Finite Group Databasebeta

← #10592659⋊C3 #13142509

Subgroups — #10592659⋊C3 #13142509

Order 1728.

1259 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#638⋊C3C212yesmaximal, complemented
2C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4C2×#3752⋊C3C212yesmaximal, complemented
5#63658⋊C3C211yesmaximal, characteristic, complemented
6#10592659C311yesmaximal, characteristic, complemented
7C3×#4960—31nomaximal
8C24×H(3,𝔽3)C2211yescharacteristic
9C23×(C3⋊D3)⋊C3C2212yes—
10C22×C4×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3—24no—
13C2×#638⋊C3—24no—
14C2×#638⋊C3—24no—
15C22×(C3⋊Dic3)⋊C3C2212yes—
16C2×C4⋊((C3⋊D3)⋊C3)—24no—
17C2×C4⋊((C3⋊D3)⋊C3)—22no—
18C22⋊C4×H(3,𝔽3)—22no—
19#3752⋊C3—24no—
20C22×#638C2×C312yescomplemented
21C22×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
22C22×C3×#67—32no—
23C22×C3×D12—31no—
24C2×C32×C22⋊C4—21no—
25C2×C32×C22⋊C4D311yescharacteristic, complemented
26C2×C32×C22⋊C4C2×C311yescharacteristic, complemented
27C2×#3752C2×C312yescomplemented
28C2×C3×(C2×D3)⋊C4—32no—
29#63658C2×C311yescharacteristic, complemented
30C3×C3⋊#95—31no—
31C23×H(3,𝔽3)—21no—
32C23×H(3,𝔽3)—22no—
33C23×H(3,𝔽3)—22no—
34C23×H(3,𝔽3)—21no—
35C23×H(3,𝔽3)D412yescomplemented
36C23×H(3,𝔽3)C2311yescharacteristic
37C22×(C3⋊D3)⋊C3—22no—
38C22×(C3⋊D3)⋊C3—22no—
39C22×(C3⋊D3)⋊C3—22no—
40C22×(C3⋊D3)⋊C3—22no—
41C22×(C3⋊D3)⋊C3—22no—
42C22×(C3⋊D3)⋊C3—22no—
43C22×(C3⋊D3)⋊C3D414yescomplemented
44C2×C4×H(3,𝔽3)—22no—
45C2×C4×H(3,𝔽3)—21no—
46C2×C4×H(3,𝔽3)—22no—
47C2×C4×H(3,𝔽3)D412yescomplemented
48#638⋊C3—216no—
49C2×(C3⋊Dic3)⋊C3—22no—
50C2×(C3⋊Dic3)⋊C3—22no—
51C2×(C3⋊Dic3)⋊C3D414yescomplemented
52C4⋊((C3⋊D3)⋊C3)—28no—
53C3×#392—91no—
54#4960—31no—
55#4960—91no—
56C24×C32—21no—
57C24×C32C2×D311yescharacteristic
58C24×C32C22×C311yescharacteristic
59C23×C3⋊D3C22×C312yes—
60C23×C3×D3—32no—
61C22×C32×C4—21no—
62C22×C32×C4—21no—
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C2×D311yescharacteristic
65C22×C32×C4C22×C311yescharacteristic
66C22×C32×C4C22×C311yescharacteristic
67C2×#638—24no—
68C2×#638—24no—
69C2×#638—24no—
70C22×C3⋊Dic3C22×C312yes—
71C2×C4⋊(C3⋊D3)—24no—
72C2×C4⋊(C3⋊D3)—22no—
73C2×C3×#67—64no—
74C2×C3×#67—64no—
75C2×C3×#67—64no—
76C22×C3×Dic3—32no—
77C2×C3×D12—64no—
78C2×C3×D12—62no—
79C32×C22⋊C4—24no—
80C32×C22⋊C4—22no—
81C32×C22⋊C4—22no—
82#3752—24no—
83C3×(C2×D3)⋊C4—64no—
84C22×H(3,𝔽3)—24no—
85C22×H(3,𝔽3)—22no—
86C22×H(3,𝔽3)—41no—
87C22×H(3,𝔽3)—42no—
88C22×H(3,𝔽3)—41no—
89C22×H(3,𝔽3)C2×D411yescharacteristic
90C22×H(3,𝔽3)C2×D412yes—
91C22×H(3,𝔽3)C2×D412yes—
