Finite Group Databasebeta

← C3×#25755 #10644362

Subgroups — C3×#25755 #10644362

Order 768.

1598 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#550C211yesmaximal, characteristic
2C2×C3×#492C211yesmaximal, characteristic, complemented
3C2×C3×#462C211yesmaximal, characteristic, complemented
4C2×C3×#452C214yesmaximal, complemented
5C3×#2542C211yesmaximal, characteristic, complemented
6C3×#2000C214yesmaximal, complemented
7C3×#1982C211yesmaximal, characteristic, complemented
8C3×#1954C211yesmaximal, characteristic, complemented
9C3×C4×#103C211yesmaximal, characteristic, complemented
10#25755C311yesmaximal, characteristic, direct product factor, Sylow
11C22×C3×#41C2212yes—
12C22×C3×Q16C2212yes—
13C22×C3×SD16C2214yes—
14C3×#550—24no—
15C3×#550—24no—
16C2×C3×C4⋊Q8C2211yescharacteristic
17C2×C3×#127—21no—
18C2×C3×#127C2211yescharacteristic
19C2×C3×C22⋊Q8C2214yes—
20C2×C3×#121—22no—
21C2×C3×#121C2212yes—
22C2×C3×C4×Q8C2211yescharacteristic
23C2×C3×C4×D4—21no—
24C2×C3×C4×D4C2211yescharacteristic
25C3×#492—24no—
26C3×#462—24no—
27C3×#452—216no—
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#103C2212yes—
30C2×C3×#103C2212yes—
31C2×C3×#102C2214yes—
32C2×C3×C22⋊C8C2214yes—
33C2×C3×C4×C8C2211yescharacteristic
34C3×#399—22no—
35C3×#390—21no—
36C3×#384—21no—
37C3×C4×C4⋊C4C2211yescharacteristic
38C3×#340C2212yes—
39C2×#550C2×C311yescharacteristic
40C2×#492C2×C311yescharacteristic, complemented
41C2×#462C2×C311yescharacteristic, complemented
42C2×#452C2×C314yescomplemented
43#2542C2×C311yescharacteristic, complemented
44#2000C2×C314yescomplemented
45#1982C2×C311yescharacteristic, complemented
46#1954C2×C311yescharacteristic, complemented
47C4×#103C2×C311yescharacteristic, complemented
48C2×C3×#41—24no—
49C2×C3×#41—24no—
50C2×C3×#41—28no—
51C2×C3×#41—44no—
52C22×C3×Q8C2312yes—
53C22×C3×Q8C2311yescharacteristic
54C22×C3×D4—22no—
55C22×C3×D4—22no—
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4D411yescharacteristic
59C22×C3×D4C2312yes—
60C23×C3×C4—22no—
61C23×C3×C4C2312yes—
62C2×C3×Q16—28no—
63C2×C3×Q16—24no—
64C2×C3×SD16—28no—
65C2×C3×SD16—28no—
66C2×C3×SD16—44no—
67C22×C3×C8C2312yes—
68C22×C3×C8C2312yes—
69C3×C4⋊Q8—22no—
70C3×C4⋊Q8—22no—
71C3×#127—28no—
72C3×#127—22no—
73C3×#127D414yes—
74C3×C22⋊Q8—28no—
75C3×C22⋊Q8—44no—
76C3×#121—216no—
77C3×#121—24no—
78C3×#121—24no—
79C3×C4×Q8—22no—
80C3×C4×Q8D414yes—
81C3×C4×D4—28no—
82C3×C4×D4—22no—
83C3×C4×D4—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—24no—
93C2×C3×C22⋊C4D412yes—
94C2×C3×C42—21no—
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97C3×C4⋊C8—22no—
98C3×#103—24no—
99C3×#103D418yescomplemented
100C3×#102—28no—
101C3×C22⋊C8—28no—
102C3×C4×C8—22no—
103C3×#95—21no—
104C22×#41C22×C312yes—
105C22×Q16C22×C312yes—
106C22×SD16C22×C314yes—
107#550—24no—
108#550—24no—
109C2×C4⋊Q8C22×C311yescharacteristic
110C2×#127—21no—
111C2×#127C22×C311yescharacteristic
112C2×C22⋊Q8C22×C314yes—
113C2×#121—22no—
114C2×#121C22×C312yes—
115C2×C4×Q8C22×C311yescharacteristic
116C2×C4×D4—21no—
117C2×C4×D4C22×C311yescharacteristic
118#492—24no—
119#462—24no—
120#452—216no—
121C2×C4⋊C8C22×C311yescharacteristic
122C2×#103C22×C312yes—
123C2×#103C22×C312yes—
124C2×#102C22×C314yes—
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—22no—
133C24×C3C2×D412yes—
134C3×#41—416no—
135C3×#41—48no—
136C3×#41—48no—
137C2×C3×Q8—21no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—24no—
141C2×C3×Q8—22no—
142C2×C3×Q8—24no—
143C2×C3×Q8C2×D414yes—
144C2×C3×Q8C2×D412yes—
