Finite Group Databasebeta

← C3×#35748 #10654355

Subgroups — C3×#35748 #10654355

Order 768.

2434 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q82C211yesmaximal, characteristic, complemented
2C2×C3×#520C211yesmaximal, characteristic
3C3×#2628C213yesmaximal, complemented
4C3×#2580C213yesmaximal, complemented
5C3×#2542C213yesmaximal, complemented
6C3×#2452C216yesmaximal, complemented
7C3×#2440C216yesmaximal, complemented
8C3×#2396C213yesmaximal, complemented
9C3×#2351C211yesmaximal, characteristic, complemented
10C3×Q8×C4⋊C4C212yesmaximal, complemented
11C3×#2341C212yesmaximal
12#35748C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C3×Q82—28no—
15C2×C3×C4⋊Q8C2213yes—
16C2×C3×C4⋊Q8C2216yes—
17C2×C3×C22⋊Q8C22112yes—
18C3×#520—28no—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2213yes—
21C2×C3×C4×Q8C2212yes—
22C2×C3×C4×Q8C2212yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2213yes—
25C2×C3×C4×D4C2213yes—
26C2×C3×#117C2213yes—
27C22×C3×C4⋊C4C2213yes—
28C3×#400C2213yes—
29C3×#399C2216yes—
30C3×#399C2213yes—
31C3×#398C2216yes—
32C3×#396C2216yes—
33C3×#395C2216yes—
34C3×#395C2211yescharacteristic
35C3×#393C2216yes—
36C3×#393C2213yes—
37C3×#390C2216yes—
38C3×#390C2216yes—
39C3×#389C2216yes—
40C3×#389C2216yes—
41C3×#389C2216yes—
42C3×#385C22112yes—
43C3×#384C2216yes—
44C3×#382C2213yes—
45C3×#380C2216yes—
46C3×C4×C4⋊C4C2216yes—
47C3×C4×C4⋊C4C2213yes—
48C3×C4×C22⋊C4C2216yes—
49C2×Q82C2×C311yescharacteristic, complemented
50C2×#520C2×C311yescharacteristic
51#2628C2×C313yescomplemented
52#2580C2×C313yescomplemented
53#2542C2×C313yescomplemented
54#2452C2×C316yescomplemented
55#2440C2×C316yescomplemented
56#2396C2×C313yescomplemented
57#2351C2×C311yescharacteristic, complemented
58Q8×C4⋊C4C2×C312yescomplemented
59#2341C2×C312yes—
60C2×C3×#41—24no—
61C2×C3×#41—24no—
62C2×C3×#41Q818yescomplemented
63C22×C3×Q8C2311yescharacteristic
64C22×C3×Q8C2316yes—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×D4C2313yes—
67C23×C3×C4C2313yes—
68C23×C3×C4C2316yes—
69C3×C4⋊Q8—224no—
70C3×C4⋊Q8—212no—
71C3×C22⋊Q8—424no—
72C3×C4×Q8—28no—
73C3×C4×Q8—24no—
74C3×C4×Q8—28no—
75C3×C4×Q8—212no—
76C3×C4×Q8D418yes—
77C3×C4×D4—212no—
78C3×C4×D4—212no—
79C3×#117—212no—
80C2×C3×C4⋊C4—212no—
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2316yes—
83C2×C3×C4⋊C4C2316yes—
84C2×C3×C4⋊C4C2316yes—
85C2×C3×C4⋊C4C2316yes—
86C2×C3×C4⋊C4C2313yes—
87C2×C3×C4⋊C4C2316yes—
88C2×C3×C4⋊C4C2313yes—
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2313yes—
91C2×C3×C4⋊C4C23112yes—
92C2×C3×C4⋊C4C2313yes—
