Finite Group Databasebeta

← C3×#72093 #11793877

Subgroups — C3×#72093 #11793877

Order 1152.

713 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×AGL(1,ℤ/8)C211yesmaximal, characteristic
2C3×#5234C211yesmaximal, characteristic, complemented
3C3×#5232C211yesmaximal, characteristic, complemented
4C2×C3×#833C211yesmaximal, characteristic, complemented
5C2×C3×#832C211yesmaximal, characteristic, complemented
6C2×C3×#822C211yesmaximal, characteristic, complemented
7C3×#4554C211yesmaximal, characteristic, complemented
8C3×#2051—31nomaximal
9#72093C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C22×C32×D4C2211yescharacteristic
12C32×AGL(1,ℤ/8)—22no—
13C32×AGL(1,ℤ/8)—22no—
14C2×C32×SD16—21no—
15C2×C32×D8—21no—
16C2×C32×M16C2211yescharacteristic
17C2×C3×C4⋊Dic3C2211yescharacteristic
18C2×C3×#208C2211yescharacteristic
19C2×C3×#208C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21C3×#835—21no—
22C3×#833—22no—
23C3×#832—21no—
24C3×#832—22no—
25C3×#830—22no—
26C3×#822—22no—
27C2×C3×AGL(1,ℤ/8)—21no—
28C2×C3×AGL(1,ℤ/8)D311yescharacteristic
29C2×C3×AGL(1,ℤ/8)C2×C311yescharacteristic
30C2×C3×#108—31no—
31C3×#417—31no—
32C2×C3×#102—31no—
33C2×C3×#100—31no—
34C3×#408—31no—
35#5234C2×C311yescharacteristic, complemented
36#5232C2×C311yescharacteristic, complemented
37C2×#833C2×C311yescharacteristic, complemented
38C2×#832C2×C311yescharacteristic, complemented
39C2×#822C2×C311yescharacteristic, complemented
40C3×#343—31no—
41#4554C2×C311yescharacteristic, complemented
42C24×C32—21no—
43C32×#41—21no—
44C32×#41—21no—
45C32×#41—21no—
46C32×#41—21no—
47C2×C32×Q8D411yescharacteristic
48C2×C32×D4—21no—
49C2×C32×D4—21no—
50C2×C32×D4—21no—
51C2×C32×D4—21no—
52C2×C32×D4—21no—
53C2×C32×D4—21no—
54C2×C32×D4D412yes—
55C2×C32×D4D411yescharacteristic
56C22×C32×C4—21no—
57C22×C32×C4C2311yescharacteristic
58C22×C3×Dic3—21no—
59C32×SD16—24no—
60C32×D8—24no—
61C32×M16—21no—
62C32×M16D412yes—
63C2×C32×C8—21no—
64C3×C4⋊Dic3—21no—
65C3×C4⋊Dic3—21no—
66C3×#208—21no—
67C3×#208—21no—
68C3×#208—21no—
69C3×#208—21no—
70C2×C3×C3⋊C8—21no—
71C2×C3×C3⋊C8—21no—
72C2×C3×C3⋊C8—21no—
73C2×C3×C3⋊C8—21no—
74C2×C3×C3⋊C8D412yes—
75#2051—31noSylow
76C2×C3×#41—21no—
77C2×C3×#41C2×D311yescharacteristic
78C2×C3×#41C22×C311yescharacteristic
79C22×C3×D4—21no—
80C22×C3×D4C2×D311yescharacteristic
81C22×C3×D4C22×C311yescharacteristic
82C3×AGL(1,ℤ/8)—24no—
83C3×AGL(1,ℤ/8)—24no—
84C3×AGL(1,ℤ/8)—22no—
85C3×AGL(1,ℤ/8)—22no—
86C3×AGL(1,ℤ/8)—22no—
87C3×AGL(1,ℤ/8)—22no—
88C2×C3×SD16—22no—
89C2×C3×SD16—21no—
90C2×C3×SD16—21no—
91C2×C3×D8—22no—
92C2×C3×D8—21no—
93C2×C3×D8—21no—
94C2×C3×M16—21no—
95C2×C3×M16—31no—
96C2×C3×M16—31no—
97C2×C3×M16C2×D311yescharacteristic
98C2×C3×M16C22×C311yescharacteristic
99C22×C3×C8—31no—
100C2×C3×C4⋊C4—31no—
101C2×C4⋊Dic3C22×C311yescharacteristic
102C2×#208C22×C311yescharacteristic
103C2×#208C22×C311yescharacteristic
104C22×C3⋊C8C22×C311yescharacteristic
105C3×#108—62no—
106C3×#103—61no—
107C3×#102—62no—
108C3×#102—61no—
109C3×#100—62no—
110C3×C22⋊C8—62no—
