Finite Group Databasebeta

← C3×#25758 #10644365

Subgroups — C3×#25758 #10644365

Order 768.

1546 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#547C211yesmaximal, characteristic
2C2×C3×#494C211yesmaximal, characteristic, complemented
3C2×C3×#461C211yesmaximal, characteristic, complemented
4C2×C3×#452C214yesmaximal, complemented
5C3×#2542C211yesmaximal, characteristic, complemented
6C3×#2001C214yesmaximal, complemented
7C3×#1981C211yesmaximal, characteristic, complemented
8C3×#1954C211yesmaximal, characteristic, complemented
9C3×C4×#102C211yesmaximal, characteristic, complemented
10#25758C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2212yes—
12C22×C3×Q16C2214yes—
13C22×C3×SD16C2212yes—
14C3×#547—24no—
15C3×#547—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#123—22no—
19C2×C3×C22⋊Q8C2212yes—
20C2×C3×C22⋊Q8C2214yes—
21C2×C3×#121—21no—
22C2×C3×C4×D4C2211yescharacteristic
23C2×C3×C4×D4C2211yescharacteristic
24C22×C3×C4⋊C4—21no—
25C3×#494—24no—
26C3×#461—24no—
27C3×#452—216no—
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#103C2212yes—
30C2×C3×#103C2214yes—
31C2×C3×#102C2212yes—
32C2×C3×C22⋊C8C2214yes—
33C2×C3×C4×C8C2211yescharacteristic
34C3×#399—22no—
35C3×#390—21no—
36C3×#384—21no—
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×#340C2212yes—
39C2×#547C2×C311yescharacteristic
40C2×#494C2×C311yescharacteristic, complemented
41C2×#461C2×C311yescharacteristic, complemented
42C2×#452C2×C314yescomplemented
43#2542C2×C311yescharacteristic, complemented
44#2001C2×C314yescomplemented
45#1981C2×C311yescharacteristic, complemented
46#1954C2×C311yescharacteristic, complemented
47C4×#102C2×C311yescharacteristic, complemented
48C2×C3×#41—24no—
49C2×C3×#41—24no—
50C2×C3×#41—28no—
51C2×C3×#41—44no—
52C22×C3×Q8C2312yes—
53C22×C3×Q8C2312yes—
54C22×C3×D4—22no—
55C22×C3×D4D411yescharacteristic
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4—22no—
58C23×C3×C4—21no—
59C23×C3×C4C2312yes—
60C2×C3×Q16—28no—
61C2×C3×Q16—28no—
62C2×C3×Q16—44no—
63C2×C3×SD16—28no—
64C2×C3×SD16—24no—
65C22×C3×C8C2312yes—
66C22×C3×C8C2312yes—
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8—22no—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8D414yes—
71C3×#123—216no—
72C3×C22⋊Q8—24no—
73C3×C22⋊Q8—24no—
74C3×C22⋊Q8—28no—
75C3×C22⋊Q8—44no—
76C3×#121—28no—
77C3×C4×D4—22no—
78C3×C4×D4—22no—
79C3×C4×D4—22no—
80C3×C4×D4D414yes—
81C2×C3×C4⋊C4—28no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—24no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—22no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—24no—
94C2×C3×C22⋊C4D412yes—
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97C3×C4⋊C8—22no—
98C3×#103—28no—
99C3×#103—24no—
100C3×#102D418yescomplemented
101C3×C22⋊C8—28no—
102C3×C4×C8—22no—
103C3×#95—21no—
104C22×#41C22×C312yes—
105C22×Q16C22×C314yes—
106C22×SD16C22×C312yes—
107#547—24no—
108#547—24no—
109C2×C4⋊Q8C22×C311yescharacteristic
110C2×C4⋊Q8C22×C311yescharacteristic
111C2×#123—22no—
112C2×C22⋊Q8C22×C312yes—
113C2×C22⋊Q8C22×C314yes—
114C2×#121—21no—
115C2×C4×D4C22×C311yescharacteristic
116C2×C4×D4C22×C311yescharacteristic
117C22×C4⋊C4—21no—
118#494—24no—
119#461—24no—
120#452—216no—
121C2×C4⋊C8C22×C311yescharacteristic
122C2×#103C22×C314yes—
123C2×#103C22×C312yes—
124C2×#102C22×C312yes—
125C2×C22⋊C8C22×C314yes—
126C2×C4×C8C22×C311yescharacteristic
127#399—22no—
128#390—21no—
129#384—21no—
130C4×C4⋊C4C22×C311yescharacteristic
131#340C22×C312yes—
132C24×C3—21no—
133C24×C3C2×D412yes—
