Finite Group Databasebeta

← C2×#79172 #11664812

Subgroups — C2×#79172 #11664812

Order 768.

1854 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5519C212yesmaximal
2C2×C3×#427C211yesmaximal, characteristic
3C22×#817C211yesmaximal, characteristic
4C2×#5101C211yesmaximal, characteristic
5C2×#4995C212yesmaximal
6#79172C2116yesmaximal, direct product factor
7C2×#4876C214yesmaximal, complemented
8C2×#4858C212yesmaximal, complemented
9C2×#4855C212yesmaximal, complemented
10C2×#3163—31nomaximal, Sylow
11C23×C3×C8C2211yescharacteristic
12C2×C3×#117C2212yes—
13C2×#926C2211yescharacteristic
14C22×C4⋊Dic3C2212yes—
15C22×#208C2211yescharacteristic
16#5519—28no—
17#5519C22116yes—
18C2×#890C2214yes—
19C2×#888C2214yes—
20C2×#878—24no—
21C2×#870—22no—
22C2×#869—22no—
23C3×#427C2218yes—
24C2×C3×#107C2212yes—
25C2×C3×#106C2211yescharacteristic
26C2×C3×#106C2211yescharacteristic
27C2×#817C2212yes—
28C2×#817C2212yes—
29C2×#817C2218yes—
30#5101C2218yes—
31C2×#815C2212yes—
32C2×C8×Dic3C2212yes—
33#4995C22116yes—
34C2×#808C2214yes—
35C2×#807C2214yes—
36#4876C22132yescomplemented
37#4858C22116yescomplemented
38#4855C22116yescomplemented
39C2×#544—32no—
40#3163—316no—
41C2×#501—34no—
42C2×#499—32no—
43C2×#498—32no—
44C2×#428—32no—
45C2×#427D311yescharacteristic
46C22×#106—31no—
47C2×#405—31no—
48C23×C3×C4C2311yescharacteristic
49C23×Dic3—22no—
50C22×Dic6—24no—
51C22×C3×C8Q818yescomplemented
52C22×C3×C8C2311yescharacteristic
53C22×C3×C8C2311yescharacteristic
54C22×C3×C8C2314yes—
55C3×#117—24no—
56C3×#117C2318yes—
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C4⋊C4C2312yes—
59C2×C3×C22⋊C4—22no—
60C2×C3×C42—22no—
61C2×#217D412yes—
62#926D414yes—
63#926C2314yes—
64C2×C4⋊Dic3—22no—
65C2×C4⋊Dic3—24no—
66C2×C4⋊Dic3—24no—
67C2×C4⋊Dic3D412yes—
68C2×C4⋊Dic3C2314yes—
69C2×C4⋊Dic3C2318yes—
70C2×Dic3⋊C4—24no—
71C2×Dic3⋊C4—24no—
72C2×C4×Dic3C2312yes—
73C2×#208Q818yescomplemented
74C2×#208C2314yes—
75C22×C3⋊C8C2312yes—
76#890—28no—
77#890C23116yes—
78#888—28no—
79#888C23116yes—
80#878—232no—
81#870—216no—
82#869—216no—
83C3×#107C2318yes—
84C3×#106C2314yes—
85C3×#106C2314yes—
86#817C2318yes—
87#817C2318yes—
88#815C2318yes—
89C8×Dic3C2318yes—
90#808C23116yes—
91#807C23116yes—
92C22×M16—31no—
93C23×C8C2×D311yescharacteristic
94#544—316no—
95#544—68no—
96C2×C4⋊Q8—34no—
97C2×C4⋊Q8—62no—
98C2×#125—34no—
99C2×#125—62no—
100C2×C22⋊Q8—64no—
101C2×#117—31no—
102C2×#117C2×D312yes—
103C22×C4⋊C4—32no—
104#501—332no—
105#499—316no—
106#498—316no—
107#428—316no—
108#427C2×D318yes—
109C2×#107—34no—
110C2×#107C2×D312yes—
111C2×#106—32no—
112C2×#106—32no—
113C2×#106—38no—
114C2×#106—34no—
