Finite Group Databasebeta

← C2×#3347 #61239

Subgroups — C2×#3347 #61239

Order 256.

981 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#555C211yesmaximal, characteristic, complemented
2C2×#553C211yesmaximal, characteristic, complemented
3#3347C2116yesmaximal, direct product factor
4C2×#501C211yesmaximal, characteristic, complemented
5C2×#482C211yesmaximal, characteristic, complemented
6C2×#482C211yesmaximal, characteristic, complemented
7C2×#482C211yesmaximal, characteristic, complemented
8C2×#480C211yesmaximal, characteristic, complemented
9C2×#478C211yesmaximal, characteristic, complemented
10C2×#474C211yesmaximal, characteristic, complemented
11C2×#469C211yesmaximal, characteristic, complemented
12C2×#466C211yesmaximal, characteristic, complemented
13C2×#442C211yesmaximal, characteristic, complemented
14C2×#435C211yesmaximal, characteristic, complemented
15C2×#428C211yesmaximal, characteristic, complemented
16C2×#427C211yesmaximal, characteristic, complemented
17C22×D8C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20#555—24no—
21#555C2218yescomplemented
22#553—24no—
23#553C2218yescomplemented
24C2×C4⋊Q8C2211yescharacteristic
25C2×#126—21no—
26C2×#125C2211yescharacteristic
27C2×#124—21no—
28C2×#124—21no—
29C2×#123—21no—
30C2×#121—21no—
31C2×#121—21no—
32C2×#121C2211yescharacteristic
33C2×#121C2211yescharacteristic
34C2×#121C2211yescharacteristic
35C2×#121C2211yescharacteristic
36C2×C4×D4—21no—
37C2×C4×D4C2211yescharacteristic
38C2×C4×D4C2211yescharacteristic
39C2×C4×D4C2211yescharacteristic
40C2×#117—21no—
41C2×#117C2211yescharacteristic
42C2×#117C2211yescharacteristic
43C2×#117C2211yescharacteristic
44C22×C4⋊C4C2211yescharacteristic
45#501C2218yescomplemented
46#482C2218yescomplemented
47#482C2218yescomplemented
48#482C2218yescomplemented
49#480C2218yescomplemented
50#478C2218yescomplemented
51#474C2218yescomplemented
52#469C2218yescomplemented
53#466C2218yescomplemented
54#442C2218yescomplemented
55#435C2218yescomplemented
56#428C2218yescomplemented
57#427C2218yescomplemented
58C2×#107C2211yescharacteristic
59C2×#107C2211yescharacteristic
60C2×#107C2211yescharacteristic
61C2×#107C2211yescharacteristic
62C2×#107C2211yescharacteristic
63C2×#106C2211yescharacteristic
64C2×#106C2211yescharacteristic
65C2×#106C2211yescharacteristic
66C2×#106C2211yescharacteristic
67C2×C4⋊C8C2211yescharacteristic
68C2×#102C2211yescharacteristic
69C2×#102C2211yescharacteristic
70C2×#102C2211yescharacteristic
71C2×#102C2211yescharacteristic
72C2×#102C2211yescharacteristic
73C2×#102C2211yescharacteristic
74C2×C22⋊C8C2211yescharacteristic
75C2×C22⋊C8C2211yescharacteristic
76C2×#97C2211yescharacteristic
77C22×Q8—21no—
78C22×D4—21no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—21no—
82C22×D4D411yescharacteristic
83C22×D4C2311yescharacteristic
84C22×D4C2311yescharacteristic
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C2×D8—24no—
91C2×D8C2314yes—
92C2×M16—24no—
93C2×M16C2314yes—
94C22×C8—24no—
95C22×C8C2311yescharacteristic
96C22×C8C2314yes—
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99C22×C8C2311yescharacteristic
100C4⋊Q8—22no—
101C4⋊Q8C2314yes—
102#126—28no—
103#125—22no—
104#125C2314yes—
105#124—28no—
106#124—28no—
107#123—28no—
108#121—28no—
109#121—28no—
110#121—22no—
111#121—22no—
112#121—22no—
113#121—22no—
114#121C2314yes—
115#121C2314yes—
116#121C2314yes—
117#121C2314yes—
118C4×D4—28no—
119C4×D4—22no—
120C4×D4—22no—
121C4×D4D414yes—
122C4×D4C2314yes—
123C4×D4C2314yes—
124C4×D4C2314yes—
125#117—22no—
126#117—22no—
127#117—22no—
128#117—28no—
129#117C2314yes—
130#117C2314yes—
131#117C2314yes—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4D411yescharacteristic
138C2×C4⋊C4C2311yescharacteristic
139C2×C4⋊C4C2311yescharacteristic
140C2×C4⋊C4C2314yes—
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2311yescharacteristic
145C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
155C2×C22⋊C4D411yescharacteristic
