Finite Group Databasebeta

← C3×#34973 #10653580

Subgroups — C3×#34973 #10653580

Order 768.

1716 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2524C214yesmaximal, complemented
2C3×#2519C214yesmaximal
3C3×#2469C214yesmaximal
4C3×#2462C214yesmaximal, complemented
5C3×#2402C211yesmaximal, characteristic, complemented
6C3×#2399C211yesmaximal, characteristic, complemented
7C3×#2396C212yesmaximal
8C3×C4×#125C214yesmaximal, complemented
9C3×C4×#126C214yesmaximal
10C3×#2298C211yesmaximal, characteristic
11C3×C42×Q8C212yesmaximal, complemented
12#34973C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×#126C2214yes—
14C2×C3×#125C2214yes—
15C2×C3×C4×Q8C2212yes—
16C2×C3×C4×Q8C2214yes—
17C2×C3×#117C2211yescharacteristic
18C3×#400C2212yes—
19C3×#400C2214yes—
20C3×#399C2212yes—
21C3×#398C2214yes—
22C3×#398C2214yes—
23C3×#398C2214yes—
24C3×#396C2214yes—
25C3×#395C2212yes—
26C3×#395C2214yes—
27C3×#393C2212yes—
28C3×#389C2214yes—
29C3×#389C2214yes—
30C3×#388C2214yes—
31C3×#388C2214yes—
32C3×#388C2214yes—
33C3×#388C2214yes—
34C3×#387C2214yes—
35C3×#387C2214yes—
36C3×#387C2214yes—
37C3×#387C2214yes—
38C3×#387C2214yes—
39C3×#387C2214yes—
40C3×#387C2214yes—
41C3×#387C2214yes—
42C3×#383C2214yes—
43C3×#382C2214yes—
44C3×C4×C4⋊C4C2212yes—
45C3×C4×C4⋊C4C2214yes—
46C3×C4×C4⋊C4C2214yes—
47C3×C4×C4⋊C4C2214yes—
48C3×C4×C4⋊C4C2214yes—
49C3×C4×C4⋊C4C2214yes—
50C3×C4×C4⋊C4C2214yes—
51C3×C4×C4⋊C4C2214yes—
52C3×C4×C4⋊C4C2212yes—
53C3×C4×C22⋊C4C2214yes—
54C3×C4×C22⋊C4C2212yes—
55C3×#376C2211yescharacteristic
56C3×C43C2211yescharacteristic
57C3×C43C2214yes—
58#2524C2×C314yescomplemented
59#2519C2×C314yes—
60#2469C2×C314yes—
61#2462C2×C314yescomplemented
62#2402C2×C311yescharacteristic, complemented
63#2399C2×C311yescharacteristic, complemented
64#2396C2×C312yes—
65C4×#125C2×C314yescomplemented
66C4×#126C2×C314yes—
67#2298C2×C311yescharacteristic
68C42×Q8C2×C312yescomplemented
69C22×C3×Q8C2312yes—
70C23×C3×C4C2311yescharacteristic
71C3×#126—216no—
72C3×#125—216no—
73C3×C4×Q8—216no—
74C3×C4×Q8—28no—
75C3×#117—24no—
76C2×C3×C4⋊C4C2314yes—
77C2×C3×C4⋊C4C2314yes—
78C2×C3×C4⋊C4C2314yes—
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2314yes—
81C2×C3×C4⋊C4C2314yes—
82C2×C3×C4⋊C4C2314yes—
83C2×C3×C4⋊C4C2314yes—
84C2×C3×C4⋊C4C2314yes—
85C2×C3×C4⋊C4C2314yes—
86C2×C3×C4⋊C4C2314yes—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C4⋊C4C2314yes—
89C2×C3×C22⋊C4C2314yes—
90C2×C3×C22⋊C4C2314yes—
91C2×C3×C22⋊C4C2314yes—
92C2×C3×C22⋊C4C2312yes—
93C2×C3×C42C2314yes—
94C2×C3×C42C2314yes—
95C2×C3×C42C2314yes—
96C2×C3×C42C2314yes—
97C2×C3×C42C2314yes—
98C2×C3×C42C2312yes—
99C2×C3×C42C2314yes—
100C2×C3×C42C2314yes—
101C2×C3×C42C2312yes—
102C2×C3×C42C2311yescharacteristic
103C2×C3×C42C2311yescharacteristic
104C3×#95C2314yes—
105C3×#95C2314yes—
106C3×#95C2314yes—
107C3×#95C2314yes—
108C3×#95C2314yes—
109C3×#95C2314yes—
110C3×#95C2312yes—
111C3×#95C2312yes—
112C3×#95C2314yes—
113C3×#95C2314yes—
114C3×#95C2314yes—
115C3×#95C2314yes—
116C3×#95C2314yes—
117C3×#95C2314yes—
