Finite Group Databasebeta

← C3×#29707 #10648314

Subgroups — C3×#29707 #10648314

Order 768.

1010 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3460C211yesmaximal, characteristic
2C3×#3080C212yesmaximal, complemented
3C3×#2566C211yesmaximal, characteristic, complemented
4C3×#2061C211yesmaximal, characteristic, complemented
5C3×#2060C211yesmaximal, characteristic, complemented
6C3×#2040C211yesmaximal, characteristic, complemented
7C3×#2039C211yesmaximal, characteristic, complemented
8C3×#2037C212yesmaximal, complemented
9C3×#2037C212yesmaximal, complemented
10C3×#1806C212yesmaximal, complemented
11C3×#1800C211yesmaximal, characteristic, complemented
12#29707C311yesmaximal, characteristic, direct product factor, Sylow
13C3×#544C2211yescharacteristic
14C3×#542—22no—
15C3×#540—22no—
16C3×#538—22no—
17C3×#537C2211yescharacteristic
18C3×#535C2212yes—
19C3×#532—22no—
20C3×#529—22no—
21C2×C3×#125C2211yescharacteristic
22C3×#525—22no—
23C3×C4×#41C2212yes—
24C2×C3×C4×D4C2211yescharacteristic
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×#100C2212yes—
37C3×#396—22no—
38C3×#390C2211yescharacteristic
39C3×#390C2211yescharacteristic
40C3×#385—22no—
41C3×#384C2211yescharacteristic
42C3×#382C2211yescharacteristic
43C3×C4×C4⋊C4C2211yescharacteristic
44C3×#343C2211yescharacteristic
45C3×#343C2211yescharacteristic
46C3×#339C2212yes—
47C3×#339C2212yes—
48#3460C2×C311yescharacteristic
49#3080C2×C312yescomplemented
50#2566C2×C311yescharacteristic, complemented
51#2061C2×C311yescharacteristic, complemented
52#2060C2×C311yescharacteristic, complemented
53#2040C2×C311yescharacteristic, complemented
54#2039C2×C311yescharacteristic, complemented
55#2037C2×C312yescomplemented
56#2037C2×C312yescomplemented
57#1806C2×C312yescomplemented
58#1800C2×C311yescharacteristic, complemented
59C2×C3×#41—22no—
60C2×C3×#41C2312yes—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4D412yes—
63C2×C3×M16C2312yes—
64C2×C3×M16C2312yes—
65C3×C4⋊Q8—22no—
66C3×#127—22no—
67C3×#126—24no—
68C3×#125—22no—
69C3×#125—22no—
70C3×#125—22no—
71C3×#124—24no—
72C3×#124—22no—
73C3×#123—24no—
74C3×#123—24no—
75C3×C22⋊Q8—22no—
76C3×C22⋊Q8—22no—
77C3×C22⋊Q8—22no—
78C3×#121—22no—
79C3×#121—24no—
80C3×#121—22no—
81C3×#121—22no—
82C3×#121—22no—
83C3×#121—22no—
84C3×#120—24no—
85C3×C4×Q8D412yes—
86C3×C4×Q8D412yes—
87C3×C4×D4—24no—
88C3×C4×D4—22no—
89C3×C4×D4—22no—
90C3×C4×D4—22no—
91C3×C4×D4—22no—
92C3×C4×D4D412yes—
93C3×C4×D4D412yes—
94C3×#117—22no—
95C3×#117—22no—
96C3×#117C2312yes—
97C2×C3×C4⋊C4—21no—
98C2×C3×C4⋊C4—21no—
99C2×C3×C4⋊C4—21no—
100C2×C3×C4⋊C4—22no—
101C2×C3×C4⋊C4C2311yescharacteristic
102C2×C3×C4⋊C4C2311yescharacteristic
103C2×C3×C4⋊C4C2311yescharacteristic
104C2×C3×C22⋊C4—22no—
105C2×C3×C22⋊C4—22no—
106C2×C3×C22⋊C4D412yes—
107C2×C3×C42—22no—
108C2×C3×C42C2311yescharacteristic
109C2×C3×C42C2311yescharacteristic
110C2×C3×C42C2311yescharacteristic
111C3×#107—22no—
112C3×#107—22no—
113C3×#106—22no—
114C3×#106—22no—
115C3×C4⋊C8—22no—
116C3×C4⋊C8—22no—
117C3×#104—42no—
118C3×#104—42no—
119C3×#103—22no—
120C3×#103—22no—
121C3×#102—22no—
122C3×#102—22no—
123C3×#100—42no—
124C3×#95—21no—
125C3×#95—22no—
126#544C22×C311yescharacteristic
127#542—22no—
128#540—22no—
129#538—22no—
130#537C22×C311yescharacteristic
131#535C22×C312yes—
132#532—22no—
133#529—22no—
134C2×#125C22×C311yescharacteristic
135#525—22no—
136C4×#41C22×C312yes—
137C2×C4×D4C22×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×#100C22×C312yes—
150#396—22no—
151#390C22×C311yescharacteristic
152#390C22×C311yescharacteristic
