Finite Group Databasebeta

← #13181407

Subgroups — #13181407

Order 1728.

1440 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 1440 conjugacy classes of subgroups — too many to draw. Only its 187 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#41⋊C3C211yesmaximal, characteristic, complemented
2#11724471C211yesmaximal, characteristic, complemented
3#11724470C212yesmaximal, complemented
4#11724469C211yesmaximal, characteristic, complemented
5#11724468C212yesmaximal, complemented
6#10596586—312nomaximal
7#10596293—31nomaximal
8#89076—41nomaximal
9C32×#41⋊C3C2211yescharacteristic
10C32×#41⋊C3C2211yescharacteristic, complemented
11C2×C32×SL(2,𝔽3)C2211yescharacteristic, complemented
12C3⋊(C3⋊GL(2,𝔽3))C2212yescomplemented
13C3⋊(C3⋊#226)C2212yescomplemented
14C2×C32×#41D311yescharacteristic, complemented
15C2×#3828—31no—
16C2×C3×#41⋊C3D3112yescomplemented
17#64267—312no—
18#64266—324no—
19#64265—312no—
20#64264—324no—
21#64160—31no—
22#64160—31no—
23#64159—32no—
24#64159—32no—
25#64158—31no—
26#64158—31no—
27#64151—32no—
28#64141—32no—
29C2×#3748—31no—
30C2×C33×C4—41no—
31#6312—42no—
32C4⋊(C3⋊(C3⋊D3))—41no—
33C4×C3⋊(C3⋊D3)—42no—
34#6308—41no—
35C32×SL(2,𝔽3)C2311yescharacteristic
36#5923—99no—
37#5822—94no—
38C32×#41—31no—
39C32×#41—31no—
40C32×#41C2×D311yescharacteristic
41C32×#41C2×D311yescharacteristic
42C2×C32×Q8C2×D311yescharacteristic
43C2×C32×D4—31no—
44C22×C32×C4—31no—
45C2×#638—61no—
46#3828—32no—
47#3828—32no—
48#3828—61no—
49#3828—61no—
50#3828—1212no—
51C2×C4⋊(C3⋊D3)—31no—
52C2×C4×C3⋊D3—61no—
53C2×#634—31no—
54C3×#41⋊C3C2×D3112yes—
55C3×#41⋊C3C2×D3112yescomplemented
56C2×C3×SL(2,𝔽3)C2×D3112yescomplemented
57C3⋊GL(2,𝔽3)—324no—
58C3⋊#226—324no—
59#3756—32no—
60#3756—32no—
61#3755—32no—
62#3755—32no—
63#3754—32no—
64#3754—32no—
65#3753—32no—
66#3753—32no—
67#3748—32no—
68#3748—31no—
69#3748—31no—
70C2×C3⋊(C3⋊C8)—32no—
71C22×C33—41no—
72C2×C3⋊(C3⋊D3)—42no—
73C33×C4—41no—
74C33×C4—41no—
75C3⋊(C3⋊Dic3)—42no—
76C2×C3×#41C3⋊D314yescomplemented
77C2×#235—94no—
78C2×#41⋊C3C3⋊D319yescomplemented
79#986—99no—
80#985—918no—
81#984—99no—
82#983—918no—
83#951—94no—
84#951—94no—
85#950—98no—
86#950—98no—
87#949—94no—
88#949—94no—
89#942—98no—
90#932—98no—
91C2×#208—94no—
92C23×C32—31no—
93C22×C3⋊D3—61no—
94C32×Q8—31no—
95C32×Q8C22×D311yescharacteristic
96C32×D4—31no—
97C32×D4—31no—
98C32×D4—61no—
99C2×C32×C4—31no—
100C2×C32×C4—31no—
101C2×C32×C4—31no—
102C2×C32×C4—31no—
