139C4×D4—22no—
140C4×D4—22no—
141C4×D4C2314yes—
142C4×D4C2314yes—
143C4×D4C2314yes—
144#117—28no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4D411yescharacteristic
151C2×C4⋊C4D411yescharacteristic
152C2×C4⋊C4D411yescharacteristic
153C2×C4⋊C4C2311yescharacteristic
154C2×C4⋊C4C2311yescharacteristic
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42C2311yescharacteristic
170C2×C42C2311yescharacteristic
171C2×C42C2311yescharacteristic
172#106C2314yes—
173C4⋊C8C2314yes—
174C4⋊C8C2314yes—
175C4⋊C8C2314yes—
176#103C2314yes—
177#103C2314yes—
178#103C2314yes—
179#102C2314yes—
180#102C2314yes—
181#102C2314yes—
182#97C2314yes—
183#97C2314yes—
184C4×C8C2314yes—
185C24—21no—
186C24—21no—
187C24—21no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—22no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—42no—
198C2×Q8C2×D414yes—
199C2×Q8C2412yes—
200C2×Q8C2412yes—
201C2×Q8C2412yes—
202C2×D4—21no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—24no—
206C2×D4—24no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—44no—
214C2×D4—82no—
215C2×D4C2412yes—
216C2×D4C2412yes—
217C2×D4C2412yes—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—44no—
227C22×C4—82no—
228C22×C4—82no—
229C22×C4C2×D411yescharacteristic
230C22×C4C2×D411yescharacteristic
231C22×C4C2×D411yescharacteristic
232C22×C4C2×D411yescharacteristic
233C22×C4C2×D411yescharacteristic
234C22×C4C2×D411yescharacteristic
235C22×C4C2411yescharacteristic
236Q16—28no—
237Q16—44no—
238SD16—28no—
239SD16—44no—
240SD16—44no—
241SD16—44no—
242SD16—44no—
243D8—28no—
244D8—44no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—22no—
248C2×C8#4114yes—
249C2×C8C2412yes—
250C2×C8C2412yes—
251C2×C8C2412yes—
252C2×C8C2412yes—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4C2×D414yes—
261C4⋊C4C2×D414yes—
262C4⋊C4C2×D414yes—
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⋊C4—24no—
276C42—24no—
277C42—24no—
278C42—24no—
279C42C2×D412yes—
280C42C2×D412yes—
281C42C2×D412yes—
282C42C2412yes—
283C42C2412yes—
284C42C2412yes—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23C22×D411yescharacteristic
298Q8—22no—
299Q8—22no—
300Q8—22no—
301Q8—22no—
302Q8—22no—
303Q8—22no—
304Q8—44no—
305Q8—42no—
306Q8—42no—
307Q8—42no—
308D4—22no—
309D4—22no—
310D4—22no—
311D4—22no—
312D4—22no—
313D4—22no—
314D4—48no—
315D4—42no—
316D4—42no—
317D4—42no—
318D4—84no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—48no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4C22×D412yes—
339C2×C4C22×D412yes—
340C2×C4C22×D412yes—
341C2×C4C22×D412yes—
342C2×C4C22×D412yes—
343C2×C4C22×D412yes—
344C2×C4C22×D411yescharacteristic
345C2×C4C2×#4112yes—
346C2×C4H(5,𝔽2)12yes—
347C2×C4C2511yescharacteristic
348C8—22no—
349C8—22no—
350C8—22no—
351C8C2×#4114yes—
352C22—42no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—84no—
362C22—84no—
363C22—84no—
364C22#55012yes—
365C22#57512yes—
366C22#57712yes—
367C22C23×D411yescharacteristic
368C4—22no—
369C4—22no—
370C4—22no—
371C4—22no—
372C4—22no—
373C4—22no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—42no—
379C4#55012yes—
380C4C22×#4111yescharacteristic
381C4C2×H(5,𝔽2)11yescharacteristic
382C2—82no—
383C2—82no—
384C2—82no—
385C2#340114yesdirect product factor
386C2C2×#55011yescharacteristic
387C2C2×#57511yescharacteristic
388C2C2×#57711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC2×#496C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#559C2semidirect
2. C2×#556C2semidirect
3. #3401C2direct (N × H)
4. C2×#497C2semidirect
6. C2×#495C2semidirect
7. C2×#464C2semidirect
8. C2×#464C2semidirect
9. C2×#460C2semidirect
10. C2×#463C2semidirect
11. C2×#463C2semidirect
12. C2×#461C2semidirect
13. C2×#446C2semidirect
14. C2×#442C2semidirect
15. C2×#441C2semidirect
16. C2×C4×SD16C2semidirect
27. #559C22semidirect
29. #556C22semidirect
51. #497C22semidirect
53. #495C22semidirect
54. #464C22semidirect
55. #464C22semidirect
56. #463C22semidirect
57. #463C22semidirect
58. #461C22semidirect
59. #460C22semidirect
60. #446C22semidirect
61. #442C22semidirect
62. #441C22semidirect
63. C4×SD16C22semidirect
385. C2#3401direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5592
C2×#5562
#34012
C2×#4972
C2×#4962
C2×#4952
C2×#4642
C2×#4602
C2×#4632
C2×#4612
C2×#4462
C2×#4422
C2×#4412
C2×C4×SD162