Finite Group Databasebeta

← C3×#14757 #10633364

Subgroups — C3×#14757 #10633364

Order 768.

3510 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#121C213yesmaximal, complemented
2C2×C3×#392C211yesmaximal, characteristic, complemented
3C2×C3×#386C213yesmaximal, complemented
4C22×C3×#95C211yesmaximal, characteristic, complemented
5C3×#1444C211yesmaximal, characteristic, complemented
6C3×#1439C213yesmaximal, complemented
7C3×#1438C213yesmaximal, complemented
8#14757C311yesmaximal, characteristic, direct product factor, Sylow
9C23×C3×D4C2213yes—
10C23×C3×D4C2213yes—
11C24×C3×C4C2213yes—
12C2×C3×#121—224no—
13C2×C3×#121—212no—
14C22×C3×C4⋊C4C2213yes—
15C22×C3×C22⋊C4C2216yes—
16C22×C3×C22⋊C4C2213yes—
17C22×C3×C22⋊C4C2211yescharacteristic
18C22×C3×C22⋊C4C2213yes—
19C3×#392—24no—
20C3×#386—212no—
21C2×C3×#95—24no—
22C2×C3×#95C2213yes—
23C2×C3×#95C2213yes—
24C2×C3×#95C2213yes—
25C2×C3×#95C2211yescharacteristic
26C22×#121C2×C313yescomplemented
27C2×#392C2×C311yescharacteristic, complemented
28C2×#386C2×C313yescomplemented
29C22×#95C2×C311yescharacteristic, complemented
30#1444C2×C311yescharacteristic, complemented
31#1439C2×C313yescomplemented
32#1438C2×C313yescomplemented
33C25×C3C2313yes—
34C25×C3C2311yescharacteristic
35C22×C3×D4—26no—
36C22×C3×D4—212no—
37C22×C3×D4—212no—
38C22×C3×D4—212no—
39C22×C3×D4—26no—
40C22×C3×D4—26no—
41C22×C3×D4—26no—
42C22×C3×D4—26no—
43C22×C3×D4—46no—
44C22×C3×D4D4112yescomplemented
45C23×C3×C4—212no—
46C23×C3×C4—26no—
47C23×C3×C4—26no—
48C23×C3×C4—46no—
49C23×C3×C4D4112yescomplemented
50C23×C3×C4C2311yescharacteristic
51C23×C3×C4C2311yescharacteristic
52C23×C3×C4C2313yes—
53C23×C3×C4C2313yes—
54C23×C3×C4C2313yes—
55C3×#121—296no—
56C2×C3×C4⋊C4—212no—
57C2×C3×C4⋊C4—26no—
58C2×C3×C22⋊C4—212no—
59C2×C3×C22⋊C4—212no—
60C2×C3×C22⋊C4—212no—
61C2×C3×C22⋊C4—26no—
62C2×C3×C22⋊C4—26no—
63C2×C3×C22⋊C4—26no—
64C2×C3×C22⋊C4—26no—
65C2×C3×C22⋊C4—26no—
66C2×C3×C22⋊C4—46no—
67C3×#95—26no—
68C3×#95—26no—
69C3×#95—26no—
70C3×#95D414yescomplemented
71C23×D4C22×C313yes—
72C23×D4C22×C313yes—
73C24×C4C22×C313yes—
74C2×#121—224no—
75C2×#121—212no—
76C22×C4⋊C4C22×C313yes—
77C22×C22⋊C4C22×C313yes—
78C22×C22⋊C4C22×C316yes—
79C22×C22⋊C4C22×C311yescharacteristic
80C22×C22⋊C4C22×C313yes—
81#392—24no—
82#386—212no—
83C2×#95—24no—
84C2×#95C22×C313yes—
85C2×#95C22×C313yes—
86C2×#95C22×C313yes—
87C2×#95C22×C311yescharacteristic
88C24×C3—21no—
89C24×C3—23no—
90C24×C3—23no—
91C24×C3—21no—
92C24×C3—23no—
93C24×C3—26no—
94C24×C3—26no—
95C24×C3—23no—
96C24×C3—23no—
97C24×C3—23no—
98C24×C3—26no—
99C24×C3—26no—
100C24×C3—23no—
