Finite Group Databasebeta

← #10596142⋊C3 #13153112

Subgroups — #10596142⋊C3 #13153112

Order 1728.

818 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#124C211yesmaximal, characteristic
2Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3#64153⋊C3C211yesmaximal, characteristic, complemented
4#64145⋊C3C211yesmaximal, characteristic
5D4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic
6#64088⋊C3C212yesmaximal, complemented
7#64088⋊C3C212yesmaximal, complemented
8#64087⋊C3C212yesmaximal, complemented
9#64086⋊C3C212yesmaximal, complemented
10C4×#634⋊C3C211yesmaximal, characteristic, complemented
11C4×#634⋊C3C211yesmaximal, characteristic, complemented
12#10596142C311yesmaximal, characteristic, complemented
13C3×#5671—31nomaximal
14C2×Q8×H(3,𝔽3)C2211yescharacteristic
15C2×D4×H(3,𝔽3)C2211yescharacteristic
16C22×(C3⋊Dic3)⋊C3C2212yes—
17C2×#634⋊C3C2211yescharacteristic
18C2×#634⋊C3C2211yescharacteristic
19C22⋊C4×H(3,𝔽3)C2212yes—
20C22⋊C4×H(3,𝔽3)C2212yes—
21C42×H(3,𝔽3)C2211yescharacteristic
22#3757⋊C3C2212yes—
23#3757⋊C3C2212yes—
24#3757⋊C3C2212yes—
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
27C4⋊((C3⋊Dic3)⋊C3)C2212yes—
28#3750⋊C3C2211yescharacteristic
29#3750⋊C3C2212yes—
30#3750⋊C3C2211yescharacteristic
31#3750⋊C3C2212yes—
32#3750⋊C3C2212yes—
33C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
34C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
35C4×(C3⋊Dic3)⋊C3C2212yes—
36C4×(C3⋊Dic3)⋊C3C2212yes—
37C32×#124—21no—
38C32×#124D311yescharacteristic
39C32×#124C2×C311yescharacteristic
40Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
41#64153C2×C311yescharacteristic, complemented
42#64145C2×C311yescharacteristic
43D4×C3⋊Dic3C2×C311yescharacteristic
44#64088C2×C312yescomplemented
45#64088C2×C312yescomplemented
46#64087C2×C312yescomplemented
47#64086C2×C312yescomplemented
48C4×#634C2×C311yescharacteristic, complemented
49C4×#634C2×C311yescharacteristic, complemented
50C3×Q8×Dic3—31no—
51C3×#944—31no—
52C3×#936—31no—
53C3×D4×Dic3—31no—
54C3×#879—32no—
55C3×#879—32no—
56C3×#878—32no—
57C3×#877—32no—
58C3×C4×Dic6—31no—
59C3×C4×Dic6—31no—
60C23×H(3,𝔽3)C2312yes—
61Q8×H(3,𝔽3)—22no—
62D4×H(3,𝔽3)—22no—
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)⋊C3—24no—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74#634⋊C3—22no—
75#634⋊C3—22no—
76C3×#556—91no—
77#5671—31no—
78#5671—91no—
79C2×C32×Q8—21no—
80C2×C32×Q8C2×D311yescharacteristic
81C2×C32×Q8C22×C311yescharacteristic
82C2×C32×D4—21no—
83C2×C32×D4C2×D311yescharacteristic
