Finite Group Databasebeta

← C3×#35983 #10654590

Subgroups — C3×#35983 #10654590

Order 768.

2384 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#548C211yesmaximal, characteristic
2C2×C3×#520C211yesmaximal, characteristic, complemented
3C3×#2625C213yesmaximal, complemented
4C3×#2503C213yesmaximal, complemented
5C3×#2487C213yesmaximal, complemented
6C3×#2447C216yesmaximal, complemented
7C3×#2420C216yesmaximal, complemented
8C3×#2395C213yesmaximal, complemented
9C3×#2351C212yesmaximal, complemented
10C3×#2341C211yesmaximal, characteristic, complemented
11C3×Q8×C22⋊C4C212yesmaximal, complemented
12#35983C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×#41C2211yescharacteristic
14C23×C3×Q8C2211yescharacteristic
15C3×#548—28no—
16C2×C3×#124C2213yes—
17C2×C3×C22⋊Q8C2213yes—
18C2×C3×#121C2213yes—
19C3×#520—28no—
20C2×C3×C4×Q8C2212yes—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×Q8C2212yes—
23C2×C3×C4×D4C2216yes—
24C2×C3×C4×D4C2213yes—
25C2×C3×#117C2213yes—
26C22×C3×C4⋊C4C2213yes—
27C3×#400C2216yes—
28C3×#400C2213yes—
29C3×#399C2213yes—
30C3×#398C2213yes—
31C3×#397C22112yes—
32C3×#397C2211yescharacteristic
33C3×#397C2216yes—
34C3×#394C2216yes—
35C3×#391C2216yes—
36C3×#391C2216yes—
37C3×#390C2216yes—
38C3×#389C2216yes—
39C3×#389C2216yes—
40C3×#386C2216yes—
41C3×#385C2216yes—
42C3×#382C2216yes—
43C3×#381C22112yes—
44C3×#380C2213yes—
45C3×#380C2216yes—
46C3×C4×C4⋊C4C2216yes—
47C3×C4×C22⋊C4C2216yes—
48C3×C4×C22⋊C4C2213yes—
49C2×#548C2×C311yescharacteristic
50C2×#520C2×C311yescharacteristic, complemented
51#2625C2×C313yescomplemented
52#2503C2×C313yescomplemented
53#2487C2×C313yescomplemented
54#2447C2×C316yescomplemented
55#2420C2×C316yescomplemented
56#2395C2×C313yescomplemented
57#2351C2×C312yescomplemented
58#2341C2×C311yescharacteristic, complemented
59Q8×C22⋊C4C2×C312yescomplemented
60C2×C3×#41—28no—
61C2×C3×#41D418yes—
62C22×C3×Q8—24no—
63C22×C3×Q8—28no—
64C22×C3×Q8C2313yes—
65C22×C3×Q8C2311yescharacteristic
66C22×C3×D4C2313yes—
67C22×C3×D4C2313yes—
68C23×C3×C4C2313yes—
69C23×C3×C4C2313yes—
70C23×C3×C4C2316yes—
71C3×#124—212no—
72C3×C22⋊Q8—212no—
73C3×#121—212no—
74C3×C4×Q8—28no—
75C3×C4×Q8—24no—
76C3×C4×Q8—28no—
77C3×C4×D4—224no—
78C3×C4×D4—212no—
79C3×#117—212no—
80C2×C3×C4⋊C4—212no—
81C2×C3×C4⋊C4C2316yes—
82C2×C3×C4⋊C4C2316yes—
83C2×C3×C4⋊C4C2313yes—
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2313yes—
86C2×C3×C4⋊C4C2313yes—
87C2×C3×C4⋊C4C2316yes—
88C2×C3×C4⋊C4C2316yes—
89C2×C3×C22⋊C4C2312yes—
90C2×C3×C22⋊C4C2316yes—
91C2×C3×C22⋊C4C2313yes—
92C2×C3×C22⋊C4C2316yes—
