Finite Group Databasebeta

← C3×#14654 #10633261

Subgroups — C3×#14654 #10633261

Order 768.

1030 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#400C211yesmaximal, characteristic
2C3×#1445C211yesmaximal, characteristic
3C3×#1442C212yesmaximal
4C3×#1442C212yesmaximal
5C3×#1442C212yesmaximal
6C3×#1442C212yesmaximal
7C3×#1441C212yesmaximal
8C3×#1436C211yesmaximal, characteristic
9C3×#1436C212yesmaximal
10#14654C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×C4⋊C4C2212yes—
12C22×C3×C4⋊C4C2212yes—
13C22×C3×C4⋊C4C2211yescharacteristic
14C22×C3×C4⋊C4C2211yescharacteristic
15C22×C3×C42C2211yescharacteristic
16C3×#400C418yescomplemented
17C2×C3×#95C2212yes—
18C2×C3×#95C2212yes—
19C2×C3×#95C2212yes—
20C2×C3×#95C2212yes—
21C2×C3×#95C2212yes—
22C2×C3×#95C2211yescharacteristic
23C2×C3×#95C2212yes—
24C2×C3×#95C2212yes—
25C2×C3×#95C2212yes—
26C2×C3×#95C2212yes—
27C2×C3×#95C2212yes—
28C2×C3×#95C2211yescharacteristic
29C2×C3×#95C2211yescharacteristic
30C2×C3×#95C2212yes—
31C2×C3×#95C2211yescharacteristic
32C2×C3×#95C2212yes—
33C2×#400C2×C311yescharacteristic
34#1445C2×C311yescharacteristic
35#1442C2×C312yes—
36#1442C2×C312yes—
37#1442C2×C312yes—
38#1442C2×C312yes—
39#1441C2×C312yes—
40#1436C2×C312yes—
41#1436C2×C311yescharacteristic
42C23×C3×C4C2312yes—
43C23×C3×C4C2312yes—
44C23×C3×C4C2312yes—
45C23×C3×C4C2311yescharacteristic
46C23×C3×C4C2312yes—
47C23×C3×C4C2311yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C23×C3×C4C2312yes—
50C23×C3×C4C2312yes—
51C2×C3×C4⋊C4—24no—
52C2×C3×C4⋊C4—41no—
53C2×C3×C4⋊C4—41no—
54C2×C3×C4⋊C4—41no—
55C2×C3×C4⋊C4—41no—
56C2×C3×C4⋊C4—42no—
57C2×C3×C4⋊C4—42no—
58C2×C3×C4⋊C4—42no—
59C2×C3×C4⋊C4—42no—
60C2×C3×C4⋊C4—42no—
61C2×C3×C4⋊C4C2×C414yes—
62C2×C3×C4⋊C4C2×C414yes—
63C2×C3×C42—42no—
64C2×C3×C42—41no—
65C3×#95—24no—
66C3×#95—24no—
67C3×#95—42no—
68C3×#95—42no—
69C3×#95—42no—
70C3×#95—42no—
71C3×#95—42no—
72C3×#95—42no—
73C3×#95—41no—
74C3×#95—41no—
75C3×#95—42no—
76C3×#95—42no—
77C3×#95—41no—
78C3×#95C2×C414yes—
79C3×#95C2×C418yes—
80C3×#95C2×C418yes—
81C22×C4⋊C4C22×C312yes—
82C22×C4⋊C4C22×C312yes—
83C22×C4⋊C4C22×C311yescharacteristic
84C22×C4⋊C4C22×C311yescharacteristic
85C22×C42C22×C311yescharacteristic
86#400C3×C418yescomplemented
87C2×#95C22×C312yes—
88C2×#95C22×C312yes—
89C2×#95C22×C312yes—
90C2×#95C22×C312yes—
91C2×#95C22×C312yes—
92C2×#95C22×C311yescharacteristic
93C2×#95C22×C312yes—
94C2×#95C22×C312yes—
95C2×#95C22×C312yes—
96C2×#95C22×C312yes—
97C2×#95C22×C312yes—
98C2×#95C22×C311yescharacteristic
99C2×#95C22×C311yescharacteristic
100C2×#95C22×C312yes—
101C2×#95C22×C311yescharacteristic
102C2×#95C22×C312yes—
103C24×C3C2411yescharacteristic
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—22no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—22no—
114C22×C3×C4—21no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—22no—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4C22×C414yes—
156C22×C3×C4C22×C412yes—
157C22×C3×C4C22×C412yes—
158C22×C3×C4C22×C414yes—
159C22×C3×C4C22×C412yes—
160C22×C3×C4#4114yes—
161C3×C4⋊C4—44no—
162C3×C4⋊C4—44no—
163C3×C4⋊C4—48no—
164C3×C4⋊C4—48no—
165C3×C42—44no—
166C23×C4C23×C312yes—
167C23×C4C23×C312yes—
168C23×C4C23×C312yes—
169C23×C4C23×C311yescharacteristic
170C23×C4C23×C312yes—
171C23×C4C23×C311yescharacteristic
172C23×C4C23×C312yes—
173C23×C4C23×C311yescharacteristic
174C23×C4C23×C312yes—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4C2×C3×C414yes—
186C2×C4⋊C4C2×C3×C414yes—
187C2×C42—42no—
188C2×C42—41no—
189#95—24no—
190#95—24no—
191#95—42no—
