Finite Group Databasebeta

← C3×#29793 #10648400

Subgroups — C3×#29793 #10648400

Order 768.

712 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
1C3×#3095C211yesmaximal, characteristic, complemented
2C3×#3094C211yesmaximal, characteristic
3C2×C3×#468C211yesmaximal, characteristic
4C3×#2905C211yesmaximal, characteristic, complemented
5C3×#2089C212yesmaximal, complemented
6C3×#2064C212yesmaximal, complemented
7C3×#2063C212yesmaximal, complemented
8C3×#2046C212yesmaximal, complemented
9C3×#1901C211yesmaximal, characteristic, complemented
10C3×#1900C211yesmaximal, characteristic
11C3×#1811C211yesmaximal, characteristic, complemented
12#29793C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2211yescharacteristic
14C22×C3×M16C2211yescharacteristic
15C22×C3×M16C2211yescharacteristic
16C2×C3×C22⋊Q8C2211yescharacteristic
17C2×C3×#121C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C3×#485—22no—
20C3×#484—21no—
21C3×#483—21no—
22C3×#482—22no—
23C3×#481—21no—
24C3×#480—21no—
25C3×#468—22no—
26C3×#468—22no—
27C2×C3×#108C2212yes—
28C3×#428C2212yes—
29C3×#428C2212yes—
30C2×C3×#107C2211yescharacteristic
31C3×#424—22no—
32C3×#424C2212yes—
33C3×#424C2212yes—
34C3×#423—22no—
35C3×#419C2212yes—
36C3×#418C2212yes—
37C2×C3×#103C2211yescharacteristic
38C2×C3×#102C2211yescharacteristic
39C2×C3×#101C2212yes—
40C2×C3×#100C2212yes—
41C3×#407C2211yescharacteristic
42C3×#407C2211yescharacteristic
43C3×#343C2212yes—
44C3×#343C2212yes—
45C3×#341C2212yes—
46#3095C2×C311yescharacteristic, complemented
47#3094C2×C311yescharacteristic
48C2×#468C2×C311yescharacteristic
49#2905C2×C311yescharacteristic, complemented
50#2089C2×C312yescomplemented
51#2064C2×C312yescomplemented
52#2063C2×C312yescomplemented
53#2046C2×C312yescomplemented
54#1901C2×C311yescharacteristic, complemented
55#1900C2×C311yescharacteristic
56#1811C2×C311yescharacteristic, complemented
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4—21no—
59C22×C3×D4C2311yescharacteristic
60C23×C3×C4C2311yescharacteristic
61C2×C3×SD16—22no—
62C2×C3×SD16—42no—
63C2×C3×M16—21no—
64C2×C3×M16—42no—
65C2×C3×M16—42no—
66C2×C3×M16D412yes—
67C2×C3×M16D412yes—
68C2×C3×M16C2312yes—
69C2×C3×M16C2312yes—
70C2×C3×M16C2312yes—
71C22×C3×C8—21no—
72C22×C3×C8C2311yescharacteristic
73C22×C3×C8C2311yescharacteristic
74C3×C22⋊Q8—21no—
75C3×C22⋊Q8—21no—
76C3×C22⋊Q8—21no—
77C3×C22⋊Q8—21no—
78C3×#121—21no—
79C3×#121—21no—
80C3×#121—21no—
81C3×#121—21no—
82C3×#117—21no—
83C3×#117—21no—
84C3×#117D412yes—
85C3×#117C2312yes—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4D411yescharacteristic
88C2×C3×C4⋊C4D411yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C42—21no—
95C3×#108—42no—
96C3×#107—22no—
97C3×#107—22no—
98C3×#107—22no—
99C3×#106—22no—
100C3×#106—22no—
101C3×C4⋊C8—24no—
102C3×C4⋊C8—24no—
103C3×#103—22no—
104C3×#103—22no—
105C3×#103—22no—
106C3×#102—22no—
107C3×#102—22no—
108C3×#102—22no—
109C3×#101—42no—
110C3×#100—42no—
111C3×C22⋊C8—21no—
112C3×C22⋊C8—21no—
113C3×C22⋊C8—21no—
114C3×C22⋊C8—21no—
115C22×SD16C22×C311yescharacteristic
116C22×M16C22×C311yescharacteristic
117C22×M16C22×C311yescharacteristic
118C2×C22⋊Q8C22×C311yescharacteristic
119C2×#121C22×C311yescharacteristic
120C2×#117C22×C311yescharacteristic
121#485—22no—
122#484—21no—
123#483—21no—
124#482—22no—