103C2×C32×C4—31no—
104C2×C32×C4—412no—
105C2×C32×C4S411yescharacteristic
106#638—62no—
107#638—62no—
108#638—1224no—
109C2×C3⋊Dic3—61no—
110C4⋊(C3⋊D3)—32no—
111C4⋊(C3⋊D3)—61no—
112C4⋊(C3⋊D3)—1212no—
113C4×C3⋊D3—62no—
114C4×C3⋊D3—62no—
115C4×C3⋊D3—1224no—
116#634—32no—
117#634—61no—
118#634—1212no—
119C3×SL(2,𝔽3)C22×D3112yes—
120C3⋊(C3⋊C8)—32no—
121C3⋊(C3⋊C8)—32no—
122#575—271noSylow
123C2×C33—41no—
124C2×C33—81no—
125C3⋊(C3⋊D3)—82no—
126C3×#41—34no—
127C3×#41—34no—
128C3×#41C2×C3⋊D314yes—
129C3×#41C2×C3⋊D314yes—
130C2×C3×Q8C2×C3⋊D314yes—
131C2×C3×D4—34no—
132C22×C3×C4—34no—
133C2×#67—184no—
134#235—98no—
135#235—98no—
136#235—184no—
137#235—184no—
138#235—369no—
139C2×D12—94no—
140C2×C4×D3—184no—
141C2×Dic6—94no—
142#41⋊C3C2×C3⋊D319yes—
143#41⋊C3C2×C3⋊D319yescomplemented
144C2×SL(2,𝔽3)C2×C3⋊D319yescomplemented
145GL(2,𝔽3)—918no—
146#226—918no—
147#216—98no—
148#216—98no—
149#215—98no—
150#215—98no—
151#214—98no—
152#214—98no—
153#213—98no—
154#213—98no—
155#208—98no—
156#208—94no—
157#208—94no—
158C2×C3⋊C8—98no—
159C22×C32—31no—
160C22×C32—31no—
161C22×C32—412no—
162C22×C32—121no—
163C22×C32C2×S411yescharacteristic
164C2×C3⋊D3—62no—
165C2×C3⋊D3—1224no—
166C2×C3⋊D3—122no—
167C32×C4—31no—
168C32×C4—31no—
169C32×C4—412no—
170C32×C4—412no—
171C32×C4C2×S411yescharacteristic
172C32×C4C2×S411yescharacteristic
173C3⋊Dic3—62no—
174C3⋊Dic3—1224no—
175C2×#41—271no—
176C2×#41C3⋊(C3⋊D3)11yescharacteristic, complemented
177#137—271no—
178#137—271no—
179#137—272no—
180AGL(1,ℤ/8)—272no—
181AGL(1,ℤ/8)—271no—
182AGL(1,ℤ/8)—271no—
183#135—272no—
184#135—272no—
185C2×M16—271no—
186C33—41noSylow
187C23×C3—34no—
188C22×D3—184no—
189C3×Q8—34no—
190C3×Q8C22×C3⋊D314yes—
191C3×D4—34no—
192C3×D4—34no—
193C3×D4—64no—
194C2×C3×C4—34no—
195C2×C3×C4—34no—
196C2×C3×C4—34no—
197C2×C3×C4—34no—
198C2×C3×C4—34no—
199C2×C3×C4—49no—
200C2×C3×C4C3⋊S414yes—
201#67—188no—
202#67—188no—
203#67—3618no—
204C2×Dic3—184no—
205D12—98no—
206D12—184no—
207D12—369no—
208C4×D3—188no—
209C4×D3—188no—
210C4×D3—3618no—
211Dic6—98no—
212Dic6—184no—
213Dic6—369no—
214SL(2,𝔽3)C22×C3⋊D319yes—
215C3⋊C8—98no—
216C3⋊C8—98no—
217C2×C32—21no—
218C2×C32—412no—
219C2×C32—61no—
220C2×C32—61no—
221C2×C32—812no—
222C2×C32C22×S411yescharacteristic
223C3⋊D3—122no—
224C3⋊D3—2424no—
225#41—31no—
226#41—31no—
227#41—272no—
228#41—272no—
229#41—541no—
230#41—541no—
231#41C2×C3⋊(C3⋊D3)11yescharacteristic
232#41C2×C3⋊(C3⋊D3)11yescharacteristic
233C2×Q8—271no—