145C2×C3×D4—28no—
146C2×C3×D4—28no—
147C2×C3×D4—22no—
148C2×C3×D4—24no—
149C2×C3×D4—24no—
150C2×C3×D4—22no—
151C2×C3×D4—22no—
152C2×C3×D4—24no—
153C2×C3×D4—22no—
154C2×C3×D4—24no—
155C2×C3×D4—48no—
156C2×C3×D4—42no—
157C2×C3×D4—42no—
158C2×C3×D4—82no—
159C2×C3×D4—84no—
160C2×C3×D4C2×D414yes—
161C22×C3×C4—21no—
162C22×C3×C4—28no—
163C22×C3×C4—22no—
164C22×C3×C4—21no—
165C22×C3×C4—24no—
166C22×C3×C4—22no—
167C22×C3×C4—22no—
168C22×C3×C4—24no—
169C22×C3×C4—22no—
170C22×C3×C4—48no—
171C22×C3×C4C2×D412yes—
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2×D411yescharacteristic
174C22×C3×C4C2411yescharacteristic
175C3×Q16—48no—
176C3×SD16—416no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—24no—
180C2×C3×C8—22no—
181C2×C3×C8C2×D414yes—
182C3×C4⋊C4—24no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—22no—
186C3×C4⋊C4—48no—
187C3×C4⋊C4—42no—
188C3×C4⋊C4—42no—
189C3×C4⋊C4C2×D414yes—
190C3×C4⋊C4C2×D412yes—
191C3×C22⋊C4—28no—
192C3×C22⋊C4—24no—
193C3×C22⋊C4—48no—
194C3×C42—24no—
195C3×C42—42no—
196C3×C42C2×D412yes—
197C3×C42C2×D412yes—
198C2×#41—24no—
199C2×#41—24no—
200C2×#41—28no—
201C2×#41—44no—
202C22×Q8C23×C312yes—
203C22×Q8C23×C311yescharacteristic
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—21no—
207C22×D4—21no—
208C22×D4C3×D411yescharacteristic
209C22×D4C23×C312yes—
210C23×C4—22no—
211C23×C4C23×C312yes—
212C2×Q16—28no—
213C2×Q16—24no—
214C2×SD16—28no—
215C2×SD16—28no—
216C2×SD16—44no—
217C22×C8C23×C312yes—
218C22×C8C23×C312yes—
219C4⋊Q8—22no—
220C4⋊Q8—22no—
221#127—28no—
222#127—22no—
223#127C3×D414yes—
224C22⋊Q8—28no—
225C22⋊Q8—44no—
226#121—216no—
227#121—24no—
228#121—24no—
229C4×Q8—22no—
230C4×Q8C3×D414yes—
231C4×D4—28no—
232C4×D4—22no—
233C4×D4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4C3×D411yescharacteristic
239C2×C4⋊C4C23×C311yescharacteristic
240C2×C4⋊C4C23×C312yes—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4C3×D412yes—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42C23×C311yescharacteristic
247C4⋊C8—22no—
248#103—24no—
249#103C3×D418yescomplemented
250#102—28no—
251C22⋊C8—28no—
252C4×C8—22no—
253#95—21no—
254C23×C3—22no—
255C23×C3—24no—
256C23×C3—24no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—24no—
260C23×C3—44no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—44no—
264C23×C3C22×D411yescharacteristic
265C3×Q8—22no—
266C3×Q8—24no—
267C3×Q8—42no—
268C3×Q8—42no—
269C3×Q8—44no—
270C3×Q8#12714yes—
271C3×D4—28no—
272C3×D4—416no—
273C3×D4—42no—
274C3×D4—42no—
275C3×D4—44no—
276C3×D4—48no—
277C3×D4—84no—
278C3×D4—88no—
279C2×C3×C4—22no—
280C2×C3×C4—24no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—24no—
286C2×C3×C4—416no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—44no—
293C2×C3×C4—42no—
294C2×C3×C4—44no—
295C2×C3×C4#12012yes—
296C2×C3×C4#12012yes—
297C2×C3×C4#13512yes—
298C2×C3×C4#13712yes—
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C22×D412yes—
301C2×C3×C4C22×D411yescharacteristic
302C2×C3×C4H(5,𝔽2)12yes—
303C3×C8—24no—
304C3×C8—42no—
305C24—22no—
306C24C2×C3×D412yes—
307#41—416no—
308#41—48no—
309#41—48no—
310C2×Q8—21no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—24no—
314C2×Q8—22no—
315C2×Q8—24no—
316C2×Q8C2×C3×D414yes—
317C2×Q8C2×C3×D412yes—
318C2×D4—28no—
319C2×D4—28no—
320C2×D4—22no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—24no—
326C2×D4—22no—
327C2×D4—24no—
328C2×D4—48no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—82no—
332C2×D4—84no—
333C2×D4C2×C3×D414yes—
