Finite Group Databasebeta

← C3×#70296 #11793976

Subgroups — C3×#70296 #11793976

Order 1152.

607 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5822C211yesmaximal, characteristic
2C32×#416C211yesmaximal, characteristic, complemented
3C3×#5240C211yesmaximal, characteristic, complemented
4C3×#5113C211yesmaximal, characteristic, complemented
5C3×#5102C211yesmaximal, characteristic, complemented
6C3×#5006C211yesmaximal, characteristic, complemented
7C3×#4531C211yesmaximal, characteristic, complemented
8C3×#2053—31nomaximal
9#70296C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C22×C32×C8C2211yescharacteristic
13C32×#117C2211yescharacteristic
14C3×#951—21no—
15C3×#950—22no—
16C3×#949—21no—
17C3×#942—21no—
18C3×#932—21no—
19C3×#926C2211yescharacteristic
20C3×#926C2211yescharacteristic
21C2×C3×#208C2211yescharacteristic
22C32×#103—21no—
23C32×#102—21no—
24C3×#834—22no—
25C3×#821—21no—
26C3×#816—21no—
27C3×Dic3⋊C8—22no—
28C3×#806—22no—
29C3×#575—31no—
30#5822C2×C311yescharacteristic
31C3×#424—31no—
32C3×#416—21no—
33C3×#416—31no—
34C3×#416D311yescharacteristic, complemented
35C3×#416C2×C311yescharacteristic, complemented
36C3×#410—31no—
37C3×#410—31no—
38#5240C2×C311yescharacteristic, complemented
39#5113C2×C311yescharacteristic, complemented
40#5102C2×C311yescharacteristic, complemented
41#5006C2×C311yescharacteristic, complemented
42C3×H(3,ℤ/4)—31no—
43#4531C2×C311yescharacteristic, complemented
44C32×#41—21no—
45C32×#41—21no—
46C32×#41—21no—
47C32×#41—21no—
48C2×C32×Q8D411yescharacteristic
49C2×C32×D4—21no—
50C2×C32×D4D411yescharacteristic
51C22×C32×C4—21no—
52C22×C32×C4C2311yescharacteristic
53C2×C3×#67—21no—
54C3×#235—21no—
55C3×#235—21no—
56C3×#235—21no—
57C3×#235—21no—
58C2×C3×D12D411yescharacteristic
59C2×C3×C4×D3—21no—
60C2×C3×Dic6D411yescharacteristic
61C2×C32×C8—21no—
62C2×C32×C8—21no—
63C2×C32×C8—21no—
64C32×C4⋊C4—21no—
65C32×C22⋊C4—21no—
66C32×C42—21no—
67C3×#217—21no—
68C3×#217D412yes—
69C3×#216—22no—
70C3×#215—22no—
71C3×#214—22no—
72C3×#213—22no—
73C3×C4⋊Dic3—21no—
74C3×C4⋊Dic3—21no—
75C3×C4×Dic3—21no—
76C3×C4×Dic3—21no—
77C3×#208—21no—
78C3×#208—21no—
79C2×C3×C3⋊C8D412yes—
80#2053—31noSylow
81C2×C3×#41—21no—
82C2×C3×#41—31no—
83C2×C3×#41C2×D311yescharacteristic
84C2×C3×#41C22×C311yescharacteristic
85C2×#235C22×C311yescharacteristic
86C3×#137—61no—
87C3×#137—61no—
88C3×AGL(1,ℤ/8)—61no—
89C3×AGL(1,ℤ/8)—61no—
90C3×#135—62no—
91C2×C3×M16—31no—
92C22×C3×C8—21no—
93C22×C3×C8C2×D311yescharacteristic
94C22×C3×C8C22×C311yescharacteristic
95C3×#117—21no—
96C3×#117—31no—
97C3×#117—31no—
98C3×#117C2×D311yescharacteristic
99C3×#117C22×C311yescharacteristic
100#951—21no—
101#950—22no—
102#949—21no—
103#942—21no—
104#932—21no—
105#926C22×C311yescharacteristic
106#926C22×C311yescharacteristic
107C2×#208C22×C311yescharacteristic
108C3×C4⋊C8—62no—
109C3×#103—22no—
110C3×#103—21no—
111C3×#103—21no—
112C3×#103—61no—
113C3×#102—22no—
114C3×#102—21no—
115C3×#102—21no—
