Finite Group Databasebeta

← C3×#34978 #10653585

Subgroups — C3×#34978 #10653585

Order 768.

1834 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2612C214yesmaximal, complemented
2C3×#2571C211yesmaximal, characteristic, complemented
3C3×#2552C212yesmaximal, complemented
4C3×#2551C211yesmaximal, characteristic, complemented
5C3×#2441C214yesmaximal
6C3×#2439C214yesmaximal
7C3×#2396C214yesmaximal
8C3×C4×C4⋊Q8C214yesmaximal, complemented
9C3×C4×#125C214yesmaximal, complemented
10C3×#2285C211yesmaximal, characteristic
11C3×C4×#117C212yesmaximal
12#34978C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8C2214yes—
14C2×C3×#125C2214yes—
15C2×C3×C4×Q8C2214yes—
16C2×C3×#117C2212yes—
17C2×C3×#117C2212yes—
18C2×C3×#117C2212yes—
19C22×C3×C42C2211yescharacteristic
20C3×#400C2214yes—
21C3×#399C2214yes—
22C3×#398C2218yes—
23C3×#396C2218yes—
24C3×#395C2214yes—
25C3×#393C2214yes—
26C3×#391C2218yes—
27C3×#389C2214yes—
28C3×#389C2214yes—
29C3×#389C2214yes—
30C3×#389C2214yes—
31C3×#389C2218yes—
32C3×#387C2218yes—
33C3×#387C2218yes—
34C3×#384C2214yes—
35C3×#384C2211yescharacteristic
36C3×#384C2211yescharacteristic
37C3×#382C2214yes—
38C3×#382C2212yes—
39C3×#380C2214yes—
40C3×C4×C4⋊C4C2214yes—
41C3×C4×C4⋊C4C2218yes—
42C3×C4×C4⋊C4C2212yes—
43C3×C4×C4⋊C4C2212yes—
44C3×C4×C4⋊C4C2212yes—
45C3×C4×C4⋊C4C2212yes—
46C3×C4×C4⋊C4C2214yes—
47C3×C4×C22⋊C4C2214yes—
48C3×C4×C22⋊C4C2214yes—
49C3×#376C2214yes—
50C3×C43C2214yes—
51#2612C2×C314yescomplemented
52#2571C2×C311yescharacteristic, complemented
53#2552C2×C312yescomplemented
54#2551C2×C311yescharacteristic, complemented
55#2441C2×C314yes—
56#2439C2×C314yes—
57#2396C2×C314yes—
58C4×C4⋊Q8C2×C314yescomplemented
59C4×#125C2×C314yescomplemented
60#2285C2×C311yescharacteristic
61C4×#117C2×C312yes—
62C22×C3×Q8C2314yes—
63C23×C3×C4C2312yes—
64C23×C3×C4C2311yescharacteristic
65C3×C4⋊Q8—216no—
66C3×#125—216no—
67C3×C4×Q8—48no—
68C3×#117—28no—
69C3×#117—28no—
70C3×#117—44no—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C4⋊C4C2312yes—
73C2×C3×C4⋊C4C2314yes—
74C2×C3×C4⋊C4C2314yes—
75C2×C3×C4⋊C4C2318yes—
76C2×C3×C4⋊C4C2318yes—
77C2×C3×C4⋊C4C2318yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2318yes—
81C2×C3×C4⋊C4C2314yes—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C22⋊C4C2314yes—
85C2×C3×C22⋊C4C2314yes—
86C2×C3×C22⋊C4C2314yes—
87C2×C3×C42Q818yescomplemented
88C2×C3×C42C2314yes—
89C2×C3×C42C2314yes—
90C2×C3×C42C2314yes—
91C2×C3×C42C2314yes—
92C2×C3×C42C2314yes—
93C2×C3×C42C2314yes—
94C2×C3×C42C2314yes—
95C2×C3×C42C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2312yes—
98C3×#95C2314yes—
99C3×#95C2314yes—
100C3×#95C2314yes—
101C3×#95C2314yes—
102C3×#95C2318yes—
103C3×#95C2314yes—
104C3×#95C2318yes—
105C3×#95C2314yes—
106C3×#95C2318yes—
107C2×C4⋊Q8C22×C314yes—
108C2×#125C22×C314yes—
109C2×C4×Q8C22×C314yes—
110C2×#117C22×C312yes—
111C2×#117C22×C312yes—
112C2×#117C22×C312yes—
113C22×C42C22×C311yescharacteristic
114#400C22×C314yes—
115#399C22×C314yes—
116#398C22×C318yes—
117#396C22×C318yes—
118#395C22×C314yes—
119#393C22×C314yes—
120#391C22×C318yes—
121#389C22×C314yes—
122#389C22×C314yes—
123#389C22×C314yes—
124#389C22×C318yes—
125#389C22×C314yes—
126#387C22×C318yes—
127#387C22×C318yes—
