Finite Group Databasebeta

← C2×#3316 #61130

Subgroups — C2×#3316 #61130

Order 256.

1079 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×#554C211yesmaximal, characteristic, complemented
2C2×#548C211yesmaximal, characteristic
3#3316C2116yesmaximal, direct product factor
4C2×#489C211yesmaximal, characteristic, complemented
5C2×#481C211yesmaximal, characteristic, complemented
6C2×#483C211yesmaximal, characteristic, complemented
7C2×#475C211yesmaximal, characteristic, complemented
8C2×#470C211yesmaximal, characteristic, complemented
9C2×#465C211yesmaximal, characteristic, complemented
10C2×#464C211yesmaximal, characteristic, complemented
11C2×#452C211yesmaximal, characteristic, complemented
12C2×#451C211yesmaximal, characteristic, complemented
13C2×#441C211yesmaximal, characteristic, complemented
14C2×#435C211yesmaximal, characteristic, complemented
15C2×#417C211yesmaximal, characteristic, complemented
16C22×#103C211yesmaximal, characteristic, complemented
17C22×#41C2211yescharacteristic
18C23×Q8C2211yescharacteristic
19C22×Q16C2211yescharacteristic
20C22×Q16C2211yescharacteristic
21C22×SD16C2211yescharacteristic
22C22×M16C2211yescharacteristic
23C23×C8C2211yescharacteristic
24#554—24no—
25#554C2218yescomplemented
26#548—24no—
27#548C2218yes—
28C2×C4⋊Q8C2211yescharacteristic
29C2×#125—21no—
30C2×#124—21no—
31C2×#124C2211yescharacteristic
32C2×C22⋊Q8—21no—
33C2×C22⋊Q8—21no—
34C2×C22⋊Q8—21no—
35C2×C22⋊Q8C2211yescharacteristic
36C2×C22⋊Q8C2211yescharacteristic
37C2×C22⋊Q8C2211yescharacteristic
38C2×#121—21no—
39C2×#121C2211yescharacteristic
40C2×C4×Q8C2211yescharacteristic
41C2×C4×Q8C2211yescharacteristic
42C2×C4×D4—21no—
43C2×C4×D4—21no—
44C2×C4×D4C2211yescharacteristic
45C22×C4⋊C4C2211yescharacteristic
46C22×C4⋊C4C2211yescharacteristic
47#489C2218yescomplemented
48#483C2218yescomplemented
49#481C2218yescomplemented
50#475C2218yescomplemented
51#470C2218yescomplemented
52#465C2218yescomplemented
53#464C2218yescomplemented
54#452C2218yescomplemented
55#451C2218yescomplemented
56#441C2218yescomplemented
57#435C2218yescomplemented
58C2×#106C2211yescharacteristic
59C2×#106C2211yescharacteristic
60C2×#106C2211yescharacteristic
61C2×C4⋊C8C2211yescharacteristic
62#417C2218yescomplemented
63C2×#103C2211yescharacteristic
64C2×#103C2211yescharacteristic
65C2×#103C2211yescharacteristic
66C2×#103C2211yescharacteristic
67C2×#103C2211yescharacteristic
68C2×#103C2211yescharacteristic
69C2×#103C2211yescharacteristic
70C2×#103C2218yescomplemented
71C2×#102C2211yescharacteristic
72C2×#102C2211yescharacteristic
73C2×#102C2211yescharacteristic
74C2×C22⋊C8C2211yescharacteristic
75C2×C22⋊C8C2211yescharacteristic
76C2×#97C2211yescharacteristic
77C2×#41—22no—
78C2×#41—24no—
79C2×#41—42no—
80C2×#41C2314yes—
81C22×Q8—21no—
82C22×Q8—22no—
83C22×Q8—21no—
84C22×Q8—24no—
85C22×Q8—42no—
86C22×Q8C2311yescharacteristic
87C22×Q8C2311yescharacteristic
88C22×Q8C2314yes—
89C22×Q8C2311yescharacteristic
90C22×D4—21no—
91C22×D4—21no—
92C22×D4D411yescharacteristic
93C22×D4C2311yescharacteristic
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4C2311yescharacteristic
99C23×C4C2311yescharacteristic
100C2×Q16—24no—
101C2×Q16—42no—
102C2×Q16C2314yes—
103C2×Q16C2314yes—
104C2×SD16—42no—
105C2×SD16C2314yes—
106C2×M16—42no—
107C2×M16C2314yes—
108C22×C8—42no—
109C22×C8C2311yescharacteristic
110C22×C8C2311yescharacteristic
111C22×C8C2311yescharacteristic
112C22×C8C2311yescharacteristic
113C22×C8C2314yes—
114C4⋊Q8—22no—
115C4⋊Q8C2314yes—
116#125—28no—
117#124—28no—
118#124—22no—
119#124C2314yes—
120C22⋊Q8—22no—
121C22⋊Q8—22no—
122C22⋊Q8—22no—
123C22⋊Q8—28no—
124C22⋊Q8—28no—
125C22⋊Q8—28no—
126C22⋊Q8C2314yes—
127C22⋊Q8C2314yes—
128C22⋊Q8C2314yes—
129#121—28no—
130#121—22no—
131#121C2314yes—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8C2314yes—
135C4×Q8C2314yes—
136C4×D4—28no—
137C4×D4—28no—
138C4×D4D414yes—
139C4×D4C2314yes—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
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⋊C4C2311yescharacteristic
152C2×C4⋊C4C2311yescharacteristic
153C2×C4⋊C4C2311yescharacteristic
154C2×C4⋊C4C2314yes—
155C2×C4⋊C4C2314yes—
156C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
