Finite Group Databasebeta

← C3×#78692 #11865401

Subgroups — C3×#78692 #11865401

Order 1152.

1511 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5964C211yesmaximal, characteristic
2C3×#5825C211yesmaximal, characteristic, complemented
3C3×#5526C212yesmaximal, complemented
4C32×#410C211yesmaximal, characteristic, complemented
5C3×#5006C212yesmaximal, complemented
6C3×#4742C218yesmaximal, complemented
7C3×#3028—31nomaximal
8#78692C311yesmaximal, characteristic, direct product factor
9C2×C32×#41C2211yescharacteristic
10C3×#1010—22no—
11C3×#1009—22no—
12C3×D3×#41—24no—
13C3×#1007—21no—
14C3×#1004—21no—
15C3×#1004C2214yes—
16C2×C3×#235—22no—
17C2×C3×#235C2212yes—
18C22×C3×Dic6C2211yescharacteristic, complemented
19C32×#117C2212yescomplemented
20C3×#944—21no—
21C3×#936—21no—
22C3×#935C2214yescomplemented
23C3×#926C2211yescharacteristic, complemented
24C3×#924—22no—
25C3×#897—24no—
26C3×#886C2218yescomplemented
27C3×#875—22no—
28C3×#869—22no—
29C32×#99C2214yescomplemented
30C3×#806C2218yescomplemented
31C3×#585—31no—
32#5964C2×C311yescharacteristic
33C3×#536—31no—
34C3×#536—32no—
35#5825C2×C311yescharacteristic, complemented
36#5526C2×C312yescomplemented
37C3×#458—38no—
38C3×#410—21no—
39C3×#410—32no—
40C3×#410D311yescharacteristic, complemented
41C3×#410C2×C311yescharacteristic, complemented
42#5006C2×C312yescomplemented
43#4742C2×C318yescomplemented
44C32×#41—22no—
45C32×#41—22no—
46C2×C32×Q8D411yescharacteristic
47C2×C32×D4D411yescharacteristic
48C2×C32×D4C2312yes—
49C22×C32×C4D412yes—
50C22×C32×C4C2311yescharacteristic
51C2×C3×#67—24no—
52C2×C3×#67C2314yes—
53C22×C3×Dic3C2312yes—
54C3×#239—24no—
55C3×D3×Q8—24no—
56C3×#237—24no—
57C3×#237—28no—
58C3×D3×D4—24no—
59C3×D3×D4—28no—
60C3×#235—24no—
61C3×#235—24no—
62C3×#235—22no—
63C3×#235—24no—
64C3×#235—22no—
65C3×#235—22no—
66C3×#235—22no—
67C2×C3×D12—22no—
68C2×C3×D12D412yes—
69C2×C3×C4×D3—24no—
70C2×C3×C4×D3D414yes—
71C2×C3×Dic6—22no—
72C2×C3×Dic6—22no—
73C2×C3×Dic6—21no—
74C2×C3×Dic6—21no—
75C2×C3×Dic6—22no—
76C2×C3×Dic6D412yes—
77C32×C4⋊C4—22no—
78C32×C22⋊C4C2314yes—
79C32×C42—22no—
80C3×#217—24no—
81C3×#217C2312yes—
82C3×(C2×D3)⋊C4—28no—
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3—28no—
85C3×Dic3⋊C4—24no—
86C3×C4×Dic3—21no—
87#3028—31noSylow
88C3×2−1+4—62no—
89C3×2−1+4—61no—
90C3×H(5,𝔽2)—34no—
91C3×H(5,𝔽2)—62no—
92C3×H(5,𝔽2)—61no—
93C2×C3×#41—21no—
94C2×C3×#41—32no—
95C2×C3×#41—64no—
96C2×C3×#41—62no—
97C2×C3×#41C2×D311yescharacteristic
98C2×C3×#41C22×C311yescharacteristic
99C22×C3×Q8—31no—
100#1010—22no—
101#1009—22no—
102D3×#41—24no—
103#1007—21no—
104#1004—21no—
105#1004C22×C314yes—
106C2×#235—22no—
107C2×#235C22×C312yes—
108C22×Dic6C22×C311yescharacteristic, complemented
109C3×C4⋊Q8—62no—
110C3×C4⋊Q8—61no—
111C3×#124—62no—
112C3×#124—61no—
113C3×#123—34no—
114C3×#123—38no—
115C3×C22⋊Q8—64no—
116C3×C22⋊Q8—62no—
117C3×#117—22no—
118C3×#117—31no—
119C3×#117C2×D312yescomplemented
120C3×#117C22×C312yescomplemented
121#944—21no—
122#936—21no—
123#935C22×C314yescomplemented
124#926C22×C311yescharacteristic, complemented
125#924—22no—
126#897—24no—
127#886C22×C318yescomplemented
128#875—22no—
129#869—22no—
130C3×#99—24no—
131C3×#99—38no—
132C3×#99C2×D314yescomplemented
133C3×#99C22×C314yescomplemented
134#806C22×C318yescomplemented
135C23×C32C2×D412yes—
136C23×C32C2411yescharacteristic
137C22×C3×D3—24no—
138C22×C3×D3C2×D414yes—
139C32×Q8—22no—
140C32×D4—24no—
141C32×D4—22no—
142C2×C32×C4—22no—
143C2×C32×C4—22no—
144C2×C32×C4—24no—
145C2×C32×C4C2×D412yes—
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4C2×D412yes—
149C2×C32×C4C2×D412yes—
150C3×#67—28no—
151C3×#67—24no—
152C3×#67—24no—
153C2×C3×Dic3—24no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—24no—
157C2×C3×Dic3C2×D414yes—
158C3×D12—24no—
159C3×D12—22no—