111#835—21no—
112#833—22no—
113#832—21no—
114#832—22no—
115#830—22no—
116#822—22no—
117C23×C32—21no—
118C23×C32—22no—
119C23×C32—41no—
120C23×C32—41no—
121C23×C32—81no—
122C23×C32—81no—
123C23×C32#4111yescharacteristic
124C32×Q8—21no—
125C32×Q8—21no—
126C32×D4—21no—
127C32×D4—22no—
128C32×D4—22no—
129C32×D4—21no—
130C32×D4—41no—
131C32×D4—41no—
132C32×D4—41no—
133C32×D4—41no—
134C2×C32×C4—21no—
135C2×C32×C4—41no—
136C2×C32×C4—41no—
137C2×C32×C4C2×D411yescharacteristic
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4#4111yescharacteristic
142C2×C32×C4#4111yescharacteristic
143C2×C3×Dic3—41no—
144C2×C3×Dic3—81no—
145C32×C8—22no—
146C3×C3⋊C8—22no—
147C3×C3⋊C8—41no—
148C3×C3⋊C8—41no—
149C2×AGL(1,ℤ/8)C3×D311yescharacteristic
150C2×#108—31no—
151#417—31no—
152C2×#102—31no—
153C2×#100—31no—
154#408—31no—
155#343—31no—
156C24×C3—21no—
157C24×C3—21no—
158C24×C3—41no—
159C3×#41—22no—
160C3×#41—22no—
161C3×#41—21no—
162C3×#41—21no—
163C3×#41—21no—
164C3×#41—21no—
165C3×#41—21no—
166C3×#41—21no—
167C3×#41—21no—
168C3×#41—21no—
169C3×#41—41no—
170C3×#41—41no—
171C2×C3×Q8—21no—
172C2×C3×Q8#6711yescharacteristic
173C2×C3×Q8C3×D411yescharacteristic
174C2×C3×D4—22no—
175C2×C3×D4—22no—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4—21no—
180C2×C3×D4—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—22no—
190C2×C3×D4—41no—
191C2×C3×D4—41no—
192C2×C3×D4—41no—
193C2×C3×D4—41no—
194C2×C3×D4#6712yes—
195C2×C3×D4#6711yescharacteristic
196C2×C3×D4C3×D411yescharacteristic
197C2×C3×D4C3×D412yes—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—41no—
202C22×C3×C4—61no—
203C22×C3×C4C22×D311yescharacteristic
204C22×C3×C4C23×C311yescharacteristic
205C22×Dic3—21no—
206C3×SD16—28no—
207C3×SD16—24no—
208C3×SD16—24no—
209C3×D8—28no—
210C3×D8—24no—
211C3×D8—24no—
212C3×M16—22no—
213C3×M16—21no—
214C3×M16—21no—
215C3×M16—22no—
216C3×M16—61no—
217C3×M16—61no—
218C3×M16—61no—
219C3×M16—61no—
220C3×M16#6712yes—
221C3×M16C3×D412yes—
222C2×C3×C8—22no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—32no—
226C2×C3×C8—61no—
227C2×C3×C8—61no—
228C2×C3×C8—61no—
229C2×C3×C8—61no—
230C3×C4⋊C4—61no—
231C3×C4⋊C4—61no—
232C4⋊Dic3—21no—
233C4⋊Dic3—21no—
234#208—21no—
235#208—21no—
236#208—21no—
237#208—21no—
238C2×C3⋊C8—21no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—21no—
241C2×C3⋊C8—21no—
242C2×C3⋊C8C3×D412yes—
243C22×C32—21no—
244C22×C32—21no—
245C22×C32—41no—
246C22×C32—42no—
247C22×C32—41no—
248C22×C32—41no—
249C22×C32—82no—
250C22×C32—82no—
251C22×C32—81no—
252C22×C32—81no—
253C22×C32#12111yescharacteristic
254C22×C32#12311yescharacteristic
255C22×C32#12411yescharacteristic
256C32×C4—41no—
257C32×C4#12011yescharacteristic
258C32×C4#12111yescharacteristic
259C32×C4#12111yescharacteristic
260C32×C4#12311yescharacteristic
261C3×Dic3—81no—
262C2×#41C2×C3×D311yescharacteristic
263C22×D4C2×C3×D311yescharacteristic
264AGL(1,ℤ/8)—22no—
265AGL(1,ℤ/8)—22no—
266C2×SD16—21no—
267C2×D8—21no—
268C2×M16—31no—
269C2×M16—31no—
270C2×M16C2×C3×D311yescharacteristic
