Finite Group Databasebeta

← #10592356⋊C3 #13141344

Subgroups — #10592356⋊C3 #13141344

Order 1728.

601 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic
2C2×#3750⋊C3C211yesmaximal, characteristic
3C2×#3750⋊C3C211yesmaximal, characteristic
4C2×#3750⋊C3C211yesmaximal, characteristic
5H(3,𝔽3)×#95C211yesmaximal, characteristic
6#63658⋊C3C211yesmaximal, characteristic
7#63658⋊C3C211yesmaximal, characteristic
8#10592356C311yesmaximal, characteristic, complemented
9C3×#4657—31nomaximal
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C22×C4×H(3,𝔽3)C2211yescharacteristic
13C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C4⋊((C3⋊Dic3)⋊C3)—22no—
18#3750⋊C3—22no—
19#3750⋊C3—22no—
20#3750⋊C3—22no—
21C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic
22C2×#3750C2×C311yescharacteristic
23C2×#3750C2×C311yescharacteristic
24C2×#3750C2×C311yescharacteristic
25C2×C3×C4⋊Dic3—31no—
26C2×C3×Dic3⋊C4—31no—
27C2×C3×Dic3⋊C4—31no—
28C2×C3×Dic3⋊C4—31no—
29C32×#95—21no—
30C32×#95D311yescharacteristic
31C32×#95C2×C311yescharacteristic
32#63658C2×C311yescharacteristic
33#63658C2×C311yescharacteristic
34C3×C3⋊#95—31no—
35C3×C3⋊#95—31no—
36C23×H(3,𝔽3)C2311yescharacteristic
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)—21no—
43C2×C4×H(3,𝔽3)—21no—
44C2×C4×H(3,𝔽3)—21no—
45C2×C4×H(3,𝔽3)Q812yes—
46C2×(C3⋊Dic3)⋊C3—21no—
47C2×(C3⋊Dic3)⋊C3—21no—
48C2×(C3⋊Dic3)⋊C3—21no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3—21no—
52C2×(C3⋊Dic3)⋊C3—21no—
53C2×(C3⋊Dic3)⋊C3—21no—
54C2×(C3⋊Dic3)⋊C3—21no—
55C2×(C3⋊Dic3)⋊C3D412yes—
56C2×(C3⋊Dic3)⋊C3D412yes—
57C2×(C3⋊Dic3)⋊C3Q812yes—
58C3×#398—91no—
59#4657—31no—
60#4657—91no—
61C22×C32×C4—21no—
62C22×C32×C4—21no—
63C22×C32×C4—21no—
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C22×C311yescharacteristic
68C22×C32×C4C22×C311yescharacteristic
69C22×C32×C4C22×C311yescharacteristic
70C22×C3⋊Dic3C22×C311yescharacteristic
71C22×C3⋊Dic3C22×C311yescharacteristic
72C22×C3⋊Dic3C22×C311yescharacteristic
73C22×C3⋊Dic3C22×C311yescharacteristic
74C22×C3×Dic3—31no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—31no—
77C22×C3×Dic3—31no—
78C4⋊(C3⋊Dic3)—22no—
79#3750—22no—
80#3750—22no—
81#3750—22no—
82C3×C4⋊Dic3—62no—
83C3×Dic3⋊C4—62no—
84C3×Dic3⋊C4—62no—
85C3×Dic3⋊C4—62no—
86C22×H(3,𝔽3)C2×D411yescharacteristic
87C22×H(3,𝔽3)C2×D411yescharacteristic
88C22×H(3,𝔽3)C2×Q811yescharacteristic
89C22×H(3,𝔽3)C2×Q811yescharacteristic
90C22×H(3,𝔽3)#4111yescharacteristic
91C22×H(3,𝔽3)#4111yescharacteristic
92C22×H(3,𝔽3)#4111yescharacteristic
93C4×H(3,𝔽3)—22no—
94C4×H(3,𝔽3)—41no—
95C4×H(3,𝔽3)—41no—
96(C3⋊Dic3)⋊C3—22no—
97(C3⋊Dic3)⋊C3—22no—
98(C3⋊Dic3)⋊C3—22no—
99(C3⋊Dic3)⋊C3—41no—
100C2×C3×C4⋊C4—91no—
101C2×C3×C4⋊C4—91no—
102C2×C3×C4⋊C4—91no—
103C2×C3×C4⋊C4—91no—
104C2×C4⋊Dic3—31no—
