Finite Group Databasebeta

← C3×#29674 #10648281

Subgroups — C3×#29674 #10648281

Order 768.

1032 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic
2C2×C3×#462C212yesmaximal, complemented
3C3×#2621C211yesmaximal, characteristic
4C3×#2026C212yesmaximal, complemented
5C3×#2016C212yesmaximal, complemented
6C3×#2001C214yesmaximal
7C3×#1805C212yesmaximal, complemented
8C3×#1798C211yesmaximal, characteristic
9#29674C311yesmaximal, characteristic, direct product factor, Sylow
10C22×C3×Q16C2214yes—
11C3×#541—28no—
12C2×C3×C4⋊Q8C2211yescharacteristic
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×#126—22no—
15C2×C3×#125—22no—
16C2×C3×#125C2211yescharacteristic
17C2×C3×C22⋊Q8C2212yes—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×C4×Q8C2212yes—
20C2×C3×#117C2211yescharacteristic
21C3×#462—28no—
22C2×C3×C4⋊C8C2212yes—
23C2×C3×#103C2214yes—
24C2×C3×#103C2214yes—
25C2×C3×C22⋊C8C2212yes—
26C3×#398—22no—
27C3×#397—22no—
28C3×#391C2212yes—
29C3×#382C2211yescharacteristic
30C3×#340C2214yes—
31C3×#336C2212yes—
32C2×#541C2×C311yescharacteristic
33C2×#462C2×C312yescomplemented
34#2621C2×C311yescharacteristic
35#2026C2×C312yescomplemented
36#2016C2×C312yescomplemented
37#2001C2×C314yes—
38#1805C2×C312yescomplemented
39#1798C2×C311yescharacteristic
40C22×C3×Q8C2312yes—
41C22×C3×Q8C2312yes—
42C23×C3×C4C2311yescharacteristic
43C2×C3×Q16—28no—
44C2×C3×Q16—44no—
45C2×C3×Q16—44no—
46C22×C3×C8C2314yes—
47C3×C4⋊Q8—24no—
48C3×C4⋊Q8—24no—
49C3×#126—216no—
50C3×#125—216no—
51C3×#125—24no—
52C3×C22⋊Q8—24no—
53C3×C22⋊Q8—24no—
54C3×C22⋊Q8—44no—
55C3×C4×Q8—24no—
56C3×C4×Q8D418yes—
57C3×#117—24no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4—22no—
64C2×C3×C4⋊C4D412yes—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4D412yes—
70C2×C3×C42—22no—
71C2×C3×C42C2311yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C3×C4⋊C8—24no—
74C3×#103—28no—
75C3×#103—44no—
76C3×C22⋊C8—42no—
77C3×#95—22no—
78C3×#95—24no—
79C22×Q16C22×C314yes—
80#541—28no—
81C2×C4⋊Q8C22×C311yescharacteristic
82C2×C4⋊Q8C22×C311yescharacteristic
83C2×#126—22no—
84C2×#125—22no—
85C2×#125C22×C311yescharacteristic
86C2×C22⋊Q8C22×C312yes—
87C2×C22⋊Q8C22×C312yes—
88C2×C4×Q8C22×C312yes—
89C2×#117C22×C311yescharacteristic
90#462—28no—
91C2×C4⋊C8C22×C312yes—
92C2×#103C22×C314yes—
93C2×#103C22×C314yes—
94C2×C22⋊C8C22×C312yes—
95#398—22no—
96#397—22no—
97#391C22×C312yes—
98#382C22×C311yescharacteristic
99#340C22×C314yes—
100#336C22×C312yes—
101C24×C3C2×D411yescharacteristic
102C2×C3×Q8—22no—
103C2×C3×Q8—24no—
104C2×C3×Q8—22no—
105C2×C3×Q8—22no—
106C2×C3×Q8—22no—
107C2×C3×Q8—24no—
108C2×C3×Q8—22no—
109C2×C3×Q8—22no—
110C2×C3×Q8—22no—
111C2×C3×Q8C2×D414yes—
112C22×C3×C4—22no—
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—21no—
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4C2×D412yes—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D412yes—
123C22×C3×C4C2411yescharacteristic
124C3×Q16—416no—
125C2×C3×C8—24no—
126C2×C3×C8—24no—
127C2×C3×C8—24no—
128C3×C4⋊C4—28no—
129C3×C4⋊C4—28no—
130C3×C4⋊C4—28no—
131C3×C4⋊C4—28no—
132C3×C4⋊C4—24no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—24no—
135C3×C4⋊C4—44no—
136C3×C4⋊C4—44no—
137C3×C4⋊C4C2×D414yes—
138C3×C22⋊C4—28no—
139C3×C22⋊C4—24no—
140C3×C22⋊C4—44no—
141C3×C42—28no—
142C3×C42—22no—
143C3×C42C2×D414yes—
144C22×Q8C23×C312yes—
145C22×Q8C23×C312yes—
146C23×C4C23×C311yescharacteristic
147C2×Q16—28no—
148C2×Q16—44no—
149C2×Q16—44no—
150C22×C8C23×C314yes—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153#126—216no—