192#95—42no—
193#95—42no—
194#95—42no—
195#95—42no—
196#95—42no—
197#95—41no—
198#95—41no—
199#95—42no—
200#95—42no—
201#95—41no—
202#95C2×C3×C414yes—
203#95C2×C3×C418yes—
204#95C2×C3×C418yes—
205C23×C3C23×C411yescharacteristic
206C23×C3C2×#4112yes—
207C23×C3H(5,𝔽2)11yescharacteristic
208C23×C3H(5,𝔽2)11yescharacteristic
209C23×C3H(5,𝔽2)12yes—
210C23×C3H(5,𝔽2)12yes—
211C23×C32−1+412yes—
212C23×C32−1+412yes—
213C23×C32−1+412yes—
214C2×C3×C4—22no—
215C2×C3×C4—24no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—22no—
219C2×C3×C4—24no—
220C2×C3×C4—22no—
221C2×C3×C4—24no—
222C2×C3×C4—24no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—24no—
226C2×C3×C4—24no—
227C2×C3×C4—24no—
228C2×C3×C4—24no—
229C2×C3×C4—24no—
230C2×C3×C4—42no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C24C24×C311yescharacteristic
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4C22×C3×C414yes—
324C22×C4C22×C3×C412yes—
325C22×C4C22×C3×C414yes—
326C22×C4C22×C3×C412yes—
327C22×C4C22×C3×C412yes—
328C22×C4C3×#4114yes—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—48no—
332C4⋊C4—48no—
333C42—44no—
334C22×C3C4×#4111yescharacteristic
335C22×C3#51811yescharacteristic
336C22×C3#51911yescharacteristic
337C22×C3#52012yes—
338C22×C3#52012yes—
339C22×C3#53812yes—
340C22×C3#53912yes—
341C22×C3#53912yes—
342C22×C3#54112yes—
343C22×C3#54212yes—
344C22×C3#54212yes—
345C22×C3#56011yescharacteristic
346C22×C3H(3,𝔽4)11yescharacteristic
347C22×C3#56212yes—
348C22×C3#56212yes—
349C22×C3#56212yes—
350C22×C3#56312yes—
351C22×C3#56312yes—
352C22×C3#56311yescharacteristic
353C22×C3#56311yescharacteristic
354C22×C3#56412yes—
355C3×C4—44no—
356C3×C4—42no—
357C3×C4—44no—
358C3×C4—42no—
359C3×C4—42no—
360C3×C4—44no—
361C3×C4—82no—
362C3×C4—82no—
363C3×C4—82no—
364C23C23×C3×C411yescharacteristic
365C23C2×C3×#4112yes—
366C23C3×H(5,𝔽2)11yescharacteristic
367C23C3×H(5,𝔽2)12yes—
368C23C3×H(5,𝔽2)11yescharacteristic
369C23C3×H(5,𝔽2)12yes—
370C23C3×2−1+412yes—
371C23C3×2−1+412yes—
372C23C3×2−1+412yes—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C3#233211yescharacteristic
431C2×C3#233412yes—
432C2×C3#237611yescharacteristic
433C2×C3#237911yescharacteristic
434C2×C3#238212yes—
435C2×C3#282712yes—
436C2×C3#283212yes—
437C2×C3#283912yes—
438C2×C3#283912yes—
439C22C3×C4×#4111yescharacteristic
440C22C3×#51811yescharacteristic
441C22C3×#51911yescharacteristic
442C22C3×#52012yes—
443C22C3×#52012yes—
444C22C3×#53812yes—
445C22C3×#53912yes—
446C22C3×#53912yes—
447C22C3×#54112yes—
448C22C3×#54212yes—
449C22C3×#54212yes—
450C22C3×#56011yescharacteristic
451C22C3×H(3,𝔽4)11yescharacteristic
452C22C3×#56212yes—
453C22C3×#56212yes—
454C22C3×#56212yes—
455C22C3×#56311yescharacteristic
456C22C3×#56311yescharacteristic
457C22C3×#56312yes—
458C22C3×#56312yes—
459C22C3×#56412yes—
460C4—42no—
461C4—42no—
462C4—44no—
463C4—44no—
464C4—42no—
465C4—44no—
466C4—82no—
467C4—82no—
468C4—82no—
469C3#1465411yescharacteristic, direct product factor, Sylow
470C2C3×#233211yescharacteristic
471C2C3×#233412yes—
472C2C3×#237611yescharacteristic
473C2C3×#237911yescharacteristic
474C2C3×#238212yes—
475C2C3×#282712yes—
476C2C3×#283212yes—
477C2C3×#283912yes—
478C2C3×#283912yes—
subgroupG/H
center, socle, prime radicalC24×C3C24
commutatorC23C23×C3×C4
Frattini, cosocleC24C24×C3
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
10. #14654C3direct (N × H)
16. C3×#400C4semidirect
86. #400C3×C4semidirect
469. C3#14654direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4002
C3×#14452
C3×#14422
C3×#14412
C3×#14362
#146543