105C2×C4⋊Dic3—91no—
106C2×Dic3⋊C4—31no—
107C2×Dic3⋊C4—31no—
108C2×Dic3⋊C4—31no—
109C2×Dic3⋊C4—91no—
110C2×Dic3⋊C4—91no—
111C2×Dic3⋊C4—91no—
112C3×#95—31no—
113C3×#95—31no—
114C3×#95—61no—
115C3×#95—91no—
116C3×#95—91no—
117C3×#95C3×D311yescharacteristic
118C3⋊#95—31no—
119C3⋊#95—31no—
120C3⋊#95—91no—
121C3⋊#95—91no—
122C23×C32—21no—
123C23×C32C22×D311yescharacteristic
124C23×C32C23×C311yescharacteristic
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—21no—
135C2×C32×C4—21no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—22no—
144C2×C32×C4—41no—
145C2×C32×C4—41no—
146C2×C32×C4—41no—
147C2×C32×C4—41no—
148C2×C32×C4—41no—
149C2×C32×C4—41no—
150C2×C32×C4Dic612yes—
151C2×C32×C4C3×Q812yes—
152C2×C3⋊Dic3—21no—
153C2×C3⋊Dic3—21no—
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3—21no—
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—21no—
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3C3×D412yes—
162C2×C3⋊Dic3C3×D412yes—
163C2×C3⋊Dic3C3×Q812yes—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176#398—91noSylow
177C2×H(3,𝔽3)#12111yescharacteristic
178C2×H(3,𝔽3)C22⋊Q811yescharacteristic
179C2×H(3,𝔽3)C22⋊Q811yescharacteristic
180C2×H(3,𝔽3)#12311yescharacteristic
181C2×H(3,𝔽3)#12511yescharacteristic
182C2×H(3,𝔽3)#12511yescharacteristic
183C2×H(3,𝔽3)C4⋊Q811yescharacteristic
184C22×C3×C4—31no—
185C22×C3×C4—31no—
186C22×C3×C4—31no—
187C22×C3×C4—31no—
188C22×C3×C4—31no—
189C22×C3×C4—31no—
190C22×C3×C4—61no—
191C22×C3×C4—61no—
192C22×C3×C4—61no—
193C22×C3×C4—91no—
194C22×C3×C4—91no—
195C22×C3×C4—91no—
196C22×C3×C4—91no—
197C22×C3×C4C2×C3×D311yescharacteristic
198C22×C3×C4C2×C3×D311yescharacteristic
199C22×C3×C4C2×C3×D311yescharacteristic
200C22×Dic3—31no—
201C22×Dic3—31no—
202C22×Dic3—31no—
203C22×Dic3—31no—
204C22×Dic3—91no—
205C22×Dic3—91no—
206C22×Dic3—91no—
207C22×Dic3—91no—
208C3×C4⋊C4—182no—
209C3×C4⋊C4—182no—
210C3×C4⋊C4—182no—
211C3×C4⋊C4—182no—
212C4⋊Dic3—62no—
213C4⋊Dic3—182no—
214Dic3⋊C4—62no—
215Dic3⋊C4—62no—
216Dic3⋊C4—62no—
217Dic3⋊C4—182no—
218Dic3⋊C4—182no—
219Dic3⋊C4—182no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32—21no—
224C22×C32—21no—
225C22×C32—21no—
226C22×C32—21no—
227C22×C32C2×Dic611yescharacteristic
228C22×C32#23511yescharacteristic
229C22×C32#23511yescharacteristic
230C22×C32D3×D411yescharacteristic
231C22×C32D3×D411yescharacteristic
232C22×C32#23711yescharacteristic
233C22×C32D3×Q811yescharacteristic
234C22×C32C2×C3×D411yescharacteristic
235C22×C32C2×C3×D411yescharacteristic
236C22×C32C2×C3×Q811yescharacteristic
237C22×C32C2×C3×Q811yescharacteristic
238C22×C32C3×#4111yescharacteristic
239C22×C32C3×#4111yescharacteristic
240C22×C32C3×#4111yescharacteristic
241C32×C4—22no—
242C32×C4—22no—
243C32×C4—42no—
244C32×C4—42no—
245C32×C4—42no—
