Finite Group Databasebeta

← #10592675⋊C3 #13142611

Subgroups — #10592675⋊C3 #13142611

Order 1728.

687 conjugacy classes of subgroups.

The subgroup lattice has 514 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×#3750⋊C3C211yesmaximal, characteristic
4C2×#3750⋊C3C211yesmaximal, characteristic
5C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
6#63658⋊C3C211yesmaximal, characteristic
7#63658⋊C3C211yesmaximal, characteristic
8#10592675C311yesmaximal, characteristic, complemented
9C3×#4976—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⋊C3—22no—
19#3750⋊C3—22no—
20#3750⋊C3C414yescomplemented
21C4×(C3⋊Dic3)⋊C3—22no—
22C2×C32×C4⋊C4—21no—
23C2×C32×C4⋊C4D311yescharacteristic
24C2×C32×C4⋊C4C2×C311yescharacteristic
25C2×#3750C2×C311yescharacteristic
26C2×#3750C2×C311yescharacteristic
27C2×#3750C2×C311yescharacteristic
28C2×C4×C3⋊Dic3C2×C311yescharacteristic
29C2×C3×Dic3⋊C4—31no—
30C2×C3×Dic3⋊C4—31no—
31C2×C3×Dic3⋊C4—31no—
32C2×C3×C4×Dic3—31no—
33#63658C2×C311yescharacteristic
34#63658C2×C311yescharacteristic
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)D412yes—
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)⋊C3Q812yes—
58C2×(C3⋊Dic3)⋊C3Q812yes—
59C3×#389—91no—
60#4976—31no—
61#4976—91no—
62C22×C32×C4—21no—
63C22×C32×C4—21no—
64C22×C32×C4—21no—
65C22×C32×C4C2×D311yescharacteristic
66C22×C32×C4C2×D311yescharacteristic
67C22×C32×C4C2×D311yescharacteristic
68C22×C32×C4C22×C311yescharacteristic
69C22×C32×C4C22×C311yescharacteristic
70C22×C32×C4C22×C311yescharacteristic
71C22×C3⋊Dic3C22×C311yescharacteristic
72C22×C3⋊Dic3C22×C311yescharacteristic
73C22×C3⋊Dic3C22×C311yescharacteristic
74C22×C3⋊Dic3C22×C311yescharacteristic
75C22×C3×Dic3—31no—
76C22×C3×Dic3—31no—
77C22×C3×Dic3—31no—
78C22×C3×Dic3—31no—
79C32×C4⋊C4—24no—
80C32×C4⋊C4—22no—
81C32×C4⋊C4—22no—
82#3750—22no—
83#3750—22no—
84#3750C3×C414yescomplemented
85C4×C3⋊Dic3—22no—
86C3×Dic3⋊C4—34no—
87C3×Dic3⋊C4—62no—
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)⋊C3C4⋊C414yescomplemented
104C2×C3×C4⋊C4—31no—
105C2×C3×C4⋊C4—31no—
106C2×C3×C4⋊C4—61no—
107C2×C3×C4⋊C4—91no—
108C2×C3×C4⋊C4—91no—
109C2×C3×C4⋊C4—91no—
110C2×C3×C4⋊C4C3×D311yescharacteristic
111C2×C3×C42—91no—
112C2×Dic3⋊C4—31no—
113C2×Dic3⋊C4—31no—
114C2×Dic3⋊C4—31no—
115C2×Dic3⋊C4—91no—
116C2×Dic3⋊C4—91no—
117C2×Dic3⋊C4—91no—
118C2×C4×Dic3—31no—
119C2×C4×Dic3—91no—
120C3×#95—91no—
121C3×#95—91no—
122C3⋊#95—31no—
123C3⋊#95—31no—
124C3⋊#95—91no—
125C3⋊#95—91no—
126C23×C32—21no—
127C23×C32C22×D311yescharacteristic
128C23×C32C23×C311yescharacteristic
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—22no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4—41no—
149C2×C32×C4—41no—
150C2×C32×C4—41no—
151C2×C32×C4—41no—
152C2×C32×C4C4×D312yes—
153C2×C32×C4#6712yes—
154C2×C32×C4C2×C3×C412yes—
155C2×C32×C4C3×D412yes—
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×Q812yes—
167C2×C3⋊Dic3C3×Q812yes—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180#389—91noSylow
181C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
182C2×H(3,𝔽3)C4×D411yescharacteristic
183C2×H(3,𝔽3)C4×Q811yescharacteristic
184C2×H(3,𝔽3)#12111yescharacteristic
185C2×H(3,𝔽3)C22⋊Q811yescharacteristic
186C2×H(3,𝔽3)#12511yescharacteristic
187C2×H(3,𝔽3)C4⋊Q811yescharacteristic
188C22×C3×C4—31no—
