Finite Group Databasebeta

← #10592677⋊C3 #13142613

Subgroups — #10592677⋊C3 #13142613

Order 1728.

631 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#3750⋊C3C211yesmaximal, characteristic
3C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
4#63658⋊C3C211yesmaximal, characteristic
5#63658⋊C3C211yesmaximal, characteristic
6#63658⋊C3C211yesmaximal, characteristic
7#63658⋊C3C211yesmaximal, characteristic
8#10592677C311yesmaximal, characteristic, complemented
9C3×#4978—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⋊C4×H(3,𝔽3)—22no—
18#3750⋊C3C414yescomplemented
19C4×(C3⋊Dic3)⋊C3—22no—
20C2×C32×C4⋊C4—21no—
21C2×C32×C4⋊C4D311yescharacteristic
22C2×C32×C4⋊C4C2×C311yescharacteristic
23C2×#3750C2×C311yescharacteristic
24C2×C4×C3⋊Dic3C2×C311yescharacteristic
25C2×C3×Dic3⋊C4—31no—
26C2×C3×C4×Dic3—31no—
27#63658C2×C311yescharacteristic
28#63658C2×C311yescharacteristic
29#63658C2×C311yescharacteristic
30#63658C2×C311yescharacteristic
31C3×C3⋊#95—31no—
32C3×C3⋊#95—31no—
33C3×C3⋊#95—31no—
34C3×C3⋊#95—31no—
35C23×H(3,𝔽3)C2311yescharacteristic
36C2×C4×H(3,𝔽3)—21no—
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)C2×C412yes—
44C2×C4×H(3,𝔽3)D412yes—
45C2×(C3⋊Dic3)⋊C3—21no—
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)⋊C3C2×C412yes—
55C2×(C3⋊Dic3)⋊C3C2×C412yes—
56C2×(C3⋊Dic3)⋊C3Q812yes—
57C3×#387—91no—
58#4978—31no—
59#4978—91no—
60C22×C32×C4—21no—
61C22×C32×C4—21no—
62C22×C32×C4—21no—
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C22×C311yescharacteristic
67C22×C32×C4C22×C311yescharacteristic
68C22×C32×C4C22×C311yescharacteristic
69C22×C3⋊Dic3C22×C311yescharacteristic
70C22×C3⋊Dic3C22×C311yescharacteristic
71C22×C3⋊Dic3C22×C311yescharacteristic
72C22×C3⋊Dic3C22×C311yescharacteristic
73C22×C3×Dic3—31no—
74C22×C3×Dic3—31no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—31no—
77C32×C4⋊C4—24no—
78C32×C4⋊C4—22no—
79C32×C4⋊C4—22no—
80#3750C3×C414yescomplemented
81C4×C3⋊Dic3—22no—
82C3×Dic3⋊C4—34no—
83C3×C4×Dic3—62no—
84C22×H(3,𝔽3)C22×C411yescharacteristic
85C22×H(3,𝔽3)C2×D411yescharacteristic
86C22×H(3,𝔽3)C2×Q811yescharacteristic
87C22×H(3,𝔽3)#4111yescharacteristic
88C22×H(3,𝔽3)#4111yescharacteristic
89C22×H(3,𝔽3)#4111yescharacteristic
90C22×H(3,𝔽3)#4111yescharacteristic
91C4×H(3,𝔽3)—22no—
92C4×H(3,𝔽3)—22no—
93C4×H(3,𝔽3)—41no—
94(C3⋊Dic3)⋊C3—22no—
95(C3⋊Dic3)⋊C3—22no—
96(C3⋊Dic3)⋊C3—22no—
97(C3⋊Dic3)⋊C3—41no—
98C2×C3×C4⋊C4—31no—
99C2×C3×C4⋊C4—31no—
100C2×C3×C4⋊C4—61no—
101C2×C3×C4⋊C4—91no—
102C2×C3×C4⋊C4C3×D311yescharacteristic
103C2×C3×C42—91no—
104C2×Dic3⋊C4—31no—
105C2×Dic3⋊C4—91no—
106C2×C4×Dic3—31no—
107C2×C4×Dic3—91no—
108C3×#95—91no—
109C3×#95—91no—
110C3×#95—91no—
111C3×#95—91no—
112C3⋊#95—31no—