246C32×C4—41no—
247C32×C4—41no—
248C32×C4—41no—
249C32×C4—41no—
250C3⋊Dic3—22no—
251C3⋊Dic3—22no—
252C3⋊Dic3—22no—
253C3⋊Dic3—41no—
254C3×Dic3—62no—
255C3×Dic3—62no—
256C3×Dic3—62no—
257C3×Dic3—121no—
258C2×C4⋊C4—91no—
259C2×C4⋊C4—91no—
260C2×C4⋊C4—91no—
261C2×C4⋊C4—91no—
262#95—91no—
263#95—91no—
264#95(C3⋊D3)⋊C311yescharacteristic
265H(3,𝔽3)#39811yescharacteristic, complemented, Sylow
266C23×C3—31no—
267C23×C3—31no—
268C23×C3—61no—
269C23×C3C22×C3×D311yescharacteristic
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—32no—
279C2×C3×C4—32no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—61no—
286C2×C3×C4—61no—
287C2×C3×C4—61no—
288C2×C3×C4—61no—
289C2×C3×C4—61no—
290C2×C3×C4—61no—
291C2×C3×C4—61no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—62no—
297C2×C3×C4—62no—
298C2×C3×C4—62no—
299C2×C3×C4—92no—
300C2×C3×C4—92no—
301C2×C3×C4—92no—
302C2×C3×C4—121no—
303C2×C3×C4—121no—
304C2×C3×C4—121no—
305C2×C3×C4—121no—
306C2×C3×C4—121no—
307C2×C3×C4—121no—
308C2×C3×C4—181no—
309C2×C3×C4—181no—
310C2×C3×C4—181no—
311C2×C3×C4—181no—
312C2×C3×C4—181no—
313C2×C3×C4—181no—
314C2×C3×C4—181no—
315C2×C3×C4—181no—
316C2×C3×C4—181no—
317C2×C3×C4C3×Dic612yes—
318C2×Dic3—32no—
319C2×Dic3—32no—
320C2×Dic3—32no—
321C2×Dic3—61no—
322C2×Dic3—61no—
323C2×Dic3—61no—
324C2×Dic3—61no—
325C2×Dic3—61no—
326C2×Dic3—61no—
327C2×Dic3—61no—
328C2×Dic3—61no—
329C2×Dic3—61no—
330C2×Dic3—92no—
331C2×Dic3—92no—
332C2×Dic3—92no—
333C2×Dic3—181no—
334C2×Dic3—181no—
335C2×Dic3—181no—
336C2×Dic3—181no—
337C2×Dic3—181no—
338C2×Dic3—181no—
339C2×Dic3—181no—
340C2×Dic3—181no—
341C2×Dic3—181no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32—21no—
349C2×C32#87011yescharacteristic
350C2×C32#87811yescharacteristic
351C2×C32#88311yescharacteristic
352C2×C32#88411yescharacteristic
353C2×C32#88811yescharacteristic
354C2×C32#88911yescharacteristic
355C2×C32#89611yescharacteristic
356C2×C32C3×#12111yescharacteristic
357C2×C32C3×C22⋊Q811yescharacteristic
358C2×C32C3×C22⋊Q811yescharacteristic
359C2×C32C3×#12311yescharacteristic
360C2×C32C3×#12511yescharacteristic
361C2×C32C3×#12511yescharacteristic
362C2×C32C3×C4⋊Q811yescharacteristic
363C22×C4—91no—
364C22×C4—91no—
365C22×C4—91no—
366C22×C4—91no—
367C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
368C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
369C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
370C4⋊C4—182no—
371C4⋊C4—182no—
372C4⋊C4—182no—
373C4⋊C4—182no—
374C22×C3—31no—
375C22×C3—31no—
376C22×C3—31no—
377C22×C3—31no—
378C22×C3—31no—
379C22×C3—31no—
380C22×C3—31no—
381C22×C3—31no—
382C22×C3—31no—
383C22×C3—31no—
384C22×C3—31no—
385C22×C3—31no—
386C22×C3—31no—
387C22×C3—31no—
388C22×C3—61no—
389C22×C3—61no—
390C22×C3—61no—
391C22×C3—61no—
392C22×C3—61no—
