Finite Group Databasebeta

← #10595981⋊C3 #13152441

Subgroups — #10595981⋊C3 #13152441

Order 1728.

1121 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C212yesmaximal, complemented
2C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#64139⋊C3C214yesmaximal, complemented
4#64106⋊C3C214yesmaximal, complemented
5#64104⋊C3C214yesmaximal, complemented
6#10595981C311yesmaximal, characteristic, complemented
7C3×#5510—31nomaximal
8C22×C4×H(3,𝔽3)C2212yes—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#638⋊C3C2214yes—
11#3828⋊C3—28no—
12C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
13C2×C4⋊((C3⋊D3)⋊C3)C2214yes—
14C2×C4×(C3⋊D3)⋊C3C2214yes—
15C2×#634⋊C3C2212yes—
16C4⋊C4×H(3,𝔽3)C2214yes—
17#3752⋊C3C2218yes—
18C4⋊((C3⋊Dic3)⋊C3)C2214yes—
19C2×#3828C2×C312yescomplemented
20C2×C3×#235—32no—
21C2×C32×C4⋊C4—21no—
22C2×C32×C4⋊C4D311yescharacteristic, complemented
23C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
24#64139C2×C314yescomplemented
25#64106C2×C314yescomplemented
26#64104C2×C314yescomplemented
27C3×#930—34no—
28C3×#897—34no—
29C3×#895—34no—
30C23×H(3,𝔽3)C2311yescharacteristic
31C22×(C3⋊D3)⋊C3C2314yes—
32C2×C4×H(3,𝔽3)—42no—
33C2×C4×H(3,𝔽3)D414yescomplemented
34C2×C4×H(3,𝔽3)C2314yes—
35C2×C4×H(3,𝔽3)C2312yes—
36#638⋊C3—28no—
37C2×(C3⋊Dic3)⋊C3C2314yes—
38C4⋊((C3⋊D3)⋊C3)—24no—
39C4⋊((C3⋊D3)⋊C3)—44no—
40C4×(C3⋊D3)⋊C3—28no—
41#634⋊C3—24no—
42C3×#537—91no—
43#5510—31no—
44#5510—91no—
45C22×C32×C4—21no—
46C22×C32×C4—22no—
47C22×C32×C4C2×D312yes—
48C22×C32×C4C2×D311yescharacteristic
49C22×C32×C4C22×C312yes—
50C22×C32×C4C22×C311yescharacteristic
51C2×#638C22×C314yes—
52#3828—28no—
53C2×C4⋊(C3⋊D3)C22×C312yes—
54C2×C4⋊(C3⋊D3)C22×C314yes—
55C2×C4×C3⋊D3C22×C314yes—
56C2×#634C22×C312yes—
57C2×C3×#67—34no—
58C3×#235—68no—
59C2×C3×D12—32no—
60C2×C3×D12—34no—
61C2×C3×C4×D3—34no—
62C2×C3×Dic6—32no—
63C32×C4⋊C4—24no—
64C32×C4⋊C4C2×D314yes—
65C32×C4⋊C4C22×C314yes—
66#3752C22×C318yes—
67C4⋊(C3⋊Dic3)C22×C314yes—
68C3×(C2×D3)⋊C4—38no—
69C3×C4⋊Dic3—34no—
70C22×H(3,𝔽3)—21no—
71C22×H(3,𝔽3)—21no—
72C22×H(3,𝔽3)C2×D412yes—
73C22×H(3,𝔽3)C2411yescharacteristic
74C2×(C3⋊D3)⋊C3—24no—
75C2×(C3⋊D3)⋊C3—24no—
76C2×(C3⋊D3)⋊C3—24no—
77C4×H(3,𝔽3)—24no—
78C4×H(3,𝔽3)C2×D414yes—
79(C3⋊Dic3)⋊C3—24no—
80C2×C3×#41—92no—
81C2×#235—32no—
82C2×#235—92no—
83C3×C22⋊Q8—94no—
84C3×#121—94no—
85C3×#121—94no—
86C2×C3×C4⋊C4—31no—
87C2×C3×C4⋊C4—31no—
88C2×C3×C4⋊C4—61no—
89C2×C3×C4⋊C4C3×D311yescharacteristic
90#930—34no—
91#930—94no—
92#897—34no—
93#897—94no—
94#895—34no—
95#895—94no—
96C23×C32—21no—
97C23×C32C22×D311yescharacteristic
98C23×C32C23×C311yescharacteristic
99C22×C3⋊D3C23×C314yes—
100C22×C3×D3—34no—
101C2×C32×C4—24no—
102C2×C32×C4—22no—
103C2×C32×C4—24no—
104C2×C32×C4—42no—
105C2×C32×C4—42no—
106C2×C32×C4—44no—
107C2×C32×C4D1214yescomplemented
108C2×C32×C4C3×D414yescomplemented
109C2×C32×C4C22×D314yes—