128#384C22×C314yes—
129#384C22×C311yescharacteristic
130#384C22×C311yescharacteristic
131#382C22×C314yes—
132#382C22×C312yes—
133#380C22×C314yes—
134C4×C4⋊C4C22×C314yes—
135C4×C4⋊C4C22×C312yes—
136C4×C4⋊C4C22×C318yes—
137C4×C4⋊C4C22×C314yes—
138C4×C4⋊C4C22×C312yes—
139C4×C4⋊C4C22×C312yes—
140C4×C4⋊C4C22×C312yes—
141C4×C22⋊C4C22×C314yes—
142C4×C22⋊C4C22×C314yes—
143#376C22×C314yes—
144C43C22×C314yes—
145C24×C3C2411yescharacteristic
146C2×C3×Q8—216no—
147C2×C3×Q8—44no—
148C22×C3×C4—24no—
149C22×C3×C4—24no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4C2×Q818yes—
153C22×C3×C4C2×Q814yes—
154C22×C3×C4C2414yes—
155C22×C3×C4C2414yes—
156C22×C3×C4C2414yes—
157C22×C3×C4C2414yes—
158C22×C3×C4C2412yes—
159C22×C3×C4C2411yescharacteristic
160C22×C3×C4C2414yes—
161C22×C3×C4C2414yes—
162C22×C3×C4C2412yes—
163C22×C3×C4C2411yescharacteristic
164C3×C4⋊C4—24no—
165C3×C4⋊C4—24no—
166C3×C4⋊C4—216no—
167C3×C4⋊C4—216no—
168C3×C4⋊C4—28no—
169C3×C4⋊C4—216no—
170C3×C4⋊C4—28no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—42no—
173C3×C4⋊C4—48no—
174C3×C4⋊C4—44no—
175C3×C4⋊C4#4118yes—
176C3×C4⋊C4#4118yes—
177C3×C22⋊C4—28no—
178C3×C22⋊C4—28no—
179C3×C22⋊C4—44no—
180C3×C42—28no—
181C3×C42—28no—
182C3×C42—28no—
183C3×C42—28no—
184C3×C42—28no—
185C3×C42—28no—
186C3×C42—28no—
187C3×C42C2×Q814yes—
188C3×C42C2×Q814yes—
189C3×C42C2×Q818yes—
190C22×Q8C23×C314yes—
191C23×C4C23×C311yescharacteristic
192C23×C4C23×C312yes—
193C4⋊Q8—216no—
194#125—216no—
195C4×Q8—48no—
196#117—28no—
197#117—28no—
198#117—44no—
199C2×C4⋊C4C23×C312yes—
200C2×C4⋊C4C23×C312yes—
201C2×C4⋊C4C23×C314yes—
202C2×C4⋊C4C23×C318yes—
203C2×C4⋊C4C23×C318yes—
204C2×C4⋊C4C23×C314yes—
205C2×C4⋊C4C23×C318yes—
206C2×C4⋊C4C23×C312yes—
207C2×C4⋊C4C23×C312yes—
208C2×C4⋊C4C23×C318yes—
209C2×C4⋊C4C23×C314yes—
210C2×C4⋊C4C23×C312yes—
211C2×C4⋊C4C23×C312yes—
212C2×C22⋊C4C23×C314yes—
213C2×C22⋊C4C23×C314yes—
214C2×C22⋊C4C23×C314yes—
215C2×C42C3×Q818yescomplemented
216C2×C42C23×C314yes—
217C2×C42C23×C314yes—
218C2×C42C23×C314yes—
219C2×C42C23×C314yes—
220C2×C42C23×C314yes—
221C2×C42C23×C314yes—
222C2×C42C23×C314yes—
223C2×C42C23×C312yes—
224C2×C42C23×C311yescharacteristic
225C2×C42C23×C311yescharacteristic
226#95C23×C314yes—
227#95C23×C314yes—
228#95C23×C314yes—
229#95C23×C314yes—
230#95C23×C318yes—
231#95C23×C314yes—
232#95C23×C318yes—
233#95C23×C314yes—
234#95C23×C318yes—
235C23×C3—22no—
236C23×C3—22no—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3C22×Q812yes—
240C23×C3C2511yescharacteristic
241C3×Q8—416no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—24no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—21no—
248C2×C3×C4—24no—
249C2×C3×C4—21no—
250C2×C3×C4—216no—
251C2×C3×C4—28no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—24no—
256C2×C3×C4—28no—
257C2×C3×C4—28no—
258C2×C3×C4—24no—
259C2×C3×C4—24no—
260C2×C3×C4—24no—
261C2×C3×C4—24no—
262C2×C3×C4C22×Q814yes—
263C2×C3×C4C22×Q812yes—
264C2×C3×C4C22×Q814yes—
265C2×C3×C4C22×Q812yes—
266C2×C3×C4C2×#4118yes—
267C2×C3×C4C2×#4118yes—
268C2×C3×C4C2×#4114yes—
269C2×C3×C4C2×#4114yes—
270C2×C3×C4H(5,𝔽2)12yes—
271C2×C3×C42−1+412yes—
272C24C24×C311yescharacteristic
273C2×Q8—216no—
274C2×Q8—44no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4C2×C3×Q818yes—
