Finite Group Databasebeta

← C3×#24692 #10643299

Subgroups — C3×#24692 #10643299

Order 768.

884 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×#476C211yesmaximal, characteristic, complemented
2C2×C3×#442C211yesmaximal, characteristic
3C3×#2542C211yesmaximal, characteristic, complemented
4C3×#2357C211yesmaximal, characteristic, complemented
5C3×#2071C212yesmaximal, complemented
6C3×#2054C212yesmaximal
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1948C212yesmaximal, complemented
9C3×#1940C211yesmaximal, characteristic, complemented
10C3×#1926C212yesmaximal, complemented
11C3×#1804C211yesmaximal, characteristic, complemented
12#24692C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C22⋊Q8—22no—
16C2×C3×C4×D4C2211yescharacteristic
17C2×C3×C4×D4C2211yescharacteristic
18C3×#476—24no—
19C3×#442—24no—
20C2×C3×#106C2211yescharacteristic
21C2×C3×#106C2212yes—
22C2×C3×C4⋊C8C2212yes—
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#102C2212yes—
25C2×C3×#102C2212yes—
26C2×C3×#102C2211yescharacteristic
27C2×C3×#102C2211yescharacteristic
28C2×C3×#97C2211yescharacteristic
29C3×#399—21no—
30C3×#399—21no—
31C3×#390C2212yes—
32C3×#390C2212yes—
33C3×#388—22no—
34C3×#384C2212yes—
35C3×#382C2211yescharacteristic
36C3×#381—21no—
37C3×C4×C4⋊C4C2212yes—
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×C4×C22⋊C4—21no—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#476C2×C311yescharacteristic, complemented
45C2×#442C2×C311yescharacteristic
46#2542C2×C311yescharacteristic, complemented
47#2357C2×C311yescharacteristic, complemented
48#2071C2×C312yescomplemented
49#2054C2×C312yes—
50#1993C2×C311yescharacteristic, complemented
51#1948C2×C312yescomplemented
52#1940C2×C311yescharacteristic, complemented
53#1926C2×C312yescomplemented
54#1804C2×C311yescharacteristic, complemented
55C22×C3×Q8—21no—
56C22×C3×D4C2311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4—21no—
60C2×C3×D8—22no—
61C2×C3×D8—22no—
62C2×C3×D8—22no—
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2312yes—
65C22×C3×C8C2311yescharacteristic
66C3×C4⋊Q8—22no—
67C3×C4⋊Q8—22no—
68C3×C22⋊Q8—48no—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C3×C4×D4—22no—
72C3×C4×D4D414yes—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4D412yes—
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2312yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C42—22no—
88C2×C3×C42C2312yes—
89C2×C3×C42C2311yescharacteristic
90C3×#106—22no—
91C3×#106—24no—
92C3×C4⋊C8—24no—
93C3×C4⋊C8—22no—
94C3×#102—22no—
95C3×#102—24no—
96C3×#102—24no—
97C3×#102—22no—
98C3×#97—22no—
99C3×#95—21no—
100C3×#95—22no—
101C3×#95—21no—
102C3×#95D411yescharacteristic
103C3×#95D411yescharacteristic
104C22×D8C22×C311yescharacteristic
105C2×C4⋊Q8C22×C311yescharacteristic
106C2×C22⋊Q8—22no—
107C2×C4×D4C22×C311yescharacteristic
108C2×C4×D4C22×C311yescharacteristic
109#476—24no—
110#442—24no—
111C2×#106C22×C311yescharacteristic
112C2×#106C22×C312yes—
113C2×C4⋊C8C22×C312yes—
114C2×C4⋊C8C22×C311yescharacteristic
115C2×#102C22×C312yes—
116C2×#102C22×C312yes—
117C2×#102C22×C311yescharacteristic
118C2×#102C22×C311yescharacteristic
119C2×#97C22×C311yescharacteristic
120#399—21no—
121#399—21no—
