Finite Group Databasebeta

← C3×#35337 #10653944

Subgroups — C3×#35337 #10653944

Order 768.

1554 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2862C211yesmaximal, characteristic, complemented
2C3×#2859C211yesmaximal, characteristic, complemented
3C3×#2851C212yesmaximal
4C3×#2647C212yesmaximal, complemented
5C3×#2628C212yesmaximal
6C3×#2589C218yesmaximal
7C3×#2588C218yesmaximal, complemented
8C3×#2331C212yesmaximal, complemented
9C3×#2314C212yesmaximal
10C3×C4×#125C212yesmaximal, complemented
11C3×C4×#117C211yesmaximal, characteristic
12#35337C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8C2212yes—
14C2×C3×#125C2212yes—
15C2×C3×#117C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C22×C3×C42C2211yescharacteristic
18C3×#400C2214yes—
19C3×#400C2218yes—
20C3×#399C2214yes—
21C3×#398C2218yes—
22C3×#395C2214yes—
23C3×#393C2214yes—
24C3×#391C2218yes—
25C3×#391C2218yes—
26C3×#389C2214yes—
27C3×#389C2214yes—
28C3×#389C2218yes—
29C3×#388C2218yes—
30C3×#387C2218yes—
31C3×#387C2218yes—
32C3×#387C2218yes—
33C3×#387C2218yes—
34C3×#385C2218yes—
35C3×#384C2212yes—
36C3×#382C2212yes—
37C3×#382C2212yes—
38C3×#382C2212yes—
39C3×#380C2214yes—
40C3×C4×C4⋊C4C2214yes—
41C3×C4×C4⋊C4C2214yes—
42C3×C4×C4⋊C4C2218yes—
43C3×C4×C22⋊C4C2214yes—
44C3×#376C2211yescharacteristic
45C3×#376C2211yescharacteristic
46C3×C43C2211yescharacteristic
47C3×C43C2211yescharacteristic
48#2862C2×C311yescharacteristic, complemented
49#2859C2×C311yescharacteristic, complemented
50#2851C2×C312yes—
51#2647C2×C312yescomplemented
52#2628C2×C312yes—
53#2589C2×C318yes—
54#2588C2×C318yescomplemented
55#2331C2×C312yescomplemented
56#2314C2×C312yes—
57C4×#125C2×C312yescomplemented
58C4×#117C2×C311yescharacteristic
59C22×C3×Q8C2314yes—
60C23×C3×C4C2311yescharacteristic
61C23×C3×C4C2312yes—
62C3×C4⋊Q8—28no—
63C3×#125—28no—
64C3×#117—24no—
65C3×#117Q818yescomplemented
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2314yes—
68C2×C3×C4⋊C4C2314yes—
69C2×C3×C4⋊C4C2318yes—
70C2×C3×C4⋊C4C2314yes—
71C2×C3×C4⋊C4C2318yes—
72C2×C3×C4⋊C4C2318yes—
73C2×C3×C4⋊C4C2318yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C22⋊C4C2312yes—
76C2×C3×C22⋊C4C2318yes—
77C2×C3×C22⋊C4C2312yes—
78C2×C3×C42—24no—
79C2×C3×C42C2311yescharacteristic
80C2×C3×C42C2311yescharacteristic
81C2×C3×C42C2312yes—
82C2×C3×C42C2314yes—
83C2×C3×C42C2314yes—
84C2×C3×C42C2318yes—
85C2×C3×C42C2311yescharacteristic
86C2×C3×C42C2311yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C3×#95C2314yes—
90C3×#95C2314yes—
91C3×#95C2318yes—
92C3×#95C2318yes—
93C3×#95C2318yes—
94C3×#95C2318yes—
95C3×#95C2314yes—
96C3×#95C2318yes—
97C3×#95C2314yes—
98C3×#95C2318yes—
99C2×C4⋊Q8C22×C312yes—
100C2×#125C22×C312yes—
101C2×#117C22×C311yescharacteristic
102C2×#117C22×C311yescharacteristic
103C22×C42C22×C311yescharacteristic
104#400C22×C314yes—
105#400C22×C318yes—
106#399C22×C314yes—
107#398C22×C318yes—
108#395C22×C314yes—
109#393C22×C314yes—
