Finite Group Databasebeta

← #10596056⋊C3 #13152726

Subgroups — #10596056⋊C3 #13152726

Order 1728.

896 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64156⋊C3C212yesmaximal, complemented
3#64103⋊C3C214yesmaximal, complemented
4#64102⋊C3C212yesmaximal, complemented
5#64084⋊C3C212yesmaximal, complemented
6#11721385C211yesmaximal, characteristic
7C4×C4⋊((C3⋊D3)⋊C3)C212yesmaximal, complemented
8C4×#634⋊C3C211yesmaximal, characteristic, complemented
9#10596056C311yesmaximal, characteristic, complemented
10C3×#5585—31nomaximal
11C2×Q8×H(3,𝔽3)C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
13C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
14C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
15C2×C4×(C3⋊D3)⋊C3C2214yes—
16C2×#634⋊C3C2211yescharacteristic
17C4⋊C4×H(3,𝔽3)C2211yescharacteristic
18C4⋊C4×H(3,𝔽3)C2212yes—
19C42×H(3,𝔽3)C2211yescharacteristic
20C42×H(3,𝔽3)C2212yes—
21#3752⋊C3C2214yes—
22#3752⋊C3C2214yes—
23#3752⋊C3C2214yes—
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25#3750⋊C3C2212yes—
26C4×(C3⋊Dic3)⋊C3C2212yes—
27C32×C4×Q8—21no—
28C32×C4×Q8D311yescharacteristic, complemented
29C32×C4×Q8C2×C311yescharacteristic, complemented
30#64156C2×C312yescomplemented
31#64103C2×C314yescomplemented
32#64102C2×C312yescomplemented
33#64084C2×C312yescomplemented
34#64083C2×C311yescharacteristic
35C4×C4⋊(C3⋊D3)C2×C312yescomplemented
36C4×#634C2×C311yescharacteristic, complemented
37C3×#947—32no—
38C3×#894—34no—
39C3×#893—32no—
40C3×#875—32no—
41C3×#874—31no—
42C3×C4×D12—32no—
43C3×C4×Dic6—31no—
44C22×(C3⋊D3)⋊C3C2314yes—
45Q8×H(3,𝔽3)—22no—
46C2×C4×H(3,𝔽3)C2312yes—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×C4×H(3,𝔽3)C2312yes—
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51C2×(C3⋊Dic3)⋊C3C2312yes—
52C2×(C3⋊Dic3)⋊C3C2312yes—
53C4⋊((C3⋊D3)⋊C3)—24no—
54C4⋊((C3⋊D3)⋊C3)—24no—
55C4⋊((C3⋊D3)⋊C3)—42no—
56C4×(C3⋊D3)⋊C3—44no—
57#634⋊C3—22no—
58C3×#559—91no—
59#5585—31no—
60#5585—91no—
61C2×C32×Q8—21no—
62C2×C32×Q8C2×D311yescharacteristic
63C2×C32×Q8C22×C311yescharacteristic
64C2×C4⋊(C3⋊D3)C22×C312yes—
65C2×C4⋊(C3⋊D3)C22×C312yes—
66C2×C4⋊(C3⋊D3)C22×C312yes—
67C2×C4×C3⋊D3C22×C314yes—
68C2×#634C22×C311yescharacteristic
69C2×C3×D12—32no—
70C2×C3×D12—32no—
71C2×C3×D12—32no—
72C2×C3×C4×D3—34no—
73C2×C3×Dic6—31no—
74C32×C4⋊C4—21no—
75C32×C4⋊C4—22no—
76C32×C4⋊C4C2×D311yescharacteristic
77C32×C4⋊C4C2×D312yes—
78C32×C4⋊C4C22×C311yescharacteristic
79C32×C4⋊C4C22×C312yes—
80C32×C42—21no—
81C32×C42—22no—
82C32×C42C2×D311yescharacteristic
83C32×C42C2×D312yes—
84C32×C42C22×C311yescharacteristic
85C32×C42C22×C312yes—
86#3752C22×C314yes—
