Finite Group Databasebeta

← #10592540⋊C3 #13142062

Subgroups — #10592540⋊C3 #13142062

Order 1728.

458 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3744⋊C3C212yesmaximal
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
3#64078⋊C3C211yesmaximal, characteristic, complemented
4H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic, complemented
5#63643⋊C3C212yesmaximal, complemented
6#10592540C311yesmaximal, characteristic, complemented
7C3×#4841—31nomaximal
8C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
9C2×C4×(C3⋊D3)⋊C3—21no—
10C2×#634⋊C3C2212yes—
11C2×C8×H(3,𝔽3)C2212yes—
12C42×H(3,𝔽3)C2211yescharacteristic
13#3752⋊C3—21no—
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)—21no—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3744⋊C3—24no—
18C2×#3744C2×C312yes—
19C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
20#64078C2×C311yescharacteristic, complemented
21C2×C3×#204—32no—
22C3×C4×D12—31no—
23C3×#869—31no—
24C32×C4⋊C8—21no—
25C32×C4⋊C8D311yescharacteristic, complemented
26C32×C4⋊C8C2×C311yescharacteristic, complemented
27#63643C2×C312yescomplemented
28C3×#816—32no—
29C22×(C3⋊D3)⋊C3—21no—
30C2×C4×H(3,𝔽3)D411yescharacteristic
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—21no—
35C4⋊((C3⋊D3)⋊C3)—21no—
36C4⋊((C3⋊D3)⋊C3)D412yes—
37C4×(C3⋊D3)⋊C3—42no—
38#634⋊C3—22no—
39#634⋊C3—22no—
40C8×H(3,𝔽3)—22no—
41C3×#461—91no—
42#4841—31no—
43#4841—91no—
44C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
45C2×C4×C3⋊D3—21no—
46C2×#634C22×C312yes—
47C2×C3×D12—31no—
48C2×C3×C4×D3—61no—
49C2×C3×Dic6—32no—
50C2×C32×C8—22no—
51C2×C32×C8C2×D312yes—
52C2×C32×C8C22×C312yes—
53C32×C42—21no—
54C32×C42C2×D311yescharacteristic
55C32×C42C22×C311yescharacteristic
56#3752—21no—
57C4⋊(C3⋊Dic3)—21no—
58C4⋊(C3⋊Dic3)—21no—
59C4⋊(C3⋊Dic3)C22×C311yescharacteristic
60#3744—24no—
61C3×(C2×D3)⋊C4—61no—
62C3×C4⋊Dic3—31no—
63C3×C4⋊Dic3—61no—
64C3×C4⋊Dic3—61no—
65C3×#204—64no—
66C22×H(3,𝔽3)C2×D411yescharacteristic
67C2×(C3⋊D3)⋊C3—22no—
68C2×(C3⋊D3)⋊C3—41no—
69C2×(C3⋊D3)⋊C3—41no—
70C4×H(3,𝔽3)—21no—
71C4×H(3,𝔽3)SD1612yescomplemented
72C4×H(3,𝔽3)C2×D411yescharacteristic
73C4×H(3,𝔽3)#4111yescharacteristic
74(C3⋊Dic3)⋊C3—42no—
75(C3⋊Dic3)⋊C3—41no—
76C2×C3×SD16—92no—
77C3×C4⋊Q8—91no—
78C3×C4×D4—91no—
79C2×#204—32no—
80C2×#204—92no—
81C4×D12—31no—
82C4×D12—91no—
83#869—31no—
84#869—91no—
85C3×C4⋊C8—31no—
86C3×C4⋊C8—31no—
87C3×C4⋊C8—61no—
88C3×C4⋊C8C3×D311yescharacteristic
89C3×#103—92no—
90#816—32no—
91#816—92no—
92C22×C3⋊D3—21no—
93C22×C3×D3—61no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—21no—
97C2×C32×C4D1211yescharacteristic
98C2×C32×C4D1211yescharacteristic
99C2×C32×C4C3×D411yescharacteristic
100C2×C32×C4C3×D411yescharacteristic
101C2×C32×C4C22×D311yescharacteristic
102C2×C32×C4C23×C311yescharacteristic
