Finite Group Databasebeta

← #10596146⋊C3 #13153117

Subgroups — #10596146⋊C3 #13153117

Order 1728.

1021 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C2×#3830⋊C3C211yesmaximal, characteristic
3H(3,𝔽3)×#124C211yesmaximal, characteristic, complemented
4#64153⋊C3C211yesmaximal, characteristic, complemented
5#64145⋊C3C211yesmaximal, characteristic, complemented
6#64092⋊C3C214yesmaximal, complemented
7#64087⋊C3C214yesmaximal, complemented
8#64081⋊C3C211yesmaximal, characteristic, complemented
9#64078⋊C3C211yesmaximal, characteristic, complemented
10#10596146C311yesmaximal, characteristic, complemented
11C3×#5675—31nomaximal
12C2×Q8×H(3,𝔽3)C2211yescharacteristic
13C2×D4×H(3,𝔽3)C2211yescharacteristic
14C2×#638⋊C3C2212yes—
15C22×(C3⋊Dic3)⋊C3C2212yes—
16Q8×(C3⋊D3)⋊C3—24no—
17#3830⋊C3—24no—
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
20C2×#634⋊C3C2211yescharacteristic
21C2×#634⋊C3C2211yescharacteristic
22C2×#634⋊C3C2212yes—
23C22⋊C4×H(3,𝔽3)C2214yes—
24C42×H(3,𝔽3)C2211yescharacteristic
25#3757⋊C3C2214yes—
26#3752⋊C3C2212yes—
27C4⋊((C3⋊Dic3)⋊C3)C2214yes—
28#3750⋊C3C2212yes—
29#3750⋊C3C2214yes—
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
31C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
32C2×#3830C2×C311yescharacteristic
33C2×C3×D3×Q8—31no—
34C2×C3×#237—31no—
35C32×#124—21no—
36C32×#124D311yescharacteristic, complemented
37C32×#124C2×C311yescharacteristic, complemented
38#64153C2×C311yescharacteristic, complemented
39#64145C2×C311yescharacteristic, complemented
40#64092C2×C314yescomplemented
41#64087C2×C314yescomplemented
42#64081C2×C311yescharacteristic, complemented
43#64078C2×C311yescharacteristic, complemented
44C3×#944—31no—
45C3×#936—31no—
46C3×#883—34no—
47C3×#878—34no—
48C3×#872—31no—
49C3×#869—31no—
50C23×H(3,𝔽3)C2312yes—
51C22×(C3⋊D3)⋊C3C2311yescharacteristic
52Q8×H(3,𝔽3)—22no—
53D4×H(3,𝔽3)—22no—
54C2×C4×H(3,𝔽3)C2312yes—
55C2×C4×H(3,𝔽3)C2312yes—
56C2×C4×H(3,𝔽3)C2311yescharacteristic
57#638⋊C3—24no—
58C2×(C3⋊Dic3)⋊C3—24no—
59C2×(C3⋊Dic3)⋊C3C2312yes—
60C2×(C3⋊Dic3)⋊C3C2312yes—
61C2×(C3⋊Dic3)⋊C3C2312yes—
62C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
63C4×(C3⋊D3)⋊C3—24no—
64C4×(C3⋊D3)⋊C3D414yes—
65#634⋊C3—22no—
66#634⋊C3—22no—
67#634⋊C3—24no—
68C3×#536—91no—
69#5675—31no—
70#5675—91no—
71C2×C32×Q8—21no—
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C22×C311yescharacteristic
74C2×C32×D4—21no—
75C2×C32×D4C2×D311yescharacteristic
76C2×C32×D4C22×C311yescharacteristic
77C2×#638C22×C312yes—
78C22×C3⋊Dic3C22×C312yes—
79Q8×C3⋊D3—24no—
80#3830—24no—
81C2×C4×C3⋊D3C22×C312yes—
82C2×C4×C3⋊D3C22×C311yescharacteristic
83C2×#634C22×C311yescharacteristic
84C2×#634C22×C311yescharacteristic
85C2×#634C22×C312yes—
86C2×C3×#67—32no—
87C22×C3×Dic3—32no—
88C3×D3×Q8—64no—
89C3×#237—64no—
90C2×C3×C4×D3—31no—
91C2×C3×C4×D3—32no—
92C2×C3×Dic6—32no—
93C2×C3×Dic6—31no—
94C2×C3×Dic6—31no—
