Finite Group Databasebeta

← C3×#24345 #10642952

Subgroups — C3×#24345 #10642952

Order 768.

1020 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Q82C211yesmaximal, characteristic
2C2×C3×#475C211yesmaximal, characteristic
3C2×C3×#441C211yesmaximal, characteristic
4C3×#2581C211yesmaximal, characteristic
5C3×#2058C212yesmaximal
6C3×#2024C212yesmaximal
7C3×#1993C211yesmaximal, characteristic
8C3×#1985C211yesmaximal, characteristic
9C3×#1878C212yesmaximal
10C3×#1865C212yesmaximal
11C3×#1805C211yesmaximal, characteristic
12#24345C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×Q16C2211yescharacteristic
14C3×Q82—24no—
15C3×Q82—24no—
16C2×C3×C4⋊Q8—22no—
17C2×C3×C4⋊Q8—21no—
18C2×C3×C4⋊Q8C2212yes—
19C2×C3×C4⋊Q8C2211yescharacteristic
20C2×C3×C4×Q8—21no—
21C2×C3×C4×Q8C2212yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×Q8C2211yescharacteristic
25C3×#475—24no—
26C3×#441—24no—
27C2×C3×#106C2211yescharacteristic
28C2×C3×#106C2212yes—
29C2×C3×C4⋊C8C2211yescharacteristic
30C2×C3×#103C2211yescharacteristic
31C2×C3×#103C2212yes—
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2212yes—
34C2×C3×#97C2211yescharacteristic
35C3×#400—21no—
36C3×#395—21no—
37C3×#391C2212yes—
38C3×#389C2212yes—
39C3×#387—22no—
40C3×#382C2211yescharacteristic
41C3×#340C2212yes—
42C3×#340C2212yes—
43C3×#340C2212yes—
44C3×#336C2212yes—
45C3×#336C2212yes—
46C2×Q82C2×C311yescharacteristic
47C2×#475C2×C311yescharacteristic
48C2×#441C2×C311yescharacteristic
49#2581C2×C311yescharacteristic
50#2058C2×C312yes—
51#2024C2×C312yes—
52#1993C2×C311yescharacteristic
53#1985C2×C311yescharacteristic
54#1878C2×C312yes—
55#1865C2×C312yes—
56#1805C2×C311yescharacteristic
57C22×C3×Q8—22no—
58C22×C3×Q8—21no—
59C22×C3×Q8D411yescharacteristic
60C22×C3×Q8C2311yescharacteristic
61C22×C3×Q8C2311yescharacteristic
62C2×C3×Q16—22no—
63C2×C3×Q16—22no—
64C2×C3×Q16Q814yescomplemented
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2312yes—
67C22×C3×C8C2311yescharacteristic
68C3×C4⋊Q8—216no—
69C3×C4⋊Q8—28no—
70C3×C4⋊Q8—24no—
71C3×C4⋊Q8—24no—
72C3×C4⋊Q8—22no—
73C3×C4⋊Q8D414yes—
74C3×C4×Q8—28no—
75C3×C4×Q8—24no—
76C3×C4×Q8—24no—
77C3×C4×Q8—22no—
78C3×C4×Q8—22no—
79C3×C4×Q8—22no—
80C3×C4×Q8—22no—
81C3×C4×Q8—22no—
82C3×C4×Q8D414yes—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—22no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4D412yes—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2312yes—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2312yes—
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C42—22no—
96C2×C3×C42—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2312yes—
99C2×C3×C42C2311yescharacteristic
100C3×#106—22no—
101C3×#106—24no—
102C3×C4⋊C8—22no—
103C3×#103—22no—
104C3×#103—24no—
105C3×#103—22no—
106C3×#103—24no—
107C3×#97—22no—
108C3×#95—21no—
109C3×#95—21no—
110C3×#95—22no—
111C3×#95—22no—
112C3×#95—22no—
113C22×Q16C22×C311yescharacteristic
114Q82—24no—
115Q82—24no—
116C2×C4⋊Q8—22no—
117C2×C4⋊Q8—21no—
118C2×C4⋊Q8C22×C312yes—
119C2×C4⋊Q8C22×C311yescharacteristic
120C2×C4×Q8—21no—
121C2×C4×Q8C22×C311yescharacteristic
122C2×C4×Q8C22×C312yes—
123C2×C4×Q8C22×C311yescharacteristic
