Finite Group Databasebeta

← #117×(C3⋊D3)⋊C3 #13152514

Subgroups — #117×(C3⋊D3)⋊C3 #13152514

Order 1728.

1231 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
3#64135⋊C3C211yesmaximal, characteristic, complemented
4#64101⋊C3C212yesmaximal, complemented
5C4⋊C4×(C3⋊D3)⋊C3C212yesmaximal, complemented
6C22⋊C4×(C3⋊D3)⋊C3C212yesmaximal
7#64086⋊C3C212yesmaximal, complemented
8#64081⋊C3C212yesmaximal, complemented
9C42×(C3⋊D3)⋊C3C212yesmaximal, complemented
10C3⋊D3×#117C311yesmaximal, characteristic, complemented
11C3×D3×#117—31nomaximal
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C418yescomplemented
16C2×C4×(C3⋊D3)⋊C3C2214yes—
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
19C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
20C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
21C4⋊C4×H(3,𝔽3)C2212yescomplemented
22C22⋊C4×H(3,𝔽3)C2212yes—
23C42×H(3,𝔽3)C2212yescomplemented
24#3757⋊C3C2212yes—
25#3752⋊C3C2214yes—
26C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
27#3750⋊C3C2214yescomplemented
28C4×(C3⋊Dic3)⋊C3C2212yescomplemented
29C4×(C3⋊Dic3)⋊C3C2214yescomplemented
30C22×C4×C3⋊D3C2×C311yescharacteristic
31C22×C3×C4×D3—31no—
32C32×#117—21no—
33C32×#117D311yescharacteristic, complemented
34C32×#117C2×C311yescharacteristic, complemented
35#64135C2×C311yescharacteristic, complemented
36#64101C2×C312yescomplemented
37C4⋊C4×C3⋊D3C2×C312yescomplemented
38C22⋊C4×C3⋊D3C2×C312yes—
39#64086C2×C312yescomplemented
40#64081C2×C312yescomplemented
41C42×C3⋊D3C2×C312yescomplemented
42C3×#926—31no—
43C3×#892—32no—
44C3×D3×C4⋊C4—32no—
45C3×D3×C22⋊C4—32no—
46C3×#877—32no—
47C3×#872—32no—
48C3×C42×D3—32no—
49C23×H(3,𝔽3)C2311yescharacteristic
50C22×(C3⋊D3)⋊C3—24no—
51C22×(C3⋊D3)⋊C3C2×C414yes—
52C22×(C3⋊D3)⋊C3C2311yescharacteristic
53C22×(C3⋊D3)⋊C3C2311yescharacteristic
54C2×C4×H(3,𝔽3)C2×C414yescomplemented
55C2×C4×H(3,𝔽3)C2314yes—
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57C2×C4×H(3,𝔽3)C2311yescharacteristic
58C2×(C3⋊Dic3)⋊C3C2×C414yescomplemented
59C2×(C3⋊Dic3)⋊C3C2314yes—
60C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
61C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
62C4×(C3⋊D3)⋊C3—28no—
63C4×(C3⋊D3)⋊C3C2×C414yes—
64C4×(C3⋊D3)⋊C3C2×C414yes—
65C4×(C3⋊D3)⋊C3C2×C414yescomplemented
66C4×(C3⋊D3)⋊C3C2×C414yescomplemented
67C2×C3×#117—91no—
68D3×#117—31no—
69D3×#117—91no—
70C23×C3⋊D3C22×C311yescharacteristic
71C23×C3×D3—31no—
72C22×C32×C4—21no—
73C22×C32×C4C2×D311yescharacteristic
74C22×C32×C4C22×C311yescharacteristic
75C22×C3⋊Dic3C22×C311yescharacteristic
76C2×C4×C3⋊D3C3×C418yescomplemented
77C2×C4×C3⋊D3C22×C314yes—
78C2×C4×C3⋊D3C22×C311yescharacteristic
79C2×C4×C3⋊D3C22×C311yescharacteristic
