Finite Group Databasebeta

← #10596122⋊C3 #13153042

Subgroups — #10596122⋊C3 #13153042

Order 1728.

764 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×#123C211yesmaximal, characteristic
2#64145⋊C3C211yesmaximal, characteristic
3#64144⋊C3C211yesmaximal, characteristic
4#64133⋊C3C212yesmaximal, complemented
5#64098⋊C3C212yesmaximal, complemented
6#64097⋊C3C212yesmaximal, complemented
7#64088⋊C3C212yesmaximal, complemented
8#64088⋊C3C212yesmaximal, complemented
9#64087⋊C3C212yesmaximal, complemented
10#10596122C311yesmaximal, characteristic, complemented
11C3×#5651—31nomaximal
12C2×D4×H(3,𝔽3)C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
15C2×#634⋊C3C2212yes—
16C2×#634⋊C3C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)C2212yes—
18C22⋊C4×H(3,𝔽3)C2212yes—
19C22⋊C4×H(3,𝔽3)C2211yescharacteristic
20#3757⋊C3C2212yes—
21#3757⋊C3C2212yes—
22#3757⋊C3C2211yescharacteristic
23#3757⋊C3C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2212yes—
25C4⋊((C3⋊Dic3)⋊C3)C2212yes—
26#3750⋊C3C2212yes—
27#3750⋊C3C2212yes—
28#3750⋊C3C2212yes—
29#3750⋊C3C2212yes—
30#3750⋊C3C2212yes—
31C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
32C4×(C3⋊Dic3)⋊C3C2212yes—
33C32×#123—21no—
34C32×#123D311yescharacteristic
35C32×#123C2×C311yescharacteristic
36#64145C2×C311yescharacteristic
37#64144C2×C311yescharacteristic
38#64133C2×C312yescomplemented
39#64098C2×C312yescomplemented
40#64097C2×C312yescomplemented
41#64088C2×C312yescomplemented
42#64088C2×C312yescomplemented
43#64087C2×C312yescomplemented
44C3×#936—31no—
45C3×#935—31no—
46C3×#924—32no—
47C3×#889—32no—
48C3×#888—32no—
49C3×#879—32no—
50C3×#879—32no—
51C3×#878—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C23×H(3,𝔽3)C2311yescharacteristic
54D4×H(3,𝔽3)—41no—
55C2×C4×H(3,𝔽3)—41no—
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57C2×C4×H(3,𝔽3)C2312yes—
58C2×C4×H(3,𝔽3)C2312yes—
59C2×(C3⋊Dic3)⋊C3—41no—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2312yes—
63C2×(C3⋊Dic3)⋊C3C2312yes—
64#634⋊C3—41no—
65#634⋊C3—42no—
66C3×#563—91no—
67#5651—31no—
68#5651—91no—
69C2×C32×D4—21no—
70C2×C32×D4C2×D311yescharacteristic
71C2×C32×D4C22×C311yescharacteristic
72C22×C32×C4—21no—
73C22×C32×C4C2×D311yescharacteristic
74C22×C32×C4C22×C311yescharacteristic
75C22×C3⋊Dic3C22×C311yescharacteristic
76C2×#634C22×C312yes—
77C2×#634C22×C311yescharacteristic
78C22×C3×Dic3—31no—
79C2×C3×Dic6—32no—
80C2×C3×Dic6—31no—
81C32×C4⋊C4—22no—
82C32×C4⋊C4C2×D312yes—
83C32×C4⋊C4C22×C312yes—
84C32×C22⋊C4—21no—
85C32×C22⋊C4—22no—
86C32×C22⋊C4C2×D312yes—
87C32×C22⋊C4C2×D311yescharacteristic
88C32×C22⋊C4C22×C312yes—
89C32×C22⋊C4C22×C311yescharacteristic
90#3757C22×C312yes—
91#3757C22×C311yescharacteristic
92#3757C22×C312yes—
93#3757C22×C312yes—
94C4⋊(C3⋊Dic3)C22×C312yes—
