Finite Group Databasebeta

← C3×#24212 #10642819

Subgroups — C3×#24212 #10642819

Order 768.

1430 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×C3×#462C212yesmaximal, complemented
3C2×C3×#451C212yesmaximal
4C3×C4×C22⋊Q8C211yesmaximal, characteristic
5C3×#2043C212yesmaximal, complemented
6C3×#2016C212yesmaximal, complemented
7C3×#1981C212yesmaximal, complemented
8C3×#1797C211yesmaximal, characteristic
9C3×#1787C212yesmaximal
10#24212C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2212yes—
12C22×C3×Q16C2214yes—
13C2×C3×C4⋊Q8—28no—
14C2×C3×C4⋊Q8—21no—
15C2×C3×C4⋊Q8—21no—
16C2×C3×C4⋊Q8C2212yes—
17C2×C3×C4⋊Q8C2212yes—
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C22×C3×C4⋊C4—21no—
21C22×C3×C4⋊C4—21no—
22C22×C3×C42C2211yescharacteristic
23C3×#462—28no—
24C3×#451—28no—
25C2×C3×C4⋊C8C2212yes—
26C2×C3×#103C2214yes—
27C2×C3×#103C2214yes—
28C2×C3×#103C2214yes—
29C2×C3×C22⋊C8C2212yes—
30C3×#391C2211yescharacteristic
31C3×#389C2211yescharacteristic
32C3×#387—21no—
33C3×#385—21no—
34C3×#380C2211yescharacteristic
35C3×C4×C4⋊C4—21no—
36C3×C4×C4⋊C4C2211yescharacteristic
37C3×C4×C22⋊C4—21no—
38C3×#336C2212yes—
39C22×C4⋊Q8C2×C311yescharacteristic
40C2×#462C2×C312yescomplemented
41C2×#451C2×C312yes—
42C4×C22⋊Q8C2×C311yescharacteristic
43#2043C2×C312yescomplemented
44#2016C2×C312yescomplemented
45#1981C2×C312yescomplemented
46#1797C2×C311yescharacteristic
47#1787C2×C312yes—
48C22×C3×Q8—28no—
49C22×C3×Q8—22no—
50C22×C3×Q8—24no—
51C22×C3×Q8—24no—
52C22×C3×Q8—44no—
53C22×C3×Q8C2311yescharacteristic
54C22×C3×Q8C2314yes—
55C23×C3×C4—22no—
56C23×C3×C4D411yescharacteristic
57C23×C3×C4D411yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C2×C3×Q16—28no—
60C2×C3×Q16—28no—
61C2×C3×Q16—44no—
62C22×C3×C8C2314yes—
63C3×C4⋊Q8—28no—
64C3×C4⋊Q8—28no—
65C3×C4⋊Q8—28no—
66C3×C4⋊Q8—24no—
67C3×C4⋊Q8D418yes—
68C3×C22⋊Q8—22no—
69C3×C22⋊Q8D414yes—
70C3×C4×Q8—24no—
71C2×C3×C4⋊C4—28no—
72C2×C3×C4⋊C4—28no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4—22no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4—21no—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C22⋊C4—21no—
84C2×C3×C22⋊C4—21no—
85C2×C3×C42—24no—
86C2×C3×C42—21no—
87C2×C3×C42—21no—
88C2×C3×C42D411yescharacteristic
89C2×C3×C42D411yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C3×C4⋊C8—24no—
93C3×#103—28no—
94C3×#103—28no—
95C3×#103—44no—
96C3×C22⋊C8—24no—
97C3×#95—21no—
98C3×#95—21no—
99C3×#95—21no—
100C3×#95—21no—
101C3×#95—21no—
102C23×Q8C22×C312yes—
103C22×Q16C22×C314yes—
104C2×C4⋊Q8—28no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8—21no—
107C2×C4⋊Q8C22×C312yes—
108C2×C4⋊Q8C22×C312yes—
109C2×C22⋊Q8C22×C311yescharacteristic
110C2×C4×Q8C22×C311yescharacteristic
111C22×C4⋊C4—21no—
112C22×C4⋊C4—21no—
113C22×C42C22×C311yescharacteristic
114#462—28no—
115#451—28no—
116C2×C4⋊C8C22×C312yes—
117C2×#103C22×C314yes—
118C2×#103C22×C314yes—
119C2×#103C22×C314yes—
120C2×C22⋊C8C22×C312yes—
121#391C22×C311yescharacteristic
122#389C22×C311yescharacteristic
123#387—21no—
124#385—21no—
125#380C22×C311yescharacteristic
126C4×C4⋊C4—21no—
127C4×C4⋊C4C22×C311yescharacteristic
128C4×C22⋊C4—21no—
129#336C22×C312yes—
130C24×C3C2×D411yescharacteristic
131C2×C3×Q8—28no—
132C2×C3×Q8—24no—
133C2×C3×Q8—21no—
134C2×C3×Q8—24no—
135C2×C3×Q8—22no—
136C2×C3×Q8—24no—
137C2×C3×Q8—24no—
138C2×C3×Q8—22no—
139C2×C3×Q8—24no—
140C2×C3×Q8—416no—
141C2×C3×Q8—416no—
142C2×C3×Q8—44no—