122#390C22×C312yes—
123#390C22×C312yes—
124#388—22no—
125#384C22×C312yes—
126#382C22×C311yescharacteristic
127#381—21no—
128C4×C4⋊C4C22×C312yes—
129C4×C4⋊C4C22×C311yescharacteristic
130C4×C22⋊C4—21no—
131#340C22×C312yes—
132#340C22×C312yes—
133#340C22×C312yes—
134#336C22×C312yes—
135C24×C3—21no—
136C24×C3—21no—
137C2×C3×Q8—44no—
138C2×C3×Q8—42no—
139C2×C3×D4—21no—
140C2×C3×D4—21no—
141C2×C3×D4—21no—
142C2×C3×D4—21no—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—21no—
148C2×C3×D4—21no—
149C2×C3×D4C2×D412yes—
150C2×C3×D4#4112yes—
151C22×C3×C4—21no—
152C22×C3×C4—24no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—44no—
157C22×C3×C4—42no—
158C22×C3×C4—82no—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2411yescharacteristic
164C3×D8—28no—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C2×C3×C8—22no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8#4112yes—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—44no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4—44no—
185C3×C4⋊C4C2×D412yes—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—44no—
188C3×C22⋊C4—42no—
189C3×C22⋊C4—44no—
190C3×C22⋊C4—44no—
191C3×C22⋊C4—82no—
192C3×C42—22no—
193C3×C42—22no—
194C3×C42—82no—
195C3×C42C2×D412yes—
196C3×C42#4112yes—
197C22×Q8—21no—
198C22×D4C23×C311yescharacteristic
199C22×D4C23×C311yescharacteristic
200C23×C4—21no—
201C23×C4—21no—
202C2×D8—22no—
203C2×D8—22no—
204C2×D8—22no—
205C22×C8C23×C311yescharacteristic
206C22×C8C23×C312yes—
207C22×C8C23×C311yescharacteristic
208C4⋊Q8—22no—
209C4⋊Q8—22no—
210C22⋊Q8—48no—
211C4×D4—22no—
212C4×D4—22no—
213C4×D4—22no—
214C4×D4C3×D414yes—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C3×D412yes—
219C2×C4⋊C4C23×C311yescharacteristic
220C2×C4⋊C4C23×C312yes—
221C2×C4⋊C4C23×C312yes—
222C2×C4⋊C4C23×C311yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C42—22no—
230C2×C42C23×C312yes—
231C2×C42C23×C311yescharacteristic
232#106—22no—
233#106—24no—
234C4⋊C8—24no—
235C4⋊C8—22no—
236#102—22no—
237#102—24no—
238#102—24no—
239#102—22no—
240#97—22no—
241#95—21no—
242#95—22no—
243#95—21no—
244#95C3×D411yescharacteristic
245#95C3×D411yescharacteristic
246C23×C3—22no—
247C23×C3—22no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3C22×D411yescharacteristic
257C3×Q8—84no—
258C3×D4—22no—
259C3×D4—22no—
260C3×D4—22no—
261C3×D4—22no—
262C3×D4—41no—
263C3×D4—41no—
264C3×D4—41no—
265C3×D4—41no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—42no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—84no—
284C2×C3×C4—84no—
285C2×C3×C4#12112yes—
286C2×C3×C4#12112yes—
287C2×C3×C4AGL(1,ℤ/8)14yes—
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C4C2×#4111yescharacteristic
291C2×C3×C4C2×#4111yescharacteristic
292C2×C3×C42−1+411yescharacteristic
293C2×C3×C42−1+411yescharacteristic
294C3×C8—22no—
295C3×C8—42no—
296C3×C8—41no—
297C24—21no—
298C24—21no—
299C2×Q8—44no—
300C2×Q8—42no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4C2×C3×D412yes—
312C2×D4C3×#4112yes—
313C22×C4—21no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—44no—
319C22×C4—42no—
320C22×C4—82no—
321C22×C4C2×C3×D412yes—
322C22×C4C2×C3×D411yescharacteristic
323C22×C4C2×C3×D411yescharacteristic