93C2×C3×C22⋊C4C2316yes—
94C2×C3×C22⋊C4C2316yes—
95C2×C3×C22⋊C4C2313yes—
96C2×C3×C22⋊C4C2313yes—
97C2×C3×C22⋊C4C23112yes—
98C2×C3×C42C2316yes—
99C2×C3×C42C2316yes—
100C2×C3×C42C2313yes—
101C2×C3×C42C2313yes—
102C3×#95C2312yes—
103C3×#95C2316yes—
104C3×#95C2311yescharacteristic
105C3×#95C2313yes—
106C3×#95C2316yes—
107C3×#95C2316yes—
108C3×#95C2316yes—
109C22×#41C22×C311yescharacteristic
110Q82—28no—
111C2×C4⋊Q8C22×C313yes—
112C2×C4⋊Q8C22×C316yes—
113C2×C22⋊Q8C22×C3112yes—
114#520—28no—
115C2×C4×Q8C22×C313yes—
116C2×C4×Q8C22×C312yes—
117C2×C4×Q8C22×C312yes—
118C2×C4×Q8C22×C311yescharacteristic
119C2×C4×Q8C22×C311yescharacteristic
120C2×C4×D4C22×C313yes—
121C2×C4×D4C22×C313yes—
122C2×#117C22×C313yes—
123C22×C4⋊C4C22×C313yes—
124#400C22×C313yes—
125#399C22×C316yes—
126#399C22×C313yes—
127#398C22×C316yes—
128#396C22×C316yes—
129#395C22×C316yes—
130#395C22×C311yescharacteristic
131#393C22×C313yes—
132#393C22×C316yes—
133#390C22×C316yes—
134#390C22×C316yes—
135#389C22×C316yes—
136#389C22×C316yes—
137#389C22×C316yes—
138#385C22×C3112yes—
139#384C22×C316yes—
140#382C22×C313yes—
141#380C22×C316yes—
142C4×C4⋊C4C22×C316yes—
143C4×C4⋊C4C22×C313yes—
144C4×C22⋊C4C22×C316yes—
145C24×C3C2413yes—
146C3×#41—232no—
147C2×C3×Q8—22no—
148C2×C3×Q8—212no—
149C2×C3×Q8—24no—
150C2×C3×Q8—412no—
151C2×C3×Q8C2×D414yes—
152C2×C3×Q8C2×Q814yes—
153C2×C3×Q8#4114yes—
154C2×C3×D4—26no—
155C2×C3×D4—26no—
156C2×C3×D4C2×Q8112yes—
157C22×C3×C4—26no—
158C22×C3×C4—26no—
159C22×C3×C4—212no—
160C22×C3×C4—412no—
161C22×C3×C4C2×Q8112yes—
162C22×C3×C4C2411yescharacteristic
163C22×C3×C4C2416yes—
164C22×C3×C4C2416yes—
165C22×C3×C4C2413yes—
166C22×C3×C4C2411yescharacteristic
167C22×C3×C4C2413yes—
168C22×C3×C4C2411yescharacteristic
169C22×C3×C4C2412yes—
170C22×C3×C4C2412yes—
171C22×C3×C4C2413yes—
172C3×C4⋊C4—26no—
173C3×C4⋊C4—26no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—212no—
176C3×C4⋊C4—26no—
177C3×C4⋊C4—212no—
178C3×C4⋊C4—212no—
179C3×C4⋊C4—212no—
180C3×C4⋊C4—212no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—412no—
183C3×C4⋊C4C2×D4112yes—
184C3×C22⋊C4—26no—
185C3×C22⋊C4—26no—
186C3×C22⋊C4—212no—
187C3×C22⋊C4—212no—
188C3×C22⋊C4—412no—
189C3×C42—26no—
190C3×C42—212no—
191C3×C42—212no—
192C3×C42C2×D4112yes—
193C2×#41—24no—
194C2×#41—24no—
195C2×#41C3×Q818yescomplemented
196C22×Q8C23×C311yescharacteristic
197C22×Q8C23×C316yes—
198C22×Q8C23×C311yescharacteristic
199C22×D4C23×C313yes—
200C23×C4C23×C313yes—
201C23×C4C23×C316yes—
202C4⋊Q8—224no—
203C4⋊Q8—212no—
204C22⋊Q8—424no—
205C4×Q8—24no—
206C4×Q8—28no—
207C4×Q8—28no—
208C4×Q8—212no—