84C2×C32×D4C22×C311yescharacteristic
85C22×C3⋊Dic3C22×C312yes—
86C2×#634C22×C311yescharacteristic
87C2×#634C22×C311yescharacteristic
88C22×C3×Dic3—32no—
89C2×C3×Dic6—31no—
90C2×C3×Dic6—31no—
91C32×C22⋊C4—22no—
92C32×C22⋊C4—22no—
93C32×C22⋊C4C2×D312yes—
94C32×C22⋊C4C2×D312yes—
95C32×C22⋊C4C22×C312yes—
96C32×C22⋊C4C22×C312yes—
97C32×C42—21no—
98C32×C42C2×D311yescharacteristic
99C32×C42C22×C311yescharacteristic
100#3757C22×C312yes—
101#3757C22×C312yes—
102#3757C22×C312yes—
103C4⋊(C3⋊Dic3)C22×C311yescharacteristic
104C4⋊(C3⋊Dic3)C22×C311yescharacteristic
105C4⋊(C3⋊Dic3)C22×C312yes—
106#3750C22×C311yescharacteristic
107#3750C22×C312yes—
108#3750C22×C311yescharacteristic
109#3750C22×C312yes—
110#3750C22×C312yes—
111C4×C3⋊Dic3C22×C312yes—
112C4×C3⋊Dic3C22×C311yescharacteristic
113C4×C3⋊Dic3C22×C311yescharacteristic
114C4×C3⋊Dic3C22×C312yes—
115C3×#217—32no—
116C3×#217—32no—
117C3×#217—32no—
118C3×C4⋊Dic3—32no—
119C3×C4⋊Dic3—31no—
120C3×C4⋊Dic3—31no—
121C3×Dic3⋊C4—31no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×Dic3⋊C4—32no—
125C3×Dic3⋊C4—31no—
126C3×C4×Dic3—32no—
127C3×C4×Dic3—32no—
128C3×C4×Dic3—31no—
129C3×C4×Dic3—31no—
130C22×H(3,𝔽3)—22no—
131C22×H(3,𝔽3)—22no—
132C22×H(3,𝔽3)—22no—
133C22×H(3,𝔽3)C2411yescharacteristic
134C4×H(3,𝔽3)—21no—
135C4×H(3,𝔽3)—22no—
136C4×H(3,𝔽3)—21no—
137C4×H(3,𝔽3)#4112yes—
138(C3⋊Dic3)⋊C3—22no—
139(C3⋊Dic3)⋊C3—21no—
140(C3⋊Dic3)⋊C3—22no—
141(C3⋊Dic3)⋊C3—21no—
142(C3⋊Dic3)⋊C3—21no—
143(C3⋊Dic3)⋊C3#4112yes—
144C3×C4⋊Q8—91no—
145C3×#126—92no—
146C3×#126—92no—
147C3×#124—31no—
148C3×#124—31no—
149C3×#124—61no—
150C3×#124—91no—
151C3×#124C3×D311yescharacteristic
152C3×C22⋊Q8—92no—
153C3×C4×Q8—91no—
154C3×C4×Q8—91no—
155C3×C4×Q8—91no—
156C3×C4×D4—91no—
157C3×#117—92no—
158Q8×Dic3—31no—
159Q8×Dic3—91no—
160#944—31no—
161#944—91no—
162#936—31no—
163#936—91no—
164D4×Dic3—31no—
165D4×Dic3—91no—
166#879—32no—
167#879—32no—
168#879—92no—
169#879—92no—
170#878—32no—
171#878—92no—
172#877—32no—
173#877—92no—
174C4×Dic6—31no—
175C4×Dic6—31no—
176C4×Dic6—91no—
177C4×Dic6—91no—
178C23×C32—22no—
179C23×C32C22×D312yes—
180C23×C32C23×C312yes—
181C32×Q8—22no—
182C32×Q8—22no—
183C32×Q8—42no—
184C32×D4—22no—
185C32×D4—22no—
186C32×D4—42no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4—22no—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D312yes—
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C32×C4C23×C311yescharacteristic
196C2×C32×C4C23×C312yes—
197C2×C32×C4C23×C311yescharacteristic
198C2×C32×C4C23×C311yescharacteristic
199C2×C3⋊Dic3—24no—