271C22×C8—31no—
272C2×C4⋊C4—31no—
273#108—62no—
274#103—61no—
275#102—62no—
276#102—61no—
277#100—62no—
278C22⋊C8—62no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—21no—
284C23×C3—41no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—42no—
288C23×C3—41no—
289C23×C3—41no—
290C23×C3—41no—
291C23×C3—41no—
292C23×C3—81no—
293C23×C3—81no—
294C23×C3—81no—
295C23×C3—81no—
296C23×C3—82no—
297C23×C3—82no—
298C23×C3#23711yescharacteristic
299C23×C3C3×#4111yescharacteristic
300C3×Q8—22no—
301C3×Q8—21no—
302C3×Q8—21no—
303C3×Q8—21no—
304C3×Q8—21no—
305C3×Q8—41no—
306C3×D4—22no—
307C3×D4—24no—
308C3×D4—21no—
309C3×D4—22no—
310C3×D4—21no—
311C3×D4—22no—
312C3×D4—21no—
313C3×D4—22no—
314C3×D4—22no—
315C3×D4—21no—
316C3×D4—42no—
317C3×D4—41no—
318C3×D4—42no—
319C3×D4—42no—
320C3×D4—42no—
321C3×D4—42no—
322C3×D4—41no—
323C3×D4—41no—
324C3×D4—41no—
325C3×D4—41no—
326C3×D4—41no—
327C3×D4—41no—
328C3×D4—41no—
329C3×D4—41no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—41no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—121no—
346C2×C3×C4—241no—
347C2×C3×C4D3×D411yescharacteristic
348C2×C3×C4#23711yescharacteristic
349C2×C3×C4#23711yescharacteristic
350C2×C3×C4C2×#6711yescharacteristic
351C2×C3×C4C2×#6711yescharacteristic
352C2×C3×C4C2×#6711yescharacteristic
353C2×C3×C4C2×C3×D411yescharacteristic
354C2×C3×C4C2×C3×D411yescharacteristic
355C2×C3×C4C2×C3×D411yescharacteristic
356C2×C3×C4C2×C3×D411yescharacteristic
357C2×C3×C4C3×#4111yescharacteristic
358C2×C3×C4C3×#4111yescharacteristic
359C2×Dic3—41no—
360C2×Dic3—81no—
361C3×C8—24no—
362C3×C8—22no—
363C3×C8—22no—
364C3×C8—62no—
365C3×C8—121no—
366C3×C8—121no—
367C3⋊C8—22no—
368C3⋊C8—41no—
369C3⋊C8—41no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—81no—
373C2×C32—82no—
374C2×C32#39411yescharacteristic
375C2×C32#47111yescharacteristic
376C2×C32#47211yescharacteristic
377C24—21no—
378#41—21no—
379#41—21no—
380#41—21no—
381#41—21no—
382C2×Q8C3×#6711yescharacteristic
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4C3×#6712yes—
390C2×D4C3×#6711yescharacteristic
391C22×C4—21no—
392C22×C4—61no—
393C22×C4C22×C3×D311yescharacteristic
394SD16—24no—
395D8—24no—
396M16—21no—
397M16—61no—
398M16—61no—
399M16—61no—
400M16—61no—
401M16C3×#6712yes—
402C2×C8—21no—
403C2×C8—32no—
404C2×C8—61no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—61no—
408C4⋊C4—61no—
409C4⋊C4—61no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—41no—
418C22×C3—41no—
419C22×C3—42no—
420C22×C3—44no—
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—41no—
424C22×C3—42no—
425C22×C3—41no—
426C22×C3—42no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3—81no—
430C22×C3—81no—
431C22×C3—82no—
432C22×C3—82no—
433C22×C3—82no—
434C22×C3—82no—
435C22×C3—82no—
436C22×C3—82no—
437C22×C3—84no—
438C22×C3—84no—
439C22×C3—81no—
440C22×C3—81no—
441C22×C3—81no—
442C22×C3—81no—
443C22×C3#93511yescharacteristic
444C22×C3#93611yescharacteristic
445C22×C3#93911yescharacteristic
