Finite Group Databasebeta

← C3×#34547 #10653154

Subgroups — C3×#34547 #10653154

Order 768.

2938 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×#41C212yesmaximal, complemented
2C3×#2399C211yesmaximal, characteristic, complemented
3C3×#2398C212yesmaximal, complemented
4C3×#2397C211yesmaximal, characteristic, complemented
5C3×#2396C214yesmaximal, complemented
6C3×#2395C214yesmaximal, complemented
7C3×#2392C214yesmaximal, complemented
8C3×#2392C214yesmaximal, complemented
9C3×#2388C214yesmaximal, complemented
10C3×C42×D4C214yesmaximal, complemented
11C3×C4×#117C211yesmaximal, characteristic, complemented
12#34547C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2212yes—
14C3×C4×#41—216no—
15C2×C3×C4×Q8C2212yes—
16C2×C3×C4×Q8C2212yes—
17C2×C3×C4×D4C2214yes—
18C2×C3×C4×D4C2214yes—
19C2×C3×C4×D4C2212yes—
20C2×C3×C4×D4C2212yes—
21C2×C3×C4×D4C2218yes—
22C2×C3×C4×D4C2214yes—
23C2×C3×#117C2214yes—
24C2×C3×#117C2212yes—
25C2×C3×#117C2212yes—
26C22×C3×C42C2214yes—
27C22×C3×C42C2211yescharacteristic
28C3×#400C2214yes—
29C3×#399C2214yes—
30C3×#398C2214yes—
31C3×#398C2214yes—
32C3×#397C2214yes—
33C3×#397C2214yes—
34C3×#396C2218yes—
35C3×#395C2214yes—
36C3×#394C2218yes—
37C3×#394C2214yes—
38C3×#394C2214yes—
39C3×#393C2214yes—
40C3×#392C2214yes—
41C3×#392C2214yes—
42C3×C4×C4⋊C4C2214yes—
43C3×C4×C4⋊C4C2214yes—
44C3×C4×C4⋊C4C2218yes—
45C3×C4×C22⋊C4C2214yes—
46C3×C4×C22⋊C4C2218yes—
47C3×C4×C22⋊C4C2218yes—
48C3×C4×C22⋊C4C2218yes—
49C3×#376C2211yescharacteristic
50C3×#376C2211yescharacteristic
51C3×C43C2212yes—
52C2×C4×#41C2×C312yescomplemented
53#2399C2×C311yescharacteristic, complemented
54#2398C2×C312yescomplemented
55#2397C2×C311yescharacteristic, complemented
56#2396C2×C314yescomplemented
57#2395C2×C314yescomplemented
58#2392C2×C314yescomplemented
59#2392C2×C314yescomplemented
60#2388C2×C314yescomplemented
61C42×D4C2×C314yescomplemented
62C4×#117C2×C311yescharacteristic, complemented
63C2×C3×#41—28no—
64C2×C3×#41—216no—
65C22×C3×Q8C2312yes—
66C22×C3×D4C2312yes—
67C22×C3×D4C2314yes—
68C22×C3×D4C2314yes—
69C23×C3×C4C2314yes—
70C23×C3×C4C2314yes—
71C23×C3×C4C2312yes—
72C23×C3×C4C2311yescharacteristic
73C23×C3×C4C2314yes—
74C3×C4×Q8—28no—
75C3×C4×Q8—28no—
76C3×C4×D4—216no—
77C3×C4×D4—216no—
78C3×C4×D4—232no—
79C3×C4×D4—28no—
80C3×C4×D4—28no—
81C3×C4×D4—216no—
82C3×#117—28no—
83C3×#117—216no—
84C3×#117—28no—
85C2×C3×C4⋊C4C2318yes—
86C2×C3×C4⋊C4C2314yes—
87C2×C3×C4⋊C4C2314yes—
88C2×C3×C4⋊C4C2314yes—
89C2×C3×C4⋊C4C2314yes—
90C2×C3×C4⋊C4C2312yes—
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C22⋊C4C2318yes—
93C2×C3×C22⋊C4C2314yes—
94C2×C3×C22⋊C4C2318yes—
95C2×C3×C22⋊C4C2318yes—
96C2×C3×C22⋊C4C2314yes—
97C2×C3×C22⋊C4C2314yes—
