Finite Group Databasebeta

← C2×#83159 #11664454

Subgroups — C2×#83159 #11664454

Order 768.

2806 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#127C211yesmaximal, characteristic
2C2×#5470C211yesmaximal, characteristic, complemented
3C2×#5215C214yesmaximal, complemented
4#83159C2116yesmaximal, direct product factor
5C2×#5071C214yesmaximal, complemented
6C2×#5070C214yesmaximal, complemented
7C2×C4×#208C211yesmaximal, characteristic, complemented
8C2×#3138—31nomaximal, Sylow
9C23×C3×D4—22no—
10C23×C3×D4C2212yes—
11C2×C3×#127—22no—
12C2×C3×#127—24no—
13C2×C3×#127C2212yes—
14C2×C3×#127C2212yes—
15C2×C3×#127C2218yes—
16C22×C3×C42C2211yescharacteristic
17C2×#926C2212yes—
18C2×#924—22no—
19C22×#208C2212yes—
20#5470—24no—
21#5470C2218yescomplemented
22C2×#870C2212yescomplemented
23C2×#869C2212yescomplemented
24C2×C4×Dic6—22no—
25#5215C22132yescomplemented
26C2×#832C22116yescomplemented
27#5071C22132yescomplemented
28#5070C22132yescomplemented
29C4×#208C2218yescomplemented
30C2×#803C2212yescomplemented
31C2×C4×C3⋊C8C2212yescomplemented
32C2×#533—31no—
33C22×#127D311yescharacteristic
34#3138—316no—
35C2×#490—34no—
36C2×#486—34no—
37C2×#418—34no—
38C2×C4×M16—31no—
39C25×C3—22no—
40C22×C3×D4—24no—
41C22×C3×D4—216no—
42C22×C3×D4—24no—
43C22×C3×D4—22no—
44C22×C3×D4—24no—
45C22×C3×D4—216no—
46C22×C3×D4—416no—
47C22×C3×D4C2314yes—
48C22×C3×D4C2318yes—
49C23×C3×C4D412yes—
50C23×C3×C4C2311yescharacteristic
51C22×Dic6—22no—
52C3×#127—216no—
53C3×#127—24no—
54C3×#127—24no—
55C3×#127C2318yes—
56C3×#127C2318yes—
57C2×C3×C42D412yes—
58C2×C3×C42D414yes—
59C2×C3×C42C2311yescharacteristic
60C2×C3×C42C2311yescharacteristic
61C2×C3×C42C2314yes—
62C2×#217—22no—
63#926—24no—
64#926C2318yes—
65C2×C4⋊Dic3—22no—
66C2×C4⋊Dic3C2314yes—
67#924—216no—
68C2×Dic3⋊C4—24no—
69C2×C4×Dic3—22no—
70C2×#208—28no—
71C2×#208C2318yes—
72C22×C3⋊C8C2314yes—
73#870—24no—
74#870C2318yescomplemented
75#869—24no—
76#869C2318yescomplemented
77C4×Dic6—216no—
78#832C23164yescomplemented
79#803C2318yescomplemented
80C4×C3⋊C8C2318yescomplemented
81C23×D4—22no—
82C23×D4C2×D312yes—
83C22×M16—32no—
84#533—38no—
85#533—64no—
86C2×C4⋊Q8—32no—
87C2×#127—22no—
88C2×#127—24no—
89C2×#127C2×D312yes—
90C2×#127C2×D312yes—
91C2×#127C2×D318yes—
92C2×#125—32no—
93C2×C22⋊Q8—62no—
94C2×C4×Q8—62no—
95C2×#117—32no—
96C22×C42C2×D311yescharacteristic
97#490—332no—
98#486—332no—
99#418—332no—
100C2×#102—316no—
101C4×M16—38no—
102C2×#97—32no—
103C2×C4×C8—32no—
104C24×C3—28no—
105C24×C3—24no—
106C24×C3—416no—
107C24×C3—48no—
108C24×C3C2×D411yescharacteristic
109C2×C3×D4—28no—
110C2×C3×D4—24no—
111C2×C3×D4—216no—
112C2×C3×D4—216no—
113C2×C3×D4—216no—
