Finite Group Databasebeta

← C3×#21861 #10640468

Subgroups — C3×#21861 #10640468

Order 768.

1166 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#529C211yesmaximal, characteristic
2C2×C3×#431C211yesmaximal, characteristic, complemented
3C2×C3×#382C211yesmaximal, characteristic, complemented
4C3×#1991C212yesmaximal, complemented
5C3×#1958C212yesmaximal, complemented
6C3×#1874C212yesmaximal
7C3×#1872C212yesmaximal, complemented
8C3×#1805C212yesmaximal, complemented
9C3×#1804C212yesmaximal, complemented
10#21861C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#529—24no—
12C3×#529C418yescomplemented
13C2×C3×#126—21no—
14C2×C3×#125C2211yescharacteristic
15C2×C3×#124C2211yescharacteristic
16C2×C3×#123—21no—
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#121C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4—21no—
21C2×C3×C4×D4C2211yescharacteristic
22C2×C3×#117—21no—
23C22×C3×C4⋊C4C2212yes—
24C22×C3×C42C2211yescharacteristic
25C3×#431—24no—
26C2×C3×C4⋊C8C2212yes—
27C2×C3×#103C2214yes—
28C2×C3×#102C2214yes—
29C2×C3×C22⋊C8C2212yes—
30C2×C3×C4×C8C2212yes—
31C3×#382—21no—
32C3×#382—21no—
33C3×#382C2212yes—
34C3×#382C2212yes—
35C2×C3×#95—21no—
36C2×C3×#95—21no—
37C3×#340C2214yes—
38C3×#340C2214yes—
39C2×#529C2×C311yescharacteristic
40C2×#431C2×C311yescharacteristic, complemented
41C2×#382C2×C311yescharacteristic, complemented
42#1991C2×C312yescomplemented
43#1958C2×C312yescomplemented
44#1874C2×C312yes—
45#1872C2×C312yescomplemented
46#1805C2×C312yescomplemented
47#1804C2×C312yescomplemented
48C22×C3×Q8C2311yescharacteristic
49C22×C3×D4—21no—
50C22×C3×D4C2311yescharacteristic
51C23×C3×C4—21no—
52C23×C3×C4—22no—
53C23×C3×C4D411yescharacteristic
54C23×C3×C4D411yescharacteristic
55C23×C3×C4C2311yescharacteristic
56C22×C3×C8C2314yes—
57C3×#126—28no—
58C3×#125—22no—
59C3×#125C2×C414yescomplemented
60C3×#124—22no—
61C3×#124C2×C414yescomplemented
62C3×#123—28no—
63C3×C22⋊Q8—22no—
64C3×C22⋊Q8C2×C414yes—
65C3×#121—22no—
66C3×#121C2×C414yes—
67C3×C4×Q8—22no—
68C3×C4×Q8C2×C414yescomplemented
69C3×C4×D4—28no—
70C3×C4×D4—22no—
71C3×C4×D4C2×C414yescomplemented
72C3×#117—28no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—44no—
79C2×C3×C4⋊C4C2314yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42—21no—
88C2×C3×C42C2×C414yes—
89C2×C3×C42D411yescharacteristic
90C2×C3×C42D411yescharacteristic
91C2×C3×C42D414yes—
92C2×C3×C42C2311yescharacteristic
93C2×C3×C42C2311yescharacteristic
94C3×C4⋊C8—24no—
95C3×#103—44no—
96C3×#102—44no—
97C3×C22⋊C8—24no—
98C3×C4×C8—24no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95—22no—
103#529—24no—
104#529C3×C418yescomplemented
105C2×#126—21no—
106C2×#125C22×C311yescharacteristic
107C2×#124C22×C311yescharacteristic
108C2×#123—21no—
109C2×C22⋊Q8C22×C311yescharacteristic
110C2×#121C22×C311yescharacteristic
111C2×C4×Q8C22×C311yescharacteristic
112C2×C4×D4—21no—
113C2×C4×D4C22×C311yescharacteristic
114C2×#117—21no—
115C22×C4⋊C4C22×C312yes—
116C22×C42C22×C311yescharacteristic
117#431—24no—
118C2×C4⋊C8C22×C312yes—
119C2×#103C22×C314yes—
120C2×#102C22×C314yes—
121C2×C22⋊C8C22×C312yes—
122C2×C4×C8C22×C312yes—
123#382—21no—
124#382—21no—
125#382C22×C312yes—
126#382C22×C312yes—
127C2×#95—21no—
128C2×#95—21no—
129#340C22×C314yes—
130#340C22×C314yes—
131C24×C3—21no—
132C24×C3C2×D411yescharacteristic
133C2×C3×Q8—21no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8C22×C412yes—
137C2×C3×D4—24no—