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3—21no—
105C4⋊(C3⋊D3)—21no—
106C4⋊(C3⋊D3)C3×D412yes—
107C4×C3⋊D3—42no—
108#634—22no—
109#634—22no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—62no—
112C3×D12—32no—
113C3×D12—61no—
114C3×C4×D3—122no—
115C3×Dic6—62no—
116C3×Dic6—62no—
117C32×C8—22no—
118C32×C8—22no—
119C32×C8—42no—
120#461—91noSylow
121C2×H(3,𝔽3)#12111yescharacteristic
122C2×H(3,𝔽3)C2×SD1611yescharacteristic
123C2×H(3,𝔽3)#13711yescharacteristic
124(C3⋊D3)⋊C3—42no—
125C2×C3×Q8—92no—
126C2×C3×D4—91no—
127C22×C3×C4—181no—
128C2×D12—31no—
129C2×D12—91no—
130C2×C4×D3—61no—
131C2×C4×D3—181no—
132C2×Dic6—32no—
133C2×Dic6—92no—
134C3×SD16—184no—
135C2×C3×C8—32no—
136C2×C3×C8—32no—
137C2×C3×C8—62no—
138C2×C3×C8C2×C3×D312yes—
139C3×C4⋊C4—91no—
140C3×C4⋊C4—181no—
141C3×C4⋊C4—181no—
142C3×C22⋊C4—181no—
143C3×C42—31no—
144C3×C42—31no—
145C3×C42—61no—
146C3×C42C2×C3×D311yescharacteristic
147(C2×D3)⋊C4—61no—
148(C2×D3)⋊C4—181no—
149C4⋊Dic3—31no—
150C4⋊Dic3—61no—
151C4⋊Dic3—61no—
152C4⋊Dic3—91no—
153C4⋊Dic3—181no—
154C4⋊Dic3—181no—
155#204—64no—
156#204—184no—
157C22×C32—21no—
158C22×C32C2×D1211yescharacteristic
159C22×C32C2×C3×D411yescharacteristic
160C2×C3⋊D3—22no—
161C2×C3⋊D3—41no—
162C2×C3⋊D3—41no—
163C2×C3×D3—62no—
164C2×C3×D3—121no—
165C2×C3×D3—121no—
166C32×C4—21no—
167C32×C4—21no—
168C32×C4—22no—
169C32×C4—21no—
170C32×C4—21no—
171C32×C4—41no—
172C32×C4#20412yescomplemented
173C32×C4C3×SD1612yescomplemented
174C32×C4D3×D411yescharacteristic
175C32×C4#23911yescharacteristic
176C32×C4C2×C3×D411yescharacteristic
177C32×C4C3×#4111yescharacteristic
178C3⋊Dic3—42no—
179C3⋊Dic3—41no—
180C3×Dic3—122no—
181C3×Dic3—121no—
182C2×SD16—92no—
183C4⋊Q8—91no—
184C4×D4—91no—
185C4⋊C8(C3⋊D3)⋊C311yescharacteristic, complemented
186#103—92no—
187H(3,𝔽3)#46111yescharacteristic, complemented, Sylow
188C23×C3—181no—
189C22×D3—61no—
190C22×D3—181no—
191C3×Q8—182no—
192C3×Q8—182no—
193C3×D4—92no—
194C3×D4—181no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—61no—
202C2×C3×C4—61no—
203C2×C3×C4—61no—
204C2×C3×C4—181no—
205C2×C3×C4—182no—
206C2×C3×C4—362no—
207C2×C3×C4C3×D1211yescharacteristic
208C2×C3×C4C3×D1211yescharacteristic
209C2×C3×C4C22×C3×D311yescharacteristic
210C2×Dic3—61no—
211C2×Dic3—62no—
212C2×Dic3—181no—
213C2×Dic3—182no—
214D12—32no—
215D12—61no—
216D12—92no—
217D12—181no—
218C4×D3—122no—
219C4×D3—362no—
220Dic6—62no—
221Dic6—62no—
222Dic6—182no—
223Dic6—182no—
224C3×C8—22no—
225C3×C8—62no—
226C3×C8—62no—
227C3×C8—122no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32#89511yescharacteristic
232C2×C32C2×#20411yescharacteristic
233C2×C32#90911yescharacteristic
234C2×C32C3×#12111yescharacteristic
235C2×C32C2×C3×SD1611yescharacteristic
236C2×C32C3×#13711yescharacteristic
237C3⋊D3—42no—
238C3×D3—122no—
