Finite Group Databasebeta

← #10592657⋊C3 #13142507

Subgroups — #10592657⋊C3 #13142507

Order 1728.

1133 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#638⋊C3C211yesmaximal, characteristic
2C23×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4C2×#3752⋊C3C212yesmaximal, complemented
5#63658⋊C3C212yesmaximal, complemented
6#10592657C311yesmaximal, characteristic, complemented
7C3×#4958—31nomaximal
8C24×H(3,𝔽3)C2211yescharacteristic
9C23×(C3⋊D3)⋊C3C2211yescharacteristic
10C22×C4×H(3,𝔽3)C2212yes—
11C2×#638⋊C3—22no—
12C2×#638⋊C3—22no—
13C2×#638⋊C3C414yescomplemented
14C22×(C3⋊Dic3)⋊C3—22no—
15C22×(C3⋊Dic3)⋊C3—24no—
16C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
17C22×(C3⋊Dic3)⋊C3C2212yes—
18C22⋊C4×H(3,𝔽3)—22no—
19#3752⋊C3—24no—
20C22×#638C2×C311yescharacteristic
21C23×C3⋊Dic3C2×C311yescharacteristic, complemented
22C22×C3×#67—31no—
23C23×C3×Dic3—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×C312yescomplemented
30C3×C3⋊#95—32no—
31C23×H(3,𝔽3)—22no—
32C23×H(3,𝔽3)—22no—
33C23×H(3,𝔽3)C2×C412yes—
34C23×H(3,𝔽3)D412yes—
35C23×H(3,𝔽3)D412yes—
36C23×H(3,𝔽3)C2311yescharacteristic
37C22×(C3⋊D3)⋊C3—21no—
38C22×(C3⋊D3)⋊C3—21no—
39C22×(C3⋊D3)⋊C3—22no—
40C22×(C3⋊D3)⋊C3—22no—
41C22×(C3⋊D3)⋊C3C2×C412yescomplemented
42C2×C4×H(3,𝔽3)—22no—
43C2×C4×H(3,𝔽3)—22no—
44C2×C4×H(3,𝔽3)—22no—
45#638⋊C3—28no—
46C2×(C3⋊Dic3)⋊C3—28no—
47C2×(C3⋊Dic3)⋊C3—22no—
48C2×(C3⋊Dic3)⋊C3—22no—
49C2×(C3⋊Dic3)⋊C3—21no—
50C2×(C3⋊Dic3)⋊C3—21no—
51C2×(C3⋊Dic3)⋊C3C2×C412yescomplemented
52C2×(C3⋊Dic3)⋊C3D414yescomplemented
53C3×#386—91no—
54#4958—31no—
55#4958—91no—
56C24×C32—21no—
57C24×C32C2×D311yescharacteristic
58C24×C32C22×C311yescharacteristic
59C23×C3⋊D3C22×C311yescharacteristic
60C23×C3×D3—31no—
61C22×C32×C4—22no—
62C22×C32×C4C2×D312yes—
63C22×C32×C4C22×C312yes—
64C2×#638—22no—
65C2×#638—22no—
66C2×#638C3×C414yescomplemented
67C22×C3⋊Dic3—22no—
68C22×C3⋊Dic3—24no—
69C22×C3⋊Dic3C22×C311yescharacteristic
70C22×C3⋊Dic3C22×C312yes—
71C2×C3×#67—34no—
72C2×C3×#67—62no—
73C2×C3×#67—62no—
74C22×C3×Dic3—32no—
75C22×C3×Dic3—31no—
76C22×C3×Dic3—64no—
77C22×C3×Dic3—62no—
78C32×C22⋊C4—24no—
79C32×C22⋊C4—22no—
80C32×C22⋊C4—22no—
81#3752—24no—
82C3×(C2×D3)⋊C4—64no—
83C22×H(3,𝔽3)—24no—
84C22×H(3,𝔽3)—22no—
85C22×H(3,𝔽3)—24no—
86C22×H(3,𝔽3)—22no—
87C22×H(3,𝔽3)C22⋊C414yescomplemented
88C22×H(3,𝔽3)C22×C411yescharacteristic
89C22×H(3,𝔽3)C2×D411yescharacteristic
90C22×H(3,𝔽3)C2×D411yescharacteristic
91C22×H(3,𝔽3)C2×D412yes—
92C22×H(3,𝔽3)#4112yes—