138C2×C3×D4—21no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—44no—
142C2×C3×D4C22×C412yes—
143C22×C3×C4—24no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—24no—
150C22×C3×C4—24no—
151C22×C3×C4—42no—
152C22×C3×C4—44no—
153C22×C3×C4—44no—
154C22×C3×C4—44no—
155C22×C3×C4—44no—
156C22×C3×C4C22⋊C414yes—
157C22×C3×C4C22⋊C414yes—
158C22×C3×C4C22×C412yes—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2411yescharacteristic
166C2×C3×C8—24no—
167C2×C3×C8—24no—
168C2×C3×C8—24no—
169C3×C4⋊C4—24no—
170C3×C4⋊C4—24no—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—24no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—24no—
177C3×C4⋊C4—44no—
178C3×C4⋊C4C22×C412yes—
179C3×C4⋊C4C22×C412yes—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—24no—
182C3×C22⋊C4—24no—
183C3×C22⋊C4—24no—
184C3×C22⋊C4—24no—
185C3×C42—24no—
186C3×C42C22⋊C414yes—
187C3×C42C22⋊C414yes—
188C3×C42C22×C412yes—
189C3×C42C22×C412yes—
190C3×C42C2×D412yes—
191C3×C42C2×D412yes—
192C22×Q8C23×C311yescharacteristic
193C22×D4—21no—
194C22×D4C23×C311yescharacteristic
195C23×C4—21no—
196C23×C4—22no—
197C23×C4C3×D411yescharacteristic
198C23×C4C3×D411yescharacteristic
199C23×C4C23×C311yescharacteristic
200C22×C8C23×C314yes—
201#126—28no—
202#125—22no—
203#125C2×C3×C414yescomplemented
204#124—22no—
205#124C2×C3×C414yescomplemented
206#123—28no—
207C22⋊Q8—22no—
208C22⋊Q8C2×C3×C414yes—
209#121—22no—
210#121C2×C3×C414yes—
211C4×Q8—22no—
212C4×Q8C2×C3×C414yescomplemented
213C4×D4—28no—
214C4×D4—22no—
215C4×D4C2×C3×C414yescomplemented
216#117—28no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4C23×C314yes—
224C2×C4⋊C4C23×C311yescharacteristic
225C2×C4⋊C4C23×C311yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C42—21no—
232C2×C42C2×C3×C414yes—
233C2×C42C3×D411yescharacteristic
234C2×C42C3×D411yescharacteristic
235C2×C42C3×D414yes—
236C2×C42C23×C311yescharacteristic
237C2×C42C23×C311yescharacteristic
238C4⋊C8—24no—
239#103—44no—
240#102—44no—
241C22⋊C8—24no—
242C4×C8—24no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247C23×C3—21no—
248C23×C3—21no—
249C23×C3—22no—
250C23×C3—22no—
251C23×C3—41no—
252C23×C3—41no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3C2×C22⋊C412yes—
256C23×C3#13514yes—
257C23×C3C22×D411yescharacteristic
258C3×Q8—24no—
259C3×Q8—42no—
260C3×D4—24no—
261C3×D4—48no—
262C3×D4—42no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—28no—
270C2×C3×C4—24no—
271C2×C3×C4—24no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—44no—
276C2×C3×C4—48no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—44no—
282C2×C3×C4—44no—
283C2×C3×C4—44no—
284C2×C3×C4—88no—
285C2×C3×C4C2×C22⋊C412yes—
286C2×C3×C4C2×C22⋊C412yes—
287C2×C3×C4C2×C22⋊C412yes—
288C2×C3×C4C2×C22⋊C412yes—
289C2×C3×C4C2×C22⋊C412yes—
290C2×C3×C4C23×C411yescharacteristic
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4H(5,𝔽2)12yes—
293C2×C3×C42−1+412yes—
294C3×C8—44no—
295C24—21no—
296C24C2×C3×D411yescharacteristic
297C2×Q8—21no—
298C2×Q8—22no—
299C2×Q8—22no—
300C2×Q8C22×C3×C412yes—
301C2×D4—24no—
302C2×D4—21no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—44no—
306C2×D4C22×C3×C412yes—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—42no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4C3×C22⋊C414yes—
321C22×C4C3×C22⋊C414yes—
322C22×C4C22×C3×C412yes—
323C22×C4C2×C3×D411yescharacteristic
324C22×C4C2×C3×D411yescharacteristic
325C22×C4C2×C3×D411yescharacteristic
326C22×C4C2×C3×D411yescharacteristic