124C2×C4×Q8C22×C311yescharacteristic
125#475—24no—
126#441—24no—
127C2×#106C22×C312yes—
128C2×#106C22×C311yescharacteristic
129C2×C4⋊C8C22×C311yescharacteristic
130C2×#103C22×C312yes—
131C2×#103C22×C312yes—
132C2×#103C22×C311yescharacteristic
133C2×#103C22×C311yescharacteristic
134C2×#97C22×C311yescharacteristic
135#400—21no—
136#395—21no—
137#391C22×C312yes—
138#389C22×C312yes—
139#387—22no—
140#382C22×C311yescharacteristic
141#340C22×C312yes—
142#340C22×C312yes—
143#340C22×C312yes—
144#336C22×C312yes—
145#336C22×C312yes—
146C2×C3×Q8—28no—
147C2×C3×Q8—22no—
148C2×C3×Q8—24no—
149C2×C3×Q8—24no—
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—82no—
159C2×C3×Q8—84no—
160C2×C3×Q8C2×D412yes—
161C2×C3×Q8C2×Q812yes—
162C2×C3×Q8C2×Q812yes—
163C2×C3×Q8#4112yes—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—22no—
167C22×C3×C4—21no—
168C22×C3×C4—22no—
169C22×C3×C4—21no—
170C22×C3×C4C2×D412yes—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D412yes—
174C22×C3×C4C2411yescharacteristic
175C3×Q16—28no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C2×C3×C8—22no—
180C2×C3×C8—22no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—22no—
184C2×C3×C8C2×Q812yes—
185C3×C4⋊C4—28no—
186C3×C4⋊C4—28no—
187C3×C4⋊C4—24no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—24no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—42no—
198C3×C4⋊C4—44no—
199C3×C4⋊C4C2×D414yes—
200C3×C4⋊C4C2×D412yes—
201C3×C42—28no—
202C3×C42—24no—
203C3×C42—22no—
204C3×C42—22no—
205C3×C42—42no—
206C3×C42C2×D412yes—
207C3×C42C2×D412yes—
208C22×Q8—22no—
209C22×Q8—21no—
210C22×Q8C3×D411yescharacteristic
211C22×Q8C23×C311yescharacteristic
212C22×Q8C23×C311yescharacteristic
213C2×Q16—22no—
214C2×Q16—22no—
215C2×Q16C3×Q814yescomplemented
216C22×C8C23×C311yescharacteristic
217C22×C8C23×C312yes—
218C22×C8C23×C311yescharacteristic
219C4⋊Q8—216no—
220C4⋊Q8—28no—
221C4⋊Q8—24no—
222C4⋊Q8—24no—
223C4⋊Q8—22no—
224C4⋊Q8C3×D414yes—
225C4×Q8—28no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—22no—
233C4×Q8C3×D414yes—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4C3×D412yes—
241C2×C4⋊C4C3×D411yescharacteristic
242C2×C4⋊C4C23×C311yescharacteristic
243C2×C4⋊C4C23×C312yes—
244C2×C4⋊C4C23×C312yes—
245C2×C4⋊C4C23×C311yescharacteristic
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42C23×C312yes—
250C2×C42C23×C311yescharacteristic
251#106—22no—
252#106—24no—
253C4⋊C8—22no—
254#103—22no—
255#103—24no—
256#103—22no—
257#103—24no—
258#97—22no—
259#95—21no—
260#95—21no—
261#95—22no—
262#95—22no—
263#95—22no—
264C23×C3C22×D411yescharacteristic
265C3×Q8—28no—
266C3×Q8—22no—
267C3×Q8—22no—
268C3×Q8—22no—
269C3×Q8—22no—
270C3×Q8—22no—
271C3×Q8—41no—
272C3×Q8—41no—
273C3×Q8—84no—
274C3×Q8—88no—
275C3×Q8C22⋊Q814yes—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—24no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—44no—
291C2×C3×C4—42no—
292C2×C3×C4—41no—
293C2×C3×C4—42no—
294C2×C3×C4—42no—
295C2×C3×C4—42no—
296C2×C3×C4—42no—
297C2×C3×C4#12012yes—
298C2×C3×C4#12012yes—
299C2×C3×C4#13714yes—
300C2×C3×C4C22×D411yescharacteristic
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4C22×Q811yescharacteristic