80C2×C4×C3⋊D3C22×C311yescharacteristic
81C2×C4×C3⋊D3C22×C311yescharacteristic
82C22×C3×Dic3—31no—
83C2×C3×C4×D3—38no—
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—31no—
88C2×C3×C4×D3—34no—
89C32×C4⋊C4—22no—
90C32×C4⋊C4C2×D312yescomplemented
91C32×C4⋊C4C22×C312yescomplemented
92C32×C22⋊C4—22no—
93C32×C22⋊C4C2×D312yes—
94C32×C22⋊C4C22×C312yes—
95C32×C42—22no—
96C32×C42C2×D312yescomplemented
97C32×C42C22×C312yescomplemented
98#3757C22×C312yes—
99#3752C22×C314yes—
100C4⋊(C3⋊Dic3)C22×C312yescomplemented
101#3750C22×C314yescomplemented
102C4×C3⋊Dic3C22×C312yescomplemented
103C4×C3⋊Dic3C22×C314yescomplemented
104C3×#217—32no—
105C3×(C2×D3)⋊C4—34no—
106C3×C4⋊Dic3—32no—
107C3×Dic3⋊C4—34no—
108C3×C4×Dic3—34no—
109C3×C4×Dic3—32no—
110C22×H(3,𝔽3)—22no—
111C22×H(3,𝔽3)C22×C412yes—
112C22×H(3,𝔽3)C2411yescharacteristic
113C2×(C3⋊D3)⋊C3—22no—
114C2×(C3⋊D3)⋊C3—28no—
115C2×(C3⋊D3)⋊C3C22×C412yes—
116C2×(C3⋊D3)⋊C3C22×C412yes—
117C2×(C3⋊D3)⋊C3#4114yes—
118C4×H(3,𝔽3)—24no—
119C4×H(3,𝔽3)C22×C412yes—
120C4×H(3,𝔽3)C22×C412yes—
121(C3⋊Dic3)⋊C3—24no—
122(C3⋊Dic3)⋊C3C22×C412yes—
123(C3⋊Dic3)⋊C3C22×C412yes—
124C23×C3×C4—91no—
125C22×C4×D3—31no—
126C22×C4×D3—91no—
127C3×#117—31no—
128C3×#117—31no—
129C3×#117—61no—
130C3×#117—92no—
131C3×#117—92no—
132C3×#117—92no—
133C3×#117—91no—
134C3×#117C3×D311yescharacteristic
135C2×C3×C4⋊C4—92no—
136C2×C3×C22⋊C4—92no—
137C2×C3×C42—92no—
138#926—31no—
139#926—91no—
140#892—32no—
141#892—92no—
142D3×C4⋊C4—32no—
143D3×C4⋊C4—92no—
144D3×C22⋊C4—32no—
145D3×C22⋊C4—92no—
146#877—32no—
147#877—92no—
148#872—32no—
149#872—92no—
150C42×D3—32no—
151C42×D3—92no—
152C23×C32—21no—
153C23×C32C22×D311yescharacteristic
154C23×C32C23×C311yescharacteristic
155C22×C3⋊D3—24no—
156C22×C3⋊D3C2×C3×C414yes—
157C22×C3⋊D3C23×C311yescharacteristic
158C22×C3⋊D3C23×C311yescharacteristic
159C22×C3×D3—31no—
160C22×C3×D3—34no—
161C22×C3×D3—31no—
162C22×C3×D3—64no—
163C2×C32×C4—24no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—24no—
167C2×C32×C4C4×D314yescomplemented
168C2×C32×C4C2×C3×C414yescomplemented
169C2×C32×C4C22×D314yes—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C23×C314yes—
173C2×C32×C4C23×C311yescharacteristic
174C2×C32×C4C23×C311yescharacteristic
175C2×C3⋊Dic3C2×C3×C414yescomplemented
176C2×C3⋊Dic3C23×C314yes—
177C2×C3⋊Dic3C23×C311yescharacteristic
178C2×C3⋊Dic3C23×C311yescharacteristic
179C4×C3⋊D3—28no—
180C4×C3⋊D3C2×C3×C414yes—
181C4×C3⋊D3C2×C3×C414yes—
182C4×C3⋊D3C2×C3×C414yescomplemented
183C4×C3⋊D3C2×C3×C414yescomplemented
184C2×C3×Dic3—34no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—31no—
187C2×C3×Dic3—34no—
188C3×C4×D3—34no—
189C3×C4×D3—34no—
190C3×C4×D3—34no—
191C3×C4×D3—34no—
192C3×C4×D3—68no—