200C2×C3⋊Dic3C23×C311yescharacteristic
201C2×C3⋊Dic3C23×C312yes—
202C2×C3⋊Dic3C23×C311yescharacteristic
203C2×C3⋊Dic3C23×C312yes—
204C2×C3⋊Dic3C23×C311yescharacteristic
205C2×C3⋊Dic3C23×C311yescharacteristic
206#634—22no—
207#634—22no—
208C2×C3×Dic3—31no—
209C2×C3×Dic3—31no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—31no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—31no—
214C2×C3×Dic3—64no—
215C3×Dic6—62no—
216C3×Dic6—62no—
217#556—91noSylow
218C2×H(3,𝔽3)—42no—
219C2×H(3,𝔽3)C2×#4111yescharacteristic
220C2×H(3,𝔽3)C2×#4111yescharacteristic
221C2×H(3,𝔽3)2−1+411yescharacteristic
222C2×C3×Q8—31no—
223C2×C3×Q8—31no—
224C2×C3×Q8—61no—
225C2×C3×Q8—91no—
226C2×C3×Q8—91no—
227C2×C3×Q8C2×C3×D311yescharacteristic
228C2×C3×D4—31no—
229C2×C3×D4—31no—
230C2×C3×D4—61no—
231C2×C3×D4C2×C3×D311yescharacteristic
232C22×C3×C4—92no—
233C22×Dic3—32no—
234C22×Dic3—92no—
235C2×Dic6—31no—
236C2×Dic6—31no—
237C2×Dic6—91no—
238C2×Dic6—91no—
239C3×C4⋊C4—91no—
240C3×C4⋊C4—92no—
241C3×C4⋊C4—92no—
242C3×C4⋊C4—91no—
243C3×C4⋊C4—92no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—92no—
246C3×C4⋊C4—91no—
247C3×C22⋊C4—32no—
248C3×C22⋊C4—32no—
249C3×C22⋊C4—32no—
250C3×C22⋊C4—32no—
251C3×C22⋊C4—62no—
252C3×C22⋊C4—62no—
253C3×C22⋊C4—92no—
254C3×C22⋊C4—92no—
255C3×C22⋊C4—92no—
256C3×C22⋊C4C2×C3×D312yes—
257C3×C22⋊C4C2×C3×D312yes—
258C3×C42—31no—
259C3×C42—31no—
260C3×C42—61no—
261C3×C42—92no—
262C3×C42—92no—
263C3×C42—91no—
264C3×C42—91no—
265C3×C42C2×C3×D311yescharacteristic
266#217—32no—
267#217—32no—
268#217—32no—
269#217—92no—
270#217—92no—
271#217—92no—
272C4⋊Dic3—31no—
273C4⋊Dic3—31no—
274C4⋊Dic3—32no—
275C4⋊Dic3—91no—
276C4⋊Dic3—91no—
277C4⋊Dic3—92no—
278Dic3⋊C4—31no—
279Dic3⋊C4—32no—
280Dic3⋊C4—31no—
281Dic3⋊C4—32no—
282Dic3⋊C4—32no—
283Dic3⋊C4—91no—
284Dic3⋊C4—92no—
285Dic3⋊C4—91no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288C4×Dic3—32no—
289C4×Dic3—31no—
290C4×Dic3—31no—
291C4×Dic3—32no—
292C4×Dic3—92no—
293C4×Dic3—91no—
294C4×Dic3—91no—
295C4×Dic3—92no—
296C22×C32—22no—
297C22×C32—22no—
298C22×C32—22no—
299C22×C32—22no—
300C22×C32—22no—
301C22×C32—22no—
302C22×C32—21no—
303C22×C32—42no—
304C22×C32—42no—
305C22×C32—42no—
306C22×C32C23×D311yescharacteristic
307C22×C32C24×C311yescharacteristic
308C32×C4—22no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—22no—
313C32×C4—21no—
314C32×C4—22no—
315C32×C4—41no—
316C32×C4—41no—
317C32×C4—42no—
318C32×C4#23712yes—
319C32×C4C3×#4112yes—
320C3⋊Dic3—22no—
321C3⋊Dic3—21no—
322C3⋊Dic3—22no—
323C3⋊Dic3—21no—
324C3⋊Dic3—21no—