153#385—22no—
154#384C22×C311yescharacteristic
155#382C22×C311yescharacteristic
156C4×C4⋊C4C22×C311yescharacteristic
157#343C22×C311yescharacteristic
158#343C22×C311yescharacteristic
159#339C22×C312yes—
160#339C22×C312yes—
161C24×C3C2×D412yes—
162C3×#41—22no—
163C3×#41—22no—
164C3×#41—22no—
165C3×#41—22no—
166C3×#41—84no—
167C2×C3×Q8—22no—
168C2×C3×Q8C2×D412yes—
169C2×C3×D4—22no—
170C2×C3×D4—24no—
171C2×C3×D4—24no—
172C2×C3×D4—22no—
173C2×C3×D4—22no—
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4—22no—
177C2×C3×D4C2×D412yes—
178C22×C3×C4—21no—
179C22×C3×C4—22no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—22no—
183C22×C3×C4—22no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—82no—
188C22×C3×C4C2×D412yes—
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2411yescharacteristic
192C3×M16—22no—
193C3×M16—22no—
194C3×M16—22no—
195C3×M16—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—24no—
201C3×C4⋊C4—22no—
202C3×C4⋊C4—22no—
203C3×C4⋊C4—22no—
204C3×C4⋊C4—22no—
205C3×C4⋊C4—22no—
206C3×C4⋊C4—42no—
207C3×C4⋊C4—41no—
208C3×C4⋊C4—42no—
209C3×C4⋊C4—41no—
210C3×C4⋊C4—82no—
211C3×C4⋊C4C2×D412yes—
212C3×C4⋊C4C2×D412yes—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—22no—
217C3×C22⋊C4—24no—
218C3×C22⋊C4—22no—
219C3×C22⋊C4—22no—
220C3×C22⋊C4—44no—
221C3×C22⋊C4—82no—
222C3×C42—22no—
223C3×C42—22no—
224C3×C42—22no—
225C3×C42—21no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42C2×D412yes—
230C3×C42C2×D412yes—
231C2×#41—22no—
232C2×#41C23×C312yes—
233C22×D4C23×C311yescharacteristic
234C23×C4C3×D412yes—
235C2×M16C23×C312yes—
236C2×M16C23×C312yes—
237C4⋊Q8—22no—
238#127—22no—
239#126—24no—
240#125—22no—
241#125—22no—
242#125—22no—
243#124—24no—
244#124—22no—
245#123—24no—
246#123—24no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8—22no—
250#121—22no—
251#121—24no—
252#121—22no—
253#121—22no—
254#121—22no—
255#121—22no—
256#120—24no—
257C4×Q8C3×D412yes—
258C4×Q8C3×D412yes—
259C4×D4—24no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4—22no—
263C4×D4—22no—
264C4×D4C3×D412yes—
265C4×D4C3×D412yes—
266#117—22no—
267#117—22no—
268#117C23×C312yes—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4C23×C311yescharacteristic
274C2×C4⋊C4C23×C311yescharacteristic
275C2×C4⋊C4C23×C311yescharacteristic
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4C3×D412yes—
279C2×C42—22no—
280C2×C42C23×C311yescharacteristic
281C2×C42C23×C311yescharacteristic
282C2×C42C23×C311yescharacteristic
283#107—22no—
284#107—22no—
285#106—22no—
286#106—22no—
287C4⋊C8—22no—
288C4⋊C8—22no—
289#104—42no—
290#104—42no—
291#103—22no—
292#103—22no—
293#102—22no—
294#102—22no—
295#100—42no—
296#95—21no—
297#95—22no—
298C23×C3—22no—
299C23×C3—22no—
300C23×C3—22no—
301C23×C3—22no—
302C23×C3—42no—
303C23×C3—42no—
304C23×C3C22×D411yescharacteristic
305C3×Q8—22no—
306C3×Q8—22no—
307C3×Q8—82no—
308C3×D4—22no—
309C3×D4—22no—
310C3×D4—42no—
311C3×D4—42no—
312C3×D4—42no—
313C3×D4—42no—
314C3×D4—84no—
315C3×D4—82no—
316C3×D4—84no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—44no—
325C2×C3×C4—44no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4—42no—
329C2×C3×C4—42no—
330C2×C3×C4—42no—
331C2×C3×C4—42no—
332C2×C3×C4—41no—
333C2×C3×C4—42no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—41no—
337C2×C3×C4—41no—
338C2×C3×C4—82no—
339C2×C3×C4—82no—
340C2×C3×C4#12011yescharacteristic
341C2×C3×C4#12012yes—
342C2×C3×C4#12011yescharacteristic
343C2×C3×C4C22×D412yes—
344C2×C3×C4H(5,𝔽2)11yescharacteristic
345C2×C3×C42−1+411yescharacteristic
346C2×C3×C42−1+412yes—