446C22×C3C3×#12111yescharacteristic
447C22×C3C3×#12311yescharacteristic
448C22×C3C3×#12411yescharacteristic
449C3×C4—21no—
450C3×C4—21no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—42no—
454C3×C4—41no—
455C3×C4—41no—
456C3×C4—241no—
457C3×C4#93511yescharacteristic
458C3×C4#93811yescharacteristic
459C3×C4#93911yescharacteristic
460C3×C4#95311yescharacteristic
461C3×C4C3×#12011yescharacteristic
462C3×C4C3×#12111yescharacteristic
463C3×C4C3×#12111yescharacteristic
464C3×C4C3×#12311yescharacteristic
465Dic3—81no—
466C32#205111yescharacteristic, complemented, Sylow
467C23—21no—
468C23—22no—
469C23—41no—
470C23—41no—
471C23—81no—
472C23—81no—
473C23C3×#23711yescharacteristic
474Q8—21no—
475Q8—21no—
476D4—21no—
477D4—22no—
478D4—21no—
479D4—22no—
480D4—41no—
481D4—41no—
482D4—41no—
483D4—41no—
484C2×C4—21no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—121no—
488C2×C4—241no—
489C2×C4C3×D3×D411yescharacteristic
490C2×C4C3×#23711yescharacteristic
491C2×C4C3×#23711yescharacteristic
492C2×C4C2×C3×#6711yescharacteristic
493C2×C4C2×C3×#6711yescharacteristic
494C2×C4C2×C3×#6711yescharacteristic
495C8—22no—
496C8—62no—
497C8—121no—
498C8—121no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—84no—
509C2×C3—82no—
510C2×C3—81no—
511C2×C3—82no—
512C2×C3—81no—
513C2×C3—82no—
514C2×C3#519911yescharacteristic
515C2×C3#520011yescharacteristic
516C2×C3#522211yescharacteristic
517C2×C3C3×#39411yescharacteristic
518C2×C3C3×#47111yescharacteristic
519C2×C3C3×#47211yescharacteristic
520C22—21no—
521C22—21no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—42no—
526C22—82no—
527C22—82no—
528C22—81no—
529C22—81no—
530C22C3×#93511yescharacteristic
531C22C3×#93611yescharacteristic
532C22C3×#93911yescharacteristic
533C4—41no—
534C4—241no—
535C4C3×#93511yescharacteristic
536C4C3×#93811yescharacteristic
537C4C3×#93911yescharacteristic
538C4C3×#95311yescharacteristic
539C3—21no—
540C3#7209311yescharacteristic, direct product factor
541C3C3×#205111yescharacteristic, complemented
542C2—21no—
543C2—21no—
544C2—81no—
545C2—82no—
546C2C3×#519911yescharacteristic
547C2C3×#520011yescharacteristic
548C2C3×#522211yescharacteristic
subgroupG/H
centerC22×C3#939
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
Fitting, prime radicalC2×C32×AGL(1,ℤ/8)C2
socleC22×C32#121
hypercenterC2×C3×AGL(1,ℤ/8)D3
hypocenterC3C3×#2051
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
2. C3×#5234C2semidirect
3. C3×#5232C2semidirect
4. C2×C3×#833C2semidirect
5. C2×C3×#832C2semidirect
6. C2×C3×#822C2semidirect
7. C3×#4554C2semidirect
9. #72093C3direct (N × H)
35. #5234C2×C3semidirect
36. #5232C2×C3semidirect
37. C2×#833C2×C3semidirect
38. C2×#832C2×C3semidirect
39. C2×#822C2×C3semidirect
41. #4554C2×C3semidirect
466. C32#2051semidirect
540. C3#72093direct (N × H)
541. C3C3×#2051semidirect

Maximal subgroups

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

subgroupindex
C2×C32×AGL(1,ℤ/8)2
C3×#52342
C3×#52322
C2×C3×#8332
C2×C3×#8322
C2×C3×#8222
C3×#45542
C3×#20513
#720933