Finite Group Databasebeta

← #10596096⋊C3 #13152950

Subgroups — #10596096⋊C3 #13152950

Order 1728.

746 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×C22⋊Q8C211yesmaximal, characteristic
2Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
3#64153⋊C3C211yesmaximal, characteristic, complemented
4#64135⋊C3C211yesmaximal, characteristic
5#64133⋊C3C211yesmaximal, characteristic
6#64099⋊C3C211yesmaximal, characteristic, complemented
7#64099⋊C3C212yesmaximal, complemented
8#64098⋊C3C212yesmaximal, complemented
9#64096⋊C3C211yesmaximal, characteristic, complemented
10#64088⋊C3C212yesmaximal
11#64088⋊C3C212yesmaximal
12#10596096C311yesmaximal, characteristic, complemented
13C3×#5625—31nomaximal
14C2×Q8×H(3,𝔽3)C2211yescharacteristic
15C22×C4×H(3,𝔽3)C2211yescharacteristic
16C2×#634⋊C3C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2212yes—
19C22⋊C4×H(3,𝔽3)C2212yes—
20#3757⋊C3C2212yes—
21#3757⋊C3C2211yescharacteristic
22#3757⋊C3C2211yescharacteristic
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26C4⋊((C3⋊Dic3)⋊C3)C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2212yes—
28#3750⋊C3C2212yes—
29#3750⋊C3C2212yes—
30#3750⋊C3C2212yes—
31#3750⋊C3C2212yes—
32#3750⋊C3C2212yes—
33C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
34C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
35C4×(C3⋊Dic3)⋊C3C2212yes—
36C4×(C3⋊Dic3)⋊C3C2212yes—
37C32×C22⋊Q8—21no—
38C32×C22⋊Q8D311yescharacteristic
39C32×C22⋊Q8C2×C311yescharacteristic
40Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
41#64153C2×C311yescharacteristic, complemented
42#64135C2×C311yescharacteristic
43#64133C2×C311yescharacteristic
44#64099C2×C312yescomplemented
45#64099C2×C311yescharacteristic, complemented
46#64098C2×C312yescomplemented
47#64096C2×C311yescharacteristic, complemented
48#64088C2×C312yes—
49#64088C2×C312yes—
50C3×Q8×Dic3—31no—
51C3×#944—31no—
52C3×#926—31no—
53C3×#924—31no—
54C3×#890—32no—
55C3×#890—31no—
56C3×#889—32no—
57C3×#887—31no—
58C3×#879—32no—
59C3×#879—32no—
60C23×H(3,𝔽3)C2311yescharacteristic
61Q8×H(3,𝔽3)—22no—
62C2×C4×H(3,𝔽3)—22no—
63C2×C4×H(3,𝔽3)C2312yes—
64C2×C4×H(3,𝔽3)C2312yes—
65C2×C4×H(3,𝔽3)C2311yescharacteristic
66C2×C4×H(3,𝔽3)C2311yescharacteristic
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2312yes—
73#634⋊C3—22no—
74C3×#541—91no—
75#5625—31no—
76#5625—91no—
77C2×C32×Q8—21no—
78C2×C32×Q8C2×D311yescharacteristic
79C2×C32×Q8C22×C311yescharacteristic
80C22×C32×C4—21no—
81C22×C32×C4C2×D311yescharacteristic
82C22×C32×C4C22×C311yescharacteristic
83C2×#634C22×C311yescharacteristic
84C2×C3×Dic6—31no—
85C32×C4⋊C4—21no—
86C32×C4⋊C4—22no—
87C32×C4⋊C4C2×D311yescharacteristic
88C32×C4⋊C4C2×D312yes—
89C32×C4⋊C4C22×C311yescharacteristic
90C32×C4⋊C4C22×C312yes—
91C32×C22⋊C4—22no—
92C32×C22⋊C4C2×D312yes—
93C32×C22⋊C4C22×C312yes—
94#3757C22×C312yes—
95#3757C22×C311yescharacteristic
96#3757C22×C311yescharacteristic
97C4⋊(C3⋊Dic3)C22×C312yes—
98C4⋊(C3⋊Dic3)C22×C312yes—
99C4⋊(C3⋊Dic3)C22×C311yescharacteristic
100C4⋊(C3⋊Dic3)C22×C311yescharacteristic
101C4⋊(C3⋊Dic3)C22×C311yescharacteristic