393C22×C3—61no—
394C22×C3—61no—
395C22×C3C2×C3×Dic611yescharacteristic
396C22×C3C3×#23511yescharacteristic
397C22×C3C3×#23511yescharacteristic
398C22×C3C3×D3×D411yescharacteristic
399C22×C3C3×D3×D411yescharacteristic
400C22×C3C3×#23711yescharacteristic
401C22×C3C3×D3×Q811yescharacteristic
402C3×C4—22no—
403C3×C4—41no—
404C3×C4—41no—
405C3×C4—62no—
406C3×C4—62no—
407C3×C4—122no—
408C3×C4—121no—
409C3×C4—121no—
410C3×C4—121no—
411C3×C4—121no—
412C3×C4—122no—
413C3×C4—122no—
414C3×C4—182no—
415C3×C4—182no—
416C3×C4—182no—
417C3×C4—361no—
418Dic3—62no—
419Dic3—62no—
420Dic3—62no—
421Dic3—121no—
422Dic3—182no—
423Dic3—182no—
424Dic3—182no—
425Dic3—361no—
426C32—21no—
427C32#465711yescharacteristic, complemented
428C32C3×#39811yescharacteristic, complemented
429C23C22×(C3⋊D3)⋊C311yescharacteristic
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—92no—
439C2×C4—92no—
440C2×C4—92no—
441C2×C4—181no—
442C2×C4—181no—
443C2×C4—181no—
444C2×C4—181no—
445C2×C4—181no—
446C2×C4—181no—
447C2×C4—181no—
448C2×C4—181no—
449C2×C4—181no—
450C2×C4#634⋊C312yes—
451C2×C3—31no—
452C2×C3—31no—
453C2×C3—31no—
454C2×C3—31no—
455C2×C3—31no—
456C2×C3—31no—
457C2×C3—31no—
458C2×C3—31no—
459C2×C3—31no—
460C2×C3—31no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—31no—
464C2×C3—31no—
465C2×C3—61no—
466C2×C3—61no—
467C2×C3—61no—
468C2×C3—61no—
469C2×C3—61no—
470C2×C3—61no—
471C2×C3—61no—
472C2×C3C3×#87011yescharacteristic
473C2×C3C3×#87811yescharacteristic
474C2×C3C3×#88311yescharacteristic
475C2×C3C3×#88411yescharacteristic
476C2×C3C3×#88811yescharacteristic
477C2×C3C3×#88911yescharacteristic
478C2×C3C3×#89611yescharacteristic
479C22C2×#634⋊C311yescharacteristic
480C22#3828⋊C311yescharacteristic
481C22#3828⋊C311yescharacteristic
482C22D4×(C3⋊D3)⋊C311yescharacteristic
483C22D4×(C3⋊D3)⋊C311yescharacteristic
484C22#3830⋊C311yescharacteristic
485C22Q8×(C3⋊D3)⋊C311yescharacteristic
486C4—22no—
487C4—41no—
488C4—41no—
489C4—182no—
490C4—182no—
491C4—182no—
492C4—361no—
493C3—31no—
494C3—31no—
495C3—61no—
496C3C3×#465711yescharacteristic
497C2#64079⋊C311yescharacteristic
498C2#64087⋊C311yescharacteristic
499C2#64092⋊C311yescharacteristic
500C2#64093⋊C311yescharacteristic
501C2#64097⋊C311yescharacteristic
502C2#64098⋊C311yescharacteristic
503C2#64105⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC23×C32C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingH(3,𝔽3)×#95C2
prime radicalC23×H(3,𝔽3)C23
hypercenter#95(C3⋊D3)⋊C3
hypocenterC32C3×#398
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
8. #10592356C3semidirect
265. H(3,𝔽3)#398semidirect
427. C32#4657semidirect
428. C32C3×#398semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊((C3⋊Dic3)⋊C3)2
C2×#3750⋊C32
H(3,𝔽3)×#952
#63658⋊C32
#105923563
C3×#46573