Finite Group Databasebeta

← #10596054⋊C3 #13152724

Subgroups — #10596054⋊C3 #13152724

Order 1728.

742 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64155⋊C3C211yesmaximal, characteristic
3#64153⋊C3C211yesmaximal, characteristic, complemented
4#64107⋊C3C212yesmaximal
5#64106⋊C3C211yesmaximal, characteristic
6#64099⋊C3C211yesmaximal, characteristic, complemented
7#64098⋊C3C212yesmaximal, complemented
8#11721387C212yesmaximal
9#64081⋊C3C211yesmaximal, characteristic
10#64079⋊C3C212yesmaximal, complemented
11C4×#634⋊C3C211yesmaximal, characteristic, complemented
12#10596054C311yesmaximal, characteristic, complemented
13C3×#5583—31nomaximal
14C2×Q8×H(3,𝔽3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#634⋊C3C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)C2212yes—
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C42×H(3,𝔽3)C2211yescharacteristic
20C42×H(3,𝔽3)C2212yes—
21#3752⋊C3C2211yescharacteristic
22#3752⋊C3C2212yes—
23#3752⋊C3C2212yes—
24#3752⋊C3C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2212yes—
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2212yes—
28#3750⋊C3C2212yes—
29#3750⋊C3C2212yes—
30#3750⋊C3C2211yescharacteristic
31#3750⋊C3C2212yes—
32#3750⋊C3C2211yescharacteristic
33#3750⋊C3C2212yes—
34#3750⋊C3C2212yes—
35C4×(C3⋊Dic3)⋊C3C2212yes—
36C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
37C32×C4×Q8—21no—
38C32×C4×Q8D311yescharacteristic, complemented
39C32×C4×Q8C2×C311yescharacteristic, complemented
40#64155C2×C311yescharacteristic
41#64153C2×C311yescharacteristic, complemented
42#64107C2×C312yes—
43#64106C2×C311yescharacteristic
44#64099C2×C311yescharacteristic, complemented
45#64098C2×C312yescomplemented
46#64085C2×C312yes—
47#64081C2×C311yescharacteristic
48#64079C2×C312yescomplemented
49C4×#634C2×C311yescharacteristic, complemented
50C3×#946—31no—
51C3×#944—31no—
52C3×#898—32no—
53C3×#897—31no—
54C3×#890—31no—
55C3×#889—32no—
56C3×#876—32no—
57C3×#872—31no—
58C3×#870—32no—
59C3×C4×Dic6—31no—
60C22×(C3⋊D3)⋊C3C2311yescharacteristic
61Q8×H(3,𝔽3)—22no—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2312yes—
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2312yes—
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C4×(C3⋊D3)⋊C3—22no—
73#634⋊C3—22no—
74C3×#541—91no—
75#5583—31no—
76#5583—91no—
77C2×C32×Q8—21no—
78C2×C32×Q8C2×D311yescharacteristic
79C2×C32×Q8C22×C311yescharacteristic
80C2×C4×C3⋊D3C22×C311yescharacteristic
81C2×#634C22×C311yescharacteristic
82C2×C3×C4×D3—31no—
83C2×C3×Dic6—31no—
84C32×C4⋊C4—22no—
85C32×C4⋊C4—21no—
86C32×C4⋊C4C2×D312yes—
87C32×C4⋊C4C2×D311yescharacteristic
88C32×C4⋊C4C22×C312yes—
89C32×C4⋊C4C22×C311yescharacteristic
90C32×C42—21no—
91C32×C42—22no—
92C32×C42C2×D311yescharacteristic
93C32×C42C2×D312yes—
94C32×C42C22×C311yescharacteristic
95C32×C42C22×C312yes—
96#3752C22×C311yescharacteristic
97#3752C22×C311yescharacteristic
98#3752C22×C312yes—
99#3752C22×C312yes—
100C4⋊(C3⋊Dic3)C22×C311yescharacteristic
101C4⋊(C3⋊Dic3)C22×C312yes—
102C4⋊(C3⋊Dic3)C22×C312yes—
103#3750C22×C312yes—
