Finite Group Databasebeta

← #10596039⋊C3 #13152709

Subgroups — #10596039⋊C3 #13152709

Order 1728.

812 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
1C4×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64156⋊C3C211yesmaximal, characteristic
3#64153⋊C3C211yesmaximal, characteristic, complemented
4#64107⋊C3C212yesmaximal, complemented
5#64105⋊C3C212yesmaximal, complemented
6#64102⋊C3C211yesmaximal, characteristic
7#64096⋊C3C211yesmaximal, characteristic, complemented
8#11721387C212yesmaximal, complemented
9#64084⋊C3C211yesmaximal, characteristic
10#64081⋊C3C212yesmaximal, complemented
11C4×#634⋊C3C211yesmaximal, characteristic, complemented
12#10596039C311yesmaximal, characteristic, complemented
13C3×#5568—31nomaximal
14C2×Q8×H(3,𝔽3)C2211yescharacteristic
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2212yes—
17C2×#634⋊C3C2211yescharacteristic
18C2×#634⋊C3C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2212yes—
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21C42×H(3,𝔽3)C2211yescharacteristic
22C42×H(3,𝔽3)C2212yes—
23#3752⋊C3C2212yes—
24#3752⋊C3C2212yes—
25#3752⋊C3C2212yes—
26#3752⋊C3C2212yes—
27#3752⋊C3C2212yes—
28C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
29#3750⋊C3C2212yes—
30#3750⋊C3C2211yescharacteristic
31#3750⋊C3C2211yescharacteristic
32#3750⋊C3C2212yes—
33#3750⋊C3C2212yes—
34C4×(C3⋊Dic3)⋊C3C2212yes—
35C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
36C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
37C32×C4×Q8—21no—
38C32×C4×Q8D311yescharacteristic, complemented
39C32×C4×Q8C2×C311yescharacteristic, complemented
40#64156C2×C311yescharacteristic
41#64153C2×C311yescharacteristic, complemented
42#64107C2×C312yescomplemented
43#64105C2×C312yescomplemented
44#64102C2×C311yescharacteristic
45#64096C2×C311yescharacteristic, complemented
46#64085C2×C312yescomplemented
47#64084C2×C311yescharacteristic
48#64081C2×C312yescomplemented
49C4×#634C2×C311yescharacteristic, complemented
50C3×#947—31no—
51C3×#944—31no—
52C3×#898—32no—
53C3×#896—32no—
54C3×#893—31no—
55C3×#887—31no—
56C3×#876—32no—
57C3×#875—31no—
58C3×#872—32no—
59C3×C4×Dic6—31no—
60C22×(C3⋊D3)⋊C3C2312yes—
61Q8×H(3,𝔽3)—22no—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2312yes—
64C2×C4×H(3,𝔽3)C2312yes—
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C4⋊((C3⋊D3)⋊C3)—22no—
73C4×(C3⋊D3)⋊C3—24no—
74#634⋊C3—22no—
75#634⋊C3—22no—
76C3×#556—91no—
77#5568—31no—
78#5568—91no—
79C2×C32×Q8—21no—
80C2×C32×Q8C2×D311yescharacteristic
81C2×C32×Q8C22×C311yescharacteristic
82C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
83C2×C4×C3⋊D3C22×C312yes—
84C2×#634C22×C311yescharacteristic
85C2×#634C22×C311yescharacteristic
86C2×C3×D12—31no—
87C2×C3×C4×D3—32no—
88C2×C3×Dic6—31no—
89C2×C3×Dic6—31no—
90C32×C4⋊C4—21no—
91C32×C4⋊C4—22no—
92C32×C4⋊C4C2×D311yescharacteristic
93C32×C4⋊C4C2×D312yes—
94C32×C4⋊C4C22×C311yescharacteristic
95C32×C4⋊C4C22×C312yes—
96C32×C42—21no—
97C32×C42—22no—
98C32×C42C2×D311yescharacteristic
99C32×C42C2×D312yes—
100C32×C42C22×C311yescharacteristic
101C32×C42C22×C312yes—
102#3752C22×C312yes—
103#3752C22×C312yes—
104#3752C22×C312yes—
