Finite Group Databasebeta

← C3×#14910 #10633517

Subgroups — C3×#14910 #10633517

Order 768.

1626 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#399C211yesmaximal, characteristic
2C2×C3×#389C213yesmaximal, complemented
3C2×C3×#388C213yesmaximal
4C2×C3×#376C211yesmaximal, characteristic, complemented
5C3×#1445C213yesmaximal, complemented
6C3×#1444C211yesmaximal, characteristic, complemented
7C3×#1439C213yesmaximal
8#14910C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×C4⋊C4C2213yes—
10C22×C3×C4⋊C4C2213yes—
11C22×C3×C4⋊C4C2213yes—
12C22×C3×C22⋊C4C2213yes—
13C22×C3×C22⋊C4C2211yescharacteristic
14C22×C3×C22⋊C4C2213yes—
15C22×C3×C42C2213yes—
16C3×#399—24no—
17C3×#389—212no—
18C3×#388—212no—
19C3×#376—24no—
20C2×C3×#95C2213yes—
21C2×C3×#95C2213yes—
22C2×C3×#95C2213yes—
23C2×C3×#95C2216yes—
24C2×C3×#95C2211yescharacteristic
25C2×#399C2×C311yescharacteristic
26C2×#389C2×C313yescomplemented
27C2×#388C2×C313yes—
28C2×#376C2×C311yescharacteristic, complemented
29#1445C2×C313yescomplemented
30#1444C2×C311yescharacteristic, complemented
31#1439C2×C313yes—
32C25×C3C2311yescharacteristic
33C23×C3×C4C2311yescharacteristic
34C23×C3×C4C2313yes—
35C23×C3×C4C2313yes—
36C23×C3×C4C2313yes—
37C23×C3×C4C2311yescharacteristic
38C23×C3×C4C2313yes—
39C2×C3×C4⋊C4—26no—
40C2×C3×C4⋊C4—212no—
41C2×C3×C4⋊C4—26no—
42C2×C3×C4⋊C4—26no—
43C2×C3×C4⋊C4—212no—
44C2×C3×C4⋊C4—26no—
45C2×C3×C4⋊C4—26no—
46C2×C3×C22⋊C4—26no—
47C2×C3×C22⋊C4—26no—
48C2×C3×C22⋊C4—26no—
49C2×C3×C22⋊C4—26no—
50C2×C3×C22⋊C4—26no—
51C2×C3×C22⋊C4—26no—
52C2×C3×C22⋊C4—26no—
53C2×C3×C42—26no—
54C2×C3×C42—26no—
55C2×C3×C42D4112yes—
56C3×#95—26no—
57C3×#95—26no—
58C3×#95—26no—
59C3×#95—212no—
60C3×#95D414yes—
61C22×C4⋊C4C22×C313yes—
62C22×C4⋊C4C22×C313yes—
63C22×C4⋊C4C22×C313yes—
64C22×C22⋊C4C22×C313yes—
65C22×C22⋊C4C22×C313yes—
66C22×C22⋊C4C22×C311yescharacteristic
67C22×C42C22×C313yes—
68#399—24no—
69#389—212no—
70#388—212no—
71#376—24no—
72C2×#95C22×C313yes—
73C2×#95C22×C316yes—
74C2×#95C22×C313yes—
75C2×#95C22×C313yes—
76C2×#95C22×C311yescharacteristic
77C24×C3—23no—
78C24×C3—21no—
79C24×C3—23no—
80C24×C3—23no—
81C24×C3—21no—
82C24×C3—23no—
83C24×C3#4112yes—
84C24×C3C2411yescharacteristic
85C22×C3×C4—21no—
86C22×C3×C4—23no—
87C22×C3×C4—26no—
88C22×C3×C4—26no—
89C22×C3×C4—23no—
90C22×C3×C4—23no—
91C22×C3×C4—26no—
92C22×C3×C4—23no—
93C22×C3×C4—23no—
94C22×C3×C4—23no—
95C22×C3×C4—23no—
96C22×C3×C4—26no—
97C22×C3×C4—23no—
98C22×C3×C4—23no—
99C22×C3×C4—23no—
100C22×C3×C4—23no—
101C22×C3×C4—26no—
102C22×C3×C4—23no—
103C22×C3×C4—23no—
104C22×C3×C4—26no—
105C22×C3×C4—23no—