93C2×(C3⋊D3)⋊C3—22no—
94C2×(C3⋊D3)⋊C3—24no—
95C2×(C3⋊D3)⋊C3—41no—
96C2×(C3⋊D3)⋊C3—42no—
97C2×(C3⋊D3)⋊C3—41no—
98C4×H(3,𝔽3)—42no—
99(C3⋊Dic3)⋊C3—22no—
100(C3⋊Dic3)⋊C3—24no—
101C22×C3×D4—91no—
102C23×C3×C4—91no—
103C22×#67—31no—
104C22×#67—91no—
105C23×Dic3—31no—
106C23×Dic3—91no—
107C2×C3×C22⋊C4—31no—
108C2×C3×C22⋊C4—31no—
109C2×C3×C22⋊C4—61no—
110C2×C3×C22⋊C4—92no—
111C2×C3×C22⋊C4C3×D311yescharacteristic
112C2×(C2×D3)⋊C4—32no—
113C2×(C2×D3)⋊C4—92no—
114C3×#95—92no—
115C3⋊#95—32no—
116C3⋊#95—92no—
117C23×C32—22no—
118C23×C32—22no—
119C23×C32—22no—
120C23×C32—22no—
121C23×C32—22no—
122C23×C32—22no—
123C23×C32—22no—
124C23×C32—21no—
125C23×C32—42no—
126C23×C32—42no—
127C23×C32C4×D312yes—
128C23×C32D1212yes—
129C23×C32#6712yes—
130C23×C32C2×C3×C412yes—
131C23×C32C3×D412yes—
132C23×C32C3×D412yes—
133C23×C32C22×D311yescharacteristic
134C23×C32C23×C311yescharacteristic
135C22×C3⋊D3—22no—
136C22×C3⋊D3—21no—
137C22×C3⋊D3—21no—
138C22×C3⋊D3—22no—
139C22×C3⋊D3C2×C3×C412yescomplemented
140C22×C3×D3—32no—
141C22×C3×D3—61no—
142C22×C3×D3—62no—
143C22×C3×D3—62no—
144C22×C3×D3—61no—
145C2×C32×C4—24no—
146C2×C32×C4—22no—
147C2×C32×C4—22no—
148C2×C32×C4—22no—
149C2×C32×C4—22no—
150C2×C32×C4—22no—
151C2×C32×C4—22no—
152C2×C32×C4—42no—
153C2×C32×C4—42no—
154#638—28no—
155C2×C3⋊Dic3—28no—
156C2×C3⋊Dic3—22no—
157C2×C3⋊Dic3—22no—
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—21no—
160C2×C3⋊Dic3C2×C3×C412yescomplemented
161C2×C3⋊Dic3C3×D414yescomplemented
162C3×#67—68no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—34no—
165C2×C3×Dic3—68no—
166C2×C3×Dic3—62no—
167C2×C3×Dic3—61no—
168C2×C3×Dic3—62no—
169C2×C3×Dic3—61no—
170#386—91noSylow
171C2×H(3,𝔽3)—24no—
172C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
173C2×H(3,𝔽3)C4×D412yes—
174C2×H(3,𝔽3)#12011yescharacteristic
175C2×H(3,𝔽3)#12112yes—
176C2×H(3,𝔽3)#12311yescharacteristic
177(C3⋊D3)⋊C3—42no—
178C24×C3—31no—
179C24×C3—31no—
180C24×C3—61no—
181C24×C3—91no—
182C24×C3C2×C3×D311yescharacteristic
183C23×D3—31no—
184C23×D3—91no—
185C2×C3×D4—94no—
186C2×C3×D4—182no—
187C2×C3×D4—182no—
188C22×C3×C4—32no—
189C22×C3×C4—32no—
190C22×C3×C4—62no—
191C22×C3×C4—91no—
192C22×C3×C4—92no—
193C22×C3×C4—184no—
194C22×C3×C4—182no—
195C22×C3×C4C2×C3×D312yes—
196C2×#67—34no—
197C2×#67—62no—
198C2×#67—62no—
199C2×#67—94no—
200C2×#67—182no—
201C2×#67—182no—
202C22×Dic3—31no—
203C22×Dic3—32no—
204C22×Dic3—64no—
205C22×Dic3—62no—
206C22×Dic3—91no—
207C22×Dic3—92no—
208C22×Dic3—184no—