95C4⋊(C3⋊Dic3)C22×C312yes—
96#3750C22×C312yes—
97#3750C22×C312yes—
98#3750C22×C312yes—
99#3750C22×C312yes—
100#3750C22×C312yes—
101C4×C3⋊Dic3C22×C312yes—
102C4×C3⋊Dic3C22×C311yescharacteristic
103C3×#217—32no—
104C3×#217—31no—
105C3×#217—32no—
106C3×#217—32no—
107C3×C4⋊Dic3—32no—
108C3×C4⋊Dic3—32no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—32no—
114C3×C4×Dic3—32no—
115C3×C4×Dic3—31no—
116C22×H(3,𝔽3)—21no—
117C22×H(3,𝔽3)—21no—
118C22×H(3,𝔽3)—22no—
119C22×H(3,𝔽3)—22no—
120C22×H(3,𝔽3)C2411yescharacteristic
121C4×H(3,𝔽3)—22no—
122C4×H(3,𝔽3)—21no—
123C4×H(3,𝔽3)—22no—
124(C3⋊Dic3)⋊C3—22no—
125(C3⋊Dic3)⋊C3—22no—
126(C3⋊Dic3)⋊C3—22no—
127(C3⋊Dic3)⋊C3—22no—
128C3×C4⋊Q8—92no—
129C3×#126—92no—
130C3×#126—92no—
131C3×#125—92no—
132C3×#124—91no—
133C3×#123—31no—
134C3×#123—31no—
135C3×#123—61no—
136C3×#123—91no—
137C3×#123C3×D311yescharacteristic
138C3×C22⋊Q8—92no—
139C3×C22⋊Q8—92no—
140#936—31no—
141#936—91no—
142#935—31no—
143#935—91no—
144#924—32no—
145#924—92no—
146#889—32no—
147#889—92no—
148#888—32no—
149#888—92no—
150#879—32no—
151#879—32no—
152#879—92no—
153#879—92no—
154#878—32no—
155#878—92no—
156C23×C32—21no—
157C23×C32—21no—
158C23×C32C22×D311yescharacteristic
159C23×C32C22×D311yescharacteristic
160C23×C32C23×C311yescharacteristic
161C23×C32C23×C311yescharacteristic
162C32×D4—41no—
163C32×D4—41no—
164C32×D4—42no—
165C2×C32×C4—22no—
166C2×C32×C4—22no—
167C2×C32×C4—21no—
168C2×C32×C4—41no—
169C2×C32×C4—41no—
170C2×C32×C4—42no—
171C2×C32×C4C22×D312yes—
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D312yes—
174C2×C32×C4C23×C312yes—
175C2×C32×C4C23×C311yescharacteristic
176C2×C32×C4C23×C312yes—
177C2×C3⋊Dic3—41no—
178C2×C3⋊Dic3C23×C312yes—
179C2×C3⋊Dic3C23×C312yes—
180C2×C3⋊Dic3C23×C312yes—
181C2×C3⋊Dic3C23×C312yes—
182#634—41no—
183#634—42no—
184C2×C3×Dic3—32no—
185C2×C3×Dic3—32no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—121no—
189C3×Dic6—121no—
190C3×Dic6—122no—
191#563—91noSylow
192C2×H(3,𝔽3)—41no—
193C2×H(3,𝔽3)—41no—
194C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
195C2×H(3,𝔽3)2−1+412yes—
196C2×C3×Q8—92no—
197C2×C3×Q8—91no—
198C2×C3×D4—31no—
199C2×C3×D4—31no—
200C2×C3×D4—61no—
201C2×C3×D4C2×C3×D311yescharacteristic
202C22×C3×C4—31no—
203C22×C3×C4—31no—
204C22×C3×C4—61no—
205C22×C3×C4—91no—
206C22×C3×C4C2×C3×D311yescharacteristic
207C22×Dic3—31no—
208C22×Dic3—91no—
209C2×Dic6—32no—
210C2×Dic6—31no—
211C2×Dic6—92no—
212C2×Dic6—91no—
213C3×C4⋊C4—32no—
214C3×C4⋊C4—32no—
215C3×C4⋊C4—62no—
216C3×C4⋊C4—92no—
217C3×C4⋊C4—92no—
218C3×C4⋊C4—92no—
219C3×C4⋊C4—92no—
220C3×C4⋊C4—92no—
221C3×C4⋊C4—92no—
222C3×C4⋊C4—92no—
223C3×C4⋊C4C2×C3×D312yes—
224C3×C22⋊C4—32no—
225C3×C22⋊C4—31no—
226C3×C22⋊C4—32no—
227C3×C22⋊C4—31no—
228C3×C22⋊C4—62no—
229C3×C22⋊C4—61no—