193C2×#117—91noSylow
194C2×H(3,𝔽3)—22no—
195C2×H(3,𝔽3)C23×C411yescharacteristic
196C2×H(3,𝔽3)C2×#4112yes—
197(C3⋊D3)⋊C3—22no—
198(C3⋊D3)⋊C3#11714yesdirect product factor
199C24×C3—91no—
200C23×D3—31no—
201C23×D3—91no—
202C22×C3×C4—31no—
203C22×C3×C4—31no—
204C22×C3×C4—61no—
205C22×C3×C4—91no—
206C22×C3×C4—91no—
207C22×C3×C4—91no—
208C22×C3×C4—91no—
209C22×C3×C4—98no—
210C22×C3×C4—94no—
211C22×C3×C4—91no—
212C22×C3×C4C2×C3×D311yescharacteristic
213C22×Dic3—31no—
214C22×Dic3—91no—
215C2×C4×D3—38no—
216C2×C4×D3—31no—
217C2×C4×D3—31no—
218C2×C4×D3—31no—
219C2×C4×D3—31no—
220C2×C4×D3—34no—
221C2×C4×D3—91no—
222C2×C4×D3—91no—
223C2×C4×D3—91no—
224C2×C4×D3—91no—
225C2×C4×D3—98no—
226C2×C4×D3—94no—
227C3×C4⋊C4—32no—
228C3×C4⋊C4—32no—
229C3×C4⋊C4—62no—
230C3×C4⋊C4—94no—
231C3×C4⋊C4—92no—
232C3×C4⋊C4C2×C3×D312yes—
233C3×C22⋊C4—32no—
234C3×C22⋊C4—32no—
235C3×C22⋊C4—62no—
236C3×C22⋊C4—94no—
237C3×C22⋊C4—92no—
238C3×C22⋊C4C2×C3×D312yes—
239C3×C42—32no—
240C3×C42—32no—
241C3×C42—62no—
242C3×C42—94no—
243C3×C42—92no—
244C3×C42C2×C3×D312yes—
245#217—32no—
246#217—92no—
247(C2×D3)⋊C4—34no—
248(C2×D3)⋊C4—94no—
249C4⋊Dic3—32no—
250C4⋊Dic3—92no—
251Dic3⋊C4—34no—
252Dic3⋊C4—94no—
253C4×Dic3—32no—
254C4×Dic3—34no—
255C4×Dic3—92no—
256C4×Dic3—94no—
257C22×C32—22no—
258C22×C32—22no—
259C22×C32—21no—
260C22×C32—22no—
261C22×C32—42no—
262C22×C32C2×C4×D312yes—
263C22×C32C22×C3×C412yes—
264C22×C32C23×D311yescharacteristic
265C22×C32C24×C311yescharacteristic
266C2×C3⋊D3—22no—
267C2×C3⋊D3—28no—
268C2×C3⋊D3C22×C3×C412yes—
269C2×C3⋊D3C22×C3×C412yes—
270C2×C3⋊D3C3×#4114yes—
271C2×C3×D3—32no—
272C2×C3×D3—34no—
273C2×C3×D3—32no—
274C2×C3×D3—68no—
275C2×C3×D3—62no—
276C32×C4—24no—
277C32×C4—24no—
278C32×C4—22no—
279C32×C4—22no—
280C32×C4—44no—
281C32×C4C2×C4×D312yes—
282C32×C4C2×C4×D312yes—
283C32×C4C22×C3×C412yes—
284C32×C4C22×C3×C412yes—
285C3⋊Dic3—24no—
286C3⋊Dic3C22×C3×C412yes—
287C3⋊Dic3C22×C3×C412yes—
288C3×Dic3—32no—
289C3×Dic3—32no—
290C3×Dic3—64no—
291C23×C4—91no—
292#117—91no—
293#117—92no—
294#117—92no—
295#117—92no—
296#117(C3⋊D3)⋊C311yescharacteristic, direct product factor
297C2×C4⋊C4—92no—
298C2×C22⋊C4—92no—
299C2×C42—92no—
300H(3,𝔽3)C2×#11711yescharacteristic, complemented, Sylow
301C23×C3—31no—
302C23×C3—31no—
303C23×C3—61no—
304C23×C3—91no—
305C23×C3—91no—
306C23×C3—94no—
307C23×C3—184no—
308C23×C3C22×C3×D311yescharacteristic
309C22×D3—31no—
310C22×D3—31no—
311C22×D3—34no—
312C22×D3—64no—
313C22×D3—91no—
314C22×D3—94no—
315C22×D3—91no—
316C22×D3—184no—
317C2×C3×C4—34no—
318C2×C3×C4—34no—
319C2×C3×C4—31no—
320C2×C3×C4—31no—
321C2×C3×C4—34no—
322C2×C3×C4—34no—
323C2×C3×C4—31no—
324C2×C3×C4—31no—
325C2×C3×C4—64no—
326C2×C3×C4—61no—
327C2×C3×C4—64no—