92C22×H(3,𝔽3)C2×D411yescharacteristic
93C22×H(3,𝔽3)#4111yescharacteristic
94C2×(C3⋊D3)⋊C3—24no—
95C2×(C3⋊D3)⋊C3—24no—
96C2×(C3⋊D3)⋊C3—24no—
97C2×(C3⋊D3)⋊C3—42no—
98C2×(C3⋊D3)⋊C3—42no—
99C2×(C3⋊D3)⋊C3—42no—
100C2×(C3⋊D3)⋊C3—42no—
101C4×H(3,𝔽3)—22no—
102C4×H(3,𝔽3)—41no—
103(C3⋊Dic3)⋊C3—24no—
104C22×C3×D4—91no—
105C22×C3×D4—92no—
106C22×#67—32no—
107C22×#67—92no—
108C22×D12—31no—
109C22×D12—91no—
110C2×C3×C22⋊C4—31no—
111C2×C3×C22⋊C4—31no—
112C2×C3×C22⋊C4—61no—
113C2×C3×C22⋊C4—92no—
114C2×C3×C22⋊C4C3×D311yescharacteristic
115C2×(C2×D3)⋊C4—32no—
116C2×(C2×D3)⋊C4—92no—
117C3×#95—91no—
118C3⋊#95—31no—
119C3⋊#95—91no—
120C23×C32—22no—
121C23×C32—22no—
122C23×C32—21no—
123C23×C32—22no—
124C23×C32—22no—
125C23×C32—21no—
126C23×C32—21no—
127C23×C32—22no—
128C23×C32—22no—
129C23×C32—21no—
130C23×C32—22no—
131C23×C32—21no—
132C23×C32—42no—
133C23×C32—42no—
134C23×C32D1212yescomplemented
135C23×C32C3×D412yescomplemented
136C23×C32C22×D311yescharacteristic
137C23×C32C23×C311yescharacteristic
138C22×C3⋊D3—22no—
139C22×C3⋊D3—22no—
140C22×C3⋊D3—22no—
141C22×C3⋊D3—22no—
142C22×C3⋊D3—22no—
143C22×C3⋊D3—22no—
144C22×C3⋊D3C3×D414yescomplemented
145C22×C3×D3—34no—
146C22×C3×D3—62no—
147C22×C3×D3—62no—
148C22×C3×D3—62no—
149C22×C3×D3—62no—
150C22×C3×D3—62no—
151C22×C3×D3—62no—
152C2×C32×C4—22no—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—22no—
156C2×C32×C4—21no—
157C2×C32×C4—22no—
158C2×C32×C4—21no—
159C2×C32×C4—22no—
160C2×C32×C4—42no—
161C2×C32×C4—42no—
162C2×C32×C4#6712yescomplemented
163C2×C32×C4C3×D412yescomplemented
164#638—216no—
165C2×C3⋊Dic3—22no—
166C2×C3⋊Dic3—22no—
167C2×C3⋊Dic3C3×D414yescomplemented
168C4⋊(C3⋊D3)—28no—
169C3×#67—616no—
170C2×C3×Dic3—34no—
171C2×C3×Dic3—62no—
172C2×C3×Dic3—62no—
173C3×D12—68no—
174#392—91noSylow
175C2×H(3,𝔽3)—42no—
176C2×H(3,𝔽3)#12012yes—
177C2×H(3,𝔽3)#12011yescharacteristic
178C2×H(3,𝔽3)#12112yes—
179C2×H(3,𝔽3)#12111yescharacteristic
180C2×H(3,𝔽3)#12711yescharacteristic
181(C3⋊D3)⋊C3—44no—
182C24×C3—31no—
183C24×C3—31no—
184C24×C3—61no—
185C24×C3—92no—
186C24×C3C2×C3×D311yescharacteristic
187C23×D3—32no—
188C23×D3—92no—
189C2×C3×D4—182no—
190C2×C3×D4—184no—
191C2×C3×D4—184no—
192C2×C3×D4—184no—
193C2×C3×D4—184no—
194C22×C3×C4—31no—
195C22×C3×C4—31no—
196C22×C3×C4—31no—
197C22×C3×C4—31no—
198C22×C3×C4—61no—
199C22×C3×C4—61no—
200C22×C3×C4—92no—
201C22×C3×C4C2×C3×D311yescharacteristic
202C22×C3×C4C2×C3×D311yescharacteristic
203C2×#67—64no—
204C2×#67—64no—
205C2×#67—64no—
206C2×#67—184no—
207C2×#67—184no—
208C2×#67—184no—
209C22×Dic3—32no—