230C3×C22⋊C4—92no—
231C3×C22⋊C4—92no—
232C3×C22⋊C4—92no—
233C3×C22⋊C4—91no—
234C3×C22⋊C4C2×C3×D312yes—
235C3×C22⋊C4C2×C3×D311yescharacteristic
236C3×C42—91no—
237C3×C42—92no—
238#217—32no—
239#217—31no—
240#217—32no—
241#217—32no—
242#217—92no—
243#217—91no—
244#217—92no—
245#217—92no—
246C4⋊Dic3—32no—
247C4⋊Dic3—32no—
248C4⋊Dic3—92no—
249C4⋊Dic3—92no—
250Dic3⋊C4—32no—
251Dic3⋊C4—32no—
252Dic3⋊C4—32no—
253Dic3⋊C4—32no—
254Dic3⋊C4—32no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4—92no—
260C4×Dic3—32no—
261C4×Dic3—31no—
262C4×Dic3—92no—
263C4×Dic3—91no—
264C22×C32—22no—
265C22×C32—21no—
266C22×C32—21no—
267C22×C32—22no—
268C22×C32—22no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—22no—
272C22×C32—22no—
273C22×C32—21no—
274C22×C32—42no—
275C22×C32—41no—
276C22×C32—42no—
277C22×C32C23×D311yescharacteristic
278C22×C32C24×C311yescharacteristic
279C32×C4—22no—
280C32×C4—21no—
281C32×C4—22no—
282C32×C4—22no—
283C32×C4—21no—
284C32×C4—22no—
285C32×C4—42no—
286C32×C4—42no—
287C32×C4—41no—
288C3⋊Dic3—22no—
289C3⋊Dic3—22no—
290C3⋊Dic3—22no—
291C3⋊Dic3—22no—
292C3×Dic3—62no—
293C3×Dic3—62no—
294C3×Dic3—62no—
295C3×Dic3—62no—
296C4⋊Q8—92no—
297#126—92no—
298#126—92no—
299#125—92no—
300#124—91no—
301#123—91no—
302#123(C3⋊D3)⋊C311yescharacteristic
303C22⋊Q8—92no—
304C22⋊Q8—92no—
305H(3,𝔽3)#56311yescharacteristic, complemented, Sylow
306C23×C3—31no—
307C23×C3—31no—
308C23×C3—31no—
309C23×C3—31no—
310C23×C3—61no—
311C23×C3—61no—
312C23×C3C22×C3×D311yescharacteristic
313C23×C3C22×C3×D311yescharacteristic
314C3×Q8—362no—
315C3×Q8—361no—
316C3×D4—41no—
317C3×D4—121no—
318C3×D4—121no—
319C3×D4—122no—
320C2×C3×C4—32no—
321C2×C3×C4—31no—
322C2×C3×C4—32no—
323C2×C3×C4—32no—
324C2×C3×C4—31no—
325C2×C3×C4—32no—
326C2×C3×C4—41no—
327C2×C3×C4—62no—
328C2×C3×C4—62no—
329C2×C3×C4—61no—
330C2×C3×C4—92no—
331C2×C3×C4—92no—
332C2×C3×C4—92no—
333C2×C3×C4—92no—
334C2×C3×C4—121no—
335C2×C3×C4—121no—
336C2×C3×C4—122no—
337C2×C3×C4—361no—
338C2×C3×C4C22×C3×D312yes—
339C2×C3×C4C22×C3×D311yescharacteristic
340C2×C3×C4C22×C3×D312yes—
341C2×Dic3—32no—
342C2×Dic3—32no—
343C2×Dic3—32no—
344C2×Dic3—32no—
345C2×Dic3—92no—
346C2×Dic3—92no—
347C2×Dic3—92no—
348C2×Dic3—92no—
349C2×Dic3—121no—
350C2×Dic3—361no—
351Dic6—121no—
352Dic6—122no—
353Dic6—361no—
354Dic6—362no—
355C2×C32—21no—
356C2×C32—22no—
357C2×C32—42no—
358C2×C32—41no—
359C2×C32—41no—
360C2×C32—41no—
361C2×C32—41no—
362C2×C32—81no—
363C2×C32#100411yescharacteristic
364C2×C32#101012yes—
365C2×C32C3×H(5,𝔽2)11yescharacteristic
366C2×C32C3×2−1+412yes—
367C2×Q8—92no—
368C2×Q8—91no—
369C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
370C22×C4—91no—
371C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
372C4⋊C4—92no—
373C4⋊C4—92no—
374C4⋊C4—92no—
