Finite Group Databasebeta

← #10596055⋊C3 #13152725

Subgroups — #10596055⋊C3 #13152725

Order 1728.

812 conjugacy classes of subgroups.

The subgroup lattice has 415 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#64155⋊C3C212yesmaximal, complemented
3#64107⋊C3C214yesmaximal, complemented
4#64101⋊C3C212yesmaximal, complemented
5#64084⋊C3C212yesmaximal
6C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
7#64078⋊C3C211yesmaximal, characteristic, complemented
8C4×#634⋊C3C212yesmaximal, complemented
9#10596055C311yesmaximal, characteristic, complemented
10C3×#5584—31nomaximal
11C2×Q8×H(3,𝔽3)C2211yescharacteristic
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2212yes—
14C2×#634⋊C3C2212yes—
15C4⋊C4×H(3,𝔽3)C2212yes—
16C4⋊C4×H(3,𝔽3)C2211yescharacteristic
17C42×H(3,𝔽3)C2212yes—
18C42×H(3,𝔽3)C2211yescharacteristic
19#3752⋊C3C2212yes—
20#3752⋊C3C2214yes—
21#3752⋊C3C2214yes—
22C4⋊((C3⋊Dic3)⋊C3)C2212yes—
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24C4⋊((C3⋊Dic3)⋊C3)C2212yes—
25#3750⋊C3C2214yes—
26C4×(C3⋊Dic3)⋊C3C2214yes—
27C32×C4×Q8—21no—
28C32×C4×Q8D311yescharacteristic, complemented
29C32×C4×Q8C2×C311yescharacteristic, complemented
30#64155C2×C312yescomplemented
31#64107C2×C314yescomplemented
32#64101C2×C312yescomplemented
33#64084C2×C312yes—
34C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
35#64078C2×C311yescharacteristic, complemented
36C4×#634C2×C312yescomplemented
37C3×#946—32no—
38C3×#898—34no—
39C3×#892—32no—
40C3×#875—32no—
41C3×C4×D12—31no—
42C3×#869—31no—
43C3×C4×Dic6—32no—
44C22×(C3⋊D3)⋊C3C2312yes—
45Q8×H(3,𝔽3)—22no—
46C2×C4×H(3,𝔽3)C2312yes—
47C2×C4×H(3,𝔽3)C2312yes—
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×C4×H(3,𝔽3)C2311yescharacteristic
51C2×(C3⋊Dic3)⋊C3C2314yes—
52C2×(C3⋊Dic3)⋊C3C2312yes—
53C4⋊((C3⋊D3)⋊C3)—22no—
54C4×(C3⋊D3)⋊C3—24no—
55#634⋊C3—24no—
56C3×#556—91no—
57#5584—31no—
58#5584—91no—
59C2×C32×Q8—21no—
60C2×C32×Q8C2×D311yescharacteristic
61C2×C32×Q8C22×C311yescharacteristic
62C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
63C2×C4×C3⋊D3C22×C312yes—
64C2×#634C22×C312yes—
65C2×C3×D12—31no—
66C2×C3×C4×D3—32no—
67C2×C3×Dic6—32no—
68C32×C4⋊C4—21no—
69C32×C4⋊C4—22no—
70C32×C4⋊C4C2×D311yescharacteristic
71C32×C4⋊C4C2×D312yes—
72C32×C4⋊C4C22×C311yescharacteristic
73C32×C4⋊C4C22×C312yes—
74C32×C42—22no—
75C32×C42—21no—
76C32×C42C2×D312yes—
77C32×C42C2×D311yescharacteristic
78C32×C42C22×C312yes—
79C32×C42C22×C311yescharacteristic
80#3752C22×C312yes—
81#3752C22×C314yes—
82#3752C22×C314yes—