115C2×#106C2×D311yescharacteristic
116C2×#106C2×D311yescharacteristic
117#405—38no—
118C2×#97—32no—
119C2×C4×C8—32no—
120C24×C3C2×D411yescharacteristic
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D412yes—
125C22×C3×C4C2×Q814yes—
126C22×C3×C4C2×Q814yes—
127C22×C3×C4C2411yescharacteristic
128C22×C3×C4C2412yes—
129C22×Dic3—24no—
130C22×Dic3—28no—
131C22×Dic3—84no—
132C22×Dic3C2×D414yes—
133C2×Dic6—216no—
134C2×Dic6—88no—
135C2×C3×C8C2×Q814yes—
136C2×C3×C8C2×Q814yes—
137C2×C3×C8C2×Q8116yescomplemented
138C2×C3×C8C2412yes—
139C2×C3×C8C2412yes—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4C2414yes—
143C3×C4⋊C4C2414yes—
144C3×C22⋊C4—28no—
145C3×C42—28no—
146#217C2×D418yes—
147C4⋊Dic3—28no—
148C4⋊Dic3—24no—
149C4⋊Dic3—216no—
150C4⋊Dic3C2×D418yes—
151C4⋊Dic3C2418yes—
152Dic3⋊C4—216no—
153Dic3⋊C4—216no—
154C4×Dic3C2×D414yes—
155C4×Dic3C2414yes—
156#208C2×Q8116yescomplemented
157C2×C3⋊C8C2×Q818yes—
158C2×C3⋊C8C2414yes—
159C22×Q8—64no—
160C23×C4—62no—
161C23×C4C22×D311yescharacteristic
162C2×M16—38no—
163C2×M16—34no—
164C22×C8—32no—
165C22×C8Dic618yescomplemented
166C22×C8C22×D311yescharacteristic
167C22×C8C22×D311yescharacteristic
168C22×C8C22×D314yes—
169C4⋊Q8—316no—
170C4⋊Q8—616no—
171C4⋊Q8—68no—
172#125—316no—
173#125—616no—
174#125—68no—
175C22⋊Q8—632no—
176#117—24no—
177#117—34no—
178#117—34no—
179#117C22×D318yes—
180C2×C4⋊C4—32no—
181C2×C4⋊C4—34no—
182C2×C4⋊C4—38no—
183C2×C4⋊C4—64no—
184C2×C4⋊C4—64no—
185C2×C4⋊C4—64no—
186C2×C4⋊C4—62no—
187C2×C4⋊C4—64no—
188C2×C4⋊C4C22×D312yes—
189C2×C4⋊C4C22×D312yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—32no—
192C2×C42—22no—
193C2×C42—32no—
194#107—316no—
195#107C22×D318yes—
196#106—38no—
197#106—38no—
198#106—316no—
199#106C22×D314yes—
200#106C22×D314yes—
201#97—38no—
202C4×C8—38no—
203C23×C3—24no—
204C23×C3C4⋊Q814yes—
205C23×C3C22×D412yes—
206C23×C3C22×D411yescharacteristic
207C2×C3×C4—24no—
208C2×C3×C4—24no—
209C2×C3×C4—44no—
210C2×C3×C4—44no—
211C2×C3×C4C4⋊Q818yes—
212C2×C3×C4C4⋊Q814yes—
213C2×C3×C4C22×D411yescharacteristic
214C2×C3×C4C22×D412yes—
215C2×C3×C4C22×Q812yes—
216C2×C3×C4C22×Q814yes—
217C2×C3×C4C22×Q812yes—
218C2×C3×C4C22×Q814yes—
219C2×C3×C4C2511yescharacteristic
220C2×Dic3—28no—
221C2×Dic3—28no—
222C2×Dic3—48no—
223C2×Dic3—88no—
224C2×Dic3C22×D418yes—
225Dic6—816no—
226C3×C8C22×Q814yes—
227C3×C8C22×Q814yes—
228C3⋊C8C22×Q818yes—
229C24D3×D411yescharacteristic
230C2×Q8—616no—
231C2×Q8—248no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—34no—
235C22×C4—64no—
236C22×C4—68no—
237C22×C4—244no—
238C22×C4C2×Dic614yes—
239C22×C4D3×D412yes—
240C22×C4D3×D411yescharacteristic
241C22×C4D3×Q814yes—
