Finite Group Databasebeta

← C2×#3178 #60683

Subgroups — C2×#3178 #60683

Order 256.

953 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#542C211yesmaximal, characteristic
2C2×#541C211yesmaximal, characteristic, complemented
3#3178C2116yesmaximal, direct product factor
4C2×#501C211yesmaximal, characteristic, complemented
5C2×#497C211yesmaximal, characteristic, complemented
6C2×#488C211yesmaximal, characteristic, complemented
7C2×#487C211yesmaximal, characteristic, complemented
8C2×#470C211yesmaximal, characteristic, complemented
9C2×#468C211yesmaximal, characteristic
10C2×#467C211yesmaximal, characteristic, complemented
11C2×#465C211yesmaximal, characteristic
12C2×#441C211yesmaximal, characteristic, complemented
13C2×#440C211yesmaximal, characteristic, complemented
14C2×C4×Q16C211yesmaximal, characteristic, complemented
15C2×C4×SD16C211yesmaximal, characteristic, complemented
16C2×#405C211yesmaximal, characteristic, complemented
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×M16C2211yescharacteristic
22C23×C8C2211yescharacteristic
23#542—24no—
24#542C2218yes—
25#541—24no—
26#541C2218yescomplemented
27C2×C4⋊Q8C2211yescharacteristic
28C2×C4⋊Q8C2211yescharacteristic
29C2×#126—21no—
30C2×#126—21no—
31C2×#126—21no—
32C2×#125—21no—
33C2×#125—21no—
34C2×#125C2211yescharacteristic
35C2×#124—21no—
36C2×#124C2211yescharacteristic
37C2×#123—21no—
38C2×C22⋊Q8—21no—
39C2×C22⋊Q8C2211yescharacteristic
40C2×C22⋊Q8C2211yescharacteristic
41C2×C22⋊Q8C2211yescharacteristic
42C2×#121C2211yescharacteristic
43C2×C4×Q8C2211yescharacteristic
44C2×C4×Q8C2211yescharacteristic
45C2×C4×Q8C2211yescharacteristic
46C2×C4×D4C2211yescharacteristic
47C2×#117C2211yescharacteristic
48#501C2218yescomplemented
49#497C2218yescomplemented
50#488C2218yescomplemented
51#487C2218yescomplemented
52#470C2218yescomplemented
53#468C2218yes—
54#467C2218yescomplemented
55#465C2218yes—
56#441C2218yescomplemented
57#440C2218yescomplemented
58C4×Q16C2218yescomplemented
59C4×SD16C2218yescomplemented
60C2×#107C2211yescharacteristic
61C2×#107C2211yescharacteristic
62C2×#106C2211yescharacteristic
63C2×#106C2211yescharacteristic
64C2×#103C2211yescharacteristic
65C2×#103C2211yescharacteristic
66C2×#103C2211yescharacteristic
67C2×#103C2211yescharacteristic
68C2×#103C2211yescharacteristic
69C2×#103C2211yescharacteristic
70C2×#102C2211yescharacteristic
71C2×#102C2211yescharacteristic
72#405C2218yescomplemented
73C2×#97C2211yescharacteristic
74C2×#97C2211yescharacteristic
75C2×C4×C8C2211yescharacteristic
76C2×C4×C8C2211yescharacteristic
77C22×Q8—21no—
78C22×Q8C2311yescharacteristic
79C22×Q8C2311yescharacteristic
80C22×Q8C2311yescharacteristic
81C22×D4—21no—
82C22×D4C2311yescharacteristic
83C23×C4—21no—
84C23×C4C2311yescharacteristic
85C2×Q16—24no—
86C2×Q16—24no—
87C2×Q16C2314yes—
88C2×Q16C2314yes—
89C2×SD16—24no—
90C2×SD16—24no—
91C2×SD16C2314yes—
92C2×SD16C2314yes—
93C2×M16—24no—
94C2×M16C2314yes—
95C22×C8—24no—
96C22×C8C2311yescharacteristic
97C22×C8C2314yes—
98C22×C8C2311yescharacteristic
99C22×C8C2311yescharacteristic
100C22×C8C2311yescharacteristic
101C4⋊Q8—22no—
102C4⋊Q8—22no—
103C4⋊Q8C2314yes—
104C4⋊Q8C2314yes—
105#126—28no—
106#126—28no—
107#126—28no—
108#125—28no—
109#125—28no—
110#125—22no—
111#125C2314yes—
112#124—28no—
113#124—22no—
114#124C2314yes—
115#123—28no—
116C22⋊Q8—22no—
117C22⋊Q8—22no—
118C22⋊Q8—22no—
119C22⋊Q8—28no—
120C22⋊Q8C2314yes—
121C22⋊Q8C2314yes—
122C22⋊Q8C2314yes—
123#121—22no—
124#121C2314yes—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8C2314yes—
129C4×Q8C2314yes—
130C4×Q8C2314yes—
131C4×D4—22no—
132C4×D4C2314yes—
133#117D414yes—
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⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4D411yescharacteristic
