Finite Group Databasebeta

← C3×#32994 #10651601

Subgroups — C3×#32994 #10651601

Order 768.

1212 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#519C211yesmaximal, characteristic
2C3×#3095C212yesmaximal
3C3×#2434C211yesmaximal, characteristic
4C3×#1886C214yesmaximal, complemented
5C3×#1857C214yesmaximal, complemented
6C3×#1826C212yesmaximal
7C3×#1784C211yesmaximal, characteristic
8#32994C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2212yes—
11C2×C3×C22⋊Q8C2211yescharacteristic
12C3×#519—22no—
13C3×#519—24no—
14C3×#519C2214yes—
15C2×C3×C4×Q8—22no—
16C2×C3×C4×Q8—22no—
17C2×C3×#117—22no—
18C2×C3×#117C2212yes—
19C3×#485—24no—
20C3×#484—22no—
21C3×#483—22no—
22C3×#428C2214yes—
23C3×#419C2214yes—
24C2×C3×#101C2214yes—
25C3×#407C2212yes—
26C3×#395—22no—
27C3×#391C2212yes—
28C3×#385—22no—
29C3×C4×C22⋊C4C2211yescharacteristic
30C3×#339C2218yes—
31C2×#519C2×C311yescharacteristic
32#3095C2×C312yes—
33#2434C2×C311yescharacteristic
34#1886C2×C314yescomplemented
35#1857C2×C314yescomplemented
36#1826C2×C312yes—
37#1784C2×C311yescharacteristic
38C22×C3×Q8—28no—
39C22×C3×Q8—22no—
40C22×C3×Q8—22no—
41C22×C3×Q8D412yes—
42C22×C3×Q8C2311yescharacteristic
43C22×C3×Q8C2312yes—
44C23×C3×C4D412yes—
45C23×C3×C4C2311yescharacteristic
46C2×C3×M16C2314yes—
47C3×C4⋊Q8—28no—
48C3×C22⋊Q8—22no—
49C3×C22⋊Q8—22no—
50C3×C4×Q8—28no—
51C3×C4×Q8—28no—
52C3×C4×Q8—28no—
53C3×C4×Q8—28no—
54C3×#117—28no—
55C3×#117—28no—
56C3×#117—22no—
57C3×#117—22no—
58C3×#117D414yes—
59C3×#117C2314yes—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4D412yes—
65C2×C3×C4⋊C4D412yes—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—21no—
70C2×C3×C42—24no—
71C2×C3×C42—22no—
72C2×C3×C42C2312yes—
73C3×#107—24no—
74C3×#106—24no—
75C3×#103—24no—
76C3×#103—24no—
77C3×#101—28no—
78C3×C22⋊C8—22no—
79C3×C22⋊C8—22no—
80C3×#95—21no—
81C3×#95—24no—
82C23×Q8C22×C311yescharacteristic
83C2×C4⋊Q8C22×C312yes—
84C2×C22⋊Q8C22×C311yescharacteristic
85#519—22no—
86#519—24no—
87#519C22×C314yes—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×#117—22no—
91C2×#117C22×C312yes—
92#485—24no—
93#484—22no—
94#483—22no—
95#428C22×C314yes—
96#419C22×C314yes—
97C2×#101C22×C314yes—
98#407C22×C312yes—
99#395—22no—
100#391C22×C312yes—
101#385—22no—
102C4×C22⋊C4C22×C311yescharacteristic
103#339C22×C318yes—
104C24×C3C2×D411yescharacteristic
105C2×C3×Q8—216no—
106C2×C3×Q8—216no—
107C2×C3×Q8—24no—
108C2×C3×Q8—28no—
109C2×C3×Q8—24no—
110C2×C3×Q8—24no—
111C2×C3×Q8—22no—
112C2×C3×Q8—24no—
113C2×C3×Q8#4118yes—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—28no—
117C22×C3×C4—24no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—21no—
122C22×C3×C4—21no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D412yes—
125C22×C3×C4C2×D412yes—
126C22×C3×C4C2×D414yes—
127C22×C3×C4C2411yescharacteristic
128C3×M16—28no—
129C2×C3×C8—24no—
130C3×C4⋊C4—28no—
131C3×C4⋊C4—28no—
132C3×C4⋊C4—22no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—42no—
136C3×C4⋊C4—48no—
137C3×C4⋊C4C2×D414yes—
138C3×C4⋊C4C2×D414yes—
139C3×C22⋊C4—24no—
140C3×C22⋊C4—24no—
141C3×C22⋊C4—24no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42—216no—
145C3×C42—24no—
146C3×C42—24no—
147C3×C42—24no—
148C22×Q8—28no—
149C22×Q8—22no—
150C22×Q8—22no—
151C22×Q8C3×D412yes—
152C22×Q8C23×C311yescharacteristic
153C22×Q8C23×C312yes—
154C23×C4C3×D412yes—
155C23×C4C23×C311yescharacteristic
156C2×M16C23×C314yes—
157C4⋊Q8—28no—
158C22⋊Q8—22no—
159C22⋊Q8—22no—
