Finite Group Databasebeta

← #10596165⋊C3 #13153185

Subgroups — #10596165⋊C3 #13153185

Order 1728.

805 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#125C211yesmaximal, characteristic, complemented
2#64107⋊C3C212yesmaximal, complemented
3#64106⋊C3C211yesmaximal, characteristic, complemented
4#64106⋊C3C211yesmaximal, characteristic, complemented
5#64103⋊C3C212yesmaximal
6#64101⋊C3C211yesmaximal, characteristic, complemented
7#64101⋊C3C212yesmaximal, complemented
8C4⋊C4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
9#64099⋊C3C212yesmaximal, complemented
10C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
11C4×#634⋊C3C211yesmaximal, characteristic, complemented
12#10596165C311yesmaximal, characteristic, complemented
13C3×#5694—31nomaximal
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2212yes—
18C2×#634⋊C3C2211yescharacteristic
19C4⋊C4×H(3,𝔽3)C2212yes—
20C4⋊C4×H(3,𝔽3)C2212yes—
21C4⋊C4×H(3,𝔽3)C2211yescharacteristic
22C4⋊C4×H(3,𝔽3)C2211yescharacteristic
23C42×H(3,𝔽3)C2211yescharacteristic
24#3752⋊C3C2211yescharacteristic
25#3752⋊C3C2212yes—
26#3752⋊C3C2212yes—
27#3752⋊C3C2211yescharacteristic
28#3752⋊C3C2212yes—
29C4⋊((C3⋊Dic3)⋊C3)C2212yes—
30C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
31C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
32C4⋊((C3⋊Dic3)⋊C3)C2212yes—
33#3750⋊C3C2212yes—
34#3750⋊C3C2212yes—
35C4×(C3⋊Dic3)⋊C3C2212yes—
36C4×(C3⋊Dic3)⋊C3C2212yes—
37C32×#125—21no—
38C32×#125D311yescharacteristic, complemented
39C32×#125C2×C311yescharacteristic, complemented
40#64107C2×C312yescomplemented
41#64106C2×C311yescharacteristic, complemented
42#64106C2×C311yescharacteristic, complemented
43#64103C2×C312yes—
44#64101C2×C312yescomplemented
45#64101C2×C311yescharacteristic, complemented
46C4⋊C4×C3⋊D3C2×C311yescharacteristic, complemented
47#64099C2×C312yescomplemented
48C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
49C4×#634C2×C311yescharacteristic, complemented
50C3×#898—32no—
51C3×#897—31no—
52C3×#897—31no—
53C3×#894—32no—
54C3×#892—31no—
55C3×#892—32no—
56C3×D3×C4⋊C4—31no—
57C3×#890—32no—
58C3×C4×D12—31no—
59C3×C4×Dic6—31no—
60C22×(C3⋊D3)⋊C3C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2312yes—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×C4×H(3,𝔽3)C2312yes—
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C4⋊((C3⋊D3)⋊C3)—22no—
72C4×(C3⋊D3)⋊C3—24no—
73C4×(C3⋊D3)⋊C3—22no—
74C4×(C3⋊D3)⋊C3—22no—
75#634⋊C3—22no—
76C3×#552—91no—
77#5694—31no—
78#5694—91no—
79C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
80C2×C4×C3⋊D3C22×C311yescharacteristic
81C2×C4×C3⋊D3C22×C311yescharacteristic
82C2×C4×C3⋊D3C22×C312yes—
83C2×#634C22×C311yescharacteristic
