Finite Group Databasebeta

← C3×#15761 #10634368

Subgroups — C3×#15761 #10634368

Order 768.

2234 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic, complemented
2C22×C3×#124C211yesmaximal, characteristic
3C2×C3×#394C212yesmaximal
4C2×C3×#393C212yesmaximal, complemented
5C3×#1446C211yesmaximal, characteristic, complemented
6C3×#1439C214yesmaximal, complemented
7C3×#1437C212yesmaximal, complemented
8C3×#1436C212yesmaximal, complemented
9#15761C311yesmaximal, characteristic, direct product factor, Sylow
10C23×C3×Q8C2211yescharacteristic
11C23×C3×Q8C2211yescharacteristic
12C23×C3×D4C2211yescharacteristic
13C2×C3×C4⋊Q8—24no—
14C2×C3×C4⋊Q8—28no—
15C2×C3×#124—24no—
16C2×C3×#124—28no—
17C22×C3×C4⋊C4C2212yes—
18C22×C3×C4⋊C4C2212yes—
19C22×C3×C22⋊C4C2212yes—
20C22×C3×C22⋊C4C2214yes—
21C22×C3×C22⋊C4C2214yes—
22C22×C3×C22⋊C4C2212yes—
23C22×C3×C42C2211yescharacteristic
24C22×C3×C42C2211yescharacteristic
25C3×#394—28no—
26C3×#393—28no—
27C2×C3×#95C2214yes—
28C2×C3×#95C2212yes—
29C2×C3×#95C2212yes—
30C2×C3×#95C2214yes—
31C2×C3×#95C2212yes—
32C22×C4⋊Q8C2×C311yescharacteristic, complemented
33C22×#124C2×C311yescharacteristic
34C2×#394C2×C312yes—
35C2×#393C2×C312yescomplemented
36#1446C2×C311yescharacteristic, complemented
37#1439C2×C314yescomplemented
38#1437C2×C312yescomplemented
39#1436C2×C312yescomplemented
40C25×C3C2312yes—
41C22×C3×Q8—24no—
42C22×C3×Q8—24no—
43C22×C3×Q8—24no—
44C22×C3×Q8—22no—
45C22×C3×Q8—24no—
46C22×C3×Q8—42no—
47C22×C3×Q8—42no—
48C22×C3×Q8D414yes—
49C22×C3×D4—24no—
50C22×C3×D4—22no—
51C22×C3×D4—24no—
52C22×C3×D4—42no—
53C23×C3×C4C2314yes—
54C23×C3×C4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×C3×C4C2312yes—
57C23×C3×C4C2311yescharacteristic
58C23×C3×C4C2311yescharacteristic
59C23×C3×C4C2312yes—
60C3×C4⋊Q8—232no—
61C3×#124—232no—
62C2×C3×C4⋊C4—24no—
63C2×C3×C4⋊C4—24no—
64C2×C3×C4⋊C4—24no—
65C2×C3×C4⋊C4—24no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4D418yes—
68C2×C3×C22⋊C4—24no—
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4—24no—
71C2×C3×C22⋊C4—24no—
72C2×C3×C22⋊C4—24no—
73C2×C3×C22⋊C4—28no—
74C2×C3×C22⋊C4—28no—
75C2×C3×C22⋊C4—28no—
76C2×C3×C22⋊C4—28no—
77C2×C3×C22⋊C4—28no—
78C2×C3×C22⋊C4—44no—
79C2×C3×C22⋊C4D418yes—
80C2×C3×C42—24no—
81C2×C3×C42—22no—
82C2×C3×C42—22no—
83C2×C3×C42—24no—
84C3×#95—28no—
85C3×#95—24no—
86C3×#95—24no—
87C3×#95—28no—
88C3×#95—42no—
89C23×Q8C22×C311yescharacteristic
90C23×Q8C22×C311yescharacteristic
91C23×D4C22×C311yescharacteristic
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—28no—
94C2×#124—24no—
95C2×#124—28no—
96C22×C4⋊C4C22×C312yes—
97C22×C4⋊C4C22×C312yes—
98C22×C22⋊C4C22×C314yes—
99C22×C22⋊C4C22×C312yes—
100C22×C22⋊C4C22×C314yes—
101C22×C22⋊C4C22×C312yes—
102C22×C42C22×C311yescharacteristic
103C22×C42C22×C311yescharacteristic
104#394—28no—
105#393—28no—
106C2×#95C22×C314yes—
107C2×#95C22×C312yes—
108C2×#95C22×C312yes—
109C2×#95C22×C314yes—
110C2×#95C22×C312yes—
111C24×C3—22no—
112C24×C3—22no—
113C24×C3—24no—
114C24×C3—24no—
115C24×C3—24no—
116C24×C3—24no—
117C24×C3—24no—
118C24×C3—24no—
119C24×C3C2×D414yes—
120C24×C3C2411yescharacteristic
121C2×C3×Q8—28no—
122C2×C3×Q8—28no—
123C2×C3×Q8—28no—