93C2×C3×C22⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4C2316yes—
95C2×C3×C22⋊C4C2316yes—
96C2×C3×C22⋊C4C2316yes—
97C2×C3×C42C2316yes—
98C2×C3×C42C2316yes—
99C2×C3×C42C2313yes—
100C3×#95C2311yescharacteristic
101C3×#95C2313yes—
102C3×#95C2316yes—
103C3×#95C2316yes—
104C3×#95C2312yes—
105C3×#95C2316yes—
106C3×#95C23112yes—
107C3×#95C23112yes—
108C22×#41C22×C311yescharacteristic
109C23×Q8C22×C311yescharacteristic
110#548—28no—
111C2×#124C22×C313yes—
112C2×C22⋊Q8C22×C313yes—
113C2×#121C22×C313yes—
114#520—28no—
115C2×C4×Q8C22×C312yes—
116C2×C4×Q8C22×C312yes—
117C2×C4×Q8C22×C311yescharacteristic
118C2×C4×D4C22×C313yes—
119C2×C4×D4C22×C316yes—
120C2×#117C22×C313yes—
121C22×C4⋊C4C22×C313yes—
122#400C22×C316yes—
123#400C22×C313yes—
124#399C22×C313yes—
125#398C22×C313yes—
126#397C22×C3112yes—
127#397C22×C311yescharacteristic
128#397C22×C316yes—
129#394C22×C316yes—
130#391C22×C316yes—
131#391C22×C316yes—
132#390C22×C316yes—
133#389C22×C316yes—
134#389C22×C316yes—
135#386C22×C316yes—
136#385C22×C316yes—
137#382C22×C316yes—
138#381C22×C3112yes—
139#380C22×C316yes—
140#380C22×C313yes—
141C4×C4⋊C4C22×C316yes—
142C4×C22⋊C4C22×C316yes—
143C4×C22⋊C4C22×C313yes—
144C24×C3C2413yes—
145C24×C3C2411yescharacteristic
146C3×#41—232no—
147C2×C3×Q8—26no—
148C2×C3×Q8—212no—
149C2×C3×Q8—416no—
150C2×C3×Q8C2×D414yes—
151C2×C3×Q8#4118yes—
152C2×C3×D4—26no—
153C2×C3×D4—26no—
154C2×C3×D4—212no—
155C2×C3×D4—43no—
156C2×C3×D4C2×D4112yes—
157C22×C3×C4—212no—
158C22×C3×C4—212no—
159C22×C3×C4—26no—
160C22×C3×C4—212no—
161C22×C3×C4—412no—
162C22×C3×C4C2×D4112yes—
163C22×C3×C4C2412yes—
164C22×C3×C4C2413yes—
165C22×C3×C4C2416yes—
166C22×C3×C4C2413yes—
167C22×C3×C4C2416yes—
168C22×C3×C4C2412yes—
169C22×C3×C4C2411yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C22×C3×C4C2413yes—
172C3×C4⋊C4—26no—
173C3×C4⋊C4—212no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—26no—
176C3×C4⋊C4—26no—
177C3×C4⋊C4—212no—
178C3×C4⋊C4—46no—
179C3×C4⋊C4—46no—
180C3×C22⋊C4—212no—
181C3×C22⋊C4—26no—
182C3×C22⋊C4—212no—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—22no—
185C3×C22⋊C4—46no—
186C3×C22⋊C4—46no—
187C3×C22⋊C4—46no—
188C3×C42—212no—
189C3×C42—26no—
190C3×C42—212no—
191C2×#41—28no—
192C2×#41C3×D418yes—
193C22×Q8—24no—
194C22×Q8—28no—
195C22×Q8C23×C313yes—
196C22×Q8C23×C311yescharacteristic
197C22×D4C23×C313yes—
198C22×D4C23×C313yes—
199C23×C4C23×C313yes—
200C23×C4C23×C313yes—
201C23×C4C23×C316yes—
202#124—212no—
203C22⋊Q8—212no—
204#121—212no—
205C4×Q8—28no—
206C4×Q8—24no—
207C4×Q8—28no—
208C4×D4—224no—
209C4×D4—212no—
210#117—212no—
211C2×C4⋊C4—212no—
212C2×C4⋊C4C23×C313yes—
