Finite Group Databasebeta

← C3×#27666 #10646273

Subgroups — C3×#27666 #10646273

Order 768.

840 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×C3×#435C211yesmaximal, characteristic
2C22×C3×#107C211yesmaximal, characteristic, complemented
3C3×#2487C212yesmaximal
4C3×#2086C212yesmaximal, complemented
5C3×#2075C212yesmaximal, complemented
6C3×#1944C212yesmaximal, complemented
7C3×#1824C212yesmaximal, complemented
8C3×#1809C212yesmaximal, complemented
9C3×#1807C211yesmaximal, characteristic, complemented
10#27666C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×M16C2211yescharacteristic
12C23×C3×C8C2211yescharacteristic
13C2×C3×C4×D4C2211yescharacteristic
14C22×C3×C4⋊C4C2212yes—
15C3×#435—24no—
16C2×C3×#107—24no—
17C2×C3×#107C2212yes—
18C2×C3×#107C2212yes—
19C2×C3×#106C2212yes—
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×C22⋊C8C2211yescharacteristic
22C2×C3×C22⋊C8C2211yescharacteristic
23C2×C3×#97C2211yescharacteristic
24C3×#400—22no—
25C3×#399—22no—
26C3×#389C2214yes—
27C3×#386—22no—
28C3×#385—22no—
29C3×#380C2212yes—
30C3×#380C2214yes—
31C3×#340C2212yes—
32C3×#340C2214yes—
33C3×#340C2214yes—
34C2×#435C2×C311yescharacteristic
35C22×#107C2×C311yescharacteristic, complemented
36#2487C2×C312yes—
37#2086C2×C312yescomplemented
38#2075C2×C312yescomplemented
39#1944C2×C312yescomplemented
40#1824C2×C312yescomplemented
41#1809C2×C312yescomplemented
42#1807C2×C311yescharacteristic, complemented
43C22×C3×D4D411yescharacteristic
44C23×C3×C4—22no—
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C2×C3×M16—22no—
48C2×C3×M16—24no—
49C22×C3×C8—24no—
50C22×C3×C8D414yes—
51C22×C3×C8C2311yescharacteristic
52C22×C3×C8C2311yescharacteristic
53C22×C3×C8C2311yescharacteristic
54C22×C3×C8C2311yescharacteristic
55C3×C4×D4—22no—
56C3×C4×D4—22no—
57C2×C3×C4⋊C4—24no—
58C2×C3×C4⋊C4—24no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—24no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C4⋊C4C2314yes—
63C2×C3×C4⋊C4C2314yes—
64C2×C3×C22⋊C4—22no—
65C2×C3×C22⋊C4—24no—
66C2×C3×C22⋊C4D411yescharacteristic
67C2×C3×C22⋊C4D411yescharacteristic
68C2×C3×C42C2311yescharacteristic
69C3×#107—24no—
70C3×#107—24no—
71C3×#106—24no—
72C3×C4⋊C8—22no—
73C3×C22⋊C8—22no—
74C3×C22⋊C8—22no—
75C3×#97—22no—
76C3×#95—24no—
77C3×#95—22no—
78C3×#95—24no—
79C22×M16C22×C311yescharacteristic
80C23×C8C22×C311yescharacteristic
81C2×C4×D4C22×C311yescharacteristic
82C22×C4⋊C4C22×C312yes—
83#435—24no—
84C2×#107—24no—
85C2×#107C22×C312yes—
86C2×#107C22×C312yes—
87C2×#106C22×C312yes—
88C2×C4⋊C8C22×C311yescharacteristic
89C2×C22⋊C8C22×C311yescharacteristic
90C2×C22⋊C8C22×C311yescharacteristic
91C2×#97C22×C311yescharacteristic
92#400—22no—
93#399—22no—
94#389C22×C314yes—
95#386—22no—
96#385—22no—
97#380C22×C312yes—
98#380C22×C314yes—
99#340C22×C312yes—
100#340C22×C314yes—
101#340C22×C314yes—
102C24×C3C2×D411yescharacteristic
103C24×C3C2×D411yescharacteristic
104C2×C3×D4—22no—
105C2×C3×D4—24no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—21no—
109C22×C3×C4—21no—
110C22×C3×C4—22no—
111C22×C3×C4—22no—
112C22×C3×C4—24no—
113C22×C3×C4—24no—
114C22×C3×C4—44no—
115C22×C3×C4—84no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D412yes—
121C22×C3×C4#4112yes—
122C22×C3×C4C2411yescharacteristic
123C3×M16—44no—
124C2×C3×C8—22no—
125C2×C3×C8—28no—
126C2×C3×C8—21no—
127C2×C3×C8—21no—
128C2×C3×C8—22no—
129C2×C3×C8—22no—
130C2×C3×C8C2×D412yes—
131C2×C3×C8C2×D412yes—
132C2×C3×C8#4114yes—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—24no—
135C3×C4⋊C4—24no—
136C3×C4⋊C4—24no—
137C3×C4⋊C4—24no—
138C3×C4⋊C4—44no—
139C3×C4⋊C4—84no—
140C3×C22⋊C4—22no—
141C3×C22⋊C4—22no—
142C3×C22⋊C4—44no—
143C3×C22⋊C4—84no—
144C3×C42—21no—
145C3×C42—21no—
146C22×D4C3×D411yescharacteristic
147C23×C4—22no—
148C23×C4C23×C311yescharacteristic
149C23×C4C23×C311yescharacteristic
150C2×M16—22no—
151C2×M16—24no—
152C22×C8—24no—
153C22×C8C3×D414yes—
154C22×C8C23×C311yescharacteristic
155C22×C8C23×C311yescharacteristic
156C22×C8C23×C311yescharacteristic
157C22×C8C23×C311yescharacteristic
158C4×D4—22no—
