Finite Group Databasebeta

← C3×#32671 #10651278

Subgroups — C3×#32671 #10651278

Order 768.

764 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2592C212yesmaximal, complemented
2C3×#2056C211yesmaximal, characteristic, complemented
3C3×#2054C211yesmaximal, characteristic
4C3×#2015C212yesmaximal, complemented
5C3×#2012C212yesmaximal
6C3×#1984C211yesmaximal, characteristic, complemented
7C3×#1968C212yesmaximal, complemented
8C3×#1927C212yesmaximal, complemented
9C3×#1926C212yesmaximal, complemented
10#32671C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×Q16C2211yescharacteristic
12C22×C3×SD16C2212yes—
13C22×C3×D8C2211yescharacteristic
14C2×C3×#124C2212yes—
15C2×C3×C4×Q8C2211yescharacteristic
16C2×C3×#106C2212yes—
17C2×C3×C4⋊C8C2211yescharacteristic
18C2×C3×#103C2212yes—
19C2×C3×#103C2212yes—
20C2×C3×#102C2212yes—
21C2×C3×#102C2212yes—
22C2×C3×C4×C8C2211yescharacteristic
23C3×#397—22no—
24C3×#393—22no—
25C3×#391C2212yes—
26C3×#391C2212yes—
27C3×#390C2212yes—
28C3×#390C2212yes—
29C3×#388—22no—
30C3×#388—22no—
31C3×C4×C4⋊C4C2212yes—
32C3×#340C2211yescharacteristic
33C3×#340C2211yescharacteristic
34C3×#336C2212yes—
35C3×#336C2212yes—
36#2592C2×C312yescomplemented
37#2056C2×C311yescharacteristic, complemented
38#2054C2×C311yescharacteristic
39#2015C2×C312yescomplemented
40#2012C2×C312yes—
41#1984C2×C311yescharacteristic, complemented
42#1968C2×C312yescomplemented
43#1927C2×C312yescomplemented
44#1926C2×C312yescomplemented
45C22×C3×Q8D411yescharacteristic
46C22×C3×Q8C2312yes—
47C22×C3×D4C2312yes—
48C2×C3×Q16—42no—
49C2×C3×Q16—41no—
50C2×C3×SD16—24no—
51C2×C3×SD16—42no—
52C2×C3×SD16—42no—
53C2×C3×D8—42no—
54C2×C3×D8—41no—
55C22×C3×C8C2311yescharacteristic
56C22×C3×C8C2311yescharacteristic
57C22×C3×C8C2312yes—
58C3×#124—42no—
59C3×#124—42no—
60C3×C4×Q8—41no—
61C3×C4×Q8—41no—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C4⋊C4D412yes—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C42—22no—
73C2×C3×C42C2312yes—
74C2×C3×C42C2311yescharacteristic
75C3×#106—42no—
76C3×C4⋊C8—41no—
77C3×#103—24no—
78C3×#103—42no—
79C3×#102—24no—
80C3×#102—42no—
81C3×C4×C8—41no—
82C3×#95—22no—
83C3×#95—22no—
84C3×#95—22no—
85C3×#95—22no—
86C3×#95—22no—
87C22×Q16C22×C311yescharacteristic
88C22×SD16C22×C312yes—
89C22×D8C22×C311yescharacteristic
90C2×#124C22×C312yes—
91C2×C4×Q8C22×C311yescharacteristic
92C2×#106C22×C312yes—
93C2×C4⋊C8C22×C311yescharacteristic
94C2×#103C22×C312yes—
95C2×#103C22×C312yes—
96C2×#102C22×C312yes—
97C2×#102C22×C312yes—
98C2×C4×C8C22×C311yescharacteristic
99#397—22no—
100#393—22no—
101#391C22×C312yes—
102#391C22×C312yes—
103#390C22×C312yes—
104#390C22×C312yes—
105#388—22no—
106#388—22no—
107C4×C4⋊C4C22×C312yes—
108#340C22×C311yescharacteristic
109#340C22×C311yescharacteristic
110#336C22×C312yes—
111#336C22×C312yes—
112C24×C3—22no—
113C2×C3×Q8—22no—
114C2×C3×Q8—22no—
115C2×C3×Q8—22no—
116C2×C3×Q8—22no—
117C2×C3×Q8—22no—
118C2×C3×Q8—22no—
119C2×C3×Q8—22no—
120C2×C3×Q8—41no—
121C2×C3×Q8—41no—
122C2×C3×D4—22no—
123C2×C3×D4—22no—
124C2×C3×D4—22no—
125C2×C3×D4—22no—
126C2×C3×D4—22no—
127C2×C3×D4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D412yes—
133C22×C3×C4C2×D412yes—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2411yescharacteristic
136C3×Q16—44no—
137C3×SD16—48no—
138C3×D8—44no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—22no—
143C2×C3×C8—21no—
144C2×C3×C8—21no—
145C2×C3×C8#4114yes—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—41no—
151C3×C4⋊C4—42no—
152C3×C22⋊C4—44no—
153C3×C22⋊C4—44no—
154C3×C22⋊C4—44no—
155C3×C22⋊C4—44no—
156C3×C22⋊C4—82no—
157C3×C22⋊C4—82no—
158C3×C42—21no—
159C3×C42—21no—
160C3×C42—22no—
161C3×C42—22no—
162C3×C42—82no—
163C22×Q8C3×D411yescharacteristic
164C22×Q8C23×C312yes—
165C22×D4C23×C312yes—
166C2×Q16—42no—
167C2×Q16—41no—
168C2×SD16—24no—
169C2×SD16—42no—
170C2×SD16—42no—
171C2×D8—42no—
172C2×D8—41no—
