Finite Group Databasebeta

← C3×#24695 #10643302

Subgroups — C3×#24695 #10643302

Order 768.

780 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#475C211yesmaximal, characteristic, complemented
2C2×C3×#440C211yesmaximal, characteristic
3C3×#2563C211yesmaximal, characteristic
4C3×#2356C211yesmaximal, characteristic, complemented
5C3×#2074C212yesmaximal, complemented
6C3×#2055C212yesmaximal
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1949C212yesmaximal, complemented
9C3×#1940C211yesmaximal, characteristic, complemented
10C3×#1926C212yesmaximal
11C3×#1804C211yesmaximal, characteristic
12#24695C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C3×#475—24no—
18C3×#440—24no—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#106C2212yes—
21C2×C3×C4⋊C8C2212yes—
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#103C2212yes—
24C2×C3×#103C2211yescharacteristic
25C2×C3×#102C2212yes—
26C2×C3×#102C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C3×#393—21no—
29C3×#393—21no—
30C3×#391C2212yes—
31C3×#390C2212yes—
32C3×#387—22no—
33C3×#383—21no—
34C3×#382C2211yescharacteristic
35C3×#382C2212yes—
36C3×#380—22no—
37C3×C4×C4⋊C4C2212yes—
38C3×C4×C4⋊C4C2211yescharacteristic
39C3×#376—21no—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#336C2212yes—
44C2×#475C2×C311yescharacteristic, complemented
45C2×#440C2×C311yescharacteristic
46#2563C2×C311yescharacteristic
47#2356C2×C311yescharacteristic, complemented
48#2074C2×C312yescomplemented
49#2055C2×C312yes—
50#1993C2×C311yescharacteristic, complemented
51#1949C2×C312yescomplemented
52#1940C2×C311yescharacteristic, complemented
53#1926C2×C312yes—
54#1804C2×C311yescharacteristic
55C22×C3×Q8—21no—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×D4C2311yescharacteristic
58C23×C3×C4—21no—
59C2×C3×SD16—22no—
60C2×C3×SD16—22no—
61C2×C3×SD16—22no—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2312yes—
64C22×C3×C8C2311yescharacteristic
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8—22no—
67C3×C4×Q8—22no—
68C3×C4×Q8D414yes—
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4D412yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C22⋊C4—21no—
79C2×C3×C22⋊C4—22no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C42—21no—
82C2×C3×C42—22no—
83C2×C3×C42C2312yes—
84C2×C3×C42C2311yescharacteristic
85C3×#107—22no—
86C3×#106—24no—
87C3×C4⋊C8—24no—
88C3×C4⋊C8—22no—
89C3×#103—22no—
90C3×#103—24no—
91C3×#102—24no—
92C3×#102—22no—
93C3×#97—22no—
94C3×#95—21no—
95C3×#95—22no—
96C3×#95—22no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—21no—
100C3×#95—22no—
101C3×#95D411yescharacteristic
102C3×#95D411yescharacteristic
103C22×SD16C22×C311yescharacteristic
104C2×C4⋊Q8C22×C311yescharacteristic
105C2×C4×Q8C22×C311yescharacteristic
106C2×C4×D4C22×C311yescharacteristic
107#475—24no—
108#440—24no—
109C2×#107C22×C311yescharacteristic
110C2×#106C22×C312yes—
111C2×C4⋊C8C22×C311yescharacteristic
112C2×C4⋊C8C22×C312yes—
113C2×#103C22×C312yes—
114C2×#103C22×C311yescharacteristic
115C2×#102C22×C312yes—
116C2×#102C22×C311yescharacteristic
117C2×#97C22×C311yescharacteristic
118#393—21no—
119#393—21no—
120#391C22×C312yes—
121#390C22×C312yes—
122#387—22no—
123#383—21no—
124#382C22×C312yes—
125#382C22×C311yescharacteristic
126#380—22no—
127C4×C4⋊C4C22×C312yes—
128C4×C4⋊C4C22×C311yescharacteristic
129#376—21no—
130#340C22×C312yes—
131#340C22×C312yes—
132#340C22×C312yes—
133#336C22×C312yes—
134C24×C3—21no—
135C2×C3×Q8—21no—
136C2×C3×Q8—21no—
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×Q8—21no—
140C2×C3×Q8—42no—
141C2×C3×Q8—82no—
142C2×C3×Q8C2×D412yes—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—21no—
148C2×C3×D4#4112yes—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—22no—
154C22×C3×C4—44no—
155C22×C3×C4—42no—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2411yescharacteristic
161C3×SD16—28no—
162C2×C3×C8—21no—
163C2×C3×C8—21no—
164C2×C3×C8—21no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—21no—
169C2×C3×C8—21no—
170C2×C3×C8#4112yes—
171C3×C4⋊C4—24no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—22no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4C2×D412yes—
182C3×C22⋊C4—42no—
183C3×C22⋊C4—82no—
184C3×C22⋊C4—82no—
185C3×C42—24no—
186C3×C42—22no—
187C3×C42—22no—
188C3×C42—44no—
189C3×C42C2×D412yes—
190C3×C42#4112yes—
191C22×Q8—21no—
192C22×Q8C23×C311yescharacteristic
193C22×D4C23×C311yescharacteristic
194C23×C4—21no—
195C2×SD16—22no—
196C2×SD16—22no—
197C2×SD16—22no—
198C22×C8C23×C311yescharacteristic
199C22×C8C23×C312yes—
