Finite Group Databasebeta

← C3×#33038 #10651645

Subgroups — C3×#33038 #10651645

Order 768.

960 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#478C211yesmaximal, characteristic, complemented
2C2×C3×#459C211yesmaximal, characteristic
3C3×#2641C211yesmaximal, characteristic, complemented
4C3×#2275C211yesmaximal, characteristic, complemented
5C3×#1989C211yesmaximal, characteristic, complemented
6C3×#1982C211yesmaximal, characteristic, complemented
7C3×#1926C212yesmaximal, complemented
8C3×#1925C212yesmaximal, complemented
9C3×#1879C212yesmaximal, complemented
10C3×#1879C212yesmaximal, complemented
11C3×#1832C211yesmaximal, characteristic, complemented
12#33038C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×D8C2212yes—
14C2×C3×#127C2211yescharacteristic
15C2×C3×#125C2211yescharacteristic
16C2×C3×C4×D4C2212yes—
17C2×C3×C4×D4C2211yescharacteristic
18C3×#478—24no—
19C3×#459—24no—
20C2×C3×#107C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×#102C2212yes—
29C3×#394—21no—
30C3×#394—22no—
31C3×#394—21no—
32C3×#390C2212yes—
33C3×#390C2212yes—
34C3×C4×C4⋊C4C2211yescharacteristic
35C3×C4×C4⋊C4C2212yes—
36C3×C4×C22⋊C4—22no—
37C3×C4×C22⋊C4—21no—
38C3×#376C2211yescharacteristic
39C2×C3×#95—21no—
40C3×#340C2212yes—
41C3×#340C2212yes—
42C3×#336C2212yes—
43C3×#336C2212yes—
44C2×#478C2×C311yescharacteristic, complemented
45C2×#459C2×C311yescharacteristic
46#2641C2×C311yescharacteristic, complemented
47#2275C2×C311yescharacteristic, complemented
48#1989C2×C311yescharacteristic, complemented
49#1982C2×C311yescharacteristic, complemented
50#1926C2×C312yescomplemented
51#1925C2×C312yescomplemented
52#1879C2×C312yescomplemented
53#1879C2×C312yescomplemented
54#1832C2×C311yescharacteristic, complemented
55C22×C3×D4—21no—
56C22×C3×D4—21no—
57C22×C3×D4C2312yes—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—22no—
60C23×C3×C4—21no—
61C2×C3×D8—24no—
62C2×C3×D8—24no—
63C2×C3×D8—24no—
64C22×C3×C8C2312yes—
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C3×#127—22no—
68C3×#127—22no—
69C3×#125—22no—
70C3×#125—22no—
71C3×C4×D4—24no—
72C3×C4×D4—24no—
73C3×C4×D4—22no—
74C3×C4×D4D414yes—
75C2×C3×C4⋊C4—21no—
76C2×C3×C4⋊C4—21no—
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C4⋊C4C2311yescharacteristic
80C2×C3×C4⋊C4C2312yes—
81C2×C3×C22⋊C4—22no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—22no—
84C2×C3×C22⋊C4—22no—
85C2×C3×C22⋊C4—22no—
86C2×C3×C22⋊C4—22no—
87C2×C3×C42—22no—
88C2×C3×C42—21no—
89C2×C3×C42C2312yes—
90C2×C3×C42C2311yescharacteristic
91C3×#107—22no—
92C3×#106—22no—
93C3×C4⋊C8—22no—
94C3×C4⋊C8—22no—
95C3×#102—22no—
96C3×#102—22no—
97C3×#102—24no—
98C3×#102—24no—
99C3×#102—24no—
100C3×#95—22no—
101C3×#95—21no—
102C3×#95D411yescharacteristic
103C3×#95D412yes—
104C3×#95D411yescharacteristic
105C22×D8C22×C312yes—
106C2×#127C22×C311yescharacteristic
107C2×#125C22×C311yescharacteristic
108C2×C4×D4C22×C312yes—
109C2×C4×D4C22×C311yescharacteristic
110#478—24no—
111#459—24no—
112C2×#107C22×C311yescharacteristic
113C2×#106C22×C311yescharacteristic
114C2×C4⋊C8C22×C311yescharacteristic
115C2×C4⋊C8C22×C311yescharacteristic
116C2×#102C22×C311yescharacteristic
117C2×#102C22×C311yescharacteristic
118C2×#102C22×C312yes—
119C2×#102C22×C312yes—
120C2×#102C22×C312yes—
121#394—21no—
122#394—22no—
123#394—21no—