125#481—21no—
126#480—21no—
127#468—22no—
128#468—22no—
129C2×#108C22×C312yes—
130#428C22×C312yes—
131#428C22×C312yes—
132C2×#107C22×C311yescharacteristic
133#424—22no—
134#424C22×C312yes—
135#424C22×C312yes—
136#423—22no—
137#419C22×C312yes—
138#418C22×C312yes—
139C2×#103C22×C311yescharacteristic
140C2×#102C22×C311yescharacteristic
141C2×#101C22×C312yes—
142C2×#100C22×C312yes—
143#407C22×C311yescharacteristic
144#407C22×C311yescharacteristic
145#343C22×C312yes—
146#343C22×C312yes—
147#341C22×C312yes—
148C24×C3—21no—
149C24×C3#4111yescharacteristic
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—22no—
153C2×C3×Q8—22no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—41no—
159C2×C3×D4—41no—
160C2×C3×D4—82no—
161C22×C3×C4—22no—
162C22×C3×C4—22no—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4#4111yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C3×SD16—44no—
175C3×M16—22no—
176C3×M16—22no—
177C3×M16—22no—
178C3×M16—22no—
179C3×M16—22no—
180C3×M16—22no—
181C3×M16—42no—
182C3×M16—42no—
183C3×M16—42no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—42no—
190C2×C3×C8C2×D412yes—
191C2×C3×C8C2×D412yes—
192C3×C4⋊C4—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—21no—
197C3×C4⋊C4—21no—
198C3×C4⋊C4—41no—
199C3×C4⋊C4—41no—
200C3×C4⋊C4C2×D412yes—
201C3×C4⋊C4C2×D412yes—
202C3×C22⋊C4—22no—
203C3×C22⋊C4—22no—
204C3×C22⋊C4—41no—
205C3×C22⋊C4—41no—
206C3×C22⋊C4—41no—
207C3×C22⋊C4—41no—
208C3×C42—22no—
209C3×C42—22no—
210C22×Q8C23×C311yescharacteristic
211C22×D4—21no—
212C22×D4C23×C311yescharacteristic
213C23×C4C23×C311yescharacteristic
214C2×SD16—22no—
215C2×SD16—42no—
216C2×M16—21no—
217C2×M16—42no—
218C2×M16—42no—
219C2×M16C3×D412yes—
220C2×M16C3×D412yes—
221C2×M16C23×C312yes—
222C2×M16C23×C312yes—
223C2×M16C23×C312yes—
224C22×C8—21no—
225C22×C8C23×C311yescharacteristic
226C22×C8C23×C311yescharacteristic
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231#121—21no—
232#121—21no—
233#121—21no—
234#121—21no—
235#117—21no—
236#117—21no—
237#117C3×D412yes—
238#117C23×C312yes—
239C2×C4⋊C4—21no—
240C2×C4⋊C4C3×D411yescharacteristic
241C2×C4⋊C4C3×D411yescharacteristic
242C2×C4⋊C4C23×C311yescharacteristic
243C2×C4⋊C4C23×C311yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C42—21no—
248#108—42no—
249#107—22no—
250#107—22no—
251#107—22no—
252#106—22no—
253#106—22no—
254C4⋊C8—24no—
255C4⋊C8—24no—
256#103—22no—
257#103—22no—
258#103—22no—
259#102—22no—
260#102—22no—
261#102—22no—
262#101—42no—
263#100—42no—
264C22⋊C8—21no—
265C22⋊C8—21no—
266C22⋊C8—21no—
267C22⋊C8—21no—
268C23×C3—22no—
269C23×C3—41no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—42no—
273C23×C3—81no—
274C23×C3—81no—
275C23×C3#12111yescharacteristic
276C23×C3#12111yescharacteristic
277C23×C3C2×#4111yescharacteristic
278C3×Q8—41no—
279C3×Q8—41no—
280C3×Q8—42no—
281C3×D4—41no—
282C3×D4—41no—
283C3×D4—42no—
284C3×D4—82no—
285C3×D4—82no—
286C2×C3×C4—22no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—22no—
292C2×C3×C4—22no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—41no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—42no—
299C2×C3×C4—81no—
300C2×C3×C4—81no—
301C2×C3×C4—81no—
302C2×C3×C4—81no—
303C2×C3×C4#12111yescharacteristic
304C2×C3×C4#12111yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4H(5,𝔽2)12yes—