87#3752C22×C314yes—
88#3752C22×C314yes—
89C4⋊(C3⋊Dic3)C22×C311yescharacteristic
90#3750C22×C312yes—
91C4×C3⋊Dic3C22×C312yes—
92C3×(C2×D3)⋊C4—34no—
93C3×(C2×D3)⋊C4—34no—
94C3×(C2×D3)⋊C4—34no—
95C3×C4⋊Dic3—31no—
96C3×Dic3⋊C4—32no—
97C3×C4×Dic3—32no—
98C22×H(3,𝔽3)C2411yescharacteristic
99C2×(C3⋊D3)⋊C3—24no—
100C2×(C3⋊D3)⋊C3—24no—
101C2×(C3⋊D3)⋊C3—24no—
102C4×H(3,𝔽3)—22no—
103C4×H(3,𝔽3)—22no—
104C4×H(3,𝔽3)—21no—
105C4×H(3,𝔽3)#4112yes—
106C4×H(3,𝔽3)#4112yes—
107(C3⋊Dic3)⋊C3—22no—
108(C3⋊Dic3)⋊C3—22no—
109C3×#127—91no—
110C3×#124—92no—
111C3×#124—92no—
112C3×#123—94no—
113C3×C4×Q8—31no—
114C3×C4×Q8—31no—
115C3×C4×Q8—61no—
116C3×C4×Q8—91no—
117C3×C4×Q8C3×D311yescharacteristic
118C3×C4×D4—92no—
119C3×C4×D4—92no—
120#947—32no—
121#947—92no—
122#894—34no—
123#894—94no—
124#893—32no—
125#893—92no—
126#875—32no—
127#875—92no—
128#874—31no—
129#874—91no—
130C4×D12—32no—
131C4×D12—92no—
132C4×Dic6—31no—
133C4×Dic6—91no—
134C22×C3⋊D3C23×C314yes—
135C22×C3×D3—34no—
136C32×Q8—24no—
137C32×Q8—22no—
138C32×Q8—22no—
139C2×C32×C4—21no—
140C2×C32×C4—22no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—22no—
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D312yes—
146C2×C32×C4C22×D312yes—
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C23×C311yescharacteristic
150C2×C32×C4C23×C312yes—
151C2×C32×C4C23×C312yes—
152C2×C32×C4C23×C311yescharacteristic
153C2×C32×C4C23×C311yescharacteristic
154C2×C3⋊Dic3C23×C312yes—
155C2×C3⋊Dic3C23×C312yes—
156C4⋊(C3⋊D3)—24no—
157C4⋊(C3⋊D3)—24no—
158C4⋊(C3⋊D3)—42no—
159C4×C3⋊D3—44no—
160#634—22no—
161C2×C3×Dic3—32no—
162C2×C3×Dic3—32no—
163C3×D12—64no—
164C3×D12—64no—
165C3×D12—122no—
166C3×C4×D3—124no—
167C3×Dic6—62no—
168#559—91noSylow
169C2×H(3,𝔽3)C2×#4111yescharacteristic
170C2×H(3,𝔽3)C2×#4111yescharacteristic
171C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
172(C3⋊D3)⋊C3—44no—
173C2×C3×Q8—31no—
174C2×C3×Q8—31no—
175C2×C3×Q8—61no—
176C2×C3×Q8—91no—
177C2×C3×Q8C2×C3×D311yescharacteristic
178C2×C3×D4—92no—
179C2×C3×D4—92no—
180C2×C3×D4—92no—
181C22×C3×C4—94no—
182C2×D12—32no—
183C2×D12—32no—
184C2×D12—32no—
185C2×D12—92no—
186C2×D12—92no—
187C2×D12—92no—
188C2×C4×D3—34no—
189C2×C4×D3—94no—
190C2×Dic6—31no—
191C2×Dic6—91no—
192C3×C4⋊C4—31no—
193C3×C4⋊C4—32no—
194C3×C4⋊C4—31no—
195C3×C4⋊C4—32no—
196C3×C4⋊C4—61no—
197C3×C4⋊C4—62no—
198C3×C4⋊C4—92no—
199C3×C4⋊C4—91no—
200C3×C4⋊C4C2×C3×D311yescharacteristic
201C3×C4⋊C4C2×C3×D312yes—
202C3×C22⋊C4—94no—
203C3×C22⋊C4—94no—
204C3×C22⋊C4—94no—
205C3×C42—31no—
206C3×C42—32no—
207C3×C42—32no—
208C3×C42—31no—
209C3×C42—62no—
210C3×C42—61no—
211C3×C42—92no—
212C3×C42C2×C3×D311yescharacteristic