116C3×#102—61no—
117C3×#99—62no—
118C3×#99—62no—
119#834—22no—
120#821—21no—
121#816—21no—
122Dic3⋊C8—22no—
123#806—22no—
124C23×C32—21no—
125C23×C32C2×D411yescharacteristic
126C22×C3×D3—21no—
127C32×Q8—21no—
128C32×Q8—21no—
129C32×D4—21no—
130C32×D4—21no—
131C32×D4—41no—
132C32×D4—41no—
133C2×C32×C4—21no—
134C2×C32×C4—41no—
135C2×C32×C4—41no—
136C2×C32×C4—41no—
137C2×C32×C4C2×D411yescharacteristic
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4#4111yescharacteristic
141C2×C32×C4#4111yescharacteristic
142C2×C32×C4#4111yescharacteristic
143C3×#67—41no—
144C3×#67—41no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—21no—
147C2×C3×Dic3—41no—
148C3×D12—21no—
149C3×D12—21no—
150C3×C4×D3—41no—
151C3×C4×D3—41no—
152C3×Dic6—21no—
153C3×Dic6—21no—
154C32×C8—41no—
155C3×C3⋊C8—22no—
156#575—31no—
157#424—31no—
158#416—31no—
159#416C3×D311yescharacteristic, complemented
160#410—31no—
161#410—31no—
162H(3,ℤ/4)—31no—
163C3×#41—22no—
164C3×#41—22no—
165C3×#41—21no—
166C3×#41—21no—
167C3×#41—21no—
168C3×#41—21no—
169C3×#41—21no—
170C3×#41—21no—
171C3×#41—21no—
172C3×#41—21no—
173C3×#41—41no—
174C3×#41—41no—
175C3×#41—61no—
176C3×#41—61no—
177C3×#41—61no—
178C3×#41—61no—
179C2×C3×Q8—21no—
180C2×C3×Q8—31no—
181C2×C3×Q8#6711yescharacteristic
182C2×C3×Q8C3×D411yescharacteristic
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—31no—
187C2×C3×D4—41no—
188C2×C3×D4—61no—
189C2×C3×D4#6711yescharacteristic
190C2×C3×D4C3×D411yescharacteristic
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—41no—
195C22×C3×C4—61no—
196C22×C3×C4C22×D311yescharacteristic
197C22×C3×C4C23×C311yescharacteristic
198C2×#67—21no—
199#235—21no—
200#235—21no—
201#235—21no—
202#235—21no—
203C2×D12C3×D411yescharacteristic
204C2×C4×D3—21no—
205C2×Dic6C3×D411yescharacteristic
206C3×Q16—62no—
207C3×SD16—62no—
208C3×SD16—62no—
209C3×D8—62no—
210C3×M16—61no—
211C3×M16—61no—
212C2×C3×C8—22no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—32no—
220C2×C3×C8—41no—
221C2×C3×C8—41no—
222C3×C4⋊C4—22no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—61no—
226C3×C4⋊C4—61no—
227C3×C22⋊C4—21no—
228C3×C22⋊C4—21no—
229C3×C22⋊C4—32no—
230C3×C22⋊C4—41no—
231C3×C22⋊C4—61no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—41no—
235C3×C42—61no—
236C3×C42—61no—
237#217—21no—
238#217C3×D412yes—
239#216—22no—
240#215—22no—
241#214—22no—
242#213—22no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4×Dic3—21no—
246C4×Dic3—21no—
247#208—21no—
248#208—21no—
249C2×C3⋊C8C3×D412yes—
250C22×C32—41no—
251C22×C32—41no—
252C22×C32—81no—
253C22×C32#12111yescharacteristic
254C22×C32#12111yescharacteristic
255C22×C32#12111yescharacteristic
256C2×C3×D3—41no—
257C2×C3×D3—81no—
258C32×C4—41no—
259C32×C4—81no—
260C32×C4#12011yescharacteristic
261C32×C4#12311yescharacteristic
262C32×C4#12311yescharacteristic
263C32×C4#12411yescharacteristic
264C3×Dic3—42no—
265C3×Dic3—41no—
266C3×Dic3—81no—
267C2×#41—31no—