104#3750C22×C311yescharacteristic
105#3750C22×C312yes—
106#3750C22×C312yes—
107#3750C22×C311yescharacteristic
108#3750C22×C312yes—
109#3750C22×C312yes—
110C4×C3⋊Dic3C22×C312yes—
111C4×C3⋊Dic3C22×C311yescharacteristic
112C3×(C2×D3)⋊C4—32no—
113C3×(C2×D3)⋊C4—31no—
114C3×(C2×D3)⋊C4—32no—
115C3×(C2×D3)⋊C4—31no—
116C3×C4⋊Dic3—32no—
117C3×C4⋊Dic3—32no—
118C3×C4⋊Dic3—31no—
119C3×Dic3⋊C4—31no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—31no—
126C3×C4×Dic3—32no—
127C3×C4×Dic3—31no—
128C22×H(3,𝔽3)C2411yescharacteristic
129C2×(C3⋊D3)⋊C3—21no—
130C2×(C3⋊D3)⋊C3—21no—
131C2×(C3⋊D3)⋊C3—21no—
132C4×H(3,𝔽3)—22no—
133C4×H(3,𝔽3)—21no—
134C4×H(3,𝔽3)—21no—
135C4×H(3,𝔽3)—22no—
136C4×H(3,𝔽3)#4112yes—
137(C3⋊Dic3)⋊C3—22no—
138(C3⋊Dic3)⋊C3—21no—
139(C3⋊Dic3)⋊C3—22no—
140(C3⋊Dic3)⋊C3—21no—
141(C3⋊Dic3)⋊C3—21no—
142C3×C4⋊Q8—91no—
143C3×#126—92no—
144C3×#126—92no—
145C3×#125—92no—
146C3×#125—92no—
147C3×#125—91no—
148C3×C22⋊Q8—91no—
149C3×C22⋊Q8—91no—
150C3×C4×Q8—31no—
151C3×C4×Q8—31no—
152C3×C4×Q8—61no—
153C3×C4×Q8—91no—
154C3×C4×Q8C3×D311yescharacteristic
155C3×#117—91no—
156#946—31no—
157#946—91no—
158#944—31no—
159#944—91no—
160#898—32no—
161#898—92no—
162#897—31no—
163#897—91no—
164#890—31no—
165#890—91no—
166#889—32no—
167#889—92no—
168#876—32no—
169#876—92no—
170#872—31no—
171#872—91no—
172#870—32no—
173#870—92no—
174C4×Dic6—31no—
175C4×Dic6—91no—
176C22×C3⋊D3C23×C311yescharacteristic
177C22×C3×D3—31no—
178C32×Q8—24no—
179C32×Q8—22no—
180C32×Q8—22no—
181C2×C32×C4—21no—
182C2×C32×C4—22no—
183C2×C32×C4—21no—
184C2×C32×C4—21no—
185C2×C32×C4—22no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D312yes—
189C2×C32×C4C22×D312yes—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C23×C311yescharacteristic
192C2×C32×C4C23×C311yescharacteristic
193C2×C32×C4C23×C312yes—
194C2×C32×C4C23×C312yes—
195C2×C32×C4C23×C311yescharacteristic
196C2×C3⋊Dic3C23×C311yescharacteristic
197C2×C3⋊Dic3C23×C312yes—
198C2×C3⋊Dic3C23×C311yescharacteristic
199C2×C3⋊Dic3C23×C312yes—
200C2×C3⋊Dic3C23×C311yescharacteristic
201C4×C3⋊D3—22no—
202#634—22no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—31no—
208C3×C4×D3—62no—
209C3×Dic6—62no—
210#541—91noSylow
211C2×H(3,𝔽3)C2×#4111yescharacteristic
212C2×H(3,𝔽3)2−1+411yescharacteristic
213C2×H(3,𝔽3)2−1+411yescharacteristic
214(C3⋊D3)⋊C3—41no—
215C2×C3×Q8—31no—
216C2×C3×Q8—31no—
217C2×C3×Q8—61no—
218C2×C3×Q8—91no—
219C2×C3×Q8C2×C3×D311yescharacteristic
220C22×C3×C4—91no—
221C2×C4×D3—31no—
222C2×C4×D3—91no—
223C2×Dic6—31no—
224C2×Dic6—91no—
225C3×C4⋊C4—32no—
226C3×C4⋊C4—31no—
227C3×C4⋊C4—31no—
228C3×C4⋊C4—32no—
229C3×C4⋊C4—61no—
230C3×C4⋊C4—62no—
231C3×C4⋊C4—91no—
232C3×C4⋊C4—91no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—91no—
239C3×C4⋊C4—92no—
240C3×C4⋊C4—92no—
241C3×C4⋊C4C2×C3×D312yes—
242C3×C4⋊C4C2×C3×D311yescharacteristic
243C3×C22⋊C4—91no—
244C3×C22⋊C4—91no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4—92no—
247C3×C42—31no—
248C3×C42—32no—
249C3×C42—31no—
250C3×C42—32no—
251C3×C42—61no—
252C3×C42—62no—
253C3×C42—91no—