189C22×C3×C4—31no—
190C22×C3×C4—31no—
191C22×C3×C4—31no—
192C22×C3×C4—31no—
193C22×C3×C4—31no—
194C22×C3×C4—61no—
195C22×C3×C4—61no—
196C22×C3×C4—61no—
197C22×C3×C4—91no—
198C22×C3×C4—91no—
199C22×C3×C4—91no—
200C22×C3×C4—91no—
201C22×C3×C4C2×C3×D311yescharacteristic
202C22×C3×C4C2×C3×D311yescharacteristic
203C22×C3×C4C2×C3×D311yescharacteristic
204C22×Dic3—31no—
205C22×Dic3—31no—
206C22×Dic3—31no—
207C22×Dic3—31no—
208C22×Dic3—91no—
209C22×Dic3—91no—
210C22×Dic3—91no—
211C22×Dic3—91no—
212C3×C4⋊C4—22no—
213C3×C4⋊C4—62no—
214C3×C4⋊C4—62no—
215C3×C4⋊C4—64no—
216C3×C4⋊C4—94no—
217C3×C4⋊C4—182no—
218C3×C4⋊C4—182no—
219C3×C42—182no—
220Dic3⋊C4—34no—
221Dic3⋊C4—62no—
222Dic3⋊C4—62no—
223Dic3⋊C4—94no—
224Dic3⋊C4—182no—
225Dic3⋊C4—182no—
226C4×Dic3—62no—
227C4×Dic3—182no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32—21no—
233C22×C32—21no—
234C22×C32—21no—
235C22×C32C2×C4×D311yescharacteristic
236C22×C32#23511yescharacteristic
237C22×C32D3×D411yescharacteristic
238C22×C32#23711yescharacteristic
239C22×C32D3×Q811yescharacteristic
240C22×C32D3×Q811yescharacteristic
241C22×C32C2×#6711yescharacteristic
242C22×C32C22×C3×C411yescharacteristic
243C22×C32C2×C3×D411yescharacteristic
244C22×C32C2×C3×D411yescharacteristic
245C22×C32C2×C3×Q811yescharacteristic
246C22×C32C2×C3×Q811yescharacteristic
247C22×C32C3×#4111yescharacteristic
248C22×C32C3×#4111yescharacteristic
249C32×C4—24no—
250C32×C4—22no—
251C32×C4—22no—
252C32×C4—22no—
253C32×C4—22no—
254C32×C4—42no—
255C32×C4—42no—
256C32×C4—41no—
257C32×C4—41no—
258C3⋊Dic3—22no—
259C3⋊Dic3—22no—
260C3⋊Dic3—41no—
261C3⋊Dic3C3×C4⋊C414yescomplemented
262C3×Dic3—34no—
263C3×Dic3—62no—
264C3×Dic3—62no—
265C3×Dic3—121no—
266C2×C4⋊C4—91no—
267C2×C4⋊C4—91no—
268C2×C4⋊C4—91no—
269C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic
270C2×C42—91no—
271#95—91no—
272#95—91no—
273H(3,𝔽3)#38911yescharacteristic, complemented, Sylow
274C23×C3—31no—
275C23×C3—31no—
276C23×C3—61no—
277C23×C3C22×C3×D311yescharacteristic
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—32no—
286C2×C3×C4—32no—
287C2×C3×C4—32no—
288C2×C3×C4—32no—
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—61no—
297C2×C3×C4—62no—
298C2×C3×C4—62no—
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—181no—
318C2×C3×C4—181no—
319C2×C3×C4—181no—
320C2×C3×C4—181no—
321C2×C3×C4—181no—
322C2×C3×C4—181no—
323C2×C3×C4—181no—
324C2×C3×C4C3×C4×D312yes—
325C2×C3×C4C3×#6712yes—
326C2×Dic3—32no—
327C2×Dic3—32no—
328C2×Dic3—32no—
329C2×Dic3—32no—
330C2×Dic3—32no—
331C2×Dic3—61no—
332C2×Dic3—61no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—92no—
339C2×Dic3—92no—
340C2×Dic3—92no—
341C2×Dic3—92no—
342C2×Dic3—92no—
343C2×Dic3—181no—
344C2×Dic3—181no—
345C2×Dic3—181no—
346C2×Dic3—181no—
347C2×Dic3—181no—
348C2×Dic3—181no—
349C2×Dic3—181no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32—21no—
356C2×C32—21no—
357C2×C32#88711yescharacteristic
358C2×C32#88911yescharacteristic
359C2×C32D3×C4⋊C411yescharacteristic
360C2×C32#89611yescharacteristic
361C2×C32C4×#6711yescharacteristic
362C2×C32#93911yescharacteristic
363C2×C32#94411yescharacteristic
364C2×C32C2×C3×C4⋊C411yescharacteristic
365C2×C32C3×C4×D411yescharacteristic
366C2×C32C3×C4×Q811yescharacteristic
367C2×C32C3×#12111yescharacteristic
