Finite Group Databasebeta

← #10592345⋊C3 #13141334

Subgroups — #10592345⋊C3 #13141334

Order 1728.

727 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#634⋊C3C211yesmaximal, characteristic
2C2×#3750⋊C3C211yesmaximal, characteristic
3C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
4H(3,𝔽3)×#95C211yesmaximal, characteristic
5#63658⋊C3C211yesmaximal, characteristic
6#63658⋊C3C211yesmaximal, characteristic
7#63658⋊C3C211yesmaximal, characteristic
8#10592345C311yesmaximal, characteristic, complemented
9C3×#4646—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
17C2×#634⋊C3—22no—
18C2×#634⋊C3—22no—
19C2×#634⋊C3C414yescomplemented
20#3750⋊C3—22no—
21C4×(C3⋊Dic3)⋊C3—22no—
22C22×#634C2×C311yescharacteristic
23C22×C3×Dic6—31no—
24C2×#3750C2×C311yescharacteristic
25C2×C4×C3⋊Dic3C2×C311yescharacteristic
26C2×C3×Dic3⋊C4—31no—
27C2×C3×C4×Dic3—31no—
28C32×#95—21no—
29C32×#95D311yescharacteristic
30C32×#95C2×C311yescharacteristic
31#63658C2×C311yescharacteristic
32#63658C2×C311yescharacteristic
33#63658C2×C311yescharacteristic
34C3×C3⋊#95—31no—
35C3×C3⋊#95—31no—
36C3×C3⋊#95—31no—
37C23×H(3,𝔽3)C2311yescharacteristic
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)C2×C412yes—
46C2×C4×H(3,𝔽3)Q812yescomplemented
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)⋊C3C2×C412yes—
55C2×(C3⋊Dic3)⋊C3C2×C412yes—
56C2×(C3⋊Dic3)⋊C3D412yes—
57C2×(C3⋊Dic3)⋊C3D412yes—
58C2×(C3⋊Dic3)⋊C3Q812yescomplemented
59#634⋊C3—28no—
60C3×#391—91no—
61#4646—31no—
62#4646—91no—
63C22×C32×C4—21no—
64C22×C32×C4—21no—
65C22×C32×C4—21no—
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C2×D311yescharacteristic
69C22×C32×C4C22×C311yescharacteristic
70C22×C32×C4C22×C311yescharacteristic
71C22×C32×C4C22×C311yescharacteristic
72C22×C3⋊Dic3C22×C311yescharacteristic
73C22×C3⋊Dic3C22×C311yescharacteristic
74C22×C3⋊Dic3C22×C311yescharacteristic
75C22×C3⋊Dic3C22×C311yescharacteristic
76C2×#634—22no—
77C2×#634—22no—
78C2×#634C3×C414yescomplemented
79C22×C3×Dic3—31no—
80C22×C3×Dic3—31no—
81C22×C3×Dic3—31no—
82C22×C3×Dic3—31no—
83C2×C3×Dic6—34no—
84C2×C3×Dic6—62no—
85C2×C3×Dic6—62no—
86#3750—22no—
87C4×C3⋊Dic3—22no—
88C3×Dic3⋊C4—62no—
89C3×C4×Dic3—62no—
90C22×H(3,𝔽3)C22×C411yescharacteristic
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C22×H(3,𝔽3)C2×D411yescharacteristic
93C22×H(3,𝔽3)C2×Q811yescharacteristic
94C22×H(3,𝔽3)C2×Q811yescharacteristic
95C22×H(3,𝔽3)#4111yescharacteristic
96C22×H(3,𝔽3)#4111yescharacteristic
97C4×H(3,𝔽3)—22no—
98C4×H(3,𝔽3)—22no—
99C4×H(3,𝔽3)—41no—
100(C3⋊Dic3)⋊C3—22no—
101(C3⋊Dic3)⋊C3—22no—
102(C3⋊Dic3)⋊C3—41no—
103(C3⋊Dic3)⋊C3C22⋊C414yes—
104C22×C3×Q8—91no—
105C22×Dic6—31no—
106C22×Dic6—91no—
107C2×C3×C4⋊C4—91no—
108C2×C3×C42—91no—
109C2×Dic3⋊C4—31no—
110C2×Dic3⋊C4—91no—
111C2×C4×Dic3—31no—
112C2×C4×Dic3—91no—
113C3×#95—31no—
114C3×#95—31no—
115C3×#95—61no—
116C3×#95—91no—
117C3×#95—91no—
118C3×#95—91no—
119C3×#95C3×D311yescharacteristic
120C3⋊#95—31no—