110C2×C32×C4C22×D312yes—
111C2×C32×C4C23×C314yes—
112C2×C32×C4C23×C312yes—
113#638—28no—
114C2×C3⋊Dic3C23×C314yes—
115C4⋊(C3⋊D3)—24no—
116C4⋊(C3⋊D3)—44no—
117C4×C3⋊D3—28no—
118#634—24no—
119C3×#67—68no—
120C2×C3×Dic3—34no—
121C3×D12—64no—
122C3×D12—124no—
123C3×C4×D3—68no—
124C3×Dic6—64no—
125#537—91noSylow
126C2×H(3,𝔽3)—22no—
127C2×H(3,𝔽3)C22×D411yescharacteristic
128C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
129C2×H(3,𝔽3)2−1+411yescharacteristic
130(C3⋊D3)⋊C3—44no—
131C3×#41—188no—
132C2×C3×Q8—92no—
133C2×C3×D4—92no—
134C2×C3×D4—94no—
135C2×C3×D4—94no—
136C22×C3×C4—31no—
137C22×C3×C4—31no—
138C22×C3×C4—32no—
139C22×C3×C4—32no—
140C22×C3×C4—61no—
141C22×C3×C4—62no—
142C22×C3×C4—94no—
143C22×C3×C4C2×C3×D312yes—
144C22×C3×C4C2×C3×D311yescharacteristic
145C2×#67—34no—
146C2×#67—94no—
147#235—68no—
148#235—188no—
149C2×D12—32no—
150C2×D12—34no—
151C2×D12—92no—
152C2×D12—94no—
153C2×C4×D3—34no—
154C2×C4×D3—94no—
155C2×Dic6—32no—
156C2×Dic6—92no—
157C3×C4⋊C4—34no—
158C3×C4⋊C4—34no—
159C3×C4⋊C4—64no—
160C3×C4⋊C4—94no—
161C3×C4⋊C4C2×C3×D314yes—
162C3×C22⋊C4—98no—
163(C2×D3)⋊C4—38no—
164(C2×D3)⋊C4—98no—
165C4⋊Dic3—34no—
166C4⋊Dic3—94no—
167C22×C32—22no—
168C22×C32—22no—
169C22×C32—21no—
170C22×C32—21no—
171C22×C32—21no—
172C22×C32—21no—
173C22×C32—21no—
174C22×C32—22no—
175C22×C32C2×D1212yes—
176C22×C32C2×C3×D412yes—
177C22×C32C23×D311yescharacteristic
178C22×C32C24×C311yescharacteristic
179C2×C3⋊D3—24no—
180C2×C3⋊D3—24no—
181C2×C3⋊D3—24no—
182C2×C3×D3—64no—
183C2×C3×D3—64no—
184C2×C3×D3—64no—
185C32×C4—24no—
186C32×C4—24no—
187C32×C4—24no—
188C32×C4—44no—
189C32×C4C2×D1214yes—
190C32×C4C2×C3×D414yes—
191C3⋊Dic3—24no—
192C3×Dic3—64no—
193C2×#41—92no—
194C22⋊Q8—94no—
195#121—94no—
196#121—94no—
197C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
198H(3,𝔽3)#53711yescharacteristic, complemented, Sylow
199C23×C3—31no—
200C23×C3—31no—
201C23×C3—61no—
202C23×C3—94no—
203C23×C3C22×C3×D311yescharacteristic
204C22×D3—34no—
205C22×D3—94no—
206C3×Q8—184no—
207C3×D4—184no—
208C3×D4—188no—
209C3×D4—364no—
210C2×C3×C4—34no—
211C2×C3×C4—34no—
212C2×C3×C4—32no—
213C2×C3×C4—32no—
214C2×C3×C4—34no—
215C2×C3×C4—34no—
216C2×C3×C4—42no—
217C2×C3×C4—64no—
218C2×C3×C4—62no—
219C2×C3×C4—64no—
220C2×C3×C4—94no—
221C2×C3×C4—122no—
222C2×C3×C4—122no—
223C2×C3×C4—124no—
224C2×C3×C4—188no—
225C2×C3×C4C3×D1214yes—
226C2×C3×C4C22×C3×D314yes—
227C2×C3×C4C22×C3×D312yes—
228#67—68no—
229#67—188no—
230C2×Dic3—34no—
231C2×Dic3—94no—
232D12—64no—
233D12—124no—
234D12—184no—
235D12—364no—
236C4×D3—68no—
237C4×D3—188no—
238Dic6—64no—
239Dic6—184no—
240C2×C32—24no—
241C2×C32—22no—
242C2×C32—22no—
243C2×C32—21no—
244C2×C32—21no—
245C2×C32—21no—
246C2×C32C22×D1211yescharacteristic
247C2×C32#100411yescharacteristic
248C2×C32#100711yescharacteristic
249C2×C32C22×C3×D411yescharacteristic