209C4×Q8C3×D418yes—
210C4×D4—212no—
211C4×D4—212no—
212#117—212no—
213C2×C4⋊C4—212no—
214C2×C4⋊C4C23×C313yes—
215C2×C4⋊C4C23×C313yes—
216C2×C4⋊C4C23×C311yescharacteristic
217C2×C4⋊C4C23×C316yes—
218C2×C4⋊C4C23×C316yes—
219C2×C4⋊C4C23×C313yes—
220C2×C4⋊C4C23×C316yes—
221C2×C4⋊C4C23×C316yes—
222C2×C4⋊C4C23×C316yes—
223C2×C4⋊C4C23×C312yes—
224C2×C4⋊C4C23×C3112yes—
225C2×C4⋊C4C23×C313yes—
226C2×C22⋊C4C23×C313yes—
227C2×C22⋊C4C23×C313yes—
228C2×C22⋊C4C23×C316yes—
229C2×C22⋊C4C23×C316yes—
230C2×C22⋊C4C23×C3112yes—
231C2×C42C23×C313yes—
232C2×C42C23×C316yes—
233C2×C42C23×C316yes—
234C2×C42C23×C313yes—
235#95C23×C312yes—
236#95C23×C316yes—
237#95C23×C311yescharacteristic
238#95C23×C313yes—
239#95C23×C316yes—
240#95C23×C316yes—
241#95C23×C316yes—
242C23×C3—23no—
243C23×C3—23no—
244C23×C3—26no—
245C23×C3—26no—
246C23×C3C22×Q816yes—
247C23×C3C2511yescharacteristic
248C3×Q8—28no—
249C3×Q8—812no—
250C3×Q8C22⋊Q8116yescomplemented
251C3×D4—224no—
252C2×C3×C4—22no—
253C2×C3×C4—21no—
254C2×C3×C4—22no—
255C2×C3×C4—21no—
256C2×C3×C4—26no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—224no—
261C2×C3×C4—21no—
262C2×C3×C4—23no—
263C2×C3×C4—26no—
264C2×C3×C4—26no—
265C2×C3×C4—26no—
266C2×C3×C4—26no—
267C2×C3×C4—26no—
268C2×C3×C4—26no—
269C2×C3×C4—26no—
270C2×C3×C4—22no—
271C2×C3×C4—812no—
272C2×C3×C4C22×D412yes—
273C2×C3×C4C22×D416yes—
274C2×C3×C4C22×D416yes—
275C2×C3×C4C22×Q812yes—
276C2×C3×C4C22×Q816yes—
277C2×C3×C4C2×#4116yes—
278C2×C3×C4H(5,𝔽2)14yes—
279C2×C3×C42−1+414yes—
280C24C24×C313yes—
281#41—232no—
282C2×Q8—22no—
283C2×Q8—212no—
284C2×Q8—24no—
285C2×Q8—412no—
286C2×Q8C2×C3×D414yes—
287C2×Q8C2×C3×Q814yes—
288C2×Q8C3×#4114yes—
289C2×D4—26no—
290C2×D4—26no—
291C2×D4C2×C3×Q8112yes—
292C22×C4—26no—
293C22×C4—26no—
294C22×C4—212no—
295C22×C4—412no—
296C22×C4C2×C3×Q8112yes—
297C22×C4C24×C311yescharacteristic
298C22×C4C24×C316yes—
299C22×C4C24×C316yes—
300C22×C4C24×C313yes—
301C22×C4C24×C311yescharacteristic
302C22×C4C24×C313yes—
303C22×C4C24×C311yescharacteristic
304C22×C4C24×C312yes—
305C22×C4C24×C312yes—
306C22×C4C24×C313yes—
307C4⋊C4—26no—
308C4⋊C4—26no—
309C4⋊C4—22no—
310C4⋊C4—212no—
311C4⋊C4—26no—
312C4⋊C4—212no—
313C4⋊C4—212no—
314C4⋊C4—212no—
315C4⋊C4—212no—
316C4⋊C4—24no—
317C4⋊C4—412no—
318C4⋊C4C2×C3×D4112yes—
319C22⋊C4—26no—
320C22⋊C4—26no—
321C22⋊C4—212no—
322C22⋊C4—212no—
323C22⋊C4—412no—
324C42—26no—
325C42—212no—
326C42—212no—
327C42C2×C3×D4112yes—
328C22×C3—26no—
329C22×C3—212no—
330C22×C3—43no—
331C22×C3—43no—
332C22×C3—46no—
333C22×C3C23×D411yescharacteristic
334C22×C3C23×Q811yescharacteristic
