Finite Group Databasebeta

← #10592836⋊C3 #13143209

Subgroups — #10592836⋊C3 #13143209

Order 1728.

453 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 453 conjugacy classes of subgroups — too many to draw. Only its 102 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×M16×H(3,𝔽3)C211yesmaximal, characteristic
2#64133⋊C3C212yesmaximal
3C2×#3746⋊C3C211yesmaximal, characteristic, complemented
4#63644⋊C3C211yesmaximal, characteristic, complemented
5#63643⋊C3C212yesmaximal, complemented
6#10592836C311yesmaximal, characteristic, complemented
7C3×#5137—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10M16×H(3,𝔽3)—22no—
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13#3757⋊C3—22no—
14C4⋊((C3⋊Dic3)⋊C3)C2212yes—
15#3750⋊C3—22no—
16#3746⋊C3—22no—
17C2×C32×M16—21no—
18C2×C32×M16D311yescharacteristic
19C2×C32×M16C2×C311yescharacteristic
20#64133C2×C312yes—
21C2×#3746C2×C311yescharacteristic, complemented
22C3×#924—32no—
23C2×C3×Dic12—31no—
24#63644C2×C311yescharacteristic, complemented
25#63643C2×C312yescomplemented
26C3×#817—31no—
27C3×#816—32no—
28C23×H(3,𝔽3)D411yescharacteristic
29C2×C4×H(3,𝔽3)—21no—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33C2×(C3⋊Dic3)⋊C3—22no—
34C2×(C3⋊Dic3)⋊C3—22no—
35#634⋊C3—22no—
36#634⋊C3—22no—
37C8×H(3,𝔽3)—21no—
38C8×H(3,𝔽3)D412yes—
39C3×#470—91no—
40#5137—31no—
41#5137—91no—
42C22×C32×C4—21no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C32×C4C22×C311yescharacteristic
45C2×#634C22×C312yes—
46C2×C3×Dic6—32no—
47C32×M16—24no—
48C32×M16—22no—
49C32×M16—22no—
50C2×C32×C8—21no—
51C2×C32×C8—21no—
52C2×C32×C8C2×D311yescharacteristic
53C2×C32×C8C2×D311yescharacteristic
54C2×C32×C8C22×C311yescharacteristic
55C2×C32×C8C22×C311yescharacteristic
56#3757—22no—
57C4⋊(C3⋊Dic3)C22×C312yes—
58#3750—22no—
59#3746—22no—
60C3×#217—62no—
61C3×C4⋊Dic3—32no—
62C3×Dic3⋊C4—62no—
63C3×Dic12—62no—
64C22×H(3,𝔽3)—21no—
65C22×H(3,𝔽3)—22no—
66C22×H(3,𝔽3)C2×D411yescharacteristic
67C4×H(3,𝔽3)—21no—
68C4×H(3,𝔽3)C2×D411yescharacteristic
69C4×H(3,𝔽3)#4111yescharacteristic
70(C3⋊Dic3)⋊C3—42no—
71(C3⋊Dic3)⋊C3—42no—
72C2×C3×Q16—91no—
73C2×C3×M16—31no—
74C2×C3×M16—31no—
75C2×C3×M16—61no—
76C2×C3×M16C3×D311yescharacteristic
77C3×C22⋊Q8—92no—
78#924—32no—
79#924—92no—
80C2×Dic12—31no—
81C2×Dic12—91no—
82C3×#106—91no—
83C3×#103—92no—
84#817—31no—
85#817—91no—
86#816—32no—
87#816—92no—
88C23×C32—21no—
89C23×C32D1211yescharacteristic
90C23×C32C3×D411yescharacteristic
91C2×C32×C4—22no—
92C2×C32×C4—22no—
93C2×C32×C4—21no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4—21no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4D1211yescharacteristic
100C2×C32×C4C3×D411yescharacteristic
101C2×C32×C4C22×D311yescharacteristic
102C2×C32×C4C23×C311yescharacteristic
103C2×C3⋊Dic3—22no—
104C2×C3⋊Dic3—22no—
105#634—22no—
106#634—22no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C3×Dic6—62no—
110C3×Dic6—62no—
111C32×C8—22no—
112C32×C8—21no—
113C32×C8—21no—
114C32×C8—22no—
115C32×C8#6712yes—
116C32×C8C3×D412yes—
117#470—91noSylow
118C2×H(3,𝔽3)—41no—
119C2×H(3,𝔽3)#12111yescharacteristic
120C2×H(3,𝔽3)#13712yes—
121C2×C3×Q8—92no—
122C22×C3×C4—31no—
123C22×C3×C4—31no—
124C22×C3×C4—61no—
125C22×C3×C4C2×C3×D311yescharacteristic
126C2×Dic6—32no—
127C2×Dic6—92no—
128C3×Q16—182no—
129C3×M16—22no—
130C3×M16—62no—
131C3×M16—62no—
132C3×M16—64no—
133C2×C3×C8—31no—
134C2×C3×C8—31no—
135C2×C3×C8—31no—
136C2×C3×C8—31no—
137C2×C3×C8—61no—
138C2×C3×C8—61no—
139C2×C3×C8C2×C3×D311yescharacteristic
140C2×C3×C8C2×C3×D311yescharacteristic
141C3×C4⋊C4—92no—
142C3×C4⋊C4—182no—
143C3×C22⋊C4—182no—
144#217—62no—
145#217—182no—
146C4⋊Dic3—32no—
147C4⋊Dic3—92no—
148Dic3⋊C4—62no—
149Dic3⋊C4—182no—
150Dic12—62no—
151Dic12—182no—
152C22×C32—22no—
153C22×C32—22no—
154C22×C32—21no—
155C22×C32—21no—
156C22×C32—22no—