143C2×C3×Q8—44no—
144C2×C3×Q8C2×D418yes—
145C2×C3×Q8C2×D412yes—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—28no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—24no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—44no—
160C22×C3×C4—44no—
161C22×C3×C4Q1614yescomplemented
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D411yescharacteristic
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4#4112yes—
169C22×C3×C4C2411yescharacteristic
170C3×Q16—416no—
171C2×C3×C8—24no—
172C2×C3×C8—24no—
173C2×C3×C8—24no—
174C3×C4⋊C4—28no—
175C3×C4⋊C4—28no—
176C3×C4⋊C4—28no—
177C3×C4⋊C4—28no—
178C3×C4⋊C4—24no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4—42no—
184C3×C4⋊C4C2×D412yes—
185C3×C22⋊C4—24no—
186C3×C22⋊C4—42no—
187C3×C42—22no—
188C3×C42—22no—
189C3×C42—22no—
190C3×C42—42no—
191C3×C42—42no—
192C3×C42C2×D414yes—
193C22×Q8—28no—
194C22×Q8—22no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—44no—
198C22×Q8C23×C311yescharacteristic
199C22×Q8C23×C314yes—
200C23×C4—22no—
201C23×C4C3×D411yescharacteristic
202C23×C4C3×D411yescharacteristic
203C23×C4C23×C311yescharacteristic
204C2×Q16—28no—
205C2×Q16—28no—
206C2×Q16—44no—
207C22×C8C23×C314yes—
208C4⋊Q8—28no—
209C4⋊Q8—28no—
210C4⋊Q8—28no—
211C4⋊Q8—24no—
212C4⋊Q8C3×D418yes—
213C22⋊Q8—22no—
214C22⋊Q8C3×D414yes—
215C4×Q8—24no—
216C2×C4⋊C4—28no—
217C2×C4⋊C4—28no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C23×C311yescharacteristic
226C2×C4⋊C4C23×C311yescharacteristic
227C2×C4⋊C4C23×C311yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—24no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42C3×D411yescharacteristic
234C2×C42C3×D411yescharacteristic
235C2×C42C23×C311yescharacteristic
236C2×C42C23×C311yescharacteristic
237C4⋊C8—24no—
238#103—28no—
239#103—28no—
240#103—44no—
241C22⋊C8—24no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247C23×C3—21no—
248C23×C3—21no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3#12112yes—
252C23×C3C2×Q1612yes—
253C23×C3#13712yes—
254C23×C3C22×D411yescharacteristic
255C3×Q8—24no—
256C3×Q8—28no—
257C3×Q8—28no—
258C3×Q8—416no—
259C3×Q8—44no—
260C3×Q8—42no—
261C3×Q8—44no—
262C2×C3×C4—28no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—28no—
274C2×C3×C4—28no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—416no—
278C2×C3×C4—41no—
279C2×C3×C4—44no—
280C2×C3×C4—44no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4#12012yes—
291C2×C3×C4#12012yes—
292C2×C3×C4#12112yes—
293C2×C3×C4C2×Q1612yes—
294C2×C3×C4C2×Q1612yes—
295C2×C3×C4#13712yes—
296C2×C3×C4#13712yes—
297C2×C3×C4C22×D411yescharacteristic
298C2×C3×C4C22×D412yes—
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C4H(5,𝔽2)12yes—
301C3×C8—44no—
302C24C2×C3×D411yescharacteristic
303C2×Q8—28no—
304C2×Q8—24no—
305C2×Q8—21no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—22no—
311C2×Q8—24no—
312C2×Q8—416no—
313C2×Q8—416no—
314C2×Q8—44no—
315C2×Q8—44no—
316C2×Q8C2×C3×D418yes—
317C2×Q8C2×C3×D412yes—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—28no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—44no—
332C22×C4—44no—
333C22×C4C3×Q1614yescomplemented
334C22×C4C2×C3×D411yescharacteristic
335C22×C4C2×C3×D411yescharacteristic
336C22×C4C2×C3×D412yes—
337C22×C4C2×C3×D411yescharacteristic
338C22×C4C2×C3×D411yescharacteristic
339C22×C4C2×C3×D411yescharacteristic
340C22×C4C3×#4112yes—
341C22×C4C24×C311yescharacteristic