303C2×C3×C4C2×#4111yescharacteristic
304C2×C3×C42−1+411yescharacteristic
305C2×C3×C42−1+411yescharacteristic
306C3×C8—22no—
307C3×C8—41no—
308C3×C8—42no—
309C2×Q8—28no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—24no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—82no—
322C2×Q8—84no—
323C2×Q8C2×C3×D412yes—
324C2×Q8C2×C3×Q812yes—
325C2×Q8C2×C3×Q812yes—
326C2×Q8C3×#4112yes—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4C2×C3×D412yes—
334C22×C4C2×C3×D411yescharacteristic
335C22×C4C2×C3×D411yescharacteristic
336C22×C4C2×C3×D412yes—
337C22×C4C24×C311yescharacteristic
338Q16—28no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—22no—
347C2×C8C2×C3×Q812yes—
348C4⋊C4—28no—
349C4⋊C4—28no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C4⋊C4—44no—
362C4⋊C4C2×C3×D414yes—
363C4⋊C4C2×C3×D412yes—
364C42—28no—
365C42—24no—
366C42—22no—
367C42—22no—
368C42—42no—
369C42C2×C3×D412yes—
370C42C2×C3×D412yes—
371C22×C3C2×#12011yescharacteristic
372C22×C3#54811yescharacteristic
373C22×C3D4×Q811yescharacteristic
374C22×C3C2×#13712yes—
375C22×C3#57712yes—
376C3×C4—24no—
377C3×C4—22no—
378C3×C4—24no—
379C3×C4—22no—
380C3×C4—42no—
381C3×C4—42no—
382C3×C4—44no—
383C3×C4—42no—
384C3×C4—82no—
385C3×C4—81no—
386C3×C4C2×C22⋊Q811yescharacteristic
387C3×C4#53611yescharacteristic
388C3×C4#54811yescharacteristic
389C3×C4D4×Q811yescharacteristic
390C23C22×C3×D411yescharacteristic
391Q8—28no—
392Q8—22no—
393Q8—22no—
394Q8—22no—
395Q8—22no—
396Q8—22no—
397Q8—41no—
398Q8—41no—
399Q8—84no—
400Q8—88no—
401Q8C3×C22⋊Q814yes—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4C3×#12012yes—
424C2×C4C3×#12012yes—
425C2×C4C3×#13714yes—
426C2×C4C22×C3×D411yescharacteristic
427C2×C4C22×C3×D411yescharacteristic
428C2×C4C22×C3×Q811yescharacteristic
429C2×C4C2×C3×#4111yescharacteristic
430C2×C4C3×2−1+411yescharacteristic
431C2×C4C3×2−1+411yescharacteristic
432C8—22no—
433C8—41no—
434C8—42no—
435C2×C3#240911yescharacteristic
436C2×C3#300411yescharacteristic
437C2×C3#300911yescharacteristic
438C2×C3#336912yes—
439C2×C3#338712yes—
440C22C2×C3×#12011yescharacteristic
441C22C3×#54811yescharacteristic
442C22C3×D4×Q811yescharacteristic
443C22C2×C3×#13712yes—
444C22C3×#57712yes—
445C4—24no—
446C4—22no—
447C4—24no—
448C4—22no—
449C4—42no—
450C4—42no—
451C4—44no—
452C4—42no—
453C4—81no—
454C4—82no—
455C4C2×C3×C22⋊Q811yescharacteristic
456C4C3×#53611yescharacteristic
457C4C3×#54811yescharacteristic
458C4C3×D4×Q811yescharacteristic
459C3#2434511yescharacteristic, direct product factor, Sylow
460C2C3×#240911yescharacteristic
461C2C3×#300411yescharacteristic
462C2C3×#300911yescharacteristic
463C2C3×#336912yes—
464C2C3×#338712yes—
subgroupG/H
center, socle, prime radicalC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8 ◁ G

Complements

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

NH ≅ G/Ntype
12. #24345C3direct (N × H)
64. C2×C3×Q16Q8semidirect
215. C2×Q16C3×Q8semidirect
459. C3#24345direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Q822
C2×C3×#4752
C2×C3×#4412
C3×#25812
C3×#20582
C3×#20242
C3×#19932
C3×#19852
C3×#18782
C3×#18652
C3×#18052
#243453