98C2×C3×C22⋊C4C2314yes—
99C2×C3×C42—216no—
100C2×C3×C42D418yescomplemented
101C2×C3×C42C2312yes—
102C2×C3×C42C2318yes—
103C2×C3×C42C2314yes—
104C2×C3×C42C2314yes—
105C2×C3×C42C2314yes—
106C2×C3×C42C2311yescharacteristic
107C2×C3×C42C2312yes—
108C2×C3×C42C2312yes—
109C2×C3×C42C2311yescharacteristic
110C3×#95C2314yes—
111C3×#95C2314yes—
112C3×#95C2314yes—
113C3×#95C2318yes—
114C3×#95C2314yes—
115C3×#95C2318yes—
116C22×#41C22×C312yes—
117C4×#41—216no—
118C2×C4×Q8C22×C312yes—
119C2×C4×Q8C22×C312yes—
120C2×C4×D4C22×C318yes—
121C2×C4×D4C22×C314yes—
122C2×C4×D4C22×C314yes—
123C2×C4×D4C22×C312yes—
124C2×C4×D4C22×C314yes—
125C2×C4×D4C22×C312yes—
126C2×#117C22×C312yes—
127C2×#117C22×C314yes—
128C2×#117C22×C312yes—
129C22×C42C22×C314yes—
130C22×C42C22×C311yescharacteristic
131#400C22×C314yes—
132#399C22×C314yes—
133#398C22×C314yes—
134#398C22×C314yes—
135#397C22×C314yes—
136#397C22×C314yes—
137#396C22×C318yes—
138#395C22×C314yes—
139#394C22×C318yes—
140#394C22×C314yes—
141#394C22×C314yes—
142#393C22×C314yes—
143#392C22×C314yes—
144#392C22×C314yes—
145C4×C4⋊C4C22×C314yes—
146C4×C4⋊C4C22×C314yes—
147C4×C4⋊C4C22×C318yes—
148C4×C22⋊C4C22×C314yes—
149C4×C22⋊C4C22×C318yes—
150C4×C22⋊C4C22×C318yes—
151C4×C22⋊C4C22×C318yes—
152#376C22×C311yescharacteristic
153#376C22×C311yescharacteristic
154C43C22×C312yes—
155C24×C3C2411yescharacteristic
156C24×C3C2414yes—
157C3×#41—264no—
158C2×C3×Q8—24no—
159C2×C3×Q8—28no—
160C2×C3×D4—28no—
161C2×C3×D4—216no—
162C2×C3×D4—28no—
163C2×C3×D4—28no—
164C2×C3×D4—24no—
165C2×C3×D4—28no—
166C2×C3×D4—28no—
167C22×C3×C4—28no—
168C22×C3×C4—24no—
169C22×C3×C4—28no—
170C22×C3×C4—28no—
171C22×C3×C4—28no—
172C22×C3×C4—28no—
173C22×C3×C4—216no—
174C22×C3×C4—216no—
175C22×C3×C4C2×D418yes—
176C22×C3×C4C2×D414yes—
177C22×C3×C4#41116yescomplemented
178C22×C3×C4C2414yes—
179C22×C3×C4C2414yes—
180C22×C3×C4C2414yes—
181C22×C3×C4C2412yes—
182C22×C3×C4C2411yescharacteristic
183C22×C3×C4C2414yes—
184C22×C3×C4C2414yes—
185C22×C3×C4C2412yes—
186C22×C3×C4C2411yescharacteristic
187C3×C4⋊C4—28no—
188C3×C4⋊C4—28no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—24no—
191C3×C4⋊C4—28no—
192C3×C4⋊C4—216no—
193C3×C4⋊C4—28no—
194C3×C22⋊C4—28no—
195C3×C22⋊C4—216no—
196C3×C22⋊C4—28no—
197C3×C22⋊C4—28no—
198C3×C22⋊C4—28no—
199C3×C22⋊C4—216no—
200C3×C22⋊C4—216no—
201C3×C42—28no—
202C3×C42—28no—
203C3×C42—28no—
204C3×C42—24no—
205C3×C42—216no—
206C3×C42C2×D414yes—
207C3×C42C2×D418yes—
208C3×C42C2×D414yes—
209C3×C42#4118yescomplemented
210C2×#41—28no—
211C2×#41—216no—
212C22×Q8C23×C312yes—
213C22×D4C23×C314yes—
214C22×D4C23×C312yes—
215C22×D4C23×C314yes—
216C23×C4C23×C314yes—
217C23×C4C23×C314yes—
218C23×C4C23×C311yescharacteristic
