Finite Group Databasebeta

← C3×#25615 #10644222

Subgroups — C3×#25615 #10644222

Order 768.

1116 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3511C211yesmaximal, characteristic
2C3×#3152C211yesmaximal, characteristic, complemented
3C3×#3039C211yesmaximal, characteristic, complemented
4C3×#2862C211yesmaximal, characteristic, complemented
5C3×#2007C214yesmaximal, complemented
6C3×#1997C211yesmaximal, characteristic, complemented
7C3×#1962C211yesmaximal, characteristic, complemented
8C3×#1882C214yesmaximal, complemented
9C3×C4×#104C211yesmaximal, characteristic, complemented
10#25615C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×#137C2214yes—
12C2×C3×#137C2212yes—
13C3×#556—21no—
14C3×#555—21no—
15C3×#554—21no—
16C3×#553—22no—
17C3×#548—22no—
18C3×#547—21no—
19C3×#542—22no—
20C3×#537C2212yes—
21C3×#533—21no—
22C3×#532—21no—
23C3×#532C2211yescharacteristic
24C2×C3×C4⋊Q8C2211yescharacteristic
25C2×C3×C4⋊Q8C2211yescharacteristic
26C3×#520C2212yes—
27C3×C4×#41C2211yescharacteristic
28C3×#497—22no—
29C3×#494—21no—
30C3×#492—21no—
31C3×#462—21no—
32C3×#462—21no—
33C3×#461—21no—
34C3×#461—21no—
35C3×#423C2211yescharacteristic
36C2×C3×#104C2212yes—
37C2×C3×#104C2214yes—
38C3×#419C2212yes—
39C3×C4×M16C2211yescharacteristic
40C3×#391—22no—
41C3×#391C2214yes—
42C3×#382—22no—
43C3×C43C2211yescharacteristic
44C3×#343C2214yes—
45C3×#339C2212yes—
46#3511C2×C311yescharacteristic
47#3152C2×C311yescharacteristic, complemented
48#3039C2×C311yescharacteristic, complemented
49#2862C2×C311yescharacteristic, complemented
50#2007C2×C314yescomplemented
51#1997C2×C311yescharacteristic, complemented
52#1962C2×C311yescharacteristic, complemented
53#1882C2×C314yescomplemented
54C4×#104C2×C311yescharacteristic, complemented
55C2×C3×#41—22no—
56C2×C3×#41C2311yescharacteristic
57C2×C3×#41C2312yes—
58C22×C3×Q8D412yes—
59C22×C3×Q8C2312yes—
60C3×#137—28no—
61C3×#137—44no—
62C3×#137—44no—
63C2×C3×Q16—22no—
64C2×C3×Q16—24no—
65C2×C3×SD16—22no—
66C2×C3×SD16—24no—
67C2×C3×M16C2312yes—
68C2×C3×M16C2312yes—
69C3×C4⋊Q8—21no—
70C3×C4⋊Q8—21no—
71C3×C4⋊Q8—21no—
72C3×C4⋊Q8—21no—
73C3×C4⋊Q8—21no—
74C3×C4⋊Q8—21no—
75C3×C4⋊Q8—21no—
76C3×C4⋊Q8—21no—
77C3×C4⋊Q8D411yescharacteristic
78C3×#127D411yescharacteristic
79C3×#126—24no—
80C3×#125—22no—
81C3×#124—22no—
82C3×#124—24no—
83C3×#124D412yes—
84C3×#123—24no—
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8—24no—
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8—22no—
89C3×#121—22no—
90C3×#121—22no—
91C3×#121—22no—
92C3×#121—21no—
93C3×#121—21no—
94C3×#121—24no—
95C3×C4×Q8—22no—
96C3×C4×Q8—22no—
97C3×C4×Q8D411yescharacteristic
98C3×C4×Q8D411yescharacteristic
99C3×C4×D4—22no—
100C3×C4×D4—22no—
101C3×C4×D4—21no—
102C3×C4×D4—21no—
103C3×C4×D4—24no—
104C3×C4×D4—21no—
105C3×C4×D4—21no—
106C3×C4×D4D411yescharacteristic
107C3×C4×D4D411yescharacteristic
108C3×#117—22no—
109C3×#117—22no—
110C3×#117—21no—
111C3×#117C2311yescharacteristic
