Finite Group Databasebeta

← #10592701⋊C3 #13142696

Subgroups — #10592701⋊C3 #13142696

Order 1728.

629 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
3C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4#63636⋊C3C212yesmaximal, complemented
5#63637⋊C3C212yesmaximal, complemented
6#10592701C311yesmaximal, characteristic, complemented
7C3×#5002—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#638⋊C3—21no—
10#3828⋊C3—22no—
11#3828⋊C3C414yescomplemented
12C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
13C2×C4×(C3⋊D3)⋊C3—21no—
14C2×#634⋊C3C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2212yes—
16C22⋊C4×H(3,𝔽3)—21no—
17C42×H(3,𝔽3)—21no—
18C2×(C3⋊(C3⋊C8))⋊C3—22no—
19C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
20C2×#3828C2×C311yescharacteristic
21C2×C3×#235—31no—
22C32×#117—21no—
23C32×#117D311yescharacteristic, complemented
24C32×#117C2×C311yescharacteristic, complemented
25C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
26C22×C3×C3⋊C8—31no—
27#63637C2×C312yescomplemented
28#63636C2×C312yescomplemented
29C3×#810—32no—
30C3×#809—32no—
31C23×H(3,𝔽3)D411yescharacteristic
32C22×(C3⋊D3)⋊C3—21no—
33C2×C4×H(3,𝔽3)—22no—
34C2×C4×H(3,𝔽3)C2×C412yes—
35C2×C4×H(3,𝔽3)D411yescharacteristic
36C2×C4×H(3,𝔽3)D412yes—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38#638⋊C3—24no—
39C2×(C3⋊Dic3)⋊C3—21no—
40C4⋊((C3⋊D3)⋊C3)—21no—
41C4⋊((C3⋊D3)⋊C3)C2×C412yes—
42C4×(C3⋊D3)⋊C3—24no—
43#634⋊C3—21no—
44#634⋊C3C2×C412yes—
45(C3⋊(C3⋊C8))⋊C3—22no—
46C3×#416—91no—
47#5002—31no—
48#5002—91no—
49C22×C32×C4—21no—
50C22×C32×C4C2×D311yescharacteristic
51C22×C32×C4C22×C311yescharacteristic
52C2×#638—21no—
53#3828—22no—
54#3828C3×C414yescomplemented
55C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
56C2×C4×C3⋊D3—21no—
57C2×#634C22×C311yescharacteristic
58C2×C3×#67—61no—
59C3×#235—34no—
60C3×#235—62no—
61C2×C3×D12—31no—
62C2×C3×C4×D3—61no—
63C2×C3×Dic6—31no—
64C32×C4⋊C4—22no—
65C32×C4⋊C4C2×D312yes—
66C32×C4⋊C4C22×C312yes—
67C32×C22⋊C4—22no—
68C32×C22⋊C4—21no—
69C32×C22⋊C4—21no—
70C32×C42—22no—
71C32×C42—21no—
72C32×C42—21no—
73C2×C3⋊(C3⋊C8)—22no—
74C2×C3⋊(C3⋊C8)C22×C312yes—
75C2×C3×C3⋊C8—32no—
76C2×C3×C3⋊C8—62no—
77C22×H(3,𝔽3)—22no—
78C22×H(3,𝔽3)C22⋊C412yes—
79C22×H(3,𝔽3)C2×D411yescharacteristic
80C2×(C3⋊D3)⋊C3—22no—
81C2×(C3⋊D3)⋊C3—42no—
82C4×H(3,𝔽3)—42no—
83C4×H(3,𝔽3)C22⋊C412yes—
84C4×H(3,𝔽3)C22×C411yescharacteristic
85C4×H(3,𝔽3)C2×D411yescharacteristic
86(C3⋊Dic3)⋊C3—22no—
87C2×C3×#41—91no—
88C2×#235—31no—
89C2×#235—91no—
90C22×C3×C8—91no—
91C3×#117—31no—
92C3×#117—31no—
93C3×#117—61no—
94C3×#117C3×D311yescharacteristic
95C22×C3⋊C8—31no—
96C22×C3⋊C8—91no—
97C3×#103—92no—
98C3×#102—92no—
99#810—32no—
100#810—92no—
101#809—32no—
102#809—92no—
103C23×C32—21no—
104C23×C32#6711yescharacteristic
105C23×C32C3×D411yescharacteristic
106C22×C3⋊D3—21no—
107C22×C3×D3—61no—
108C2×C32×C4—24no—
109C2×C32×C4—22no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—22no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4C4×D312yes—
116C2×C32×C4D1212yes—
117C2×C32×C4#6711yescharacteristic