84C2×C3×D12—31no—
85C2×C3×C4×D3—32no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—31no—
88C2×C3×Dic6—31no—
89C32×C4⋊C4—21no—
90C32×C4⋊C4—21no—
91C32×C4⋊C4—22no—
92C32×C4⋊C4—22no—
93C32×C4⋊C4C2×D312yes—
94C32×C4⋊C4C2×D311yescharacteristic
95C32×C4⋊C4C2×D311yescharacteristic
96C32×C4⋊C4C2×D312yes—
97C32×C4⋊C4C22×C312yes—
98C32×C4⋊C4C22×C311yescharacteristic
99C32×C4⋊C4C22×C311yescharacteristic
100C32×C4⋊C4C22×C312yes—
101C32×C42—21no—
102C32×C42C2×D311yescharacteristic
103C32×C42C22×C311yescharacteristic
104#3752C22×C311yescharacteristic
105#3752C22×C311yescharacteristic
106#3752C22×C312yes—
107#3752C22×C312yes—
108#3752C22×C312yes—
109C4⋊(C3⋊Dic3)C22×C311yescharacteristic
110C4⋊(C3⋊Dic3)C22×C311yescharacteristic
111C4⋊(C3⋊Dic3)C22×C312yes—
112C4⋊(C3⋊Dic3)C22×C312yes—
113#3750C22×C312yes—
114#3750C22×C312yes—
115C4×C3⋊Dic3C22×C312yes—
116C4×C3⋊Dic3C22×C312yes—
117C3×(C2×D3)⋊C4—31no—
118C3×(C2×D3)⋊C4—32no—
119C3×(C2×D3)⋊C4—32no—
120C3×(C2×D3)⋊C4—32no—
121C3×(C2×D3)⋊C4—31no—
122C3×C4⋊Dic3—31no—
123C3×C4⋊Dic3—32no—
124C3×C4⋊Dic3—32no—
125C3×C4⋊Dic3—31no—
126C3×Dic3⋊C4—32no—
127C3×Dic3⋊C4—32no—
128C3×C4×Dic3—32no—
129C3×C4×Dic3—32no—
130C22×H(3,𝔽3)C2411yescharacteristic
131C2×(C3⋊D3)⋊C3—21no—
132C2×(C3⋊D3)⋊C3—21no—
133C2×(C3⋊D3)⋊C3—21no—
134C2×(C3⋊D3)⋊C3—21no—
135C2×(C3⋊D3)⋊C3—21no—
136C2×(C3⋊D3)⋊C3#4112yes—
137C4×H(3,𝔽3)—22no—
138C4×H(3,𝔽3)—21no—
139C4×H(3,𝔽3)—21no—
140C4×H(3,𝔽3)—22no—
141C4×H(3,𝔽3)#4112yes—
142(C3⋊Dic3)⋊C3—22no—
143(C3⋊Dic3)⋊C3—21no—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3—22no—
146C3×#126—92no—
147C3×#125—31no—
148C3×#125—31no—
149C3×#125—61no—
150C3×#125—92no—
151C3×#125C3×D311yescharacteristic
152C3×#123—92no—
153C3×C22⋊Q8—91no—
154C3×C22⋊Q8—91no—
155C3×C4×Q8—91no—
156C3×C4×D4—91no—
157C3×#117—91no—
158C3×#117—92no—
159C2×C3×C4⋊C4—91no—
160#898—32no—
161#898—92no—
162#897—31no—
163#897—31no—
164#897—91no—
165#897—91no—
166#894—32no—
167#894—92no—
168#892—32no—
169#892—31no—
170#892—92no—
171#892—91no—
172D3×C4⋊C4—31no—
173D3×C4⋊C4—91no—
174#890—32no—
175#890—92no—
176C4×D12—31no—
177C4×D12—91no—
178C4×Dic6—31no—
179C4×Dic6—91no—
180C22×C3⋊D3C23×C311yescharacteristic
181C22×C3⋊D3C23×C311yescharacteristic
182C22×C3×D3—31no—
183C22×C3×D3—31no—
184C2×C32×C4—21no—
185C2×C32×C4—21no—
186C2×C32×C4—21no—
187C2×C32×C4—22no—
188C2×C32×C4—22no—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D312yes—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D312yes—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C23×C311yescharacteristic
195C2×C32×C4C23×C312yes—
196C2×C32×C4C23×C311yescharacteristic
197C2×C32×C4C23×C312yes—
198C2×C32×C4C23×C311yescharacteristic
199C2×C3⋊Dic3C23×C312yes—