112C2×C3×C4⋊C4—22no—
113C2×C3×C4⋊C4—21no—
114C2×C3×C4⋊C4—21no—
115C2×C3×C4⋊C4D412yes—
116C2×C3×C4⋊C4C2312yes—
117C2×C3×C42—21no—
118C2×C3×C42—21no—
119C2×C3×C42—21no—
120C2×C3×C42C2312yes—
121C2×C3×C42C2311yescharacteristic
122C3×C4⋊C8—21no—
123C3×C4⋊C8—21no—
124C3×#104—24no—
125C3×#104—44no—
126C3×#103—22no—
127C3×#103—22no—
128C3×#97—21no—
129C3×C4×C8—21no—
130C3×#95—28no—
131C2×#137C22×C314yes—
132C2×#137C22×C312yes—
133#556—21no—
134#555—21no—
135#554—21no—
136#553—22no—
137#548—22no—
138#547—21no—
139#542—22no—
140#537C22×C312yes—
141#533—21no—
142#532—21no—
143#532C22×C311yescharacteristic
144C2×C4⋊Q8C22×C311yescharacteristic
145C2×C4⋊Q8C22×C311yescharacteristic
146#520C22×C312yes—
147C4×#41C22×C311yescharacteristic
148#497—22no—
149#494—21no—
150#492—21no—
151#462—21no—
152#462—21no—
153#461—21no—
154#461—21no—
155#423C22×C311yescharacteristic
156C2×#104C22×C312yes—
157C2×#104C22×C314yes—
158#419C22×C312yes—
159C4×M16C22×C311yescharacteristic
160#391—22no—
161#391C22×C314yes—
162#382—22no—
163C43C22×C311yescharacteristic
164#343C22×C314yes—
165#339C22×C312yes—
166C3×#41—22no—
167C3×#41—24no—
168C3×#41—24no—
169C3×#41—22no—
170C3×#41—84no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—24no—
176C2×C3×Q8—24no—
177C2×C3×Q8—44no—
178C2×C3×Q8C2×D412yes—
179C2×C3×Q8C2×D411yescharacteristic
180C2×C3×D4—21no—
181C2×C3×D4—22no—
182C2×C3×D4—24no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4C2×D412yes—
186C2×C3×D4C2×D411yescharacteristic
187C22×C3×C4—22no—
188C22×C3×C4—24no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—22no—
193C22×C3×C4—22no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D414yes—
197C22×C3×C4C2411yescharacteristic
198C3×Q16—48no—
199C3×Q16—44no—
200C3×SD16—48no—
201C3×SD16—44no—
202C3×M16—22no—
203C3×M16—24no—
204C3×M16#4114yescomplemented
205C2×C3×C8—22no—
206C2×C3×C8—22no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—22no—
210C3×C4⋊C4—22no—
211C3×C4⋊C4—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—22no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—41no—
218C3×C4⋊C4—41no—
219C3×C4⋊C4—41no—
220C3×C4⋊C4—41no—
221C3×C4⋊C4C2×D411yescharacteristic
222C3×C4⋊C4C2×D411yescharacteristic
223C3×C22⋊C4—24no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—24no—
226C3×C22⋊C4—21no—
227C3×C22⋊C4—21no—
228C3×C22⋊C4—21no—
229C3×C22⋊C4—22no—
230C3×C22⋊C4—22no—
231C3×C42—24no—
232C3×C42—24no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—22no—
237C3×C42—22no—
238C3×C42—42no—
239C3×C42C2×D411yescharacteristic
240C3×C42C2×D411yescharacteristic
241C3×C42C2×D411yescharacteristic
242C3×C42C2×D411yescharacteristic
243C2×#41—22no—
244C2×#41C23×C311yescharacteristic
245C2×#41C23×C312yes—
246C22×Q8C3×D412yes—
247C22×Q8C23×C312yes—
248#137—28no—
249#137—44no—
250#137—44no—
251C2×Q16—22no—
252C2×Q16—24no—
253C2×SD16—22no—
254C2×SD16—24no—
255C2×M16C23×C312yes—
256C2×M16C23×C312yes—
257C4⋊Q8—21no—
258C4⋊Q8—21no—
