Finite Group Databasebeta

← C3×#35283 #10653890

Subgroups — C3×#35283 #10653890

Order 768.

1738 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2868C211yesmaximal, characteristic, complemented
2C3×#2862C211yesmaximal, characteristic, complemented
3C3×#2857C212yesmaximal
4C3×#2647C212yesmaximal, complemented
5C3×#2628C212yesmaximal
6C3×#2511C214yesmaximal, complemented
7C3×#2492C214yesmaximal
8C3×#2451C214yesmaximal, complemented
9C3×#2423C214yesmaximal
10C3×#2331C212yesmaximal, complemented
11C3×#2314C212yesmaximal
12C3×C4×C4⋊Q8C212yesmaximal, complemented
13C3×C4×#117C211yesmaximal, characteristic
14#35283C311yesmaximal, characteristic, direct product factor, Sylow
15C2×C3×C4⋊Q8C2212yes—
16C2×C3×C4⋊Q8C2212yes—
17C2×C3×C4⋊Q8C2214yes—
18C2×C3×C22⋊Q8C2214yes—
19C2×C3×C4×Q8C2214yes—
20C2×C3×#117C2211yescharacteristic
21C2×C3×#117C2211yescharacteristic
22C22×C3×C42C2211yescharacteristic
23C3×#400C2214yes—
24C3×#400C2218yes—
25C3×#399C2214yes—
26C3×#398C2218yes—
27C3×#395C2214yes—
28C3×#393C2214yes—
29C3×#391C2214yes—
30C3×#391C2218yes—
31C3×#391C2214yes—
32C3×#391C2214yes—
33C3×#391C2214yes—
34C3×#389C2214yes—
35C3×#389C2214yes—
36C3×#389C2214yes—
37C3×#389C2214yes—
38C3×#388C2218yes—
39C3×#387C2218yes—
40C3×#387C2218yes—
41C3×#384C2212yes—
42C3×#384C2214yes—
43C3×#384C2212yes—
44C3×#382C2212yes—
45C3×#382C2212yes—
46C3×#380C2214yes—
47C3×#380C2214yes—
48C3×C4×C4⋊C4C2212yes—
49C3×C4×C4⋊C4C2214yes—
50C3×C4×C4⋊C4C2212yes—
51C3×C4×C4⋊C4C2214yes—
52C3×C4×C22⋊C4C2212yes—
53C3×C4×C22⋊C4C2212yes—
54C3×#376C2211yescharacteristic
55C3×#376C2211yescharacteristic
56C3×C43C2211yescharacteristic
57C3×C43C2211yescharacteristic
58#2868C2×C311yescharacteristic, complemented
59#2862C2×C311yescharacteristic, complemented
60#2857C2×C312yes—
61#2647C2×C312yescomplemented
62#2628C2×C312yes—
63#2511C2×C314yescomplemented
64#2492C2×C314yes—
65#2451C2×C314yescomplemented
66#2423C2×C314yes—
67#2331C2×C312yescomplemented
68#2314C2×C312yes—
69C4×C4⋊Q8C2×C312yescomplemented
70C4×#117C2×C311yescharacteristic
71C22×C3×Q8C2314yes—
72C22×C3×Q8C2314yes—
73C23×C3×C4C2312yes—
74C23×C3×C4C2311yescharacteristic
75C3×C4⋊Q8—28no—
76C3×C4⋊Q8—28no—
77C3×C4⋊Q8—48no—
78C3×C22⋊Q8—48no—
79C3×C4×Q8—48no—
80C3×#117—24no—
81C3×#117Q818yescomplemented
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2314yes—
85C2×C3×C4⋊C4C2314yes—
86C2×C3×C4⋊C4C2314yes—
87C2×C3×C4⋊C4C2314yes—
88C2×C3×C4⋊C4C2314yes—
89C2×C3×C4⋊C4C2314yes—
90C2×C3×C4⋊C4C2318yes—
91C2×C3×C4⋊C4C2318yes—
92C2×C3×C4⋊C4C2318yes—
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C22⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4C2318yes—
97C2×C3×C22⋊C4C2312yes—