134C3×#41—416no—
135C3×#41—48no—
136C3×#41—48no—
137C2×C3×Q8—22no—
138C2×C3×Q8—24no—
139C2×C3×Q8—22no—
140C2×C3×Q8—24no—
141C2×C3×Q8—22no—
142C2×C3×Q8—24no—
143C2×C3×Q8—24no—
144C2×C3×Q8C2×D414yes—
145C2×C3×D4—28no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—22no—
149C2×C3×D4—21no—
150C2×C3×D4—22no—
151C2×C3×D4—48no—
152C2×C3×D4C2×D414yes—
153C2×C3×D4C2×D412yes—
154C22×C3×C4—28no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—24no—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—24no—
163C22×C3×C4—22no—
164C22×C3×C4—48no—
165C22×C3×C4—42no—
166C22×C3×C4—42no—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2411yescharacteristic
171C3×Q16—416no—
172C3×SD16—48no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C2×C3×C8—24no—
176C2×C3×C8—22no—
177C2×C3×C8C2×D414yes—
178C3×C4⋊C4—28no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—24no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—48no—
185C3×C4⋊C4—42no—
186C3×C4⋊C4—42no—
187C3×C4⋊C4C2×D414yes—
188C3×C4⋊C4C2×D412yes—
189C3×C22⋊C4—28no—
190C3×C22⋊C4—24no—
191C3×C22⋊C4—24no—
192C3×C22⋊C4—48no—
193C3×C42—42no—
194C3×C42C2×D412yes—
195C3×C42C2×D412yes—
196C2×#41—24no—
197C2×#41—24no—
198C2×#41—28no—
199C2×#41—44no—
200C22×Q8C23×C312yes—
201C22×Q8C23×C312yes—
202C22×D4—22no—
203C22×D4C3×D411yescharacteristic
204C22×D4C23×C311yescharacteristic
205C23×C4—21no—
206C23×C4—22no—
207C23×C4C23×C312yes—
208C2×Q16—28no—
209C2×Q16—28no—
210C2×Q16—44no—
211C2×SD16—28no—
212C2×SD16—24no—
213C22×C8C23×C312yes—
214C22×C8C23×C312yes—
215C4⋊Q8—22no—
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218C4⋊Q8C3×D414yes—
219#123—216no—
220C22⋊Q8—24no—
221C22⋊Q8—24no—
222C22⋊Q8—28no—
223C22⋊Q8—44no—
224#121—28no—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4C3×D414yes—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C3×D411yescharacteristic
237C2×C4⋊C4C23×C311yescharacteristic
238C2×C4⋊C4C23×C312yes—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4C3×D412yes—
243C2×C42—21no—
244C2×C42C23×C311yescharacteristic
245C4⋊C8—22no—
246#103—28no—
247#103—24no—
248#102C3×D418yescomplemented
249C22⋊C8—28no—
250C4×C8—22no—
251#95—21no—
252C23×C3—22no—
253C23×C3—24no—
254C23×C3—22no—
255C23×C3—24no—
256C23×C3—22no—
257C23×C3—22no—
258C23×C3—24no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3—42no—
262C23×C3C22×D411yescharacteristic
263C3×Q8—28no—
264C3×Q8—42no—
265C3×Q8—42no—
266C3×Q8—42no—
267C3×Q8—42no—
268C3×Q8—44no—
269C3×D4—22no—
270C3×D4—24no—
271C3×D4—416no—
272C3×D4—44no—
273C3×D4#12714yes—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—24no—
278C2×C3×C4—24no—
279C2×C3×C4—24no—
280C2×C3×C4—416no—
281C2×C3×C4—44no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—44no—
289C2×C3×C4—48no—
290C2×C3×C4—42no—
291C2×C3×C4—44no—
292C2×C3×C4#12012yes—
293C2×C3×C4#12012yes—
294C2×C3×C4#13512yes—
295C2×C3×C4#13712yes—
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C22×D412yes—
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4H(5,𝔽2)12yes—
300C3×C8—24no—
301C3×C8—42no—
302C24—21no—
303C24C2×C3×D412yes—
304#41—416no—
305#41—48no—
306#41—48no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—22no—
310C2×Q8—24no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8C2×C3×D414yes—
315C2×D4—28no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—21no—
320C2×D4—22no—
321C2×D4—48no—
322C2×D4C2×C3×D414yes—
323C2×D4C2×C3×D412yes—
324C22×C4—28no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—48no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4C2×C3×D412yes—