113C3⋊#95—31no—
114C3⋊#95—31no—
115C3⋊#95—31no—
116C3⋊#95—91no—
117C3⋊#95—91no—
118C3⋊#95—91no—
119C3⋊#95—91no—
120C23×C32—21no—
121C23×C32C22×D311yescharacteristic
122C23×C32C23×C311yescharacteristic
123C2×C32×C4—22no—
124C2×C32×C4—22no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—22no—
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—22no—
141C2×C32×C4—22no—
142C2×C32×C4—41no—
143C2×C32×C4—41no—
144C2×C32×C4—41no—
145C2×C32×C4—41no—
146C2×C32×C4C4×D312yes—
147C2×C32×C4#6712yes—
148C2×C32×C4C2×C3×C412yes—
149C2×C32×C4C3×D412yes—
150C2×C3⋊Dic3—21no—
151C2×C3⋊Dic3—21no—
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⋊Dic3C2×C3×C412yes—
160C2×C3⋊Dic3C2×C3×C412yes—
161C2×C3⋊Dic3C3×Q812yes—
162C2×C3×Dic3—32no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—61no—
166C2×C3×Dic3—61no—
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—
174#387—91noSylow
175C2×H(3,𝔽3)#11711yescharacteristic
176C2×H(3,𝔽3)C4×D411yescharacteristic
177C2×H(3,𝔽3)C4×Q811yescharacteristic
178C2×H(3,𝔽3)C22⋊Q811yescharacteristic
179C2×H(3,𝔽3)#12311yescharacteristic
180C2×H(3,𝔽3)#12511yescharacteristic
181C2×H(3,𝔽3)#12611yescharacteristic
182C22×C3×C4—31no—
183C22×C3×C4—31no—
184C22×C3×C4—31no—
185C22×C3×C4—31no—
186C22×C3×C4—31no—
187C22×C3×C4—31no—
188C22×C3×C4—61no—
189C22×C3×C4—61no—
190C22×C3×C4—61no—
191C22×C3×C4—91no—
192C22×C3×C4—91no—
193C22×C3×C4—91no—
194C22×C3×C4—91no—
195C22×C3×C4C2×C3×D311yescharacteristic
196C22×C3×C4C2×C3×D311yescharacteristic
197C22×C3×C4C2×C3×D311yescharacteristic
198C22×Dic3—31no—
199C22×Dic3—31no—
200C22×Dic3—31no—
201C22×Dic3—31no—
202C22×Dic3—91no—
203C22×Dic3—91no—
204C22×Dic3—91no—
205C22×Dic3—91no—
206C3×C4⋊C4—22no—
207C3×C4⋊C4—62no—
208C3×C4⋊C4—62no—
209C3×C4⋊C4—64no—
210C3×C4⋊C4—94no—
211C3×C42—182no—
212Dic3⋊C4—34no—
213Dic3⋊C4—94no—
214C4×Dic3—62no—
215C4×Dic3—182no—
216C22×C32—21no—
217C22×C32—21no—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32—21no—
221C22×C32—21no—
222C22×C32—21no—
223C22×C32C2×C4×D311yescharacteristic
224C22×C32#23511yescharacteristic
225C22×C32#23711yescharacteristic
226C22×C32#23711yescharacteristic
227C22×C32D3×Q811yescharacteristic
228C22×C32#23911yescharacteristic
229C22×C32C2×#6711yescharacteristic
230C22×C32C22×C3×C411yescharacteristic
231C22×C32C2×C3×D411yescharacteristic
232C22×C32C2×C3×Q811yescharacteristic
233C22×C32C3×#4111yescharacteristic
234C22×C32C3×#4111yescharacteristic
235C22×C32C3×#4111yescharacteristic
236C22×C32C3×#4111yescharacteristic
237C32×C4—24no—
238C32×C4—22no—
239C32×C4—22no—
240C32×C4—22no—
241C32×C4—22no—
242C32×C4—42no—
243C32×C4—42no—
244C32×C4—41no—
245C32×C4—41no—
246C3⋊Dic3—22no—
247C3⋊Dic3—22no—
248C3⋊Dic3—22no—
