Finite Group Databasebeta

← C3×#72141 #11784963

Subgroups — C3×#72141 #11784963

Order 1152.

1625 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
1C2×C32×H(5,𝔽2)C211yesmaximal, characteristic
2C3×#5240C211yesmaximal, characteristic, complemented
3C3×#5237C211yesmaximal, characteristic, complemented
4C3×#5236C212yesmaximal, complemented
5C3×#5215C212yesmaximal, complemented
6C3×#1795—31nomaximal
7#72141C311yesmaximal, characteristic, direct product factor
8C32×H(5,𝔽2)—24no—
9C32×H(5,𝔽2)—24no—
10C2×C32×#41—21no—
11C2×C32×#41C412yescomplemented
12C2×C32×#41C2211yescharacteristic
13C2×C32×#41C2211yescharacteristic
14C22×C32×D4—21no—
15C22×C32×D4—22no—
16C22×C32×D4C412yescomplemented
17C22×C32×D4C2211yescharacteristic
18C3×#926C2212yescomplemented
19C2×C3×#208C2212yescomplemented
20C3×#837—24no—
21C3×#836—21no—
22C3×#834—22no—
23C3×#833—21no—
24C3×#832—24no—
25C2×C3×H(5,𝔽2)—21no—
26C2×C3×H(5,𝔽2)D311yescharacteristic
27C2×C3×H(5,𝔽2)C2×C311yescharacteristic
28C3×#421—32no—
29C3×#418—32no—
30C3×#413—31no—
31C3×#410—31no—
32#5240C2×C311yescharacteristic, complemented
33#5237C2×C311yescharacteristic, complemented
34#5236C2×C312yescomplemented
35#5215C2×C312yescomplemented
36C24×C32—22no—
37C24×C32—21no—
38C32×#41—28no—
39C32×#41—24no—
40C32×#41—24no—
41C32×#41—22no—
42C32×#41—22no—
43C32×#41—22no—
44C32×#41D414yes—
45C2×C32×Q8C2×C411yescharacteristic
46C2×C32×Q8D411yescharacteristic
47C2×C32×D4—216no—
48C2×C32×D4—28no—
49C2×C32×D4—22no—
50C2×C32×D4—21no—
51C2×C32×D4—24no—
52C2×C32×D4—24no—
53C2×C32×D4—22no—
54C2×C32×D4—21no—
55C2×C32×D4—22no—
56C2×C32×D4C2×C411yescharacteristic
57C2×C32×D4C2×C412yes—
58C2×C32×D4D412yes—
59C2×C32×D4D411yescharacteristic
60C2×C32×D4D414yes—
61C22×C32×C4—22no—
62C22×C32×C4—21no—
63C22×C32×C4C2×C412yes—
64C22×C32×C4C2311yescharacteristic
65C3×#217—22no—
66C3×C4⋊Dic3—22no—
67C3×C4×Dic3—22no—
68C3×#208—22no—
69C3×#208—22no—
70C2×C3×C3⋊C8—22no—
71#1795—31noSylow
72C3×H(5,𝔽2)—28no—
73C3×H(5,𝔽2)—28no—
74C3×H(5,𝔽2)—24no—
75C3×H(5,𝔽2)—24no—
76C3×H(5,𝔽2)—24no—
77C3×H(5,𝔽2)—24no—
78C2×C3×#41—22no—
79C2×C3×#41—21no—
80C2×C3×#41—21no—
81C2×C3×#41—22no—
82C2×C3×#41—21no—
83C2×C3×#41—21no—
84C2×C3×#41Dic312yescomplemented
85C2×C3×#41C3×C412yescomplemented
86C2×C3×#41C2×D311yescharacteristic
87C2×C3×#41C2×D311yescharacteristic
88C2×C3×#41C22×C311yescharacteristic
89C2×C3×#41C22×C311yescharacteristic
90C22×C3×D4—24no—
91C22×C3×D4—22no—
92C22×C3×D4—22no—
93C22×C3×D4—21no—
94C22×C3×D4—22no—
95C22×C3×D4—21no—