124#390C22×C312yes—
125#390C22×C312yes—
126C4×C4⋊C4C22×C312yes—
127C4×C4⋊C4C22×C311yescharacteristic
128C4×C22⋊C4—22no—
129C4×C22⋊C4—21no—
130#376C22×C311yescharacteristic
131C2×#95—21no—
132#340C22×C312yes—
133#340C22×C312yes—
134#336C22×C312yes—
135#336C22×C312yes—
136C24×C3—21no—
137C24×C3—22no—
138C2×C3×D4—22no—
139C2×C3×D4—21no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—42no—
147C2×C3×D4—42no—
148C2×C3×D4—81no—
149C2×C3×D4—82no—
150C2×C3×D4—81no—
151C2×C3×D4C2×D412yes—
152C2×C3×D4#4114yes—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—28no—
156C22×C3×C4—24no—
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—82no—
160C22×C3×C4—82no—
161C22×C3×C4—82no—
162C22×C3×C4C2×D412yes—
163C22×C3×C4C2×D412yes—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C3×D8—48no—
168C2×C3×C8—21no—
169C2×C3×C8—21no—
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—21no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4—42no—
183C3×C4⋊C4C2×D412yes—
184C3×C4⋊C4#4114yes—
185C3×C22⋊C4—28no—
186C3×C22⋊C4—24no—
187C3×C22⋊C4—82no—
188C3×C22⋊C4—82no—
189C3×C22⋊C4—82no—
190C3×C22⋊C4—82no—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42—81no—
194C3×C42—82no—
195C3×C42C2×D412yes—
196C3×C42#4112yes—
197C22×D4—21no—
198C22×D4—21no—
199C22×D4C23×C312yes—
200C22×D4C23×C311yescharacteristic
201C23×C4—22no—
202C23×C4—21no—
203C2×D8—24no—
204C2×D8—24no—
205C2×D8—24no—
206C22×C8C23×C312yes—
207C22×C8C23×C311yescharacteristic
208C22×C8C23×C311yescharacteristic
209#127—22no—
210#127—22no—
211#125—22no—
212#125—22no—
213C4×D4—24no—
214C4×D4—24no—
215C4×D4—22no—
216C4×D4C3×D414yes—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C23×C311yescharacteristic
220C2×C4⋊C4C23×C311yescharacteristic
221C2×C4⋊C4C23×C311yescharacteristic
222C2×C4⋊C4C23×C312yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42C23×C312yes—
232C2×C42C23×C311yescharacteristic
233#107—22no—
234#106—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237#102—22no—
238#102—22no—
239#102—24no—
240#102—24no—
241#102—24no—
242#95—22no—
243#95—21no—
244#95C3×D411yescharacteristic
245#95C3×D412yes—
246#95C3×D411yescharacteristic
247C23×C3—22no—
248C23×C3—24no—
249C23×C3—42no—
250C23×C3—41no—
251C23×C3—41no—
252C23×C3—42no—
253C23×C3—42no—
254C23×C3—42no—
255C23×C3—42no—
256C23×C3—42no—
257C23×C3—42no—
258C23×C3—42no—
259C23×C3C22×D411yescharacteristic
260C3×D4—24no—
261C3×D4—22no—
262C3×D4—42no—
263C3×D4—42no—
264C3×D4—42no—
265C3×D4—42no—
266C3×D4—84no—
267C3×D4—84no—
268C3×D4#12414yes—
269C2×C3×C4—21no—
270C2×C3×C4—22no—
271C2×C3×C4—21no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—24no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—22no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—42no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—84no—
289C2×C3×C4—88no—
290C2×C3×C4#12112yes—
291C2×C3×C4#12112yes—
292C2×C3×C4C22×D411yescharacteristic
293C2×C3×C4C2×#4112yes—
294C2×C3×C4C2×#4111yescharacteristic
295C2×C3×C4C2×#4112yes—
296C3×C8—42no—
297C3×C8—41no—
298C3×C8—41no—
299C24—21no—
300C24—22no—
301C2×D4—22no—
302C2×D4—21no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—81no—
312C2×D4—82no—
313C2×D4—81no—
314C2×D4C2×C3×D412yes—
315C2×D4C3×#4114yes—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—28no—