118C3×#95C2314yes—
119C2×#126C22×C314yes—
120C2×#125C22×C314yes—
121C2×C4×Q8C22×C312yes—
122C2×C4×Q8C22×C314yes—
123C2×#117C22×C311yescharacteristic
124#400C22×C312yes—
125#400C22×C314yes—
126#399C22×C312yes—
127#398C22×C314yes—
128#398C22×C314yes—
129#398C22×C314yes—
130#396C22×C314yes—
131#395C22×C312yes—
132#395C22×C314yes—
133#393C22×C312yes—
134#389C22×C314yes—
135#389C22×C314yes—
136#388C22×C314yes—
137#388C22×C314yes—
138#388C22×C314yes—
139#388C22×C314yes—
140#387C22×C314yes—
141#387C22×C314yes—
142#387C22×C314yes—
143#387C22×C314yes—
144#387C22×C314yes—
145#387C22×C314yes—
146#387C22×C314yes—
147#387C22×C314yes—
148#383C22×C314yes—
149#382C22×C314yes—
150C4×C4⋊C4C22×C314yes—
151C4×C4⋊C4C22×C314yes—
152C4×C4⋊C4C22×C314yes—
153C4×C4⋊C4C22×C314yes—
154C4×C4⋊C4C22×C312yes—
155C4×C4⋊C4C22×C312yes—
156C4×C4⋊C4C22×C314yes—
157C4×C4⋊C4C22×C314yes—
158C4×C4⋊C4C22×C314yes—
159C4×C22⋊C4C22×C314yes—
160C4×C22⋊C4C22×C312yes—
161#376C22×C311yescharacteristic
162C43C22×C314yes—
163C43C22×C311yescharacteristic
164C24×C3C2411yescharacteristic
165C2×C3×Q8—28no—
166C2×C3×Q8—24no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4C2411yescharacteristic
170C22×C3×C4C2414yes—
171C22×C3×C4C2414yes—
172C22×C3×C4C2414yes—
173C22×C3×C4C2412yes—
174C22×C3×C4C2414yes—
175C22×C3×C4C2414yes—
176C22×C3×C4C2414yes—
177C22×C3×C4C2412yes—
178C22×C3×C4C2411yescharacteristic
179C3×C4⋊C4—24no—
180C3×C4⋊C4—28no—
181C3×C4⋊C4—28no—
182C3×C4⋊C4—28no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4—28no—
185C3×C4⋊C4—28no—
186C3×C4⋊C4—28no—
187C3×C4⋊C4—28no—
188C3×C4⋊C4—28no—
189C3×C4⋊C4—28no—
190C3×C4⋊C4—28no—
191C3×C4⋊C4#41116yes—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—28no—
194C3×C22⋊C4—28no—
195C3×C22⋊C4—28no—
196C3×C42—22no—
197C3×C42—28no—
198C3×C42—28no—
199C3×C42—28no—
200C3×C42—28no—
201C3×C42—28no—
202C3×C42—28no—
203C3×C42—24no—
204C3×C42—28no—
205C3×C42—24no—
206C3×C42#4114yes—
207C22×Q8C23×C312yes—
208C23×C4C23×C311yescharacteristic
209#126—216no—
210#125—216no—
211C4×Q8—216no—
212C4×Q8—28no—
213#117—24no—
214C2×C4⋊C4C23×C312yes—
215C2×C4⋊C4C23×C314yes—
216C2×C4⋊C4C23×C314yes—
217C2×C4⋊C4C23×C314yes—
218C2×C4⋊C4C23×C312yes—
219C2×C4⋊C4C23×C314yes—
220C2×C4⋊C4C23×C314yes—
221C2×C4⋊C4C23×C314yes—
222C2×C4⋊C4C23×C314yes—
223C2×C4⋊C4C23×C314yes—
224C2×C4⋊C4C23×C314yes—
225C2×C4⋊C4C23×C314yes—
226C2×C4⋊C4C23×C314yes—
227C2×C22⋊C4C23×C312yes—
228C2×C22⋊C4C23×C314yes—
229C2×C22⋊C4C23×C314yes—
230C2×C22⋊C4C23×C314yes—
231C2×C42C23×C311yescharacteristic
232C2×C42C23×C314yes—
233C2×C42C23×C314yes—
234C2×C42C23×C314yes—
235C2×C42C23×C314yes—
236C2×C42C23×C314yes—
237C2×C42C23×C314yes—
238C2×C42C23×C312yes—
239C2×C42C23×C314yes—
240C2×C42C23×C312yes—
241C2×C42C23×C311yescharacteristic
242#95C23×C314yes—
243#95C23×C314yes—
244#95C23×C314yes—
245#95C23×C314yes—
246#95C23×C314yes—
247#95C23×C314yes—
248#95C23×C312yes—
249#95C23×C312yes—
250#95C23×C314yes—
251#95C23×C314yes—
252#95C23×C314yes—
253#95C23×C314yes—
254#95C23×C314yes—
255#95C23×C314yes—
256#95C23×C314yes—