200C22×C8C23×C311yescharacteristic
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203C4×Q8—22no—
204C4×Q8C3×D414yes—
205C4×D4—22no—
206C4×D4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4C3×D412yes—
210C2×C4⋊C4C23×C311yescharacteristic
211C2×C4⋊C4C23×C312yes—
212C2×C4⋊C4C23×C312yes—
213C2×C4⋊C4C23×C311yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42C23×C312yes—
220C2×C42C23×C311yescharacteristic
221#107—22no—
222#106—24no—
223C4⋊C8—24no—
224C4⋊C8—22no—
225#103—22no—
226#103—24no—
227#102—24no—
228#102—22no—
229#97—22no—
230#95—21no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—21no—
236#95—22no—
237#95C3×D411yescharacteristic
238#95C3×D411yescharacteristic
239C23×C3—22no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3C22×D411yescharacteristic
245C3×Q8—22no—
246C3×Q8—22no—
247C3×Q8—41no—
248C3×Q8—41no—
249C3×Q8—84no—
250C3×D4—22no—
251C3×D4—22no—
252C3×D4—41no—
253C3×D4—41no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—84no—
276C2×C3×C4#12112yes—
277C2×C3×C4#12112yes—
278C2×C3×C4C22×D411yescharacteristic
279C2×C3×C4C2×#4111yescharacteristic
280C2×C3×C4C2×#4111yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C42−1+411yescharacteristic
283C2×C3×C42−1+411yescharacteristic
284C3×C8—22no—
285C3×C8—42no—
286C3×C8—41no—
287C24—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—42no—
294C2×Q8—82no—
295C2×Q8C2×C3×D412yes—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4C3×#4112yes—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—44no—
308C22×C4—42no—
309C22×C4C2×C3×D412yes—
310C22×C4C2×C3×D412yes—
311C22×C4C2×C3×D411yescharacteristic
312C22×C4C2×C3×D411yescharacteristic
313C22×C4C24×C311yescharacteristic
314SD16—28no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8C3×#4112yes—
324C4⋊C4—24no—
325C4⋊C4—21no—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4C2×C3×D412yes—
335C22⋊C4—42no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C42—24no—
339C42—22no—
340C42—22no—
341C42—44no—
342C42C2×C3×D412yes—
343C42C3×#4112yes—
344C22×C3—44no—
345C22×C3—42no—
346C22×C3—82no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3C2×#12111yescharacteristic
350C22×C3#54711yescharacteristic
351C22×C3#54811yescharacteristic
352C22×C3#57512yes—
353C22×C3#57712yes—
354C3×C4—22no—
355C3×C4—22no—
356C3×C4—42no—
357C3×C4—42no—
358C3×C4—42no—
359C3×C4—42no—
360C3×C4—82no—
361C3×C4—81no—
362C3×C4—82no—
363C3×C4#52511yescharacteristic
364C3×C4#53611yescharacteristic
365C3×C4#55211yescharacteristic
366C3×C4#55611yescharacteristic
367C23—22no—
368C23—42no—
369C23—42no—
370C23—41no—
371C23—41no—
372C23C22×C3×D411yescharacteristic
373Q8—22no—
374Q8—22no—
375Q8—41no—
376Q8—41no—
377Q8—84no—
378D4—22no—
379D4—22no—
380D4—41no—
381D4—41no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—84no—
404C2×C4C3×#12112yes—
405C2×C4C3×#12112yes—
406C2×C4C22×C3×D411yescharacteristic
407C2×C4C2×C3×#4111yescharacteristic
408C2×C4C2×C3×#4111yescharacteristic
409C2×C4C2×C3×#4111yescharacteristic
410C2×C4C3×2−1+411yescharacteristic
411C2×C4C3×2−1+411yescharacteristic
412C8—22no—
413C8—42no—
414C8—41no—
415C2×C3—82no—
416C2×C3#241511yescharacteristic
417C2×C3#304011yescharacteristic
418C2×C3#304311yescharacteristic
419C2×C3#334812yes—
420C2×C3#337612yes—
421C22—42no—
422C22—44no—
423C22—81no—
424C22—81no—
425C22—82no—
426C22C2×C3×#12111yescharacteristic
427C22C3×#54711yescharacteristic
428C22C3×#54811yescharacteristic
429C22C3×#57512yes—
430C22C3×#57712yes—
431C4—22no—
432C4—22no—
433C4—42no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—81no—
438C4—82no—
439C4—82no—
440C4C3×#52511yescharacteristic
441C4C3×#53611yescharacteristic
442C4C3×#55211yescharacteristic
443C4C3×#55611yescharacteristic
444C3#2469511yescharacteristic, direct product factor, Sylow
445C2—82no—
446C2C3×#241511yescharacteristic
447C2C3×#304011yescharacteristic
448C2C3×#304311yescharacteristic
449C2C3×#334812yes—
450C2C3×#337612yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
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×#475C2semidirect
4. C3×#2356C2semidirect
5. C3×#2074C2semidirect
7. C3×#1993C2semidirect
8. C3×#1949C2semidirect
9. C3×#1940C2semidirect
12. #24695C3direct (N × H)
44. C2×#475C2×C3semidirect
47. #2356C2×C3semidirect
48. #2074C2×C3semidirect
50. #1993C2×C3semidirect
51. #1949C2×C3semidirect
52. #1940C2×C3semidirect
444. C3#24695direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4752
C2×C3×#4402
C3×#25632
C3×#23562
C3×#20742
C3×#20552
C3×#19932
C3×#19492
C3×#19402
C3×#19262
C3×#18042
#246953