Finite Group Databasebeta

← C3×#61820 #10680427

Subgroups — C3×#61820 #10680427

Order 768.

1966 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3497C211yesmaximal, characteristic
2C2×C3×Q82C211yesmaximal, characteristic, complemented
3C3×Q8×SD16C218yesmaximal, complemented
4C3×#3362C218yesmaximal, complemented
5C3×#3227C216yesmaximal, complemented
6C3×#3103C216yesmaximal, complemented
7C3×#2982C211yesmaximal, characteristic, complemented
8#61820C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×2−1+4—21no—
10C3×Q82—24no—
11C3×Q82C2218yes—
12C3×#550C2214yes—
13C3×D4×Q8C2214yes—
14C3×#548—24no—
15C3×#537—23no—
16C3×#532C2216yes—
17C2×C3×C4⋊Q8—23no—
18C2×C3×C4⋊Q8C2213yes—
19C2×C3×C4×Q8—21no—
20C2×C3×C4×Q8C2213yes—
21C2×C3×C4×Q8C2211yescharacteristic
22C3×#498C22112yes—
23C3×#486C22112yes—
24C3×#481C22112yes—
25C3×#476C22124yes—
26C3×#475C22112yes—
27C3×#460C22112yes—
28C3×#448C2214yes—
29C3×C8×Q8C2218yes—
30C3×C4×SD16C22124yes—
31C3×#431C2216yes—
32#3497C2×C311yescharacteristic
33C2×Q82C2×C311yescharacteristic, complemented
34Q8×SD16C2×C318yescomplemented
35#3362C2×C318yescomplemented
36#3227C2×C316yescomplemented
37#3103C2×C316yescomplemented
38#2982C2×C311yescharacteristic, complemented
39C3×2−1+4—48no—
40C2×C3×#41—24no—
41C2×C3×#41—26no—
42C22×C3×Q8—24no—
43C22×C3×Q8—23no—
44C22×C3×Q8D411yescharacteristic
45C22×C3×Q8C2311yescharacteristic
46C2×C3×SD16C2318yes—
47C3×C4⋊Q8—224no—
48C3×C4⋊Q8—26no—
49C3×C4⋊Q8C23112yes—
50C3×C4⋊Q8C2316yes—
51C3×#127C2316yes—
52C3×#124—26no—
53C3×C22⋊Q8—212no—
54C3×#121—26no—
55C3×#121—212no—
56C3×#121C2314yes—
57C3×C4×Q8—28no—
58C3×C4×Q8—26no—
59C3×C4×Q8D414yes—
60C3×C4×Q8C23112yes—
61C3×C4×Q8C2314yes—
62C3×C4×D4—26no—
63C3×C4×D4C23112yes—
64C2×C3×C4⋊C4—23no—
65C2×C3×C4⋊C4—23no—
66C2×C3×C4⋊C4D413yes—
67C2×C3×C4⋊C4C2313yes—
68C2×C3×C42—23no—
69C2×C3×C42C2313yes—
70C3×#106C23124yes—
71C3×C4⋊C8C23112yes—
72C3×#103C2318yes—
73C3×#102C23124yes—
74C3×C22⋊C8C2314yes—
75C3×C4×C8C23112yes—
76C2×2−1+4—21no—
77Q82—24no—
78Q82C22×C318yes—
79#550C22×C314yes—
80D4×Q8C22×C314yes—
81#548—24no—
82#537—23no—
83#532C22×C316yes—
84C2×C4⋊Q8—23no—
85C2×C4⋊Q8C22×C313yes—
86C2×C4×Q8—21no—
87C2×C4×Q8C22×C313yes—
88C2×C4×Q8C22×C311yescharacteristic
89#498C22×C3112yes—
90#486C22×C3112yes—
91#481C22×C3112yes—
92#476C22×C3124yes—
93#475C22×C3112yes—
94#460C22×C3112yes—
95#448C22×C314yes—
96C8×Q8C22×C318yes—
97C4×SD16C22×C3124yes—
98#431C22×C316yes—
99C3×#41—416no—
100C3×#41—424no—
101C2×C3×Q8—24no—
102C2×C3×Q8—21no—
103C2×C3×Q8—26no—
104C2×C3×Q8—212no—
105C2×C3×Q8—416no—
106C2×C3×Q8—86no—
107C2×C3×Q8SD1618yescomplemented
108C2×C3×Q8C2×D414yes—
109C2×C3×Q8C2412yes—
110C2×C3×D4—24no—
111C2×C3×D4—26no—
112C2×C3×D4—26no—
113C2×C3×D4C2414yes—
114C22×C3×C4—21no—
115C22×C3×C4—23no—
116C22×C3×C4—212no—
117C22×C3×C4C2×D413yes—
118C22×C3×C4C2×D413yes—
119C22×C3×C4C2411yescharacteristic
120C3×SD16—216no—
121C2×C3×C8C2418yes—
122C3×C4⋊C4—212no—
123C3×C4⋊C4—212no—
124C3×C4⋊C4—23no—
125C3×C4⋊C4—26no—
126C3×C4⋊C4C2×D4112yes—
127C3×C4⋊C4C2416yes—
128C3×C4⋊C4C2414yes—
129C3×C22⋊C4—24no—
130C3×C22⋊C4—212no—
131C3×C42—212no—
132C3×C42C2×D416yes—
133C3×C42C2416yes—
1342−1+4—48no—
135C2×#41—24no—
136C2×#41—26no—
137C22×Q8—24no—
138C22×Q8—23no—
139C22×Q8C3×D411yescharacteristic
140C22×Q8C23×C311yescharacteristic
141C2×SD16C23×C318yes—
142C4⋊Q8—224no—
143C4⋊Q8—26no—
144C4⋊Q8C23×C3112yes—
145C4⋊Q8C23×C316yes—
146#127C23×C316yes—
147#124—26no—
148C22⋊Q8—212no—
149#121—26no—
150#121—212no—
151#121C23×C314yes—
152C4×Q8—28no—
153C4×Q8—26no—