234C2×Q8C2×C3⋊(C3⋊D3)11yescharacteristic
235C2×D4—31no—
236C2×D4—271no—
237C2×D4—541no—
238C22×C4—31no—
239C22×C4—541no—
240Q16—272no—
241Q16—272no—
242SD16—272no—
243SD16—272no—
244SD16—272no—
245SD16—272no—
246D8—272no—
247D8—272no—
248M16—272no—
249M16—271no—
250M16—271no—
251C2×C8—272no—
252C22×C3—34no—
253C22×C3—34no—
254C22×C3—49no—
255C22×C3—124no—
256C22×C3C2×C3⋊S414yes—
257C2×D3—188no—
258C2×D3—3618no—
259C2×D3—368no—
260C3×C4—34no—
261C3×C4—34no—
262C3×C4—49no—
263C3×C4—49no—
264C3×C4C2×C3⋊S414yes—
265C3×C4C2×C3⋊S414yes—
266Dic3—188no—
267Dic3—3618no—
268C32—412no—
269C32#592311yescharacteristic, complemented
270C23—31no—
271C23—541no—
272Q8—31no—
273Q8—272no—
274Q8—541no—
275Q8C22×C3⋊(C3⋊D3)11yescharacteristic
276D4—31no—
277D4—31no—
278D4—61no—
279D4—272no—
280D4—542no—
281D4—542no—
282D4—541no—
283C2×C4—31no—
284C2×C4—31no—
285C2×C4—31no—
286C2×C4—31no—
287C2×C4—31no—
288C2×C4—542no—
289C2×C4—541no—
290C2×C4—542no—
291C2×C4C3⋊(C3⋊S4)11yescharacteristic
292C8—272no—
293C8—272no—
294C2×C3—24no—
295C2×C3—49no—
296C2×C3—64no—
297C2×C3—64no—
298C2×C3—89no—
299C2×C3C22×C3⋊S414yes—
300D3—368no—
301D3—7218no—
302C22—31no—
303C22—31no—
304C22—121no—
305C22—542no—
306C22—1082no—
307C22C2×C3⋊(C3⋊S4)11yescharacteristic
308C4—31no—
309C4—31no—
310C4—542no—
311C4C2×C3⋊(C3⋊S4)11yescharacteristic
312C4C2×C3⋊(C3⋊S4)11yescharacteristic
313C3—49no—
314C3#1059658614yescomplemented
315C2—21no—
316C2—61no—
317C2—61no—
318C2—1082no—
319C2C22×C3⋊(C3⋊S4)11yescharacteristic
subgroupG/H
centerC4C2×C3⋊(C3⋊S4)
commutator, cosocle, hypocenterC32×SL(2,𝔽3)C23
FrattiniC2C22×C3⋊(C3⋊S4)
FittingC2×C32×#41D3
socleC2×C32C22×S4
hypercenterC2×C4C3⋊(C3⋊S4)
LCSG ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C32×#41⋊C3C2semidirect
2. #11724471C2semidirect
3. #11724470C2semidirect
4. #11724469C2semidirect
5. #11724468C2semidirect
10. C32×#41⋊C3C22semidirect
11. C2×C32×SL(2,𝔽3)C22semidirect
12. C3⋊(C3⋊GL(2,𝔽3))C22semidirect
13. C3⋊(C3⋊#226)C22semidirect
14. C2×C32×#41D3semidirect
16. C2×C3×#41⋊C3D3semidirect
55. C3×#41⋊C3C2×D3semidirect
56. C2×C3×SL(2,𝔽3)C2×D3semidirect
76. C2×C3×#41C3⋊D3semidirect
78. C2×#41⋊C3C3⋊D3semidirect
143. #41⋊C3C2×C3⋊D3semidirect
144. C2×SL(2,𝔽3)C2×C3⋊D3semidirect
176. C2×#41C3⋊(C3⋊D3)semidirect
269. C32#5923semidirect
314. C3#10596586semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#41⋊C32
#117244712
#117244702
#117244692
#117244682
#105965863
#105962933
#890764