Finite Group Databasebeta

← C3×#29597 #10648204

Subgroups — C3×#29597 #10648204

Order 768.

1442 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C22⋊Q8C211yesmaximal, characteristic
2C2×C3×#484C211yesmaximal, characteristic, complemented
3C2×C3×#451C211yesmaximal, characteristic, complemented
4C3×#2848C211yesmaximal, characteristic
5C3×#2036C214yesmaximal, complemented
6C3×#2001C214yesmaximal, complemented
7C3×#1824C211yesmaximal, characteristic, complemented
8C3×#1787C211yesmaximal, characteristic, complemented
9C3×#1779C211yesmaximal, characteristic
10#29597C311yesmaximal, characteristic, direct product factor, Sylow
11C23×C3×Q8C2211yescharacteristic
12C24×C3×C4C2211yescharacteristic
13C22×C3×Q16C2212yescomplemented
14C2×C3×C22⋊Q8—24no—
15C2×C3×C22⋊Q8—22no—
16C2×C3×C22⋊Q8—24no—
17C2×C3×C22⋊Q8C2214yes—
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×C3×C22⋊Q8C2212yes—
20C22×C3×C4⋊C4—21no—
21C22×C3×C4⋊C4C2211yescharacteristic
22C22×C3×C22⋊C4—21no—
23C3×#484—24no—
24C3×#451—24no—
25C2×C3×#107C2212yescomplemented
26C2×C3×#103C2212yescomplemented
27C2×C3×#103C2212yescomplemented
28C2×C3×#103C2214yescomplemented
29C2×C3×C22⋊C8C2211yescharacteristic
30C2×C3×C22⋊C8C2211yescharacteristic
31C2×C3×C22⋊C8C2214yes—
32C3×#385—21no—
33C3×#385—22no—
34C3×#380C2212yes—
35C3×#380C2211yescharacteristic
36C3×#379—21no—
37C3×#340C2214yescomplemented
38C22×C22⋊Q8C2×C311yescharacteristic
39C2×#484C2×C311yescharacteristic, complemented
40C2×#451C2×C311yescharacteristic, complemented
41#2848C2×C311yescharacteristic
42#2036C2×C314yescomplemented
43#2001C2×C314yescomplemented
44#1824C2×C311yescharacteristic, complemented
45#1787C2×C311yescharacteristic, complemented
46#1779C2×C311yescharacteristic
47C25×C3D411yescharacteristic
48C22×C3×Q8—22no—
49C22×C3×Q8—21no—
50C22×C3×Q8—44no—
51C22×C3×Q8D414yes—
52C22×C3×Q8C2311yescharacteristic
53C22×C3×Q8C2312yes—
54C23×C3×C4—21no—
55C23×C3×C4—21no—
56C23×C3×C4—22no—
57C23×C3×C4—22no—
58C23×C3×C4—44no—
59C23×C3×C4D411yescharacteristic
60C23×C3×C4D412yes—
61C23×C3×C4C2312yes—
62C23×C3×C4C2311yescharacteristic
63C2×C3×Q16—24no—
64C2×C3×Q16—44no—
65C22×C3×C8C2312yes—
66C22×C3×C8C2312yes—
67C3×C22⋊Q8—216no—
68C3×C22⋊Q8—28no—
69C3×C22⋊Q8—28no—
70C3×C22⋊Q8—22no—
71C3×C22⋊Q8—22no—
72C3×C22⋊Q8—44no—
73C2×C3×C4⋊C4—28no—
74C2×C3×C4⋊C4—24no—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—22no—
77C2×C3×C4⋊C4—22no—
78C2×C3×C4⋊C4—21no—
79C2×C3×C4⋊C4D414yes—
80C2×C3×C4⋊C4C2311yescharacteristic
81C2×C3×C4⋊C4C2312yes—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C22⋊C4—28no—
84C2×C3×C22⋊C4—24no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4—21no—
89C3×#107—24no—
90C3×#103—24no—
91C3×#103—24no—
92C3×#103—44no—
93C3×C22⋊C8—22no—
94C3×C22⋊C8—22no—
95C3×C22⋊C8—44no—
96C3×#95—22no—
97C3×#95—21no—
98C23×Q8C22×C311yescharacteristic
99C24×C4C22×C311yescharacteristic
100C22×Q16C22×C312yescomplemented
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8C22×C311yescharacteristic
105C2×C22⋊Q8C22×C314yes—
106C2×C22⋊Q8C22×C312yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4C22×C311yescharacteristic
109C22×C22⋊C4—21no—
110#484—24no—
111#451—24no—
112C2×#107C22×C312yescomplemented
113C2×#103C22×C312yescomplemented
114C2×#103C22×C312yescomplemented
115C2×#103C22×C314yescomplemented
116C2×C22⋊C8C22×C311yescharacteristic
117C2×C22⋊C8C22×C311yescharacteristic
118C2×C22⋊C8C22×C314yes—
119#385—21no—
120#385—22no—
121#380C22×C311yescharacteristic
122#380C22×C312yes—