106C22×C3×C4—21no—
107C22×C3×C4C2×D416yes—
108C22×C3×C4C2×D416yes—
109C22×C3×C4C2×D416yes—
110C22×C3×C4C2×D416yes—
111C22×C3×C4#4116yes—
112C22×C3×C4#4116yes—
113C3×C4⋊C4—224no—
114C3×C4⋊C4—412no—
115C3×C4⋊C4—412no—
116C3×C22⋊C4—224no—
117C3×C22⋊C4—412no—
118C3×C22⋊C4—44no—
119C3×C42—224no—
120C25C23×C311yescharacteristic
121C23×C4C23×C313yes—
122C23×C4C23×C311yescharacteristic
123C23×C4C23×C313yes—
124C23×C4C23×C313yes—
125C23×C4C23×C311yescharacteristic
126C23×C4C23×C313yes—
127C2×C4⋊C4—26no—
128C2×C4⋊C4—212no—
129C2×C4⋊C4—26no—
130C2×C4⋊C4—26no—
131C2×C4⋊C4—212no—
132C2×C4⋊C4—26no—
133C2×C4⋊C4—26no—
134C2×C22⋊C4—26no—
135C2×C22⋊C4—26no—
136C2×C22⋊C4—26no—
137C2×C22⋊C4—26no—
138C2×C22⋊C4—26no—
139C2×C22⋊C4—26no—
140C2×C22⋊C4—26no—
141C2×C42—26no—
142C2×C42—26no—
143C2×C42C3×D4112yes—
144#95—26no—
145#95—26no—
146#95—26no—
147#95—212no—
148#95C3×D414yes—
149C23×C3—26no—
150C23×C3—22no—
151C23×C3—26no—
152C23×C3—43no—
153C23×C3—43no—
154C23×C3—41no—
155C23×C3—43no—
156C23×C3—43no—
157C23×C3—43no—
158C23×C3—43no—
159C23×C3—46no—
160C23×C3—43no—
161C23×C3C22×D411yescharacteristic
162C23×C3C22×D413yes—
163C23×C3C2×#4113yes—
164C23×C3C2×#4113yes—
165C23×C3C2×#4111yescharacteristic
166C23×C32−1+413yes—
167C23×C32−1+411yescharacteristic
168C2×C3×C4—26no—
169C2×C3×C4—26no—
170C2×C3×C4—26no—
171C2×C3×C4—26no—
172C2×C3×C4—212no—
173C2×C3×C4—212no—
174C2×C3×C4—26no—
175C2×C3×C4—26no—
176C2×C3×C4—26no—
177C2×C3×C4—212no—
178C2×C3×C4—26no—
179C2×C3×C4—43no—
180C2×C3×C4—46no—
181C2×C3×C4—43no—
182C2×C3×C4—43no—
183C2×C3×C4—43no—
184C2×C3×C4—43no—
185C2×C3×C4—43no—
186C2×C3×C4—43no—
187C2×C3×C4—46no—
188C2×C3×C4—41no—
189C2×C3×C4—43no—
190C2×C3×C4—43no—
191C2×C3×C4—46no—
192C2×C3×C4—43no—
193C2×C3×C4—43no—
194C2×C3×C4#121112yes—
195C2×C3×C4#12714yes—
196C24—23no—
197C24—21no—
198C24—23no—
199C24—23no—
200C24—21no—
201C24—23no—
202C24C3×#4112yes—
203C24C24×C311yescharacteristic
204C22×C4—21no—
205C22×C4—23no—
206C22×C4—26no—
207C22×C4—26no—
208C22×C4—23no—
209C22×C4—23no—
210C22×C4—26no—
211C22×C4—23no—
212C22×C4—23no—
213C22×C4—23no—
214C22×C4—23no—
215C22×C4—26no—
216C22×C4—23no—
217C22×C4—23no—
218C22×C4—23no—
219C22×C4—23no—
220C22×C4—26no—
221C22×C4—23no—
222C22×C4—23no—
223C22×C4—26no—
224C22×C4—23no—
225C22×C4—21no—
226C22×C4C2×C3×D416yes—
227C22×C4C2×C3×D416yes—
228C22×C4C2×C3×D416yes—
229C22×C4C2×C3×D416yes—
230C22×C4C3×#4116yes—
231C22×C4C3×#4116yes—
232C4⋊C4—224no—
233C4⋊C4—412no—
234C4⋊C4—412no—
235C22⋊C4—224no—
236C22⋊C4—412no—
237C22⋊C4—44no—
238C42—224no—
239C22×C3—46no—
240C22×C3—42no—
241C22×C3—46no—
