Finite Group Databasebeta

← C3×#27621 #10646228

Subgroups — C3×#27621 #10646228

Order 768.

670 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2316C211yesmaximal, characteristic
2C3×#2314C211yesmaximal, characteristic
3C3×#2101C212yesmaximal, complemented
4C3×#2084C212yesmaximal, complemented
5C3×#1920C212yesmaximal, complemented
6C3×#1919C212yesmaximal, complemented
7C3×#1853C212yesmaximal
8C3×#1825C212yesmaximal
9C3×#1818C211yesmaximal, characteristic
10#27621C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#117C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C22×C3×C4⋊C4C2211yescharacteristic
14C2×C3×#106C2212yes—
15C2×C3×#106C2212yes—
16C2×C3×C22⋊C8C2212yes—
17C3×#389—22no—
18C3×#389C2212yes—
19C3×#389C2212yes—
20C3×#388—22no—
21C3×#387—22no—
22C3×#384C2211yescharacteristic
23C3×#382C2211yescharacteristic
24C3×#380—22no—
25C3×#380C2212yes—
26C3×C4×C4⋊C4C2212yes—
27C3×#340C2212yes—
28C3×#340C2212yes—
29C3×#340C2212yes—
30C3×#340C2212yes—
31C3×#340C2212yes—
32C3×#340C2212yes—
33C3×#336C2212yes—
34C3×#336C2211yescharacteristic
35C3×#336C2211yescharacteristic
36#2316C2×C311yescharacteristic
37#2314C2×C311yescharacteristic
38#2101C2×C312yescomplemented
39#2084C2×C312yescomplemented
40#1920C2×C312yescomplemented
41#1919C2×C312yescomplemented
42#1853C2×C312yes—
43#1825C2×C312yes—
44#1818C2×C311yescharacteristic
45C23×C3×C4—21no—
46C23×C3×C4C2311yescharacteristic
47C22×C3×C8C2312yes—
48C22×C3×C8C2312yes—
49C3×#117—24no—
50C3×#117—24no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4D412yes—
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C4⋊C4C2312yes—
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C4⋊C4C2312yes—
61C2×C3×C22⋊C4—21no—
62C2×C3×C22⋊C4—22no—
63C2×C3×C22⋊C4D412yes—
64C2×C3×C42—21no—
65C2×C3×C42—22no—
66C2×C3×C42C2311yescharacteristic
67C2×C3×C42C2311yescharacteristic
68C3×#106—24no—
69C3×#106—24no—
70C3×C22⋊C8—24no—
71C3×#95—22no—
72C3×#95—22no—
73C3×#95—22no—
74C3×#95—22no—
75C3×#95—22no—
76C2×#117C22×C311yescharacteristic
77C2×#117C22×C311yescharacteristic
78C22×C4⋊C4C22×C311yescharacteristic
79C2×#106C22×C312yes—
80C2×#106C22×C312yes—
81C2×C22⋊C8C22×C312yes—
82#389—22no—
83#389C22×C312yes—
84#389C22×C312yes—
85#388—22no—
86#387—22no—
87#384C22×C311yescharacteristic
88#382C22×C311yescharacteristic
89#380—22no—
90#380C22×C312yes—
91C4×C4⋊C4C22×C312yes—
92#340C22×C312yes—
93#340C22×C312yes—
94#340C22×C312yes—
95#340C22×C312yes—
96#340C22×C312yes—
97#340C22×C312yes—
98#336C22×C312yes—
99#336C22×C311yescharacteristic
100#336C22×C311yescharacteristic
101C24×C3C2×D411yescharacteristic
102C22×C3×C4—22no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—22no—
106C22×C3×C4—24no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—82no—
111C22×C3×C4C2×D412yes—
112C22×C3×C4C2×D412yes—
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4#4114yes—
115C22×C3×C4C2411yescharacteristic
116C2×C3×C8—22no—
117C2×C3×C8—22no—
118C2×C3×C8—22no—
119C2×C3×C8—22no—
120C2×C3×C8—22no—
121C2×C3×C8—22no—
122C3×C4⋊C4—24no—
123C3×C4⋊C4—24no—
124C3×C4⋊C4—22no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—24no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—22no—
130C3×C4⋊C4—82no—
131C3×C4⋊C4—82no—
132C3×C4⋊C4—82no—
133C3×C4⋊C4#4114yes—
134C3×C22⋊C4—24no—
135C3×C22⋊C4—42no—
136C3×C22⋊C4—82no—
137C3×C42—22no—
138C3×C42—22no—
139C3×C42—42no—
140C3×C42—82no—
141C23×C4—21no—
142C23×C4C23×C311yescharacteristic
143C22×C8C23×C312yes—
144C22×C8C23×C312yes—
145#117—24no—