101C24×C3—23no—
102C24×C3C2×D416yes—
103C24×C3C2×D416yes—
104C24×C3C2×D416yes—
105C24×C3#4112yes—
106C24×C3C2411yescharacteristic
107C2×C3×D4—224no—
108C2×C3×D4—248no—
109C2×C3×D4—224no—
110C2×C3×D4—424no—
111C2×C3×D4—424no—
112C2×C3×D4—412no—
113C2×C3×D4—412no—
114C2×C3×D4—424no—
115C2×C3×D4—412no—
116C2×C3×D4—412no—
117C22×C3×C4—26no—
118C22×C3×C4—26no—
119C22×C3×C4—224no—
120C22×C3×C4—23no—
121C22×C3×C4—23no—
122C22×C3×C4—26no—
123C22×C3×C4—26no—
124C22×C3×C4—26no—
125C22×C3×C4—23no—
126C22×C3×C4—23no—
127C22×C3×C4—26no—
128C22×C3×C4—224no—
129C22×C3×C4—23no—
130C22×C3×C4—23no—
131C22×C3×C4—21no—
132C22×C3×C4—23no—
133C22×C3×C4—23no—
134C22×C3×C4—21no—
135C22×C3×C4—412no—
136C22×C3×C4—424no—
137C22×C3×C4—424no—
138C22×C3×C4C2×D416yes—
139C22×C3×C4C2×D416yes—
140C22×C3×C4C2×D416yes—
141C22×C3×C4C2×D416yes—
142C22×C3×C4#4116yes—
143C3×C4⋊C4—224no—
144C3×C22⋊C4—248no—
145C3×C22⋊C4—412no—
146C3×C22⋊C4—44no—
147C3×C22⋊C4—412no—
148C25C23×C313yes—
149C25C23×C311yescharacteristic
150C22×D4—212no—
151C22×D4—212no—
152C22×D4—212no—
153C22×D4—26no—
154C22×D4—26no—
155C22×D4—26no—
156C22×D4—26no—
157C22×D4—26no—
158C22×D4—46no—
159C22×D4C3×D4112yescomplemented
160C23×C4—212no—
161C23×C4—26no—
162C23×C4—26no—
163C23×C4—46no—
164C23×C4C3×D4112yescomplemented
165C23×C4C23×C311yescharacteristic
166C23×C4C23×C311yescharacteristic
167C23×C4C23×C313yes—
168C23×C4C23×C313yes—
169C23×C4C23×C313yes—
170#121—296no—
171C2×C4⋊C4—212no—
172C2×C4⋊C4—26no—
173C2×C22⋊C4—212no—
174C2×C22⋊C4—212no—
175C2×C22⋊C4—212no—
176C2×C22⋊C4—26no—
177C2×C22⋊C4—26no—
178C2×C22⋊C4—26no—
179C2×C22⋊C4—26no—
180C2×C22⋊C4—26no—
181C2×C22⋊C4—46no—
182#95—26no—
183#95—26no—
184#95—26no—
185#95C3×D414yescomplemented
186C23×C3—26no—
187C23×C3—26no—
188C23×C3—26no—
189C23×C3—26no—
190C23×C3—26no—
191C23×C3—212no—
192C23×C3—26no—
193C23×C3—22no—
194C23×C3—26no—
195C23×C3—212no—
196C23×C3—26no—
197C23×C3—26no—
198C23×C3—212no—
199C23×C3—26no—
200C23×C3—43no—
201C23×C3—43no—
202C23×C3—46no—
203C23×C3—46no—
204C23×C3—43no—
205C23×C3—46no—
206C23×C3—43no—
207C23×C3—43no—
208C23×C3—46no—
209C23×C3—43no—
210C23×C3—46no—
211C23×C3—46no—
212C23×C3—46no—
213C23×C3—46no—
214C23×C3—46no—
215C23×C3—43no—
216C23×C3—43no—
217C23×C3—46no—
218C23×C3#120112yes—
219C23×C3#121112yes—
220C23×C3#12714yes—
221C23×C3C22×D413yes—
222C23×C3C22×D413yes—
223C23×C3C22×D411yescharacteristic
224C23×C3C2×#4111yescharacteristic
225C23×C3C2×#4113yes—
226C23×C3H(5,𝔽2)13yes—
227C23×C32−1+411yescharacteristic
228C3×D4—448no—
229C3×D4—448no—
230C2×C3×C4—26no—
231C2×C3×C4—26no—
232C2×C3×C4—212no—