338C22×C4C2×C3×D411yescharacteristic
339C22×C4C2×C3×D411yescharacteristic
340C22×C4C24×C311yescharacteristic
341Q16—416no—
342SD16—48no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—24no—
346C2×C8—22no—
347C2×C8C2×C3×D414yes—
348C4⋊C4—28no—
349C4⋊C4—22no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—48no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4C2×C3×D414yes—
358C4⋊C4C2×C3×D412yes—
359C22⋊C4—28no—
360C22⋊C4—24no—
361C22⋊C4—24no—
362C22⋊C4—48no—
363C42—42no—
364C42C2×C3×D412yes—
365C42C2×C3×D412yes—
366C22×C3—24no—
367C22×C3—42no—
368C22×C3—42no—
369C22×C3—44no—
370C22×C3—44no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—44no—
375C22×C3—44no—
376C22×C3C2×#12011yescharacteristic
377C22×C3D4212yes—
378C22×C3C2×#13511yescharacteristic
379C22×C3C2×#13711yescharacteristic
380C22×C3#57812yes—
381C3×C4—22no—
382C3×C4—42no—
383C3×C4—44no—
384C3×C4—42no—
385C3×C4—44no—
386C3×C4—82no—
387C3×C4#45214yescomplemented
388C3×C4C2×#12711yescharacteristic
389C3×C4#53511yescharacteristic
390C3×C4D4212yes—
391C23—22no—
392C23—24no—
393C23—22no—
394C23—22no—
395C23—24no—
396C23—22no—
397C23—24no—
398C23—41no—
399C23—41no—
400C23—42no—
401C23C22×C3×D411yescharacteristic
402Q8—28no—
403Q8—42no—
404Q8—42no—
405Q8—42no—
406Q8—42no—
407Q8—44no—
408D4—22no—
409D4—24no—
410D4—416no—
411D4—44no—
412D4C3×#12714yes—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—416no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—48no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4C3×#12012yes—
432C2×C4C3×#12012yes—
433C2×C4C3×#13512yes—
434C2×C4C3×#13712yes—
435C2×C4C22×C3×D411yescharacteristic
436C2×C4C22×C3×D412yes—
437C2×C4C22×C3×D411yescharacteristic
438C2×C4C3×H(5,𝔽2)12yes—
439C8—24no—
440C8—42no—
441C2×C3—44no—
442C2×C3—82no—
443C2×C3#243311yescharacteristic
444C2×C3C2×#45211yescharacteristic
445C2×C3#301111yescharacteristic
446C2×C3#327612yes—
447C2×C3#328312yes—
448C22—24no—
449C22—42no—
450C22—42no—
451C22—44no—
452C22—44no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—44no—
457C22—44no—
458C22C2×C3×#12011yescharacteristic
459C22C3×D4212yes—
460C22C2×C3×#13511yescharacteristic
461C22C2×C3×#13711yescharacteristic
462C22C3×#57812yes—
463C4—22no—
464C4—42no—
465C4—44no—
466C4—42no—
467C4—44no—
468C4—82no—
469C4C3×#45214yescomplemented
470C4C2×C3×#12711yescharacteristic
471C4C3×#53511yescharacteristic
472C4C3×D4212yes—
473C3#2575811yescharacteristic, direct product factor, Sylow
474C2—44no—
475C2—82no—
476C2C3×#243311yescharacteristic
477C2C2×C3×#45211yescharacteristic
478C2C3×#301111yescharacteristic
479C2C3×#327612yes—
480C2C3×#328312yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×D4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#494C2semidirect
3. C2×C3×#461C2semidirect
4. C2×C3×#452C2semidirect
5. C3×#2542C2semidirect
6. C3×#2001C2semidirect
7. C3×#1981C2semidirect
8. C3×#1954C2semidirect
9. C3×C4×#102C2semidirect
10. #25758C3direct (N × H)
40. C2×#494C2×C3semidirect
41. C2×#461C2×C3semidirect
42. C2×#452C2×C3semidirect
43. #2542C2×C3semidirect
44. #2001C2×C3semidirect
45. #1981C2×C3semidirect
46. #1954C2×C3semidirect
47. C4×#102C2×C3semidirect
100. C3×#102D4semidirect
248. #102C3×D4semidirect
387. C3×C4#452semidirect
469. C4C3×#452semidirect
473. C3#25758direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5472
C2×C3×#4942
C2×C3×#4612
C2×C3×#4522
C3×#25422
C3×#20012
C3×#19812
C3×#19542
C3×C4×#1022
#257583