213C2×C4⋊C4C23×C316yes—
214C2×C4⋊C4C23×C311yescharacteristic
215C2×C4⋊C4C23×C313yes—
216C2×C4⋊C4C23×C313yes—
217C2×C4⋊C4C23×C316yes—
218C2×C4⋊C4C23×C316yes—
219C2×C4⋊C4C23×C316yes—
220C2×C22⋊C4C23×C316yes—
221C2×C22⋊C4C23×C311yescharacteristic
222C2×C22⋊C4C23×C313yes—
223C2×C22⋊C4C23×C316yes—
224C2×C22⋊C4C23×C312yes—
225C2×C22⋊C4C23×C316yes—
226C2×C22⋊C4C23×C316yes—
227C2×C22⋊C4C23×C316yes—
228C2×C42C23×C316yes—
229C2×C42C23×C313yes—
230C2×C42C23×C316yes—
231#95C23×C311yescharacteristic
232#95C23×C313yes—
233#95C23×C316yes—
234#95C23×C316yes—
235#95C23×C312yes—
236#95C23×C316yes—
237#95C23×C3112yes—
238#95C23×C3112yes—
239C23×C3—21no—
240C23×C3—22no—
241C23×C3—26no—
242C23×C3—23no—
243C23×C3—22no—
244C23×C3—23no—
245C23×C3—26no—
246C23×C3—21no—
247C23×C3C22×D416yes—
248C23×C32−1+412yes—
249C23×C3C2511yescharacteristic
250C3×Q8—412no—
251C3×Q8#123116yes—
252C3×D4—224no—
253C3×D4—412no—
254C2×C3×C4—22no—
255C2×C3×C4—224no—
256C2×C3×C4—21no—
257C2×C3×C4—22no—
258C2×C3×C4—23no—
259C2×C3×C4—26no—
260C2×C3×C4—23no—
261C2×C3×C4—22no—
262C2×C3×C4—26no—
263C2×C3×C4—26no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—23no—
267C2×C3×C4—22no—
268C2×C3×C4—26no—
269C2×C3×C4—26no—
270C2×C3×C4—26no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—23no—
274C2×C3×C4—26no—
275C2×C3×C4—412no—
276C2×C3×C4—812no—
277C2×C3×C4C22×D412yes—
278C2×C3×C4C22×D416yes—
279C2×C3×C4C2×#41112yes—
280C24C24×C313yes—
281C24C24×C311yescharacteristic
282#41—232no—
283C2×Q8—26no—
284C2×Q8—212no—
285C2×Q8—416no—
286C2×Q8C2×C3×D414yes—
287C2×Q8C3×#4118yes—
288C2×D4—26no—
289C2×D4—26no—
290C2×D4—212no—
291C2×D4—43no—
292C2×D4C2×C3×D4112yes—
293C22×C4—212no—
294C22×C4—212no—
295C22×C4—26no—
296C22×C4—212no—
297C22×C4—412no—
298C22×C4C2×C3×D4112yes—
299C22×C4C24×C312yes—
300C22×C4C24×C313yes—
301C22×C4C24×C316yes—
302C22×C4C24×C313yes—
303C22×C4C24×C316yes—
304C22×C4C24×C312yes—
305C22×C4C24×C311yescharacteristic
306C22×C4C24×C311yescharacteristic
307C22×C4C24×C313yes—
308C4⋊C4—26no—
309C4⋊C4—212no—
310C4⋊C4—22no—
311C4⋊C4—26no—
312C4⋊C4—26no—
313C4⋊C4—212no—
314C4⋊C4—46no—
315C4⋊C4—46no—
316C22⋊C4—212no—
317C22⋊C4—26no—
318C22⋊C4—212no—
319C22⋊C4—24no—
320C22⋊C4—22no—
321C22⋊C4—46no—
322C22⋊C4—46no—
323C22⋊C4—46no—
324C42—212no—
325C42—26no—
326C42—212no—
327C22×C3—26no—
328C22×C3—22no—
329C22×C3—26no—
330C22×C3—26no—
331C22×C3—24no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—46no—
335C22×C3—42no—
336C22×C3—46no—
337C22×C3C23×D411yescharacteristic
338C22×C3C22×#4112yes—
339C22×C3C2×2−1+411yescharacteristic
340C22×C3#58511yescharacteristic