96C22×C3×D4—22no—
97C22×C3×D4—21no—
98C22×C3×D4Dic312yescomplemented
99C22×C3×D4C3×C412yescomplemented
100C22×C3×D4C2×D311yescharacteristic
101C22×C3×D4C22×C311yescharacteristic
102C2×C3×M16—32no—
103C3×#117—32no—
104#926C22×C312yescomplemented
105C2×#208C22×C312yescomplemented
106C3×#104—64no—
107C3×#102—64no—
108C3×#101—61no—
109C3×#100—61no—
110C3×#99—62no—
111#837—24no—
112#836—21no—
113#834—22no—
114#833—21no—
115#832—24no—
116C23×C32—28no—
117C23×C32—21no—
118C23×C32—22no—
119C23×C32—24no—
120C23×C32—82no—
121C23×C32—84no—
122C23×C32C22⋊C412yes—
123C23×C32C2×D411yescharacteristic
124C32×Q8—21no—
125C32×Q8—22no—
126C32×Q8C22⋊C412yes—
127C32×D4—28no—
128C32×D4—28no—
129C32×D4—24no—
130C32×D4—24no—
131C32×D4—22no—
132C32×D4—21no—
133C32×D4—22no—
134C32×D4—24no—
135C32×D4C22⋊C412yes—
136C32×D4C22⋊C414yes—
137C2×C32×C4—28no—
138C2×C32×C4—21no—
139C2×C32×C4—22no—
140C2×C32×C4—22no—
141C2×C32×C4—24no—
142C2×C32×C4C22⋊C412yes—
143C2×C32×C4C22×C411yescharacteristic
144C2×C32×C4C2×D412yes—
145C2×C32×C4C2×D412yes—
146C2×C32×C4C2×D411yescharacteristic
147C2×C3×Dic3—42no—
148C3×C3⋊C8—42no—
149C2×H(5,𝔽2)C3×D311yescharacteristic
150#421—32no—
151#418—32no—
152#413—31no—
153#410—31no—
154C24×C3—24no—
155C24×C3—22no—
156C24×C3—22no—
157C24×C3—22no—
158C24×C3—21no—
159C24×C3—21no—
160C3×#41—216no—
161C3×#41—28no—
162C3×#41—28no—
163C3×#41—28no—
164C3×#41—28no—
165C3×#41—24no—
166C3×#41—24no—
167C3×#41—24no—
168C3×#41—24no—
169C3×#41—24no—
170C3×#41—24no—
171C3×#41—24no—
172C3×#41—22no—
173C3×#41—22no—
174C3×#41—22no—
175C3×#41—22no—
176C3×#41—22no—
177C3×#41—22no—
178C3×#41—24no—
179C3×#41#6714yes—
180C3×#41C3×D414yes—
181C2×C3×Q8—21no—
182C2×C3×Q8—21no—
183C2×C3×Q8C2×Dic311yescharacteristic
184C2×C3×Q8#6711yescharacteristic
185C2×C3×Q8C2×C3×C411yescharacteristic
186C2×C3×Q8C3×D411yescharacteristic
187C2×C3×D4—24no—
188C2×C3×D4—22no—
189C2×C3×D4—24no—
190C2×C3×D4—22no—
191C2×C3×D4—232no—
192C2×C3×D4—216no—
193C2×C3×D4—216no—
194C2×C3×D4—216no—
195C2×C3×D4—28no—
196C2×C3×D4—28no—
197C2×C3×D4—28no—
198C2×C3×D4—28no—
199C2×C3×D4—24no—
200C2×C3×D4—22no—
201C2×C3×D4—22no—
202C2×C3×D4—21no—
203C2×C3×D4—21no—
204C2×C3×D4—24no—
205C2×C3×D4—24no—
206C2×C3×D4—24no—
207C2×C3×D4—24no—
208C2×C3×D4—22no—
209C2×C3×D4—22no—
210C2×C3×D4—21no—
211C2×C3×D4—22no—
212C2×C3×D4—21no—
213C2×C3×D4—22no—
214C2×C3×D4—21no—
215C2×C3×D4—22no—
216C2×C3×D4—22no—
217C2×C3×D4—21no—
218C2×C3×D4—24no—
219C2×C3×D4C2×Dic311yescharacteristic
220C2×C3×D4C2×Dic312yes—
221C2×C3×D4#6712yes—