200C2×C3⋊Dic3C23×C311yescharacteristic
201C2×C3⋊Dic3C23×C312yes—
202C2×C3⋊Dic3C23×C311yescharacteristic
203C4⋊(C3⋊D3)—22no—
204C4×C3⋊D3—24no—
205C4×C3⋊D3—22no—
206C4×C3⋊D3—22no—
207#634—22no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—31no—
212C3×D12—62no—
213C3×C4×D3—64no—
214C3×C4×D3—62no—
215C3×C4×D3—62no—
216C3×Dic6—62no—
217#552—91noSylow
218C2×H(3,𝔽3)C2×#4111yescharacteristic
219C2×H(3,𝔽3)C2×#4111yescharacteristic
220C2×H(3,𝔽3)2−1+411yescharacteristic
221(C3⋊D3)⋊C3—22no—
222(C3⋊D3)⋊C3—41no—
223C2×C3×Q8—91no—
224C2×C3×D4—91no—
225C22×C3×C4—92no—
226C22×C3×C4—91no—
227C22×C3×C4—91no—
228C2×D12—31no—
229C2×D12—91no—
230C2×C4×D3—31no—
231C2×C4×D3—31no—
232C2×C4×D3—32no—
233C2×C4×D3—91no—
234C2×C4×D3—91no—
235C2×C4×D3—92no—
236C2×Dic6—31no—
237C2×Dic6—91no—
238C3×C4⋊C4—32no—
239C3×C4⋊C4—31no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—32no—
242C3×C4⋊C4—31no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—32no—
245C3×C4⋊C4—32no—
246C3×C4⋊C4—61no—
247C3×C4⋊C4—61no—
248C3×C4⋊C4—62no—
249C3×C4⋊C4—62no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4—92no—
252C3×C4⋊C4—91no—
253C3×C4⋊C4—92no—
254C3×C4⋊C4—91no—
255C3×C4⋊C4—92no—
256C3×C4⋊C4C2×C3×D312yes—
257C3×C4⋊C4C2×C3×D311yescharacteristic
258C3×C4⋊C4C2×C3×D311yescharacteristic
259C3×C4⋊C4C2×C3×D312yes—
260C3×C22⋊C4—91no—
261C3×C22⋊C4—92no—
262C3×C22⋊C4—92no—
263C3×C22⋊C4—91no—
264C3×C22⋊C4—92no—
265C3×C42—31no—
266C3×C42—31no—
267C3×C42—61no—
268C3×C42—92no—
269C3×C42—92no—
270C3×C42C2×C3×D311yescharacteristic
271(C2×D3)⋊C4—31no—
272(C2×D3)⋊C4—31no—
273(C2×D3)⋊C4—32no—
274(C2×D3)⋊C4—32no—
275(C2×D3)⋊C4—32no—
276(C2×D3)⋊C4—91no—
277(C2×D3)⋊C4—91no—
278(C2×D3)⋊C4—92no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4—92no—
281C4⋊Dic3—31no—
282C4⋊Dic3—31no—
283C4⋊Dic3—32no—
284C4⋊Dic3—32no—
285C4⋊Dic3—91no—
286C4⋊Dic3—91no—
287C4⋊Dic3—92no—
288C4⋊Dic3—92no—
289Dic3⋊C4—32no—
290Dic3⋊C4—32no—
291Dic3⋊C4—92no—
292Dic3⋊C4—92no—
293C4×Dic3—32no—
294C4×Dic3—32no—
295C4×Dic3—92no—
296C4×Dic3—92no—
297C22×C32—21no—
298C22×C32C23×D311yescharacteristic
299C22×C32C24×C311yescharacteristic
300C2×C3⋊D3—21no—
301C2×C3⋊D3—21no—
302C2×C3⋊D3—21no—
303C2×C3⋊D3—21no—
304C2×C3⋊D3—21no—
305C2×C3⋊D3C3×#4112yes—
306C2×C3×D3—32no—
307C2×C3×D3—61no—
308C2×C3×D3—61no—
309C2×C3×D3—61no—
310C2×C3×D3—61no—
311C2×C3×D3—61no—
312C32×C4—22no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—22no—
316C32×C4—22no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—22no—
320C32×C4—22no—
321C32×C4—42no—
322C32×C4—41no—
323C32×C4—41no—
324C32×C4—42no—
325C32×C4#23912yes—
326C32×C4C3×#4112yes—
327C3⋊Dic3—22no—
328C3⋊Dic3—21no—
329C3⋊Dic3—21no—
330C3⋊Dic3—22no—
331C3×Dic3—62no—
