Finite Group Databasebeta

← #10596195⋊C3 #13153321

Subgroups — #10596195⋊C3 #13153321

Order 1728.

1013 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
1C2×#88722C211yesmaximal, characteristic
2C2×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
4#64156⋊C3C211yesmaximal, characteristic, complemented
5#64153⋊C3C211yesmaximal, characteristic, complemented
6#64105⋊C3C214yesmaximal, complemented
7#64103⋊C3C214yesmaximal, complemented
8#64084⋊C3C211yesmaximal, characteristic, complemented
9#64081⋊C3C211yesmaximal, characteristic, complemented
10#10596195C311yesmaximal, characteristic, complemented
11C3×#5724—31nomaximal
12C2×Q8×H(3,𝔽3)C2211yescharacteristic
13C2×Q8×H(3,𝔽3)C2211yescharacteristic
14#88722—24no—
15Q8×(C3⋊D3)⋊C3—24no—
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
18C2×C4×(C3⋊D3)⋊C3C2212yes—
19C2×C4×(C3⋊D3)⋊C3C2212yes—
20C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
21C2×#634⋊C3C2211yescharacteristic
22C2×#634⋊C3C2212yes—
23C4⋊C4×H(3,𝔽3)C2214yes—
24C42×H(3,𝔽3)C2211yescharacteristic
25#3752⋊C3C2212yes—
26#3752⋊C3C2214yes—
27#3752⋊C3C2214yes—
28#3750⋊C3C2212yes—
29#3750⋊C3C2214yes—
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
31C2×#3832C2×C311yescharacteristic
32C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
33C2×C3×#239—31no—
34C2×C3×D3×Q8—31no—
35C32×C4⋊Q8—21no—
36C32×C4⋊Q8D311yescharacteristic, complemented
37C32×C4⋊Q8C2×C311yescharacteristic, complemented
38#64156C2×C311yescharacteristic, complemented
39#64153C2×C311yescharacteristic, complemented
40#64105C2×C314yescomplemented
41#64103C2×C314yescomplemented
42#64084C2×C311yescharacteristic, complemented
43#64081C2×C311yescharacteristic, complemented
44C3×#947—31no—
45C3×#944—31no—
46C3×#896—34no—
47C3×#894—34no—
48C3×#875—31no—
49C3×#872—31no—
50C22×(C3⋊D3)⋊C3C2312yes—
51C22×(C3⋊D3)⋊C3C2311yescharacteristic
52Q8×H(3,𝔽3)—22no—
53Q8×H(3,𝔽3)—22no—
54C2×C4×H(3,𝔽3)C2312yes—
55C2×C4×H(3,𝔽3)C2312yes—
56C2×C4×H(3,𝔽3)C2312yes—
57C2×C4×H(3,𝔽3)C2311yescharacteristic
58C2×(C3⋊Dic3)⋊C3C2312yes—
59C2×(C3⋊Dic3)⋊C3C2312yes—
60C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
61C4⋊((C3⋊D3)⋊C3)—24no—
62C4⋊((C3⋊D3)⋊C3)—22no—
63C4×(C3⋊D3)⋊C3—24no—
64C4×(C3⋊D3)⋊C3—24no—
65C4×(C3⋊D3)⋊C3D414yes—
66#634⋊C3—22no—
67#634⋊C3—24no—
68C3×#536—91no—
69#5724—31no—
70#5724—91no—
71C2×C32×Q8—21no—
72C2×C32×Q8—21no—
73C2×C32×Q8C2×D311yescharacteristic
74C2×C32×Q8C2×D311yescharacteristic
75C2×C32×Q8C22×C311yescharacteristic
76C2×C32×Q8C22×C311yescharacteristic
77#3832—24no—
78Q8×C3⋊D3—24no—
79C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
80C2×C4⋊(C3⋊D3)C22×C312yes—
81C2×C4×C3⋊D3C22×C312yes—
82C2×C4×C3⋊D3C22×C312yes—
83C2×C4×C3⋊D3C22×C311yescharacteristic
84C2×#634C22×C311yescharacteristic
85C2×#634C22×C312yes—
86C3×#239—64no—
87C3×D3×Q8—64no—
88C2×C3×D12—32no—
89C2×C3×D12—31no—
90C2×C3×C4×D3—31no—