114C2×C3×D4—232no—
115C2×C3×D4—416no—
116C2×C3×D4—464no—
117C2×C3×D4—416no—
118C2×C3×D4—48no—
119C2×C3×D4—416no—
120C2×C3×D4C2418yes—
121C22×C3×C4—28no—
122C22×C3×C4C2×D412yes—
123C22×C3×C4C2×D418yes—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D412yes—
126C22×C3×C4C2×D412yes—
127C22×C3×C4#4118yes—
128C22×C3×C4C2411yescharacteristic
129C22×C3×C4C2412yes—
130C22×Dic3—24no—
131C2×Dic6—28no—
132C2×Dic6—48no—
133C3×C42C2×D418yes—
134C3×C42C2×D412yes—
135C3×C42C2×D412yes—
136C3×C42C2412yes—
137C3×C42C2412yes—
138#217—28no—
139C4⋊Dic3—28no—
140C4⋊Dic3—24no—
141C4⋊Dic3C2418yes—
142Dic3⋊C4—216no—
143C4×Dic3—28no—
144#208—216no—
145C2×C3⋊C8—28no—
146C2×C3⋊C8C2418yes—
147C25—22no—
148C22×Q8—62no—
149C22×D4—216no—
150C22×D4—24no—
151C22×D4—22no—
152C22×D4—216no—
153C22×D4—24no—
154C22×D4—24no—
155C22×D4—416no—
156C22×D4C22×D314yes—
157C22×D4C22×D318yes—
158C23×C4#6712yes—
159C23×C4C22×D311yescharacteristic
160C2×M16—38no—
161C2×M16—68no—
162C22×C8—34no—
163C4⋊Q8—38no—
164C4⋊Q8—64no—
165#127—216no—
166#127—24no—
167#127—24no—
168#127C22×D318yes—
169#127C22×D318yes—
170#125—38no—
171#125—64no—
172C22⋊Q8—616no—
173C4×Q8—616no—
174#117—38no—
175#117—64no—
176C2×C4⋊C4—34no—
177C2×C4⋊C4—64no—
178C2×C4⋊C4—62no—
179C2×C22⋊C4—62no—
180C2×C42—62no—
181C2×C42#6712yes—
182C2×C42#6714yes—
183C2×C42C22×D311yescharacteristic
184C2×C42C22×D311yescharacteristic
185C2×C42C22×D314yes—
186#102—364no—
187#97—38no—
188C4×C8—38no—
189C23×C3—24no—
190C23×C3—28no—
191C23×C3—432no—
192C23×C3—416no—
193C23×C3—48no—
194C23×C3—416no—
195C23×C3—832no—
196C23×C3#12414yes—
197C23×C3C22×D412yes—
198C23×C3C22×D411yescharacteristic
199C3×D4—216no—
200C3×D4—432no—
201C3×D4—416no—
202C3×D4—48no—
203C3×D4—432no—
204C2×C3×C4—24no—
205C2×C3×C4—216no—
206C2×C3×C4#12418yes—
207C2×C3×C4#12414yes—
208C2×C3×C4AGL(1,ℤ/8)18yes—
209C2×C3×C4C22×D414yes—
210C2×C3×C4C22×D412yes—
211C2×C3×C4C22×D411yescharacteristic
212C2×C3×C4C22×D414yes—
213C2×C3×C4C2×#4114yes—
214C2×C3×C4C2×#4118yes—
215C2×C3×C4C2511yescharacteristic
216C2×Dic3—28no—
217C2×Dic3—48no—
218Dic6—416no—
219C3⋊C8—28no—
220C24—28no—
221C24—24no—
222C24—416no—
223C24—48no—
224C24C2×#6711yescharacteristic
225C2×Q8—68no—
226C2×Q8—128no—
227C2×D4—216no—
228C2×D4—232no—
229C2×D4—28no—
230C2×D4—24no—
231C2×D4—216no—
232C2×D4—216no—
233C2×D4—416no—
234C2×D4—464no—
235C2×D4—416no—
236C2×D4—416no—
237C2×D4—48no—
238C2×D4C23×D318yes—
239C22×C4—28no—
240C22×C4—64no—
241C22×C4#23718yes—
242C22×C4C2×#6718yes—
243C22×C4C2×#6712yes—
244C22×C4C2×#6712yes—
245C22×C4C2×#6712yes—
246C22×C4C2×#6711yescharacteristic
247C22×C4C23×D312yes—
248C22×C4C23×D311yescharacteristic
249M16—616no—
250C2×C8—38no—
