Finite Group Databasebeta

← C3×#24696 #10643303

Subgroups — C3×#24696 #10643303

Order 768.

796 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#474C211yesmaximal, characteristic
2C2×C3×#440C211yesmaximal, characteristic
3C3×#2567C211yesmaximal, characteristic, complemented
4C3×#2357C211yesmaximal, characteristic
5C3×#2071C212yesmaximal
6C3×#2055C212yesmaximal
7C3×#1993C211yesmaximal, characteristic, complemented
8C3×#1947C212yesmaximal, complemented
9C3×#1940C211yesmaximal, characteristic, complemented
10C3×#1928C212yesmaximal, complemented
11C3×#1805C211yesmaximal, characteristic, complemented
12#24696C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C3×#474—24no—
18C3×#440—24no—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#107C2212yes—
21C2×C3×C4⋊C8C2212yes—
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×#103C2212yes—
24C2×C3×#103C2211yescharacteristic
25C2×C3×#102C2212yes—
26C2×C3×#102C2211yescharacteristic
27C2×C3×#97C2211yescharacteristic
28C3×#395—21no—
29C3×#395—21no—
30C3×#391C2212yes—
31C3×#390C2212yes—
32C3×#389—22no—
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×#474C2×C311yescharacteristic
45C2×#440C2×C311yescharacteristic
46#2567C2×C311yescharacteristic, complemented
47#2357C2×C311yescharacteristic
48#2071C2×C312yes—
49#2055C2×C312yes—
50#1993C2×C311yescharacteristic, complemented
51#1947C2×C312yescomplemented
52#1940C2×C311yescharacteristic, complemented
53#1928C2×C312yescomplemented
54#1805C2×C311yescharacteristic, complemented
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×Q8—22no—
69C3×C4×D4—22no—
70C3×C4×D4D414yes—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4D412yes—
76C2×C3×C4⋊C4C2312yes—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—22no—
83C2×C3×C42—21no—
84C2×C3×C42—22no—
85C2×C3×C42C2312yes—
86C2×C3×C42C2311yescharacteristic
87C3×#107—22no—
88C3×#107—24no—
89C3×C4⋊C8—24no—
90C3×C4⋊C8—22no—
91C3×#103—24no—
92C3×#103—22no—
93C3×#102—22no—
94C3×#102—24no—
95C3×#97—22no—
96C3×#95—21no—
97C3×#95—22no—
98C3×#95—21no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95D411yescharacteristic
102C3×#95D411yescharacteristic
103C22×SD16C22×C311yescharacteristic
104C2×C4⋊Q8C22×C311yescharacteristic
105C2×C4×Q8C22×C311yescharacteristic
106C2×C4×D4C22×C311yescharacteristic
107#474—24no—
108#440—24no—
109C2×#107C22×C311yescharacteristic
110C2×#107C22×C312yes—
111C2×C4⋊C8C22×C312yes—
112C2×C4⋊C8C22×C311yescharacteristic
113C2×#103C22×C312yes—
114C2×#103C22×C311yescharacteristic
115C2×#102C22×C312yes—
116C2×#102C22×C311yescharacteristic
117C2×#97C22×C311yescharacteristic
118#395—21no—
119#395—21no—
120#391C22×C312yes—
121#390C22×C312yes—
122#389—22no—
123#388—22no—
124#384C22×C312yes—
125#382C22×C311yescharacteristic
126#381—21no—
127C4×C4⋊C4C22×C312yes—
128C4×C4⋊C4C22×C311yescharacteristic
129C4×C22⋊C4—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×Q8#4112yes—
143C2×C3×D4—21no—
144C2×C3×D4—21no—
145C2×C3×D4—21no—
146C2×C3×D4—21no—
147C2×C3×D4—21no—
148C2×C3×D4C2×D412yes—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—24no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—21no—
155C22×C3×C4—82no—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D412yes—
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—21no—
174C3×C4⋊C4—22no—
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⋊C4—82no—
182C3×C4⋊C4—82no—
183C3×C4⋊C4C2×D412yes—
184C3×C22⋊C4—24no—
185C3×C22⋊C4—44no—
186C3×C22⋊C4—82no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42—42no—
190C3×C42—82no—
191C3×C42C2×D412yes—
192C3×C42#4112yes—
193C22×Q8—21no—
194C22×Q8C23×C311yescharacteristic
195C22×D4C23×C311yescharacteristic
196C23×C4—21no—
197C2×SD16—22no—
198C2×SD16—22no—
199C2×SD16—22no—
200C22×C8C23×C311yescharacteristic
201C22×C8C23×C312yes—
202C22×C8C23×C311yescharacteristic
203C4⋊Q8—22no—
204C4⋊Q8—22no—
205C4×Q8—22no—
206C4×Q8—22no—
207C4×D4—22no—
208C4×D4C3×D414yes—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4C3×D412yes—
214C2×C4⋊C4C23×C311yescharacteristic
215C2×C4⋊C4C23×C312yes—
216C2×C4⋊C4C23×C312yes—
217C2×C4⋊C4C23×C311yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C42—21no—
222C2×C42—22no—
223C2×C42C23×C312yes—
224C2×C42C23×C311yescharacteristic