110#391C22×C318yes—
111#391C22×C318yes—
112#389C22×C314yes—
113#389C22×C314yes—
114#389C22×C318yes—
115#388C22×C318yes—
116#387C22×C318yes—
117#387C22×C318yes—
118#387C22×C318yes—
119#387C22×C318yes—
120#385C22×C318yes—
121#384C22×C312yes—
122#382C22×C312yes—
123#382C22×C312yes—
124#382C22×C312yes—
125#380C22×C314yes—
126C4×C4⋊C4C22×C314yes—
127C4×C4⋊C4C22×C318yes—
128C4×C4⋊C4C22×C314yes—
129C4×C22⋊C4C22×C314yes—
130#376C22×C311yescharacteristic
131#376C22×C311yescharacteristic
132C43C22×C311yescharacteristic
133C43C22×C311yescharacteristic
134C24×C3C2411yescharacteristic
135C2×C3×Q8—28no—
136C2×C3×Q8—48no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—24no—
140C22×C3×C4—44no—
141C22×C3×C4C2×Q814yes—
142C22×C3×C4C2412yes—
143C22×C3×C4C2418yes—
144C22×C3×C4C2412yes—
145C22×C3×C4C2418yes—
146C22×C3×C4C2411yescharacteristic
147C22×C3×C4C2412yes—
148C22×C3×C4C2412yes—
149C22×C3×C4C2412yes—
150C22×C3×C4C2412yes—
151C22×C3×C4C2411yescharacteristic
152C3×C4⋊C4—24no—
153C3×C4⋊C4—216no—
154C3×C4⋊C4—28no—
155C3×C4⋊C4—28no—
156C3×C4⋊C4—28no—
157C3×C4⋊C4—48no—
158C3×C4⋊C4—48no—
159C3×C4⋊C4—48no—
160C3×C4⋊C4C2×Q818yes—
161C3×C22⋊C4—24no—
162C3×C22⋊C4—48no—
163C3×C22⋊C4C2×Q818yes—
164C3×C42—24no—
165C3×C42—22no—
166C3×C42—22no—
167C3×C42—44no—
168C3×C42—48no—
169C3×C42—44no—
170C3×C42C2×Q814yes—
171C3×C42C2×Q814yes—
172C3×C42#4114yes—
173C3×C42#4114yes—
174C22×Q8C23×C314yes—
175C23×C4C23×C311yescharacteristic
176C23×C4C23×C312yes—
177C4⋊Q8—28no—
178#125—28no—
179#117—24no—
180#117C3×Q818yescomplemented
181C2×C4⋊C4C23×C312yes—
182C2×C4⋊C4C23×C314yes—
183C2×C4⋊C4C23×C318yes—
184C2×C4⋊C4C23×C314yes—
185C2×C4⋊C4C23×C314yes—
186C2×C4⋊C4C23×C318yes—
187C2×C4⋊C4C23×C318yes—
188C2×C4⋊C4C23×C318yes—
189C2×C4⋊C4C23×C312yes—
190C2×C22⋊C4C23×C312yes—
191C2×C22⋊C4C23×C318yes—
192C2×C22⋊C4C23×C312yes—
193C2×C42—24no—
194C2×C42C23×C311yescharacteristic
195C2×C42C23×C311yescharacteristic
196C2×C42C23×C312yes—
197C2×C42C23×C314yes—
198C2×C42C23×C314yes—
199C2×C42C23×C318yes—
200C2×C42C23×C311yescharacteristic
201C2×C42C23×C311yescharacteristic
202C2×C42C23×C311yescharacteristic
203C2×C42C23×C311yescharacteristic
204#95C23×C314yes—
205#95C23×C314yes—
206#95C23×C318yes—
207#95C23×C318yes—
208#95C23×C318yes—
209#95C23×C318yes—
210#95C23×C314yes—
211#95C23×C318yes—
212#95C23×C314yes—
213#95C23×C318yes—
214C23×C3—21no—
215C23×C3—21no—
216C23×C3—24no—
217C23×C3C22×Q812yes—
218C23×C3C2511yescharacteristic
219C3×Q8—88no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—24no—
226C2×C3×C4—28no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—28no—
230C2×C3×C4—28no—
231C2×C3×C4—28no—
232C2×C3×C4—28no—
233C2×C3×C4—24no—
234C2×C3×C4—28no—
235C2×C3×C4—24no—
236C2×C3×C4—28no—
237C2×C3×C4—48no—
238C2×C3×C4C22×Q814yes—
239C2×C3×C4C22×Q814yes—
240C2×C3×C4C22×Q812yes—
241C2×C3×C4C22×Q812yes—
242C2×C3×C4C2×#4114yes—
243C2×C3×C4C2×#4114yes—
244C2×C3×C4C2×#4112yes—