233C2×C3×C4—212no—
234C2×C3×C4—26no—
235C2×C3×C4—212no—
236C2×C3×C4—26no—
237C2×C3×C4—26no—
238C2×C3×C4—26no—
239C2×C3×C4—448no—
240C2×C3×C4—43no—
241C2×C3×C4—46no—
242C2×C3×C4—43no—
243C2×C3×C4—46no—
244C2×C3×C4—46no—
245C2×C3×C4—43no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—43no—
249C2×C3×C4—43no—
250C2×C3×C4—43no—
251C2×C3×C4#121112yescomplemented
252C24—21no—
253C24—23no—
254C24—23no—
255C24—21no—
256C24—23no—
257C24—26no—
258C24—26no—
259C24—23no—
260C24—23no—
261C24—23no—
262C24—26no—
263C24—26no—
264C24—23no—
265C24—23no—
266C24C2×C3×D416yes—
267C24C2×C3×D416yes—
268C24C2×C3×D416yes—
269C24C3×#4112yes—
270C24C24×C311yescharacteristic
271C2×D4—224no—
272C2×D4—248no—
273C2×D4—224no—
274C2×D4—424no—
275C2×D4—424no—
276C2×D4—412no—
277C2×D4—412no—
278C2×D4—424no—
279C2×D4—412no—
280C2×D4—412no—
281C22×C4—26no—
282C22×C4—26no—
283C22×C4—224no—
284C22×C4—23no—
285C22×C4—23no—
286C22×C4—26no—
287C22×C4—26no—
288C22×C4—26no—
289C22×C4—23no—
290C22×C4—23no—
291C22×C4—26no—
292C22×C4—224no—
293C22×C4—23no—
294C22×C4—23no—
295C22×C4—21no—
296C22×C4—23no—
297C22×C4—23no—
298C22×C4—21no—
299C22×C4—412no—
300C22×C4—424no—
301C22×C4—424no—
302C22×C4C2×C3×D416yes—
303C22×C4C2×C3×D416yes—
304C22×C4C2×C3×D416yes—
305C22×C4C2×C3×D416yes—
306C22×C4C3×#4116yes—
307C4⋊C4—224no—
308C22⋊C4—248no—
309C22⋊C4—412no—
310C22⋊C4—44no—
311C22⋊C4—412no—
312C22×C3—24no—
313C22×C3—24no—
314C22×C3—212no—
315C22×C3—212no—
316C22×C3—212no—
317C22×C3—212no—
318C22×C3—46no—
319C22×C3—412no—
320C22×C3—46no—
321C22×C3—412no—
322C22×C3—46no—
323C22×C3—83no—
324C22×C3—83no—
325C22×C3—83no—
326C22×C3—86no—
327C22×C3—83no—
328C22×C3—86no—
329C22×C3#39218yescomplemented
330C22×C3C2×#12013yes—
331C22×C3C2×#12113yes—
332C22×C3C2×#12113yes—
333C22×C3C2×#12711yescharacteristic
334C22×C3#53213yes—
335C22×C3#53413yes—
336C22×C3#53713yes—
337C22×C3D4213yes—
338C22×C3#54616yes—
339C22×C3#54713yes—
340C22×C3#54811yescharacteristic
341C22×C3#55313yes—
342C3×C4—212no—
343C3×C4—46no—
344C3×C4—46no—
345C3×C4—81no—
346C3×C4—81no—
347C23—26no—
348C23—26no—
349C23—26no—
350C23—26no—
351C23—26no—
352C23—212no—
353C23—26no—
354C23—22no—
355C23—26no—
356C23—212no—
357C23—26no—
358C23—26no—
359C23—212no—
360C23—26no—
361C23—43no—
362C23—43no—
363C23—46no—
364C23—46no—
365C23—43no—
366C23—46no—
367C23—43no—
368C23—43no—
369C23—46no—
370C23—43no—
371C23—46no—
372C23—46no—
373C23—46no—
374C23—46no—
375C23—46no—
376C23—43no—
377C23—43no—
378C23—46no—
379C23C3×#120112yes—
380C23C3×#121112yes—
381C23C3×#12714yes—
382C23C22×C3×D413yes—
383C23C22×C3×D413yes—
384C23C22×C3×D411yescharacteristic