209C22×Dic3—182no—
210C3×C22⋊C4—22no—
211C3×C22⋊C4—62no—
212C3×C22⋊C4—62no—
213C3×C22⋊C4—64no—
214C3×C22⋊C4—184no—
215(C2×D3)⋊C4—64no—
216(C2×D3)⋊C4—184no—
217C22×C32—24no—
218C22×C32—22no—
219C22×C32—22no—
220C22×C32—24no—
221C22×C32—24no—
222C22×C32—22no—
223C22×C32—24no—
224C22×C32—22no—
225C22×C32—24no—
226C22×C32—21no—
227C22×C32—21no—
228C22×C32—22no—
229C22×C32—22no—
230C22×C32—21no—
231C22×C32—44no—
232C22×C32—42no—
233C22×C32—42no—
234C22×C32—44no—
235C22×C32(C2×D3)⋊C414yescomplemented
236C22×C32C3×C22⋊C414yescomplemented
237C22×C32C2×C4×D311yescharacteristic
238C22×C32C2×D1211yescharacteristic
239C22×C32D3×D412yes—
240C22×C32#23712yes—
241C22×C32C2×#6711yescharacteristic
242C22×C32C22×C3×C411yescharacteristic
243C22×C32C2×C3×D411yescharacteristic
244C22×C32C2×C3×D412yes—
245C22×C32C2×C3×D411yescharacteristic
246C22×C32C3×#4112yes—
247C2×C3⋊D3—22no—
248C2×C3⋊D3—24no—
249C2×C3⋊D3—41no—
250C2×C3⋊D3—42no—
251C2×C3⋊D3—41no—
252C2×C3×D3—64no—
253C2×C3×D3—62no—
254C2×C3×D3—121no—
255C2×C3×D3—121no—
256C2×C3×D3—122no—
257C32×C4—44no—
258C32×C4—42no—
259C32×C4—42no—
260C3⋊Dic3—22no—
261C3⋊Dic3—24no—
262C3×Dic3—64no—
263C3×Dic3—62no—
264C22×D4—91no—
265C23×C4—91no—
266C2×C22⋊C4—92no—
267C2×C22⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
268#95—92no—
269H(3,𝔽3)#38611yescharacteristic, complemented, Sylow
270C23×C3—22no—
271C23×C3—22no—
272C23×C3—32no—
273C23×C3—32no—
274C23×C3—32no—
275C23×C3—32no—
276C23×C3—32no—
277C23×C3—32no—
278C23×C3—31no—
279C23×C3—31no—
280C23×C3—62no—
281C23×C3—62no—
282C23×C3—62no—
283C23×C3—62no—
284C23×C3—62no—
285C23×C3—62no—
286C23×C3—62no—
287C23×C3—61no—
288C23×C3—92no—
289C23×C3—122no—
290C23×C3—122no—
291C23×C3—182no—
292C23×C3—181no—
293C23×C3—181no—
294C23×C3—182no—
295C23×C3C3×C4×D312yes—
296C23×C3C3×D1212yes—
297C23×C3C3×#6712yes—
298C23×C3C22×C3×D311yescharacteristic
299C22×D3—32no—
300C22×D3—61no—
301C22×D3—62no—
302C22×D3—62no—
303C22×D3—61no—
304C22×D3—92no—
305C22×D3—182no—
306C22×D3—181no—
307C22×D3—181no—
308C22×D3—182no—
309C3×D4—188no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—62no—
314C2×C3×C4—62no—
315C2×C3×C4—62no—
316C2×C3×C4—62no—
317C2×C3×C4—62no—
318C2×C3×C4—62no—
319C2×C3×C4—64no—
320C2×C3×C4—94no—
321C2×C3×C4—92no—
322C2×C3×C4—122no—
323C2×C3×C4—122no—
324C2×C3×C4—182no—
325C2×C3×C4—182no—
326C2×C3×C4—181no—
327C2×C3×C4—181no—
328C2×C3×C4—188no—
329#67—68no—
330#67—188no—
331C2×Dic3—32no—
332C2×Dic3—34no—
333C2×Dic3—68no—