210C22×Dic3—92no—
211C2×D12—64no—
212C2×D12—62no—
213C2×D12—182no—
214C2×D12—184no—
215C3×C22⋊C4—22no—
216C3×C22⋊C4—62no—
217C3×C22⋊C4—62no—
218C3×C22⋊C4—64no—
219C3×C22⋊C4—184no—
220(C2×D3)⋊C4—64no—
221(C2×D3)⋊C4—184no—
222C22×C32—24no—
223C22×C32—24no—
224C22×C32—22no—
225C22×C32—24no—
226C22×C32—22no—
227C22×C32—21no—
228C22×C32—21no—
229C22×C32—22no—
230C22×C32—22no—
231C22×C32—21no—
232C22×C32—44no—
233C22×C32—44no—
234C22×C32—42no—
235C22×C32—42no—
236C22×C32—42no—
237C22×C32—41no—
238C22×C32—41no—
239C22×C32—41no—
240C22×C32—42no—
241C22×C32—41no—
242C22×C32C2×D1211yescharacteristic
243C22×C32#23511yescharacteristic
244C22×C32D3×D412yes—
245C22×C32D3×D412yes—
246C22×C32C2×#6711yescharacteristic
247C22×C32C2×C3×D411yescharacteristic
248C22×C32C2×C3×D412yes—
249C22×C32C2×C3×D412yes—
250C22×C32C2×C3×D411yescharacteristic
251C22×C32C3×#4111yescharacteristic
252C2×C3⋊D3—24no—
253C2×C3⋊D3—24no—
254C2×C3⋊D3—24no—
255C2×C3⋊D3—42no—
256C2×C3⋊D3—42no—
257C2×C3⋊D3—42no—
258C2×C3⋊D3—42no—
259C2×C3×D3—64no—
260C2×C3×D3—64no—
261C2×C3×D3—64no—
262C2×C3×D3—122no—
263C2×C3×D3—122no—
264C2×C3×D3—122no—
265C2×C3×D3—122no—
266C32×C4—22no—
267C32×C4—22no—
268C32×C4—42no—
269C32×C4—42no—
270C32×C4—41no—
271C32×C4—41no—
272C3⋊Dic3—24no—
273C3×Dic3—64no—
274C22×D4—92no—
275C22×D4—91no—
276C2×C22⋊C4—92no—
277C2×C22⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
278#95—91no—
279H(3,𝔽3)#39211yescharacteristic, complemented, Sylow
280C23×C3—21no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—21no—
284C23×C3—32no—
285C23×C3—32no—
286C23×C3—31no—
287C23×C3—31no—
288C23×C3—61no—
289C23×C3—61no—
290C23×C3—62no—
291C23×C3—62no—
292C23×C3—62no—
293C23×C3—62no—
294C23×C3—61no—
295C23×C3—61no—
296C23×C3—62no—
297C23×C3—62no—
298C23×C3—62no—
299C23×C3—61no—
300C23×C3—94no—
301C23×C3—122no—
302C23×C3—122no—
303C23×C3—182no—
304C23×C3—182no—
305C23×C3—182no—
306C23×C3—182no—
307C23×C3—182no—
308C23×C3—182no—
309C23×C3C3×D1212yes—
310C23×C3C22×C3×D311yescharacteristic
311C22×D3—34no—
312C22×D3—62no—
313C22×D3—62no—
314C22×D3—62no—
315C22×D3—62no—
316C22×D3—62no—
317C22×D3—62no—
318C22×D3—94no—
319C22×D3—182no—
320C22×D3—182no—
321C22×D3—182no—
322C22×D3—182no—
323C22×D3—182no—
324C22×D3—182no—
325C3×D4—188no—
326C3×D4—1816no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—32no—
331C2×C3×C4—32no—
332C2×C3×C4—62no—
333C2×C3×C4—62no—
334C2×C3×C4—62no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—62no—
338C2×C3×C4—62no—
339C2×C3×C4—62no—
340C2×C3×C4—94no—
341C2×C3×C4—122no—
342C2×C3×C4—122no—