368C2×C32C3×C22⋊Q811yescharacteristic
369C2×C32C3×#12511yescharacteristic
370C2×C32C3×C4⋊Q811yescharacteristic
371C22×C4—91no—
372C22×C4—91no—
373C22×C4—91no—
374C22×C4—91no—
375C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
376C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
377C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
378C4⋊C4—22no—
379C4⋊C4—94no—
380C4⋊C4—182no—
381C4⋊C4—182no—
382C42—182no—
383C22×C3—31no—
384C22×C3—31no—
385C22×C3—31no—
386C22×C3—31no—
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—61no—
398C22×C3—61no—
399C22×C3—61no—
400C22×C3—61no—
401C22×C3—61no—
402C22×C3—61no—
403C22×C3—61no—
404C22×C3C2×C3×C4×D311yescharacteristic
405C22×C3C3×#23511yescharacteristic
406C22×C3C3×D3×D411yescharacteristic
407C22×C3C3×#23711yescharacteristic
408C22×C3C3×D3×Q811yescharacteristic
409C22×C3C3×D3×Q811yescharacteristic
410C22×C3C2×C3×#6711yescharacteristic
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—41no—
414C3×C4—62no—
415C3×C4—62no—
416C3×C4—62no—
417C3×C4—62no—
418C3×C4—64no—
419C3×C4—94no—
420C3×C4—122no—
421C3×C4—121no—
422C3×C4—121no—
423C3×C4—122no—
424C3×C4—182no—
425C3×C4—182no—
426C3×C4—361no—
427Dic3—34no—
428Dic3—62no—
429Dic3—62no—
430Dic3—94no—
431Dic3—121no—
432Dic3—182no—
433Dic3—182no—
434Dic3—361no—
435C32—21no—
436C32#497611yescharacteristic, complemented
437C32C3×#38911yescharacteristic, complemented
438C23C22×(C3⋊D3)⋊C311yescharacteristic
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—92no—
447C2×C4—92no—
448C2×C4—92no—
449C2×C4—92no—
450C2×C4—92no—
451C2×C4—181no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4—181no—
455C2×C4—181no—
456C2×C4—181no—
457C2×C4—181no—
458C2×C4C4×(C3⋊D3)⋊C312yes—
459C2×C4#638⋊C312yes—
460C2×C3—31no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—31no—
464C2×C3—31no—
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—61no—
475C2×C3—61no—
476C2×C3—61no—
477C2×C3—61no—
478C2×C3—61no—
479C2×C3—61no—
480C2×C3—61no—
481C2×C3C3×#88711yescharacteristic
482C2×C3C3×#88911yescharacteristic
483C2×C3C3×D3×C4⋊C411yescharacteristic
484C2×C3C3×#89611yescharacteristic
485C2×C3C3×C4×#6711yescharacteristic
486C2×C3C3×#93911yescharacteristic
487C2×C3C3×#94411yescharacteristic
488C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
489C22#3828⋊C311yescharacteristic
490C22D4×(C3⋊D3)⋊C311yescharacteristic
491C22#3830⋊C311yescharacteristic
492C22Q8×(C3⋊D3)⋊C311yescharacteristic
493C22Q8×(C3⋊D3)⋊C311yescharacteristic
494C22C2×#638⋊C311yescharacteristic
495C4—22no—
496C4—22no—
497C4—41no—
498C4—94no—
499C4—182no—
500C4—182no—
501C4—361no—
502C3—31no—
503C3—31no—
504C3—61no—
505C3C3×#497611yescharacteristic
506C2#64096⋊C311yescharacteristic
507C2#64098⋊C311yescharacteristic
508C2C4⋊C4×(C3⋊D3)⋊C311yescharacteristic
509C2#64105⋊C311yescharacteristic
510C2C4×#638⋊C311yescharacteristic
511C2#64148⋊C311yescharacteristic
512C2#64153⋊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×#389
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
8. #10592675C3semidirect
20. #3750⋊C3C4semidirect
84. #3750C3×C4semidirect
103. (C3⋊Dic3)⋊C3C4⋊C4semidirect
261. C3⋊Dic3C3×C4⋊C4semidirect
273. H(3,𝔽3)#389semidirect
436. C32#4976semidirect
437. C32C3×#389semidirect

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
#105926753
C3×#49763