124C2×C3×Q8—44no—
125C2×C3×Q8—416no—
126C2×C3×Q8—416no—
127C2×C3×Q8—44no—
128C2×C3×Q8—44no—
129C2×C3×D4—28no—
130C2×C3×D4—416no—
131C2×C3×D4—44no—
132C2×C3×D4—44no—
133C22×C3×C4—22no—
134C22×C3×C4—22no—
135C22×C3×C4—22no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—24no—
140C22×C3×C4—24no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4—22no—
146C22×C3×C4—22no—
147C22×C3×C4—24no—
148C22×C3×C4—22no—
149C22×C3×C4—24no—
150C22×C3×C4—24no—
151C22×C3×C4—24no—
152C22×C3×C4—24no—
153C22×C3×C4—24no—
154C22×C3×C4—24no—
155C22×C3×C4—24no—
156C22×C3×C4—24no—
157C22×C3×C4—24no—
158C22×C3×C4C2×D414yes—
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D414yes—
161C22×C3×C4C2×D414yes—
162C22×C3×C4C2×D414yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D414yes—
166C22×C3×C4#4118yes—
167C3×C4⋊C4—216no—
168C3×C4⋊C4—216no—
169C3×C22⋊C4—216no—
170C3×C22⋊C4—216no—
171C3×C22⋊C4—416no—
172C3×C22⋊C4—416no—
173C3×C42—28no—
174C3×C42—28no—
175C25C23×C312yes—
176C22×Q8—24no—
177C22×Q8—24no—
178C22×Q8—24no—
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—42no—
182C22×Q8—42no—
183C22×Q8C3×D414yes—
184C22×D4—24no—
185C22×D4—22no—
186C22×D4—24no—
187C22×D4—42no—
188C23×C4C23×C314yes—
189C23×C4C23×C312yes—
190C23×C4C23×C311yescharacteristic
191C23×C4C23×C312yes—
192C23×C4C23×C311yescharacteristic
193C23×C4C23×C311yescharacteristic
194C23×C4C23×C312yes—
195C4⋊Q8—232no—
196#124—232no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4C3×D418yes—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4C3×D418yes—
215C2×C42—24no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—24no—
219#95—28no—
220#95—24no—
221#95—24no—
222#95—28no—
223#95—42no—
224C23×C3—24no—
225C23×C3—24no—
226C23×C3—24no—
227C23×C3—28no—
228C23×C3—28no—
229C23×C3—42no—
230C23×C3—44no—
231C23×C3—42no—
232C23×C3—42no—
233C23×C3—44no—
234C23×C3—42no—
235C23×C3—44no—
236C23×C3—44no—
237C23×C3—44no—
238C23×C3—44no—
239C23×C3—44no—
240C23×C3—44no—
241C23×C3—44no—
242C23×C3—44no—
243C23×C3—44no—
244C23×C3—44no—
245C23×C3C22×D411yescharacteristic
246C23×C3C22×D412yes—
247C23×C3C22×D412yes—
248C23×C3C2×#4114yes—
249C23×C3H(5,𝔽2)12yes—
250C23×C32−1+414yes—
251C3×Q8—416no—
252C3×Q8—416no—
253C3×D4—416no—
254C2×C3×C4—24no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—24no—
260C2×C3×C4—28no—
261C2×C3×C4—28no—
262C2×C3×C4—28no—
263C2×C3×C4—28no—
264C2×C3×C4—24no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—24no—
268C2×C3×C4—22no—
269C2×C3×C4—28no—
270C2×C3×C4—28no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—44no—
276C2×C3×C4—44no—
277C2×C3×C4—44no—
278C2×C3×C4—44no—
279C2×C3×C4—44no—
280C2×C3×C4—44no—
281C2×C3×C4—44no—
282C2×C3×C4—44no—
283C2×C3×C4—44no—
284C2×C3×C4—44no—
285C2×C3×C4#12014yes—
286C2×C3×C4#12014yes—
287C2×C3×C4#12418yescomplemented
288C24—22no—
289C24—22no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24C2×C3×D414yes—
297C24C24×C311yescharacteristic
298C2×Q8—28no—
299C2×Q8—28no—
300C2×Q8—28no—
301C2×Q8—44no—
302C2×Q8—416no—
303C2×Q8—416no—
304C2×Q8—44no—
305C2×Q8—44no—
306C2×D4—28no—
307C2×D4—416no—
308C2×D4—44no—
309C2×D4—44no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4C2×C3×D414yes—