219C23×C4C23×C312yes—
220C23×C4C23×C314yes—
221C4×Q8—28no—
222C4×Q8—28no—
223C4×D4—216no—
224C4×D4—216no—
225C4×D4—232no—
226C4×D4—28no—
227C4×D4—28no—
228C4×D4—216no—
229#117—28no—
230#117—216no—
231#117—28no—
232C2×C4⋊C4C23×C314yes—
233C2×C4⋊C4C23×C314yes—
234C2×C4⋊C4C23×C312yes—
235C2×C4⋊C4C23×C312yes—
236C2×C4⋊C4C23×C314yes—
237C2×C4⋊C4C23×C318yes—
238C2×C4⋊C4C23×C314yes—
239C2×C22⋊C4C23×C314yes—
240C2×C22⋊C4C23×C314yes—
241C2×C22⋊C4C23×C314yes—
242C2×C22⋊C4C23×C314yes—
243C2×C22⋊C4C23×C318yes—
244C2×C22⋊C4C23×C318yes—
245C2×C22⋊C4C23×C318yes—
246C2×C42—216no—
247C2×C42C3×D418yescomplemented
248C2×C42C23×C314yes—
249C2×C42C23×C312yes—
250C2×C42C23×C314yes—
251C2×C42C23×C314yes—
252C2×C42C23×C318yes—
253C2×C42C23×C312yes—
254C2×C42C23×C311yescharacteristic
255C2×C42C23×C311yescharacteristic
256C2×C42C23×C312yes—
257#95C23×C314yes—
258#95C23×C314yes—
259#95C23×C314yes—
260#95C23×C318yes—
261#95C23×C314yes—
262#95C23×C318yes—
263C23×C3—24no—
264C23×C3—22no—
265C23×C3—28no—
266C23×C3—28no—
267C23×C3—24no—
268C23×C3—24no—
269C23×C3C22×D412yes—
270C23×C3C2×#4118yes—
271C23×C3C2511yescharacteristic
272C3×Q8—216no—
273C3×D4—232no—
274C3×D4—216no—
275C3×D4—232no—
276C2×C3×C4—24no—
277C2×C3×C4—28no—
278C2×C3×C4—22no—
279C2×C3×C4—24no—
280C2×C3×C4—24no—
281C2×C3×C4—216no—
282C2×C3×C4—232no—
283C2×C3×C4—28no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—28no—
287C2×C3×C4—24no—
288C2×C3×C4—28no—
289C2×C3×C4—28no—
290C2×C3×C4—416no—
291C2×C3×C4—416no—
292C2×C3×C4C22×D412yes—
293C2×C3×C4C22×D414yes—
294C2×C3×C4C22×D414yes—
295C2×C3×C4C22×D412yes—
296C2×C3×C4C2×#4118yes—
297C2×C3×C4C2×#4118yes—
298C2×C3×C4C2×#4118yes—
299C2×C3×C4C2×#4114yes—
300C24C24×C311yescharacteristic
301C24C24×C314yes—
302#41—264no—
303C2×Q8—24no—
304C2×Q8—28no—
305C2×D4—28no—
306C2×D4—216no—
307C2×D4—28no—
308C2×D4—28no—
309C2×D4—24no—
310C2×D4—28no—
311C2×D4—28no—
312C22×C4—28no—
313C22×C4—24no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—216no—
319C22×C4—216no—
320C22×C4C2×C3×D418yes—
321C22×C4C2×C3×D414yes—
322C22×C4C3×#41116yescomplemented
323C22×C4C24×C314yes—
324C22×C4C24×C314yes—
325C22×C4C24×C314yes—
326C22×C4C24×C312yes—
327C22×C4C24×C311yescharacteristic
328C22×C4C24×C314yes—
329C22×C4C24×C314yes—
330C22×C4C24×C312yes—
331C22×C4C24×C311yescharacteristic
332C4⋊C4—28no—
333C4⋊C4—28no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—28no—
337C4⋊C4—216no—
338C4⋊C4—28no—
339C22⋊C4—28no—
340C22⋊C4—216no—
341C22⋊C4—28no—
342C22⋊C4—28no—
343C22⋊C4—28no—
344C22⋊C4—216no—
345C22⋊C4—216no—
346C42—28no—
347C42—28no—
348C42—28no—
349C42—24no—
350C42—216no—
351C42C2×C3×D414yes—
352C42C2×C3×D418yes—
353C42C2×C3×D414yes—
