Finite Group Databasebeta

← #10596203⋊C3 #13153328

Subgroups — #10596203⋊C3 #13153328

Order 1728.

810 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1H(3,𝔽3)×C4⋊Q8C211yesmaximal, characteristic, complemented
2#64156⋊C3C212yesmaximal
3Q8×(C3⋊Dic3)⋊C3C212yesmaximal, complemented
4#64107⋊C3C214yesmaximal, complemented
5#64106⋊C3C212yesmaximal, complemented
6#64101⋊C3C212yesmaximal, complemented
7C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
8C4×#634⋊C3C211yesmaximal, characteristic, complemented
9#10596203C311yesmaximal, characteristic, complemented
10C3×#5732—31nomaximal
11C2×Q8×H(3,𝔽3)C2212yes—
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3C2212yes—
14C2×#634⋊C3C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2212yes—
16C4⋊C4×H(3,𝔽3)C2212yes—
17C42×H(3,𝔽3)C2211yescharacteristic
18#3752⋊C3C2212yes—
19#3752⋊C3C2214yes—
20#3752⋊C3C2214yes—
21C4⋊((C3⋊Dic3)⋊C3)C2214yes—
22C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24#3750⋊C3C2212yes—
25C4×(C3⋊Dic3)⋊C3C2214yes—
26C4×(C3⋊Dic3)⋊C3C2212yes—
27C32×C4⋊Q8—21no—
28C32×C4⋊Q8D311yescharacteristic, complemented
29C32×C4⋊Q8C2×C311yescharacteristic, complemented
30#64156C2×C312yes—
31Q8×C3⋊Dic3C2×C312yescomplemented
32#64107C2×C314yescomplemented
33#64106C2×C312yescomplemented
34#64101C2×C312yescomplemented
35C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
36C4×#634C2×C311yescharacteristic, complemented
37C3×#947—32no—
38C3×Q8×Dic3—32no—
39C3×#898—34no—
40C3×#897—32no—
41C3×#892—32no—
42C3×C4×D12—31no—
43C3×C4×Dic6—31no—
44C22×(C3⋊D3)⋊C3C2312yes—
45Q8×H(3,𝔽3)—24no—
46C2×C4×H(3,𝔽3)C2314yes—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×(C3⋊Dic3)⋊C3C2312yes—
51C2×(C3⋊Dic3)⋊C3C2312yes—
52C2×(C3⋊Dic3)⋊C3C2312yes—
53C4⋊((C3⋊D3)⋊C3)—22no—
54C4×(C3⋊D3)⋊C3—24no—
55#634⋊C3—22no—
56C3×#556—91no—
57#5732—31no—
58#5732—91no—
59C2×C32×Q8—22no—
60C2×C32×Q8C2×D312yes—
61C2×C32×Q8C22×C312yes—
62C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
63C2×C4×C3⋊D3C22×C312yes—
64C2×#634C22×C311yescharacteristic
65C2×C3×D12—31no—
66C2×C3×C4×D3—32no—
67C2×C3×Dic6—31no—
68C32×C4⋊C4—22no—
69C32×C4⋊C4—22no—
70C32×C4⋊C4C2×D312yes—
71C32×C4⋊C4C2×D312yes—
72C32×C4⋊C4C22×C312yes—
73C32×C4⋊C4C22×C312yes—
74C32×C42—21no—
75C32×C42C2×D311yescharacteristic
76C32×C42C22×C311yescharacteristic
77#3752C22×C312yes—
78#3752C22×C314yes—
79#3752C22×C314yes—
80C4⋊(C3⋊Dic3)C22×C311yescharacteristic
81C4⋊(C3⋊Dic3)C22×C311yescharacteristic