343C2×C3×C4—182no—
344C2×C3×C4—182no—
345C2×C3×C4C3×#6712yes—
346#67—616no—
347#67—1816no—
348C2×Dic3—34no—
349C2×Dic3—62no—
350C2×Dic3—62no—
351C2×Dic3—94no—
352C2×Dic3—182no—
353C2×Dic3—182no—
354D12—68no—
355D12—188no—
356C2×C32—22no—
357C2×C32—22no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—44no—
362C2×C32—42no—
363C2×C32—42no—
364C2×C32#88212yes—
365C2×C32#88412yes—
366C2×C32#93011yescharacteristic
367C2×C32#93711yescharacteristic
368C2×C32#94011yescharacteristic
369C2×C32C3×#12012yes—
370C2×C32C3×#12011yescharacteristic
371C2×C32C3×#12112yes—
372C2×C32C3×#12111yescharacteristic
373C2×C32C3×#12711yescharacteristic
374C3⋊D3—44no—
375C3×D3—124no—
376C24—92no—
377C24C2×(C3⋊D3)⋊C311yescharacteristic
378C2×D4—184no—
379C2×D4—184no—
380C2×D4—184no—
381C2×D4—184no—
382C2×D4—182no—
383C22×C4—92no—
384C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
385C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
386C22⋊C4—22no—
387C22⋊C4—184no—
388C22×C3—24no—
389C22×C3—22no—
390C22×C3—31no—
391C22×C3—31no—
392C22×C3—31no—
393C22×C3—31no—
394C22×C3—32no—
395C22×C3—32no—
396C22×C3—32no—
397C22×C3—32no—
398C22×C3—31no—
399C22×C3—31no—
400C22×C3—42no—
401C22×C3—41no—
402C22×C3—41no—
403C22×C3—64no—
404C22×C3—64no—
405C22×C3—62no—
406C22×C3—62no—
407C22×C3—64no—
408C22×C3—61no—
409C22×C3—61no—
410C22×C3—62no—
411C22×C3—62no—
412C22×C3—61no—
413C22×C3—124no—
414C22×C3—121no—
415C22×C3—121no—
416C22×C3—121no—
417C22×C3—121no—
418C22×C3—122no—
419C22×C3—122no—
420C22×C3—122no—
421C22×C3—122no—
422C22×C3—124no—
423C22×C3—184no—
424C22×C3—184no—
425C22×C3—184no—
426C22×C3—362no—
427C22×C3—362no—
428C22×C3—362no—
429C22×C3—362no—
430C22×C3C2×C3×D1211yescharacteristic
431C22×C3C3×#23511yescharacteristic
432C22×C3C3×D3×D412yes—
433C22×C3C3×D3×D412yes—
434C22×C3C2×C3×#6711yescharacteristic
435C2×D3—64no—
436C2×D3—64no—
437C2×D3—64no—
438C2×D3—122no—
439C2×D3—122no—
440C2×D3—122no—
441C2×D3—122no—
442C2×D3—184no—
443C2×D3—184no—
444C2×D3—184no—
445C2×D3—362no—
446C2×D3—362no—
447C2×D3—362no—
448C2×D3—362no—
449C3×C4—22no—
450C3×C4—41no—
451C3×C4—62no—
452C3×C4—62no—
453C3×C4—122no—
454C3×C4—121no—
455C3×C4—121no—
456C3×C4—122no—
457C3×C4—184no—
458Dic3—64no—
459Dic3—184no—
460C32—21no—
461C32#496011yescharacteristic, complemented
462C32C3×#39211yescharacteristic, complemented
463C23—21no—
464C23—22no—
465C23—22no—
466C23—21no—
467C23—94no—
468C23—182no—
469C23—182no—
470C23—182no—
471C23—182no—
472C23—182no—
473C23—182no—
474C23C4⋊((C3⋊D3)⋊C3)12yescomplemented