123#379—21no—
124#340C22×C314yescomplemented
125C24×C3—22no—
126C24×C3—22no—
127C24×C3—42no—
128C24×C3—42no—
129C24×C3Q1614yescomplemented
130C24×C3C2×D412yes—
131C24×C3C2×D411yescharacteristic
132C2×C3×Q8—24no—
133C2×C3×Q8—22no—
134C2×C3×Q8—21no—
135C2×C3×Q8—21no—
136C2×C3×Q8—24no—
137C2×C3×Q8—24no—
138C2×C3×Q8—24no—
139C2×C3×Q8—48no—
140C2×C3×Q8—44no—
141C2×C3×Q8—48no—
142C2×C3×Q8C2×D414yes—
143C22×C3×C4—24no—
144C22×C3×C4—24no—
145C22×C3×C4—24no—
146C22×C3×C4—22no—
147C22×C3×C4—22no—
148C22×C3×C4—22no—
149C22×C3×C4—24no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—24no—
153C22×C3×C4—24no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—21no—
157C22×C3×C4—42no—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4—48no—
161C22×C3×C4—48no—
162C22×C3×C4—44no—
163C22×C3×C4—44no—
164C22×C3×C4C2×D412yes—
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2411yescharacteristic
169C3×Q16—48no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—24no—
173C2×C3×C8—24no—
174C3×C4⋊C4—216no—
175C3×C4⋊C4—24no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—24no—
180C3×C4⋊C4—44no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4C2×D414yes—
183C3×C22⋊C4—216no—
184C3×C22⋊C4—44no—
185C3×C22⋊C4—44no—
186C3×C22⋊C4—42no—
187C3×C22⋊C4—42no—
188C25C3×D411yescharacteristic
189C22×Q8—22no—
190C22×Q8—21no—
191C22×Q8—44no—
192C22×Q8C3×D414yes—
193C22×Q8C23×C311yescharacteristic
194C22×Q8C23×C312yes—
195C23×C4—21no—
196C23×C4—21no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—44no—
200C23×C4C3×D411yescharacteristic
201C23×C4C3×D412yes—
202C23×C4C23×C312yes—
203C23×C4C23×C311yescharacteristic
204C2×Q16—24no—
205C2×Q16—44no—
206C22×C8C23×C312yes—
207C22×C8C23×C312yes—
208C22⋊Q8—216no—
209C22⋊Q8—28no—
210C22⋊Q8—28no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—44no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C3×D414yes—
221C2×C4⋊C4C23×C312yes—
222C2×C4⋊C4C23×C311yescharacteristic
223C2×C4⋊C4C23×C312yes—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230#107—24no—
231#103—24no—
232#103—24no—
233#103—44no—
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—44no—
237#95—22no—
238#95—21no—
239C23×C3—22no—
240C23×C3—22no—
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—22no—
244C23×C3—22no—
245C23×C3—22no—
246C23×C3—22no—
247C23×C3—28no—
248C23×C3—48no—
249C23×C3—44no—
250C23×C3—48no—
251C23×C3—44no—
252C23×C3#12012yes—
253C23×C3C2×Q1614yes—
254C23×C3C2×Q1612yes—
255C23×C3#13712yes—
256C23×C3C22×D411yescharacteristic
257C3×Q8—24no—
258C3×Q8—24no—
259C3×Q8—48no—
260C3×Q8—44no—
261C3×Q8—42no—
262C3×Q8—42no—
263C2×C3×C4—22no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—24no—
267C2×C3×C4—24no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—48no—
272C2×C3×C4—48no—
273C2×C3×C4—44no—
274C2×C3×C4—48no—
275C2×C3×C4—44no—
276C2×C3×C4—44no—
277C2×C3×C4—44no—
278C2×C3×C4—44no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4#12012yes—
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4H(5,𝔽2)14yes—
290C3×C8—42no—
291C3×C8—42no—
292C24—22no—
293C24—22no—
294C24—42no—
295C24—42no—
296C24C3×Q1614yescomplemented
297C24C2×C3×D412yes—
298C24C2×C3×D411yescharacteristic
299C2×Q8—24no—
300C2×Q8—22no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—24no—
304C2×Q8—24no—
305C2×Q8—24no—
306C2×Q8—48no—
307C2×Q8—44no—
308C2×Q8—48no—
309C2×Q8C2×C3×D414yes—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—48no—
328C22×C4—48no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4C2×C3×D412yes—