335C22×C3C22×#4111yescharacteristic
336C22×C3C2×H(5,𝔽2)12yes—
337C22×C3C2×2−1+412yes—
338C3×C4—22no—
339C3×C4—22no—
340C3×C4—24no—
341C3×C4—26no—
342C3×C4—24no—
343C3×C4—41no—
344C3×C4—43no—
345C3×C4—46no—
346C3×C4—46no—
347C3×C4C2×C22⋊Q8112yes—
348C23—23no—
349C23—23no—
350C23—26no—
351C23—26no—
352C23C22×C3×Q816yes—
353C23C25×C311yescharacteristic
354Q8—28no—
355Q8—812no—
356Q8C3×C22⋊Q8116yescomplemented
357D4—224no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—26no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—224no—
367C2×C4—21no—
368C2×C4—23no—
369C2×C4—26no—
370C2×C4—26no—
371C2×C4—26no—
372C2×C4—26no—
373C2×C4—26no—
374C2×C4—26no—
375C2×C4—26no—
376C2×C4—22no—
377C2×C4—812no—
378C2×C4C22×C3×D412yes—
379C2×C4C22×C3×D416yes—
380C2×C4C22×C3×D416yes—
381C2×C4C22×C3×Q812yes—
382C2×C4C22×C3×Q816yes—
383C2×C4C2×C3×#4116yes—
384C2×C4C3×H(5,𝔽2)14yes—
385C2×C4C3×2−1+414yes—
386C2×C3—46no—
387C2×C3C22×C22⋊Q811yescharacteristic
388C2×C3#348611yescharacteristic
389C2×C3#349811yescharacteristic
390C2×C3#350312yes—
391C2×C3#352212yes—
392C22—26no—
393C22—212no—
394C22—43no—
395C22—43no—
396C22—46no—
397C22C23×C3×D411yescharacteristic
398C22C23×C3×Q811yescharacteristic
399C22C22×C3×#4111yescharacteristic
400C22C2×C3×H(5,𝔽2)12yes—
401C22C2×C3×2−1+412yes—
402C4—22no—
403C4—22no—
404C4—24no—
405C4—26no—
406C4—24no—
407C4—41no—
408C4—43no—
409C4—46no—
410C4—46no—
411C4C2×C3×C22⋊Q8112yes—
412C3#3574811yescharacteristic, direct product factor, Sylow
413C2—46no—
414C2C22×C3×C22⋊Q811yescharacteristic
415C2C3×#348611yescharacteristic
416C2C3×#349811yescharacteristic
417C2C3×#350312yes—
418C2C3×#352212yes—
subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
prime radicalC22×C3×#41C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q82C2semidirect
3. C3×#2628C2semidirect
4. C3×#2580C2semidirect
5. C3×#2542C2semidirect
6. C3×#2452C2semidirect
7. C3×#2440C2semidirect
8. C3×#2396C2semidirect
9. C3×#2351C2semidirect
10. C3×Q8×C4⋊C4C2semidirect
12. #35748C3direct (N × H)
49. C2×Q82C2×C3semidirect
51. #2628C2×C3semidirect
52. #2580C2×C3semidirect
53. #2542C2×C3semidirect
54. #2452C2×C3semidirect
55. #2440C2×C3semidirect
56. #2396C2×C3semidirect
57. #2351C2×C3semidirect
58. Q8×C4⋊C4C2×C3semidirect
62. C2×C3×#41Q8semidirect
195. C2×#41C3×Q8semidirect
250. C3×Q8C22⋊Q8semidirect
356. Q8C3×C22⋊Q8semidirect
412. C3#35748direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q822
C2×C3×#5202
C3×#26282
C3×#25802
C3×#25422
C3×#24522
C3×#24402
C3×#23962
C3×#23512
C3×Q8×C4⋊C42
C3×#23412
#357483