173C22×C8C23×C311yescharacteristic
174C22×C8C23×C311yescharacteristic
175C22×C8C23×C312yes—
176#124—42no—
177#124—42no—
178C4×Q8—41no—
179C4×Q8—41no—
180C2×C4⋊C4C3×D411yescharacteristic
181C2×C4⋊C4C3×D412yes—
182C2×C4⋊C4C23×C312yes—
183C2×C4⋊C4C23×C312yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C42—22no—
191C2×C42C23×C312yes—
192C2×C42C23×C311yescharacteristic
193#106—42no—
194C4⋊C8—41no—
195#103—24no—
196#103—42no—
197#102—24no—
198#102—42no—
199C4×C8—41no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205C23×C3—42no—
206C23×C3—42no—
207C23×C3—42no—
208C23×C3—42no—
209C23×C3—42no—
210C23×C3—42no—
211C23×C3—42no—
212C23×C3C22×D411yescharacteristic
213C3×Q8—44no—
214C3×Q8—42no—
215C3×Q8—42no—
216C3×Q8—42no—
217C3×Q8—42no—
218C3×D4—42no—
219C3×D4—42no—
220C3×D4—42no—
221C3×D4—42no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—22no—
225C2×C3×C4—22no—
226C2×C3×C4—21no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4#13512yes—
242C2×C3×C4C2×#4112yes—
243C2×C3×C4H(5,𝔽2)11yescharacteristic
244C2×C3×C4H(5,𝔽2)12yes—
245C2×C3×C4H(5,𝔽2)11yescharacteristic
246C3×C8—24no—
247C3×C8—41no—
248C3×C8—41no—
249C24—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4C2×C3×D411yescharacteristic
269C22×C4C2×C3×D412yes—
270C22×C4C2×C3×D412yes—
271C22×C4C2×C3×D411yescharacteristic
272C22×C4C24×C311yescharacteristic
273Q16—44no—
274SD16—48no—
275D8—44no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8C3×#4114yes—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—82no—
294C22⋊C4—82no—
295C42—21no—
296C42—21no—
297C42—22no—
298C42—22no—
299C42—82no—
300C22×C3—82no—
301C22×C3—82no—
302C22×C3—82no—
303C22×C3—82no—
304C22×C3—82no—
305C22×C3—82no—
306C22×C3—82no—
307C22×C3#53411yescharacteristic
308C22×C3#55012yes—
309C22×C3C2×#13511yescharacteristic
310C22×C3#57511yescharacteristic
311C22×C3#57712yes—
312C3×C4—22no—
313C3×C4—42no—
314C3×C4—42no—
315C3×C4—41no—
316C3×C4—82no—
317C3×C4—82no—
318C3×C4—82no—
319C3×C4#53812yes—
320C3×C4#55511yescharacteristic
321C3×C4H(3,𝔽4)11yescharacteristic
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23C22×C3×D411yescharacteristic
330Q8—44no—
331Q8—42no—
332Q8—42no—
333Q8—42no—
334Q8—42no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—42no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—21no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4C3×#13512yes—
359C2×C4C2×C3×#4112yes—
360C2×C4C3×H(5,𝔽2)12yes—
361C2×C4C3×H(5,𝔽2)11yescharacteristic
362C2×C4C3×H(5,𝔽2)11yescharacteristic
363C8—24no—
364C8—41no—
365C8—41no—
366C2×C3—162no—
367C2×C3#273311yescharacteristic
368C2×C3#319111yescharacteristic
369C2×C3#320411yescharacteristic
370C2×C3#339412yes—
371C2×C3#340112yes—
372C22—82no—
373C22—82no—
374C22—82no—
375C22—82no—
376C22—82no—
377C22—82no—
378C22—82no—
379C22C3×#53411yescharacteristic
380C22C3×#55012yes—
381C22C2×C3×#13511yescharacteristic
382C22C3×#57511yescharacteristic
383C22C3×#57712yes—
384C4—22no—
385C4—42no—
386C4—42no—
387C4—41no—
388C4—82no—
389C4—82no—
390C4—82no—
391C4C3×#53812yes—
392C4C3×#55511yescharacteristic
393C4C3×H(3,𝔽4)11yescharacteristic
394C3#3267111yescharacteristic, direct product factor, Sylow
395C2—162no—
396C2C3×#273311yescharacteristic
397C2C3×#319111yescharacteristic
398C2C3×#320411yescharacteristic
399C2C3×#339412yes—
400C2C3×#340112yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2592C2semidirect
2. C3×#2056C2semidirect
4. C3×#2015C2semidirect
6. C3×#1984C2semidirect
7. C3×#1968C2semidirect
8. C3×#1927C2semidirect
9. C3×#1926C2semidirect
10. #32671C3direct (N × H)
36. #2592C2×C3semidirect
37. #2056C2×C3semidirect
39. #2015C2×C3semidirect
41. #1984C2×C3semidirect
42. #1968C2×C3semidirect
43. #1927C2×C3semidirect
44. #1926C2×C3semidirect
394. C3#32671direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25922
C3×#20562
C3×#20542
C3×#20152
C3×#20122
C3×#19842
C3×#19682
C3×#19272
C3×#19262
#326713