242C22×C3—81no—
243C22×C3—83no—
244C22×C3—81no—
245C22×C3—83no—
246C22×C3C2×#12113yes—
247C22×C3#52513yes—
248C22×C3#52513yes—
249C22×C3C2×#12711yescharacteristic
250C22×C3#53213yes—
251C22×C3#53613yes—
252C22×C3#53613yes—
253C22×C3#54713yes—
254C22×C3#54811yescharacteristic
255C22×C3#55216yes—
256C22×C3#55213yes—
257C22×C3#55613yes—
258C3×C4—24no—
259C3×C4—212no—
260C3×C4—46no—
261C3×C4—46no—
262C3×C4—83no—
263C3×C4—81no—
264C23—26no—
265C23—22no—
266C23—26no—
267C23—43no—
268C23—43no—
269C23—41no—
270C23—43no—
271C23—43no—
272C23—43no—
273C23—43no—
274C23—46no—
275C23—43no—
276C23C22×C3×D411yescharacteristic
277C23C22×C3×D413yes—
278C23C2×C3×#4113yes—
279C23C2×C3×#4113yes—
280C23C2×C3×#4111yescharacteristic
281C23C3×2−1+413yes—
282C23C3×2−1+411yescharacteristic
283C2×C4—26no—
284C2×C4—26no—
285C2×C4—26no—
286C2×C4—26no—
287C2×C4—212no—
288C2×C4—212no—
289C2×C4—26no—
290C2×C4—26no—
291C2×C4—26no—
292C2×C4—212no—
293C2×C4—26no—
294C2×C4—43no—
295C2×C4—46no—
296C2×C4—43no—
297C2×C4—43no—
298C2×C4—43no—
299C2×C4—43no—
300C2×C4—43no—
301C2×C4—43no—
302C2×C4—46no—
303C2×C4—41no—
304C2×C4—43no—
305C2×C4—43no—
306C2×C4—46no—
307C2×C4—43no—
308C2×C4—43no—
309C2×C4C3×#121112yes—
310C2×C4C3×#12714yes—
311C2×C3—82no—
312C2×C3#239711yescharacteristic
313C2×C3#241313yes—
314C2×C3#241513yes—
315C2×C3#260413yes—
316C2×C3#261113yes—
317C2×C3#265811yescharacteristic
318C2×C3#285611yescharacteristic
319C22—42no—
320C22—46no—
321C22—46no—
322C22—81no—
323C22—83no—
324C22—81no—
325C22—83no—
326C22C2×C3×#12113yes—
327C22C3×#52513yes—
328C22C3×#52513yes—
329C22C2×C3×#12711yescharacteristic
330C22C3×#53213yes—
331C22C3×#53613yes—
332C22C3×#53613yes—
333C22C3×#54713yes—
334C22C3×#54811yescharacteristic
335C22C3×#55216yes—
336C22C3×#55213yes—
337C22C3×#55613yes—
338C4—212no—
339C4—24no—
340C4—46no—
341C4—46no—
342C4—81no—
343C4—83no—
344C3#1491011yescharacteristic, direct product factor, Sylow
345C2—82no—
346C2C3×#239711yescharacteristic
347C2C3×#241313yes—
348C2C3×#241513yes—
349C2C3×#260413yes—
350C2C3×#261113yes—
351C2C3×#265811yescharacteristic
352C2C3×#285611yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC25×C3C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#389C2semidirect
4. C2×C3×#376C2semidirect
5. C3×#1445C2semidirect
6. C3×#1444C2semidirect
8. #14910C3direct (N × H)
26. C2×#389C2×C3semidirect
28. C2×#376C2×C3semidirect
29. #1445C2×C3semidirect
30. #1444C2×C3semidirect
344. C3#14910direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3992
C2×C3×#3892
C2×C3×#3882
C2×C3×#3762
C3×#14452
C3×#14442
C3×#14392
#149103