83C4⋊(C3⋊Dic3)C22×C311yescharacteristic
84C4⋊(C3⋊Dic3)C22×C312yes—
85C4⋊(C3⋊Dic3)C22×C312yes—
86#3750C22×C314yes—
87C4×C3⋊Dic3C22×C314yes—
88C3×(C2×D3)⋊C4—34no—
89C3×(C2×D3)⋊C4—32no—
90C3×(C2×D3)⋊C4—34no—
91C3×C4⋊Dic3—32no—
92C3×C4⋊Dic3—32no—
93C3×C4⋊Dic3—31no—
94C3×Dic3⋊C4—34no—
95C3×C4×Dic3—34no—
96C22×H(3,𝔽3)C2411yescharacteristic
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—22no—
99C2×(C3⋊D3)⋊C3—22no—
100C4×H(3,𝔽3)—22no—
101C4×H(3,𝔽3)—22no—
102C4×H(3,𝔽3)—21no—
103C4×H(3,𝔽3)#4112yes—
104C4×H(3,𝔽3)#4112yes—
105(C3⋊Dic3)⋊C3—24no—
106(C3⋊Dic3)⋊C3—22no—
107C3×C4⋊Q8—91no—
108C3×#126—94no—
109C3×#124—92no—
110C3×C22⋊Q8—92no—
111C3×C4×Q8—31no—
112C3×C4×Q8—31no—
113C3×C4×Q8—61no—
114C3×C4×Q8—92no—
115C3×C4×Q8C3×D311yescharacteristic
116C3×C4×D4—91no—
117C3×#117—92no—
118#946—32no—
119#946—92no—
120#898—34no—
121#898—94no—
122#892—32no—
123#892—92no—
124#875—32no—
125#875—92no—
126C4×D12—31no—
127C4×D12—91no—
128#869—31no—
129#869—91no—
130C4×Dic6—32no—
131C4×Dic6—92no—
132C22×C3⋊D3C23×C312yes—
133C22×C3×D3—32no—
134C32×Q8—24no—
135C32×Q8—22no—
136C32×Q8—22no—
137C2×C32×C4—21no—
138C2×C32×C4—22no—
139C2×C32×C4—21no—
140C2×C32×C4—22no—
141C2×C32×C4—21no—
142C2×C32×C4C22×D312yes—
143C2×C32×C4C22×D312yes—
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C23×C312yes—
148C2×C32×C4C23×C312yes—
149C2×C32×C4C23×C311yescharacteristic
150C2×C32×C4C23×C311yescharacteristic
151C2×C32×C4C23×C311yescharacteristic
152C2×C3⋊Dic3C23×C312yes—
153C2×C3⋊Dic3C23×C314yes—
154C4⋊(C3⋊D3)—22no—
155C4×C3⋊D3—24no—
156#634—24no—
157C2×C3×Dic3—32no—
158C2×C3×Dic3—34no—
159C3×D12—62no—
160C3×C4×D3—64no—
161C3×Dic6—64no—
162#556—91noSylow
163C2×H(3,𝔽3)C2×#4111yescharacteristic
164C2×H(3,𝔽3)C2×#4111yescharacteristic
165C2×H(3,𝔽3)2−1+411yescharacteristic
166(C3⋊D3)⋊C3—42no—
167C2×C3×Q8—31no—
168C2×C3×Q8—31no—
169C2×C3×Q8—61no—
170C2×C3×Q8—92no—
171C2×C3×Q8C2×C3×D311yescharacteristic
172C2×C3×D4—91no—
173C22×C3×C4—92no—
174C2×D12—31no—
175C2×D12—91no—
176C2×C4×D3—32no—
177C2×C4×D3—92no—
178C2×Dic6—32no—
179C2×Dic6—92no—
180C3×C4⋊C4—31no—
181C3×C4⋊C4—32no—
182C3×C4⋊C4—31no—
183C3×C4⋊C4—32no—
184C3×C4⋊C4—61no—
185C3×C4⋊C4—62no—
186C3×C4⋊C4—94no—
187C3×C4⋊C4—92no—
188C3×C4⋊C4—91no—
189C3×C4⋊C4—92no—
190C3×C4⋊C4C2×C3×D311yescharacteristic
191C3×C4⋊C4C2×C3×D312yes—
192C3×C22⋊C4—94no—
193C3×C22⋊C4—92no—
194C3×C22⋊C4—94no—
195C3×C42—32no—
196C3×C42—31no—
197C3×C42—31no—
198C3×C42—32no—
199C3×C42—61no—
200C3×C42—62no—
201C3×C42—94no—
202C3×C42C2×C3×D312yes—
203C3×C42C2×C3×D311yescharacteristic