213C3×C42C2×C3×D312yes—
214(C2×D3)⋊C4—34no—
215(C2×D3)⋊C4—34no—
216(C2×D3)⋊C4—34no—
217(C2×D3)⋊C4—94no—
218(C2×D3)⋊C4—94no—
219(C2×D3)⋊C4—94no—
220C4⋊Dic3—31no—
221C4⋊Dic3—91no—
222Dic3⋊C4—32no—
223Dic3⋊C4—92no—
224C4×Dic3—32no—
225C4×Dic3—92no—
226C22×C32—21no—
227C22×C32C23×D311yescharacteristic
228C22×C32C24×C311yescharacteristic
229C2×C3⋊D3—24no—
230C2×C3⋊D3—24no—
231C2×C3⋊D3—24no—
232C2×C3×D3—64no—
233C2×C3×D3—64no—
234C2×C3×D3—64no—
235C32×C4—24no—
236C32×C4—22no—
237C32×C4—22no—
238C32×C4—21no—
239C32×C4—22no—
240C32×C4—22no—
241C32×C4—21no—
242C32×C4—22no—
243C32×C4—22no—
244C32×C4—42no—
245C32×C4—41no—
246C32×C4#23512yes—
247C32×C4#23912yes—
248C32×C4C3×#4112yes—
249C32×C4C3×#4112yes—
250C3⋊Dic3—22no—
251C3⋊Dic3—22no—
252C3×Dic3—62no—
253C3×Dic3—62no—
254#127—91no—
255#124—92no—
256#124—92no—
257#123—94no—
258C4×Q8—91no—
259C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
260C4×D4—92no—
261C4×D4—92no—
262H(3,𝔽3)#55911yescharacteristic, complemented, Sylow
263C23×C3—94no—
264C22×D3—34no—
265C22×D3—94no—
266C3×Q8—22no—
267C3×Q8—62no—
268C3×Q8—62no—
269C3×Q8—64no—
270C3×Q8—182no—
271C3×D4—184no—
272C3×D4—184no—
273C3×D4—362no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—31no—
278C2×C3×C4—32no—
279C2×C3×C4—32no—
280C2×C3×C4—31no—
281C2×C3×C4—31no—
282C2×C3×C4—32no—
283C2×C3×C4—32no—
284C2×C3×C4—61no—
285C2×C3×C4—62no—
286C2×C3×C4—61no—
287C2×C3×C4—62no—
288C2×C3×C4—61no—
289C2×C3×C4—92no—
290C2×C3×C4—92no—
291C2×C3×C4—364no—
292C2×C3×C4C22×C3×D311yescharacteristic
293C2×C3×C4C22×C3×D312yes—
294C2×C3×C4C22×C3×D312yes—
295C2×C3×C4C22×C3×D311yescharacteristic
296C2×C3×C4C22×C3×D311yescharacteristic
297C2×Dic3—32no—
298C2×Dic3—32no—
299C2×Dic3—92no—
300C2×Dic3—92no—
301D12—64no—
302D12—64no—
303D12—122no—
304D12—184no—
305D12—184no—
306D12—362no—
307C4×D3—124no—
308C4×D3—364no—
309Dic6—62no—
310Dic6—182no—
311C2×C32—21no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32C2×#23511yescharacteristic
315C2×C32C2×#23911yescharacteristic
316C2×C32#100911yescharacteristic
317C2×C32C2×C3×#4111yescharacteristic
318C2×C32C2×C3×#4111yescharacteristic
319C2×C32C3×H(5,𝔽2)11yescharacteristic
320C3⋊D3—44no—
321C3×D3—124no—
322C2×Q8—91no—
323C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
324C2×D4—92no—
325C2×D4—92no—
326C2×D4—92no—
327C22×C4—94no—
328C4⋊C4—91no—
329C4⋊C4—92no—
330C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
331C4⋊C4C2×(C3⋊D3)⋊C312yes—
332C22⋊C4—94no—
333C22⋊C4—94no—
334C22⋊C4—94no—
335C42—92no—
336C42C2×(C3⋊D3)⋊C312yes—
337C42C2×(C3⋊D3)⋊C311yescharacteristic
338C22×C3—31no—
339C22×C3—31no—
340C22×C3—61no—
341C22×C3—184no—