336C22×C4C2×C3×D412yes—
337C22×C4C2×C3×D414yes—
338C22×C4C2×C3×D414yes—
339C22×C4C2×C3×D414yes—
340C22×C4C2×C3×D412yes—
341C22×C4C2×C3×D412yes—
342C22×C4C2×C3×D414yes—
343C22×C4C3×#4118yes—
344C4⋊C4—216no—
345C4⋊C4—216no—
346C22⋊C4—216no—
347C22⋊C4—216no—
348C22⋊C4—416no—
349C22⋊C4—416no—
350C42—28no—
351C42—28no—
352C22×C3—44no—
353C22×C3—48no—
354C22×C3—44no—
355C22×C3—44no—
356C22×C3—48no—
357C22×C3—84no—
358C22×C3—84no—
359C22×C3—84no—
360C22×C3—84no—
361C22×C3C2×#12011yescharacteristic
362C22×C3C2×#12011yescharacteristic
363C22×C3C2×#12412yes—
364C22×C3#53212yes—
365C22×C3#53411yescharacteristic
366C22×C3#53612yes—
367C22×C3#53612yes—
368C22×C3#54214yes—
369C22×C3D4214yes—
370C22×C3#54714yes—
371C22×C3#54814yes—
372C22×C3#54814yes—
373C22×C3#55512yes—
374C22×C3#56312yes—
375C3×C4—24no—
376C3×C4—24no—
377C3×C4—28no—
378C3×C4—42no—
379C3×C4—44no—
380C3×C4—44no—
381C3×C4—84no—
382C23—24no—
383C23—24no—
384C23—24no—
385C23—28no—
386C23—28no—
387C23—44no—
388C23—42no—
389C23—44no—
390C23—42no—
391C23—42no—
392C23—44no—
393C23—42no—
394C23—44no—
395C23—44no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23C22×C3×D411yescharacteristic
404C23C22×C3×D412yes—
405C23C22×C3×D412yes—
406C23C2×C3×#4114yes—
407C23C3×H(5,𝔽2)12yes—
408C23C3×2−1+414yes—
409Q8—416no—
410Q8—416no—
411D4—416no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—28no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—28no—
428C2×C4—28no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4C3×#12014yes—
444C2×C4C3×#12014yes—
445C2×C4C3×#12418yescomplemented
446C2×C3—84no—
447C2×C3#240311yescharacteristic
448C2×C3#243212yes—
449C2×C3#243612yes—
450C2×C3#249114yes—
451C2×C3#267414yes—
452C2×C3#273112yes—
453C22—44no—
454C22—48no—
455C22—44no—
456C22—44no—
457C22—48no—
458C22—84no—
459C22—84no—
460C22—84no—
461C22—84no—
462C22C2×C3×#12011yescharacteristic
463C22C2×C3×#12011yescharacteristic
464C22C2×C3×#12412yes—
465C22C3×#53212yes—
466C22C3×#53411yescharacteristic
467C22C3×#53612yes—
468C22C3×#53612yes—
469C22C3×#54214yes—
470C22C3×D4214yes—
471C22C3×#54714yes—
472C22C3×#54814yes—
473C22C3×#54814yes—
474C22C3×#55512yes—
475C22C3×#56312yes—
476C4—24no—
477C4—24no—
478C4—28no—
479C4—42no—
480C4—44no—
481C4—44no—
482C4—84no—
483C3#1576111yescharacteristic, direct product factor, Sylow
484C2—84no—
485C2C3×#240311yescharacteristic
486C2C3×#243212yes—
487C2C3×#243612yes—
488C2C3×#249114yes—
489C2C3×#267414yes—
490C2C3×#273112yes—
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
prime radicalC23×C3×D4C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊Q8C2semidirect
4. C2×C3×#393C2semidirect
5. C3×#1446C2semidirect
6. C3×#1439C2semidirect
7. C3×#1437C2semidirect
8. C3×#1436C2semidirect
9. #15761C3direct (N × H)
32. C22×C4⋊Q8C2×C3semidirect
35. C2×#393C2×C3semidirect
36. #1446C2×C3semidirect
37. #1439C2×C3semidirect
38. #1437C2×C3semidirect
39. #1436C2×C3semidirect
287. C2×C3×C4#124semidirect
445. C2×C4C3×#124semidirect
483. C3#15761direct (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
C22×C3×#1242
C2×C3×#3942
C2×C3×#3932
C3×#14462
C3×#14392
C3×#14372
C3×#14362
#157613