257C23×C3—24no—
258C23×C3—21no—
259C23×C3—22no—
260C23×C3C2511yescharacteristic
261C3×Q8—216no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—24no—
269C2×C3×C4—22no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—24no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—24no—
277C2×C3×C4—24no—
278C2×C3×C4—21no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—22no—
282C2×C3×C4—44no—
283C2×C3×C4C2×#4118yes—
284C2×C3×C4C2×#4118yes—
285C2×C3×C4C2×#4118yes—
286C2×C3×C4C2×#4114yes—
287C2×C3×C4C2×#4112yes—
288C2×C3×C42−1+414yes—
289C24C24×C311yescharacteristic
290C2×Q8—28no—
291C2×Q8—24no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4C24×C311yescharacteristic
295C22×C4C24×C314yes—
296C22×C4C24×C314yes—
297C22×C4C24×C314yes—
298C22×C4C24×C312yes—
299C22×C4C24×C314yes—
300C22×C4C24×C314yes—
301C22×C4C24×C314yes—
302C22×C4C24×C312yes—
303C22×C4C24×C311yescharacteristic
304C4⋊C4—24no—
305C4⋊C4—28no—
306C4⋊C4—28no—
307C4⋊C4—28no—
308C4⋊C4—24no—
309C4⋊C4—28no—
310C4⋊C4—28no—
311C4⋊C4—28no—
312C4⋊C4—28no—
313C4⋊C4—28no—
314C4⋊C4—28no—
315C4⋊C4—28no—
316C4⋊C4C3×#41116yes—
317C22⋊C4—24no—
318C22⋊C4—28no—
319C22⋊C4—28no—
320C22⋊C4—28no—
321C42—22no—
322C42—28no—
323C42—28no—
324C42—28no—
325C42—28no—
326C42—28no—
327C42—28no—
328C42—24no—
329C42—28no—
330C42—24no—
331C42C3×#4114yes—
332C22×C3—42no—
333C22×C3—41no—
334C22×C3—44no—
335C22×C3C22×#4111yescharacteristic
336C22×C3C22×#4114yes—
337C22×C3C2×2−1+412yes—
338C3×C4—28no—
339C3×C4—24no—
340C3×C4—28no—
341C3×C4—28no—
342C3×C4—44no—
343C3×C4—42no—
344C3×C4—41no—
345C3×C4—44no—
346C3×C4—44no—
347C3×C4#54114yes—
348C23—24no—
349C23—21no—
350C23—22no—
351C23C25×C311yescharacteristic
352Q8—216no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—21no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—44no—
374C2×C4C2×C3×#4118yes—
375C2×C4C2×C3×#4118yes—
376C2×C4C2×C3×#4118yes—
377C2×C4C2×C3×#4114yes—
378C2×C4C2×C3×#4112yes—
379C2×C4C3×2−1+414yes—
380C2×C3—81no—
381C2×C3#344412yes—
382C2×C3C2×#54111yescharacteristic
383C2×C3#348214yes—
384C22—42no—
385C22—41no—
386C22—44no—
387C22C22×C3×#4111yescharacteristic
388C22C22×C3×#4114yes—
389C22C2×C3×2−1+412yes—
390C4—28no—
391C4—24no—
392C4—28no—
393C4—28no—
394C4—44no—
395C4—42no—
396C4—41no—
397C4—44no—
398C4—44no—
399C4C3×#54114yes—
400C3#3497311yescharacteristic, direct product factor, Sylow
401C2—81no—
402C2C3×#344412yes—
403C2C2×C3×#54111yescharacteristic
404C2C3×#348214yes—
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×#2524C2semidirect
4. C3×#2462C2semidirect
5. C3×#2402C2semidirect
6. C3×#2399C2semidirect
8. C3×C4×#125C2semidirect
11. C3×C42×Q8C2semidirect
12. #34973C3direct (N × H)
58. #2524C2×C3semidirect
61. #2462C2×C3semidirect
62. #2402C2×C3semidirect
63. #2399C2×C3semidirect
65. C4×#125C2×C3semidirect
68. C42×Q8C2×C3semidirect
400. C3#34973direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#25242
C3×#25192
C3×#24692
C3×#24622
C3×#24022
C3×#23992
C3×#23962
C3×C4×#1252
C3×C4×#1262
C3×#22982
C3×C42×Q82
#349733