259C4⋊Q8—21no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—21no—
263C4⋊Q8—21no—
264C4⋊Q8—21no—
265C4⋊Q8C3×D411yescharacteristic
266#127C3×D411yescharacteristic
267#126—24no—
268#125—22no—
269#124—22no—
270#124—24no—
271#124C3×D412yes—
272#123—24no—
273C22⋊Q8—22no—
274C22⋊Q8—24no—
275C22⋊Q8—22no—
276C22⋊Q8—22no—
277#121—22no—
278#121—22no—
279#121—22no—
280#121—21no—
281#121—21no—
282#121—24no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8C3×D411yescharacteristic
286C4×Q8C3×D411yescharacteristic
287C4×D4—22no—
288C4×D4—22no—
289C4×D4—21no—
290C4×D4—21no—
291C4×D4—24no—
292C4×D4—21no—
293C4×D4—21no—
294C4×D4C3×D411yescharacteristic
295C4×D4C3×D411yescharacteristic
296#117—22no—
297#117—22no—
298#117—21no—
299#117C23×C311yescharacteristic
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4C3×D412yes—
304C2×C4⋊C4C23×C312yes—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42C23×C312yes—
309C2×C42C23×C311yescharacteristic
310C4⋊C8—21no—
311C4⋊C8—21no—
312#104—24no—
313#104—44no—
314#103—22no—
315#103—22no—
316#97—21no—
317C4×C8—21no—
318#95—28no—
319C23×C3—21no—
320C23×C3—22no—
321C23×C3C22×D411yescharacteristic
322C3×Q8—21no—
323C3×Q8—21no—
324C3×Q8—24no—
325C3×Q8—44no—
326C3×Q8—42no—
327C3×Q8—42no—
328C3×Q8—44no—
329C3×Q8—82no—
330C3×D4—21no—
331C3×D4—21no—
332C3×D4—24no—
333C3×D4—42no—
334C3×D4—44no—
335C3×D4—42no—
336C3×D4—42no—
337C3×D4—84no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—24no—
341C2×C3×C4—21no—
342C2×C3×C4—22no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—24no—
346C2×C3×C4—42no—
347C2×C3×C4—42no—
348C2×C3×C4—42no—
349C2×C3×C4—42no—
350C2×C3×C4—42no—
351C2×C3×C4—42no—
352C2×C3×C4—42no—
353C2×C3×C4—44no—
354C2×C3×C4—42no—
355C2×C3×C4—44no—
356C2×C3×C4—84no—
357C2×C3×C4#12011yescharacteristic
358C2×C3×C4#12011yescharacteristic
359C2×C3×C4#12011yescharacteristic
360C2×C3×C4#12011yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4C22×D411yescharacteristic
363C2×C3×C4C2×#4112yes—
364C2×C3×C4H(5,𝔽2)12yes—
365C3×C8—24no—
366C3×C8—42no—
367#41—22no—
368#41—24no—
369#41—24no—
370#41—22no—
371#41—84no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—24no—
377C2×Q8—24no—
378C2×Q8—44no—
379C2×Q8C2×C3×D412yes—
380C2×Q8C2×C3×D411yescharacteristic
381C2×D4—21no—
382C2×D4—22no—
383C2×D4—24no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4C2×C3×D412yes—
387C2×D4C2×C3×D411yescharacteristic
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4C2×C3×D411yescharacteristic
396C22×C4C2×C3×D411yescharacteristic
397C22×C4C2×C3×D414yes—
398C22×C4C24×C311yescharacteristic
399Q16—48no—
400Q16—44no—
401SD16—48no—
402SD16—44no—
403M16—22no—
404M16—24no—
405M16C3×#4114yescomplemented
406C2×C8—22no—
407C2×C8—22no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—21no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4C2×C3×D411yescharacteristic
423C4⋊C4C2×C3×D411yescharacteristic
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—21no—