118C2×C32×C4C2×C3×C412yes—
119C2×C32×C4C3×D412yes—
120C2×C32×C4C3×D411yescharacteristic
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C23×C311yescharacteristic
123#638—24no—
124C2×C3⋊Dic3—21no—
125C4⋊(C3⋊D3)—21no—
126C4⋊(C3⋊D3)C2×C3×C412yes—
127C4×C3⋊D3—24no—
128#634—21no—
129#634C2×C3×C412yes—
130C3×#67—64no—
131C2×C3×Dic3—61no—
132C3×D12—32no—
133C3×D12—61no—
134C3×C4×D3—64no—
135C3×Dic6—32no—
136C3×Dic6—61no—
137C3⋊(C3⋊C8)—22no—
138C3×C3⋊C8—62no—
139#416—91noSylow
140C2×H(3,𝔽3)—22no—
141C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
142C2×H(3,𝔽3)#13512yes—
143(C3⋊D3)⋊C3—42no—
144C3×#41—94no—
145C3×#41—182no—
146C2×C3×Q8—91no—
147C2×C3×D4—91no—
148C2×C3×D4—181no—
149C22×C3×C4—31no—
150C22×C3×C4—31no—
151C22×C3×C4—61no—
152C22×C3×C4—181no—
153C22×C3×C4C2×C3×D311yescharacteristic
154C2×#67—61no—
155C2×#67—181no—
156#235—34no—
157#235—62no—
158#235—94no—
159#235—182no—
160C2×D12—31no—
161C2×D12—91no—
162C2×C4×D3—61no—
163C2×C4×D3—181no—
164C2×Dic6—31no—
165C2×Dic6—91no—
166C2×C3×C8—92no—
167C2×C3×C8—182no—
168C3×C4⋊C4—32no—
169C3×C4⋊C4—32no—
170C3×C4⋊C4—62no—
171C3×C4⋊C4C2×C3×D312yes—
172C3×C22⋊C4—21no—
173C3×C22⋊C4—61no—
174C3×C22⋊C4—61no—
175C3×C22⋊C4—62no—
176C3×C42—21no—
177C3×C42—61no—
178C3×C42—61no—
179C3×C42—62no—
180C2×C3⋊C8—32no—
181C2×C3⋊C8—62no—
182C2×C3⋊C8—92no—
183C2×C3⋊C8—182no—
184C22×C32—22no—
185C22×C32—22no—
186C22×C32—22no—
187C22×C32—21no—
188C22×C32—42no—
189C22×C32(C2×D3)⋊C412yes—
190C22×C32C3×C22⋊C412yes—
191C22×C32C2×#6711yescharacteristic
192C22×C32C2×C3×D411yescharacteristic
193C2×C3⋊D3—22no—
194C2×C3⋊D3—42no—
195C2×C3×D3—62no—
196C2×C3×D3—122no—
197C32×C4—22no—
198C32×C4—21no—
199C32×C4—21no—
200C32×C4—44no—
201C32×C4—42no—
202C32×C4—42no—
203C32×C4(C2×D3)⋊C412yes—
204C32×C4C3×C22⋊C412yes—
205C32×C4C2×C4×D311yescharacteristic
206C32×C4C2×D1211yescharacteristic
207C32×C4C22×C3×C411yescharacteristic
208C32×C4C2×C3×D411yescharacteristic
209C3⋊Dic3—22no—
210C3×Dic3—62no—
211C2×#41—91no—
212C22×C8—91no—
213#117(C3⋊D3)⋊C311yescharacteristic, complemented
214#103—92no—
215#102—92no—
216H(3,𝔽3)#41611yescharacteristic, complemented, Sylow
217C23×C3—31no—
218C23×C3—31no—
219C23×C3—61no—
220C23×C3—181no—
221C23×C3C3×#6711yescharacteristic
222C22×D3—61no—
223C22×D3—181no—
224C3×Q8—92no—
225C3×Q8—181no—
226C3×D4—92no—
227C3×D4—181no—
228C3×D4—184no—
229C2×C3×C4—22no—
230C2×C3×C4—32no—
231C2×C3×C4—32no—
232C2×C3×C4—32no—
233C2×C3×C4—32no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—62no—
239C2×C3×C4—62no—
240C2×C3×C4—64no—
241C2×C3×C4—62no—
242C2×C3×C4—62no—
243C2×C3×C4—61no—
244C2×C3×C4—61no—
245C2×C3×C4—184no—
246C2×C3×C4—181no—
247C2×C3×C4C3×C4×D312yes—
248C2×C3×C4C3×D1212yes—
249C2×C3×C4C3×#6711yescharacteristic
250C2×C3×C4C22×C3×D311yescharacteristic
251#67—64no—
252#67—184no—
253C2×Dic3—61no—
254C2×Dic3—181no—
255D12—32no—
256D12—61no—
257D12—92no—
258D12—181no—
259C4×D3—64no—
260C4×D3—184no—
261Dic6—32no—
262Dic6—61no—
263Dic6—92no—
264Dic6—181no—
265C3×C8—182no—
266C3⋊C8—62no—
267C3⋊C8—182no—
268C2×C32—22no—
269C2×C32—22no—
270C2×C32—21no—