375C4⋊C4—92no—
376C4⋊C4—92no—
377C4⋊C4—92no—
378C4⋊C4—92no—
379C4⋊C4C2×(C3⋊D3)⋊C312yes—
380C22⋊C4—92no—
381C22⋊C4—92no—
382C22⋊C4—92no—
383C22⋊C4—91no—
384C22⋊C4C2×(C3⋊D3)⋊C312yes—
385C22⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
386C42—91no—
387C42—92no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—31no—
393C22×C3—31no—
394C22×C3—61no—
395C22×C3—61no—
396C22×C3—61no—
397C22×C3—61no—
398C22×C3—62no—
399C22×C3—62no—
400C22×C3—62no—
401C22×C3—62no—
402C22×C3—62no—
403C22×C3—61no—
404C22×C3—122no—
405C22×C3—121no—
406C22×C3—122no—
407C22×C3C23×C3×D311yescharacteristic
408C3×C4—22no—
409C3×C4—21no—
410C3×C4—22no—
411C3×C4—62no—
412C3×C4—61no—
413C3×C4—62no—
414C3×C4—61no—
415C3×C4—62no—
416C3×C4—62no—
417C3×C4—122no—
418C3×C4—121no—
419C3×C4—122no—
420C3×C4—182no—
421C3×C4—182no—
422C3×C4—182no—
423C3×C4—182no—
424Dic3—62no—
425Dic3—62no—
426Dic3—62no—
427Dic3—62no—
428Dic3—182no—
429Dic3—182no—
430Dic3—182no—
431Dic3—182no—
432C32—21no—
433C32#565111yescharacteristic, complemented
434C32C3×#56311yescharacteristic, complemented
435C23C22×(C3⋊D3)⋊C311yescharacteristic
436C23C22×(C3⋊D3)⋊C311yescharacteristic
437Q8—362no—
438Q8—361no—
439D4—41no—
440C2×C4—41no—
441C2×C4—92no—
442C2×C4—92no—
443C2×C4—92no—
444C2×C4—92no—
445C2×C4—361no—
446C2×C4C22×(C3⋊D3)⋊C312yes—
447C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
448C2×C4C22×(C3⋊D3)⋊C312yes—
449C2×C3—31no—
450C2×C3—31no—
451C2×C3—32no—
452C2×C3—32no—
453C2×C3—41no—
454C2×C3—41no—
455C2×C3—61no—
456C2×C3—62no—
457C2×C3—121no—
458C2×C3—121no—
459C2×C3—121no—
460C2×C3—121no—
461C2×C3—122no—
462C2×C3—241no—
463C2×C3C3×#100411yescharacteristic
464C2×C3C3×#101012yes—
465C22—21no—
466C22—21no—
467C22—22no—
468C22—22no—
469C22C23×(C3⋊D3)⋊C311yescharacteristic
470C4—22no—
471C4—21no—
472C4—22no—
473C4—182no—
474C4—182no—
475C4—182no—
476C4—182no—
477C3—31no—
478C3—31no—
479C3—61no—
480C3C3×#565111yescharacteristic
481C2—41no—
482C2—41no—
483C2#64361⋊C311yescharacteristic
484C2#1172263912yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#123C2
prime radicalC2×D4×H(3,𝔽3)C22
hypercenter#123(C3⋊D3)⋊C3
hypocenterC32C3×#563
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
4. #64133⋊C3C2semidirect
5. #64098⋊C3C2semidirect
6. #64097⋊C3C2semidirect
7. #64088⋊C3C2semidirect
8. #64088⋊C3C2semidirect
9. #64087⋊C3C2semidirect
10. #10596122C3semidirect
38. #64133C2×C3semidirect
39. #64098C2×C3semidirect
40. #64097C2×C3semidirect
41. #64088C2×C3semidirect
42. #64088C2×C3semidirect
43. #64087C2×C3semidirect
305. H(3,𝔽3)#563semidirect
433. C32#5651semidirect
434. C32C3×#563semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1232
#64145⋊C32
#64144⋊C32
#64133⋊C32
#64098⋊C32
#64097⋊C32
#64088⋊C32
#64087⋊C32
#105961223
C3×#56513