204(C2×D3)⋊C4—32no—
205(C2×D3)⋊C4—34no—
206(C2×D3)⋊C4—34no—
207(C2×D3)⋊C4—92no—
208(C2×D3)⋊C4—94no—
209(C2×D3)⋊C4—94no—
210C4⋊Dic3—31no—
211C4⋊Dic3—32no—
212C4⋊Dic3—32no—
213C4⋊Dic3—91no—
214C4⋊Dic3—92no—
215C4⋊Dic3—92no—
216Dic3⋊C4—34no—
217Dic3⋊C4—94no—
218C4×Dic3—34no—
219C4×Dic3—94no—
220C22×C32—21no—
221C22×C32C23×D311yescharacteristic
222C22×C32C24×C311yescharacteristic
223C2×C3⋊D3—22no—
224C2×C3⋊D3—22no—
225C2×C3⋊D3—22no—
226C2×C3×D3—62no—
227C2×C3×D3—62no—
228C2×C3×D3—62no—
229C32×C4—24no—
230C32×C4—22no—
231C32×C4—22no—
232C32×C4—21no—
233C32×C4—22no—
234C32×C4—22no—
235C32×C4—21no—
236C32×C4—22no—
237C32×C4—22no—
238C32×C4—42no—
239C32×C4—41no—
240C32×C4#23512yes—
241C32×C4#23912yes—
242C32×C4C3×#4112yes—
243C32×C4C3×#4112yes—
244C3⋊Dic3—24no—
245C3⋊Dic3—22no—
246C3×Dic3—62no—
247C3×Dic3—64no—
248C4⋊Q8—91no—
249#126—94no—
250#124—92no—
251C22⋊Q8—92no—
252C4×Q8—92no—
253C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
254C4×D4—91no—
255#117—92no—
256H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
257C23×C3—92no—
258C22×D3—32no—
259C22×D3—92no—
260C3×Q8—22no—
261C3×Q8—62no—
262C3×Q8—62no—
263C3×Q8—64no—
264C3×Q8—184no—
265C3×D4—182no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—32no—
271C2×C3×C4—31no—
272C2×C3×C4—32no—
273C2×C3×C4—32no—
274C2×C3×C4—32no—
275C2×C3×C4—31no—
276C2×C3×C4—61no—
277C2×C3×C4—62no—
278C2×C3×C4—61no—
279C2×C3×C4—62no—
280C2×C3×C4—61no—
281C2×C3×C4—92no—
282C2×C3×C4—94no—
283C2×C3×C4—184no—
284C2×C3×C4C22×C3×D312yes—
285C2×C3×C4C22×C3×D312yes—
286C2×C3×C4C22×C3×D311yescharacteristic
287C2×C3×C4C22×C3×D311yescharacteristic
288C2×C3×C4C22×C3×D311yescharacteristic
289C2×Dic3—32no—
290C2×Dic3—34no—
291C2×Dic3—92no—
292C2×Dic3—94no—
293D12—62no—
294D12—182no—
295C4×D3—64no—
296C4×D3—184no—
297Dic6—64no—
298Dic6—184no—
299C2×C32—21no—
300C2×C32—21no—
301C2×C32—21no—
302C2×C32C2×#23511yescharacteristic
303C2×C32C2×#23911yescharacteristic
304C2×C32#101011yescharacteristic
305C2×C32C2×C3×#4111yescharacteristic
306C2×C32C2×C3×#4111yescharacteristic
307C2×C32C3×2−1+411yescharacteristic
308C3⋊D3—42no—
309C3×D3—122no—
310C2×Q8—92no—
311C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
312C2×D4—91no—
313C22×C4—92no—
314C4⋊C4—92no—
315C4⋊C4—91no—
316C4⋊C4—94no—
317C4⋊C4—92no—
318C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
319C4⋊C4C2×(C3⋊D3)⋊C312yes—
320C22⋊C4—94no—
321C22⋊C4—92no—
322C22⋊C4—94no—
323C42—94no—
324C42C2×(C3⋊D3)⋊C311yescharacteristic
325C42C2×(C3⋊D3)⋊C312yes—
326C22×C3—31no—
327C22×C3—31no—