328C2×C3×C4—61no—
329C2×C3×C4—94no—
330C2×C3×C4—94no—
331C2×C3×C4—94no—
332C2×C3×C4—94no—
333C2×C3×C4—91no—
334C2×C3×C4—91no—
335C2×C3×C4—94no—
336C2×C3×C4—94no—
337C2×C3×C4—188no—
338C2×C3×C4C3×C4×D314yes—
339C2×C3×C4C22×C3×D314yes—
340C2×C3×C4C22×C3×D311yescharacteristic
341C2×C3×C4C22×C3×D311yescharacteristic
342C2×Dic3—31no—
343C2×Dic3—31no—
344C2×Dic3—34no—
345C2×Dic3—34no—
346C2×Dic3—94no—
347C2×Dic3—91no—
348C2×Dic3—91no—
349C2×Dic3—94no—
350C4×D3—34no—
351C4×D3—34no—
352C4×D3—34no—
353C4×D3—34no—
354C4×D3—68no—
355C4×D3—94no—
356C4×D3—94no—
357C4×D3—94no—
358C4×D3—94no—
359C4×D3—188no—
360C2×C32—22no—
361C2×C32—22no—
362C2×C32—21no—
363C2×C32—22no—
364C2×C32—42no—
365C2×C32C22×C4×D311yescharacteristic
366C2×C32D3×#4112yes—
367C2×C32C23×C3×C411yescharacteristic
368C2×C32C2×C3×#4112yes—
369C3⋊D3—22no—
370C3⋊D3C3×#11714yescomplemented
371C3×D3—34no—
372C3×D3—62no—
373C24—91no—
374C22×C4—91no—
375C22×C4—98no—
376C22×C4—91no—
377C22×C4—91no—
378C22×C4—91no—
379C22×C4—91no—
380C22×C4—94no—
381C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
382C4⋊C4—94no—
383C4⋊C4—92no—
384C4⋊C4C2×(C3⋊D3)⋊C312yescomplemented
385C22⋊C4—92no—
386C22⋊C4—94no—
387C22⋊C4C2×(C3⋊D3)⋊C312yes—
388C42—94no—
389C42—92no—
390C42C2×(C3⋊D3)⋊C312yescomplemented
391C22×C3—22no—
392C22×C3—31no—
393C22×C3—31no—
394C22×C3—32no—
395C22×C3—32no—
396C22×C3—62no—
397C22×C3—62no—
398C22×C3—61no—
399C22×C3—62no—
400C22×C3—92no—
401C22×C3—94no—
402C22×C3—92no—
403C22×C3—122no—
404C22×C3—182no—
405C22×C3—188no—
406C22×C3C2×C3×C4×D312yes—
407C22×C3C23×C3×D311yescharacteristic
408C2×D3—32no—
409C2×D3—34no—
410C2×D3—32no—
411C2×D3—68no—
412C2×D3—62no—
413C2×D3—92no—
414C2×D3—94no—
415C2×D3—92no—
416C2×D3—188no—
417C2×D3—182no—
418C3×C4—24no—
419C3×C4—32no—
420C3×C4—32no—
421C3×C4—32no—
422C3×C4—32no—
423C3×C4—64no—
424C3×C4—64no—
425C3×C4—62no—
426C3×C4—62no—
427C3×C4—92no—
428C3×C4—92no—
429C3×C4—124no—
430C3×C4—184no—
431C3×C4C2×C3×C4×D312yes—
432C3×C4C2×C3×C4×D312yes—
433Dic3—32no—
434Dic3—32no—
435Dic3—64no—
436Dic3—92no—
437Dic3—92no—
438Dic3—184no—
439C32—21no—
440C32D3×#11711yescharacteristic, complemented
441C32C2×C3×#11711yescharacteristic, complemented
442C23—91no—
443C23—91no—
444C23—94no—
445C23—184no—
446C23C22×(C3⋊D3)⋊C311yescharacteristic
447C2×C4—91no—
448C2×C4—91no—
449C2×C4—94no—
450C2×C4—94no—
451C2×C4—94no—
452C2×C4—94no—
453C2×C4—94no—
454C2×C4—94no—
455C2×C4—188no—
456C2×C4C4×(C3⋊D3)⋊C314yescomplemented
457C2×C4C22×(C3⋊D3)⋊C314yes—
458C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
459C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
460C2×C3—22no—
461C2×C3—31no—
462C2×C3—31no—
463C2×C3—32no—
464C2×C3—32no—
465C2×C3—62no—