251C2×C8—68no—
252C4⋊C4—38no—
253C4⋊C4—64no—
254C4⋊C4—616no—
255C4⋊C4—68no—
256C22⋊C4—68no—
257C42—68no—
258C42C2×#6718yes—
259C42C2×#6712yes—
260C42C2×#6712yes—
261C42C23×D312yes—
262C42C23×D312yes—
263C22×C3—28no—
264C22×C3—24no—
265C22×C3—48no—
266C22×C3—416no—
267C22×C3—832no—
268C22×C3—816no—
269C22×C3C2×#12414yes—
270C22×C3C2×#12412yes—
271C22×C3C2×AGL(1,ℤ/8)14yes—
272C22×C3C23×D411yescharacteristic
273C3×C4—24no—
274C3×C4C2×#12414yes—
275C3×C4C2×#12412yes—
276C3×C4C2×AGL(1,ℤ/8)18yes—
277C3×C4C22×#4112yes—
278Dic3—48no—
279C23—24no—
280C23—28no—
281C23—432no—
282C23—416no—
283C23—416no—
284C23—48no—
285C23—832no—
286C23#93614yes—
287C23C22×#6712yes—
288C23C22×#6711yescharacteristic
289Q8—1216no—
290D4—216no—
291D4—432no—
292D4—432no—
293D4—48no—
294D4—416no—
295C2×C4—24no—
296C2×C4—216no—
297C2×C4—68no—
298C2×C4—128no—
299C2×C4#93218yes—
300C2×C4#93618yes—
301C2×C4#93614yes—
302C2×C4C2×#23714yes—
303C2×C4C2×#23718yes—
304C2×C4C22×#6714yes—
305C2×C4C22×#6714yes—
306C2×C4C22×#6712yes—
307C2×C4C22×#6711yescharacteristic
308C2×C4C24×D311yescharacteristic
309C8—68no—
310C2×C3—24no—
311C2×C3—88no—
312C2×C3#313814yescomplemented
313C2×C3C22×#12411yescharacteristic
314C2×C3C22×AGL(1,ℤ/8)12yes—
315C22—24no—
316C22—28no—
317C22—416no—
318C22—48no—
319C22—832no—
320C22—816no—
321C22C2×#93214yes—
322C22C2×#93614yes—
323C22C2×#93612yes—
324C22C23×#6711yescharacteristic
325C4—24no—
326C4—128no—
327C4C2×#93218yes—
328C4C2×#93612yes—
329C4C2×#93614yes—
330C4C22×#23712yes—
331C3C2×#313811yescharacteristic, complemented, Sylow
332C2—24no—
333C2—88no—
334C2#8315914yesdirect product factor
335C2C22×#93212yes—
336C2C22×#93611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
Fitting, prime radicalC22×C3×#127C2
socleC23×C3C22×D4
hypercenterC22×#127D3
hypocenterC3C2×#3138
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#127

Complements

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

NH ≅ G/Ntype
2. C2×#5470C2semidirect
3. C2×#5215C2semidirect
4. #83159C2direct (N × H)
5. C2×#5071C2semidirect
6. C2×#5070C2semidirect
7. C2×C4×#208C2semidirect
21. #5470C22semidirect
22. C2×#870C22semidirect
23. C2×#869C22semidirect
25. #5215C22semidirect
26. C2×#832C22semidirect
27. #5071C22semidirect
28. #5070C22semidirect
29. C4×#208C22semidirect
30. C2×#803C22semidirect
31. C2×C4×C3⋊C8C22semidirect
74. #870C23semidirect
76. #869C23semidirect
78. #832C23semidirect
79. #803C23semidirect
80. C4×C3⋊C8C23semidirect
312. C2×C3#3138semidirect
331. C3C2×#3138semidirect
334. C2#83159direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1272
C2×#54702
C2×#52152
#831592
C2×#50712
C2×#50702
C2×C4×#2082
C2×#31383