98C2×C3×C42—24no—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2312yes—
101C2×C3×C42C2312yes—
102C2×C3×C42C2312yes—
103C2×C3×C42C2312yes—
104C2×C3×C42C2311yescharacteristic
105C2×C3×C42C2312yes—
106C2×C3×C42C2318yes—
107C2×C3×C42C2311yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C3×#95C2312yes—
112C3×#95C2312yes—
113C3×#95C2312yes—
114C3×#95C2312yes—
115C3×#95C2318yes—
116C3×#95C2318yes—
117C3×#95C2318yes—
118C3×#95C2318yes—
119C3×#95C2314yes—
120C3×#95C2318yes—
121C3×#95C2314yes—
122C2×C4⋊Q8C22×C312yes—
123C2×C4⋊Q8C22×C312yes—
124C2×C4⋊Q8C22×C314yes—
125C2×C22⋊Q8C22×C314yes—
126C2×C4×Q8C22×C314yes—
127C2×#117C22×C311yescharacteristic
128C2×#117C22×C311yescharacteristic
129C22×C42C22×C311yescharacteristic
130#400C22×C314yes—
131#400C22×C318yes—
132#399C22×C314yes—
133#398C22×C318yes—
134#395C22×C314yes—
135#393C22×C314yes—
136#391C22×C314yes—
137#391C22×C318yes—
138#391C22×C314yes—
139#391C22×C314yes—
140#391C22×C314yes—
141#389C22×C314yes—
142#389C22×C314yes—
143#389C22×C314yes—
144#389C22×C314yes—
145#388C22×C318yes—
146#387C22×C318yes—
147#387C22×C318yes—
148#384C22×C312yes—
149#384C22×C312yes—
150#384C22×C314yes—
151#382C22×C312yes—
152#382C22×C312yes—
153#380C22×C314yes—
154#380C22×C314yes—
155C4×C4⋊C4C22×C314yes—
156C4×C4⋊C4C22×C314yes—
157C4×C4⋊C4C22×C312yes—
158C4×C4⋊C4C22×C312yes—
159C4×C22⋊C4C22×C312yes—
160C4×C22⋊C4C22×C312yes—
161#376C22×C311yescharacteristic
162#376C22×C311yescharacteristic
163C43C22×C311yescharacteristic
164C43C22×C311yescharacteristic
165C24×C3C2411yescharacteristic
166C2×C3×Q8—28no—
167C2×C3×Q8—28no—
168C2×C3×Q8—28no—
169C2×C3×Q8—28no—
170C2×C3×Q8—48no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—24no—
174C22×C3×C4—24no—
175C22×C3×C4—24no—
176C22×C3×C4C2×Q814yes—
177C22×C3×C4C2411yescharacteristic
178C22×C3×C4C2412yes—
179C22×C3×C4C2418yes—
180C22×C3×C4C2418yes—
181C22×C3×C4C2411yescharacteristic
182C22×C3×C4C2411yescharacteristic
183C22×C3×C4C2412yes—
184C22×C3×C4C2412yes—
185C22×C3×C4C2411yescharacteristic
186C22×C3×C4C2411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C22×C3×C4C2412yes—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—28no—
192C3×C4⋊C4—28no—
193C3×C4⋊C4—28no—
194C3×C4⋊C4—28no—
195C3×C4⋊C4—28no—
196C3×C4⋊C4—28no—
197C3×C4⋊C4—48no—
198C3×C4⋊C4—48no—
199C3×C4⋊C4—48no—
200C3×C4⋊C4C2×Q818yes—
201C3×C22⋊C4—22no—
202C3×C22⋊C4—22no—
203C3×C22⋊C4—48no—
204C3×C22⋊C4C2×Q818yes—
205C3×C42—24no—
206C3×C42—22no—
207C3×C42—22no—
208C3×C42—24no—
209C3×C42—24no—
210C3×C42—24no—
211C3×C42—24no—
212C3×C42—48no—
213C3×C42C2×Q814yes—
214C3×C42C2×Q814yes—
215C3×C42#4114yes—
216C3×C42#4114yes—
217C22×Q8C23×C314yes—
218C22×Q8C23×C314yes—
219C23×C4C23×C312yes—
220C23×C4C23×C311yescharacteristic