102#3750C22×C312yes—
103#3750C22×C312yes—
104#3750C22×C312yes—
105#3750C22×C312yes—
106#3750C22×C312yes—
107C4×C3⋊Dic3C22×C312yes—
108C4×C3⋊Dic3C22×C312yes—
109C4×C3⋊Dic3C22×C311yescharacteristic
110C4×C3⋊Dic3C22×C311yescharacteristic
111C3×#217—32no—
112C3×#217—31no—
113C3×#217—31no—
114C3×C4⋊Dic3—32no—
115C3×C4⋊Dic3—32no—
116C3×C4⋊Dic3—31no—
117C3×C4⋊Dic3—31no—
118C3×C4⋊Dic3—31no—
119C3×Dic3⋊C4—32no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—32no—
122C3×Dic3⋊C4—32no—
123C3×Dic3⋊C4—32no—
124C3×C4×Dic3—32no—
125C3×C4×Dic3—32no—
126C3×C4×Dic3—31no—
127C3×C4×Dic3—31no—
128C22×H(3,𝔽3)—21no—
129C22×H(3,𝔽3)—21no—
130C22×H(3,𝔽3)—21no—
131C22×H(3,𝔽3)C2411yescharacteristic
132C4×H(3,𝔽3)—21no—
133C4×H(3,𝔽3)—22no—
134C4×H(3,𝔽3)—22no—
135C4×H(3,𝔽3)#4112yes—
136(C3⋊Dic3)⋊C3—22no—
137(C3⋊Dic3)⋊C3—21no—
138(C3⋊Dic3)⋊C3—21no—
139(C3⋊Dic3)⋊C3—22no—
140(C3⋊Dic3)⋊C3—21no—
141(C3⋊Dic3)⋊C3—21no—
142C3×C4⋊Q8—91no—
143C3×#126—92no—
144C3×#126—92no—
145C3×#125—92no—
146C3×#125—91no—
147C3×#125—92no—
148C3×C22⋊Q8—31no—
149C3×C22⋊Q8—31no—
150C3×C22⋊Q8—61no—
151C3×C22⋊Q8—91no—
152C3×C22⋊Q8C3×D311yescharacteristic
153C3×C4×Q8—91no—
154C3×C4×Q8—91no—
155C3×#117—91no—
156Q8×Dic3—31no—
157Q8×Dic3—91no—
158#944—31no—
159#944—91no—
160#926—31no—
161#926—91no—
162#924—31no—
163#924—91no—
164#890—32no—
165#890—31no—
166#890—92no—
167#890—91no—
168#889—32no—
169#889—92no—
170#887—31no—
171#887—91no—
172#879—32no—
173#879—32no—
174#879—92no—
175#879—92no—
176C23×C32—21no—
177C23×C32C22×D311yescharacteristic
178C23×C32C23×C311yescharacteristic
179C32×Q8—22no—
180C32×Q8—22no—
181C32×Q8—42no—
182C2×C32×C4—22no—
183C2×C32×C4—22no—
184C2×C32×C4—21no—
185C2×C32×C4—22no—
186C2×C32×C4—22no—
187C2×C32×C4—21no—
188C2×C32×C4—42no—
189C2×C32×C4C22×D312yes—
190C2×C32×C4C22×D312yes—
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C23×C312yes—
194C2×C32×C4C23×C312yes—
195C2×C32×C4C23×C311yescharacteristic
196C2×C32×C4C23×C311yescharacteristic
197C2×C3⋊Dic3C23×C311yescharacteristic
198C2×C3⋊Dic3C23×C311yescharacteristic
199C2×C3⋊Dic3C23×C311yescharacteristic
200C2×C3⋊Dic3C23×C312yes—
201C2×C3⋊Dic3C23×C312yes—
202C2×C3⋊Dic3C23×C311yescharacteristic
203#634—22no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—31no—
210C3×Dic6—62no—
211#541—91noSylow
212C2×H(3,𝔽3)—41no—
213C2×H(3,𝔽3)C2×#4111yescharacteristic
214C2×H(3,𝔽3)2−1+411yescharacteristic
215C2×H(3,𝔽3)2−1+411yescharacteristic
216C2×C3×Q8—31no—
217C2×C3×Q8—31no—
218C2×C3×Q8—61no—
219C2×C3×Q8—91no—
220C2×C3×Q8C2×C3×D311yescharacteristic
221C22×C3×C4—31no—
222C22×C3×C4—31no—
223C22×C3×C4—61no—
224C22×C3×C4C2×C3×D311yescharacteristic
225C2×Dic6—31no—
226C2×Dic6—91no—
227C3×C4⋊C4—31no—
228C3×C4⋊C4—32no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—32no—
231C3×C4⋊C4—61no—
232C3×C4⋊C4—62no—
233C3×C4⋊C4—92no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—92no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—91no—
239C3×C4⋊C4—91no—
240C3×C4⋊C4—91no—
241C3×C4⋊C4—92no—
242C3×C4⋊C4—92no—
243C3×C4⋊C4C2×C3×D311yescharacteristic