82C4⋊(C3⋊Dic3)C22×C314yes—
83#3750C22×C312yes—
84C4×C3⋊Dic3C22×C312yes—
85C4×C3⋊Dic3C22×C314yes—
86C3×(C2×D3)⋊C4—34no—
87C3×(C2×D3)⋊C4—34no—
88C3×(C2×D3)⋊C4—32no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—34no—
91C3×C4⋊Dic3—31no—
92C3×Dic3⋊C4—32no—
93C3×C4×Dic3—34no—
94C3×C4×Dic3—32no—
95C22×H(3,𝔽3)C2411yescharacteristic
96C2×(C3⋊D3)⋊C3—22no—
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—22no—
99C4×H(3,𝔽3)—24no—
100C4×H(3,𝔽3)—21no—
101C4×H(3,𝔽3)#4112yes—
102C4×H(3,𝔽3)#4112yes—
103(C3⋊Dic3)⋊C3—22no—
104(C3⋊Dic3)⋊C3—22no—
105(C3⋊Dic3)⋊C3—22no—
106C3×C4⋊Q8—31no—
107C3×C4⋊Q8—31no—
108C3×C4⋊Q8—61no—
109C3×C4⋊Q8C3×D311yescharacteristic
110C3×#126—94no—
111C3×#124—92no—
112C3×C22⋊Q8—92no—
113C3×C4×Q8—91no—
114C3×C4×Q8—92no—
115C3×C4×D4—91no—
116C3×#117—92no—
117#947—32no—
118#947—92no—
119Q8×Dic3—32no—
120Q8×Dic3—92no—
121#898—34no—
122#898—94no—
123#897—32no—
124#897—92no—
125#892—32no—
126#892—92no—
127C4×D12—31no—
128C4×D12—91no—
129C4×Dic6—31no—
130C4×Dic6—91no—
131C22×C3⋊D3C23×C312yes—
132C22×C3×D3—32no—
133C32×Q8—24no—
134C32×Q8—24no—
135C32×Q8—44no—
136C2×C32×C4—21no—
137C2×C32×C4—21no—
138C2×C32×C4—21no—
139C2×C32×C4—24no—
140C2×C32×C4C22×D314yes—
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C23×C314yes—
145C2×C32×C4C23×C311yescharacteristic
146C2×C32×C4C23×C311yescharacteristic
147C2×C32×C4C23×C311yescharacteristic
148C2×C3⋊Dic3C23×C312yes—
149C2×C3⋊Dic3C23×C312yes—
150C2×C3⋊Dic3C23×C312yes—
151C4⋊(C3⋊D3)—22no—
152C4×C3⋊D3—24no—
153#634—22no—
154C2×C3×Dic3—32no—
155C2×C3×Dic3—32no—
156C2×C3×Dic3—32no—
157C3×D12—62no—
158C3×C4×D3—64no—
159C3×Dic6—62no—
160#556—91noSylow
161C2×H(3,𝔽3)C2×#4111yescharacteristic
162C2×H(3,𝔽3)C2×#4111yescharacteristic
163C2×H(3,𝔽3)2−1+411yescharacteristic
164(C3⋊D3)⋊C3—42no—
165C2×C3×Q8—32no—
166C2×C3×Q8—32no—
167C2×C3×Q8—62no—
168C2×C3×Q8—91no—
169C2×C3×Q8C2×C3×D312yes—
170C2×C3×D4—91no—
171C22×C3×C4—92no—
172C2×D12—31no—
173C2×D12—91no—
174C2×C4×D3—32no—
175C2×C4×D3—92no—
176C2×Dic6—31no—
177C2×Dic6—91no—
178C3×C4⋊C4—32no—
179C3×C4⋊C4—32no—
180C3×C4⋊C4—32no—
181C3×C4⋊C4—32no—
182C3×C4⋊C4—62no—
183C3×C4⋊C4—62no—
184C3×C4⋊C4—92no—
185C3×C4⋊C4—91no—
186C3×C4⋊C4—91no—
187C3×C4⋊C4—94no—
188C3×C4⋊C4C2×C3×D312yes—
189C3×C4⋊C4C2×C3×D312yes—
190C3×C22⋊C4—94no—
191C3×C22⋊C4—92no—
192C3×C22⋊C4—94no—
193C3×C42—31no—
194C3×C42—31no—
195C3×C42—61no—
196C3×C42—94no—
197C3×C42—92no—
198C3×C42C2×C3×D311yescharacteristic