271C2×C32—22no—
272C2×C32—42no—
273C2×C32C2×(C2×D3)⋊C411yescharacteristic
274C2×C32#95012yes—
275C2×C32C2×C3×C22⋊C411yescharacteristic
276C2×C32C3×#13512yes—
277C3⋊D3—42no—
278C3×D3—122no—
279#41—94no—
280#41—182no—
281C2×Q8—91no—
282C2×D4—91no—
283C2×D4—181no—
284C22×C4—181no—
285C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
286C2×C8—92no—
287C2×C8—182no—
288C4⋊C4C2×(C3⋊D3)⋊C312yes—
289C22⋊C4—21no—
290C42—21no—
291C22×C3—22no—
292C22×C3—31no—
293C22×C3—31no—
294C22×C3—32no—
295C22×C3—32no—
296C22×C3—62no—
297C22×C3—62no—
298C22×C3—61no—
299C22×C3—62no—
300C22×C3—122no—
301C22×C3—182no—
302C22×C3—362no—
303C22×C3C3×(C2×D3)⋊C412yes—
304C22×C3C2×C3×#6711yescharacteristic
305C2×D3—62no—
306C2×D3—122no—
307C2×D3—182no—
308C2×D3—362no—
309C3×C4—31no—
310C3×C4—31no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—32no—
314C3×C4—32no—
315C3×C4—42no—
316C3×C4—61no—
317C3×C4—61no—
318C3×C4—62no—
319C3×C4—122no—
320C3×C4—122no—
321C3×C4—124no—
322C3×C4—182no—
323C3×C4C3×(C2×D3)⋊C412yes—
324C3×C4C2×C3×C4×D311yescharacteristic
325C3×C4C2×C3×D1211yescharacteristic
326Dic3—62no—
327Dic3—182no—
328C32—21no—
329C32#500211yescharacteristic, complemented
330C32C3×#41611yescharacteristic, complemented
331C23—181no—
332C23#638⋊C311yescharacteristic
333Q8—92no—
334Q8—181no—
335D4—92no—
336D4—184no—
337D4—181no—
338C2×C4—22no—
339C2×C4—181no—
340C2×C4—184no—
341C2×C4C4×(C3⋊D3)⋊C312yes—
342C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
343C2×C4#638⋊C311yescharacteristic
344C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
345C8—182no—
346C2×C3—22no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—32no—
350C2×C3—32no—
351C2×C3—62no—
352C2×C3—62no—
353C2×C3—61no—
354C2×C3—62no—
355C2×C3—122no—
356C2×C3—362no—
357C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
358C2×C3C3×#95012yes—
359D3—122no—
360D3—362no—
361C22—22no—
362C22—182no—
363C22—362no—
364C22#3752⋊C312yes—
365C22C2×#638⋊C311yescharacteristic
366C4—42no—
367C4—182no—
368C4#3752⋊C312yes—
369C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
370C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
371C3—31no—
372C3—31no—
373C3—61no—
374C3C3×#500211yescharacteristic
375C2—22no—
376C2—362no—
377C2C2×#3752⋊C311yescharacteristic
378C2#64159⋊C312yes—
subgroupG/H
centerC2×C4#638⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#117C2
socleC22×C3C2×C3×#67
prime radicalC2×#3828⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C3×#416
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#117C2semidirect
3. C22×(C3⋊(C3⋊C8))⋊C3C2semidirect
4. #63636⋊C3C2semidirect
5. #63637⋊C3C2semidirect
6. #10592701C3semidirect
11. #3828⋊C3C4semidirect
23. C32×#117D3semidirect
24. C32×#117C2×C3semidirect
25. C22×C3⋊(C3⋊C8)C2×C3semidirect
27. #63637C2×C3semidirect
28. #63636C2×C3semidirect
54. #3828C3×C4semidirect
213. #117(C3⋊D3)⋊C3semidirect
216. H(3,𝔽3)#416semidirect
329. C32#5002semidirect
330. C32C3×#416semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
H(3,𝔽3)×#1172
C22×(C3⋊(C3⋊C8))⋊C32
#63636⋊C32
#63637⋊C32
#105927013
C3×#50023