Finite Group Databasebeta

← C3×#25683 #10644290

Subgroups — C3×#25683 #10644290

Order 768.

960 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#541C211yesmaximal, characteristic
2C2×C3×#499C211yesmaximal, characteristic, complemented
3C2×C3×#491C211yesmaximal, characteristic, complemented
4C3×#2629C211yesmaximal, characteristic
5C3×#2036C212yesmaximal
6C3×#2035C212yesmaximal
7C3×#1992C211yesmaximal, characteristic, complemented
8C3×#1956C211yesmaximal, characteristic, complemented
9C3×#1865C212yesmaximal, complemented
10C3×#1863C212yesmaximal, complemented
11C3×#1750C211yesmaximal, characteristic
12#25683C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#541—24no—
14C3×#541—24no—
15C2×C3×C4⋊Q8C2211yescharacteristic
16C2×C3×#126—22no—
17C2×C3×#125—22no—
18C2×C3×#125C2211yescharacteristic
19C2×C3×#125C2211yescharacteristic
20C2×C3×C22⋊Q8C2212yes—
21C2×C3×C4×Q8C2212yes—
22C2×C3×#117C2211yescharacteristic
23C3×#499—24no—
24C3×#491—24no—
25C2×C3×#107C2212yes—
26C2×C3×#106C2212yes—
27C2×C3×#103C2212yes—
28C2×C3×#103C2212yes—
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×C22⋊C8C2211yescharacteristic
32C2×C3×C22⋊C8C2211yescharacteristic
33C2×C3×#97C2211yescharacteristic
34C2×C3×C4×C8C2211yescharacteristic
35C3×#398—22no—
36C3×#396—22no—
37C3×#389C2212yes—
38C3×#384C2211yescharacteristic
39C3×#380C2212yes—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#336C2211yescharacteristic
43C3×#336C2211yescharacteristic
44C2×#541C2×C311yescharacteristic
45C2×#499C2×C311yescharacteristic, complemented
46C2×#491C2×C311yescharacteristic, complemented
47#2629C2×C311yescharacteristic
48#2036C2×C312yes—
49#2035C2×C312yes—
50#1992C2×C311yescharacteristic, complemented
51#1956C2×C311yescharacteristic, complemented
52#1865C2×C312yescomplemented
53#1863C2×C312yescomplemented
54#1750C2×C311yescharacteristic
55C22×C3×Q8C2312yes—
56C23×C3×C4C2311yescharacteristic
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×C4⋊Q8—22no—
62C3×C4⋊Q8D414yes—
63C3×#126—216no—
64C3×#125—216no—
65C3×#125—22no—
66C3×#125—22no—
67C3×#125—22no—
68C3×#125D414yes—
69C3×C22⋊Q8—24no—
70C3×C22⋊Q8—24no—
71C3×C4×Q8—24no—
72C3×C4×Q8—24no—
73C3×#117—22no—
74C3×#117—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—22no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4D412yes—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C22⋊C4—22no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4D412yes—
89C2×C3×C42—22no—
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C3×#107—24no—
93C3×#106—24no—
94C3×#103—24no—
95C3×#103—24no—
96C3×#103—24no—
97C3×#103—24no—
98C3×C22⋊C8—22no—
99C3×C22⋊C8—22no—
100C3×#97—22no—
101C3×C4×C8—22no—
102C3×#95—22no—
103C3×#95—22no—
104#541—24no—
105#541—24no—
106C2×C4⋊Q8C22×C311yescharacteristic
107C2×#126—22no—
108C2×#125—22no—
109C2×#125C22×C311yescharacteristic
110C2×#125C22×C311yescharacteristic
111C2×C22⋊Q8C22×C312yes—
112C2×C4×Q8C22×C312yes—
113C2×#117C22×C311yescharacteristic
114#499—24no—
115#491—24no—
116C2×#107C22×C312yes—
117C2×#106C22×C312yes—
118C2×#103C22×C312yes—
119C2×#103C22×C312yes—
120C2×#103C22×C312yes—
121C2×#103C22×C312yes—
122C2×C22⋊C8C22×C311yescharacteristic
123C2×C22⋊C8C22×C311yescharacteristic
124C2×#97C22×C311yescharacteristic
125C2×C4×C8C22×C311yescharacteristic
126#398—22no—
127#396—22no—
128#389C22×C312yes—
129#384C22×C311yescharacteristic
130#380C22×C312yes—
131#340C22×C312yes—
132#340C22×C312yes—
133#336C22×C311yescharacteristic
134#336C22×C311yescharacteristic
135C24×C3C2×D411yescharacteristic
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8C2×D414yes—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4C2×D411yescharacteristic
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2411yescharacteristic
156C2×C3×C8—21no—
157C2×C3×C8—21no—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8—21no—
162C2×C3×C8—21no—
163C2×C3×C8—21no—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8#4112yes—
167C2×C3×C8#4112yes—
168C3×C4⋊C4—28no—
169C3×C4⋊C4—28no—
170C3×C4⋊C4—28no—
171C3×C4⋊C4—28no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—44no—
179C3×C4⋊C4—82no—
180C3×C4⋊C4—82no—
181C3×C4⋊C4C2×D414yes—
182C3×C22⋊C4—28no—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—82no—
185C3×C42—28no—
186C3×C42—21no—
187C3×C42—21no—
188C3×C42C2×D412yes—
189C3×C42C2×D412yes—
190C22×Q8C23×C312yes—
191C23×C4C23×C311yescharacteristic