199(C2×D3)⋊C4—32no—
200(C2×D3)⋊C4—34no—
201(C2×D3)⋊C4—34no—
202(C2×D3)⋊C4—92no—
203(C2×D3)⋊C4—94no—
204(C2×D3)⋊C4—94no—
205C4⋊Dic3—31no—
206C4⋊Dic3—31no—
207C4⋊Dic3—34no—
208C4⋊Dic3—91no—
209C4⋊Dic3—91no—
210C4⋊Dic3—94no—
211Dic3⋊C4—32no—
212Dic3⋊C4—92no—
213C4×Dic3—32no—
214C4×Dic3—34no—
215C4×Dic3—92no—
216C4×Dic3—94no—
217C22×C32—21no—
218C22×C32C23×D311yescharacteristic
219C22×C32C24×C311yescharacteristic
220C2×C3⋊D3—22no—
221C2×C3⋊D3—22no—
222C2×C3⋊D3—22no—
223C2×C3×D3—62no—
224C2×C3×D3—62no—
225C2×C3×D3—62no—
226C32×C4—22no—
227C32×C4—24no—
228C32×C4—21no—
229C32×C4—24no—
230C32×C4—21no—
231C32×C4—22no—
232C32×C4—22no—
233C32×C4—44no—
234C32×C4#23712yes—
235C32×C4#23912yes—
236C32×C4C3×#4112yes—
237C32×C4C3×#4112yes—
238C3⋊Dic3—22no—
239C3⋊Dic3—22no—
240C3⋊Dic3—22no—
241C3×Dic3—62no—
242C3×Dic3—62no—
243C3×Dic3—62no—
244C4⋊Q8(C3⋊D3)⋊C311yescharacteristic, complemented
245#126—94no—
246#124—92no—
247C22⋊Q8—92no—
248C4×Q8—92no—
249C4×Q8—91no—
250C4×D4—91no—
251#117—92no—
252H(3,𝔽3)#55611yescharacteristic, complemented, Sylow
253C23×C3—92no—
254C22×D3—32no—
255C22×D3—92no—
256C3×Q8—24no—
257C3×Q8—64no—
258C3×Q8—64no—
259C3×Q8—124no—
260C3×Q8—182no—
261C3×D4—182no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—34no—
268C2×C3×C4—34no—
269C2×C3×C4—31no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—61no—
273C2×C3×C4—64no—
274C2×C3×C4—92no—
275C2×C3×C4—92no—
276C2×C3×C4—92no—
277C2×C3×C4—184no—
278C2×C3×C4C22×C3×D314yes—
279C2×C3×C4C22×C3×D311yescharacteristic
280C2×C3×C4C22×C3×D311yescharacteristic
281C2×C3×C4C22×C3×D311yescharacteristic
282C2×Dic3—32no—
283C2×Dic3—32no—
284C2×Dic3—32no—
285C2×Dic3—92no—
286C2×Dic3—92no—
287C2×Dic3—92no—
288D12—62no—
289D12—182no—
290C4×D3—64no—
291C4×D3—184no—
292Dic6—62no—
293Dic6—182no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32C2×#23711yescharacteristic
298C2×C32C2×#23911yescharacteristic
299C2×C32#100711yescharacteristic
300C2×C32C2×C3×#4111yescharacteristic
301C2×C32C2×C3×#4111yescharacteristic
302C2×C32C3×2−1+411yescharacteristic
303C3⋊D3—42no—
304C3×D3—122no—
305C2×Q8—91no—
306C2×Q8C2×(C3⋊D3)⋊C312yes—
307C2×D4—91no—
308C22×C4—92no—
309C4⋊C4—91no—
310C4⋊C4—91no—
311C4⋊C4—94no—
312C4⋊C4—92no—
313C4⋊C4C2×(C3⋊D3)⋊C312yes—
314C4⋊C4C2×(C3⋊D3)⋊C312yes—
315C22⋊C4—94no—
316C22⋊C4—92no—
317C22⋊C4—94no—
318C42—94no—
319C42—92no—
320C42C2×(C3⋊D3)⋊C311yescharacteristic
321C22×C3—31no—
322C22×C3—31no—
323C22×C3—61no—
324C22×C3—182no—