332C22×C4C2×C3×D411yescharacteristic
333C22×C4C2×C3×D412yes—
334C22×C4C2×C3×D412yes—
335C22×C4C24×C311yescharacteristic
336Q16—48no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—24no—
340C2×C8—24no—
341C4⋊C4—216no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—24no—
347C4⋊C4—44no—
348C4⋊C4—42no—
349C4⋊C4C2×C3×D414yes—
350C22⋊C4—216no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22×C3—22no—
356C22×C3—24no—
357C22×C3—24no—
358C22×C3—22no—
359C22×C3—24no—
360C22×C3—24no—
361C22×C3—24no—
362C22×C3—416no—
363C22×C3—42no—
364C22×C3—42no—
365C22×C3—42no—
366C22×C3—42no—
367C22×C3#45114yes—
368C22×C3C2×#12011yescharacteristic
369C22×C3#53412yes—
370C22×C3C22×Q1611yescharacteristic
371C22×C3C2×#13711yescharacteristic
372C22×C3#57512yes—
373C3×C4—22no—
374C3×C4—42no—
375C3×C4—44no—
376C3×C4—44no—
377C3×C4—81no—
378C3×C4—82no—
379C3×C4—81no—
380C3×C4#53412yes—
381C3×C4#53512yes—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—28no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—48no—
392C23—44no—
393C23—48no—
394C23—44no—
395C23C3×#12012yes—
396C23C2×C3×Q1614yes—
397C23C2×C3×Q1612yes—
398C23C3×#13712yes—
399C23C22×C3×D411yescharacteristic
400Q8—24no—
401Q8—24no—
402Q8—48no—
403Q8—44no—
404Q8—42no—
405Q8—42no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—48no—
415C2×C4—44no—
416C2×C4—48no—
417C2×C4—48no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4C3×#12012yes—
430C2×C4C22×C3×D411yescharacteristic
431C2×C4C22×C3×D411yescharacteristic
432C2×C4C3×H(5,𝔽2)14yes—
433C8—42no—
434C8—42no—
435C2×C3—24no—
436C2×C3—44no—
437C2×C3#261311yescharacteristic
438C2×C3C2×#45111yescharacteristic
439C2×C3#300511yescharacteristic
440C2×C3#318912yes—
441C2×C3#319612yes—
442C22—24no—
443C22—24no—
444C22—22no—
445C22—22no—
446C22—24no—
447C22—24no—
448C22—24no—
449C22—416no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22C3×#45114yes—
455C22C2×C3×#12011yescharacteristic
456C22C3×#53412yes—
457C22C22×C3×Q1611yescharacteristic
458C22C2×C3×#13711yescharacteristic
459C22C3×#57512yes—
460C4—22no—
461C4—44no—
462C4—42no—
463C4—44no—
464C4—82no—
465C4—81no—
466C4—81no—
467C4C3×#53412yes—
468C4C3×#53512yes—
469C3#2959711yescharacteristic, direct product factor, Sylow
470C2—24no—
471C2—44no—
472C2C3×#261311yescharacteristic
473C2C2×C3×#45111yescharacteristic
474C2C3×#300511yescharacteristic
475C2C3×#318912yes—
476C2C3×#319612yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC25×C3D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C24×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#484C2semidirect
3. C2×C3×#451C2semidirect
5. C3×#2036C2semidirect
6. C3×#2001C2semidirect
7. C3×#1824C2semidirect
8. C3×#1787C2semidirect
10. #29597C3direct (N × H)
13. C22×C3×Q16C22semidirect
25. C2×C3×#107C22semidirect
26. C2×C3×#103C22semidirect
27. C2×C3×#103C22semidirect
28. C2×C3×#103C22semidirect
37. C3×#340C22semidirect
39. C2×#484C2×C3semidirect
40. C2×#451C2×C3semidirect
42. #2036C2×C3semidirect
43. #2001C2×C3semidirect
44. #1824C2×C3semidirect
45. #1787C2×C3semidirect
100. C22×Q16C22×C3semidirect
112. C2×#107C22×C3semidirect
113. C2×#103C22×C3semidirect
114. C2×#103C22×C3semidirect
115. C2×#103C22×C3semidirect
124. #340C22×C3semidirect
129. C24×C3Q16semidirect
296. C24C3×Q16semidirect
469. C3#29597direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C22⋊Q82
C2×C3×#4842
C2×C3×#4512
C3×#28482
C3×#20362
C3×#20012
C3×#18242
C3×#17872
C3×#17792
#295973