95C32×C22⋊C4—24no—
96C32×C22⋊C4C2×D314yes—
97C32×C22⋊C4C22×C314yes—
98C32×C42—21no—
99C32×C42C2×D311yescharacteristic
100C32×C42C22×C311yescharacteristic
101#3757C22×C314yes—
102#3752C22×C312yes—
103C4⋊(C3⋊Dic3)C22×C314yes—
104#3750C22×C312yes—
105#3750C22×C314yes—
106C4×C3⋊Dic3C22×C311yescharacteristic
107C3×#217—34no—
108C3×(C2×D3)⋊C4—32no—
109C3×C4⋊Dic3—34no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—34no—
112C3×C4×Dic3—31no—
113C22×H(3,𝔽3)—22no—
114C22×H(3,𝔽3)—24no—
115C22×H(3,𝔽3)C2411yescharacteristic
116C2×(C3⋊D3)⋊C3—22no—
117C2×(C3⋊D3)⋊C3C2×D412yes—
118C4×H(3,𝔽3)—22no—
119C4×H(3,𝔽3)—22no—
120C4×H(3,𝔽3)C2×D412yes—
121(C3⋊Dic3)⋊C3—22no—
122(C3⋊Dic3)⋊C3—22no—
123(C3⋊Dic3)⋊C3—22no—
124(C3⋊Dic3)⋊C3C2×D412yes—
125C2×C3×#41—91no—
126C22×C3×Q8—91no—
127C2×D3×Q8—31no—
128C2×D3×Q8—91no—
129C2×#237—31no—
130C2×#237—91no—
131C3×C4⋊Q8—91no—
132C3×C4⋊Q8—91no—
133C3×#124—31no—
134C3×#124—31no—
135C3×#124—61no—
136C3×#124—91no—
137C3×#124C3×D311yescharacteristic
138C3×#123—94no—
139C3×C22⋊Q8—94no—
140C3×#117—91no—
141#944—31no—
142#944—91no—
143#936—31no—
144#936—91no—
145#883—34no—
146#883—94no—
147#878—34no—
148#878—94no—
149#872—31no—
150#872—91no—
151#869—31no—
152#869—91no—
153C23×C32—22no—
154C23×C32C22×D312yes—
155C23×C32C23×C312yes—
156C22×C3⋊D3C23×C311yescharacteristic
157C22×C3×D3—31no—
158C32×Q8—22no—
159C32×Q8—22no—
160C32×Q8—42no—
161C32×D4—22no—
162C32×D4—22no—
163C32×D4—42no—
164C2×C32×C4—21no—
165C2×C32×C4—22no—
166C2×C32×C4—22no—
167C2×C32×C4C22×D312yes—
168C2×C32×C4C22×D312yes—
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C23×C312yes—
171C2×C32×C4C23×C312yes—
172C2×C32×C4C23×C311yescharacteristic
173#638—24no—
174C2×C3⋊Dic3—24no—
175C2×C3⋊Dic3C23×C312yes—
176C2×C3⋊Dic3C23×C312yes—
177C2×C3⋊Dic3C23×C312yes—
178C2×C3⋊Dic3C23×C311yescharacteristic
179C4×C3⋊D3—24no—
180C4×C3⋊D3C3×D414yes—
181#634—22no—
182#634—22no—
183#634—24no—
184C3×#67—64no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—32no—
188C2×C3×Dic3—32no—
189C2×C3×Dic3—64no—
190C3×C4×D3—34no—
191C3×C4×D3—64no—
192C3×Dic6—64no—
193C3×Dic6—62no—
194C3×Dic6—62no—
195#536—91noSylow
196C2×H(3,𝔽3)—42no—
197C2×H(3,𝔽3)C22×D411yescharacteristic
198C2×H(3,𝔽3)2−1+412yes—
199(C3⋊D3)⋊C3—22no—
200C3×#41—184no—
201C2×C3×Q8—31no—
202C2×C3×Q8—31no—
203C2×C3×Q8—61no—
204C2×C3×Q8—92no—
205C2×C3×Q8—91no—
206C2×C3×Q8—91no—
207C2×C3×Q8—184no—
208C2×C3×Q8C2×C3×D311yescharacteristic
209C2×C3×D4—31no—
210C2×C3×D4—31no—
211C2×C3×D4—61no—
212C2×C3×D4—92no—
213C2×C3×D4C2×C3×D311yescharacteristic
214C22×C3×C4—91no—
215C22×C3×C4—92no—
216C22×C3×C4—92no—
217C2×#67—32no—
218C2×#67—92no—
219C22×Dic3—32no—
220C22×Dic3—92no—
221D3×Q8—64no—
222D3×Q8—184no—
223#237—64no—
224#237—184no—
225C2×C4×D3—31no—
226C2×C4×D3—32no—
227C2×C4×D3—91no—
228C2×C4×D3—92no—
229C2×Dic6—31no—
230C2×Dic6—31no—
231C2×Dic6—32no—
232C2×Dic6—91no—
233C2×Dic6—91no—
234C2×Dic6—92no—