154C4×Q8C3×D414yes—
155C4×Q8C23×C3112yes—
156C4×Q8C23×C314yes—
157C4×D4—26no—
158C4×D4C23×C3112yes—
159C2×C4⋊C4—23no—
160C2×C4⋊C4—23no—
161C2×C4⋊C4C3×D413yes—
162C2×C4⋊C4C23×C313yes—
163C2×C42—23no—
164C2×C42C23×C313yes—
165#106C23×C3124yes—
166C4⋊C8C23×C3112yes—
167#103C23×C318yes—
168#102C23×C3124yes—
169C22⋊C8C23×C314yes—
170C4×C8C23×C3112yes—
171C23×C3—24no—
172C23×C3C22×D411yescharacteristic
173C3×Q8—24no—
174C3×Q8—22no—
175C3×Q8—412no—
176C3×Q8—812no—
177C3×Q8C2×SD16116yes—
178C3×Q82−1+414yescomplemented
179C3×D4—28no—
180C3×D4—48no—
181C3×D4—412no—
182C3×D4—86no—
183C2×C3×C4—24no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—26no—
187C2×C3×C4—24no—
188C2×C3×C4—43no—
189C2×C3×C4—46no—
190C2×C3×C4—424no—
191C2×C3×C4C2×SD16112yes—
192C2×C3×C4C22×D411yescharacteristic
193C2×C3×C4C22×D416yes—
194C2×C3×C4C22×D416yes—
195C2×C3×C4C2511yescharacteristic
196C3×C8—28no—
197#41—416no—
198#41—424no—
199C2×Q8—24no—
200C2×Q8—21no—
201C2×Q8—26no—
202C2×Q8—212no—
203C2×Q8—416no—
204C2×Q8—86no—
205C2×Q8C3×SD1618yescomplemented
206C2×Q8C2×C3×D414yes—
207C2×Q8C24×C312yes—
208C2×D4—24no—
209C2×D4—26no—
210C2×D4—26no—
211C2×D4C24×C314yes—
212C22×C4—21no—
213C22×C4—23no—
214C22×C4—212no—
215C22×C4C2×C3×D413yes—
216C22×C4C2×C3×D413yes—
217C22×C4C24×C311yescharacteristic
218SD16—216no—
219C2×C8C24×C318yes—
220C4⋊C4—212no—
221C4⋊C4—212no—
222C4⋊C4—23no—
223C4⋊C4—26no—
224C4⋊C4C2×C3×D4112yes—
225C4⋊C4C24×C316yes—
226C4⋊C4C24×C314yes—
227C22⋊C4—24no—
228C22⋊C4—212no—
229C42—212no—
230C42C2×C3×D416yes—
231C42C24×C316yes—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—44no—
235C22×C3—44no—
236C22×C3—44no—
237C22×C3C22×SD1612yes—
238C22×C3C23×D411yescharacteristic
239C3×C4—26no—
240C3×C4—24no—
241C3×C4—21no—
242C3×C4—46no—
243C3×C4—44no—
244C3×C4C22×SD16112yes—
245C3×C4C2×2−1+412yes—
246C23—24no—
247C23C22×C3×D411yescharacteristic
248Q8—24no—
249Q8—22no—
250Q8—412no—
251Q8—812no—
252Q8C2×C3×SD16116yes—
253Q8C3×2−1+414yescomplemented
254D4—28no—
255D4—48no—
256D4—412no—
257D4—86no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—26no—
262C2×C4—24no—
263C2×C4—43no—
264C2×C4—46no—
265C2×C4—424no—
266C2×C4C2×C3×SD16112yes—
267C2×C4C22×C3×D411yescharacteristic
268C2×C4C22×C3×D416yes—
269C2×C4C22×C3×D416yes—
270C2×C4C25×C311yescharacteristic
271C8—28no—
272C2×C3—22no—
273C2×C3—84no—
274C2×C3#348611yescharacteristic
275C2×C3C23×SD1611yescharacteristic
276C2×C3#358511yescharacteristic
277C22—21no—
278C22—21no—
279C22—44no—
280C22—44no—
281C22—44no—
282C22C22×C3×SD1612yes—
283C22C23×C3×D411yescharacteristic
284C4—24no—
285C4—21no—
286C4—26no—
287C4—44no—
288C4—46no—
289C4C22×C3×SD16112yes—
290C4C2×C3×2−1+412yes—
291C3#6182011yescharacteristic, direct product factor, Sylow
292C2—22no—
293C2—84no—
294C2C3×#348611yescharacteristic
295C2C23×C3×SD1611yescharacteristic
296C2C3×#358511yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC3×#3497C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×Q82C2semidirect
3. C3×Q8×SD16C2semidirect
4. C3×#3362C2semidirect
5. C3×#3227C2semidirect
6. C3×#3103C2semidirect
7. C3×#2982C2semidirect
8. #61820C3direct (N × H)
33. C2×Q82C2×C3semidirect
34. Q8×SD16C2×C3semidirect
35. #3362C2×C3semidirect
36. #3227C2×C3semidirect
37. #3103C2×C3semidirect
38. #2982C2×C3semidirect
107. C2×C3×Q8SD16semidirect
178. C3×Q82−1+4semidirect
205. C2×Q8C3×SD16semidirect
253. Q8C3×2−1+4semidirect
291. C3#61820direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34972
C2×C3×Q822
C3×Q8×SD162
C3×#33622
C3×#32272
C3×#31032
C3×#29822
#618203