327C22×C4C2×C3×D411yescharacteristic
328C22×C4C2×C3×D412yes—
329C22×C4C24×C311yescharacteristic
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—44no—
342C4⋊C4C22×C3×C412yes—
343C4⋊C4C22×C3×C412yes—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C42—24no—
350C42C3×C22⋊C414yes—
351C42C3×C22⋊C414yes—
352C42C22×C3×C412yes—
353C42C22×C3×C412yes—
354C42C2×C3×D412yes—
355C42C2×C3×D412yes—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—24no—
359C22×C3—24no—
360C22×C3—42no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3—82no—
364C22×C3—82no—
365C22×C3#41614yes—
366C22×C3C22×C22⋊C411yescharacteristic
367C22×C3#53511yescharacteristic
368C22×C3#53611yescharacteristic
369C22×C3C2×#13512yes—
370C22×C3#57512yes—
371C3×C4—22no—
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—22no—
375C3×C4—22no—
376C3×C4—42no—
377C3×C4—42no—
378C3×C4—42no—
379C3×C4—84no—
380C3×C4#52012yes—
381C3×C4#53712yes—
382C23—22no—
383C23—21no—
384C23—21no—
385C23—22no—
386C23—42no—
387C23—42no—
388C23—41no—
389C23—41no—
390C23C2×C3×C22⋊C412yes—
391C23C3×#13514yes—
392C23C22×C3×D411yescharacteristic
393Q8—24no—
394Q8—42no—
395D4—24no—
396D4—48no—
397D4—42no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—28no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—44no—
411C2×C4—48no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—88no—
420C2×C4C2×C3×C22⋊C412yes—
421C2×C4C2×C3×C22⋊C412yes—
422C2×C4C2×C3×C22⋊C412yes—
423C2×C4C2×C3×C22⋊C412yes—
424C2×C4C2×C3×C22⋊C412yes—
425C2×C4C23×C3×C411yescharacteristic
426C2×C4C22×C3×D411yescharacteristic
427C2×C4C3×H(5,𝔽2)12yes—
428C2×C4C3×2−1+412yes—
429C8—44no—
430C2×C3—24no—
431C2×C3—82no—
432C2×C3#231111yescharacteristic
433C2×C3C2×#41611yescharacteristic
434C2×C3#289611yescharacteristic
435C2×C3#321112yes—
436C2×C3#323212yes—
437C22—22no—
438C22—24no—
439C22—24no—
440C22—22no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—82no—
445C22—82no—
446C22C3×#41614yes—
447C22C22×C3×C22⋊C411yescharacteristic
448C22C3×#53511yescharacteristic
449C22C3×#53611yescharacteristic
450C22C2×C3×#13512yes—
451C22C3×#57512yes—
452C4—22no—
453C4—22no—
454C4—22no—
455C4—22no—
456C4—22no—
457C4—42no—
458C4—42no—
459C4—42no—
460C4—84no—
461C4C3×#52012yes—
462C4C3×#53712yes—
463C3#2186111yescharacteristic, direct product factor, Sylow
464C2—24no—
465C2—82no—
466C2C3×#231111yescharacteristic
467C2C2×C3×#41611yescharacteristic
468C2C3×#289611yescharacteristic
469C2C3×#321112yes—
470C2C3×#323212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutatorC2×C4C23×C3×C4
Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C22×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#431C2semidirect
3. C2×C3×#382C2semidirect
4. C3×#1991C2semidirect
5. C3×#1958C2semidirect
7. C3×#1872C2semidirect
8. C3×#1805C2semidirect
9. C3×#1804C2semidirect
10. #21861C3direct (N × H)
12. C3×#529C4semidirect
40. C2×#431C2×C3semidirect
41. C2×#382C2×C3semidirect
42. #1991C2×C3semidirect
43. #1958C2×C3semidirect
45. #1872C2×C3semidirect
46. #1805C2×C3semidirect
47. #1804C2×C3semidirect
59. C3×#125C2×C4semidirect
61. C3×#124C2×C4semidirect
68. C3×C4×Q8C2×C4semidirect
71. C3×C4×D4C2×C4semidirect
104. #529C3×C4semidirect
203. #125C2×C3×C4semidirect
205. #124C2×C3×C4semidirect
212. C4×Q8C2×C3×C4semidirect
215. C4×D4C2×C3×C4semidirect
463. C3#21861direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C2×C3×#4312
C2×C3×#3822
C3×#19912
C3×#19582
C3×#18742
C3×#18722
C3×#18052
C3×#18042
#218613