342C22×C3—184no—
343C22×C3—184no—
344C22×C3C23×C3×D311yescharacteristic
345C2×D3—64no—
346C2×D3—64no—
347C2×D3—64no—
348C2×D3—184no—
349C2×D3—184no—
350C2×D3—184no—
351C3×C4—22no—
352C3×C4—22no—
353C3×C4—21no—
354C3×C4—32no—
355C3×C4—32no—
356C3×C4—32no—
357C3×C4—32no—
358C3×C4—62no—
359C3×C4—61no—
360C3×C4—62no—
361C3×C4—61no—
362C3×C4—62no—
363C3×C4—62no—
364C3×C4—64no—
365C3×C4—62no—
366C3×C4—62no—
367C3×C4—122no—
368C3×C4—121no—
369C3×C4—182no—
370C3×C4—182no—
371C3×C4C3×#23512yes—
372C3×C4C3×#23912yes—
373Dic3—62no—
374Dic3—62no—
375Dic3—182no—
376Dic3—182no—
377C32—21no—
378C32#558511yescharacteristic, complemented
379C32C3×#55911yescharacteristic, complemented
380C23—94no—
381Q8—22no—
382Q8—182no—
383D4—184no—
384D4—184no—
385D4—362no—
386C2×C4—92no—
387C2×C4—92no—
388C2×C4—364no—
389C2×C4C22×(C3⋊D3)⋊C312yes—
390C2×C4C22×(C3⋊D3)⋊C312yes—
391C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
392C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
393C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
394C2×C3—31no—
395C2×C3—31no—
396C2×C3—31no—
397C2×C3—31no—
398C2×C3—31no—
399C2×C3—31no—
400C2×C3—61no—
401C2×C3—61no—
402C2×C3—61no—
403C2×C3—364no—
404C2×C3C2×C3×#23511yescharacteristic
405C2×C3C2×C3×#23911yescharacteristic
406C2×C3C3×#100911yescharacteristic
407D3—124no—
408D3—364no—
409C22—184no—
410C22—184no—
411C22—184no—
412C22C23×(C3⋊D3)⋊C311yescharacteristic
413C4—22no—
414C4—22no—
415C4—21no—
416C4—182no—
417C4—182no—
418C4#3828⋊C312yes—
419C4#8872212yes—
420C3—31no—
421C3—31no—
422C3—61no—
423C3C3×#558511yescharacteristic
424C2—364no—
425C2C2×#3828⋊C311yescharacteristic
426C2C2×#8872211yescharacteristic
427C2#1172263811yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC4×Q8×H(3,𝔽3)C2
prime radical#11721385C2
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#559
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C4×Q8×H(3,𝔽3)C2semidirect
2. #64156⋊C3C2semidirect
3. #64103⋊C3C2semidirect
4. #64102⋊C3C2semidirect
5. #64084⋊C3C2semidirect
7. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
8. C4×#634⋊C3C2semidirect
9. #10596056C3semidirect
28. C32×C4×Q8D3semidirect
29. C32×C4×Q8C2×C3semidirect
30. #64156C2×C3semidirect
31. #64103C2×C3semidirect
32. #64102C2×C3semidirect
33. #64084C2×C3semidirect
35. C4×C4⋊(C3⋊D3)C2×C3semidirect
36. C4×#634C2×C3semidirect
259. C4×Q8(C3⋊D3)⋊C3semidirect
262. H(3,𝔽3)#559semidirect
378. C32#5585semidirect
379. C32C3×#559semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×H(3,𝔽3)2
#64156⋊C32
#64103⋊C32
#64102⋊C32
#64084⋊C32
#117213852
C4×C4⋊((C3⋊D3)⋊C3)2
C4×#634⋊C32
#105960563
C3×#55853