222C2×C3×D4#6711yescharacteristic
223C2×C3×D4#6714yes—
224C2×C3×D4C2×C3×C411yescharacteristic
225C2×C3×D4C2×C3×C412yes—
226C2×C3×D4C3×D412yes—
227C2×C3×D4C3×D411yescharacteristic
228C2×C3×D4C3×D414yes—
229C22×C3×C4—24no—
230C22×C3×C4—22no—
231C22×C3×C4—22no—
232C22×C3×C4—22no—
233C22×C3×C4—21no—
234C22×C3×C4—21no—
235C22×C3×C4—22no—
236C22×C3×C4—21no—
237C22×C3×C4C2×Dic312yes—
238C22×C3×C4C2×C3×C412yes—
239C22×C3×C4C22×D311yescharacteristic
240C22×C3×C4C23×C311yescharacteristic
241C3×M16—62no—
242C3×M16—62no—
243C2×C3×C8—62no—
244C3×C4⋊C4—62no—
245C3×C22⋊C4—62no—
246C3×C42—62no—
247#217—22no—
248C4⋊Dic3—22no—
249C4×Dic3—22no—
250#208—22no—
251#208—22no—
252C2×C3⋊C8—22no—
253C22×C32—22no—
254C22×C32—24no—
255C22×C32—22no—
256C22×C32—41no—
257C22×C32—42no—
258C22×C32—44no—
259C22×C32—42no—
260C22×C32—84no—
261C22×C32—88no—
262C22×C32C2×C22⋊C411yescharacteristic
263C22×C32#12012yes—
264C32×C4—22no—
265C32×C4—22no—
266C32×C4C2×C22⋊C411yescharacteristic
267C32×C4C2×C22⋊C411yescharacteristic
268C32×C4#12012yes—
269C3×Dic3—82no—
270H(5,𝔽2)—24no—
271H(5,𝔽2)—24no—
272C2×#41—21no—
273C2×#41C3×Dic312yescomplemented
274C2×#41C2×C3×D311yescharacteristic
275C2×#41C2×C3×D311yescharacteristic
276C22×D4—22no—
277C22×D4—21no—
278C22×D4C3×Dic312yescomplemented
279C22×D4C2×C3×D311yescharacteristic
280C2×M16—32no—
281#117—32no—
282#104—64no—
283#102—64no—
284#101—61no—
285#100—61no—
286#99—62no—
287C23×C3—24no—
288C23×C3—22no—
289C23×C3—216no—
290C23×C3—28no—
291C23×C3—28no—
292C23×C3—28no—
293C23×C3—22no—
294C23×C3—21no—
295C23×C3—22no—
296C23×C3—21no—
297C23×C3—24no—
298C23×C3—24no—
299C23×C3—22no—
300C23×C3—21no—
301C23×C3—82no—
302C23×C3—82no—
303C23×C3—84no—
304C23×C3—84no—
305C23×C3—84no—
306C23×C3—88no—
307C23×C3#21712yes—
308C23×C3C3×C22⋊C412yes—
309C23×C3C2×#6711yescharacteristic
310C23×C3C2×C3×D411yescharacteristic
311C3×Q8—22no—
312C3×Q8—24no—
313C3×Q8—21no—
314C3×Q8—22no—
315C3×Q8—21no—
316C3×Q8—22no—
317C3×Q8—22no—
318C3×Q8#21712yes—
319C3×Q8C3×C22⋊C412yes—
320C3×D4—216no—
321C3×D4—216no—
322C3×D4—28no—
323C3×D4—28no—
324C3×D4—28no—
325C3×D4—28no—
326C3×D4—28no—
327C3×D4—28no—
328C3×D4—24no—
329C3×D4—22no—
330C3×D4—28no—
331C3×D4—24no—
332C3×D4—24no—
333C3×D4—24no—
334C3×D4—24no—
335C3×D4—24no—
336C3×D4—22no—
337C3×D4—21no—
338C3×D4—22no—
339C3×D4—24no—
340C3×D4—22no—
341C3×D4—21no—
342C3×D4—22no—
343C3×D4—24no—
344C3×D4—22no—
345C3×D4—24no—
346C3×D4#21712yes—
347C3×D4#21714yes—
348C3×D4C3×C22⋊C412yes—
349C3×D4C3×C22⋊C414yes—
350C2×C3×C4—22no—
