Finite Group Databasebeta

← C2×#3344 #61236

Subgroups — C2×#3344 #61236

Order 256.

939 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#554C211yesmaximal, characteristic
2C2×#552C211yesmaximal, characteristic
3#3344C2116yesmaximal, direct product factor
4C2×#501C211yesmaximal, characteristic, complemented
5C2×#484C211yesmaximal, characteristic, complemented
6C2×#485C211yesmaximal, characteristic, complemented
7C2×#483C211yesmaximal, characteristic, complemented
8C2×#483C211yesmaximal, characteristic, complemented
9C2×#479C211yesmaximal, characteristic, complemented
10C2×#475C211yesmaximal, characteristic, complemented
11C2×#470C211yesmaximal, characteristic, complemented
12C2×#467C211yesmaximal, characteristic, complemented
13C2×#441C211yesmaximal, characteristic, complemented
14C2×#435C211yesmaximal, characteristic
15C2×#428C211yesmaximal, characteristic, complemented
16C22×#106C211yesmaximal, characteristic, complemented
17C22×Q16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20#554—24no—
21#554C2218yes—
22#552—24no—
23#552C2218yes—
24C2×C4⋊Q8C2211yescharacteristic
25C2×#126—21no—
26C2×#125—21no—
27C2×#125—21no—
28C2×#125C2211yescharacteristic
29C2×#123—21no—
30C2×C22⋊Q8—21no—
31C2×C22⋊Q8—21no—
32C2×C22⋊Q8C2211yescharacteristic
33C2×C22⋊Q8C2211yescharacteristic
34C2×C22⋊Q8C2211yescharacteristic
35C2×C22⋊Q8C2211yescharacteristic
36C2×C4×Q8C2211yescharacteristic
37C2×C4×Q8C2211yescharacteristic
38C2×C4×D4—21no—
39C2×C4×D4C2211yescharacteristic
40C2×#117—21no—
41C2×#117C2211yescharacteristic
42C22×C4⋊C4C2211yescharacteristic
43C22×C4⋊C4C2211yescharacteristic
44C22×C4⋊C4C2211yescharacteristic
45#501C2218yescomplemented
46#485C2218yescomplemented
47#484C2218yescomplemented
48#483C2218yescomplemented
49#483C2218yescomplemented
50#479C2218yescomplemented
51#475C2218yescomplemented
52#470C2218yescomplemented
53#467C2218yescomplemented
54#441C2218yescomplemented
55#435C2218yes—
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×#103C2211yescharacteristic
69C2×#103C2211yescharacteristic
70C2×#103C2211yescharacteristic
71C2×#103C2211yescharacteristic
72C2×#103C2211yescharacteristic
73C2×#103C2211yescharacteristic
74C2×C22⋊C8C2211yescharacteristic
75C2×C22⋊C8C2211yescharacteristic
76C2×#97C2211yescharacteristic
77C22×Q8—21no—
78C22×Q8C2311yescharacteristic
79C22×Q8C2311yescharacteristic
80C22×D4D411yescharacteristic
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4C2311yescharacteristic
85C23×C4C2311yescharacteristic
86C2×Q16—24no—
87C2×Q16C2314yes—
88C2×M16—24no—
89C2×M16C2314yes—
90C22×C8—24no—
91C22×C8C2311yescharacteristic
92C22×C8C2314yes—
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C4⋊Q8—22no—
97C4⋊Q8C2314yes—
98#126—28no—
99#125—28no—
100#125—28no—
101#125—22no—
102#125C2314yes—
103#123—28no—
104C22⋊Q8—22no—
105C22⋊Q8—22no—
106C22⋊Q8—22no—
107C22⋊Q8—22no—
108C22⋊Q8—28no—
109C22⋊Q8—28no—
110C22⋊Q8C2314yes—
111C22⋊Q8C2314yes—
112C22⋊Q8C2314yes—
113C22⋊Q8C2314yes—
114C4×Q8—22no—
115C4×Q8—22no—
116C4×Q8C2314yes—
117C4×Q8C2314yes—
118C4×D4—28no—
119C4×D4D414yes—
120C4×D4C2314yes—
121#117—22no—
122#117—28no—
123#117C2314yes—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4D411yescharacteristic
140C2×C4⋊C4C2311yescharacteristic
141C2×C4⋊C4C2311yescharacteristic
142C2×C4⋊C4C2311yescharacteristic
143C2×C4⋊C4C2311yescharacteristic
144C2×C4⋊C4C2311yescharacteristic
145C2×C4⋊C4C2314yes—
146C2×C4⋊C4C2314yes—
147C2×C4⋊C4C2314yes—