221C4⋊Q8—28no—
222C4⋊Q8—28no—
223C4⋊Q8—48no—
224C22⋊Q8—48no—
225C4×Q8—48no—
226#117—24no—
227#117C3×Q818yescomplemented
228C2×C4⋊C4C23×C311yescharacteristic
229C2×C4⋊C4C23×C311yescharacteristic
230C2×C4⋊C4C23×C314yes—
231C2×C4⋊C4C23×C314yes—
232C2×C4⋊C4C23×C314yes—
233C2×C4⋊C4C23×C314yes—
234C2×C4⋊C4C23×C314yes—
235C2×C4⋊C4C23×C314yes—
236C2×C4⋊C4C23×C318yes—
237C2×C4⋊C4C23×C318yes—
238C2×C4⋊C4C23×C318yes—
239C2×C4⋊C4C23×C312yes—
240C2×C22⋊C4C23×C311yescharacteristic
241C2×C22⋊C4C23×C311yescharacteristic
242C2×C22⋊C4C23×C318yes—
243C2×C22⋊C4C23×C312yes—
244C2×C42—24no—
245C2×C42C23×C311yescharacteristic
246C2×C42C23×C312yes—
247C2×C42C23×C312yes—
248C2×C42C23×C312yes—
249C2×C42C23×C312yes—
250C2×C42C23×C311yescharacteristic
251C2×C42C23×C312yes—
252C2×C42C23×C318yes—
253C2×C42C23×C311yescharacteristic
254C2×C42C23×C311yescharacteristic
255C2×C42C23×C311yescharacteristic
256C2×C42C23×C311yescharacteristic
257#95C23×C312yes—
258#95C23×C312yes—
259#95C23×C312yes—
260#95C23×C312yes—
261#95C23×C318yes—
262#95C23×C318yes—
263#95C23×C318yes—
264#95C23×C318yes—
265#95C23×C314yes—
266#95C23×C318yes—
267#95C23×C314yes—
268C23×C3—21no—
269C23×C3—21no—
270C23×C3—22no—
271C23×C3—22no—
272C23×C3C22×Q812yes—
273C23×C3C2511yescharacteristic
274C3×Q8—416no—
275C3×Q8—88no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—28no—
285C2×C3×C4—22no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—28no—
289C2×C3×C4—28no—
290C2×C3×C4—22no—
291C2×C3×C4—28no—
292C2×C3×C4—28no—
293C2×C3×C4—28no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—28no—
298C2×C3×C4—48no—
299C2×C3×C4C22×Q814yes—
300C2×C3×C4C22×Q814yes—
301C2×C3×C4C22×Q812yes—
302C2×C3×C4C22×Q812yes—
303C2×C3×C4C2×#4112yes—
304C2×C3×C4C2×#4112yes—
305C2×C3×C4C2×#4112yes—
306C2×C3×C4C2×#4112yes—
307C2×C3×C4C2×#4114yes—
308C2×C3×C4H(5,𝔽2)14yes—
309C2×C3×C42−1+414yes—
310C24C24×C311yescharacteristic
311C2×Q8—28no—
312C2×Q8—28no—
313C2×Q8—28no—
314C2×Q8—28no—
315C2×Q8—48no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4C2×C3×Q814yes—
322C22×C4C24×C311yescharacteristic
323C22×C4C24×C312yes—
324C22×C4C24×C318yes—
325C22×C4C24×C318yes—
326C22×C4C24×C311yescharacteristic
327C22×C4C24×C311yescharacteristic
328C22×C4C24×C312yes—
329C22×C4C24×C312yes—
330C22×C4C24×C311yescharacteristic
331C22×C4C24×C311yescharacteristic
332C22×C4C24×C311yescharacteristic
333C22×C4C24×C312yes—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—28no—
337C4⋊C4—28no—
338C4⋊C4—28no—
339C4⋊C4—28no—
340C4⋊C4—28no—
341C4⋊C4—28no—
342C4⋊C4—48no—
343C4⋊C4—48no—
344C4⋊C4—48no—
345C4⋊C4C2×C3×Q818yes—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—48no—
349C22⋊C4C2×C3×Q818yes—
350C42—24no—
351C42—22no—
