Finite Group Databasebeta

← C3×#32991 #10651598

Subgroups — C3×#32991 #10651598

Order 768.

1238 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#519C211yesmaximal, characteristic, complemented
2C3×#3094C212yesmaximal
3C3×#2436C211yesmaximal, characteristic
4C3×#1888C214yesmaximal, complemented
5C3×#1857C214yesmaximal, complemented
6C3×#1826C212yesmaximal, complemented
7C3×#1784C211yesmaximal, characteristic, complemented
8#32991C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×Q8C2211yescharacteristic
10C2×C3×#124C2212yes—
11C2×C3×#121C2211yescharacteristic
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×#482—24no—
20C3×#481—22no—
21C3×#480—22no—
22C3×#428C2214yes—
23C3×#418C2214yes—
24C2×C3×#101C2214yes—
25C3×#407C2212yes—
26C3×#393—22no—
27C3×#391C2212yes—
28C3×#386—22no—
29C3×C4×C22⋊C4C2211yescharacteristic
30C3×#339C2218yes—
31C2×#519C2×C311yescharacteristic, complemented
32#3094C2×C312yes—
33#2436C2×C311yescharacteristic
34#1888C2×C314yescomplemented
35#1857C2×C314yescomplemented
36#1826C2×C312yescomplemented
37#1784C2×C311yescharacteristic, complemented
38C22×C3×Q8—28no—
39C22×C3×Q8—22no—
40C22×C3×Q8—22no—
41C22×C3×Q8D412yes—
42C22×C3×Q8C2312yes—
43C22×C3×D4—21no—
44C22×C3×D4C2311yescharacteristic
45C23×C3×C4D412yes—
46C23×C3×C4C2311yescharacteristic
47C2×C3×M16C2314yes—
48C3×#124—28no—
49C3×#121—22no—
50C3×#121—22no—
51C3×C4×Q8—28no—
52C3×C4×Q8—28no—
53C3×C4×Q8—28no—
54C3×C4×Q8—28no—
55C3×#117—28no—
56C3×#117—28no—
57C3×#117—22no—
58C3×#117—22no—
59C3×#117D414yes—
60C3×#117C2314yes—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4—22no—
63C2×C3×C4⋊C4D412yes—
64C2×C3×C4⋊C4D412yes—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—24no—
69C2×C3×C22⋊C4—21no—
70C2×C3×C42—24no—
71C2×C3×C42—22no—
72C2×C3×C42C2312yes—
73C3×#107—24no—
74C3×#106—24no—
75C3×#102—24no—
76C3×#102—24no—
77C3×#101—28no—
78C3×C22⋊C8—22no—
79C3×C22⋊C8—22no—
80C3×#95—21no—
81C3×#95—24no—
82C23×Q8C22×C311yescharacteristic
83C2×#124C22×C312yes—
84C2×#121C22×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#482—24no—
93#481—22no—
94#480—22no—
95#428C22×C314yes—
96#418C22×C314yes—
97C2×#101C22×C314yes—
98#407C22×C312yes—
99#393—22no—
100#391C22×C312yes—
101#386—22no—
102C4×C22⋊C4C22×C311yescharacteristic
103#339C22×C318yes—
104C24×C3—21no—
105C24×C3C2×D411yescharacteristic
106C2×C3×Q8—216no—
107C2×C3×Q8—216no—
108C2×C3×Q8—24no—
109C2×C3×Q8—28no—
110C2×C3×Q8—24no—
111C2×C3×Q8—24no—
112C2×C3×Q8#4118yes—
113C2×C3×D4—22no—
114C2×C3×D4—24no—
115C2×C3×D4—44no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—28no—
121C22×C3×C4—24no—
122C22×C3×C4—22no—
123C22×C3×C4—22no—
124C22×C3×C4—21no—
125C22×C3×C4—22no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D412yes—
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2×D414yes—
130C22×C3×C4C2411yescharacteristic
131C3×M16—28no—
132C2×C3×C8—24no—
133C3×C4⋊C4—28no—
134C3×C4⋊C4—28no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4C2×D414yes—
139C3×C4⋊C4C2×D414yes—
140C3×C22⋊C4—24no—
141C3×C22⋊C4—24no—
142C3×C22⋊C4—42no—
143C3×C22⋊C4—42no—
144C3×C22⋊C4—48no—
145C3×C42—216no—
146C3×C42—24no—
147C3×C42—24no—
148C3×C42—24no—
149C22×Q8—28no—
150C22×Q8—22no—
151C22×Q8—22no—
152C22×Q8C3×D412yes—
153C22×Q8C23×C312yes—
154C22×D4—21no—
155C22×D4C23×C311yescharacteristic
156C23×C4C3×D412yes—
157C23×C4C23×C311yescharacteristic
158C2×M16C23×C314yes—
159#124—28no—
160#121—22no—
161#121—22no—
162C4×Q8—28no—
163C4×Q8—28no—
164C4×Q8—28no—
165C4×Q8—28no—
166#117—28no—
167#117—28no—
168#117—22no—
169#117—22no—
170#117C3×D414yes—
171#117C23×C314yes—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