244C3×C4⋊C4C2×C3×D312yes—
245C3×C22⋊C4—32no—
246C3×C22⋊C4—32no—
247C3×C22⋊C4—62no—
248C3×C22⋊C4—91no—
249C3×C22⋊C4—91no—
250C3×C22⋊C4—92no—
251C3×C22⋊C4C2×C3×D312yes—
252C3×C42—91no—
253C3×C42—91no—
254C3×C42—92no—
255C3×C42—92no—
256#217—32no—
257#217—31no—
258#217—31no—
259#217—92no—
260#217—91no—
261#217—91no—
262C4⋊Dic3—32no—
263C4⋊Dic3—32no—
264C4⋊Dic3—31no—
265C4⋊Dic3—31no—
266C4⋊Dic3—31no—
267C4⋊Dic3—92no—
268C4⋊Dic3—92no—
269C4⋊Dic3—91no—
270C4⋊Dic3—91no—
271C4⋊Dic3—91no—
272Dic3⋊C4—32no—
273Dic3⋊C4—32no—
274Dic3⋊C4—32no—
275Dic3⋊C4—32no—
276Dic3⋊C4—32no—
277Dic3⋊C4—92no—
278Dic3⋊C4—92no—
279Dic3⋊C4—92no—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282C4×Dic3—32no—
283C4×Dic3—32no—
284C4×Dic3—31no—
285C4×Dic3—31no—
286C4×Dic3—92no—
287C4×Dic3—92no—
288C4×Dic3—91no—
289C4×Dic3—91no—
290C22×C32—22no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32—41no—
299C22×C32—41no—
300C22×C32C23×D311yescharacteristic
301C22×C32C24×C311yescharacteristic
302C32×C4—21no—
303C32×C4—22no—
304C32×C4—22no—
305C32×C4—21no—
306C32×C4—22no—
307C32×C4—22no—
308C32×C4—22no—
309C32×C4—42no—
310C32×C4—41no—
311C32×C4—42no—
312C32×C4#23712yes—
313C32×C4C3×#4112yes—
314C3⋊Dic3—22no—
315C3⋊Dic3—21no—
316C3⋊Dic3—22no—
317C3⋊Dic3—21no—
318C3⋊Dic3—21no—
319C3⋊Dic3—21no—
320C3×Dic3—62no—
321C3×Dic3—61no—
322C3×Dic3—62no—
323C3×Dic3—61no—
324C3×Dic3—61no—
325C3×Dic3—61no—
326C4⋊Q8—91no—
327#126—92no—
328#126—92no—
329#125—92no—
330#125—91no—
331#125—92no—
332C22⋊Q8—91no—
333C22⋊Q8(C3⋊D3)⋊C311yescharacteristic
334C4×Q8—91no—
335C4×Q8—91no—
336#117—91no—
337H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
338C23×C3—31no—
339C23×C3—31no—
340C23×C3—61no—
341C23×C3C22×C3×D311yescharacteristic
342C3×Q8—22no—
343C3×Q8—62no—
344C3×Q8—62no—
345C3×Q8—122no—
346C3×Q8—182no—
347C2×C3×C4—22no—
348C2×C3×C4—31no—
349C2×C3×C4—31no—
350C2×C3×C4—32no—
351C2×C3×C4—32no—
352C2×C3×C4—31no—
353C2×C3×C4—32no—
354C2×C3×C4—32no—
355C2×C3×C4—31no—
356C2×C3×C4—62no—
357C2×C3×C4—62no—
358C2×C3×C4—61no—
359C2×C3×C4—62no—
360C2×C3×C4—62no—
361C2×C3×C4—61no—
362C2×C3×C4—91no—
363C2×C3×C4—91no—
364C2×C3×C4—91no—
365C2×C3×C4—91no—
366C2×C3×C4—92no—
367C2×C3×C4—92no—
368C2×C3×C4—122no—
369C2×C3×C4C22×C3×D312yes—
370C2×C3×C4C22×C3×D312yes—
371C2×C3×C4C22×C3×D311yescharacteristic
372C2×C3×C4C22×C3×D311yescharacteristic
373C2×Dic3—31no—
374C2×Dic3—31no—
375C2×Dic3—31no—
376C2×Dic3—32no—
377C2×Dic3—32no—
378C2×Dic3—31no—
379C2×Dic3—91no—
380C2×Dic3—91no—
381C2×Dic3—91no—
382C2×Dic3—92no—
383C2×Dic3—92no—
384C2×Dic3—91no—
385Dic6—62no—
386Dic6—182no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—42no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32C2×#23711yescharacteristic
394C2×C32#100711yescharacteristic
395C2×C32#101011yescharacteristic
396C2×C32C2×C3×#4111yescharacteristic
397C2×C32C3×2−1+411yescharacteristic
398C2×C32C3×2−1+411yescharacteristic
399C2×Q8—91no—
400C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
401C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
402C4⋊C4—91no—
403C4⋊C4—91no—