354C42C3×#4118yescomplemented
355C22×C3—24no—
356C22×C3—28no—
357C22×C3—22no—
358C22×C3—216no—
359C22×C3—44no—
360C22×C3—44no—
361C22×C3—44no—
362C22×C3—48no—
363C22×C3C23×D411yescharacteristic
364C22×C3C22×#4114yes—
365C22×C3C22×#4112yes—
366C3×C4—24no—
367C3×C4—28no—
368C3×C4—22no—
369C3×C4—28no—
370C3×C4—44no—
371C3×C4—44no—
372C3×C4—44no—
373C3×C4#52518yes—
374C3×C4#53214yes—
375C23—24no—
376C23—22no—
377C23—28no—
378C23—28no—
379C23—24no—
380C23—24no—
381C23C22×C3×D412yes—
382C23C2×C3×#4118yes—
383C23C25×C311yescharacteristic
384Q8—216no—
385D4—216no—
386D4—232no—
387D4—232no—
388C2×C4—24no—
389C2×C4—28no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—216no—
394C2×C4—232no—
395C2×C4—28no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—28no—
399C2×C4—24no—
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—416no—
403C2×C4—416no—
404C2×C4C22×C3×D412yes—
405C2×C4C22×C3×D414yes—
406C2×C4C22×C3×D414yes—
407C2×C4C22×C3×D412yes—
408C2×C4C2×C3×#4118yes—
409C2×C4C2×C3×#4118yes—
410C2×C4C2×C3×#4118yes—
411C2×C4C2×C3×#4114yes—
412C2×C3—42no—
413C2×C3—48no—
414C2×C3C2×#52512yes—
415C2×C3C2×#53211yescharacteristic
416C2×C3#344414yes—
417C22—24no—
418C22—28no—
419C22—22no—
420C22—216no—
421C22—44no—
422C22—44no—
423C22—44no—
424C22—48no—
425C22C23×C3×D411yescharacteristic
426C22C22×C3×#4114yes—
427C22C22×C3×#4112yes—
428C4—24no—
429C4—28no—
430C4—22no—
431C4—28no—
432C4—44no—
433C4—44no—
434C4—44no—
435C4C3×#52518yes—
436C4C3×#53214yes—
437C3#3454711yescharacteristic, direct product factor, Sylow
438C2—42no—
439C2—48no—
440C2C2×C3×#52512yes—
441C2C2×C3×#53211yescharacteristic
442C2C3×#344414yes—
subgroupG/H
centerC2×C3×C42C23
commutator, Frattini, cosocleC23C25×C3
socleC23×C3C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×#41C2semidirect
2. C3×#2399C2semidirect
3. C3×#2398C2semidirect
4. C3×#2397C2semidirect
5. C3×#2396C2semidirect
6. C3×#2395C2semidirect
7. C3×#2392C2semidirect
8. C3×#2392C2semidirect
9. C3×#2388C2semidirect
10. C3×C42×D4C2semidirect
11. C3×C4×#117C2semidirect
12. #34547C3direct (N × H)
52. C2×C4×#41C2×C3semidirect
53. #2399C2×C3semidirect
54. #2398C2×C3semidirect
55. #2397C2×C3semidirect
56. #2396C2×C3semidirect
57. #2395C2×C3semidirect
58. #2392C2×C3semidirect
59. #2392C2×C3semidirect
60. #2388C2×C3semidirect
61. C42×D4C2×C3semidirect
62. C4×#117C2×C3semidirect
100. C2×C3×C42D4semidirect
177. C22×C3×C4#41semidirect
209. C3×C42#41semidirect
247. C2×C42C3×D4semidirect
322. C22×C4C3×#41semidirect
354. C42C3×#41semidirect
437. C3#34547direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×#412
C3×#23992
C3×#23982
C3×#23972
C3×#23962
C3×#23952
C3×#23922
C3×#23882
C3×C42×D42
C3×C4×#1172
#345473