235C3×C4⋊C4—92no—
236C3×C4⋊C4—94no—
237C3×C4⋊C4—94no—
238C3×C22⋊C4—34no—
239C3×C22⋊C4—34no—
240C3×C22⋊C4—64no—
241C3×C22⋊C4—92no—
242C3×C22⋊C4—94no—
243C3×C22⋊C4C2×C3×D314yes—
244C3×C42—31no—
245C3×C42—31no—
246C3×C42—61no—
247C3×C42—91no—
248C3×C42C2×C3×D311yescharacteristic
249#217—34no—
250#217—94no—
251(C2×D3)⋊C4—32no—
252(C2×D3)⋊C4—92no—
253C4⋊Dic3—34no—
254C4⋊Dic3—94no—
255Dic3⋊C4—32no—
256Dic3⋊C4—34no—
257Dic3⋊C4—92no—
258Dic3⋊C4—94no—
259C4×Dic3—31no—
260C4×Dic3—91no—
261C22×C32—22no—
262C22×C32—24no—
263C22×C32—22no—
264C22×C32—24no—
265C22×C32—21no—
266C22×C32—42no—
267C22×C32—44no—
268C22×C32C23×D311yescharacteristic
269C22×C32C24×C311yescharacteristic
270C2×C3⋊D3—22no—
271C2×C3⋊D3C2×C3×D412yes—
272C2×C3×D3—32no—
273C2×C3×D3—62no—
274C32×C4—22no—
275C32×C4—22no—
276C32×C4—22no—
277C32×C4—22no—
278C32×C4—22no—
279C32×C4—42no—
280C32×C4—42no—
281C32×C4D3×D412yes—
282C32×C4C2×C3×D412yes—
283C3⋊Dic3—22no—
284C3⋊Dic3—22no—
285C3⋊Dic3—22no—
286C3⋊Dic3C2×C3×D412yes—
287C3×Dic3—32no—
288C3×Dic3—62no—
289C3×Dic3—62no—
290C3×Dic3—62no—
291C2×#41—91no—
292C22×Q8—91no—
293C4⋊Q8—91no—
294C4⋊Q8—91no—
295#124—91no—
296#124(C3⋊D3)⋊C311yescharacteristic, complemented
297#123—94no—
298C22⋊Q8—94no—
299#117—91no—
300H(3,𝔽3)#53611yescharacteristic, complemented, Sylow
301C23×C3—32no—
302C23×C3—32no—
303C23×C3—62no—
304C23×C3—91no—
305C23×C3C22×C3×D312yes—
306C22×D3—31no—
307C22×D3—91no—
308C3×Q8—22no—
309C3×Q8—62no—
310C3×Q8—62no—
311C3×Q8—122no—
312C3×Q8—182no—
313C3×Q8—182no—
314C3×Q8—184no—
315C3×D4—22no—
316C3×D4—62no—
317C3×D4—62no—
318C3×D4—122no—
319C3×D4—184no—
320C2×C3×C4—31no—
321C2×C3×C4—31no—
322C2×C3×C4—32no—
323C2×C3×C4—32no—
324C2×C3×C4—32no—
325C2×C3×C4—32no—
326C2×C3×C4—61no—
327C2×C3×C4—62no—
328C2×C3×C4—62no—
329C2×C3×C4—94no—
330C2×C3×C4—91no—
331C2×C3×C4—92no—
332C2×C3×C4—92no—
333C2×C3×C4—92no—
334C2×C3×C4—184no—
335C2×C3×C4—184no—
336C2×C3×C4C22×C3×D312yes—
337C2×C3×C4C22×C3×D312yes—
338C2×C3×C4C22×C3×D311yescharacteristic
339#67—64no—
340#67—184no—
341C2×Dic3—31no—
342C2×Dic3—32no—
343C2×Dic3—32no—
344C2×Dic3—32no—
345C2×Dic3—64no—
346C2×Dic3—91no—
347C2×Dic3—92no—
348C2×Dic3—92no—
349C2×Dic3—92no—
350C2×Dic3—184no—
351C4×D3—34no—
352C4×D3—64no—
353C4×D3—94no—
354C4×D3—184no—
355Dic6—64no—
356Dic6—62no—
357Dic6—62no—
358Dic6—184no—
359Dic6—182no—
360Dic6—182no—
361C2×C32—21no—
362C2×C32—22no—
363C2×C32—42no—
364C2×C32—42no—
365C2×C32—82no—
366C2×C32C2×D3×D411yescharacteristic
367C2×C32#101012yes—
368C2×C32C22×C3×D411yescharacteristic
369C2×C32C3×2−1+412yes—
370C3⋊D3—22no—
371C3×D3—62no—
372#41—184no—
373C2×Q8—92no—
374C2×Q8—91no—
375C2×Q8—91no—
376C2×Q8—184no—
377C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
378C2×D4—92no—
379C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
380C22×C4—91no—
381C22×C4—92no—
382C22×C4—92no—
383C4⋊C4—92no—
384C4⋊C4—94no—
385C4⋊C4—94no—
386C22⋊C4—94no—