160C3×D12—22no—
161C3×C4×D3—28no—
162C3×C4×D3—24no—
163C3×C4×D3—24no—
164C3×Dic6—24no—
165C3×Dic6—22no—
166C3×Dic6—22no—
167C3×Dic6—41no—
168C3×Dic6—42no—
169C3×Dic6—41no—
170#585—31no—
171#536—31no—
172#536—32no—
173#458—38no—
174#410—32no—
175#410C3×D311yescharacteristic, complemented
176C3×#41—22no—
177C3×#41—22no—
178C3×#41—22no—
179C3×#41—22no—
180C3×#41—42no—
181C3×#41—42no—
182C3×#41—64no—
183C3×#41—64no—
184C3×#41—68no—
185C3×#41—64no—
186C3×#41—64no—
187C3×#41—62no—
188C3×#41—62no—
189C3×#41—64no—
190C3×#41—62no—
191C3×#41—62no—
192C2×C3×Q8—21no—
193C2×C3×Q8—32no—
194C2×C3×Q8—62no—
195C2×C3×Q8—64no—
196C2×C3×Q8—61no—
197C2×C3×Q8—61no—
198C2×C3×Q8—62no—
199C2×C3×Q8—62no—
200C2×C3×Q8D1211yescharacteristic
201C2×C3×Q8C3×D411yescharacteristic
202C2×C3×D4—21no—
203C2×C3×D4—22no—
204C2×C3×D4—34no—
205C2×C3×D4—32no—
206C2×C3×D4—62no—
207C2×C3×D4—68no—
208C2×C3×D4—64no—
209C2×C3×D4—64no—
210C2×C3×D4D1211yescharacteristic
211C2×C3×D4C3×D411yescharacteristic
212C2×C3×D4C22×D312yes—
213C2×C3×D4C23×C312yes—
214C22×C3×C4—22no—
215C22×C3×C4—21no—
216C22×C3×C4—34no—
217C22×C3×C4—32no—
218C22×C3×C4—64no—
219C22×C3×C4D1212yes—
220C22×C3×C4C3×D412yes—
221C22×C3×C4C22×D311yescharacteristic
222C22×C3×C4C23×C311yescharacteristic
223C2×#67—24no—
224C2×#67C23×C314yes—
225C22×Dic3C23×C312yes—
226#239—24no—
227D3×Q8—24no—
228#237—24no—
229#237—28no—
230D3×D4—24no—
231D3×D4—28no—
232#235—24no—
233#235—24no—
234#235—22no—
235#235—24no—
236#235—22no—
237#235—22no—
238#235—22no—
239C2×D12—22no—
240C2×D12C3×D412yes—
241C2×C4×D3—24no—
242C2×C4×D3C3×D414yes—
243C2×Dic6—22no—
244C2×Dic6—22no—
245C2×Dic6—22no—
246C2×Dic6—21no—
247C2×Dic6—21no—
248C2×Dic6C3×D412yes—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—42no—
252C3×C4⋊C4—68no—
253C3×C4⋊C4—64no—
254C3×C4⋊C4—61no—
255C3×C22⋊C4—24no—
256C3×C22⋊C4—32no—
257C3×C22⋊C4—68no—
258C3×C22⋊C4—64no—
259C3×C22⋊C4C22×D314yes—
260C3×C22⋊C4C23×C314yes—
261C3×C42—22no—
262C3×C42—22no—
263C3×C42—42no—
264C3×C42—61no—
265#217—24no—
266#217C23×C312yes—
267(C2×D3)⋊C4—28no—
268C4⋊Dic3—21no—
269C4⋊Dic3—28no—
270Dic3⋊C4—24no—
271C4×Dic3—21no—
272C22×C32—22no—
273C22×C32—41no—
274C22×C32—42no—
275C22×C32C22×D411yescharacteristic
276C22×C32C22×D412yes—
277C2×C3×D3—24no—
278C2×C3×D3—44no—
279C2×C3×D3—84no—
280C32×C4—22no—
281C32×C4—44no—
282C32×C4#12011yescharacteristic
283C32×C4#12012yes—
284C32×C4#12011yescharacteristic
285C3×Dic3—24no—
286C3×Dic3—42no—
287C3×Dic3—42no—
2882−1+4—62no—
2892−1+4—61no—
290H(5,𝔽2)—34no—
291H(5,𝔽2)—62no—
292H(5,𝔽2)—61no—
293C2×#41—32no—
294C2×#41—64no—
295C2×#41—62no—
296C2×#41C2×C3×D311yescharacteristic
297C22×Q8—31no—
298C4⋊Q8—62no—
299C4⋊Q8—61no—
300#124—62no—
301#124—61no—
302#123—34no—
303#123—38no—
304C22⋊Q8—64no—
305C22⋊Q8—62no—
306#117—31no—
307#117C2×C3×D312yescomplemented
308#99—38no—
309#99C2×C3×D314yescomplemented
310C23×C3—21no—
311C23×C3—22no—
312C23×C3—34no—
313C23×C3—64no—
314C23×C3C2×D1212yes—
315C23×C3C2×C3×D412yes—
316C23×C3C23×D311yescharacteristic
317C23×C3C24×C311yescharacteristic
318C22×D3—24no—
319C22×D3C2×C3×D414yes—
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—42no—
323C3×Q8—64no—
324C3×Q8—62no—
325C3×Q8—62no—
326C3×Q8—121no—
327C3×Q8—121no—
328C3×Q8—122no—
329C3×D4—24no—
330C3×D4—24no—
331C3×D4—22no—
332C3×D4—22no—
333C3×D4—42no—
334C3×D4—44no—
335C3×D4—64no—
336C3×D4—68no—
337C3×D4—64no—
338C3×D4—64no—
339C3×D4—62no—
340C3×D4—62no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—24no—
346C2×C3×C4—24no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—22no—
352C2×C3×C4—34no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—44no—
356C2×C3×C4—68no—
357C2×C3×C4—64no—
358C2×C3×C4—64no—
359C2×C3×C4—64no—
360C2×C3×C4—62no—
361C2×C3×C4—62no—
362C2×C3×C4—64no—
363C2×C3×C4C2×D1212yes—
364C2×C3×C4C2×D1211yescharacteristic