242C22×C4C23×D312yes—
243C22×C4C23×D311yescharacteristic
244M16—316no—
245C2×C8—38no—
246C2×C8—34no—
247C2×C8C2×Dic614yes—
248C2×C8C2×Dic614yes—
249C2×C8C2×Dic6116yescomplemented
250C2×C8C23×D312yes—
251C2×C8C23×D312yes—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—38no—
255C4⋊C4—38no—
256C4⋊C4—616no—
257C4⋊C4—616no—
258C4⋊C4—616no—
259C4⋊C4—68no—
260C4⋊C4—64no—
261C4⋊C4C23×D314yes—
262C4⋊C4C23×D314yes—
263C22⋊C4—28no—
264C22⋊C4—38no—
265C42—28no—
266C42—34no—
267C42—34no—
268C22×C3—28no—
269C22×C3—24no—
270C22×C3C2×C4⋊Q814yes—
271C22×C3C2×C4⋊Q812yes—
272C22×C3#57714yes—
273C22×C3C23×D411yescharacteristic
274C3×C4—44no—
275C3×C4—44no—
276C3×C4C2×C4⋊Q812yes—
277C3×C4C2×C4⋊Q814yes—
278C3×C4C23×Q811yescharacteristic
279C3×C4C23×Q811yescharacteristic
280Dic3—28no—
281Dic3—48no—
282C23—24no—
283C23#88814yes—
284C23C2×D3×D412yes—
285C23C2×D3×D411yescharacteristic
286Q8—2416no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—38no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—68no—
293C2×C4—68no—
294C2×C4—128no—
295C2×C4—248no—
296C2×C4#88818yes—
297C2×C4#88814yes—
298C2×C4C22×Dic612yes—
299C2×C4C22×Dic614yes—
300C2×C4C2×D3×D412yes—
301C2×C4C2×D3×D411yescharacteristic
302C2×C4C2×D3×Q812yes—
303C2×C4C2×D3×Q814yes—
304C2×C4C24×D311yescharacteristic
305C8—38no—
306C8C22×Dic614yes—
307C8C22×Dic614yes—
308C2×C3—24no—
309C2×C3#316314yescomplemented
310C2×C3C22×C4⋊Q811yescharacteristic
311C2×C3C2×#57712yes—
312C22—28no—
313C22—24no—
314C22C2×#88814yes—
315C22C2×#88812yes—
316C22#577014yes—
317C22C22×D3×D411yescharacteristic
318C4—44no—
319C4—44no—
320C4—68no—
321C4—128no—
322C4C2×#88812yes—
323C4C2×#88814yes—
324C4C23×Dic611yescharacteristic
325C4C22×D3×Q811yescharacteristic
326C3C2×#316311yescharacteristic, complemented, Sylow
327C2—24no—
328C2#7917214yesdirect product factor
329C2C22×#88811yescharacteristic
330C2C2×#577012yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC2×C3×#427C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#427D3
hypocenterC3C2×#3163
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
6. #79172C2direct (N × H)
7. C2×#4876C2semidirect
8. C2×#4858C2semidirect
9. C2×#4855C2semidirect
36. #4876C22semidirect
37. #4858C22semidirect
38. #4855C22semidirect
51. C22×C3×C8Q8semidirect
73. C2×#208Q8semidirect
137. C2×C3×C8C2×Q8semidirect
156. #208C2×Q8semidirect
165. C22×C8Dic6semidirect
249. C2×C8C2×Dic6semidirect
309. C2×C3#3163semidirect
326. C3C2×#3163semidirect
328. C2#79172direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#55192
C2×C3×#4272
C22×#8172
C2×#51012
C2×#49952
#791722
C2×#48762
C2×#48582
C2×#48552
C2×#31633