91C2×C3×C4×D3—32no—
92C2×C3×C4×D3—32no—
93C2×C3×Dic6—32no—
94C2×C3×Dic6—31no—
95C32×C4⋊C4—24no—
96C32×C4⋊C4C2×D314yes—
97C32×C4⋊C4C22×C314yes—
98C32×C42—21no—
99C32×C42C2×D311yescharacteristic
100C32×C42C22×C311yescharacteristic
101#3752C22×C312yes—
102#3752C22×C314yes—
103#3752C22×C314yes—
104#3750C22×C312yes—
105#3750C22×C314yes—
106C4×C3⋊Dic3C22×C311yescharacteristic
107C3×(C2×D3)⋊C4—32no—
108C3×(C2×D3)⋊C4—34no—
109C3×(C2×D3)⋊C4—34no—
110C3×Dic3⋊C4—34no—
111C3×Dic3⋊C4—32no—
112C3×C4×Dic3—31no—
113C22×H(3,𝔽3)C2411yescharacteristic
114C2×(C3⋊D3)⋊C3—22no—
115C2×(C3⋊D3)⋊C3—22no—
116C2×(C3⋊D3)⋊C3—24no—
117C2×(C3⋊D3)⋊C3C2×D412yes—
118C4×H(3,𝔽3)—22no—
119C4×H(3,𝔽3)—22no—
120C4×H(3,𝔽3)—22no—
121C4×H(3,𝔽3)C2×D412yes—
122(C3⋊Dic3)⋊C3—22no—
123(C3⋊Dic3)⋊C3—22no—
124(C3⋊Dic3)⋊C3C2×D412yes—
125C2×C3×#41—91no—
126C22×C3×Q8—91no—
127C2×#239—31no—
128C2×#239—91no—
129C2×D3×Q8—31no—
130C2×D3×Q8—91no—
131C3×C4⋊Q8—31no—
132C3×C4⋊Q8—31no—
133C3×C4⋊Q8—61no—
134C3×C4⋊Q8—91no—
135C3×C4⋊Q8C3×D311yescharacteristic
136C3×#124—91no—
137C3×#124—91no—
138C3×#123—94no—
139C3×C22⋊Q8—94no—
140C3×#117—91no—
141#947—31no—
142#947—91no—
143#944—31no—
144#944—91no—
145#896—34no—
146#896—94no—
147#894—34no—
148#894—94no—
149#875—31no—
150#875—91no—
151#872—31no—
152#872—91no—
153C22×C3⋊D3C23×C312yes—
154C22×C3⋊D3C23×C311yescharacteristic
155C22×C3×D3—31no—
156C22×C3×D3—32no—
157C32×Q8—22no—
158C32×Q8—22no—
159C32×Q8—22no—
160C32×Q8—22no—
161C32×Q8—42no—
162C32×Q8—42no—
163C2×C32×C4—22no—
164C2×C32×C4—21no—
165C2×C32×C4—22no—
166C2×C32×C4—22no—
167C2×C32×C4C22×D312yes—
168C2×C32×C4C22×D312yes—
169C2×C32×C4C22×D312yes—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C23×C312yes—
172C2×C32×C4C23×C312yes—
173C2×C32×C4C23×C312yes—
174C2×C32×C4C23×C311yescharacteristic
175C2×C3⋊Dic3C23×C312yes—
176C2×C3⋊Dic3C23×C312yes—
177C2×C3⋊Dic3C23×C311yescharacteristic
178C4⋊(C3⋊D3)—24no—
179C4⋊(C3⋊D3)—22no—
180C4×C3⋊D3—24no—
181C4×C3⋊D3—24no—
182C4×C3⋊D3C3×D414yes—
183#634—22no—
184#634—24no—
185C2×C3×Dic3—31no—
186C2×C3×Dic3—32no—
187C2×C3×Dic3—32no—
188C3×D12—64no—
189C3×D12—62no—
190C3×C4×D3—34no—
191C3×C4×D3—64no—
192C3×C4×D3—64no—
193C3×Dic6—64no—
194C3×Dic6—62no—
195#536—91noSylow
196C2×H(3,𝔽3)C22×D411yescharacteristic
197C2×H(3,𝔽3)2−1+412yes—
198(C3⋊D3)⋊C3—22no—
199(C3⋊D3)⋊C3—42no—
200C3×#41—184no—
201C2×C3×Q8—31no—
202C2×C3×Q8—31no—
203C2×C3×Q8—31no—
204C2×C3×Q8—31no—
205C2×C3×Q8—61no—
206C2×C3×Q8—61no—
207C2×C3×Q8—92no—
208C2×C3×Q8—91no—
209C2×C3×Q8—184no—
210C2×C3×Q8C2×C3×D311yescharacteristic
211C2×C3×Q8C2×C3×D311yescharacteristic
212C2×C3×D4—92no—
213C2×C3×D4—91no—
214C22×C3×C4—91no—
215C22×C3×C4—92no—
216C22×C3×C4—92no—
217#239—64no—
218#239—184no—
219D3×Q8—64no—
220D3×Q8—184no—
221C2×D12—31no—
222C2×D12—32no—
223C2×D12—91no—
224C2×D12—92no—
225C2×C4×D3—31no—
226C2×C4×D3—32no—
227C2×C4×D3—32no—
228C2×C4×D3—91no—