164C2×C22⋊C4D411yescharacteristic
165C2×C42—21no—
166C2×C42—21no—
167C2×C42C2311yescharacteristic
168#106C2314yes—
169#106C2314yes—
170#106C2314yes—
171C4⋊C8C2314yes—
172#103C2314yes—
173#103C2314yes—
174#103C2314yes—
175#103C2314yes—
176#103C2314yes—
177#103C2314yes—
178#103C2314yes—
179#102C2314yes—
180#102C2314yes—
181#102C2314yes—
182C22⋊C8C2314yes—
183C22⋊C8C2314yes—
184#97C2314yes—
185C24—21no—
186C24C2×D411yescharacteristic
187C24C2×D411yescharacteristic
188#41—28no—
189#41—44no—
190#41—48no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—24no—
194C2×Q8—21no—
195C2×Q8—24no—
196C2×Q8—22no—
197C2×Q8—28no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—44no—
203C2×Q8—44no—
204C2×Q8—48no—
205C2×Q8—82no—
206C2×Q8#4114yes—
207C2×Q8C2412yes—
208C2×Q8C2412yes—
209C2×Q8C2412yes—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—21no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—44no—
217C2×D4—44no—
218C2×D4C2×D414yes—
219C2×D4C2412yes—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—82no—
238C22×C4—82no—
239C22×C4C2×D412yes—
240C22×C4C2×D412yes—
241C22×C4C2×D411yescharacteristic
242C22×C4C2×D411yescharacteristic
243C22×C4C2×D411yescharacteristic
244C22×C4C2×D411yescharacteristic
245C22×C4C2412yes—
246C22×C4C2412yes—
247C22×C4C2411yescharacteristic
248Q16—28no—
249Q16—44no—
250SD16—44no—
251M16—44no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—44no—
257C2×C8C2412yes—
258C2×C8C2412yes—
259C2×C8C2412yes—
260C2×C8C2412yes—
261C4⋊C4—21no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—21no—
273C4⋊C4C2×D414yes—
274C4⋊C4C2412yes—
275C4⋊C4C2412yes—
276C4⋊C4C2412yes—
277C4⋊C4C2412yes—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4C2×D414yes—
285C22⋊C4C2×D414yes—
286C42—24no—
287C42—24no—
288C42C2×D412yes—
289C42C2412yes—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23#13714yes—
300C23C22×D412yes—
301C23C22×D412yes—
302C23C22×D411yescharacteristic
303Q8—24no—
304Q8—22no—
305Q8—22no—
306Q8—22no—
307Q8—22no—
308Q8—22no—
309Q8—48no—
310Q8—42no—
311Q8—42no—
312Q8—84no—
313Q8C2×#4114yes—
314D4—28no—
315D4—22no—
316D4—22no—
317D4—48no—
318D4—48no—
319D4—42no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—24no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—48no—
339C2×C4—48no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4C22×D412yes—
349C2×C4C22×D412yes—
350C2×C4C22×D412yes—
351C2×C4C22×D412yes—
352C2×C4C22×D411yescharacteristic
353C2×C4C2×#4112yes—
354C2×C4H(5,𝔽2)12yes—
355C2×C4C2511yescharacteristic
356C8—22no—
357C8—22no—
358C8—22no—
359C8—22no—
360C22—22no—
361C22—28no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—44no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—84no—
371C22#54612yes—
372C22C2×#13714yes—
373C22C2×#13712yes—
374C22#57712yes—
375C22C23×D411yescharacteristic
376C4—22no—
377C4—22no—
378C4—24no—
379C4—22no—
380C4—22no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—42no—
387C4#54612yes—
388C4C22×#4111yescharacteristic
389C4C2×H(5,𝔽2)11yescharacteristic
390C2—24no—
391C2—42no—
392C2—82no—
393C2#331614yesdirect product factor
394C2C2×#54611yescharacteristic
395C2C22×#13711yescharacteristic
396C2C2×#57711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC2×#548C2
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×#554C2semidirect
3. #3316C2direct (N × H)
4. C2×#489C2semidirect
5. C2×#481C2semidirect
6. C2×#483C2semidirect
7. C2×#475C2semidirect
8. C2×#470C2semidirect
9. C2×#465C2semidirect
10. C2×#464C2semidirect
11. C2×#452C2semidirect
12. C2×#451C2semidirect
13. C2×#441C2semidirect
14. C2×#435C2semidirect
15. C2×#417C2semidirect
16. C22×#103C2semidirect
25. #554C22semidirect
47. #489C22semidirect
48. #483C22semidirect
49. #481C22semidirect
50. #475C22semidirect
51. #470C22semidirect
52. #465C22semidirect
53. #464C22semidirect
54. #452C22semidirect
55. #451C22semidirect
56. #441C22semidirect
57. #435C22semidirect
62. #417C22semidirect
70. C2×#103C22semidirect
393. C2#3316direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#5482
#33162
C2×#4892
C2×#4812
C2×#4832
C2×#4752
C2×#4702
C2×#4652
C2×#4642
C2×#4522
C2×#4512
C2×#4412
C2×#4352
C2×#4172
C22×#1032