121C3⋊#95—31no—
122C3⋊#95—31no—
123C3⋊#95—91no—
124C3⋊#95—91no—
125C3⋊#95—91no—
126C23×C32—21no—
127C23×C32C22×D311yescharacteristic
128C23×C32C23×C311yescharacteristic
129C2×C32×C4—21no—
130C2×C32×C4—21no—
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—22no—
138C2×C32×C4—21no—
139C2×C32×C4—21no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—22no—
145C2×C32×C4—22no—
146C2×C32×C4—41no—
147C2×C32×C4—41no—
148C2×C32×C4—41no—
149C2×C32×C4—41no—
150C2×C32×C4—41no—
151C2×C32×C4—41no—
152C2×C32×C4Dic612yescomplemented
153C2×C32×C4C4×D312yes—
154C2×C32×C4C2×C3×C412yes—
155C2×C32×C4C3×Q812yescomplemented
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—21no—
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3C2×C3×C412yes—
164C2×C3⋊Dic3C2×C3×C412yes—
165C2×C3⋊Dic3C3×D412yes—
166C2×C3⋊Dic3C3×D412yes—
167C2×C3⋊Dic3C3×Q812yescomplemented
168#634—28no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C3×Dic6—68no—
182#391—91noSylow
183C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
184C2×H(3,𝔽3)C4×Q811yescharacteristic
185C2×H(3,𝔽3)C4×Q811yescharacteristic
186C2×H(3,𝔽3)C22⋊Q811yescharacteristic
187C2×H(3,𝔽3)C22⋊Q811yescharacteristic
188C2×H(3,𝔽3)#12411yescharacteristic
189C2×H(3,𝔽3)C4⋊Q811yescharacteristic
190C2×C3×Q8—94no—
191C2×C3×Q8—182no—
192C2×C3×Q8—182no—
193C22×C3×C4—31no—
194C22×C3×C4—31no—
195C22×C3×C4—31no—
196C22×C3×C4—31no—
197C22×C3×C4—31no—
198C22×C3×C4—31no—
199C22×C3×C4—61no—
200C22×C3×C4—61no—
201C22×C3×C4—61no—
202C22×C3×C4—91no—
203C22×C3×C4—91no—
204C22×C3×C4—91no—
205C22×C3×C4—91no—
206C22×C3×C4C2×C3×D311yescharacteristic
207C22×C3×C4C2×C3×D311yescharacteristic
208C22×C3×C4C2×C3×D311yescharacteristic
209C22×Dic3—31no—
210C22×Dic3—31no—
211C22×Dic3—31no—
212C22×Dic3—31no—
213C22×Dic3—91no—
214C22×Dic3—91no—
215C22×Dic3—91no—
216C22×Dic3—91no—
217C2×Dic6—34no—
218C2×Dic6—62no—
219C2×Dic6—62no—
220C2×Dic6—94no—
221C2×Dic6—182no—
222C2×Dic6—182no—
223C3×C4⋊C4—182no—
224C3×C42—182no—
225Dic3⋊C4—62no—
226Dic3⋊C4—182no—
227C4×Dic3—62no—
228C4×Dic3—182no—
229C22×C32—21no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32—21no—
233C22×C32—21no—
234C22×C32—21no—
235C22×C32—21no—
236C22×C32C2×Dic611yescharacteristic
237C22×C32C2×C4×D311yescharacteristic
238C22×C32#23511yescharacteristic
239C22×C32D3×D411yescharacteristic
240C22×C32D3×D411yescharacteristic
241C22×C32#23711yescharacteristic
242C22×C32D3×Q811yescharacteristic
243C22×C32C22×C3×C411yescharacteristic
244C22×C32C2×C3×D411yescharacteristic
245C22×C32C2×C3×D411yescharacteristic
246C22×C32C2×C3×Q811yescharacteristic
247C22×C32C2×C3×Q811yescharacteristic
248C22×C32C3×#4111yescharacteristic
249C22×C32C3×#4111yescharacteristic
250C32×C4—22no—
251C32×C4—22no—
252C32×C4—22no—
253C32×C4—22no—
254C32×C4—42no—
255C32×C4—42no—
256C32×C4—42no—
257C32×C4—41no—
258C32×C4—41no—
259C3⋊Dic3—22no—
260C3⋊Dic3—22no—
261C3⋊Dic3—41no—
262C3⋊Dic3C3×C22⋊C414yes—
263C3×Dic3—34no—
264C3×Dic3—62no—