280C22×C4C2×C3×Q814yes—
281C22×C4C24×C314yes—
282C22×C4C24×C314yes—
283C22×C4C24×C314yes—
284C22×C4C24×C314yes—
285C22×C4C24×C312yes—
286C22×C4C24×C311yescharacteristic
287C22×C4C24×C314yes—
288C22×C4C24×C314yes—
289C22×C4C24×C312yes—
290C22×C4C24×C311yescharacteristic
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—216no—
294C4⋊C4—216no—
295C4⋊C4—28no—
296C4⋊C4—216no—
297C4⋊C4—28no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—48no—
301C4⋊C4—44no—
302C4⋊C4C3×#4118yes—
303C4⋊C4C3×#4118yes—
304C22⋊C4—28no—
305C22⋊C4—28no—
306C22⋊C4—44no—
307C42—28no—
308C42—28no—
309C42—28no—
310C42—28no—
311C42—28no—
312C42—28no—
313C42—28no—
314C42C2×C3×Q814yes—
315C42C2×C3×Q814yes—
316C42C2×C3×Q818yes—
317C22×C3—24no—
318C22×C3—22no—
319C22×C3—41no—
320C22×C3—42no—
321C22×C3—41no—
322C22×C3C23×Q811yescharacteristic
323C22×C3C22×#4112yes—
324C22×C3C22×#4112yes—
325C22×C3C2×H(5,𝔽2)11yescharacteristic
326C22×C3C2×2−1+411yescharacteristic
327C3×C4—24no—
328C3×C4—22no—
329C3×C4—28no—
330C3×C4—28no—
331C3×C4—44no—
332C3×C4—44no—
333C3×C4—44no—
334C3×C4—44no—
335C3×C4#53318yes—
336C3×C4#54414yes—
337C23—22no—
338C23—22no—
339C23—21no—
340C23—21no—
341C23C22×C3×Q812yes—
342C23C25×C311yescharacteristic
343Q8—416no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—21no—
350C2×C4—24no—
351C2×C4—21no—
352C2×C4—216no—
353C2×C4—28no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—28no—
359C2×C4—28no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4C22×C3×Q814yes—
365C2×C4C22×C3×Q812yes—
366C2×C4C22×C3×Q814yes—
367C2×C4C22×C3×Q812yes—
368C2×C4C2×C3×#4118yes—
369C2×C4C2×C3×#4118yes—
370C2×C4C2×C3×#4114yes—
371C2×C4C2×C3×#4114yes—
372C2×C4C3×H(5,𝔽2)12yes—
373C2×C4C3×2−1+412yes—
374C2×C3—42no—
375C2×C3C2×#53312yes—
376C2×C3C2×#54411yescharacteristic
377C2×C3#348111yescharacteristic
378C2×C3#348212yes—
379C2×C3#348311yescharacteristic
380C22—24no—
381C22—22no—
382C22—41no—
383C22—42no—
384C22—41no—
385C22C23×C3×Q811yescharacteristic
386C22C22×C3×#4112yes—
387C22C22×C3×#4112yes—
388C22C2×C3×H(5,𝔽2)11yescharacteristic
389C22C2×C3×2−1+411yescharacteristic
390C4—24no—
391C4—22no—
392C4—28no—
393C4—28no—
394C4—44no—
395C4—44no—
396C4—44no—
397C4—44no—
398C4C3×#53318yes—
399C4C3×#54414yes—
400C3#3497811yescharacteristic, direct product factor, Sylow
401C2—42no—
402C2C2×C3×#53312yes—
403C2C2×C3×#54411yescharacteristic
404C2C3×#348111yescharacteristic
405C2C3×#348212yes—
406C2C3×#348311yescharacteristic
subgroupG/H
centerC22×C3×C4C24
commutator, Frattini, cosocleC23C25×C3
socleC23×C3C25
prime radicalC24×C3C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#2612C2semidirect
2. C3×#2571C2semidirect
3. C3×#2552C2semidirect
4. C3×#2551C2semidirect
8. C3×C4×C4⋊Q8C2semidirect
9. C3×C4×#125C2semidirect
12. #34978C3direct (N × H)
51. #2612C2×C3semidirect
52. #2571C2×C3semidirect
53. #2552C2×C3semidirect
54. #2551C2×C3semidirect
58. C4×C4⋊Q8C2×C3semidirect
59. C4×#125C2×C3semidirect
87. C2×C3×C42Q8semidirect
215. C2×C42C3×Q8semidirect
400. C3#34978direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#26122
C3×#25712
C3×#25522
C3×#25512
C3×#24412
C3×#24392
C3×#23962
C3×C4×C4⋊Q82
C3×C4×#1252
C3×#22852
C3×C4×#1172
#349783