341C22×C3#58512yes—
342C3×C4—22no—
343C3×C4—43no—
344C3×C4—42no—
345C3×C4—43no—
346C3×C4—41no—
347C3×C4—46no—
348C3×C4—42no—
349C3×C4—46no—
350C3×C4C2×#123112yes—
351C23—21no—
352C23—22no—
353C23—26no—
354C23—23no—
355C23—22no—
356C23—23no—
357C23—26no—
358C23—21no—
359C23C22×C3×D416yes—
360C23C3×2−1+412yes—
361C23C25×C311yescharacteristic
362Q8—412no—
363Q8C3×#123116yes—
364D4—224no—
365D4—412no—
366C2×C4—22no—
367C2×C4—224no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—23no—
371C2×C4—26no—
372C2×C4—23no—
373C2×C4—22no—
374C2×C4—26no—
375C2×C4—26no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—23no—
379C2×C4—22no—
380C2×C4—26no—
381C2×C4—26no—
382C2×C4—26no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—23no—
386C2×C4—26no—
387C2×C4—412no—
388C2×C4—812no—
389C2×C4C22×C3×D412yes—
390C2×C4C22×C3×D416yes—
391C2×C4C2×C3×#41112yes—
392C2×C3—46no—
393C2×C3—42no—
394C2×C3C22×#12311yescharacteristic
395C2×C3#348911yescharacteristic
396C2×C3#350011yescharacteristic
397C2×C3#351612yes—
398C2×C3#352412yes—
399C22—26no—
400C22—22no—
401C22—26no—
402C22—26no—
403C22—24no—
404C22—41no—
405C22—41no—
406C22—46no—
407C22—42no—
408C22—46no—
409C22C23×C3×D411yescharacteristic
410C22C22×C3×#4112yes—
411C22C2×C3×2−1+411yescharacteristic
412C22C3×#58511yescharacteristic
413C22C3×#58512yes—
414C4—22no—
415C4—43no—
416C4—43no—
417C4—42no—
418C4—46no—
419C4—41no—
420C4—42no—
421C4—46no—
422C4C2×C3×#123112yes—
423C3#3598311yescharacteristic, direct product factor, Sylow
424C2—46no—
425C2—42no—
426C2C22×C3×#12311yescharacteristic
427C2C3×#348911yescharacteristic
428C2C3×#350011yescharacteristic
429C2C3×#351612yes—
430C2C3×#352412yes—
subgroupG/H
center, socleC23×C3C25
commutator, Frattini, cosocleC23C25×C3
prime radicalC2×C3×#548C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C3 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#520C2semidirect
3. C3×#2625C2semidirect
4. C3×#2503C2semidirect
5. C3×#2487C2semidirect
6. C3×#2447C2semidirect
7. C3×#2420C2semidirect
8. C3×#2395C2semidirect
9. C3×#2351C2semidirect
10. C3×#2341C2semidirect
11. C3×Q8×C22⋊C4C2semidirect
12. #35983C3direct (N × H)
50. C2×#520C2×C3semidirect
51. #2625C2×C3semidirect
52. #2503C2×C3semidirect
53. #2487C2×C3semidirect
54. #2447C2×C3semidirect
55. #2420C2×C3semidirect
56. #2395C2×C3semidirect
57. #2351C2×C3semidirect
58. #2341C2×C3semidirect
59. Q8×C22⋊C4C2×C3semidirect
423. C3#35983direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5482
C2×C3×#5202
C3×#26252
C3×#25032
C3×#24872
C3×#24472
C3×#24202
C3×#23952
C3×#23512
C3×#23412
C3×Q8×C22⋊C42
#359833