325C22×C3—182no—
326C22×C3—182no—
327C22×C3C23×C3×D311yescharacteristic
328C2×D3—62no—
329C2×D3—62no—
330C2×D3—62no—
331C2×D3—182no—
332C2×D3—182no—
333C2×D3—182no—
334C3×C4—24no—
335C3×C4—21no—
336C3×C4—32no—
337C3×C4—32no—
338C3×C4—32no—
339C3×C4—32no—
340C3×C4—64no—
341C3×C4—64no—
342C3×C4—61no—
343C3×C4—61no—
344C3×C4—62no—
345C3×C4—62no—
346C3×C4—62no—
347C3×C4—124no—
348C3×C4—182no—
349C3×C4—182no—
350C3×C4—182no—
351C3×C4C3×#23712yes—
352C3×C4C3×#23912yes—
353Dic3—62no—
354Dic3—62no—
355Dic3—62no—
356Dic3—182no—
357Dic3—182no—
358Dic3—182no—
359C32—21no—
360C32#573211yescharacteristic, complemented
361C32C3×#55611yescharacteristic, complemented
362C23—92no—
363Q8—24no—
364Q8—182no—
365D4—182no—
366C2×C4—92no—
367C2×C4—92no—
368C2×C4—92no—
369C2×C4—184no—
370C2×C4C22×(C3⋊D3)⋊C314yes—
371C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
372C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
373C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
374C2×C3—31no—
375C2×C3—31no—
376C2×C3—31no—
377C2×C3—31no—
378C2×C3—31no—
379C2×C3—31no—
380C2×C3—61no—
381C2×C3—61no—
382C2×C3—61no—
383C2×C3—362no—
384C2×C3C2×C3×#23711yescharacteristic
385C2×C3C2×C3×#23911yescharacteristic
386C2×C3C3×#100711yescharacteristic
387D3—122no—
388D3—362no—
389C22—182no—
390C22—182no—
391C22—182no—
392C22C23×(C3⋊D3)⋊C311yescharacteristic
393C4—24no—
394C4—21no—
395C4—182no—
396C4—182no—
397C4—182no—
398C4#3830⋊C312yes—
399C4#8872212yes—
400C3—31no—
401C3—31no—
402C3—61no—
403C3C3×#573211yescharacteristic
404C2—362no—
405C2C2×#3830⋊C311yescharacteristic
406C2C2×#8872211yescharacteristic
407C2#1172262711yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×C4⋊Q8C2
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. H(3,𝔽3)×C4⋊Q8C2semidirect
3. Q8×(C3⋊Dic3)⋊C3C2semidirect
4. #64107⋊C3C2semidirect
5. #64106⋊C3C2semidirect
6. #64101⋊C3C2semidirect
8. C4×#634⋊C3C2semidirect
9. #10596203C3semidirect
28. C32×C4⋊Q8D3semidirect
29. C32×C4⋊Q8C2×C3semidirect
31. Q8×C3⋊Dic3C2×C3semidirect
32. #64107C2×C3semidirect
33. #64106C2×C3semidirect
34. #64101C2×C3semidirect
36. C4×#634C2×C3semidirect
244. C4⋊Q8(C3⋊D3)⋊C3semidirect
252. H(3,𝔽3)#556semidirect
360. C32#5732semidirect
361. C32C3×#556semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×C4⋊Q82
#64156⋊C32
Q8×(C3⋊Dic3)⋊C32
#64107⋊C32
#64106⋊C32
#64101⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
C4×#634⋊C32
#105962033
C3×#57323