146#117—24no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4C3×D412yes—
153C2×C4⋊C4C23×C312yes—
154C2×C4⋊C4C23×C312yes—
155C2×C4⋊C4C23×C312yes—
156C2×C4⋊C4C23×C312yes—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4C3×D412yes—
160C2×C42—21no—
161C2×C42—22no—
162C2×C42C23×C311yescharacteristic
163C2×C42C23×C311yescharacteristic
164#106—24no—
165#106—24no—
166C22⋊C8—24no—
167#95—22no—
168#95—22no—
169#95—22no—
170#95—22no—
171#95—22no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—22no—
175C23×C3—22no—
176C23×C3#12312yes—
177C23×C3C22×D411yescharacteristic
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—24no—
181C2×C3×C4—24no—
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—24no—
185C2×C3×C4—22no—
186C2×C3×C4—22no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—42no—
199C2×C3×C4—42no—
200C2×C3×C4—84no—
201C2×C3×C4#12312yes—
202C2×C3×C4C2×#4112yes—
203C2×C3×C4C2×#4112yes—
204C2×C3×C42−1+411yescharacteristic
205C2×C3×C42−1+411yescharacteristic
206C3×C8—42no—
207C3×C8—42no—
208C24C2×C3×D411yescharacteristic
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—82no—
218C22×C4C2×C3×D412yes—
219C22×C4C2×C3×D412yes—
220C22×C4C2×C3×D411yescharacteristic
221C22×C4C3×#4114yes—
222C22×C4C24×C311yescharacteristic
223C2×C8—22no—
224C2×C8—22no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—22no—
229C4⋊C4—24no—
230C4⋊C4—24no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—24no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—82no—
238C4⋊C4—82no—
239C4⋊C4—82no—
240C4⋊C4C3×#4114yes—
241C22⋊C4—24no—
242C22⋊C4—42no—
243C22⋊C4—82no—
244C42—22no—
245C42—22no—
246C42—42no—
247C42—82no—
248C22×C3—24no—
249C22×C3—22no—
250C22×C3—41no—
251C22×C3—41no—
252C22×C3—42no—
253C22×C3C2×#12311yescharacteristic
254C22×C3#52511yescharacteristic
255C22×C3#53611yescharacteristic
256C22×C3#57712yes—
257C22×C3#57712yes—
258C3×C4—22no—
259C3×C4—42no—
260C3×C4—42no—
261C3×C4—44no—
262C3×C4—82no—
263C3×C4—82no—
264C3×C4—82no—
265C3×C4#55212yes—
266C3×C4#55212yes—
267C23—22no—
268C23—22no—
269C23—21no—
270C23—21no—
271C23C3×#12312yes—
272C23C22×C3×D411yescharacteristic
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—84no—
296C2×C4C3×#12312yes—
297C2×C4C2×C3×#4112yes—
298C2×C4C2×C3×#4112yes—
299C2×C4C3×2−1+411yescharacteristic
300C2×C4C3×2−1+411yescharacteristic
301C8—42no—
302C8—42no—
303C2×C3—42no—
304C2×C3#248411yescharacteristic
305C2×C3#309912yes—
306C2×C3#312311yescharacteristic
307C2×C3#312611yescharacteristic
308C2×C3#324412yes—
309C22—22no—
310C22—24no—
311C22—42no—
312C22—41no—
313C22—41no—
314C22C2×C3×#12311yescharacteristic
315C22C3×#52511yescharacteristic
316C22C3×#53611yescharacteristic
317C22C3×#57712yes—
318C22C3×#57712yes—
319C4—22no—
320C4—42no—
321C4—42no—
322C4—44no—
323C4—82no—
324C4—82no—
325C4—82no—
326C4C3×#55212yes—
327C4C3×#55212yes—
328C3#2762111yescharacteristic, direct product factor, Sylow
329C2—42no—
330C2C3×#248411yescharacteristic
331C2C3×#309912yes—
332C2C3×#312311yescharacteristic
333C2C3×#312611yescharacteristic
334C2C3×#324412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2101C2semidirect
4. C3×#2084C2semidirect
5. C3×#1920C2semidirect
6. C3×#1919C2semidirect
10. #27621C3direct (N × H)
38. #2101C2×C3semidirect
39. #2084C2×C3semidirect
40. #1920C2×C3semidirect
41. #1919C2×C3semidirect
328. C3#27621direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#23162
C3×#23142
C3×#21012
C3×#20842
C3×#19202
C3×#19192
C3×#18532
C3×#18252
C3×#18182
#276213