325C3⋊Dic3C3×#4112yes—
326C3×Dic3—32no—
327C3×Dic3—62no—
328C3×Dic3—61no—
329C3×Dic3—62no—
330C3×Dic3—61no—
331C3×Dic3—61no—
332C4⋊Q8—91no—
333#126—92no—
334#126—92no—
335#124—91no—
336#124(C3⋊D3)⋊C311yescharacteristic
337C22⋊Q8—92no—
338C4×Q8—91no—
339C4×Q8—91no—
340C4×Q8—91no—
341C4×D4—91no—
342#117—92no—
343H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
344C23×C3—32no—
345C23×C3—32no—
346C23×C3—62no—
347C23×C3C22×C3×D312yes—
348C3×Q8—22no—
349C3×Q8—62no—
350C3×Q8—62no—
351C3×Q8—122no—
352C3×Q8—182no—
353C3×Q8—182no—
354C3×D4—22no—
355C3×D4—62no—
356C3×D4—62no—
357C3×D4—122no—
358C2×C3×C4—31no—
359C2×C3×C4—31no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—32no—
363C2×C3×C4—31no—
364C2×C3×C4—32no—
365C2×C3×C4—31no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—62no—
370C2×C3×C4—91no—
371C2×C3×C4—91no—
372C2×C3×C4—92no—
373C2×C3×C4—91no—
374C2×C3×C4—92no—
375C2×C3×C4—91no—
376C2×C3×C4—184no—
377C2×C3×C4C22×C3×D311yescharacteristic
378C2×C3×C4C22×C3×D312yes—
379C2×C3×C4C22×C3×D311yescharacteristic
380C2×C3×C4C22×C3×D311yescharacteristic
381C2×Dic3—31no—
382C2×Dic3—31no—
383C2×Dic3—32no—
384C2×Dic3—31no—
385C2×Dic3—32no—
386C2×Dic3—31no—
387C2×Dic3—64no—
388C2×Dic3—91no—
389C2×Dic3—91no—
390C2×Dic3—92no—
391C2×Dic3—91no—
392C2×Dic3—92no—
393C2×Dic3—91no—
394C2×Dic3—184no—
395Dic6—62no—
396Dic6—62no—
397Dic6—182no—
398Dic6—182no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—42no—
403C2×C32—42no—
404C2×C32—82no—
405C2×C32C2×#23711yescharacteristic
406C2×C32D3×#4111yescharacteristic
407C2×C32#101011yescharacteristic
408C2×C32C2×C3×#4111yescharacteristic
409C2×C32C2×C3×#4111yescharacteristic
410C2×C32C3×2−1+411yescharacteristic
411C2×Q8—91no—
412C2×Q8—91no—
413C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
414C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
415C22×C4—92no—
416C4⋊C4—91no—
417C4⋊C4—92no—
418C4⋊C4—91no—
419C4⋊C4—92no—
420C4⋊C4—92no—
421C4⋊C4—91no—
422C4⋊C4—91no—
423C4⋊C4—92no—
424C22⋊C4—92no—
425C22⋊C4—92no—
426C22⋊C4—92no—
427C22⋊C4C2×(C3⋊D3)⋊C312yes—
428C22⋊C4C2×(C3⋊D3)⋊C312yes—
429C42—92no—
430C42—92no—
431C42—91no—
432C42—91no—
433C42C2×(C3⋊D3)⋊C311yescharacteristic
434C22×C3—22no—
435C22×C3—22no—
436C22×C3—22no—
437C22×C3—31no—
438C22×C3—31no—
439C22×C3—62no—
440C22×C3—62no—
441C22×C3—62no—
442C22×C3—62no—
443C22×C3—62no—
444C22×C3—62no—
445C22×C3—61no—
446C22×C3—122no—
447C22×C3—122no—
448C22×C3—122no—
449C22×C3C23×C3×D311yescharacteristic
450C3×C4—22no—
451C3×C4—21no—
452C3×C4—21no—
453C3×C4—32no—
454C3×C4—32no—