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×C22⋊C4—24no—
181C2×C42—24no—
182C2×C42—22no—
183C2×C42C23×C312yes—
184#107—24no—
185#106—24no—
186#102—24no—
187#102—24no—
188#101—28no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191#95—21no—
192#95—24no—
193C23×C3—22no—
194C23×C3—22no—
195C23×C3—41no—
196C23×C3—42no—
197C23×C3—42no—
198C23×C3—41no—
199C23×C3—42no—
200C23×C3#12011yescharacteristic
201C23×C3#12011yescharacteristic
202C23×C3C22×D411yescharacteristic
203C3×Q8—216no—
204C3×Q8—216no—
205C3×D4—44no—
206C3×D4—82no—
207C3×D4—82no—
208C2×C3×C4—28no—
209C2×C3×C4—28no—
210C2×C3×C4—24no—
211C2×C3×C4—22no—
212C2×C3×C4—24no—
213C2×C3×C4—24no—
214C2×C3×C4—24no—
215C2×C3×C4—24no—
216C2×C3×C4—24no—
217C2×C3×C4—44no—
218C2×C3×C4—44no—
219C2×C3×C4—42no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4—42no—
224C2×C3×C4#12012yes—
225C2×C3×C4#12318yes—
226C2×C3×C4C22×D412yes—
227C2×C3×C4C2×#4114yes—
228C3×C8—44no—
229C24—21no—
230C24C2×C3×D411yescharacteristic
231C2×Q8—216no—
232C2×Q8—216no—
233C2×Q8—24no—
234C2×Q8—28no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8C3×#4118yes—
238C2×D4—22no—
239C2×D4—24no—
240C2×D4—44no—
241C2×D4—41no—
242C2×D4—41no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—28no—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4C2×C3×D411yescharacteristic
252C22×C4C2×C3×D412yes—
253C22×C4C2×C3×D412yes—
254C22×C4C2×C3×D414yes—
255C22×C4C24×C311yescharacteristic
256M16—28no—
257C2×C8—24no—
258C4⋊C4—28no—
259C4⋊C4—28no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4C2×C3×D414yes—
264C4⋊C4C2×C3×D414yes—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—48no—
270C42—216no—
271C42—24no—
272C42—24no—
273C42—24no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—44no—
282C22×C3—84no—
283C22×C3—81no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3#45412yes—
287C22×C3C2×#12011yescharacteristic
288C22×C3C2×#12312yes—
289C3×C4—22no—
290C3×C4—28no—
291C3×C4—44no—
292C3×C4—44no—
293C3×C4—82no—
294C3×C4—81no—
295C3×C4#52514yes—
296C23—22no—
297C23—22no—
298C23—41no—
299C23—42no—
300C23—42no—
301C23—41no—
302C23—42no—
303C23C3×#12011yescharacteristic
304C23C3×#12011yescharacteristic
305C23C22×C3×D411yescharacteristic
306Q8—216no—
307Q8—216no—
308D4—44no—
309D4—82no—
310D4—82no—
311C2×C4—28no—
312C2×C4—28no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4C3×#12012yes—
328C2×C4C3×#12318yes—
329C2×C4C22×C3×D412yes—
330C2×C4C2×C3×#4114yes—
331C8—44no—
332C2×C3—22no—
333C2×C3—22no—
334C2×C3—41no—
335C2×C3—161no—
336C2×C3#284911yescharacteristic
337C2×C3C2×#45411yescharacteristic
338C2×C3#302011yescharacteristic
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22—84no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22C3×#45412yes—
352C22C2×C3×#12011yescharacteristic
353C22C2×C3×#12312yes—
354C4—22no—
355C4—28no—
356C4—44no—
357C4—44no—
358C4—82no—
359C4—81no—
360C4C3×#52514yes—
361C3#3299111yescharacteristic, direct product factor, Sylow
362C2—22no—
363C2—22no—
364C2—41no—
365C2—161no—
366C2C3×#284911yescharacteristic
367C2C2×C3×#45411yescharacteristic
368C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
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
1. C2×C3×#519C2semidirect
4. C3×#1888C2semidirect
5. C3×#1857C2semidirect
6. C3×#1826C2semidirect
7. C3×#1784C2semidirect
8. #32991C3direct (N × H)
31. C2×#519C2×C3semidirect
34. #1888C2×C3semidirect
35. #1857C2×C3semidirect
36. #1826C2×C3semidirect
37. #1784C2×C3semidirect
361. C3#32991direct (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×#30942
C3×#24362
C3×#18882
C3×#18572
C3×#18262
C3×#17842
#329913