265C3×Dic3—62no—
266C3×Dic3—121no—
267C22×Q8—91no—
268C2×C4⋊C4—91no—
269C2×C42—91no—
270#95—91no—
271#95—91no—
272#95—91no—
273#95(C3⋊D3)⋊C311yescharacteristic
274H(3,𝔽3)#39111yescharacteristic, complemented, Sylow
275C23×C3—31no—
276C23×C3—31no—
277C23×C3—61no—
278C23×C3C22×C3×D311yescharacteristic
279C3×Q8—188no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—32no—
288C2×C3×C4—32no—
289C2×C3×C4—32no—
290C2×C3×C4—32no—
291C2×C3×C4—61no—
292C2×C3×C4—61no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—61no—
296C2×C3×C4—61no—
297C2×C3×C4—61no—
298C2×C3×C4—61no—
299C2×C3×C4—61no—
300C2×C3×C4—61no—
301C2×C3×C4—61no—
302C2×C3×C4—61no—
303C2×C3×C4—61no—
304C2×C3×C4—61no—
305C2×C3×C4—62no—
306C2×C3×C4—62no—
307C2×C3×C4—62no—
308C2×C3×C4—92no—
309C2×C3×C4—92no—
310C2×C3×C4—92no—
311C2×C3×C4—92no—
312C2×C3×C4—92no—
313C2×C3×C4—121no—
314C2×C3×C4—121no—
315C2×C3×C4—121no—
316C2×C3×C4—121no—
317C2×C3×C4—121no—
318C2×C3×C4—121no—
319C2×C3×C4—181no—
320C2×C3×C4—181no—
321C2×C3×C4—181no—
322C2×C3×C4—181no—
323C2×C3×C4—181no—
324C2×C3×C4—181no—
325C2×C3×C4—181no—
326C2×C3×C4C3×Dic612yes—
327C2×C3×C4C3×C4×D312yes—
328C2×Dic3—32no—
329C2×Dic3—32no—
330C2×Dic3—32no—
331C2×Dic3—32no—
332C2×Dic3—32no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—92no—
341C2×Dic3—92no—
342C2×Dic3—92no—
343C2×Dic3—92no—
344C2×Dic3—92no—
345C2×Dic3—181no—
346C2×Dic3—181no—
347C2×Dic3—181no—
348C2×Dic3—181no—
349C2×Dic3—181no—
350C2×Dic3—181no—
351C2×Dic3—181no—
352Dic6—68no—
353Dic6—188no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32C4×Dic611yescharacteristic
362C2×C32#87811yescharacteristic
363C2×C32D3×C22⋊C411yescharacteristic
364C2×C32#88511yescharacteristic
365C2×C32#88711yescharacteristic
366C2×C32#88811yescharacteristic
367C2×C32#89611yescharacteristic
368C2×C32C2×C3×C22⋊C411yescharacteristic
369C2×C32C3×C4×Q811yescharacteristic
370C2×C32C3×C4×Q811yescharacteristic
371C2×C32C3×C22⋊Q811yescharacteristic
372C2×C32C3×C22⋊Q811yescharacteristic
373C2×C32C3×#12411yescharacteristic
374C2×C32C3×C4⋊Q811yescharacteristic
375C2×Q8—94no—
376C2×Q8—182no—
377C2×Q8—182no—
378C22×C4—91no—
379C22×C4—91no—
380C22×C4—91no—
381C22×C4—91no—
382C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
383C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
384C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
385C4⋊C4—182no—
386C42—182no—
387C22×C3—31no—
388C22×C3—31no—
389C22×C3—31no—
390C22×C3—31no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—31no—
394C22×C3—31no—
395C22×C3—31no—
396C22×C3—31no—
397C22×C3—31no—
398C22×C3—31no—
399C22×C3—31no—
400C22×C3—31no—
401C22×C3—61no—
402C22×C3—61no—
403C22×C3—61no—
404C22×C3—61no—
405C22×C3—61no—
406C22×C3—61no—
407C22×C3—61no—
408C22×C3C2×C3×Dic611yescharacteristic
409C22×C3C2×C3×C4×D311yescharacteristic
410C22×C3C3×#23511yescharacteristic
411C22×C3C3×D3×D411yescharacteristic
412C22×C3C3×D3×D411yescharacteristic
413C22×C3C3×#23711yescharacteristic