404C4⋊C4—91no—
405C4⋊C4—92no—
406C4⋊C4—92no—
407C4⋊C4—92no—
408C4⋊C4—92no—
409C4⋊C4—92no—
410C4⋊C4—92no—
411C4⋊C4—92no—
412C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
413C4⋊C4C2×(C3⋊D3)⋊C312yes—
414C22⋊C4—91no—
415C22⋊C4—91no—
416C22⋊C4—92no—
417C22⋊C4C2×(C3⋊D3)⋊C312yes—
418C42—91no—
419C42—91no—
420C42—92no—
421C42—92no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—31no—
426C22×C3—31no—
427C22×C3—61no—
428C22×C3—61no—
429C22×C3—61no—
430C22×C3—61no—
431C22×C3—61no—
432C22×C3—61no—
433C22×C3—62no—
434C22×C3—61no—
435C22×C3—121no—
436C22×C3—121no—
437C22×C3C23×C3×D311yescharacteristic
438C3×C4—21no—
439C3×C4—22no—
440C3×C4—22no—
441C3×C4—32no—
442C3×C4—32no—
443C3×C4—61no—
444C3×C4—61no—
445C3×C4—62no—
446C3×C4—62no—
447C3×C4—62no—
448C3×C4—62no—
449C3×C4—62no—
450C3×C4—121no—
451C3×C4—122no—
452C3×C4—122no—
453C3×C4—182no—
454C3×C4—181no—
455C3×C4—181no—
456C3×C4—182no—
457C3×C4—181no—
458C3×C4—181no—
459C3×C4C3×#23712yes—
460Dic3—62no—
461Dic3—61no—
462Dic3—62no—
463Dic3—61no—
464Dic3—61no—
465Dic3—61no—
466Dic3—182no—
467Dic3—181no—
468Dic3—182no—
469Dic3—181no—
470Dic3—181no—
471Dic3—181no—
472C32—21no—
473C32#562511yescharacteristic, complemented
474C32C3×#54111yescharacteristic, complemented
475C23C22×(C3⋊D3)⋊C311yescharacteristic
476Q8—22no—
477Q8—182no—
478C2×C4—22no—
479C2×C4—91no—
480C2×C4—91no—
481C2×C4—91no—
482C2×C4—92no—
483C2×C4—91no—
484C2×C4—92no—
485C2×C4C22×(C3⋊D3)⋊C312yes—
486C2×C4C22×(C3⋊D3)⋊C312yes—
487C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
488C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—31no—
492C2×C3—31no—
493C2×C3—31no—
494C2×C3—31no—
495C2×C3—41no—
496C2×C3—61no—
497C2×C3—61no—
498C2×C3—61no—
499C2×C3—121no—
500C2×C3—121no—
501C2×C3—122no—
502C2×C3C2×C3×#23711yescharacteristic
503C2×C3C3×#100711yescharacteristic
504C2×C3C3×#101011yescharacteristic
505C22—21no—
506C22—21no—
507C22—21no—
508C22C23×(C3⋊D3)⋊C311yescharacteristic
509C4—21no—
510C4—22no—
511C4—22no—
512C4—182no—
513C4—181no—
514C4—181no—
515C4—182no—
516C4—181no—
517C4—181no—
518C4#3830⋊C312yes—
519C3—31no—
520C3—31no—
521C3—61no—
522C3C3×#562511yescharacteristic
523C2—41no—
524C2C2×#3830⋊C311yescharacteristic
525C2#1172262711yescharacteristic
526C2#1172263911yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C22⋊Q8C2
prime radicalC23×H(3,𝔽3)C23
hypercenterC22⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#541
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. Q8×(C3⋊Dic3)⋊C3C2semidirect
3. #64153⋊C3C2semidirect
6. #64099⋊C3C2semidirect
7. #64099⋊C3C2semidirect
8. #64098⋊C3C2semidirect
9. #64096⋊C3C2semidirect
12. #10596096C3semidirect
40. Q8×C3⋊Dic3C2×C3semidirect
41. #64153C2×C3semidirect
44. #64099C2×C3semidirect
45. #64099C2×C3semidirect
46. #64098C2×C3semidirect
47. #64096C2×C3semidirect
337. H(3,𝔽3)#541semidirect
473. C32#5625semidirect
474. C32C3×#541semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C22⋊Q82
Q8×(C3⋊Dic3)⋊C32
#64153⋊C32
#64135⋊C32
#64133⋊C32
#64099⋊C32
#64098⋊C32
#64096⋊C32
#64088⋊C32
#105960963
C3×#56253