334C2×Dic3—62no—
335C2×Dic3—61no—
336C2×Dic3—62no—
337C2×Dic3—61no—
338C2×Dic3—92no—
339C2×Dic3—94no—
340C2×Dic3—182no—
341C2×Dic3—182no—
342C2×Dic3—181no—
343C2×Dic3—181no—
344C2×Dic3—188no—
345C2×C32—24no—
346C2×C32—24no—
347C2×C32—22no—
348C2×C32—21no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—22no—
352C2×C32—44no—
353C2×C32#67⋊C412yes—
354C2×C32#88211yescharacteristic
355C2×C32#88611yescharacteristic
356C2×C32C2×(C2×D3)⋊C411yescharacteristic
357C2×C32#93912yes—
358C2×C32C2×C3×C22⋊C411yescharacteristic
359C2×C32C3×C4×D412yes—
360C2×C32C3×#12011yescharacteristic
361C2×C32C3×#12112yes—
362C2×C32C3×#12311yescharacteristic
363C3⋊D3—42no—
364C3×D3—122no—
365C24—91no—
366C24C2×(C3⋊D3)⋊C311yescharacteristic
367C2×D4—94no—
368C2×D4—182no—
369C2×D4—182no—
370C22×C4—91no—
371C22×C4—92no—
372C22×C4—184no—
373C22×C4—182no—
374C22×C4C2×(C3⋊D3)⋊C312yes—
375C22⋊C4—22no—
376C22⋊C4—184no—
377C22×C3—24no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—24no—
381C22×C3—31no—
382C22×C3—31no—
383C22×C3—31no—
384C22×C3—31no—
385C22×C3—32no—
386C22×C3—32no—
387C22×C3—32no—
388C22×C3—32no—
389C22×C3—31no—
390C22×C3—31no—
391C22×C3—34no—
392C22×C3—34no—
393C22×C3—62no—
394C22×C3—62no—
395C22×C3—64no—
396C22×C3—64no—
397C22×C3—64no—
398C22×C3—64no—
399C22×C3—62no—
400C22×C3—62no—
401C22×C3—61no—
402C22×C3—61no—
403C22×C3—62no—
404C22×C3—62no—
405C22×C3—61no—
406C22×C3—64no—
407C22×C3—124no—
408C22×C3—124no—
409C22×C3—122no—
410C22×C3—122no—
411C22×C3—182no—
412C22×C3—184no—
413C22×C3—362no—
414C22×C3—361no—
415C22×C3—361no—
416C22×C3C3×(C2×D3)⋊C414yes—
417C22×C3C2×C3×C4×D311yescharacteristic
418C22×C3C2×C3×D1211yescharacteristic
419C22×C3C3×D3×D412yes—
420C22×C3C3×#23712yes—
421C22×C3C2×C3×#6711yescharacteristic
422C2×D3—64no—
423C2×D3—62no—
424C2×D3—121no—
425C2×D3—121no—
426C2×D3—122no—
427C2×D3—182no—
428C2×D3—184no—
429C2×D3—362no—
430C2×D3—361no—
431C2×D3—361no—
432C3×C4—42no—
433C3×C4—122no—
434C3×C4—122no—
435C3×C4—124no—
436C3×C4—184no—
437C3×C4—182no—
438Dic3—64no—
439Dic3—62no—
440Dic3—184no—
441Dic3—182no—
442C32—21no—
443C32#495811yescharacteristic, complemented
444C32C3×#38611yescharacteristic, complemented
445C23—22no—
446C23—22no—
447C23—92no—
448C23—181no—
449C23—182no—
450C23—182no—
451C23—181no—
452C23C4×(C3⋊D3)⋊C312yes—
453C23C4⋊((C3⋊D3)⋊C3)12yes—
454C23#638⋊C312yes—
455C23C22×(C3⋊D3)⋊C311yescharacteristic
456D4—188no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—92no—
461C2×C4—94no—