105#3752C22×C312yes—
106#3752C22×C312yes—
107C4⋊(C3⋊Dic3)C22×C311yescharacteristic
108#3750C22×C311yescharacteristic
109#3750C22×C311yescharacteristic
110#3750C22×C312yes—
111#3750C22×C312yes—
112#3750C22×C312yes—
113C4×C3⋊Dic3C22×C311yescharacteristic
114C4×C3⋊Dic3C22×C311yescharacteristic
115C4×C3⋊Dic3C22×C312yes—
116C3×(C2×D3)⋊C4—32no—
117C3×(C2×D3)⋊C4—32no—
118C3×(C2×D3)⋊C4—32no—
119C3×(C2×D3)⋊C4—32no—
120C3×(C2×D3)⋊C4—32no—
121C3×C4⋊Dic3—31no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—31no—
126C3×Dic3⋊C4—31no—
127C3×C4×Dic3—32no—
128C3×C4×Dic3—31no—
129C3×C4×Dic3—31no—
130C22×H(3,𝔽3)C2411yescharacteristic
131C2×(C3⋊D3)⋊C3—22no—
132C2×(C3⋊D3)⋊C3—22no—
133C2×(C3⋊D3)⋊C3—22no—
134C4×H(3,𝔽3)—22no—
135C4×H(3,𝔽3)—21no—
136C4×H(3,𝔽3)—22no—
137C4×H(3,𝔽3)—21no—
138C4×H(3,𝔽3)#4112yes—
139(C3⋊Dic3)⋊C3—22no—
140(C3⋊Dic3)⋊C3—21no—
141(C3⋊Dic3)⋊C3—21no—
142(C3⋊Dic3)⋊C3—21no—
143(C3⋊Dic3)⋊C3#4112yes—
144C3×C4⋊Q8—91no—
145C3×#126—92no—
146C3×#126—92no—
147C3×#124—91no—
148C3×#124—91no—
149C3×C22⋊Q8—92no—
150C3×C4×Q8—31no—
151C3×C4×Q8—31no—
152C3×C4×Q8—61no—
153C3×C4×Q8—91no—
154C3×C4×Q8—91no—
155C3×C4×Q8C3×D311yescharacteristic
156C3×C4×D4—91no—
157C3×#117—92no—
158#947—31no—
159#947—91no—
160#944—31no—
161#944—91no—
162#898—32no—
163#898—92no—
164#896—32no—
165#896—92no—
166#893—31no—
167#893—91no—
168#887—31no—
169#887—91no—
170#876—32no—
171#876—92no—
172#875—31no—
173#875—91no—
174#872—32no—
175#872—92no—
176C4×Dic6—31no—
177C4×Dic6—91no—
178C22×C3⋊D3C23×C312yes—
179C22×C3×D3—32no—
180C32×Q8—24no—
181C32×Q8—22no—
182C32×Q8—22no—
183C2×C32×C4—21no—
184C2×C32×C4—22no—
185C2×C32×C4—21no—
186C2×C32×C4—22no—
187C2×C32×C4—21no—
188C2×C32×C4C22×D312yes—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D312yes—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C23×C312yes—
194C2×C32×C4C23×C311yescharacteristic
195C2×C32×C4C23×C312yes—
196C2×C32×C4C23×C311yescharacteristic
197C2×C32×C4C23×C311yescharacteristic
198C2×C3⋊Dic3C23×C311yescharacteristic
199C2×C3⋊Dic3C23×C311yescharacteristic
200C2×C3⋊Dic3C23×C311yescharacteristic
201C2×C3⋊Dic3C23×C312yes—
202C2×C3⋊Dic3C23×C311yescharacteristic
203C4⋊(C3⋊D3)—22no—
204C4×C3⋊D3—24no—
205#634—22no—
206#634—22no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—31no—
212C3×D12—62no—
213C3×C4×D3—64no—
214C3×Dic6—62no—
215C3×Dic6—62no—
216#556—91noSylow
217C2×H(3,𝔽3)C2×#4111yescharacteristic
218C2×H(3,𝔽3)C2×#4111yescharacteristic
219C2×H(3,𝔽3)2−1+411yescharacteristic
220(C3⋊D3)⋊C3—42no—
221C2×C3×Q8—31no—
222C2×C3×Q8—31no—
223C2×C3×Q8—61no—
224C2×C3×Q8—91no—
225C2×C3×Q8—91no—
226C2×C3×Q8C2×C3×D311yescharacteristic
227C2×C3×D4—91no—
228C22×C3×C4—92no—
229C2×D12—31no—
230C2×D12—91no—
231C2×C4×D3—32no—
232C2×C4×D3—92no—
233C2×Dic6—31no—
234C2×Dic6—31no—
235C2×Dic6—91no—
236C2×Dic6—91no—
237C3×C4⋊C4—31no—
238C3×C4⋊C4—32no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—32no—
241C3×C4⋊C4—61no—
242C3×C4⋊C4—62no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4—92no—
245C3×C4⋊C4—92no—
246C3×C4⋊C4—91no—
247C3×C4⋊C4—92no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4C2×C3×D311yescharacteristic
250C3×C4⋊C4C2×C3×D312yes—
251C3×C22⋊C4—92no—
