Finite Group Databasebeta

← C3×#29708 #10648315

Subgroups — C3×#29708 #10648315

Order 768.

930 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3461C211yesmaximal, characteristic
2C3×#3080C212yesmaximal, complemented
3C3×#2565C211yesmaximal, characteristic, complemented
4C3×#2061C211yesmaximal, characteristic, complemented
5C3×#2060C211yesmaximal, characteristic, complemented
6C3×#2040C211yesmaximal, characteristic
7C3×#2039C211yesmaximal, characteristic
8C3×#2038C212yesmaximal, complemented
9C3×#2038C212yesmaximal, complemented
10C3×#1806C212yesmaximal, complemented
11C3×#1801C211yesmaximal, characteristic, complemented
12#29708C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#544C2211yescharacteristic
14C3×#542—22no—
15C3×#541—22no—
16C3×#539—22no—
17C3×#537C2211yescharacteristic
18C3×#536C2212yes—
19C3×#533—22no—
20C3×#529—22no—
21C3×#529—22no—
22C2×C3×#125C2211yescharacteristic
23C3×C4×#41C2212yes—
24C2×C3×C4×Q8C2211yescharacteristic
25C3×#479—22no—
26C3×#479—22no—
27C3×#478—22no—
28C3×#478—22no—
29C3×#428C2212yes—
30C3×#428C2212yes—
31C3×#423C2212yes—
32C2×C3×#104C2212yes—
33C2×C3×#104C2212yes—
34C3×#417C2212yes—
35C3×#417C2212yes—
36C2×C3×#101C2212yes—
37C3×#398—22no—
38C3×#391C2211yescharacteristic
39C3×#391C2211yescharacteristic
40C3×#387—22no—
41C3×#384C2211yescharacteristic
42C3×#382C2211yescharacteristic
43C3×C4×C4⋊C4C2211yescharacteristic
44C3×#343C2211yescharacteristic
45C3×#343C2211yescharacteristic
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3461C2×C311yescharacteristic
49#3080C2×C312yescomplemented
50#2565C2×C311yescharacteristic, complemented
51#2061C2×C311yescharacteristic, complemented
52#2060C2×C311yescharacteristic, complemented
53#2040C2×C311yescharacteristic
54#2039C2×C311yescharacteristic
55#2038C2×C312yescomplemented
56#2038C2×C312yescomplemented
57#1806C2×C312yescomplemented
58#1801C2×C311yescharacteristic, complemented
59C2×C3×#41C2312yes—
60C22×C3×Q8C2311yescharacteristic
61C2×C3×M16C2312yes—
62C2×C3×M16C2312yes—
63C3×C4⋊Q8—22no—
64C3×C4⋊Q8—22no—
65C3×#126—24no—
66C3×#126—24no—
67C3×#125—22no—
68C3×#125—24no—
69C3×#125—22no—
70C3×#125—22no—
71C3×#125—22no—
72C3×#124—22no—
73C3×#124—22no—
74C3×#123—24no—
75C3×#123—24no—
76C3×C22⋊Q8—24no—
77C3×C22⋊Q8—22no—
78C3×C22⋊Q8—22no—
79C3×C22⋊Q8—22no—
80C3×C22⋊Q8—22no—
81C3×#121—22no—
82C3×#121—22no—
83C3×C4×Q8—22no—
84C3×C4×Q8—22no—
85C3×C4×Q8—22no—
86C3×C4×Q8D412yes—
87C3×C4×Q8D412yes—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×C4×D4D412yes—
92C3×C4×D4D412yes—
93C3×#117—24no—
94C3×#117—22no—
95C3×#117C2312yes—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4D412yes—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C42—22no—
105C2×C3×C42D412yes—
106C2×C3×C42C2311yescharacteristic
107C2×C3×C42C2311yescharacteristic
108C2×C3×C42C2311yescharacteristic
109C3×#107—22no—
110C3×#107—22no—
111C3×#106—22no—
112C3×#106—22no—
113C3×C4⋊C8—22no—
114C3×C4⋊C8—22no—
115C3×#104—42no—
116C3×#104—42no—
117C3×#103—22no—
118C3×#103—22no—
119C3×#102—22no—
120C3×#102—22no—
121C3×#101—42no—
122C3×#95—21no—
123C3×#95—22no—
124C3×#95—22no—
125C3×#95—22no—
126#544C22×C311yescharacteristic
127#542—22no—
128#541—22no—
129#539—22no—
130#537C22×C311yescharacteristic
131#536C22×C312yes—
132#533—22no—
133#529—22no—
134#529—22no—
135C2×#125C22×C311yescharacteristic
136C4×#41C22×C312yes—
137C2×C4×Q8C22×C311yescharacteristic