157C22×C32—21no—
158C22×C32—42no—
159C22×C32C2×D1211yescharacteristic
160C22×C32C2×C3×D411yescharacteristic
161C32×C4—22no—
162C32×C4—21no—
163C32×C4—21no—
164C32×C4—21no—
165C32×C4—21no—
166C32×C4#23511yescharacteristic
167C32×C4C2×#6711yescharacteristic
168C32×C4C2×C3×D411yescharacteristic
169C32×C4C3×#4111yescharacteristic
170C3⋊Dic3—42no—
171C3⋊Dic3—42no—
172C3×Dic3—122no—
173C3×Dic3—122no—
174C2×Q16—91no—
175C2×M16(C3⋊D3)⋊C311yescharacteristic
176C22⋊Q8—92no—
177#106—91no—
178#103—92no—
179H(3,𝔽3)#47011yescharacteristic, complemented, Sylow
180C23×C3—31no—
181C23×C3—31no—
182C23×C3—61no—
183C23×C3C3×D1211yescharacteristic
184C3×Q8—182no—
185C3×Q8—182no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—61no—
193C2×C3×C4—61no—
194C2×C3×C4—61no—
195C2×C3×C4—61no—
196C2×C3×C4—62no—
197C2×C3×C4—62no—
198C2×C3×C4—61no—
199C2×C3×C4—61no—
200C2×C3×C4—182no—
201C2×C3×C4—182no—
202C2×C3×C4C3×D1211yescharacteristic
203C2×C3×C4C22×C3×D311yescharacteristic
204C2×Dic3—62no—
205C2×Dic3—62no—
206C2×Dic3—182no—
207C2×Dic3—182no—
208Dic6—62no—
209Dic6—62no—
210Dic6—182no—
211Dic6—182no—
212C3×C8—21no—
213C3×C8—32no—
214C3×C8—32no—
215C3×C8—61no—
216C3×C8—61no—
217C3×C8—62no—
218C3×C8—62no—
219C3×C8C3×#6712yes—
220C2×C32—22no—
221C2×C32—21no—
222C2×C32—42no—
223C2×C32—41no—
224C2×C32—41no—
225C2×C32#90912yes—
226C2×C32#93011yescharacteristic
227C2×C32C3×#12111yescharacteristic
228C2×C32C3×#13712yes—
229C2×Q8—92no—
230C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
231Q16—182no—
232M16—22no—
233C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
234C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
235C4⋊C4—92no—
236C4⋊C4—182no—
237C22⋊C4—182no—
238C22×C3—22no—
239C22×C3—21no—
240C22×C3—31no—
241C22×C3—31no—
242C22×C3—61no—
243C22×C3—61no—
244C22×C3—62no—
245C22×C3—62no—
246C22×C3—62no—
247C22×C3—61no—
248C22×C3—122no—
249C22×C3C2×C3×D1211yescharacteristic
250C3×C4—21no—
251C3×C4—31no—
252C3×C4—31no—
253C3×C4—31no—
254C3×C4—31no—
255C3×C4—61no—
256C3×C4—61no—
257C3×C4—62no—
258C3×C4—61no—
259C3×C4—61no—
260C3×C4—362no—
261C3×C4—362no—
262C3×C4C3×#23511yescharacteristic
263C3×C4C2×C3×#6711yescharacteristic
264Dic3—122no—
265Dic3—122no—
266Dic3—362no—
267Dic3—362no—
268C32—21no—
269C32#513711yescharacteristic, complemented
270C32C3×#47011yescharacteristic, complemented
271C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
272Q8—182no—
273Q8—182no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—182no—
277C2×C4—182no—
278C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
279C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
280C8—21no—
281C8#638⋊C312yes—
282C2×C3—32no—
283C2×C3—32no—
284C2×C3—31no—
285C2×C3—31no—
286C2×C3—41no—
287C2×C3—62no—
288C2×C3—61no—
289C2×C3—121no—
290C2×C3—121no—
291C2×C3—122no—
292C2×C3C3×#90912yes—
293C2×C3C3×#93011yescharacteristic
294C22—22no—
295C22—21no—
296C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
297C4—21no—
298C4—362no—
299C4—362no—
300C4#3828⋊C311yescharacteristic
301C4C2×#638⋊C311yescharacteristic
302C3—31no—
303C3—31no—
304C3—61no—
305C3C3×#513711yescharacteristic
306C2—41no—
307C2#64118⋊C312yes—
308C2#64139⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×M16×H(3,𝔽3)C2
socleC22×C3C2×C3×D12
prime radicalC23×H(3,𝔽3)D4
hypercenterC2×M16(C3⋊D3)⋊C3
hypocenterC32C3×#470
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
3. C2×#3746⋊C3C2semidirect
4. #63644⋊C3C2semidirect
5. #63643⋊C3C2semidirect
6. #10592836C3semidirect
21. C2×#3746C2×C3semidirect
24. #63644C2×C3semidirect
25. #63643C2×C3semidirect
179. H(3,𝔽3)#470semidirect
269. C32#5137semidirect
270. C32C3×#470semidirect

Maximal subgroups

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

subgroupindex
C2×M16×H(3,𝔽3)2
#64133⋊C32
C2×#3746⋊C32
#63644⋊C32
#63643⋊C32
#105928363
C3×#51373