347C3×C8—42no—
348C3×C8—42no—
349C24C2×C3×D412yes—
350#41—22no—
351#41—22no—
352#41—22no—
353#41—22no—
354#41—84no—
355C2×Q8—22no—
356C2×Q8C2×C3×D412yes—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—22no—
365C2×D4C2×C3×D412yes—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—82no—
376C22×C4C2×C3×D412yes—
377C22×C4C2×C3×D411yescharacteristic
378C22×C4C2×C3×D411yescharacteristic
379C22×C4C24×C311yescharacteristic
380M16—22no—
381M16—22no—
382M16—22no—
383M16—22no—
384C2×C8—22no—
385C2×C8—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—24no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—42no—
395C4⋊C4—41no—
396C4⋊C4—42no—
397C4⋊C4—41no—
398C4⋊C4—82no—
399C4⋊C4C2×C3×D412yes—
400C4⋊C4C2×C3×D412yes—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—22no—
405C22⋊C4—24no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—44no—
409C22⋊C4—82no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42C2×C3×D412yes—
418C42C2×C3×D412yes—
419C22×C3—21no—
420C22×C3—21no—
421C22×C3—44no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—42no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—82no—
429C22×C3C2×#12011yescharacteristic
430C22×C3#53611yescharacteristic
431C22×C3#53711yescharacteristic
432C3×C4—22no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—42no—
436C3×C4—42no—
437C3×C4—42no—
438C3×C4—42no—
439C3×C4—42no—
440C3×C4—81no—
441C3×C4—81no—
442C3×C4—81no—
443C3×C4#53612yes—
444C3×C4#53712yes—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—42no—
450C23—42no—
451C23C22×C3×D411yescharacteristic
452Q8—22no—
453Q8—22no—
454Q8—82no—
455D4—22no—
456D4—22no—
457D4—42no—
458D4—42no—
459D4—42no—
460D4—42no—
461D4—82no—
462D4—84no—
463D4—84no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4C3×#12011yescharacteristic
488C2×C4C3×#12012yes—
489C2×C4C3×#12011yescharacteristic
490C2×C4C22×C3×D412yes—
491C2×C4C3×H(5,𝔽2)11yescharacteristic
492C2×C4C3×2−1+411yescharacteristic
493C2×C4C3×2−1+412yes—
494C8—42no—
495C8—42no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—82no—
499C2×C3—82no—
500C2×C3#261611yescharacteristic
501C2×C3#301811yescharacteristic
502C2×C3#301911yescharacteristic
503C22—21no—
504C22—21no—
505C22—44no—
506C22—42no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—82no—
513C22C2×C3×#12011yescharacteristic
514C22C3×#53611yescharacteristic
515C22C3×#53711yescharacteristic
516C4—22no—
517C4—21no—
518C4—21no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—42no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4C3×#53612yes—
528C4C3×#53712yes—
529C3#2970711yescharacteristic, direct product factor, Sylow
530C2—21no—
531C2—21no—
532C2—82no—
533C2—82no—
534C2C3×#261611yescharacteristic
535C2C3×#301811yescharacteristic
536C2C3×#301911yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3460C2
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×#2566C2semidirect
4. C3×#2061C2semidirect
5. C3×#2060C2semidirect
6. C3×#2040C2semidirect
7. C3×#2039C2semidirect
8. C3×#2037C2semidirect
9. C3×#2037C2semidirect
10. C3×#1806C2semidirect
11. C3×#1800C2semidirect
12. #29707C3direct (N × H)
49. #3080C2×C3semidirect
50. #2566C2×C3semidirect
51. #2061C2×C3semidirect
52. #2060C2×C3semidirect
53. #2040C2×C3semidirect
54. #2039C2×C3semidirect
55. #2037C2×C3semidirect
56. #2037C2×C3semidirect
57. #1806C2×C3semidirect
58. #1800C2×C3semidirect
529. C3#29707direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#34602
C3×#30802
C3×#25662
C3×#20612
C3×#20602
C3×#20402
C3×#20392
C3×#20372
C3×#18062
C3×#18002
#297073