249C3⋊Dic3—41no—
250C3×Dic3—62no—
251C3×Dic3—62no—
252C3×Dic3—62no—
253C3×Dic3—121no—
254C2×C4⋊C4—91no—
255C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic
256C2×C42—91no—
257#95—91no—
258#95—91no—
259#95—91no—
260#95—91no—
261H(3,𝔽3)#38711yescharacteristic, complemented, Sylow
262C23×C3—31no—
263C23×C3—31no—
264C23×C3—61no—
265C23×C3C22×C3×D311yescharacteristic
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—32no—
274C2×C3×C4—32no—
275C2×C3×C4—32no—
276C2×C3×C4—32no—
277C2×C3×C4—61no—
278C2×C3×C4—61no—
279C2×C3×C4—61no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—62no—
286C2×C3×C4—62no—
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—62no—
294C2×C3×C4—62no—
295C2×C3×C4—62no—
296C2×C3×C4—92no—
297C2×C3×C4—92no—
298C2×C3×C4—92no—
299C2×C3×C4—121no—
300C2×C3×C4—121no—
301C2×C3×C4—121no—
302C2×C3×C4—121no—
303C2×C3×C4—181no—
304C2×C3×C4—181no—
305C2×C3×C4—181no—
306C2×C3×C4—181no—
307C2×C3×C4—181no—
308C2×C3×C4—181no—
309C2×C3×C4—181no—
310C2×C3×C4—181no—
311C2×C3×C4—181no—
312C2×C3×C4C3×C4×D312yes—
313C2×C3×C4C3×#6712yes—
314C2×Dic3—32no—
315C2×Dic3—32no—
316C2×Dic3—32no—
317C2×Dic3—61no—
318C2×Dic3—61no—
319C2×Dic3—61no—
320C2×Dic3—61no—
321C2×Dic3—61no—
322C2×Dic3—61no—
323C2×Dic3—61no—
324C2×Dic3—61no—
325C2×Dic3—61no—
326C2×Dic3—92no—
327C2×Dic3—92no—
328C2×Dic3—92no—
329C2×Dic3—181no—
330C2×Dic3—181no—
331C2×Dic3—181no—
332C2×Dic3—181no—
333C2×Dic3—181no—
334C2×Dic3—181no—
335C2×Dic3—181no—
336C2×Dic3—181no—
337C2×Dic3—181no—
338C2×C32—21no—
339C2×C32—21no—
340C2×C32—21no—
341C2×C32—21no—
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32#88711yescharacteristic
346C2×C32#88911yescharacteristic
347C2×C32#89211yescharacteristic
348C2×C32#89811yescharacteristic
349C2×C32C4×#6711yescharacteristic
350C2×C32#93511yescharacteristic
351C2×C32#94611yescharacteristic
352C2×C32C3×#11711yescharacteristic
353C2×C32C3×C4×D411yescharacteristic
354C2×C32C3×C4×Q811yescharacteristic
355C2×C32C3×C22⋊Q811yescharacteristic
356C2×C32C3×#12311yescharacteristic
357C2×C32C3×#12511yescharacteristic
358C2×C32C3×#12611yescharacteristic
359C22×C4—91no—
360C22×C4—91no—
361C22×C4—91no—
362C22×C4—91no—
363C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
364C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
365C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
366C4⋊C4—22no—
367C4⋊C4—94no—
368C42—182no—
369C22×C3—31no—
370C22×C3—31no—
371C22×C3—31no—
372C22×C3—31no—
373C22×C3—31no—
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—61no—
384C22×C3—61no—
385C22×C3—61no—
386C22×C3—61no—
387C22×C3—61no—
388C22×C3—61no—
389C22×C3—61no—
390C22×C3C2×C3×C4×D311yescharacteristic
391C22×C3C3×#23511yescharacteristic