332C3×Dic3—61no—
333C3×Dic3—61no—
334C3×Dic3—62no—
335#126—92no—
336#125—92no—
337#125(C3⋊D3)⋊C311yescharacteristic, complemented
338#123—92no—
339C22⋊Q8—91no—
340C22⋊Q8—91no—
341C4×Q8—91no—
342C4×D4—91no—
343#117—91no—
344#117—92no—
345C2×C4⋊C4—91no—
346H(3,𝔽3)#55211yescharacteristic, complemented, Sylow
347C23×C3—91no—
348C23×C3—91no—
349C22×D3—31no—
350C22×D3—31no—
351C22×D3—91no—
352C22×D3—91no—
353C3×Q8—182no—
354C3×D4—182no—
355C2×C3×C4—31no—
356C2×C3×C4—31no—
357C2×C3×C4—31no—
358C2×C3×C4—31no—
359C2×C3×C4—32no—
360C2×C3×C4—32no—
361C2×C3×C4—31no—
362C2×C3×C4—32no—
363C2×C3×C4—31no—
364C2×C3×C4—32no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—62no—
369C2×C3×C4—62no—
370C2×C3×C4—92no—
371C2×C3×C4—91no—
372C2×C3×C4—92no—
373C2×C3×C4—91no—
374C2×C3×C4—182no—
375C2×C3×C4—182no—
376C2×C3×C4—184no—
377C2×C3×C4C22×C3×D311yescharacteristic
378C2×C3×C4C22×C3×D312yes—
379C2×C3×C4C22×C3×D311yescharacteristic
380C2×C3×C4C22×C3×D312yes—
381C2×C3×C4C22×C3×D311yescharacteristic
382C2×Dic3—32no—
383C2×Dic3—31no—
384C2×Dic3—32no—
385C2×Dic3—31no—
386C2×Dic3—92no—
387C2×Dic3—91no—
388C2×Dic3—92no—
389C2×Dic3—91no—
390D12—62no—
391D12—182no—
392C4×D3—64no—
393C4×D3—62no—
394C4×D3—62no—
395C4×D3—182no—
396C4×D3—182no—
397C4×D3—184no—
398Dic6—62no—
399Dic6—182no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32C2×#23911yescharacteristic
404C2×C32D3×#4111yescharacteristic
405C2×C32#101011yescharacteristic
406C2×C32C2×C3×#4111yescharacteristic
407C2×C32C2×C3×#4111yescharacteristic
408C2×C32C3×2−1+411yescharacteristic
409C3⋊D3—22no—
410C3⋊D3—41no—
411C3×D3—62no—
412C3×D3—121no—
413C2×Q8—91no—
414C2×D4—91no—
415C22×C4—92no—
416C22×C4—91no—
417C22×C4—91no—
418C4⋊C4—91no—
419C4⋊C4—92no—
420C4⋊C4—92no—
421C4⋊C4—91no—
422C4⋊C4—92no—
423C4⋊C4—92no—
424C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
425C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
426C4⋊C4C2×(C3⋊D3)⋊C312yes—
427C4⋊C4C2×(C3⋊D3)⋊C312yes—
428C22⋊C4—91no—
429C22⋊C4—92no—
430C22⋊C4—92no—
431C22⋊C4—91no—
432C22⋊C4—92no—
433C42—92no—
434C42—92no—
435C42C2×(C3⋊D3)⋊C311yescharacteristic
436C22×C3—31no—
437C22×C3—31no—
438C22×C3—61no—
439C22×C3—92no—
440C22×C3—181no—
441C22×C3—181no—
442C22×C3—181no—
443C22×C3—181no—
444C22×C3—181no—
445C22×C3C23×C3×D311yescharacteristic
446C2×D3—32no—
447C2×D3—61no—
448C2×D3—61no—
449C2×D3—61no—
450C2×D3—61no—
451C2×D3—61no—
452C2×D3—92no—
453C2×D3—181no—
454C2×D3—181no—
455C2×D3—181no—
456C2×D3—181no—
457C2×D3—181no—
458C3×C4—22no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—22no—
462C3×C4—32no—
463C3×C4—32no—
464C3×C4—62no—
465C3×C4—61no—
466C3×C4—62no—
467C3×C4—61no—
468C3×C4—61no—
469C3×C4—62no—
470C3×C4—61no—