385C23C2×C3×#4111yescharacteristic
386C23C2×C3×#4113yes—
387C23C3×H(5,𝔽2)13yes—
388C23C3×2−1+411yescharacteristic
389D4—448no—
390D4—448no—
391C2×C4—26no—
392C2×C4—26no—
393C2×C4—212no—
394C2×C4—212no—
395C2×C4—26no—
396C2×C4—212no—
397C2×C4—26no—
398C2×C4—26no—
399C2×C4—26no—
400C2×C4—448no—
401C2×C4—43no—
402C2×C4—46no—
403C2×C4—43no—
404C2×C4—46no—
405C2×C4—46no—
406C2×C4—43no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—43no—
410C2×C4—43no—
411C2×C4—43no—
412C2×C4C3×#121112yescomplemented
413C2×C3—28no—
414C2×C3—86no—
415C2×C3C2×#39211yescharacteristic
416C2×C3#241013yes—
417C2×C3#241613yes—
418C2×C3#243013yes—
419C2×C3#254313yes—
420C2×C3#265611yescharacteristic
421C2×C3#285611yescharacteristic
422C22—24no—
423C22—212no—
424C22—212no—
425C22—212no—
426C22—212no—
427C22—24no—
428C22—46no—
429C22—412no—
430C22—46no—
431C22—412no—
432C22—46no—
433C22—83no—
434C22—83no—
435C22—86no—
436C22—83no—
437C22—86no—
438C22—83no—
439C22C3×#39218yescomplemented
440C22C2×C3×#12013yes—
441C22C2×C3×#12113yes—
442C22C2×C3×#12113yes—
443C22C2×C3×#12711yescharacteristic
444C22C3×#53213yes—
445C22C3×#53413yes—
446C22C3×#53713yes—
447C22C3×D4213yes—
448C22C3×#54616yes—
449C22C3×#54713yes—
450C22C3×#54811yescharacteristic
451C22C3×#55313yes—
452C4—212no—
453C4—46no—
454C4—46no—
455C4—81no—
456C4—81no—
457C3#1475711yescharacteristic, direct product factor, Sylow
458C2—28no—
459C2—86no—
460C2C2×C3×#39211yescharacteristic
461C2C3×#241013yes—
462C2C3×#241613yes—
463C2C3×#243013yes—
464C2C3×#254313yes—
465C2C3×#265611yescharacteristic
466C2C3×#285611yescharacteristic
subgroupG/H
center, socleC24×C3C24
commutator, Frattini, cosocleC24C24×C3
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C3 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×C3×#121C2semidirect
2. C2×C3×#392C2semidirect
3. C2×C3×#386C2semidirect
4. C22×C3×#95C2semidirect
5. C3×#1444C2semidirect
6. C3×#1439C2semidirect
7. C3×#1438C2semidirect
8. #14757C3direct (N × H)
26. C22×#121C2×C3semidirect
27. C2×#392C2×C3semidirect
28. C2×#386C2×C3semidirect
29. C22×#95C2×C3semidirect
30. #1444C2×C3semidirect
31. #1439C2×C3semidirect
32. #1438C2×C3semidirect
44. C22×C3×D4D4semidirect
49. C23×C3×C4D4semidirect
70. C3×#95D4semidirect
159. C22×D4C3×D4semidirect
164. C23×C4C3×D4semidirect
185. #95C3×D4semidirect
251. C2×C3×C4#121semidirect
329. C22×C3#392semidirect
412. C2×C4C3×#121semidirect
439. C22C3×#392semidirect
457. C3#14757direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#1212
C2×C3×#3922
C2×C3×#3862
C22×C3×#952
C3×#14442
C3×#14392
C3×#14382
#147573