342Q16—416no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C4⋊C4—28no—
347C4⋊C4—28no—
348C4⋊C4—28no—
349C4⋊C4—28no—
350C4⋊C4—24no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4C2×C3×D412yes—
357C22⋊C4—24no—
358C22⋊C4—42no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—42no—
363C42—42no—
364C42C2×C3×D414yes—
365C22×C3—22no—
366C22×C3—22no—
367C22×C3—22no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3#46214yescomplemented
372C22×C3C2×#12011yescharacteristic
373C22×C3C2×#12111yescharacteristic
374C22×C3#53511yescharacteristic
375C22×C3C22×Q1611yescharacteristic
376C22×C3C2×#13711yescharacteristic
377C22×C3C2×#13711yescharacteristic
378C22×C3C2×#13711yescharacteristic
379C3×C4—22no—
380C3×C4—22no—
381C3×C4—22no—
382C3×C4—22no—
383C3×C4—48no—
384C3×C4—42no—
385C3×C4—42no—
386C3×C4—81no—
387C3×C4—81no—
388C3×C4#45114yescomplemented
389C3×C4D4212yes—
390C3×C4#54612yes—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23C3×#12112yes—
396C23C2×C3×Q1612yes—
397C23C3×#13712yes—
398C23C22×C3×D411yescharacteristic
399Q8—24no—
400Q8—28no—
401Q8—28no—
402Q8—416no—
403Q8—44no—
404Q8—44no—
405Q8—42no—
406C2×C4—28no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—28no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—28no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—416no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4C3×#12012yes—
435C2×C4C3×#12012yes—
436C2×C4C3×#12112yes—
437C2×C4C2×C3×Q1612yes—
438C2×C4C2×C3×Q1612yes—
439C2×C4C3×#13712yes—
440C2×C4C3×#13712yes—
441C2×C4C22×C3×D411yescharacteristic
442C2×C4C22×C3×D412yes—
443C2×C4C2×C3×#4111yescharacteristic
444C2×C4C3×H(5,𝔽2)12yes—
445C8—44no—
446C2×C3—24no—
447C2×C3#240811yescharacteristic
448C2×C3C2×#45111yescharacteristic
449C2×C3#300411yescharacteristic
450C2×C3C2×#46211yescharacteristic
451C2×C3#303911yescharacteristic
452C2×C3#320911yescharacteristic
453C2×C3#321811yescharacteristic
454C22—22no—
455C22—22no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22C3×#46214yescomplemented
461C22C2×C3×#12011yescharacteristic
462C22C2×C3×#12111yescharacteristic
463C22C3×#53511yescharacteristic
464C22C22×C3×Q1611yescharacteristic
465C22C2×C3×#13711yescharacteristic
466C22C2×C3×#13711yescharacteristic
467C22C2×C3×#13711yescharacteristic
468C4—22no—
469C4—22no—
470C4—22no—
471C4—22no—
472C4—48no—
473C4—42no—
474C4—42no—
475C4—81no—
476C4—81no—
477C4C3×#45114yescomplemented
478C4C3×D4212yes—
479C4C3×#54612yes—
480C3#2421211yescharacteristic, direct product factor, Sylow
481C2—24no—
482C2C3×#240811yescharacteristic
483C2C2×C3×#45111yescharacteristic
484C2C3×#300411yescharacteristic
485C2C2×C3×#46211yescharacteristic
486C2C3×#303911yescharacteristic
487C2C3×#320911yescharacteristic
488C2C3×#321811yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
LCSG ▷ C22×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×#462C2semidirect
5. C3×#2043C2semidirect
6. C3×#2016C2semidirect
7. C3×#1981C2semidirect
10. #24212C3direct (N × H)
40. C2×#462C2×C3semidirect
43. #2043C2×C3semidirect
44. #2016C2×C3semidirect
45. #1981C2×C3semidirect
161. C22×C3×C4Q16semidirect
333. C22×C4C3×Q16semidirect
371. C22×C3#462semidirect
388. C3×C4#451semidirect
460. C22C3×#462semidirect
477. C4C3×#451semidirect
480. C3#24212direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C2×C3×#4622
C2×C3×#4512
C3×C4×C22⋊Q82
C3×#20432
C3×#20162
C3×#19812
C3×#17972
C3×#17872
#242123