160C4×Q8—28no—
161C4×Q8—28no—
162C4×Q8—28no—
163C4×Q8—28no—
164#117—28no—
165#117—28no—
166#117—22no—
167#117—22no—
168#117C3×D414yes—
169#117C23×C314yes—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4C3×D412yes—
175C2×C4⋊C4C3×D412yes—
176C2×C4⋊C4C23×C311yescharacteristic
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C42—24no—
181C2×C42—22no—
182C2×C42C23×C312yes—
183#107—24no—
184#106—24no—
185#103—24no—
186#103—24no—
187#101—28no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190#95—21no—
191#95—24no—
192C23×C3—22no—
193C23×C3—22no—
194C23×C3—41no—
195C23×C3#12011yescharacteristic
196C23×C3#12011yescharacteristic
197C23×C3C22×D411yescharacteristic
198C3×Q8—216no—
199C3×Q8—216no—
200C3×Q8—44no—
201C2×C3×C4—28no—
202C2×C3×C4—28no—
203C2×C3×C4—24no—
204C2×C3×C4—22no—
205C2×C3×C4—24no—
206C2×C3×C4—24no—
207C2×C3×C4—24no—
208C2×C3×C4—24no—
209C2×C3×C4—24no—
210C2×C3×C4—24no—
211C2×C3×C4—42no—
212C2×C3×C4—44no—
213C2×C3×C4—42no—
214C2×C3×C4—44no—
215C2×C3×C4—42no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4#12012yes—
220C2×C3×C4#12318yes—
221C2×C3×C4C22×D412yes—
222C2×C3×C4C2×#4114yes—
223C3×C8—44no—
224C24C2×C3×D411yescharacteristic
225C2×Q8—216no—
226C2×Q8—216no—
227C2×Q8—24no—
228C2×Q8—28no—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—22no—
232C2×Q8—24no—
233C2×Q8C3×#4118yes—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—28no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4C2×C3×D411yescharacteristic
244C22×C4C2×C3×D412yes—
245C22×C4C2×C3×D412yes—
246C22×C4C2×C3×D414yes—
247C22×C4C24×C311yescharacteristic
248M16—28no—
249C2×C8—24no—
250C4⋊C4—28no—
251C4⋊C4—28no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—42no—
256C4⋊C4—48no—
257C4⋊C4C2×C3×D414yes—
258C4⋊C4C2×C3×D414yes—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C42—216no—
265C42—24no—
266C42—24no—
267C42—24no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—44no—
276C22×C3#45612yes—
277C22×C3C2×#12011yescharacteristic
278C22×C3C2×#12312yes—
279C3×C4—22no—
280C3×C4—28no—
281C3×C4—44no—
282C3×C4—44no—
283C3×C4—44no—
284C3×C4—81no—
285C3×C4—81no—
286C3×C4#52514yes—
287C23—22no—
288C23—22no—
289C23—41no—
290C23C3×#12011yescharacteristic
291C23C3×#12011yescharacteristic
292C23C22×C3×D411yescharacteristic
293Q8—216no—
294Q8—216no—
295Q8—44no—
296C2×C4—28no—
297C2×C4—28no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—24no—
306C2×C4—42no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4C3×#12012yes—
315C2×C4C3×#12318yes—
316C2×C4C22×C3×D412yes—
317C2×C4C2×C3×#4114yes—
318C8—44no—
319C2×C3—22no—
320C2×C3—22no—
321C2×C3—41no—
322C2×C3#284911yescharacteristic
323C2×C3C2×#45611yescharacteristic
324C2×C3#302011yescharacteristic
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—44no—
333C22C3×#45612yes—
334C22C2×C3×#12011yescharacteristic
335C22C2×C3×#12312yes—
336C4—22no—
337C4—28no—
338C4—44no—
339C4—44no—
340C4—44no—
341C4—81no—
342C4—81no—
343C4C3×#52514yes—
344C3#3299411yescharacteristic, direct product factor, Sylow
345C2—22no—
346C2—22no—
347C2—41no—
348C2C3×#284911yescharacteristic
349C2C2×C3×#45611yescharacteristic
350C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C3×#1886C2semidirect
5. C3×#1857C2semidirect
8. #32994C3direct (N × H)
34. #1886C2×C3semidirect
35. #1857C2×C3semidirect
344. C3#32994direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5192
C3×#30952
C3×#24342
C3×#18862
C3×#18572
C3×#18262
C3×#17842
#329943