148C2×C4⋊C4C2311yescharacteristic
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#106C2314yes—
164#106C2314yes—
165#106C2314yes—
166#106C2314yes—
167#106C2314yes—
168#106C2314yes—
169C4⋊C8C2314yes—
170#103C2314yes—
171#103C2314yes—
172#103C2314yes—
173#103C2314yes—
174#103C2314yes—
175#103C2314yes—
176C22⋊C8C2314yes—
177C22⋊C8C2314yes—
178#97C2314yes—
179C24C2×D411yescharacteristic
180C24C2×D411yescharacteristic
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—24no—
184C2×Q8—22no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×Q8—82no—
189C2×Q8C2412yes—
190C2×Q8C2412yes—
191C2×D4—24no—
192C2×D4C2×D414yes—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—24no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—44no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4C2×D412yes—
212C22×C4C2×D412yes—
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2412yes—
218C22×C4C2412yes—
219C22×C4C2411yescharacteristic
220Q16—28no—
221M16—28no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—28no—
226C2×C8#4114yes—
227C2×C8C2412yes—
228C2×C8C2412yes—
229C2×C8C2412yes—
230C2×C8C2412yes—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
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⋊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⋊C4C2×D414yes—
262C22⋊C4C2×D414yes—
263C42—24no—
264C42—24no—
265C42—24no—
266C42C2×D412yes—
267C42C2412yes—
268C23—22no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23#13714yes—
274C23C22×D412yes—
275C23C22×D412yes—
276C23C22×D411yescharacteristic
277Q8—22no—
278Q8—22no—
279Q8—22no—
280Q8—22no—
281Q8—42no—
282Q8—42no—
283Q8—84no—
284D4—28no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—48no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—84no—
313C2×C4—84no—
314C2×C4C22×D412yes—
315C2×C4C22×D412yes—
316C2×C4C22×D412yes—
317C2×C4C22×D412yes—
318C2×C4C22×D411yescharacteristic
319C2×C4C2×#4112yes—
320C2×C42−1+412yes—
321C2×C4C2511yescharacteristic
322C8—22no—
323C8—22no—
324C8—22no—
325C8C2×#4114yes—
326C22—28no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—44no—
333C22#54712yes—
334C22C2×#13714yes—
335C22C2×#13712yes—
336C22#57712yes—
337C22C23×D411yescharacteristic
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—42no—
343C4—42no—
344C4—42no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—42no—
350C4#54712yes—
351C4C22×#4111yescharacteristic
352C4C2×2−1+411yescharacteristic
353C2—24no—
354C2—42no—
355C2#334414yesdirect product factor
356C2C2×#54711yescharacteristic
357C2C22×#13711yescharacteristic
358C2C2×#57711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC22×D4D4
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
3. #3344C2direct (N × H)
4. C2×#501C2semidirect
5. C2×#484C2semidirect
6. C2×#485C2semidirect
7. C2×#483C2semidirect
8. C2×#483C2semidirect
9. C2×#479C2semidirect
10. C2×#475C2semidirect
11. C2×#470C2semidirect
12. C2×#467C2semidirect
13. C2×#441C2semidirect
15. C2×#428C2semidirect
16. C22×#106C2semidirect
45. #501C22semidirect
46. #485C22semidirect
47. #484C22semidirect
48. #483C22semidirect
49. #483C22semidirect
50. #479C22semidirect
51. #475C22semidirect
52. #470C22semidirect
53. #467C22semidirect
54. #441C22semidirect
56. #428C22semidirect
66. C2×#106C22semidirect
355. C2#3344direct (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×#5522
#33442
C2×#5012
C2×#4842
C2×#4852
C2×#4832
C2×#4792
C2×#4752
C2×#4702
C2×#4672
C2×#4412
C2×#4352
C2×#4282
C22×#1062