365C2×C3×C4C2×D1211yescharacteristic
366C2×C3×C4D3×D412yes—
367C2×C3×C4D3×D412yes—
368C2×C3×C4C2×C3×D412yes—
369C2×C3×C4C2×C3×D411yescharacteristic
370C2×C3×C4C2×C3×D411yescharacteristic
371C2×C3×C4C2×C3×D412yes—
372C2×C3×C4C2×C3×D412yes—
373#67—28no—
374#67—24no—
375#67—24no—
376C2×Dic3—24no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—24no—
380C2×Dic3C2×C3×D414yes—
381D12—24no—
382D12—22no—
383D12—22no—
384C4×D3—28no—
385C4×D3—24no—
386C4×D3—24no—
387Dic6—24no—
388Dic6—22no—
389Dic6—22no—
390Dic6—41no—
391Dic6—42no—
392Dic6—41no—
393C2×C32—21no—
394C2×C32—22no—
395C2×C32—42no—
396C2×C32C2×#12011yescharacteristic
397C3×D3—44no—
398#41—22no—
399#41—22no—
400#41—64no—
401#41—64no—
402#41—68no—
403#41—64no—
404#41—64no—
405#41—62no—
406#41—62no—
407#41—64no—
408#41—62no—
409#41—62no—
410C2×Q8—32no—
411C2×Q8—62no—
412C2×Q8—64no—
413C2×Q8—61no—
414C2×Q8—61no—
415C2×Q8—62no—
416C2×Q8—62no—
417C2×Q8C3×D1211yescharacteristic
418C2×D4—34no—
419C2×D4—32no—
420C2×D4—62no—
421C2×D4—68no—
422C2×D4—64no—
423C2×D4—64no—
424C2×D4C3×D1211yescharacteristic
425C2×D4C22×C3×D312yes—
426C22×C4—34no—
427C22×C4—32no—
428C22×C4—64no—
429C22×C4C3×D1212yes—
430C22×C4C22×C3×D311yescharacteristic
431C4⋊C4—22no—
432C4⋊C4—68no—
433C4⋊C4—61no—
434C4⋊C4—64no—
435C22⋊C4—32no—
436C22⋊C4—68no—
437C22⋊C4—64no—
438C22⋊C4C22×C3×D314yes—
439C42—22no—
440C42—61no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—21no—
444C22×C3—22no—
445C22×C3—42no—
446C22×C3—41no—
447C22×C3—41no—
448C22×C3—42no—
449C22×C3—42no—
450C22×C3—64no—
451C22×C3—81no—
452C22×C3—82no—
453C22×C3—124no—
454C22×C3—244no—
455C22×C3C22×D1211yescharacteristic
456C22×C3C2×D3×D412yes—
457C22×C3C22×C3×D411yescharacteristic
458C22×C3C22×C3×D412yes—
459C2×D3—24no—
460C2×D3—44no—
461C2×D3—84no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—22no—
467C3×C4—42no—
468C3×C4—44no—
469C3×C4—44no—
470C3×C4—64no—
471C3×C4—84no—
472C3×C4—122no—
473C3×C4—122no—
474C3×C4#88211yescharacteristic
475C3×C4#88211yescharacteristic
476C3×C4#88212yes—
477C3×C4C3×#12011yescharacteristic
478C3×C4C3×#12012yes—
479C3×C4C3×#12011yescharacteristic
480Dic3—24no—
481Dic3—42no—
482Dic3—42no—
483C32#302811yescharacteristic, complemented, Sylow
484C23—34no—
485C23—64no—
486C23C2×C3×D1212yes—
487C23C23×C3×D311yescharacteristic
488Q8—22no—
489Q8—64no—
490Q8—62no—
491Q8—62no—
492Q8—121no—
493Q8—121no—
494Q8—122no—
495D4—24no—
496D4—22no—
497D4—64no—
498D4—68no—
499D4—64no—
500D4—64no—
501D4—62no—
502D4—62no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—34no—
507C2×C4—68no—
508C2×C4—64no—
509C2×C4—64no—
510C2×C4—64no—
511C2×C4—62no—
512C2×C4—62no—
513C2×C4—64no—
514C2×C4C2×C3×D1212yes—
515C2×C4C2×C3×D1211yescharacteristic
516C2×C4C2×C3×D1211yescharacteristic
517C2×C4C3×D3×D412yes—
518C2×C4C3×D3×D412yes—
519C2×C3—22no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—22no—
523C2×C3—21no—
524C2×C3—42no—
525C2×C3—41no—
526C2×C3—42no—
527C2×C3—42no—
528C2×C3—82no—
529C2×C3—124no—
530C2×C3C2×#88211yescharacteristic
531C2×C3C2×C3×#12011yescharacteristic
532D3—44no—
533C22—22no—
534C22—41no—
535C22—42no—
536C22—64no—
537C22—124no—
538C22—244no—
539C22C22×C3×D1211yescharacteristic
540C22C2×C3×D3×D412yes—
541C4—22no—
542C4—44no—
543C4—64no—
544C4—122no—
545C4—122no—
546C4C3×#88211yescharacteristic
547C4C3×#88211yescharacteristic
548C4C3×#88212yes—
549C3—21no—
550C3#7869211yescharacteristic, direct product factor
551C3C3×#302811yescharacteristic, complemented
552C2—22no—
553C2—21no—
554C2—42no—
555C2—124no—
556C2C2×C3×#88211yescharacteristic
subgroupG/H
centerC2×C3C2×#882
commutator, cosocleC23×C3C24×C3
FrattiniC23C23×C3×D3
FittingC32×#410C2
socleC2×C32C2×#120
prime radicalC3×#5964C2
hypercenterC3×#410D3
hypocenterC3C3×#3028
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#410