455C3×C4—61no—
456C3×C4—62no—
457C3×C4—62no—
458C3×C4—61no—
459C3×C4—61no—
460C3×C4—61no—
461C3×C4—62no—
462C3×C4—92no—
463C3×C4—122no—
464C3×C4—121no—
465C3×C4—121no—
466C3×C4—182no—
467C3×C4—182no—
468C3×C4—181no—
469C3×C4—181no—
470C3×C4—181no—
471C3×C4C3×#23712yes—
472Dic3—32no—
473Dic3—62no—
474Dic3—61no—
475Dic3—62no—
476Dic3—61no—
477Dic3—61no—
478Dic3—92no—
479Dic3—182no—
480Dic3—181no—
481Dic3—182no—
482Dic3—181no—
483Dic3—181no—
484C32—21no—
485C32#567111yescharacteristic, complemented
486C32C3×#55611yescharacteristic, complemented
487C23C22×(C3⋊D3)⋊C312yes—
488Q8—22no—
489Q8—182no—
490Q8—182no—
491D4—22no—
492C2×C4—91no—
493C2×C4—91no—
494C2×C4—92no—
495C2×C4—91no—
496C2×C4—92no—
497C2×C4—91no—
498C2×C4—184no—
499C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
500C2×C4C22×(C3⋊D3)⋊C312yes—
501C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
502C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
503C2×C3—31no—
504C2×C3—31no—
505C2×C3—31no—
506C2×C3—31no—
507C2×C3—31no—
508C2×C3—31no—
509C2×C3—42no—
510C2×C3—61no—
511C2×C3—61no—
512C2×C3—61no—
513C2×C3—122no—
514C2×C3—122no—
515C2×C3—242no—
516C2×C3C2×C3×#23711yescharacteristic
517C2×C3C3×D3×#4111yescharacteristic
518C2×C3C3×#101011yescharacteristic
519C22—22no—
520C22—22no—
521C22—22no—
522C22C23×(C3⋊D3)⋊C311yescharacteristic
523C4—22no—
524C4—21no—
525C4—21no—
526C4—92no—
527C4—182no—
528C4—182no—
529C4—181no—
530C4—181no—
531C4—181no—
532C4#3830⋊C312yes—
533C3—31no—
534C3—31no—
535C3—61no—
536C3C3×#567111yescharacteristic
537C2—42no—
538C2C2×#3830⋊C311yescharacteristic
539C2#41×(C3⋊D3)⋊C311yescharacteristic
540C2#1172263911yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#124C2
prime radicalC2×D4×H(3,𝔽3)C22
hypercenter#124(C3⋊D3)⋊C3
hypocenterC32C3×#556
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
2. Q8×(C3⋊Dic3)⋊C3C2semidirect
3. #64153⋊C3C2semidirect
6. #64088⋊C3C2semidirect
7. #64088⋊C3C2semidirect
8. #64087⋊C3C2semidirect
9. #64086⋊C3C2semidirect
10. C4×#634⋊C3C2semidirect
11. C4×#634⋊C3C2semidirect
12. #10596142C3semidirect
40. Q8×C3⋊Dic3C2×C3semidirect
41. #64153C2×C3semidirect
44. #64088C2×C3semidirect
45. #64088C2×C3semidirect
46. #64087C2×C3semidirect
47. #64086C2×C3semidirect
48. C4×#634C2×C3semidirect
49. C4×#634C2×C3semidirect
343. H(3,𝔽3)#556semidirect
485. C32#5671semidirect
486. C32C3×#556semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1242
Q8×(C3⋊Dic3)⋊C32
#64153⋊C32
#64145⋊C32
D4×(C3⋊Dic3)⋊C32
#64088⋊C32
#64087⋊C32
#64086⋊C32
C4×#634⋊C32
#105961423
C3×#56713