334C22×C4—21no—
335C22×C4—28no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—48no—
344C22×C4C2×C3×D412yes—
345C22×C4C2×C3×D411yescharacteristic
346C22×C4C2×C3×D411yescharacteristic
347C22×C4C24×C311yescharacteristic
348Q16—48no—
349SD16—416no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8C2×C3×D414yes—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—48no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4C2×C3×D414yes—
363C4⋊C4C2×C3×D412yes—
364C22⋊C4—28no—
365C22⋊C4—24no—
366C22⋊C4—48no—
367C42—24no—
368C42—42no—
369C42C2×C3×D412yes—
370C42C2×C3×D412yes—
371C22×C3—44no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—44no—
375C22×C3—42no—
376C22×C3—44no—
377C22×C3—44no—
378C22×C3—44no—
379C22×C3—84no—
380C22×C3—84no—
381C22×C3C2×#12011yescharacteristic
382C22×C3D4212yes—
383C22×C3C2×#13511yescharacteristic
384C22×C3C2×#13711yescharacteristic
385C22×C3#57712yes—
386C3×C4—24no—
387C3×C4—22no—
388C3×C4—42no—
389C3×C4—42no—
390C3×C4—44no—
391C3×C4—82no—
392C3×C4#45214yescomplemented
393C3×C4C2×#12711yescharacteristic
394C3×C4#53511yescharacteristic
395C3×C4D4212yes—
396C23—22no—
397C23—22no—
398C23—24no—
399C23—22no—
400C23—24no—
401C23—24no—
402C23—42no—
403C23—42no—
404C23—44no—
405C23—44no—
406C23C22×C3×D411yescharacteristic
407Q8—22no—
408Q8—24no—
409Q8—42no—
410Q8—42no—
411Q8—44no—
412Q8C3×#12714yes—
413D4—28no—
414D4—416no—
415D4—44no—
416D4—48no—
417D4—42no—
418D4—42no—
419D4—84no—
420D4—88no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—416no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4C3×#12012yes—
438C2×C4C3×#12012yes—
439C2×C4C3×#13512yes—
440C2×C4C3×#13712yes—
441C2×C4C22×C3×D411yescharacteristic
442C2×C4C22×C3×D412yes—
443C2×C4C22×C3×D411yescharacteristic
444C2×C4C3×H(5,𝔽2)12yes—
445C8—24no—
446C8—42no—
447C2×C3—82no—
448C2×C3—84no—
449C2×C3#243311yescharacteristic
450C2×C3C2×#45211yescharacteristic
451C2×C3#300911yescharacteristic
452C2×C3#327812yes—
453C2×C3#328412yes—
454C22—44no—
455C22—42no—
456C22—42no—
457C22—44no—
458C22—42no—
459C22—44no—
460C22—44no—
461C22—44no—
462C22—84no—
463C22—84no—
464C22C2×C3×#12011yescharacteristic
465C22C3×D4212yes—
466C22C2×C3×#13511yescharacteristic
467C22C2×C3×#13711yescharacteristic
468C22C3×#57712yes—
469C4—24no—
470C4—22no—
471C4—42no—
472C4—42no—
473C4—44no—
474C4—82no—
475C4C3×#45214yescomplemented
476C4C2×C3×#12711yescharacteristic
477C4C3×#53511yescharacteristic
478C4C3×D4212yes—
479C3#2575511yescharacteristic, direct product factor, Sylow
480C2—82no—
481C2—84no—
482C2C3×#243311yescharacteristic
483C2C2×C3×#45211yescharacteristic
484C2C3×#300911yescharacteristic
485C2C3×#327812yes—
486C2C3×#328412yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#550C2
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×#492C2semidirect
3. C2×C3×#462C2semidirect
4. C2×C3×#452C2semidirect
5. C3×#2542C2semidirect
6. C3×#2000C2semidirect
7. C3×#1982C2semidirect
8. C3×#1954C2semidirect
9. C3×C4×#103C2semidirect
10. #25755C3direct (N × H)
40. C2×#492C2×C3semidirect
41. C2×#462C2×C3semidirect
42. C2×#452C2×C3semidirect
43. #2542C2×C3semidirect
44. #2000C2×C3semidirect
45. #1982C2×C3semidirect
46. #1954C2×C3semidirect
47. C4×#103C2×C3semidirect
99. C3×#103D4semidirect
249. #103C3×D4semidirect
392. C3×C4#452semidirect
475. C4C3×#452semidirect
479. C3#25755direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5502
C2×C3×#4922
C2×C3×#4622
C2×C3×#4522
C3×#25422
C3×#20002
C3×#19822
C3×#19542
C3×C4×#1032
#257553