229C2×C4×D3—92no—
230C2×C4×D3—92no—
231C2×Dic6—31no—
232C2×Dic6—32no—
233C2×Dic6—91no—
234C2×Dic6—92no—
235C3×C4⋊C4—34no—
236C3×C4⋊C4—34no—
237C3×C4⋊C4—64no—
238C3×C4⋊C4—92no—
239C3×C4⋊C4—94no—
240C3×C4⋊C4C2×C3×D314yes—
241C3×C22⋊C4—92no—
242C3×C22⋊C4—94no—
243C3×C22⋊C4—94no—
244C3×C42—31no—
245C3×C42—31no—
246C3×C42—61no—
247C3×C42—91no—
248C3×C42C2×C3×D311yescharacteristic
249(C2×D3)⋊C4—32no—
250(C2×D3)⋊C4—34no—
251(C2×D3)⋊C4—34no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—94no—
254(C2×D3)⋊C4—94no—
255Dic3⋊C4—32no—
256Dic3⋊C4—34no—
257Dic3⋊C4—92no—
258Dic3⋊C4—94no—
259C4×Dic3—31no—
260C4×Dic3—91no—
261C22×C32—21no—
262C22×C32C23×D311yescharacteristic
263C22×C32C24×C311yescharacteristic
264C2×C3⋊D3—22no—
265C2×C3⋊D3—22no—
266C2×C3⋊D3—24no—
267C2×C3⋊D3C2×C3×D412yes—
268C2×C3×D3—32no—
269C2×C3×D3—62no—
270C2×C3×D3—62no—
271C2×C3×D3—64no—
272C32×C4—24no—
273C32×C4—22no—
274C32×C4—22no—
275C32×C4—22no—
276C32×C4—22no—
277C32×C4—22no—
278C32×C4—22no—
279C32×C4—22no—
280C32×C4—42no—
281C32×C4—42no—
282C32×C4D3×D412yes—
283C32×C4C2×C3×D412yes—
284C3⋊Dic3—22no—
285C3⋊Dic3—22no—
286C3⋊Dic3C2×C3×D412yes—
287C3×Dic3—32no—
288C3×Dic3—62no—
289C3×Dic3—62no—
290C2×#41—91no—
291C22×Q8—91no—
292C4⋊Q8—91no—
293C4⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
294#124—91no—
295#124—91no—
296#123—94no—
297C22⋊Q8—94no—
298#117—91no—
299H(3,𝔽3)#53611yescharacteristic, complemented, Sylow
300C23×C3—91no—
301C23×C3—92no—
302C22×D3—31no—
303C22×D3—32no—
304C22×D3—91no—
305C22×D3—92no—
306C3×Q8—22no—
307C3×Q8—22no—
308C3×Q8—62no—
309C3×Q8—62no—
310C3×Q8—62no—
311C3×Q8—62no—
312C3×Q8—122no—
313C3×Q8—122no—
314C3×Q8—182no—
315C3×Q8—184no—
316C3×D4—182no—
317C3×D4—184no—
318C2×C3×C4—31no—
319C2×C3×C4—31no—
320C2×C3×C4—32no—
321C2×C3×C4—32no—
322C2×C3×C4—32no—
323C2×C3×C4—32no—
324C2×C3×C4—32no—
325C2×C3×C4—32no—
326C2×C3×C4—62no—
327C2×C3×C4—61no—
328C2×C3×C4—62no—
329C2×C3×C4—62no—
330C2×C3×C4—94no—
331C2×C3×C4—91no—
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×D312yes—
339C2×C3×C4C22×C3×D311yescharacteristic
340C2×Dic3—31no—
341C2×Dic3—32no—
342C2×Dic3—32no—
343C2×Dic3—91no—
344C2×Dic3—92no—
345C2×Dic3—92no—
346D12—64no—
347D12—62no—
348D12—182no—
349D12—184no—
350C4×D3—34no—
351C4×D3—64no—
352C4×D3—64no—
353C4×D3—94no—
354C4×D3—184no—
355C4×D3—184no—
356Dic6—64no—
357Dic6—62no—
358Dic6—184no—
359Dic6—182no—
360C2×C32—21no—
361C2×C32—22no—
362C2×C32C2×D3×D411yescharacteristic
363C2×C32#100712yes—
364C2×C32C22×C3×D411yescharacteristic
365C2×C32C3×2−1+412yes—
366C3⋊D3—22no—
367C3⋊D3—42no—
368C3×D3—62no—
369C3×D3—122no—
370#41—184no—
371C2×Q8—92no—
372C2×Q8—91no—
373C2×Q8—184no—
374C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
375C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
376C2×D4—92no—
377C2×D4—91no—
378C22×C4—91no—
379C22×C4—92no—
380C22×C4—92no—