154#125—216no—
155#125—24no—
156C22⋊Q8—24no—
157C22⋊Q8—24no—
158C22⋊Q8—44no—
159C4×Q8—24no—
160C4×Q8C3×D418yes—
161#117—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C3×D412yes—
169C2×C4⋊C4C23×C312yes—
170C2×C4⋊C4C23×C312yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4C3×D412yes—
174C2×C42—22no—
175C2×C42C23×C311yescharacteristic
176C2×C42C23×C311yescharacteristic
177C4⋊C8—24no—
178#103—28no—
179#103—44no—
180C22⋊C8—42no—
181#95—22no—
182#95—24no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—24no—
187C23×C3C22×D411yescharacteristic
188C3×Q8—24no—
189C3×Q8—24no—
190C3×Q8—42no—
191C3×Q8—42no—
192C3×Q8—44no—
193C3×Q8—44no—
194C2×C3×C4—24no—
195C2×C3×C4—22no—
196C2×C3×C4—24no—
197C2×C3×C4—24no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—24no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4—44no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4—42no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—44no—
212C2×C3×C4#12014yes—
213C2×C3×C4C22×D412yes—
214C2×C3×C4H(5,𝔽2)11yescharacteristic
215C2×C3×C4H(5,𝔽2)12yes—
216C2×C3×C42−1+411yescharacteristic
217C3×C8—44no—
218C24C2×C3×D411yescharacteristic
219C2×Q8—22no—
220C2×Q8—24no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—24no—
225C2×Q8—22no—
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8C2×C3×D414yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4C2×C3×D412yes—
238C22×C4C2×C3×D411yescharacteristic
239C22×C4C2×C3×D412yes—
240C22×C4C24×C311yescharacteristic
241Q16—416no—
242C2×C8—24no—
243C2×C8—24no—
244C2×C8—24no—
245C4⋊C4—28no—
246C4⋊C4—28no—
247C4⋊C4—28no—
248C4⋊C4—28no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—24no—
252C4⋊C4—44no—
253C4⋊C4—44no—
254C4⋊C4C2×C3×D414yes—
255C22⋊C4—28no—
256C22⋊C4—24no—
257C22⋊C4—44no—
258C42—28no—
259C42—22no—
260C42C2×C3×D414yes—
261C22×C3—41no—
262C22×C3—42no—
263C22×C3—42no—
264C22×C3—42no—
265C22×C3C2×#12011yescharacteristic
266C22×C3#53511yescharacteristic
267C22×C3#53711yescharacteristic
268C22×C3#57814yes—
269C3×C4—24no—
270C3×C4—41no—
271C3×C4—44no—
272C3×C4—42no—
273C3×C4—44no—
274C3×C4—82no—
275C3×C4—82no—
276C3×C4#53512yes—
277C3×C4#53712yes—
278C23—24no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23C22×C3×D411yescharacteristic
283Q8—24no—
284Q8—24no—
285Q8—42no—
286Q8—42no—
287Q8—44no—
288Q8—44no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4C3×#12014yes—
308C2×C4C22×C3×D412yes—
309C2×C4C3×H(5,𝔽2)11yescharacteristic
310C2×C4C3×H(5,𝔽2)12yes—
311C2×C4C3×2−1+411yescharacteristic
312C8—44no—
313C2×C3—81no—
314C2×C3#261511yescharacteristic
315C2×C3#301112yes—
316C2×C3#322312yes—
317C2×C3#326112yes—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—41no—
322C22C2×C3×#12011yescharacteristic
323C22C3×#53511yescharacteristic
324C22C3×#53711yescharacteristic
325C22C3×#57814yes—
326C4—24no—
327C4—42no—
328C4—44no—
329C4—41no—
330C4—44no—
331C4—82no—
332C4—82no—
333C4C3×#53512yes—
334C4C3×#53712yes—
335C3#2967411yescharacteristic, direct product factor, Sylow
336C2—81no—
337C2C3×#261511yescharacteristic
338C2C3×#301112yes—
339C2C3×#322312yes—
340C2C3×#326112yes—
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
2. C2×C3×#462C2semidirect
4. C3×#2026C2semidirect
5. C3×#2016C2semidirect
7. C3×#1805C2semidirect
9. #29674C3direct (N × H)
33. C2×#462C2×C3semidirect
35. #2026C2×C3semidirect
36. #2016C2×C3semidirect
38. #1805C2×C3semidirect
335. C3#29674direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×C3×#4622
C3×#26212
C3×#20262
C3×#20162
C3×#20012
C3×#18052
C3×#17982
#296743