462C2×C4—188no—
463C2×C4—182no—
464C2×C4—181no—
465C2×C4—182no—
466C2×C4—181no—
467C2×C3—24no—
468C2×C3—31no—
469C2×C3—31no—
470C2×C3—32no—
471C2×C3—32no—
472C2×C3—32no—
473C2×C3—32no—
474C2×C3—31no—
475C2×C3—31no—
476C2×C3—31no—
477C2×C3—31no—
478C2×C3—64no—
479C2×C3—64no—
480C2×C3—61no—
481C2×C3—62no—
482C2×C3—62no—
483C2×C3—61no—
484C2×C3—61no—
485C2×C3—124no—
486C2×C3—362no—
487C2×C3C3×#67⋊C412yes—
488C2×C3C3×#88211yescharacteristic
489C2×C3C3×#88611yescharacteristic
490C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
491C2×C3C3×#93912yes—
492D3—122no—
493D3—362no—
494C22—24no—
495C22—24no—
496C22—22no—
497C22—22no—
498C22—182no—
499C22—184no—
500C22—362no—
501C22—361no—
502C22—361no—
503C22#3752⋊C314yescomplemented
504C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
505C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
506C22D4×(C3⋊D3)⋊C312yes—
507C22#3830⋊C312yes—
508C22C2×#638⋊C311yescharacteristic
509C4—42no—
510C4—182no—
511C4—184no—
512C3—31no—
513C3—31no—
514C3—61no—
515C3C3×#495811yescharacteristic
516C2—24no—
517C2—362no—
518C2#64090⋊C312yes—
519C2#64091⋊C311yescharacteristic
520C2#64095⋊C311yescharacteristic
521C2C2×#3752⋊C311yescharacteristic
522C2#64148⋊C312yes—
subgroupG/H
centerC23C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
Frattini, socleC23×C3C22×C3×D3
FittingC2×C22⋊C4×H(3,𝔽3)C2
prime radicalC22×#638⋊C3C2
cosocleC23×C32C23×C3
hypercenterC2×C22⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#386
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C23×(C3⋊Dic3)⋊C3C2semidirect
3. C2×C22⋊C4×H(3,𝔽3)C2semidirect
4. C2×#3752⋊C3C2semidirect
5. #63658⋊C3C2semidirect
6. #10592657C3semidirect
13. C2×#638⋊C3C4semidirect
21. C23×C3⋊Dic3C2×C3semidirect
25. C2×C32×C22⋊C4D3semidirect
26. C2×C32×C22⋊C4C2×C3semidirect
27. C2×#3752C2×C3semidirect
29. #63658C2×C3semidirect
41. C22×(C3⋊D3)⋊C3C2×C4semidirect
51. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
52. C2×(C3⋊Dic3)⋊C3D4semidirect
66. C2×#638C3×C4semidirect
87. C22×H(3,𝔽3)C22⋊C4semidirect
139. C22×C3⋊D3C2×C3×C4semidirect
160. C2×C3⋊Dic3C2×C3×C4semidirect
161. C2×C3⋊Dic3C3×D4semidirect
235. C22×C32(C2×D3)⋊C4semidirect
236. C22×C32C3×C22⋊C4semidirect
267. C2×C22⋊C4(C3⋊D3)⋊C3semidirect
269. H(3,𝔽3)#386semidirect
443. C32#4958semidirect
444. C32C3×#386semidirect
503. C22#3752⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×#638⋊C32
C23×(C3⋊Dic3)⋊C32
C2×C22⋊C4×H(3,𝔽3)2
C2×#3752⋊C32
#63658⋊C32
#105926573
C3×#49583