245C2×C3×C4C2×#4112yes—
246C24C24×C311yescharacteristic
247C2×Q8—28no—
248C2×Q8—48no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—44no—
253C22×C4C2×C3×Q814yes—
254C22×C4C24×C312yes—
255C22×C4C24×C318yes—
256C22×C4C24×C312yes—
257C22×C4C24×C318yes—
258C22×C4C24×C311yescharacteristic
259C22×C4C24×C312yes—
260C22×C4C24×C312yes—
261C22×C4C24×C312yes—
262C22×C4C24×C312yes—
263C22×C4C24×C311yescharacteristic
264C4⋊C4—24no—
265C4⋊C4—216no—
266C4⋊C4—28no—
267C4⋊C4—28no—
268C4⋊C4—28no—
269C4⋊C4—48no—
270C4⋊C4—48no—
271C4⋊C4—48no—
272C4⋊C4C2×C3×Q818yes—
273C22⋊C4—24no—
274C22⋊C4—48no—
275C22⋊C4C2×C3×Q818yes—
276C42—24no—
277C42—22no—
278C42—22no—
279C42—44no—
280C42—48no—
281C42—44no—
282C42C2×C3×Q814yes—
283C42C2×C3×Q814yes—
284C42C3×#4114yes—
285C42C3×#4114yes—
286C22×C3—22no—
287C22×C3—24no—
288C22×C3—42no—
289C22×C3—42no—
290C22×C3C23×Q811yescharacteristic
291C22×C3C22×#4111yescharacteristic
292C22×C3C22×#4111yescharacteristic
293C22×C3#58514yes—
294C3×C4—22no—
295C3×C4—24no—
296C3×C4—24no—
297C3×C4—24no—
298C3×C4—24no—
299C3×C4—48no—
300C3×C4—42no—
301C3×C4—42no—
302C3×C4—48no—
303C3×C4#53314yes—
304C23—21no—
305C23—21no—
306C23—24no—
307C23C22×C3×Q812yes—
308C23C25×C311yescharacteristic
309Q8—88no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—28no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—28no—
322C2×C4—28no—
323C2×C4—24no—
324C2×C4—28no—
325C2×C4—24no—
326C2×C4—28no—
327C2×C4—48no—
328C2×C4C22×C3×Q814yes—
329C2×C4C22×C3×Q814yes—
330C2×C4C22×C3×Q812yes—
331C2×C4C22×C3×Q812yes—
332C2×C4C2×C3×#4114yes—
333C2×C4C2×C3×#4114yes—
334C2×C4C2×C3×#4112yes—
335C2×C4C2×C3×#4112yes—
336C2×C3—42no—
337C2×C3C2×#53311yescharacteristic
338C2×C3#345512yes—
339C2×C3#350412yes—
340C2×C3#351012yes—
341C22—22no—
342C22—24no—
343C22—42no—
344C22—42no—
345C22C23×C3×Q811yescharacteristic
346C22C22×C3×#4111yescharacteristic
347C22C22×C3×#4111yescharacteristic
348C22C3×#58514yes—
349C4—22no—
350C4—24no—
351C4—24no—
352C4—24no—
353C4—24no—
354C4—42no—
355C4—48no—
356C4—48no—
357C4—42no—
358C4C3×#53314yes—
359C3#3533711yescharacteristic, direct product factor, Sylow
360C2—42no—
361C2C2×C3×#53311yescharacteristic
362C2C3×#345512yes—
363C2C3×#350412yes—
364C2C3×#351012yes—
subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
prime radicalC24×C3C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2862C2semidirect
2. C3×#2859C2semidirect
4. C3×#2647C2semidirect
7. C3×#2588C2semidirect
8. C3×#2331C2semidirect
10. C3×C4×#125C2semidirect
12. #35337C3direct (N × H)
48. #2862C2×C3semidirect
49. #2859C2×C3semidirect
51. #2647C2×C3semidirect
54. #2588C2×C3semidirect
55. #2331C2×C3semidirect
57. C4×#125C2×C3semidirect
65. C3×#117Q8semidirect
180. #117C3×Q8semidirect
359. C3#35337direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28622
C3×#28592
C3×#28512
C3×#26472
C3×#26282
C3×#25892
C3×#25882
C3×#23312
C3×#23142
C3×C4×#1252
C3×C4×#1172
#353373