387C22⋊C4—92no—
388C22⋊C4C2×(C3⋊D3)⋊C314yes—
389C42—91no—
390C42C2×(C3⋊D3)⋊C311yescharacteristic
391C22×C3—22no—
392C22×C3—24no—
393C22×C3—31no—
394C22×C3—31no—
395C22×C3—62no—
396C22×C3—62no—
397C22×C3—64no—
398C22×C3—64no—
399C22×C3—61no—
400C22×C3—92no—
401C22×C3—122no—
402C22×C3—124no—
403C22×C3—182no—
404C22×C3C23×C3×D311yescharacteristic
405C2×D3—32no—
406C2×D3—62no—
407C2×D3—92no—
408C2×D3—182no—
409C3×C4—22no—
410C3×C4—22no—
411C3×C4—32no—
412C3×C4—32no—
413C3×C4—62no—
414C3×C4—62no—
415C3×C4—62no—
416C3×C4—62no—
417C3×C4—62no—
418C3×C4—92no—
419C3×C4—122no—
420C3×C4—122no—
421C3×C4—182no—
422C3×C4—182no—
423C3×C4—182no—
424C3×C4C3×D3×D412yes—
425Dic3—32no—
426Dic3—62no—
427Dic3—62no—
428Dic3—62no—
429Dic3—92no—
430Dic3—182no—
431Dic3—182no—
432Dic3—182no—
433C32—21no—
434C32#567511yescharacteristic, complemented
435C32C3×#53611yescharacteristic, complemented
436C23—91no—
437C23C22×(C3⋊D3)⋊C312yes—
438Q8—22no—
439Q8—184no—
440Q8—182no—
441Q8—182no—
442D4—22no—
443D4—184no—
444C2×C4—91no—
445C2×C4—94no—
446C2×C4—92no—
447C2×C4—92no—
448C2×C4—92no—
449C2×C4—184no—
450C2×C4—184no—
451C2×C4C22×(C3⋊D3)⋊C312yes—
452C2×C4C22×(C3⋊D3)⋊C312yes—
453C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
454C2×C3—31no—
455C2×C3—31no—
456C2×C3—32no—
457C2×C3—32no—
458C2×C3—42no—
459C2×C3—61no—
460C2×C3—62no—
461C2×C3—122no—
462C2×C3—122no—
463C2×C3—182no—
464C2×C3—242no—
465C2×C3C2×C3×D3×D411yescharacteristic
466C2×C3C3×#101012yes—
467D3—62no—
468D3—182no—
469C22—22no—
470C22—24no—
471C22—92no—
472C22—182no—
473C22C23×(C3⋊D3)⋊C311yescharacteristic
474C4—22no—
475C4—22no—
476C4—92no—
477C4—182no—
478C4—182no—
479C4—182no—
480C4D4×(C3⋊D3)⋊C312yes—
481C3—31no—
482C3—31no—
483C3—61no—
484C3C3×#567511yescharacteristic
485C2—42no—
486C2—182no—
487C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
488C2#1172263912yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#124C2
prime radicalC2×#3830⋊C3C2
hypercenter#124(C3⋊D3)⋊C3
hypocenterC32C3×#536
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #124

Complements

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

NH ≅ G/Ntype
1. C2×Q8×(C3⋊D3)⋊C3C2semidirect
3. H(3,𝔽3)×#124C2semidirect
4. #64153⋊C3C2semidirect
5. #64145⋊C3C2semidirect
6. #64092⋊C3C2semidirect
7. #64087⋊C3C2semidirect
8. #64081⋊C3C2semidirect
9. #64078⋊C3C2semidirect
10. #10596146C3semidirect
31. C2×Q8×C3⋊D3C2×C3semidirect
36. C32×#124D3semidirect
37. C32×#124C2×C3semidirect
38. #64153C2×C3semidirect
39. #64145C2×C3semidirect
40. #64092C2×C3semidirect
41. #64087C2×C3semidirect
42. #64081C2×C3semidirect
43. #64078C2×C3semidirect
296. #124(C3⋊D3)⋊C3semidirect
300. H(3,𝔽3)#536semidirect
434. C32#5675semidirect
435. C32C3×#536semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×(C3⋊D3)⋊C32
C2×#3830⋊C32
H(3,𝔽3)×#1242
#64153⋊C32
#64145⋊C32
#64092⋊C32
#64087⋊C32
#64081⋊C32
#64078⋊C32
#105961463
C3×#56753