351C2×C3×C4—216no—
352C2×C3×C4—28no—
353C2×C3×C4—28no—
354C2×C3×C4—24no—
355C2×C3×C4—24no—
356C2×C3×C4—28no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—24no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—24no—
365C2×C3×C4—22no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—22no—
369C2×C3×C4—22no—
370C2×C3×C4—122no—
371C2×C3×C4#21712yes—
372C2×C3×C4C3×C22⋊C412yes—
373C2×C3×C4C22×Dic311yescharacteristic
374C2×C3×C4C2×#6712yes—
375C2×C3×C4C2×#6712yes—
376C2×C3×C4C2×#6711yescharacteristic
377C2×C3×C4C22×C3×C411yescharacteristic
378C2×C3×C4C2×C3×D412yes—
379C2×C3×C4C2×C3×D412yes—
380C2×C3×C4C2×C3×D411yescharacteristic
381C2×Dic3—42no—
382C3×C8—122no—
383C3⋊C8—42no—
384C2×C32—21no—
385C2×C32—22no—
386C2×C32—44no—
387C2×C32—42no—
388C2×C32—42no—
389C2×C32#37911yescharacteristic
390C24—22no—
391C24—21no—
392#41—28no—
393#41—24no—
394#41—24no—
395#41—22no—
396#41—22no—
397#41—22no—
398#41C3×#6714yes—
399C2×Q8C2×C3×Dic311yescharacteristic
400C2×Q8C3×#6711yescharacteristic
401C2×D4—216no—
402C2×D4—28no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—21no—
409C2×D4—22no—
410C2×D4C2×C3×Dic311yescharacteristic
411C2×D4C2×C3×Dic312yes—
412C2×D4C3×#6712yes—
413C2×D4C3×#6711yescharacteristic
414C2×D4C3×#6714yes—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4C2×C3×Dic312yes—
418C22×C4C22×C3×D311yescharacteristic
419M16—62no—
420M16—62no—
421C2×C8—62no—
422C4⋊C4—62no—
423C22⋊C4—62no—
424C42—62no—
425C22×C3—24no—
426C22×C3—28no—
427C22×C3—24no—
428C22×C3—22no—
429C22×C3—24no—
430C22×C3—22no—
431C22×C3—22no—
432C22×C3—24no—
433C22×C3—22no—
434C22×C3—21no—
435C22×C3—22no—
436C22×C3—41no—
437C22×C3—41no—
438C22×C3—42no—
439C22×C3—42no—
440C22×C3—42no—
441C22×C3—44no—
442C22×C3—48no—
443C22×C3—44no—
444C22×C3—44no—
445C22×C3—42no—
446C22×C3—44no—
447C22×C3—42no—
448C22×C3—84no—
449C22×C3—84no—
450C22×C3—88no—
451C22×C3—816no—
452C22×C3—88no—
453C22×C3—88no—
454C22×C3C2×#21711yescharacteristic
455C22×C3#95312yes—
456C22×C3C2×C3×C22⋊C411yescharacteristic
457C22×C3C3×#12012yes—
458C3×C4—24no—
459C3×C4—24no—
460C3×C4—22no—
461C3×C4—22no—
462C3×C4—22no—
463C3×C4—22no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—22no—
467C3×C4—242no—
468C3×C4C2×#21711yescharacteristic
469C3×C4C2×#21711yescharacteristic
470C3×C4#95312yes—
471C3×C4C2×C3×C22⋊C411yescharacteristic
472C3×C4C2×C3×C22⋊C411yescharacteristic
473C3×C4C3×#12012yes—
474Dic3—82no—
475C32#179511yescharacteristic, complemented, Sylow
476C23—28no—
477C23—22no—