392C22×C3C3×#23711yescharacteristic
393C22×C3C3×#23711yescharacteristic
394C22×C3C3×D3×Q811yescharacteristic
395C22×C3C3×#23911yescharacteristic
396C22×C3C2×C3×#6711yescharacteristic
397C3×C4—22no—
398C3×C4—22no—
399C3×C4—41no—
400C3×C4—62no—
401C3×C4—62no—
402C3×C4—62no—
403C3×C4—62no—
404C3×C4—64no—
405C3×C4—122no—
406C3×C4—121no—
407C3×C4—121no—
408C3×C4—122no—
409C3×C4—182no—
410C3×C4—182no—
411C3×C4—182no—
412C3×C4—361no—
413Dic3—62no—
414Dic3—62no—
415Dic3—62no—
416Dic3—121no—
417Dic3—182no—
418Dic3—182no—
419Dic3—182no—
420Dic3—361no—
421C32—21no—
422C32#497811yescharacteristic, complemented
423C32C3×#38711yescharacteristic, complemented
424C23C22×(C3⋊D3)⋊C311yescharacteristic
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—92no—
433C2×C4—92no—
434C2×C4—92no—
435C2×C4—181no—
436C2×C4—181no—
437C2×C4—181no—
438C2×C4—181no—
439C2×C4—181no—
440C2×C4—181no—
441C2×C4—181no—
442C2×C4—181no—
443C2×C4—181no—
444C2×C4C4×(C3⋊D3)⋊C312yes—
445C2×C4#638⋊C312yes—
446C2×C3—31no—
447C2×C3—31no—
448C2×C3—31no—
449C2×C3—31no—
450C2×C3—31no—
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—61no—
461C2×C3—61no—
462C2×C3—61no—
463C2×C3—61no—
464C2×C3—61no—
465C2×C3—61no—
466C2×C3—61no—
467C2×C3C3×#88711yescharacteristic
468C2×C3C3×#88911yescharacteristic
469C2×C3C3×#89211yescharacteristic
470C2×C3C3×#89811yescharacteristic
471C2×C3C3×C4×#6711yescharacteristic
472C2×C3C3×#93511yescharacteristic
473C2×C3C3×#94611yescharacteristic
474C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
475C22#3828⋊C311yescharacteristic
476C22#3830⋊C311yescharacteristic
477C22#3830⋊C311yescharacteristic
478C22Q8×(C3⋊D3)⋊C311yescharacteristic
479C22#8872211yescharacteristic
480C22C2×#638⋊C311yescharacteristic
481C4—22no—
482C4—22no—
483C4—41no—
484C4—182no—
485C4—182no—
486C4—182no—
487C4—361no—
488C3—31no—
489C3—31no—
490C3—61no—
491C3C3×#497811yescharacteristic
492C2#64096⋊C311yescharacteristic
493C2#64098⋊C311yescharacteristic
494C2#64101⋊C311yescharacteristic
495C2#64107⋊C311yescharacteristic
496C2C4×#638⋊C311yescharacteristic
497C2#64144⋊C311yescharacteristic
498C2#64155⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
Frattini, socleC23×C3C22×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
prime radicalC23×H(3,𝔽3)C23
cosocleC23×C32C23×C3
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#387
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
8. #10592677C3semidirect
18. #3750⋊C3C4semidirect
80. #3750C3×C4semidirect
261. H(3,𝔽3)#387semidirect
422. C32#4978semidirect
423. C32C3×#387semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×H(3,𝔽3)2
C2×#3750⋊C32
C2×C4×(C3⋊Dic3)⋊C32
#63658⋊C32
#105926773
C3×#49783