252C3×C22⋊C4—92no—
253C3×C22⋊C4—92no—
254C3×C22⋊C4—92no—
255C3×C22⋊C4—92no—
256C3×C42—31no—
257C3×C42—32no—
258C3×C42—32no—
259C3×C42—31no—
260C3×C42—62no—
261C3×C42—61no—
262C3×C42—92no—
263C3×C42—91no—
264C3×C42—91no—
265C3×C42C2×C3×D311yescharacteristic
266C3×C42C2×C3×D312yes—
267(C2×D3)⋊C4—32no—
268(C2×D3)⋊C4—32no—
269(C2×D3)⋊C4—32no—
270(C2×D3)⋊C4—32no—
271(C2×D3)⋊C4—32no—
272(C2×D3)⋊C4—92no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277C4⋊Dic3—31no—
278C4⋊Dic3—91no—
279Dic3⋊C4—31no—
280Dic3⋊C4—31no—
281Dic3⋊C4—32no—
282Dic3⋊C4—32no—
283Dic3⋊C4—32no—
284Dic3⋊C4—91no—
285Dic3⋊C4—91no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288Dic3⋊C4—92no—
289C4×Dic3—31no—
290C4×Dic3—31no—
291C4×Dic3—32no—
292C4×Dic3—91no—
293C4×Dic3—91no—
294C4×Dic3—92no—
295C22×C32—21no—
296C22×C32C23×D311yescharacteristic
297C22×C32C24×C311yescharacteristic
298C2×C3⋊D3—22no—
299C2×C3⋊D3—22no—
300C2×C3⋊D3—22no—
301C2×C3×D3—62no—
302C2×C3×D3—62no—
303C2×C3×D3—62no—
304C32×C4—22no—
305C32×C4—24no—
306C32×C4—22no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—22no—
310C32×C4—21no—
311C32×C4—22no—
312C32×C4—22no—
313C32×C4—21no—
314C32×C4—22no—
315C32×C4—41no—
316C32×C4—42no—
317C32×C4#23512yes—
318C32×C4C3×#4112yes—
319C3⋊Dic3—22no—
320C3⋊Dic3—21no—
321C3⋊Dic3—21no—
322C3⋊Dic3—21no—
323C3⋊Dic3C3×#4112yes—
324C3×Dic3—32no—
325C3×Dic3—62no—
326C3×Dic3—61no—
327C3×Dic3—61no—
328C3×Dic3—61no—
329C4⋊Q8—91no—
330#126—92no—
331#126—92no—
332#124—91no—
333#124—91no—
334C22⋊Q8—92no—
335C4×Q8—91no—
336C4×Q8—91no—
337C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
338C4×D4—91no—
339#117—92no—
340H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
341C23×C3—92no—
342C22×D3—32no—
343C22×D3—92no—
344C3×Q8—22no—
345C3×Q8—62no—
346C3×Q8—62no—
347C3×Q8—64no—
348C3×Q8—182no—
349C3×Q8—182no—
350C3×D4—182no—
351C2×C3×C4—31no—
352C2×C3×C4—31no—
353C2×C3×C4—31no—
354C2×C3×C4—32no—
355C2×C3×C4—31no—
356C2×C3×C4—32no—
357C2×C3×C4—32no—
358C2×C3×C4—31no—
359C2×C3×C4—32no—
360C2×C3×C4—31no—
361C2×C3×C4—61no—
362C2×C3×C4—62no—
363C2×C3×C4—61no—
364C2×C3×C4—62no—
365C2×C3×C4—61no—
366C2×C3×C4—91no—
367C2×C3×C4—91no—
368C2×C3×C4—91no—
369C2×C3×C4—92no—
370C2×C3×C4—91no—
371C2×C3×C4—184no—
372C2×C3×C4C22×C3×D312yes—
373C2×C3×C4C22×C3×D311yescharacteristic
374C2×C3×C4C22×C3×D312yes—
375C2×C3×C4C22×C3×D311yescharacteristic
376C2×C3×C4C22×C3×D311yescharacteristic
377C2×Dic3—31no—
378C2×Dic3—31no—
379C2×Dic3—31no—
380C2×Dic3—32no—
381C2×Dic3—31no—
382C2×Dic3—91no—
383C2×Dic3—91no—
384C2×Dic3—91no—
385C2×Dic3—91no—
386C2×Dic3—92no—
387D12—62no—
388D12—182no—
389C4×D3—64no—
390C4×D3—184no—
391Dic6—62no—
392Dic6—62no—
393Dic6—182no—
394Dic6—182no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32C2×#23511yescharacteristic
399C2×C32#100711yescharacteristic
400C2×C32D3×#4111yescharacteristic
401C2×C32C2×C3×#4111yescharacteristic
402C2×C32C2×C3×#4111yescharacteristic
403C2×C32C3×2−1+411yescharacteristic
404C3⋊D3—42no—
405C3×D3—122no—
406C2×Q8—91no—
407C2×Q8—91no—
408C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
409C2×D4—91no—
410C22×C4—92no—
411C4⋊C4—91no—
412C4⋊C4—92no—
413C4⋊C4—92no—