138#479—22no—
139#479—22no—
140#478—22no—
141#478—22no—
142#428C22×C312yes—
143#428C22×C312yes—
144#423C22×C312yes—
145C2×#104C22×C312yes—
146C2×#104C22×C312yes—
147#417C22×C312yes—
148#417C22×C312yes—
149C2×#101C22×C312yes—
150#398—22no—
151#391C22×C311yescharacteristic
152#391C22×C311yescharacteristic
153#387—22no—
154#384C22×C311yescharacteristic
155#382C22×C311yescharacteristic
156C4×C4⋊C4C22×C311yescharacteristic
157#343C22×C311yescharacteristic
158#343C22×C311yescharacteristic
159#339C22×C312yes—
160#339C22×C312yes—
161C3×#41—22no—
162C3×#41—22no—
163C3×#41—22no—
164C3×#41—22no—
165C2×C3×Q8—22no—
166C2×C3×Q8—22no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—21no—
170C2×C3×Q8C2×D412yes—
171C2×C3×D4—22no—
172C2×C3×D4—22no—
173C2×C3×D4C2×D412yes—
174C22×C3×C4—21no—
175C22×C3×C4—22no—
176C22×C3×C4—22no—
177C22×C3×C4—22no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—21no—
181C22×C3×C4C2×D412yes—
182C22×C3×C4C2×D412yes—
183C22×C3×C4C2×D411yescharacteristic
184C22×C3×C4C2×D411yescharacteristic
185C22×C3×C4C2411yescharacteristic
186C3×M16—22no—
187C3×M16—22no—
188C3×M16—22no—
189C3×M16—22no—
190C2×C3×C8—22no—
191C2×C3×C8—22no—
192C3×C4⋊C4—24no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—22no—
195C3×C4⋊C4—22no—
196C3×C4⋊C4—24no—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—42no—
204C3×C4⋊C4—44no—
205C3×C4⋊C4—41no—
206C3×C4⋊C4—41no—
207C3×C4⋊C4—42no—
208C3×C4⋊C4C2×D412yes—
209C3×C4⋊C4C2×D412yes—
210C3×C22⋊C4—24no—
211C3×C22⋊C4—24no—
212C3×C22⋊C4—24no—
213C3×C22⋊C4—22no—
214C3×C22⋊C4—22no—
215C3×C22⋊C4—22no—
216C3×C42—22no—
217C3×C42—22no—
218C3×C42—22no—
219C3×C42—22no—
220C3×C42—22no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42—21no—
224C3×C42—21no—
225C3×C42C2×D412yes—
226C3×C42C2×D412yes—
227C2×#41C23×C312yes—
228C22×Q8C23×C311yescharacteristic
229C2×M16C23×C312yes—
230C2×M16C23×C312yes—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233#126—24no—
234#126—24no—
235#125—22no—
236#125—24no—
237#125—22no—
238#125—22no—
239#125—22no—
240#124—22no—
241#124—22no—
242#123—24no—
243#123—24no—
244C22⋊Q8—24no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249#121—22no—
250#121—22no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×Q8—22no—
254C4×Q8C3×D412yes—
255C4×Q8C3×D412yes—
256C4×D4—22no—
257C4×D4—22no—
258C4×D4—22no—
259C4×D4C3×D412yes—
260C4×D4C3×D412yes—
261#117—24no—
262#117—22no—
263#117C23×C312yes—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4C3×D412yes—
269C2×C4⋊C4C23×C311yescharacteristic
270C2×C4⋊C4C23×C311yescharacteristic
271C2×C4⋊C4C23×C311yescharacteristic
272C2×C42—22no—
273C2×C42C3×D412yes—
274C2×C42C23×C311yescharacteristic
275C2×C42C23×C311yescharacteristic
276C2×C42C23×C311yescharacteristic
277#107—22no—
278#107—22no—
279#106—22no—
280#106—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283#104—42no—
284#104—42no—
285#103—22no—
286#103—22no—
287#102—22no—
288#102—22no—
289#101—42no—
290#95—21no—
291#95—22no—
292#95—22no—
293#95—22no—
294C23×C3—22no—
295C23×C3C22×D411yescharacteristic
296C3×Q8—22no—
297C3×Q8—22no—
298C3×Q8—42no—
299C3×Q8—42no—
300C3×Q8—82no—
301C3×D4—22no—
302C3×D4—22no—
303C3×D4—42no—
304C3×D4—42no—
305C3×D4—82no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—44no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4—42no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—41no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—42no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—41no—