225#107—22no—
226#107—24no—
227C4⋊C8—24no—
228C4⋊C8—22no—
229#103—24no—
230#103—22no—
231#102—22no—
232#102—24no—
233#97—22no—
234#95—21no—
235#95—22no—
236#95—21no—
237#95—22no—
238#95—22no—
239#95C3×D411yescharacteristic
240#95C3×D411yescharacteristic
241C23×C3—22no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—42no—
245C23×C3—42no—
246C23×C3C22×D411yescharacteristic
247C3×Q8—22no—
248C3×Q8—22no—
249C3×Q8—41no—
250C3×Q8—41no—
251C3×Q8—84no—
252C3×D4—22no—
253C3×D4—22no—
254C3×D4—41no—
255C3×D4—41no—
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—22no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—42no—
268C2×C3×C4—41no—
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—42no—
276C2×C3×C4—84no—
277C2×C3×C4#12112yes—
278C2×C3×C4#12112yes—
279C2×C3×C4#13714yes—
280C2×C3×C4C22×D411yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C4C2×#4111yescharacteristic
283C2×C3×C4C2×#4111yescharacteristic
284C2×C3×C42−1+411yescharacteristic
285C2×C3×C42−1+411yescharacteristic
286C3×C8—22no—
287C3×C8—42no—
288C3×C8—41no—
289C24—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—42no—
296C2×Q8—82no—
297C2×Q8C3×#4112yes—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4C2×C3×D412yes—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—82no—
311C22×C4C2×C3×D412yes—
312C22×C4C2×C3×D411yescharacteristic
313C22×C4C2×C3×D411yescharacteristic
314C22×C4C2×C3×D412yes—
315C22×C4C24×C311yescharacteristic
316SD16—28no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8C3×#4112yes—
326C4⋊C4—24no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4C2×C3×D412yes—
339C22⋊C4—24no—
340C22⋊C4—44no—
341C22⋊C4—82no—
342C42—22no—
343C42—22no—
344C42—42no—
345C42—82no—
346C42C2×C3×D412yes—
347C42C3×#4112yes—
348C22×C3—42no—
349C22×C3—42no—
350C22×C3—42no—
351C22×C3—82no—
352C22×C3—82no—
353C22×C3C2×#12111yescharacteristic
354C22×C3#54711yescharacteristic
355C22×C3#54811yescharacteristic
356C22×C3C2×#13712yes—
357C22×C3#57612yes—
358C3×C4—24no—
359C3×C4—22no—
360C3×C4—22no—
361C3×C4—42no—
362C3×C4—42no—
363C3×C4—42no—
364C3×C4—82no—
365C3×C4—81no—
366C3×C4—82no—
367C3×C4#52511yescharacteristic
368C3×C4#53611yescharacteristic
369C3×C4#55211yescharacteristic
370C3×C4#55611yescharacteristic
371C23—22no—
372C23—42no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23C22×C3×D411yescharacteristic
377Q8—22no—
378Q8—22no—
379Q8—41no—
380Q8—41no—
381Q8—84no—
382D4—22no—
383D4—22no—
384D4—41no—
385D4—41no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—84no—
407C2×C4C3×#12112yes—
408C2×C4C3×#12112yes—
409C2×C4C3×#13714yes—
410C2×C4C22×C3×D411yescharacteristic
411C2×C4C2×C3×#4111yescharacteristic
412C2×C4C2×C3×#4111yescharacteristic
413C2×C4C2×C3×#4111yescharacteristic
414C2×C4C3×2−1+411yescharacteristic
415C2×C4C3×2−1+411yescharacteristic
416C8—22no—
417C8—42no—
418C8—41no—
419C2×C3—82no—
420C2×C3#241511yescharacteristic
421C2×C3#303911yescharacteristic
422C2×C3#304111yescharacteristic
423C2×C3#334612yes—
424C2×C3#337512yes—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—82no—
429C22—82no—
430C22C2×C3×#12111yescharacteristic
431C22C3×#54711yescharacteristic
432C22C3×#54811yescharacteristic
433C22C2×C3×#13712yes—
434C22C3×#57612yes—
435C4—22no—
436C4—22no—
437C4—24no—
438C4—42no—
439C4—42no—
440C4—42no—
441C4—82no—
442C4—81no—
443C4—82no—
444C4C3×#52511yescharacteristic
445C4C3×#53611yescharacteristic
446C4C3×#55211yescharacteristic
447C4C3×#55611yescharacteristic
448C3#2469611yescharacteristic, direct product factor, Sylow
449C2—82no—
450C2C3×#241511yescharacteristic
451C2C3×#303911yescharacteristic
452C2C3×#304111yescharacteristic
453C2C3×#334612yes—
454C2C3×#337512yes—
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
3. C3×#2567C2semidirect
7. C3×#1993C2semidirect
8. C3×#1947C2semidirect
9. C3×#1940C2semidirect
10. C3×#1928C2semidirect
11. C3×#1805C2semidirect
12. #24696C3direct (N × H)
46. #2567C2×C3semidirect
50. #1993C2×C3semidirect
51. #1947C2×C3semidirect
52. #1940C2×C3semidirect
53. #1928C2×C3semidirect
54. #1805C2×C3semidirect
448. C3#24696direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4742
C2×C3×#4402
C3×#25672
C3×#23572
C3×#20712
C3×#20552
C3×#19932
C3×#19472
C3×#19402
C3×#19282
C3×#18052
#246963