254C3×C42—92no—
255C3×C42C2×C3×D311yescharacteristic
256C3×C42C2×C3×D312yes—
257(C2×D3)⋊C4—31no—
258(C2×D3)⋊C4—31no—
259(C2×D3)⋊C4—32no—
260(C2×D3)⋊C4—32no—
261(C2×D3)⋊C4—91no—
262(C2×D3)⋊C4—91no—
263(C2×D3)⋊C4—92no—
264(C2×D3)⋊C4—92no—
265C4⋊Dic3—31no—
266C4⋊Dic3—32no—
267C4⋊Dic3—32no—
268C4⋊Dic3—91no—
269C4⋊Dic3—92no—
270C4⋊Dic3—92no—
271Dic3⋊C4—32no—
272Dic3⋊C4—31no—
273Dic3⋊C4—32no—
274Dic3⋊C4—32no—
275Dic3⋊C4—31no—
276Dic3⋊C4—32no—
277Dic3⋊C4—32no—
278Dic3⋊C4—92no—
279Dic3⋊C4—91no—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—91no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285C4×Dic3—32no—
286C4×Dic3—31no—
287C4×Dic3—92no—
288C4×Dic3—91no—
289C22×C32—21no—
290C22×C32C23×D311yescharacteristic
291C22×C32C24×C311yescharacteristic
292C2×C3⋊D3—21no—
293C2×C3⋊D3—21no—
294C2×C3⋊D3—21no—
295C2×C3×D3—61no—
296C2×C3×D3—61no—
297C2×C3×D3—61no—
298C32×C4—22no—
299C32×C4—24no—
300C32×C4—22no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—22no—
304C32×C4—22no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—22no—
308C32×C4—22no—
309C32×C4—41no—
310C32×C4—42no—
311C32×C4#23512yes—
312C32×C4C3×#4112yes—
313C3⋊Dic3—22no—
314C3⋊Dic3—21no—
315C3⋊Dic3—22no—
316C3⋊Dic3—21no—
317C3⋊Dic3—21no—
318C3×Dic3—62no—
319C3×Dic3—61no—
320C3×Dic3—62no—
321C3×Dic3—61no—
322C3×Dic3—61no—
323C4⋊Q8—91no—
324#126—92no—
325#126—92no—
326#125—92no—
327#125—91no—
328#125—92no—
329C22⋊Q8—91no—
330C22⋊Q8—91no—
331C4×Q8—91no—
332C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
333#117—91no—
334H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
335C23×C3—91no—
336C22×D3—31no—
337C22×D3—91no—
338C3×Q8—22no—
339C3×Q8—62no—
340C3×Q8—62no—
341C3×Q8—64no—
342C3×Q8—182no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—31no—
346C2×C3×C4—31no—
347C2×C3×C4—32no—
348C2×C3×C4—32no—
349C2×C3×C4—31no—
350C2×C3×C4—31no—
351C2×C3×C4—32no—
352C2×C3×C4—32no—
353C2×C3×C4—61no—
354C2×C3×C4—62no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—62no—
358C2×C3×C4—91no—
359C2×C3×C4—91no—
360C2×C3×C4—92no—
361C2×C3×C4—92no—
362C2×C3×C4—91no—
363C2×C3×C4—182no—
364C2×C3×C4C22×C3×D311yescharacteristic
365C2×C3×C4C22×C3×D311yescharacteristic
366C2×C3×C4C22×C3×D312yes—
367C2×C3×C4C22×C3×D312yes—
368C2×C3×C4C22×C3×D311yescharacteristic
369C2×Dic3—31no—
370C2×Dic3—32no—
371C2×Dic3—31no—
372C2×Dic3—32no—
373C2×Dic3—31no—
374C2×Dic3—91no—
375C2×Dic3—92no—
376C2×Dic3—91no—
377C2×Dic3—92no—
378C2×Dic3—91no—
379C4×D3—62no—
380C4×D3—182no—
381Dic6—62no—
382Dic6—182no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32C2×#23511yescharacteristic
387C2×C32#100711yescharacteristic
388C2×C32#101011yescharacteristic
389C2×C32C2×C3×#4111yescharacteristic
390C2×C32C3×2−1+411yescharacteristic
391C2×C32C3×2−1+411yescharacteristic
392C3⋊D3—41no—
393C3×D3—121no—
394C2×Q8—91no—
395C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
396C22×C4—91no—
397C4⋊C4—91no—
398C4⋊C4—91no—
399C4⋊C4—91no—
400C4⋊C4—92no—
401C4⋊C4—92no—
402C4⋊C4—92no—
403C4⋊C4—92no—
404C4⋊C4—92no—
405C4⋊C4—92no—
406C4⋊C4—92no—