329C2×C3×C4—82no—
330C2×C3×C4#12011yescharacteristic
331C2×C3×C4#12011yescharacteristic
332C2×C3×C4#12012yes—
333C2×C3×C4C22×D412yes—
334C2×C3×C4H(5,𝔽2)11yescharacteristic
335C2×C3×C42−1+411yescharacteristic
336C2×C3×C42−1+412yes—
337C3×C8—42no—
338C3×C8—42no—
339#41—22no—
340#41—22no—
341#41—22no—
342#41—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—21no—
348C2×Q8C2×C3×D412yes—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4C2×C3×D412yes—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4C2×C3×D412yes—
360C22×C4C2×C3×D412yes—
361C22×C4C2×C3×D411yescharacteristic
362C22×C4C2×C3×D411yescharacteristic
363C22×C4C24×C311yescharacteristic
364M16—22no—
365M16—22no—
366M16—22no—
367M16—22no—
368C2×C8—22no—
369C2×C8—22no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—42no—
386C4⋊C4C2×C3×D412yes—
387C4⋊C4C2×C3×D412yes—
388C22⋊C4—24no—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42C2×C3×D412yes—
404C42C2×C3×D412yes—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—42no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3C2×#12011yescharacteristic
411C22×C3#53611yescharacteristic
412C22×C3#53711yescharacteristic
413C3×C4—22no—
414C3×C4—21no—
415C3×C4—21no—
416C3×C4—42no—
417C3×C4—42no—
418C3×C4—42no—
419C3×C4—42no—
420C3×C4—42no—
421C3×C4—42no—
422C3×C4—81no—
423C3×C4—81no—
424C3×C4—81no—
425C3×C4#53612yes—
426C3×C4#53712yes—
427C23—22no—
428C23C22×C3×D411yescharacteristic
429Q8—22no—
430Q8—22no—
431Q8—42no—
432Q8—42no—
433Q8—82no—
434D4—22no—
435D4—22no—
436D4—42no—
437D4—42no—
438D4—82no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—44no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—82no—
463C2×C4C3×#12011yescharacteristic
464C2×C4C3×#12011yescharacteristic
465C2×C4C3×#12012yes—
466C2×C4C22×C3×D412yes—
467C2×C4C3×H(5,𝔽2)11yescharacteristic
468C2×C4C3×2−1+411yescharacteristic
469C2×C4C3×2−1+412yes—
470C8—42no—
471C8—42no—
472C2×C3—21no—
473C2×C3—21no—
474C2×C3—82no—
475C2×C3#261611yescharacteristic
476C2×C3#301811yescharacteristic
477C2×C3#302011yescharacteristic
478C22—21no—
479C22—21no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22C2×C3×#12011yescharacteristic
484C22C3×#53611yescharacteristic
485C22C3×#53711yescharacteristic
486C4—22no—
487C4—21no—
488C4—21no—
489C4—42no—
490C4—42no—
491C4—42no—
492C4—42no—
493C4—42no—
494C4—42no—
495C4—81no—
496C4—81no—
497C4—81no—
498C4C3×#53612yes—
499C4C3×#53712yes—
500C3#2970811yescharacteristic, direct product factor, Sylow
501C2—21no—
502C2—21no—
503C2—82no—
504C2C3×#261611yescharacteristic
505C2C3×#301811yescharacteristic
506C2C3×#302011yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3080C2semidirect
3. C3×#2565C2semidirect
4. C3×#2061C2semidirect
5. C3×#2060C2semidirect
8. C3×#2038C2semidirect
9. C3×#2038C2semidirect
10. C3×#1806C2semidirect
11. C3×#1801C2semidirect
12. #29708C3direct (N × H)
49. #3080C2×C3semidirect
50. #2565C2×C3semidirect
51. #2061C2×C3semidirect
52. #2060C2×C3semidirect
55. #2038C2×C3semidirect
56. #2038C2×C3semidirect
57. #1806C2×C3semidirect
58. #1801C2×C3semidirect
500. C3#29708direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34612
C3×#30802
C3×#25652
C3×#20612
C3×#20602
C3×#20402
C3×#20392
C3×#20382
C3×#18062
C3×#18012
#297083