324C22×C4C2×C3×D412yes—
325C22×C4C24×C311yescharacteristic
326D8—28no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8C3×#4112yes—
336C4⋊C4—24no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C4⋊C4C2×C3×D412yes—
348C22⋊C4—24no—
349C22⋊C4—44no—
350C22⋊C4—42no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—82no—
354C42—22no—
355C42—22no—
356C42—82no—
357C42C2×C3×D412yes—
358C42C3×#4112yes—
359C22×C3—44no—
360C22×C3—42no—
361C22×C3—42no—
362C22×C3—42no—
363C22×C3—42no—
364C22×C3—82no—
365C22×C3—82no—
366C22×C3—82no—
367C22×C3—81no—
368C22×C3—81no—
369C22×C3C2×#12111yescharacteristic
370C22×C3#54711yescharacteristic
371C22×C3#54811yescharacteristic
372C22×C3C2×AGL(1,ℤ/8)12yes—
373C22×C3#57712yes—
374C3×C4—24no—
375C3×C4—22no—
376C3×C4—22no—
377C3×C4—42no—
378C3×C4—42no—
379C3×C4—82no—
380C3×C4—81no—
381C3×C4—82no—
382C3×C4#52511yescharacteristic
383C3×C4#53611yescharacteristic
384C3×C4#55211yescharacteristic
385C3×C4#55611yescharacteristic
386C23—22no—
387C23—22no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23C22×C3×D411yescharacteristic
397Q8—84no—
398D4—22no—
399D4—22no—
400D4—22no—
401D4—22no—
402D4—41no—
403D4—41no—
404D4—41no—
405D4—41no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—84no—
424C2×C4—84no—
425C2×C4C3×#12112yes—
426C2×C4C3×#12112yes—
427C2×C4C3×AGL(1,ℤ/8)14yes—
428C2×C4C22×C3×D411yescharacteristic
429C2×C4C2×C3×#4111yescharacteristic
430C2×C4C2×C3×#4111yescharacteristic
431C2×C4C2×C3×#4111yescharacteristic
432C2×C4C3×2−1+411yescharacteristic
433C2×C4C3×2−1+411yescharacteristic
434C8—22no—
435C8—42no—
436C8—41no—
437C2×C3—82no—
438C2×C3—82no—
439C2×C3#241511yescharacteristic
440C2×C3#303811yescharacteristic
441C2×C3#304211yescharacteristic
442C2×C3#334512yes—
443C2×C3#337312yes—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—44no—
449C22—81no—
450C22—81no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22C2×C3×#12111yescharacteristic
455C22C3×#54711yescharacteristic
456C22C3×#54811yescharacteristic
457C22C2×C3×AGL(1,ℤ/8)12yes—
458C22C3×#57712yes—
459C4—22no—
460C4—22no—
461C4—24no—
462C4—42no—
463C4—42no—
464C4—81no—
465C4—82no—
466C4—82no—
467C4C3×#52511yescharacteristic
468C4C3×#53611yescharacteristic
469C4C3×#55211yescharacteristic
470C4C3×#55611yescharacteristic
471C3#2469211yescharacteristic, direct product factor, Sylow
472C2—82no—
473C2—82no—
474C2C3×#241511yescharacteristic
475C2C3×#303811yescharacteristic
476C2C3×#304211yescharacteristic
477C2C3×#334512yes—
478C2C3×#337312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#476C2semidirect
3. C3×#2542C2semidirect
4. C3×#2357C2semidirect
5. C3×#2071C2semidirect
7. C3×#1993C2semidirect
8. C3×#1948C2semidirect
9. C3×#1940C2semidirect
10. C3×#1926C2semidirect
11. C3×#1804C2semidirect
12. #24692C3direct (N × H)
44. C2×#476C2×C3semidirect
46. #2542C2×C3semidirect
47. #2357C2×C3semidirect
48. #2071C2×C3semidirect
50. #1993C2×C3semidirect
51. #1948C2×C3semidirect
52. #1940C2×C3semidirect
53. #1926C2×C3semidirect
54. #1804C2×C3semidirect
471. C3#24692direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4762
C2×C3×#4422
C3×#25422
C3×#23572
C3×#20712
C3×#20542
C3×#19932
C3×#19482
C3×#19402
C3×#19262
C3×#18042
#246923