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42C2311yescharacteristic
160#107C2314yes—
161#107C2314yes—
162#107C2314yes—
163#107C2314yes—
164#107C2314yes—
165#106C2314yes—
166#106C2314yes—
167#106C2314yes—
168#106C2314yes—
169C4⋊C8C2314yes—
170#102C2314yes—
171#102C2314yes—
172#102C2314yes—
173#102C2314yes—
174#102C2314yes—
175#102C2314yes—
176C22⋊C8C2314yes—
177C22⋊C8C2314yes—
178#97C2314yes—
179C24—21no—
180C24—21no—
181C24C2×D411yescharacteristic
182C24C2×D411yescharacteristic
183C2×Q8—24no—
184C2×Q8—82no—
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—24no—
188C2×D4—24no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—42no—
196C2×D4—44no—
197C2×D4—82no—
198C2×D4—82no—
199C2×D4—82no—
200C2×D4C2×D414yes—
201C2×D4C2412yes—
202C2×D4C2412yes—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—44no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4C2×D412yes—
220C22×C4C2×D412yes—
221C22×C4C2×D411yescharacteristic
222C22×C4C2×D411yescharacteristic
223C22×C4C2×D411yescharacteristic
224C22×C4C2×D411yescharacteristic
225C22×C4C2412yes—
226C22×C4C2412yes—
227C22×C4C2411yescharacteristic
228D8—28no—
229M16—28no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—22no—
233C2×C8—28no—
234C2×C8#4114yes—
235C2×C8C2412yes—
236C2×C8C2412yes—
237C2×C8C2412yes—
238C2×C8C2412yes—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4C2×D414yes—
250C4⋊C4C2412yes—
251C4⋊C4C2412yes—
252C4⋊C4C2412yes—
253C4⋊C4C2412yes—
254C4⋊C4C2412yes—
255C4⋊C4C2412yes—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—24no—
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—
270C42C2×D412yes—
271C42C2412yes—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—22no—
279C23—22no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—42no—
286C23C22×D412yes—
287C23C22×D412yes—
288C23C22×D411yescharacteristic
289Q8—84no—
290D4—22no—
291D4—22no—
292D4—22no—
293D4—22no—
294D4—44no—
295D4—48no—
296D4—42no—
297D4—42no—
298D4—84no—
299D4—84no—
300D4—84no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—48no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—84no—
325C2×C4—84no—
326C2×C4C22×D412yes—
327C2×C4C22×D412yes—
328C2×C4C22×D412yes—
329C2×C4C22×D412yes—
330C2×C4C22×D411yescharacteristic
331C2×C4C2×#4112yes—
332C2×C42−1+412yes—
333C2×C4C2511yescharacteristic
334C8—22no—
335C8—22no—
336C8—22no—
337C8C2×#4114yes—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—44no—
345C22—44no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—84no—
353C22—84no—
354C22#54712yes—
355C22#57512yes—
356C22#57612yes—
357C22C23×D411yescharacteristic
358C4—22no—
359C4—22no—
360C4—22no—
361C4—22no—
362C4—42no—
363C4—42no—
364C4—42no—
365C4—42no—
366C4—42no—
367C4—42no—
368C4#54712yes—
369C4C22×#4111yescharacteristic
370C4C2×2−1+411yescharacteristic
371C2—42no—
372C2—42no—
373C2—82no—
374C2—82no—
375C2#334714yesdirect product factor
376C2C2×#54711yescharacteristic
377C2C2×#57511yescharacteristic
378C2C2×#57611yescharacteristic
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×#555C2semidirect
2. C2×#553C2semidirect
3. #3347C2direct (N × H)
4. C2×#501C2semidirect
5. C2×#482C2semidirect
6. C2×#482C2semidirect
7. C2×#482C2semidirect
8. C2×#480C2semidirect
9. C2×#478C2semidirect
10. C2×#474C2semidirect
11. C2×#469C2semidirect
12. C2×#466C2semidirect
13. C2×#442C2semidirect
14. C2×#435C2semidirect
15. C2×#428C2semidirect
16. C2×#427C2semidirect
21. #555C22semidirect
23. #553C22semidirect
45. #501C22semidirect
46. #482C22semidirect
47. #482C22semidirect
48. #482C22semidirect
49. #480C22semidirect
50. #478C22semidirect
51. #474C22semidirect
52. #469C22semidirect
53. #466C22semidirect
54. #442C22semidirect
55. #435C22semidirect
56. #428C22semidirect
57. #427C22semidirect
375. C2#3347direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5552
C2×#5532
#33472
C2×#5012
C2×#4822
C2×#4802
C2×#4782
C2×#4742
C2×#4692
C2×#4662
C2×#4422
C2×#4352
C2×#4282
C2×#4272