466C2×C3—62no—
467C2×C3—61no—
468C2×C3—62no—
469C2×C3—94no—
470C2×C3—122no—
471C2×C3—182no—
472C2×C3C22×C3×C4×D311yescharacteristic
473C2×C3C3×D3×#4112yes—
474D3—34no—
475D3—62no—
476D3—94no—
477D3—182no—
478C22—22no—
479C22—92no—
480C22—94no—
481C22—92no—
482C22—188no—
483C22—182no—
484C22C2×C4×(C3⋊D3)⋊C312yes—
485C22C23×(C3⋊D3)⋊C311yescharacteristic
486C4—24no—
487C4—92no—
488C4—92no—
489C4—184no—
490C4C2×C4×(C3⋊D3)⋊C312yes—
491C4C2×C4×(C3⋊D3)⋊C312yes—
492C3—31no—
493C3—31no—
494C3—61no—
495C3C3×D3×#11711yescharacteristic
496C2—22no—
497C2—94no—
498C2—182no—
499C2C22×C4×(C3⋊D3)⋊C311yescharacteristic
500C2#41×(C3⋊D3)⋊C312yes—
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC2×C32C23×C3×C4
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#117C2
prime radicalC23×(C3⋊D3)⋊C3C22
cosocleC22×C32C24×C3
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C2×C3×#117
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#117C2semidirect
3. #64135⋊C3C2semidirect
4. #64101⋊C3C2semidirect
5. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
7. #64086⋊C3C2semidirect
8. #64081⋊C3C2semidirect
9. C42×(C3⋊D3)⋊C3C2semidirect
10. C3⋊D3×#117C3semidirect
15. C2×C4×(C3⋊D3)⋊C3C4semidirect
21. C4⋊C4×H(3,𝔽3)C22semidirect
23. C42×H(3,𝔽3)C22semidirect
26. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
27. #3750⋊C3C22semidirect
28. C4×(C3⋊Dic3)⋊C3C22semidirect
29. C4×(C3⋊Dic3)⋊C3C22semidirect
33. C32×#117D3semidirect
34. C32×#117C2×C3semidirect
35. #64135C2×C3semidirect
36. #64101C2×C3semidirect
37. C4⋊C4×C3⋊D3C2×C3semidirect
39. #64086C2×C3semidirect
40. #64081C2×C3semidirect
41. C42×C3⋊D3C2×C3semidirect
54. C2×C4×H(3,𝔽3)C2×C4semidirect
58. C2×(C3⋊Dic3)⋊C3C2×C4semidirect
65. C4×(C3⋊D3)⋊C3C2×C4semidirect
66. C4×(C3⋊D3)⋊C3C2×C4semidirect
76. C2×C4×C3⋊D3C3×C4semidirect
90. C32×C4⋊C4C2×D3semidirect
91. C32×C4⋊C4C22×C3semidirect
96. C32×C42C2×D3semidirect
97. C32×C42C22×C3semidirect
100. C4⋊(C3⋊Dic3)C22×C3semidirect
101. #3750C22×C3semidirect
102. C4×C3⋊Dic3C22×C3semidirect
103. C4×C3⋊Dic3C22×C3semidirect
167. C2×C32×C4C4×D3semidirect
168. C2×C32×C4C2×C3×C4semidirect
175. C2×C3⋊Dic3C2×C3×C4semidirect
182. C4×C3⋊D3C2×C3×C4semidirect
183. C4×C3⋊D3C2×C3×C4semidirect
198. (C3⋊D3)⋊C3#117direct (N × H)
296. #117(C3⋊D3)⋊C3direct (N × H)
300. H(3,𝔽3)C2×#117semidirect
370. C3⋊D3C3×#117semidirect
384. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
390. C42C2×(C3⋊D3)⋊C3semidirect
440. C32D3×#117semidirect
441. C32C2×C3×#117semidirect
456. C2×C4C4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C4×(C3⋊D3)⋊C32
H(3,𝔽3)×#1172
#64135⋊C32
#64101⋊C32
C4⋊C4×(C3⋊D3)⋊C32
C22⋊C4×(C3⋊D3)⋊C32
#64086⋊C32
#64081⋊C32
C42×(C3⋊D3)⋊C32
C3⋊D3×#1173
C3×D3×#1173