268C2×#41C2×C3×D311yescharacteristic
269#137—61no—
270#137—61no—
271AGL(1,ℤ/8)—61no—
272AGL(1,ℤ/8)—61no—
273#135—62no—
274C2×M16—31no—
275C22×C8C2×C3×D311yescharacteristic
276#117—31no—
277#117—31no—
278#117C2×C3×D311yescharacteristic
279C4⋊C8—62no—
280#103—21no—
281#103—61no—
282#102—21no—
283#102—61no—
284#99—62no—
285#99—62no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—21no—
289C23×C3—41no—
290C23×C3—61no—
291C23×C3D3×D411yescharacteristic
292C23×C3C2×C3×D411yescharacteristic
293C22×D3—21no—
294C3×Q8—22no—
295C3×Q8—21no—
296C3×Q8—21no—
297C3×Q8—21no—
298C3×Q8—21no—
299C3×Q8—41no—
300C3×Q8—61no—
301C3×Q8—61no—
302C3×D4—22no—
303C3×D4—21no—
304C3×D4—21no—
305C3×D4—21no—
306C3×D4—21no—
307C3×D4—41no—
308C3×D4—42no—
309C3×D4—42no—
310C3×D4—41no—
311C3×D4—41no—
312C3×D4—41no—
313C3×D4—41no—
314C3×D4—61no—
315C3×D4—61no—
316C3×D4—121no—
317C3×D4—121no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—41no—
327C2×C3×C4—42no—
328C2×C3×C4—42no—
329C2×C3×C4—41no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4—42no—
334C2×C3×C4—41no—
335C2×C3×C4—41no—
336C2×C3×C4—62no—
337C2×C3×C4—61no—
338C2×C3×C4—121no—
339C2×C3×C4—121no—
340C2×C3×C4—121no—
341C2×C3×C4#23511yescharacteristic
342C2×C3×C4#23511yescharacteristic
343C2×C3×C4D3×D411yescharacteristic
344C2×C3×C4D3×D411yescharacteristic
345C2×C3×C4#23711yescharacteristic
346C2×C3×C4C2×#6711yescharacteristic
347C2×C3×C4C2×C3×D411yescharacteristic
348C2×C3×C4C2×C3×D411yescharacteristic
349C2×C3×C4C2×C3×D411yescharacteristic
350C2×C3×C4C3×#4111yescharacteristic
351C2×C3×C4C3×#4111yescharacteristic
352C2×C3×C4C3×#4111yescharacteristic
353#67—41no—
354#67—41no—
355C2×Dic3—22no—
356C2×Dic3—21no—
357C2×Dic3—41no—
358D12—21no—
359D12—21no—
360C4×D3—41no—
361C4×D3—41no—
362Dic6—21no—
363Dic6—21no—
364C3×C8—42no—
365C3×C8—41no—
366C3×C8—41no—
367C3×C8—62no—
368C3⋊C8—22no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—81no—
373C2×C32#39411yescharacteristic
374C3×D3—81no—
375#41—21no—
376#41—21no—
377#41—21no—
378#41—21no—
379#41—61no—
380#41—61no—
381#41—61no—
382#41—61no—
383C2×Q8—31no—
384C2×Q8C3×#6711yescharacteristic
385C2×D4—21no—
386C2×D4—31no—
387C2×D4—61no—
388C2×D4C3×#6711yescharacteristic
389C22×C4—21no—
390C22×C4—61no—
391C22×C4C22×C3×D311yescharacteristic
392Q16—62no—
393SD16—62no—
394SD16—62no—
395D8—62no—
396M16—61no—
397M16—61no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—32no—
402C4⋊C4—21no—
403C4⋊C4—61no—
404C4⋊C4—61no—
405C22⋊C4—21no—
406C22⋊C4—32no—
407C22⋊C4—61no—
408C42—21no—
409C42—61no—
410C42—61no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—41no—
415C22×C3—41no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—41no—
419C22×C3—41no—
420C22×C3—81no—
421C22×C3—81no—
422C22×C3—82no—
423C22×C3—121no—
424C22×C3—241no—
425C22×C3#88411yescharacteristic
426C22×C3#88411yescharacteristic
427C22×C3#93911yescharacteristic
428C22×C3C3×#12111yescharacteristic
429C22×C3C3×#12111yescharacteristic