414C4⋊C4—91no—
415C4⋊C4—91no—
416C4⋊C4—92no—
417C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
418C4⋊C4C2×(C3⋊D3)⋊C312yes—
419C22⋊C4—92no—
420C22⋊C4—92no—
421C22⋊C4—92no—
422C22⋊C4—92no—
423C22⋊C4—92no—
424C42—92no—
425C42—91no—
426C42—91no—
427C42C2×(C3⋊D3)⋊C312yes—
428C42C2×(C3⋊D3)⋊C311yescharacteristic
429C22×C3—31no—
430C22×C3—31no—
431C22×C3—61no—
432C22×C3—182no—
433C22×C3—182no—
434C22×C3—182no—
435C22×C3C23×C3×D311yescharacteristic
436C2×D3—62no—
437C2×D3—62no—
438C2×D3—62no—
439C2×D3—182no—
440C2×D3—182no—
441C2×D3—182no—
442C3×C4—22no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—22no—
446C3×C4—32no—
447C3×C4—32no—
448C3×C4—61no—
449C3×C4—62no—
450C3×C4—62no—
451C3×C4—62no—
452C3×C4—62no—
453C3×C4—61no—
454C3×C4—61no—
455C3×C4—61no—
456C3×C4—64no—
457C3×C4—62no—
458C3×C4—62no—
459C3×C4—92no—
460C3×C4—122no—
461C3×C4—121no—
462C3×C4—182no—
463C3×C4—181no—
464C3×C4—181no—
465C3×C4—181no—
466C3×C4C3×#23512yes—
467Dic3—32no—
468Dic3—62no—
469Dic3—61no—
470Dic3—61no—
471Dic3—61no—
472Dic3—92no—
473Dic3—182no—
474Dic3—181no—
475Dic3—181no—
476Dic3—181no—
477C32—21no—
478C32#556811yescharacteristic, complemented
479C32C3×#55611yescharacteristic, complemented
480C23—92no—
481Q8—22no—
482Q8—182no—
483Q8—182no—
484D4—182no—
485C2×C4—91no—
486C2×C4—91no—
487C2×C4—91no—
488C2×C4—92no—
489C2×C4—91no—
490C2×C4—184no—
491C2×C4C22×(C3⋊D3)⋊C312yes—
492C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
493C2×C4C22×(C3⋊D3)⋊C312yes—
494C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
495C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
496C2×C3—31no—
497C2×C3—31no—
498C2×C3—31no—
499C2×C3—31no—
500C2×C3—31no—
501C2×C3—31no—
502C2×C3—61no—
503C2×C3—61no—
504C2×C3—61no—
505C2×C3—362no—
506C2×C3C2×C3×#23511yescharacteristic
507C2×C3C3×#100711yescharacteristic
508C2×C3C3×D3×#4111yescharacteristic
509D3—122no—
510D3—362no—
511C22—182no—
512C22—182no—
513C22—182no—
514C22C23×(C3⋊D3)⋊C311yescharacteristic
515C4—22no—
516C4—21no—
517C4—22no—
518C4—21no—
519C4—92no—
520C4—182no—
521C4—181no—
522C4—181no—
523C4—181no—
524C4#3828⋊C312yes—
525C3—31no—
526C3—31no—
527C3—61no—
528C3C3×#556811yescharacteristic
529C2—362no—
530C2C2×#3828⋊C311yescharacteristic
531C2#1172262711yescharacteristic
532C2#41×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC4×Q8×H(3,𝔽3)C2
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#556
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
4. #64107⋊C3C2semidirect
5. #64105⋊C3C2semidirect
7. #64096⋊C3C2semidirect
8. #11721387C2semidirect
10. #64081⋊C3C2semidirect
11. C4×#634⋊C3C2semidirect
12. #10596039C3semidirect
38. C32×C4×Q8D3semidirect
39. C32×C4×Q8C2×C3semidirect
41. #64153C2×C3semidirect
42. #64107C2×C3semidirect
43. #64105C2×C3semidirect
45. #64096C2×C3semidirect
46. #64085C2×C3semidirect
48. #64081C2×C3semidirect
49. C4×#634C2×C3semidirect
337. C4×Q8(C3⋊D3)⋊C3semidirect
340. H(3,𝔽3)#556semidirect
478. C32#5568semidirect
479. C32C3×#556semidirect

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
#64156⋊C32
#64153⋊C32
#64107⋊C32
#64105⋊C32
#64102⋊C32
#64096⋊C32
#117213872
#64084⋊C32
#64081⋊C32
C4×#634⋊C32
#105960393
C3×#55683