471C3×C4—62no—
472C3×C4—62no—
473C3×C4—122no—
474C3×C4—121no—
475C3×C4—121no—
476C3×C4—122no—
477C3×C4—181no—
478C3×C4—181no—
479C3×C4—182no—
480C3×C4—182no—
481C3×C4C3×#23912yes—
482Dic3—62no—
483Dic3—61no—
484Dic3—61no—
485Dic3—62no—
486Dic3—182no—
487Dic3—181no—
488Dic3—181no—
489Dic3—182no—
490C32—21no—
491C32#569411yescharacteristic, complemented
492C32C3×#55211yescharacteristic, complemented
493C23—91no—
494C23—91no—
495Q8—182no—
496D4—182no—
497C2×C4—91no—
498C2×C4—92no—
499C2×C4—92no—
500C2×C4—91no—
501C2×C4—184no—
502C2×C4—182no—
503C2×C4—182no—
504C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
505C2×C4C22×(C3⋊D3)⋊C312yes—
506C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
507C2×C4C22×(C3⋊D3)⋊C312yes—
508C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
509C2×C3—31no—
510C2×C3—31no—
511C2×C3—31no—
512C2×C3—31no—
513C2×C3—31no—
514C2×C3—31no—
515C2×C3—61no—
516C2×C3—61no—
517C2×C3—61no—
518C2×C3—182no—
519C2×C3—361no—
520C2×C3C2×C3×#23911yescharacteristic
521C2×C3C3×D3×#4111yescharacteristic
522C2×C3C3×#101011yescharacteristic
523D3—62no—
524D3—121no—
525D3—182no—
526D3—361no—
527C22—92no—
528C22—181no—
529C22—181no—
530C22—181no—
531C22—181no—
532C22—181no—
533C22C23×(C3⋊D3)⋊C311yescharacteristic
534C4—22no—
535C4—21no—
536C4—21no—
537C4—22no—
538C4—181no—
539C4—181no—
540C4—182no—
541C4—182no—
542C4#8872212yes—
543C3—31no—
544C3—31no—
545C3—61no—
546C3C3×#569411yescharacteristic
547C2—182no—
548C2—361no—
549C2C2×#8872211yescharacteristic
550C2#41×(C3⋊D3)⋊C311yescharacteristic
551C2#1172263911yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#125C2
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenter#125(C3⋊D3)⋊C3
hypocenterC32C3×#552
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #125

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#125C2semidirect
2. #64107⋊C3C2semidirect
3. #64106⋊C3C2semidirect
4. #64106⋊C3C2semidirect
6. #64101⋊C3C2semidirect
7. #64101⋊C3C2semidirect
8. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
9. #64099⋊C3C2semidirect
11. C4×#634⋊C3C2semidirect
12. #10596165C3semidirect
38. C32×#125D3semidirect
39. C32×#125C2×C3semidirect
40. #64107C2×C3semidirect
41. #64106C2×C3semidirect
42. #64106C2×C3semidirect
44. #64101C2×C3semidirect
45. #64101C2×C3semidirect
46. C4⋊C4×C3⋊D3C2×C3semidirect
47. #64099C2×C3semidirect
49. C4×#634C2×C3semidirect
337. #125(C3⋊D3)⋊C3semidirect
346. H(3,𝔽3)#552semidirect
491. C32#5694semidirect
492. C32C3×#552semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
#64107⋊C32
#64106⋊C32
#64103⋊C32
#64101⋊C32
C4⋊C4×(C3⋊D3)⋊C32
#64099⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
C4×#634⋊C32
#105961653
C3×#56943