319C22×C4—24no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—82no—
325C22×C4C2×C3×D412yes—
326C22×C4C2×C3×D412yes—
327C22×C4C2×C3×D411yescharacteristic
328C22×C4C2×C3×D411yescharacteristic
329C22×C4C24×C311yescharacteristic
330D8—48no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4C2×C3×D412yes—
347C4⋊C4C3×#4114yes—
348C22⋊C4—28no—
349C22⋊C4—24no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C42—22no—
355C42—22no—
356C42—81no—
357C42—82no—
358C42C2×C3×D412yes—
359C42C3×#4112yes—
360C22×C3—24no—
361C22×C3—44no—
362C22×C3—44no—
363C22×C3—42no—
364C22×C3—42no—
365C22×C3—82no—
366C22×C3—82no—
367C22×C3—82no—
368C22×C3—82no—
369C22×C3—82no—
370C22×C3—82no—
371C22×C3—82no—
372C22×C3C2×#12111yescharacteristic
373C22×C3#52512yes—
374C22×C3#57612yes—
375C22×C3#57611yescharacteristic
376C22×C3#57711yescharacteristic
377C3×C4—22no—
378C3×C4—22no—
379C3×C4—42no—
380C3×C4—42no—
381C3×C4—44no—
382C3×C4—42no—
383C3×C4—81no—
384C3×C4—81no—
385C3×C4C2×#12411yescharacteristic
386C3×C4#52912yes—
387C3×C4#53211yescharacteristic
388C23—22no—
389C23—24no—
390C23—42no—
391C23—41no—
392C23—41no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23C22×C3×D411yescharacteristic
401D4—24no—
402D4—22no—
403D4—42no—
404D4—42no—
405D4—42no—
406D4—42no—
407D4—84no—
408D4—84no—
409D4C3×#12414yes—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—84no—
430C2×C4—88no—
431C2×C4C3×#12112yes—
432C2×C4C3×#12112yes—
433C2×C4C22×C3×D411yescharacteristic
434C2×C4C2×C3×#4112yes—
435C2×C4C2×C3×#4111yescharacteristic
436C2×C4C2×C3×#4112yes—
437C8—42no—
438C8—41no—
439C8—41no—
440C2×C3—44no—
441C2×C3—162no—
442C2×C3#285011yescharacteristic
443C2×C3#304111yescharacteristic
444C2×C3#304511yescharacteristic
445C2×C3#311912yes—
446C2×C3#312212yes—
447C22—24no—
448C22—42no—
449C22—42no—
450C22—44no—
451C22—44no—
452C22—82no—
453C22—82no—
454C22—82no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22C2×C3×#12111yescharacteristic
460C22C3×#52512yes—
461C22C3×#57612yes—
462C22C3×#57611yescharacteristic
463C22C3×#57711yescharacteristic
464C4—22no—
465C4—22no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—44no—
470C4—81no—
471C4—81no—
472C4C2×C3×#12411yescharacteristic
473C4C3×#52912yes—
474C4C3×#53211yescharacteristic
475C3#3303811yescharacteristic, direct product factor, Sylow
476C2—44no—
477C2—162no—
478C2C3×#285011yescharacteristic
479C2C3×#304111yescharacteristic
480C2C3×#304511yescharacteristic
481C2C3×#311912yes—
482C2C3×#312212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#459C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C3×#478C2semidirect
3. C3×#2641C2semidirect
4. C3×#2275C2semidirect
5. C3×#1989C2semidirect
6. C3×#1982C2semidirect
7. C3×#1926C2semidirect
8. C3×#1925C2semidirect
9. C3×#1879C2semidirect
10. C3×#1879C2semidirect
11. C3×#1832C2semidirect
12. #33038C3direct (N × H)
44. C2×#478C2×C3semidirect
46. #2641C2×C3semidirect
47. #2275C2×C3semidirect
48. #1989C2×C3semidirect
49. #1982C2×C3semidirect
50. #1926C2×C3semidirect
51. #1925C2×C3semidirect
52. #1879C2×C3semidirect
53. #1879C2×C3semidirect
54. #1832C2×C3semidirect
475. C3#33038direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4782
C2×C3×#4592
C3×#26412
C3×#22752
C3×#19892
C3×#19822
C3×#19262
C3×#19252
C3×#18792
C3×#18322
#330383