239C2×Q8—92no—
240C2×D4—91no—
241C22×C4—181no—
242SD16—184no—
243C2×C8C2×(C3⋊D3)⋊C312yes—
244C4⋊C4—91no—
245C4⋊C4—181no—
246C4⋊C4—181no—
247C22⋊C4—181no—
248C42C2×(C3⋊D3)⋊C311yescharacteristic
249C22×C3—31no—
250C22×C3—31no—
251C22×C3—61no—
252C22×C3—182no—
253C22×C3—361no—
254C22×C3—361no—
255C22×C3C2×C3×D1211yescharacteristic
256C2×D3—62no—
257C2×D3—121no—
258C2×D3—121no—
259C2×D3—182no—
260C2×D3—361no—
261C2×D3—361no—
262C3×C4—21no—
263C3×C4—31no—
264C3×C4—31no—
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—32no—
268C3×C4—32no—
269C3×C4—61no—
270C3×C4—61no—
271C3×C4—61no—
272C3×C4—61no—
273C3×C4—62no—
274C3×C4—121no—
275C3×C4—361no—
276C3×C4—362no—
277C3×C4C3×#20412yes—
278C3×C4C3×D3×D411yescharacteristic
279C3×C4C3×#23911yescharacteristic
280Dic3—122no—
281Dic3—121no—
282Dic3—362no—
283Dic3—361no—
284C32—21no—
285C32#484111yescharacteristic, complemented
286C32C3×#46111yescharacteristic, complemented
287C23—181no—
288Q8—182no—
289Q8—182no—
290D4—92no—
291D4—181no—
292C2×C4—181no—
293C2×C4—182no—
294C2×C4—362no—
295C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
296C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
297C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
298C8—22no—
299C2×C3—31no—
300C2×C3—31no—
301C2×C3—31no—
302C2×C3—31no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—61no—
306C2×C3—61no—
307C2×C3—61no—
308C2×C3—362no—
309C2×C3C3×#89511yescharacteristic
310C2×C3C2×C3×#20411yescharacteristic
311C2×C3C3×#90911yescharacteristic
312D3—122no—
313D3—362no—
314C22—182no—
315C22—361no—
316C22—361no—
317C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
318C4—21no—
319C4—361no—
320C4—362no—
321C4#3744⋊C312yescomplemented
322C4D4×(C3⋊D3)⋊C311yescharacteristic
323C4#8872211yescharacteristic
324C3—31no—
325C3—31no—
326C3—61no—
327C3C3×#484111yescharacteristic
328C2—362no—
329C2#64104⋊C311yescharacteristic
330C2C2×#3744⋊C311yescharacteristic
331C2#64118⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C4⋊C8C2
socleC22×C3C2×C3×D12
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenterC4⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#461
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
3. #64078⋊C3C2semidirect
4. H(3,𝔽3)×C4⋊C8C2semidirect
5. #63643⋊C3C2semidirect
6. #10592540C3semidirect
20. #64078C2×C3semidirect
25. C32×C4⋊C8D3semidirect
26. C32×C4⋊C8C2×C3semidirect
27. #63643C2×C3semidirect
71. C4×H(3,𝔽3)SD16semidirect
172. C32×C4#204semidirect
173. C32×C4C3×SD16semidirect
185. C4⋊C8(C3⋊D3)⋊C3semidirect
187. H(3,𝔽3)#461semidirect
285. C32#4841semidirect
286. C32C3×#461semidirect
321. C4#3744⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#3744⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
#64078⋊C32
H(3,𝔽3)×C4⋊C82
#63643⋊C32
#105925403
C3×#48413