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C42—24no—
433C42—24no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—22no—
438C42—22no—
439C42—42no—
440C42C2×C3×D411yescharacteristic
441C42C2×C3×D411yescharacteristic
442C42C2×C3×D411yescharacteristic
443C42C2×C3×D411yescharacteristic
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—42no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3—42no—
450C22×C3—41no—
451C22×C3—42no—
452C22×C3C2×#12011yescharacteristic
453C22×C3#55012yes—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—44no—
457C3×C4—42no—
458C3×C4—41no—
459C3×C4—41no—
460C3×C4—44no—
461C3×C4—41no—
462C3×C4—82no—
463C3×C4#45512yes—
464C3×C4#45612yes—
465C3×C4#53211yescharacteristic
466C3×C4#53511yescharacteristic
467C3×C4#54612yes—
468C23—21no—
469C23—22no—
470C23C22×C3×D411yescharacteristic
471Q8—21no—
472Q8—21no—
473Q8—24no—
474Q8—44no—
475Q8—42no—
476Q8—42no—
477Q8—44no—
478Q8—82no—
479D4—21no—
480D4—21no—
481D4—24no—
482D4—42no—
483D4—44no—
484D4—42no—
485D4—42no—
486D4—84no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—24no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—84no—
506C2×C4C3×#12011yescharacteristic
507C2×C4C3×#12011yescharacteristic
508C2×C4C3×#12011yescharacteristic
509C2×C4C3×#12011yescharacteristic
510C2×C4C22×C3×D411yescharacteristic
511C2×C4C22×C3×D411yescharacteristic
512C2×C4C2×C3×#4112yes—
513C2×C4C3×H(5,𝔽2)12yes—
514C8—24no—
515C8—42no—
516C2×C3—22no—
517C2×C3—81no—
518C2×C3—82no—
519C2×C3#243111yescharacteristic
520C2×C3C2×#45511yescharacteristic
521C2×C3C2×#45611yescharacteristic
522C22—21no—
523C22—21no—
524C22—42no—
525C22—41no—
526C22—41no—
527C22—42no—
528C22—41no—
529C22—42no—
530C22C2×C3×#12011yescharacteristic
531C22C3×#55012yes—
532C4—21no—
533C4—21no—
534C4—44no—
535C4—42no—
536C4—41no—
537C4—41no—
538C4—44no—
539C4—41no—
540C4—82no—
541C4C3×#45512yes—
542C4C3×#45612yes—
543C4C3×#53211yescharacteristic
544C4C3×#53511yescharacteristic
545C4C3×#54612yes—
546C3#2561511yescharacteristic, direct product factor, Sylow
547C2—22no—
548C2—81no—
549C2—82no—
550C2C3×#243111yescharacteristic
551C2C2×C3×#45511yescharacteristic
552C2C2×C3×#45611yescharacteristic
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC3×#3511C2
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×#3152C2semidirect
3. C3×#3039C2semidirect
4. C3×#2862C2semidirect
5. C3×#2007C2semidirect
6. C3×#1997C2semidirect
7. C3×#1962C2semidirect
8. C3×#1882C2semidirect
9. C3×C4×#104C2semidirect
10. #25615C3direct (N × H)
47. #3152C2×C3semidirect
48. #3039C2×C3semidirect
49. #2862C2×C3semidirect
50. #2007C2×C3semidirect
51. #1997C2×C3semidirect
52. #1962C2×C3semidirect
53. #1882C2×C3semidirect
54. C4×#104C2×C3semidirect
204. C3×M16#41semidirect
405. M16C3×#41semidirect
546. C3#25615direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35112
C3×#31522
C3×#30392
C3×#28622
C3×#20072
C3×#19972
C3×#19622
C3×#18822
C3×C4×#1042
#256153