250C2×C32C3×H(5,𝔽2)11yescharacteristic
251C2×C32C3×2−1+411yescharacteristic
252C3⋊D3—44no—
253C3×D3—124no—
254#41—188no—
255C2×Q8—92no—
256C2×D4—94no—
257C2×D4—92no—
258C2×D4—94no—
259C22×C4—94no—
260C22×C4C2×(C3⋊D3)⋊C312yes—
261C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
262C4⋊C4—94no—
263C4⋊C4C2×(C3⋊D3)⋊C314yes—
264C22⋊C4—98no—
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—31no—
268C22×C3—31no—
269C22×C3—32no—
270C22×C3—32no—
271C22×C3—61no—
272C22×C3—61no—
273C22×C3—61no—
274C22×C3—61no—
275C22×C3—62no—
276C22×C3—62no—
277C22×C3—61no—
278C22×C3—62no—
279C22×C3—184no—
280C22×C3—184no—
281C22×C3—184no—
282C22×C3C2×C3×D1212yes—
283C22×C3C23×C3×D311yescharacteristic
284C2×D3—64no—
285C2×D3—64no—
286C2×D3—64no—
287C2×D3—184no—
288C2×D3—184no—
289C2×D3—184no—
290C3×C4—24no—
291C3×C4—34no—
292C3×C4—34no—
293C3×C4—64no—
294C3×C4—64no—
295C3×C4—64no—
296C3×C4—124no—
297C3×C4—184no—
298C3×C4C2×C3×D1214yes—
299Dic3—64no—
300Dic3—184no—
301C32—21no—
302C32#551011yescharacteristic, complemented
303C32C3×#53711yescharacteristic, complemented
304C23—94no—
305C23C22×(C3⋊D3)⋊C311yescharacteristic
306Q8—184no—
307D4—188no—
308D4—184no—
309D4—364no—
310C2×C4—42no—
311C2×C4—94no—
312C2×C4—188no—
313C2×C4C4⋊((C3⋊D3)⋊C3)14yescomplemented
314C2×C4C22×(C3⋊D3)⋊C314yes—
315C2×C4C22×(C3⋊D3)⋊C312yes—
316C2×C3—22no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—31no—
321C2×C3—31no—
322C2×C3—31no—
323C2×C3—62no—
324C2×C3—62no—
325C2×C3—64no—
326C2×C3—61no—
327C2×C3—61no—
328C2×C3—61no—
329C2×C3—364no—
330C2×C3C22×C3×D1211yescharacteristic
331C2×C3C3×#100411yescharacteristic
332C2×C3C3×#100711yescharacteristic
333D3—124no—
334D3—364no—
335C22—21no—
336C22—21no—
337C22—184no—
338C22—184no—
339C22—184no—
340C22C2×C4⋊((C3⋊D3)⋊C3)12yes—
341C22C23×(C3⋊D3)⋊C311yescharacteristic
342C4—24no—
343C4—184no—
344C4C2×C4⋊((C3⋊D3)⋊C3)14yes—
345C3—31no—
346C3—31no—
347C3—61no—
348C3C3×#551011yescharacteristic
349C2—22no—
350C2—364no—
351C2C22×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
352C2#64361⋊C311yescharacteristic
353C2#1172262711yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#537
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#3828⋊C3C2semidirect
2. C2×C4⋊C4×H(3,𝔽3)C2semidirect
3. #64139⋊C3C2semidirect
4. #64106⋊C3C2semidirect
5. #64104⋊C3C2semidirect
6. #10595981C3semidirect
19. C2×#3828C2×C3semidirect
22. C2×C32×C4⋊C4D3semidirect
23. C2×C32×C4⋊C4C2×C3semidirect
24. #64139C2×C3semidirect
25. #64106C2×C3semidirect
26. #64104C2×C3semidirect
33. C2×C4×H(3,𝔽3)D4semidirect
107. C2×C32×C4D12semidirect
108. C2×C32×C4C3×D4semidirect
197. C2×C4⋊C4(C3⋊D3)⋊C3semidirect
198. H(3,𝔽3)#537semidirect
302. C32#5510semidirect
303. C32C3×#537semidirect
313. C2×C4C4⋊((C3⋊D3)⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C2×C4⋊C4×H(3,𝔽3)2
#64139⋊C32
#64106⋊C32
#64104⋊C32
#105959813
C3×#55103