192C22×C8C23×C311yescharacteristic
193C22×C8C23×C311yescharacteristic
194C22×C8C23×C311yescharacteristic
195C22×C8C23×C311yescharacteristic
196C4⋊Q8—22no—
197C4⋊Q8C3×D414yes—
198#126—216no—
199#125—216no—
200#125—22no—
201#125—22no—
202#125—22no—
203#125C3×D414yes—
204C22⋊Q8—24no—
205C22⋊Q8—24no—
206C4×Q8—24no—
207C4×Q8—24no—
208#117—22no—
209#117—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4C3×D412yes—
218C2×C4⋊C4C23×C312yes—
219C2×C4⋊C4C23×C312yes—
220C2×C4⋊C4C23×C312yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4C3×D412yes—
224C2×C42—22no—
225C2×C42C23×C311yescharacteristic
226C2×C42C23×C311yescharacteristic
227#107—24no—
228#106—24no—
229#103—24no—
230#103—24no—
231#103—24no—
232#103—24no—
233C22⋊C8—22no—
234C22⋊C8—22no—
235#97—22no—
236C4×C8—22no—
237#95—22no—
238#95—22no—
239C23×C3—21no—
240C23×C3—21no—
241C23×C3—21no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3C22×D411yescharacteristic
245C3×Q8—24no—
246C3×Q8—24no—
247C3×Q8—42no—
248C3×Q8—42no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—24no—
253C2×C3×C4—24no—
254C2×C3×C4—21no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4#12014yes—
274C2×C3×C4C22×D411yescharacteristic
275C2×C3×C4C22×D411yescharacteristic
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C42−1+411yescharacteristic
279C2×C3×C42−1+411yescharacteristic
280C3×C8—22no—
281C3×C8—22no—
282C3×C8—41no—
283C3×C8—41no—
284C24C2×C3×D411yescharacteristic
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×Q8C2×C3×D414yes—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4C2×C3×D411yescharacteristic
302C22×C4C2×C3×D412yes—
303C22×C4C2×C3×D412yes—
304C22×C4C24×C311yescharacteristic
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8C3×#4112yes—
316C2×C8C3×#4112yes—
317C4⋊C4—28no—
318C4⋊C4—28no—
319C4⋊C4—28no—
320C4⋊C4—28no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—44no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C4⋊C4C2×C3×D414yes—
331C22⋊C4—28no—
332C22⋊C4—24no—
333C22⋊C4—82no—
334C42—28no—
335C42—21no—
336C42—21no—
337C42C2×C3×D412yes—
338C42C2×C3×D412yes—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3C2×#12011yescharacteristic
345C22×C3#53211yescharacteristic
346C22×C3#53611yescharacteristic
347C22×C3#57712yes—
348C22×C3#57812yes—
349C3×C4—24no—
350C3×C4—41no—
351C3×C4—44no—
352C3×C4—42no—
353C3×C4—44no—
354C3×C4—82no—
355C3×C4—82no—
356C3×C4#54711yescharacteristic
357C3×C4#54711yescharacteristic
358C3×C4#54811yescharacteristic
359C3×C4#54811yescharacteristic
360C23—22no—
361C23—22no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23C22×C3×D411yescharacteristic
366Q8—24no—
367Q8—24no—
368Q8—42no—
369Q8—42no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4C3×#12014yes—
395C2×C4C22×C3×D411yescharacteristic
396C2×C4C22×C3×D411yescharacteristic
397C2×C4C2×C3×#4111yescharacteristic
398C2×C4C2×C3×#4111yescharacteristic
399C2×C4C3×2−1+411yescharacteristic
400C2×C4C3×2−1+411yescharacteristic
401C8—22no—
402C8—22no—
403C8—41no—
404C8—41no—
405C2×C3—81no—
406C2×C3#243211yescharacteristic
407C2×C3#301312yes—
408C2×C3#317411yescharacteristic
409C2×C3#317711yescharacteristic
410C2×C3#324212yes—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—42no—
415C22—42no—
416C22C2×C3×#12011yescharacteristic
417C22C3×#53211yescharacteristic
418C22C3×#53611yescharacteristic
419C22C3×#57712yes—
420C22C3×#57812yes—
421C4—24no—
422C4—41no—
423C4—44no—
424C4—42no—
425C4—44no—
426C4—82no—
427C4—82no—
428C4C3×#54711yescharacteristic
429C4C3×#54711yescharacteristic
430C4C3×#54811yescharacteristic
431C4C3×#54811yescharacteristic
432C3#2568311yescharacteristic, direct product factor, Sylow
433C2—81no—
434C2C3×#243211yescharacteristic
435C2C3×#301312yes—
436C2C3×#317411yescharacteristic
437C2C3×#317711yescharacteristic
438C2C3×#324212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#499C2semidirect
3. C2×C3×#491C2semidirect
7. C3×#1992C2semidirect
8. C3×#1956C2semidirect
9. C3×#1865C2semidirect
10. C3×#1863C2semidirect
12. #25683C3direct (N × H)
45. C2×#499C2×C3semidirect
46. C2×#491C2×C3semidirect
50. #1992C2×C3semidirect
51. #1956C2×C3semidirect
52. #1865C2×C3semidirect
53. #1863C2×C3semidirect
432. C3#25683direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5412
C2×C3×#4992
C2×C3×#4912
C3×#26292
C3×#20362
C3×#20352
C3×#19922
C3×#19562
C3×#18652
C3×#18632
C3×#17502
#256833