328C22×C3—61no—
329C22×C3—182no—
330C22×C3—182no—
331C22×C3—182no—
332C22×C3C23×C3×D311yescharacteristic
333C2×D3—62no—
334C2×D3—62no—
335C2×D3—62no—
336C2×D3—182no—
337C2×D3—182no—
338C2×D3—182no—
339C3×C4—22no—
340C3×C4—22no—
341C3×C4—21no—
342C3×C4—32no—
343C3×C4—32no—
344C3×C4—32no—
345C3×C4—32no—
346C3×C4—62no—
347C3×C4—62no—
348C3×C4—62no—
349C3×C4—61no—
350C3×C4—62no—
351C3×C4—61no—
352C3×C4—64no—
353C3×C4—62no—
354C3×C4—62no—
355C3×C4—122no—
356C3×C4—121no—
357C3×C4—182no—
358C3×C4—184no—
359C3×C4C3×#23512yes—
360C3×C4C3×#23912yes—
361Dic3—62no—
362Dic3—64no—
363Dic3—184no—
364Dic3—182no—
365C32—21no—
366C32#558411yescharacteristic, complemented
367C32C3×#55611yescharacteristic, complemented
368C23—92no—
369Q8—22no—
370Q8—184no—
371D4—182no—
372C2×C4—92no—
373C2×C4—94no—
374C2×C4—184no—
375C2×C4C22×(C3⋊D3)⋊C312yes—
376C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
377C2×C4C22×(C3⋊D3)⋊C312yes—
378C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
379C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
380C2×C3—31no—
381C2×C3—31no—
382C2×C3—31no—
383C2×C3—31no—
384C2×C3—31no—
385C2×C3—31no—
386C2×C3—61no—
387C2×C3—61no—
388C2×C3—61no—
389C2×C3—362no—
390C2×C3C2×C3×#23511yescharacteristic
391C2×C3C2×C3×#23911yescharacteristic
392C2×C3C3×#101011yescharacteristic
393D3—122no—
394D3—362no—
395C22—182no—
396C22—182no—
397C22—182no—
398C22C23×(C3⋊D3)⋊C311yescharacteristic
399C4—22no—
400C4—22no—
401C4—21no—
402C4—182no—
403C4—184no—
404C4#3828⋊C312yes—
405C4#8872212yes—
406C3—31no—
407C3—31no—
408C3—61no—
409C3C3×#558411yescharacteristic
410C2—362no—
411C2C2×#3828⋊C311yescharacteristic
412C2C2×#8872211yescharacteristic
413C2#1172263911yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC4×Q8×H(3,𝔽3)C2
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#556
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. #64155⋊C3C2semidirect
3. #64107⋊C3C2semidirect
4. #64101⋊C3C2semidirect
7. #64078⋊C3C2semidirect
8. C4×#634⋊C3C2semidirect
9. #10596055C3semidirect
28. C32×C4×Q8D3semidirect
29. C32×C4×Q8C2×C3semidirect
30. #64155C2×C3semidirect
31. #64107C2×C3semidirect
32. #64101C2×C3semidirect
35. #64078C2×C3semidirect
36. C4×#634C2×C3semidirect
253. C4×Q8(C3⋊D3)⋊C3semidirect
256. H(3,𝔽3)#556semidirect
366. C32#5584semidirect
367. C32C3×#556semidirect

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
#64155⋊C32
#64107⋊C32
#64101⋊C32
#64084⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
#64078⋊C32
C4×#634⋊C32
#105960553
C3×#55843