430C22×C3C3×#12111yescharacteristic
431C2×D3—41no—
432C2×D3—81no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—42no—
438C3×C4—41no—
439C3×C4—41no—
440C3×C4—82no—
441C3×C4—81no—
442C3×C4—81no—
443C3×C4—122no—
444C3×C4—121no—
445C3×C4—241no—
446C3×C4#88311yescharacteristic
447C3×C4#88511yescharacteristic
448C3×C4#92911yescharacteristic
449C3×C4#93711yescharacteristic
450C3×C4C3×#12011yescharacteristic
451C3×C4C3×#12311yescharacteristic
452C3×C4C3×#12311yescharacteristic
453C3×C4C3×#12411yescharacteristic
454Dic3—42no—
455Dic3—41no—
456Dic3—81no—
457C32#205311yescharacteristic, complemented, Sylow
458C23—21no—
459C23—61no—
460C23C3×D3×D411yescharacteristic
461Q8—21no—
462Q8—21no—
463Q8—61no—
464Q8—61no—
465D4—21no—
466D4—21no—
467D4—41no—
468D4—41no—
469D4—61no—
470D4—61no—
471D4—121no—
472D4—121no—
473C2×C4—21no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—62no—
478C2×C4—61no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4C3×#23511yescharacteristic
483C2×C4C3×#23511yescharacteristic
484C2×C4C3×D3×D411yescharacteristic
485C2×C4C3×D3×D411yescharacteristic
486C2×C4C3×#23711yescharacteristic
487C2×C4C2×C3×#6711yescharacteristic
488C8—41no—
489C8—62no—
490C2×C3—22no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—41no—
499C2×C3—41no—
500C2×C3—82no—
501C2×C3—81no—
502C2×C3—81no—
503C2×C3—241no—
504C2×C3#495911yescharacteristic
505C2×C3C3×#39411yescharacteristic
506D3—81no—
507C22—41no—
508C22—41no—
509C22—81no—
510C22—121no—
511C22—241no—
512C22C3×#88411yescharacteristic
513C22C3×#88411yescharacteristic
514C22C3×#93911yescharacteristic
515C4—41no—
516C4—81no—
517C4—122no—
518C4—121no—
519C4—241no—
520C4C3×#88311yescharacteristic
521C4C3×#88511yescharacteristic
522C4C3×#92911yescharacteristic
523C4C3×#93711yescharacteristic
524C3—21no—
525C3#7029611yescharacteristic, direct product factor
526C3C3×#205311yescharacteristic, complemented
527C2—21no—
528C2—21no—
529C2—21no—
530C2—81no—
531C2—241no—
532C2C3×#495911yescharacteristic
subgroupG/H
centerC3×C4#885
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#416C2
socleC2×C32#394
prime radicalC3×#5822C2
hypercenterC3×#416D3
hypocenterC3C3×#2053
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#416

Complements

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

NH ≅ G/Ntype
2. C32×#416C2semidirect
3. C3×#5240C2semidirect
4. C3×#5113C2semidirect
5. C3×#5102C2semidirect
6. C3×#5006C2semidirect
7. C3×#4531C2semidirect
9. #70296C3direct (N × H)
34. C3×#416D3semidirect
35. C3×#416C2×C3semidirect
38. #5240C2×C3semidirect
39. #5113C2×C3semidirect
40. #5102C2×C3semidirect
41. #5006C2×C3semidirect
43. #4531C2×C3semidirect
159. #416C3×D3semidirect
457. C32#2053semidirect
525. C3#70296direct (N × H)
526. C3C3×#2053semidirect

Maximal subgroups

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

subgroupindex
C3×#58222
C32×#4162
C3×#52402
C3×#51132
C3×#51022
C3×#50062
C3×#45312
C3×#20533
#702963