308C2×C3×C42−1+412yes—
309C3×C8—22no—
310C3×C8—22no—
311C3×C8—42no—
312C3×C8—42no—
313C3×C8—42no—
314C24—21no—
315C24C3×#4111yescharacteristic
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—82no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4C2×C3×D411yescharacteristic
335C22×C4C2×C3×D411yescharacteristic
336C22×C4C2×C3×D411yescharacteristic
337C22×C4C2×C3×D411yescharacteristic
338C22×C4C3×#4111yescharacteristic
339C22×C4C24×C311yescharacteristic
340SD16—44no—
341M16—22no—
342M16—22no—
343M16—22no—
344M16—22no—
345M16—22no—
346M16—22no—
347M16—42no—
348M16—42no—
349M16—42no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—42no—
356C2×C8C2×C3×D412yes—
357C2×C8C2×C3×D412yes—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4C2×C3×D412yes—
367C4⋊C4C2×C3×D412yes—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C42—22no—
375C42—22no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—82no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3—162no—
389C22×C3C2×#12111yescharacteristic
390C22×C3#53912yes—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—42no—
394C3×C4—42no—
395C3×C4—81no—
396C3×C4—81no—
397C3×C4—81no—
398C3×C4#53411yescharacteristic
399C3×C4#53611yescharacteristic
400C3×C4#53712yes—
401C23—22no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—42no—
406C23—81no—
407C23—81no—
408C23C3×#12111yescharacteristic
409C23C3×#12111yescharacteristic
410C23C2×C3×#4111yescharacteristic
411Q8—41no—
412Q8—41no—
413Q8—42no—
414D4—41no—
415D4—41no—
416D4—42no—
417D4—82no—
418D4—82no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4C3×#12111yescharacteristic
437C2×C4C3×#12111yescharacteristic
438C2×C4C22×C3×D411yescharacteristic
439C2×C4C22×C3×D411yescharacteristic
440C2×C4C3×H(5,𝔽2)12yes—
441C2×C4C3×2−1+412yes—
442C8—22no—
443C8—22no—
444C8—42no—
445C8—42no—
446C8—42no—
447C2×C3—22no—
448C2×C3—41no—
449C2×C3—41no—
450C2×C3—161no—
451C2×C3#261811yescharacteristic
452C2×C3#306811yescharacteristic
453C2×C3#306911yescharacteristic
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—82no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—162no—
467C22C2×C3×#12111yescharacteristic
468C22C3×#53912yes—
469C4—21no—
470C4—21no—
471C4—42no—
472C4—42no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4C3×#53411yescharacteristic
477C4C3×#53611yescharacteristic
478C4C3×#53712yes—
479C3#2979311yescharacteristic, direct product factor, Sylow
480C2—22no—
481C2—41no—
482C2—41no—
483C2—161no—
484C2C3×#261811yescharacteristic
485C2C3×#306811yescharacteristic
486C2C3×#306911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#121
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3095C2semidirect
4. C3×#2905C2semidirect
5. C3×#2089C2semidirect
6. C3×#2064C2semidirect
7. C3×#2063C2semidirect
8. C3×#2046C2semidirect
9. C3×#1901C2semidirect
11. C3×#1811C2semidirect
12. #29793C3direct (N × H)
46. #3095C2×C3semidirect
49. #2905C2×C3semidirect
50. #2089C2×C3semidirect
51. #2064C2×C3semidirect
52. #2063C2×C3semidirect
53. #2046C2×C3semidirect
54. #1901C2×C3semidirect
56. #1811C2×C3semidirect
479. C3#29793direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30952
C3×#30942
C2×C3×#4682
C3×#29052
C3×#20892
C3×#20642
C3×#20632
C3×#20462
C3×#19012
C3×#19002
C3×#18112
#297933