Complements

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

NH ≅ G/Ntype
2. C3×#5825C2semidirect
3. C3×#5526C2semidirect
4. C32×#410C2semidirect
5. C3×#5006C2semidirect
6. C3×#4742C2semidirect
8. #78692C3direct (N × H)
18. C22×C3×Dic6C22semidirect
19. C32×#117C22semidirect
22. C3×#935C22semidirect
23. C3×#926C22semidirect
26. C3×#886C22semidirect
29. C32×#99C22semidirect
30. C3×#806C22semidirect
35. #5825C2×C3semidirect
36. #5526C2×C3semidirect
40. C3×#410D3semidirect
41. C3×#410C2×C3semidirect
42. #5006C2×C3semidirect
43. #4742C2×C3semidirect
108. C22×Dic6C22×C3semidirect
119. C3×#117C2×D3semidirect
120. C3×#117C22×C3semidirect
123. #935C22×C3semidirect
124. #926C22×C3semidirect
127. #886C22×C3semidirect
132. C3×#99C2×D3semidirect
133. C3×#99C22×C3semidirect
134. #806C22×C3semidirect
175. #410C3×D3semidirect
307. #117C2×C3×D3semidirect
309. #99C2×C3×D3semidirect
483. C32#3028semidirect
550. C3#78692direct (N × H)
551. C3C3×#3028semidirect

Maximal subgroups

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

subgroupindex
C3×#59642
C3×#58252
C3×#55262
C32×#4102
C3×#50062
C3×#47422
C3×#30283
#786923