381C4⋊C4—92no—
382C4⋊C4—94no—
383C4⋊C4C2×(C3⋊D3)⋊C314yes—
384C22⋊C4—92no—
385C22⋊C4—94no—
386C22⋊C4—94no—
387C42—91no—
388C42C2×(C3⋊D3)⋊C311yescharacteristic
389C22×C3—31no—
390C22×C3—31no—
391C22×C3—61no—
392C22×C3—92no—
393C22×C3—182no—
394C22×C3—182no—
395C22×C3—184no—
396C22×C3C23×C3×D311yescharacteristic
397C2×D3—32no—
398C2×D3—62no—
399C2×D3—62no—
400C2×D3—64no—
401C2×D3—92no—
402C2×D3—182no—
403C2×D3—182no—
404C2×D3—184no—
405C3×C4—22no—
406C3×C4—22no—
407C3×C4—22no—
408C3×C4—32no—
409C3×C4—32no—
410C3×C4—62no—
411C3×C4—62no—
412C3×C4—62no—
413C3×C4—62no—
414C3×C4—62no—
415C3×C4—62no—
416C3×C4—64no—
417C3×C4—62no—
418C3×C4—92no—
419C3×C4—122no—
420C3×C4—122no—
421C3×C4—182no—
422C3×C4—182no—
423C3×C4C3×D3×D412yes—
424Dic3—32no—
425Dic3—62no—
426Dic3—62no—
427Dic3—92no—
428Dic3—182no—
429Dic3—182no—
430C32—21no—
431C32#572411yescharacteristic, complemented
432C32C3×#53611yescharacteristic, complemented
433C23—91no—
434C23—92no—
435Q8—22no—
436Q8—22no—
437Q8—184no—
438Q8—182no—
439D4—184no—
440D4—182no—
441C2×C4—91no—
442C2×C4—94no—
443C2×C4—92no—
444C2×C4—92no—
445C2×C4—184no—
446C2×C4—184no—
447C2×C4C22×(C3⋊D3)⋊C312yes—
448C2×C4C22×(C3⋊D3)⋊C312yes—
449C2×C4C22×(C3⋊D3)⋊C312yes—
450C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
451C2×C3—31no—
452C2×C3—31no—
453C2×C3—32no—
454C2×C3—32no—
455C2×C3—61no—
456C2×C3—62no—
457C2×C3—182no—
458C2×C3—362no—
459C2×C3C2×C3×D3×D411yescharacteristic
460C2×C3C3×#100712yes—
461D3—62no—
462D3—122no—
463D3—182no—
464D3—362no—
465C22—92no—
466C22—182no—
467C22—182no—
468C22—184no—
469C22C23×(C3⋊D3)⋊C311yescharacteristic
470C4—22no—
471C4—22no—
472C4—22no—
473C4—92no—
474C4—182no—
475C4—182no—
476C4D4×(C3⋊D3)⋊C312yes—
477C3—31no—
478C3—31no—
479C3—61no—
480C3C3×#572411yescharacteristic
481C2—182no—
482C2—362no—
483C2C2×D4×(C3⋊D3)⋊C311yescharacteristic
484C2#1172262712yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C4⋊Q8C2
prime radicalC2×#88722C2
hypercenterC4⋊Q8(C3⋊D3)⋊C3
hypocenterC32C3×#536
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×Q8×(C3⋊D3)⋊C3C2semidirect
3. H(3,𝔽3)×C4⋊Q8C2semidirect
4. #64156⋊C3C2semidirect
5. #64153⋊C3C2semidirect
6. #64105⋊C3C2semidirect
7. #64103⋊C3C2semidirect
8. #64084⋊C3C2semidirect
9. #64081⋊C3C2semidirect
10. #10596195C3semidirect
32. C2×Q8×C3⋊D3C2×C3semidirect
36. C32×C4⋊Q8D3semidirect
37. C32×C4⋊Q8C2×C3semidirect
38. #64156C2×C3semidirect
39. #64153C2×C3semidirect
40. #64105C2×C3semidirect
41. #64103C2×C3semidirect
42. #64084C2×C3semidirect
43. #64081C2×C3semidirect
293. C4⋊Q8(C3⋊D3)⋊C3semidirect
299. H(3,𝔽3)#536semidirect
431. C32#5724semidirect
432. C32C3×#536semidirect

Maximal subgroups

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

subgroupindex
C2×#887222
C2×Q8×(C3⋊D3)⋊C32
H(3,𝔽3)×C4⋊Q82
#64156⋊C32
#64153⋊C32
#64105⋊C32
#64103⋊C32
#64084⋊C32
#64081⋊C32
#105961953
C3×#57243