147C2×C4⋊C4D411yescharacteristic
148C2×C4⋊C4C2311yescharacteristic
149C2×C4⋊C4C2311yescharacteristic
150C2×C4⋊C4C2311yescharacteristic
151C2×C4⋊C4C2311yescharacteristic
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4D411yescharacteristic
160C2×C22⋊C4D411yescharacteristic
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42C2311yescharacteristic
165C2×C42C2311yescharacteristic
166#107C2314yes—
167#107C2314yes—
168#106C2314yes—
169#106C2314yes—
170#103C2314yes—
171#103C2314yes—
172#103C2314yes—
173#103C2314yes—
174#103C2314yes—
175#103C2314yes—
176#102C2314yes—
177#102C2314yes—
178#97C2314yes—
179#97C2314yes—
180C4×C8C2314yes—
181C4×C8C2314yes—
182C24—21no—
183C24C2×D411yescharacteristic
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—24no—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—82no—
195C2×Q8C2412yes—
196C2×Q8C2412yes—
197C2×Q8C2412yes—
198C2×D4—24no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—82no—
203C2×D4C2412yes—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—24no—
214C22×C4—82no—
215C22×C4C2×D412yes—
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4C2412yes—
222C22×C4C2411yescharacteristic
223Q16—28no—
224Q16—28no—
225SD16—28no—
226SD16—28no—
227M16—28no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—28no—
231C2×C8#4114yes—
232C2×C8#4114yes—
233C2×C8C2412yes—
234C2×C8C2412yes—
235C2×C8C2412yes—
236C2×C8C2412yes—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4C2×D414yes—
253C4⋊C4C2×D414yes—
254C4⋊C4C2412yes—
255C4⋊C4C2412yes—
256C4⋊C4C2412yes—
257C4⋊C4C2412yes—
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—
271C42C2×D412yes—
272C42C2412yes—
273C42C2412yes—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23C22×D412yes—
282C23C22×D411yescharacteristic
283Q8—22no—
284Q8—22no—
285Q8—22no—
286Q8—22no—
287Q8—22no—
288Q8—22no—
289Q8—42no—
290Q8—42no—
291Q8—42no—
292Q8—84no—
293D4—22no—
294D4—22no—
295D4—42no—
296D4—84no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—84no—
320C2×C4C22×D412yes—
321C2×C4C22×D412yes—
322C2×C4C22×D412yes—
323C2×C4C22×D412yes—
324C2×C4C22×D412yes—
325C2×C4C22×D411yescharacteristic
326C2×C4C2×#4112yes—
327C2×C4C2×#4112yes—
328C2×C4C2511yescharacteristic
329C8—22no—
330C8—22no—
331C8C2×#4114yes—
332C8C2×#4114yes—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—44no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—84no—
341C22#53212yes—
342C22#57712yes—
343C22#57812yes—
344C22C23×D411yescharacteristic
345C4—22no—
346C4—22no—
347C4—22no—
348C4—22no—
349C4—22no—
350C4—42no—
351C4—42no—
352C4—42no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4#53212yes—
358C4C22×#4111yescharacteristic
359C4C22×#4111yescharacteristic
360C2—42no—
361C2—82no—
362C2#317814yesdirect product factor
363C2C2×#53211yescharacteristic
364C2C2×#57711yescharacteristic
365C2C2×#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×#541C2semidirect
3. #3178C2direct (N × H)
4. C2×#501C2semidirect
5. C2×#497C2semidirect
6. C2×#488C2semidirect
7. C2×#487C2semidirect
8. C2×#470C2semidirect
10. C2×#467C2semidirect
12. C2×#441C2semidirect
13. C2×#440C2semidirect
14. C2×C4×Q16C2semidirect
15. C2×C4×SD16C2semidirect
16. C2×#405C2semidirect
26. #541C22semidirect
48. #501C22semidirect
49. #497C22semidirect
50. #488C22semidirect
51. #487C22semidirect
52. #470C22semidirect
54. #467C22semidirect
56. #441C22semidirect
57. #440C22semidirect
58. C4×Q16C22semidirect
59. C4×SD16C22semidirect
72. #405C22semidirect
362. C2#3178direct (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×#5412
#31782
C2×#5012
C2×#4972
C2×#4882
C2×#4872
C2×#4702
C2×#4682
C2×#4672
C2×#4652
C2×#4412
C2×#4402
C2×C4×Q162
C2×C4×SD162
C2×#4052