475C23C22×(C3⋊D3)⋊C311yescharacteristic
476D4—188no—
477D4—1816no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—94no—
482C2×C4—182no—
483C2×C4—182no—
484C2×C4#638⋊C312yescomplemented
485C2×C3—31no—
486C2×C3—31no—
487C2×C3—31no—
488C2×C3—31no—
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—32no—
492C2×C3—32no—
493C2×C3—32no—
494C2×C3—32no—
495C2×C3—42no—
496C2×C3—61no—
497C2×C3—61no—
498C2×C3—61no—
499C2×C3—62no—
500C2×C3—62no—
501C2×C3—122no—
502C2×C3—122no—
503C2×C3—124no—
504C2×C3—364no—
505C2×C3C3×#88212yes—
506C2×C3C3×#88412yes—
507C2×C3C3×#93011yescharacteristic
508C2×C3C3×#93711yescharacteristic
509C2×C3C3×#94011yescharacteristic
510D3—124no—
511D3—364no—
512C22—22no—
513C22—24no—
514C22—41no—
515C22—41no—
516C22—42no—
517C22—184no—
518C22—184no—
519C22—184no—
520C22—362no—
521C22—362no—
522C22—362no—
523C22—362no—
524C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
525C22#3828⋊C311yescharacteristic
526C22D4×(C3⋊D3)⋊C312yes—
527C22D4×(C3⋊D3)⋊C312yes—
528C22C2×#638⋊C311yescharacteristic
529C4—22no—
530C4—41no—
531C4—184no—
532C3—31no—
533C3—31no—
534C3—61no—
535C3C3×#496011yescharacteristic
536C2—42no—
537C2—364no—
538C2#64091⋊C312yes—
539C2#64093⋊C312yes—
540C2#64139⋊C311yescharacteristic
541C2#64146⋊C311yescharacteristic
542C2#64149⋊C311yescharacteristic
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutator, cosocleC23×C32C23×C3
Frattini, socleC23×C3C22×C3×D3
FittingC2×C22⋊C4×H(3,𝔽3)C2
hypercenterC2×C22⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#392
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#638⋊C3C2semidirect
2. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. C2×C22⋊C4×H(3,𝔽3)C2semidirect
4. C2×#3752⋊C3C2semidirect
5. #63658⋊C3C2semidirect
6. #10592659C3semidirect
20. C22×#638C2×C3semidirect
21. C22×C4⋊(C3⋊D3)C2×C3semidirect
25. C2×C32×C22⋊C4D3semidirect
26. C2×C32×C22⋊C4C2×C3semidirect
27. C2×#3752C2×C3semidirect
29. #63658C2×C3semidirect
35. C23×H(3,𝔽3)D4semidirect
43. C22×(C3⋊D3)⋊C3D4semidirect
47. C2×C4×H(3,𝔽3)D4semidirect
51. C2×(C3⋊Dic3)⋊C3D4semidirect
134. C23×C32D12semidirect
135. C23×C32C3×D4semidirect
144. C22×C3⋊D3C3×D4semidirect
162. C2×C32×C4#67semidirect
163. C2×C32×C4C3×D4semidirect
167. C2×C3⋊Dic3C3×D4semidirect
277. C2×C22⋊C4(C3⋊D3)⋊C3semidirect
279. H(3,𝔽3)#392semidirect
461. C32#4960semidirect
462. C32C3×#392semidirect
474. C23C4⋊((C3⋊D3)⋊C3)semidirect
484. C2×C4#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×#638⋊C32
C22×C4⋊((C3⋊D3)⋊C3)2
C2×C22⋊C4×H(3,𝔽3)2
C2×#3752⋊C32
#63658⋊C32
#105926593
C3×#49603