478C23—21no—
479C23—24no—
480C23—82no—
481C23—84no—
482C23C3×#21712yes—
483C23C2×C3×#6711yescharacteristic
484Q8—21no—
485Q8—22no—
486Q8C3×#21712yes—
487D4—28no—
488D4—28no—
489D4—24no—
490D4—24no—
491D4—22no—
492D4—21no—
493D4—22no—
494D4—24no—
495D4C3×#21712yes—
496D4C3×#21714yes—
497C2×C4—28no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—122no—
503C2×C4C3×#21712yes—
504C2×C4C22×C3×Dic311yescharacteristic
505C2×C4C2×C3×#6712yes—
506C2×C4C2×C3×#6712yes—
507C2×C4C2×C3×#6711yescharacteristic
508C8—122no—
509C2×C3—22no—
510C2×C3—22no—
511C2×C3—21no—
512C2×C3—22no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—42no—
516C2×C3—48no—
517C2×C3—44no—
518C2×C3—44no—
519C2×C3—44no—
520C2×C3—42no—
521C2×C3—42no—
522C2×C3—44no—
523C2×C3—42no—
524C2×C3—42no—
525C2×C3#524711yescharacteristic
526C2×C3C3×#37911yescharacteristic
527C22—22no—
528C22—24no—
529C22—22no—
530C22—41no—
531C22—42no—
532C22—44no—
533C22—42no—
534C22—84no—
535C22—88no—
536C22C2×C3×#21711yescharacteristic
537C22C3×#95312yes—
538C4—22no—
539C4—22no—
540C4—242no—
541C4C2×C3×#21711yescharacteristic
542C4C2×C3×#21711yescharacteristic
543C4C3×#95312yes—
544C3—21no—
545C3#7214111yescharacteristic, direct product factor
546C3C3×#179511yescharacteristic, complemented
547C2—22no—
548C2—21no—
549C2—44no—
550C2—42no—
551C2—42no—
552C2C3×#524711yescharacteristic
subgroupG/H
centerC2×C3#5247
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
Fitting, prime radicalC2×C32×H(5,𝔽2)C2
socleC2×C32#379
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×H(5,𝔽2)D3
hypocenterC3C3×#1795
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C2×C3×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. C3×#5240C2semidirect
3. C3×#5237C2semidirect
4. C3×#5236C2semidirect
5. C3×#5215C2semidirect
7. #72141C3direct (N × H)
11. C2×C32×#41C4semidirect
16. C22×C32×D4C4semidirect
18. C3×#926C22semidirect
19. C2×C3×#208C22semidirect
32. #5240C2×C3semidirect
33. #5237C2×C3semidirect
34. #5236C2×C3semidirect
35. #5215C2×C3semidirect
84. C2×C3×#41Dic3semidirect
85. C2×C3×#41C3×C4semidirect
98. C22×C3×D4Dic3semidirect
99. C22×C3×D4C3×C4semidirect
104. #926C22×C3semidirect
105. C2×#208C22×C3semidirect
273. C2×#41C3×Dic3semidirect
278. C22×D4C3×Dic3semidirect
475. C32#1795semidirect
545. C3#72141direct (N × H)
546. C3C3×#1795semidirect

Maximal subgroups

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

subgroupindex
C2×C32×H(5,𝔽2)2
C3×#52402
C3×#52372
C3×#52362
C3×#52152
C3×#17953
#721413