414C22×C3C3×D3×Q811yescharacteristic
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—41no—
418C3×C4—62no—
419C3×C4—62no—
420C3×C4—62no—
421C3×C4—62no—
422C3×C4—94no—
423C3×C4—122no—
424C3×C4—122no—
425C3×C4—121no—
426C3×C4—121no—
427C3×C4—122no—
428C3×C4—182no—
429C3×C4—182no—
430C3×C4—361no—
431Dic3—34no—
432Dic3—62no—
433Dic3—62no—
434Dic3—94no—
435Dic3—121no—
436Dic3—182no—
437Dic3—182no—
438Dic3—361no—
439C32—21no—
440C32#464611yescharacteristic, complemented
441C32C3×#39111yescharacteristic, complemented
442C23C22×(C3⋊D3)⋊C311yescharacteristic
443Q8—188no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—92no—
452C2×C4—92no—
453C2×C4—92no—
454C2×C4—92no—
455C2×C4—92no—
456C2×C4—181no—
457C2×C4—181no—
458C2×C4—181no—
459C2×C4—181no—
460C2×C4—181no—
461C2×C4—181no—
462C2×C4—181no—
463C2×C4#634⋊C312yescomplemented
464C2×C4C4×(C3⋊D3)⋊C312yes—
465C2×C3—31no—
466C2×C3—31no—
467C2×C3—31no—
468C2×C3—31no—
469C2×C3—31no—
470C2×C3—31no—
471C2×C3—31no—
472C2×C3—31no—
473C2×C3—31no—
474C2×C3—31no—
475C2×C3—31no—
476C2×C3—31no—
477C2×C3—31no—
478C2×C3—31no—
479C2×C3—61no—
480C2×C3—61no—
481C2×C3—61no—
482C2×C3—61no—
483C2×C3—61no—
484C2×C3—61no—
485C2×C3—61no—
486C2×C3C3×C4×Dic611yescharacteristic
487C2×C3C3×#87811yescharacteristic
488C2×C3C3×D3×C22⋊C411yescharacteristic
489C2×C3C3×#88511yescharacteristic
490C2×C3C3×#88711yescharacteristic
491C2×C3C3×#88811yescharacteristic
492C2×C3C3×#89611yescharacteristic
493C22C2×#634⋊C311yescharacteristic
494C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
495C22#3828⋊C311yescharacteristic
496C22D4×(C3⋊D3)⋊C311yescharacteristic
497C22D4×(C3⋊D3)⋊C311yescharacteristic
498C22#3830⋊C311yescharacteristic
499C22Q8×(C3⋊D3)⋊C311yescharacteristic
500C4—22no—
501C4—22no—
502C4—41no—
503C4—94no—
504C4—182no—
505C4—182no—
506C4—361no—
507C3—31no—
508C3—31no—
509C3—61no—
510C3C3×#464611yescharacteristic
511C2C4×#634⋊C311yescharacteristic
512C2#64087⋊C311yescharacteristic
513C2C22⋊C4×(C3⋊D3)⋊C311yescharacteristic
514C2#64094⋊C311yescharacteristic
515C2#64096⋊C311yescharacteristic
516C2#64097⋊C311yescharacteristic
517C2#64105⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
Frattini, socleC23×C3C22×C3×D3
FittingH(3,𝔽3)×#95C2
prime radicalC23×H(3,𝔽3)C23
cosocleC23×C32C23×C3
hypercenter#95(C3⋊D3)⋊C3
hypocenterC32C3×#391
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ #95

Complements

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

NH ≅ G/Ntype
8. #10592345C3semidirect
19. C2×#634⋊C3C4semidirect
46. C2×C4×H(3,𝔽3)Q8semidirect
58. C2×(C3⋊Dic3)⋊C3Q8semidirect
78. C2×#634C3×C4semidirect
152. C2×C32×C4Dic6semidirect
155. C2×C32×C4C3×Q8semidirect
167. C2×C3⋊Dic3C3×Q8semidirect
274. H(3,𝔽3)#391semidirect
440. C32#4646semidirect
441. C32C3×#391semidirect
463. C2×C4#634⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×#634⋊C32
C2×#3750⋊C32
C2×C4×(C3⋊Dic3)⋊C32
H(3,𝔽3)×#952
#63658⋊C32
#105923453
C3×#46463