159C4×D4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4C3×D411yescharacteristic
165C2×C4⋊C4C23×C314yes—
166C2×C4⋊C4C23×C314yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4C3×D411yescharacteristic
170C2×C22⋊C4C3×D411yescharacteristic
171C2×C42C23×C311yescharacteristic
172#107—24no—
173#107—24no—
174#106—24no—
175C4⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178#97—22no—
179#95—24no—
180#95—22no—
181#95—24no—
182C23×C3—22no—
183C23×C3—22no—
184C23×C3—21no—
185C23×C3—22no—
186C23×C3—22no—
187C23×C3—22no—
188C23×C3#12112yes—
189C23×C3#13714yes—
190C23×C3C22×D411yescharacteristic
191C3×D4—44no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—24no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—22no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—44no—
206C2×C3×C4—44no—
207C2×C3×C4—44no—
208C2×C3×C4—44no—
209C2×C3×C4—44no—
210C2×C3×C4—44no—
211C2×C3×C4—88no—
212C2×C3×C4#12112yes—
213C2×C3×C4C22×D411yescharacteristic
214C2×C3×C4C2×#4112yes—
215C2×C3×C4C2×#4111yescharacteristic
216C2×C3×C42−1+412yes—
217C3×C8—22no—
218C3×C8—41no—
219C3×C8—41no—
220C3×C8#12314yes—
221C24C2×C3×D411yescharacteristic
222C24C2×C3×D411yescharacteristic
223C2×D4—22no—
224C2×D4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—44no—
234C22×C4—84no—
235C22×C4C2×C3×D411yescharacteristic
236C22×C4C2×C3×D411yescharacteristic
237C22×C4C2×C3×D411yescharacteristic
238C22×C4C2×C3×D411yescharacteristic
239C22×C4C2×C3×D412yes—
240C22×C4C3×#4112yes—
241C22×C4C24×C311yescharacteristic
242M16—44no—
243C2×C8—22no—
244C2×C8—28no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—22no—
248C2×C8—22no—
249C2×C8C2×C3×D412yes—
250C2×C8C2×C3×D412yes—
251C2×C8C3×#4114yes—
252C4⋊C4—22no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—44no—
258C4⋊C4—84no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—44no—
262C22⋊C4—84no—
263C42—21no—
264C42—21no—
265C22×C3—28no—
266C22×C3—22no—
267C22×C3—22no—
268C22×C3—44no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3#47014yes—
273C22×C3C2×#12111yescharacteristic
274C22×C3#54712yes—
275C22×C3C2×#13712yes—
276C22×C3#57612yes—
277C3×C4—22no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4—84no—
282C3×C4—84no—
283C3×C4C2×#12311yescharacteristic
284C3×C4#53611yescharacteristic
285C3×C4#55212yes—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—21no—
291C23—22no—
292C23C3×#12112yes—
293C23C3×#13714yes—
294C23C22×C3×D411yescharacteristic
295D4—44no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—88no—
316C2×C4C3×#12112yes—
317C2×C4C22×C3×D411yescharacteristic
318C2×C4C2×C3×#4112yes—
319C2×C4C2×C3×#4111yescharacteristic
320C2×C4C3×2−1+412yes—
321C8—22no—
322C8—41no—
323C8—41no—
324C8C3×#12314yes—
325C2×C3—24no—
326C2×C3—81no—
327C2×C3#248611yescharacteristic
328C2×C3C2×#47011yescharacteristic
329C2×C3#305811yescharacteristic
330C2×C3#334614yes—
331C22—22no—
332C22—22no—
333C22—28no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—44no—
338C22C3×#47014yes—
339C22C2×C3×#12111yescharacteristic
340C22C3×#54712yes—
341C22C2×C3×#13712yes—
342C22C3×#57612yes—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—84no—
348C4—84no—
349C4C2×C3×#12311yescharacteristic
350C4C3×#53611yescharacteristic
351C4C3×#55212yes—
352C3#2766611yescharacteristic, direct product factor, Sylow
353C2—24no—
354C2—81no—
355C2C3×#248611yescharacteristic
356C2C2×C3×#47011yescharacteristic
357C2C3×#305811yescharacteristic
358C2C3×#334614yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C3×#107C2semidirect
4. C3×#2086C2semidirect
5. C3×#2075C2semidirect
6. C3×#1944C2semidirect
7. C3×#1824C2semidirect
8. C3×#1809C2semidirect
9. C3×#1807C2semidirect
10. #27666C3direct (N × H)
35. C22×#107C2×C3semidirect
37. #2086C2×C3semidirect
38. #2075C2×C3semidirect
39. #1944C2×C3semidirect
40. #1824C2×C3semidirect
41. #1809C2×C3semidirect
42. #1807C2×C3semidirect
352. C3#27666direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4352
C22×C3×#1072
C3×#24872
C3×#20862
C3×#20752
C3×#19442
C3×#18242
C3×#18092
C3×#18072
#276663