407C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
408C4⋊C4C2×(C3⋊D3)⋊C312yes—
409C22⋊C4—91no—
410C22⋊C4—91no—
411C22⋊C4—92no—
412C22⋊C4—92no—
413C42—91no—
414C42—92no—
415C42C2×(C3⋊D3)⋊C311yescharacteristic
416C42C2×(C3⋊D3)⋊C312yes—
417C22×C3—31no—
418C22×C3—31no—
419C22×C3—61no—
420C22×C3—181no—
421C22×C3—181no—
422C22×C3—181no—
423C22×C3C23×C3×D311yescharacteristic
424C2×D3—61no—
425C2×D3—61no—
426C2×D3—61no—
427C2×D3—181no—
428C2×D3—181no—
429C2×D3—181no—
430C3×C4—22no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—22no—
434C3×C4—32no—
435C3×C4—32no—
436C3×C4—62no—
437C3×C4—62no—
438C3×C4—61no—
439C3×C4—61no—
440C3×C4—61no—
441C3×C4—62no—
442C3×C4—61no—
443C3×C4—62no—
444C3×C4—64no—
445C3×C4—62no—
446C3×C4—62no—
447C3×C4—121no—
448C3×C4—122no—
449C3×C4—181no—
450C3×C4—182no—
451C3×C4—181no—
452C3×C4—182no—
453C3×C4—181no—
454C3×C4C3×#23512yes—
455Dic3—62no—
456Dic3—61no—
457Dic3—62no—
458Dic3—61no—
459Dic3—61no—
460Dic3—182no—
461Dic3—181no—
462Dic3—182no—
463Dic3—181no—
464Dic3—181no—
465C32—21no—
466C32#558311yescharacteristic, complemented
467C32C3×#54111yescharacteristic, complemented
468C23—91no—
469Q8—22no—
470Q8—182no—
471C2×C4—91no—
472C2×C4—92no—
473C2×C4—91no—
474C2×C4—92no—
475C2×C4—91no—
476C2×C4—182no—
477C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
478C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
479C2×C4C22×(C3⋊D3)⋊C312yes—
480C2×C4C22×(C3⋊D3)⋊C312yes—
481C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
482C2×C3—31no—
483C2×C3—31no—
484C2×C3—31no—
485C2×C3—31no—
486C2×C3—31no—
487C2×C3—31no—
488C2×C3—61no—
489C2×C3—61no—
490C2×C3—61no—
491C2×C3—361no—
492C2×C3C2×C3×#23511yescharacteristic
493C2×C3C3×#100711yescharacteristic
494C2×C3C3×#101011yescharacteristic
495D3—121no—
496D3—361no—
497C22—181no—
498C22—181no—
499C22—181no—
500C22C23×(C3⋊D3)⋊C311yescharacteristic
501C4—22no—
502C4—21no—
503C4—21no—
504C4—22no—
505C4—181no—
506C4—182no—
507C4—181no—
508C4—182no—
509C4—181no—
510C4#3828⋊C312yes—
511C3—31no—
512C3—31no—
513C3—61no—
514C3C3×#558311yescharacteristic
515C2—361no—
516C2C2×#3828⋊C311yescharacteristic
517C2#1172262711yescharacteristic
518C2#1172263911yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC4×Q8×H(3,𝔽3)C2
prime radicalC22×(C3⋊D3)⋊C3C23
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#541
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C4×Q8×H(3,𝔽3)C2semidirect
3. #64153⋊C3C2semidirect
6. #64099⋊C3C2semidirect
7. #64098⋊C3C2semidirect
10. #64079⋊C3C2semidirect
11. C4×#634⋊C3C2semidirect
12. #10596054C3semidirect
38. C32×C4×Q8D3semidirect
39. C32×C4×Q8C2×C3semidirect
41. #64153C2×C3semidirect
44. #64099C2×C3semidirect
45. #64098C2×C3semidirect
48. #64079C2×C3semidirect
49. C4×#634C2×C3semidirect
332. C4×Q8(C3⋊D3)⋊C3semidirect
334. H(3,𝔽3)#541semidirect
466. C32#5583semidirect
467. C32C3×#541semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×H(3,𝔽3)2
#64155⋊C32
#64153⋊C32
#64107⋊C32
#64106⋊C32
#64099⋊C32
#64098⋊C32
#117213872
#64081⋊C32
#64079⋊C32
C4×#634⋊C32
#105960543
C3×#55833