352C42—22no—
353C42—24no—
354C42—24no—
355C42—24no—
356C42—24no—
357C42—48no—
358C42C2×C3×Q814yes—
359C42C2×C3×Q814yes—
360C42C3×#4114yes—
361C42C3×#4114yes—
362C22×C3—22no—
363C22×C3—24no—
364C22×C3—42no—
365C22×C3—41no—
366C22×C3—41no—
367C22×C3C23×Q811yescharacteristic
368C22×C3C22×#4111yescharacteristic
369C22×C3C22×#4111yescharacteristic
370C22×C3C2×H(5,𝔽2)12yes—
371C22×C3C2×2−1+412yes—
372C3×C4—22no—
373C3×C4—24no—
374C3×C4—24no—
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—42no—
378C3×C4—48no—
379C3×C4—41no—
380C3×C4—48no—
381C3×C4—41no—
382C3×C4#53314yes—
383C3×C4#54218yes—
384C23—21no—
385C23—21no—
386C23—22no—
387C23—22no—
388C23C22×C3×Q812yes—
389C23C25×C311yescharacteristic
390Q8—416no—
391Q8—88no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—28no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—28no—
405C2×C4—28no—
406C2×C4—22no—
407C2×C4—28no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—28no—
414C2×C4—48no—
415C2×C4C22×C3×Q814yes—
416C2×C4C22×C3×Q814yes—
417C2×C4C22×C3×Q812yes—
418C2×C4C22×C3×Q812yes—
419C2×C4C2×C3×#4112yes—
420C2×C4C2×C3×#4112yes—
421C2×C4C2×C3×#4112yes—
422C2×C4C2×C3×#4112yes—
423C2×C4C2×C3×#4114yes—
424C2×C4C3×H(5,𝔽2)14yes—
425C2×C4C3×2−1+414yes—
426C2×C3—42no—
427C2×C3C2×#53311yescharacteristic
428C2×C3C2×#54212yes—
429C2×C3#350312yes—
430C2×C3#350811yescharacteristic
431C2×C3#350911yescharacteristic
432C22—22no—
433C22—24no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22C23×C3×Q811yescharacteristic
438C22C22×C3×#4111yescharacteristic
439C22C22×C3×#4111yescharacteristic
440C22C2×C3×H(5,𝔽2)12yes—
441C22C2×C3×2−1+412yes—
442C4—22no—
443C4—24no—
444C4—24no—
445C4—22no—
446C4—22no—
447C4—42no—
448C4—48no—
449C4—48no—
450C4—41no—
451C4—41no—
452C4C3×#53314yes—
453C4C3×#54218yes—
454C3#3528311yescharacteristic, direct product factor, Sylow
455C2—42no—
456C2C2×C3×#53311yescharacteristic
457C2C2×C3×#54212yes—
458C2C3×#350312yes—
459C2C3×#350811yescharacteristic
460C2C3×#350911yescharacteristic
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×#2868C2semidirect
2. C3×#2862C2semidirect
4. C3×#2647C2semidirect
6. C3×#2511C2semidirect
8. C3×#2451C2semidirect
10. C3×#2331C2semidirect
12. C3×C4×C4⋊Q8C2semidirect
14. #35283C3direct (N × H)
58. #2868C2×C3semidirect
59. #2862C2×C3semidirect
61. #2647C2×C3semidirect
63. #2511C2×C3semidirect
65. #2451C2×C3semidirect
67. #2331C2×C3semidirect
69. C4×C4⋊Q8C2×C3semidirect
81. C3×#117Q8semidirect
227. #117C3×Q8semidirect
454. C3#35283direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#28682
C3×#28622
C3×#28572
C3×#26472
C3×#26282
C3×#25112
C3×#24922
C3×#24512
C3×#24232
C3×#23312
C3×#23142
C3×C4×C4⋊Q82
C3×C4×#1172
#352833