Finite Group Databasebeta

← #10589318⋊C3 #13135256

Subgroups — #10589318⋊C3 #13135256

Order 1728.

356 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 70 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
1#64133⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×#95C211yesmaximal, characteristic
3#63657⋊C3C211yesmaximal, characteristic
4#10589318C311yesmaximal, characteristic, complemented
5C3×#4452—31nomaximal
6C22×C4×H(3,𝔽3)—21no—
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C2×#634⋊C3C411yescharacteristic, complemented
9#3757⋊C3—21no—
10C4⋊((C3⋊Dic3)⋊C3)C411yescharacteristic, complemented
11#3750⋊C3—21no—
12C2×(C3⋊(C3⋊C8))⋊C3—21no—
13#64133C2×C311yescharacteristic
14C3×#924—31no—
15C32×#95—21no—
16C32×#95D311yescharacteristic
17C32×#95C2×C311yescharacteristic
18#63657C2×C311yescharacteristic
19C3×#830—31no—
20C23×H(3,𝔽3)D411yescharacteristic
21C2×C4×H(3,𝔽3)—22no—
22C2×C4×H(3,𝔽3)—41no—
23C2×C4×H(3,𝔽3)—41no—
24C2×C4×H(3,𝔽3)—41no—
25C2×C4×H(3,𝔽3)C2×C411yescharacteristic
26C2×C4×H(3,𝔽3)D411yescharacteristic
27C2×(C3⋊Dic3)⋊C3—21no—
28C2×(C3⋊Dic3)⋊C3—21no—
29#634⋊C3—41no—
30(C3⋊(C3⋊C8))⋊C3—41no—
31C3×#328—91no—
32#4452—31no—
33#4452—91no—
34C22×C32×C4—22no—
35C22×C32×C4—21no—
36C22×C32×C4—21no—
37C22×C32×C4—21no—
38C22×C32×C4C2×D311yescharacteristic
39C22×C32×C4C22×C311yescharacteristic
40C2×#634C3×C411yescharacteristic, complemented
41C2×C3×Dic6—31no—
42#3757—21no—
43C4⋊(C3⋊Dic3)C3×C411yescharacteristic, complemented
44#3750—21no—
45C2×C3⋊(C3⋊C8)—21no—
46C3×#217—61no—
47C3×C4⋊Dic3—31no—
48C3×Dic3⋊C4—61no—
49C2×C3×C3⋊C8—61no—
50C22×H(3,𝔽3)—21no—
51C22×H(3,𝔽3)—21no—
52C22×H(3,𝔽3)C22⋊C411yescharacteristic
53C22×H(3,𝔽3)SD1611yescharacteristic
54C22×H(3,𝔽3)Q1611yescharacteristic
55C4×H(3,𝔽3)—21no—
56C4×H(3,𝔽3)—21no—
57C4×H(3,𝔽3)—42no—
58(C3⋊Dic3)⋊C3—41no—
59(C3⋊Dic3)⋊C3—41no—
60C3×C22⋊Q8—91no—
61#924—31no—
62#924—91no—
63C3×C22⋊C8—91no—
64C3×#95—31no—
65C3×#95—31no—
66C3×#95—61no—
67C3×#95C3×D311yescharacteristic
68#830—31no—
69#830—91no—
70C23×C32—21no—
71C23×C32#6711yescharacteristic
72C23×C32C3×D411yescharacteristic
73C2×C32×C4—24no—
74C2×C32×C4—22no—
75C2×C32×C4—22no—
76C2×C32×C4—21no—
77C2×C32×C4—21no—
78C2×C32×C4—41no—
79C2×C32×C4—41no—
80C2×C32×C4—42no—
81C2×C32×C4—42no—
82C2×C32×C4—41no—
83C2×C32×C4—41no—
84C2×C32×C4—42no—
85C2×C32×C4—41no—
86C2×C32×C4—41no—
87C2×C32×C4C4×D311yescharacteristic
88C2×C32×C4D1211yescharacteristic
89C2×C32×C4C2×C3×C411yescharacteristic
90C2×C32×C4C3×D411yescharacteristic
91C2×C3⋊Dic3—21no—
92C2×C3⋊Dic3—21no—
93#634—41no—
94C2×C3×Dic3—61no—
95C2×C3×Dic3—61no—
96C3×Dic6—121no—
97C3⋊(C3⋊C8)—41no—
98C3×C3⋊C8—121no—
99#328—91noSylow
100C2×H(3,𝔽3)—21no—
101C2×H(3,𝔽3)—21no—
102C2×H(3,𝔽3)#9911yescharacteristic
103C2×H(3,𝔽3)#10311yescharacteristic
104C2×H(3,𝔽3)#10411yescharacteristic
105C2×C3×Q8—91no—
106C22×C3×C4—21no—
107C22×C3×C4—31no—
108C22×C3×C4—31no—
109C22×C3×C4—61no—
110C22×C3×C4—61no—
111C22×C3×C4—62no—
112C22×C3×C4—61no—
113C22×C3×C4C2×C3×D311yescharacteristic
114C2×Dic6—31no—
115C2×Dic6—91no—
116C2×C3×C8—181no—
117C3×C4⋊C4—91no—
118C3×C4⋊C4—181no—
119C3×C22⋊C4—181no—
120#217—61no—
121#217—181no—
122C4⋊Dic3—31no—
123C4⋊Dic3—91no—
124Dic3⋊C4—61no—
125Dic3⋊C4—181no—
126C2×C3⋊C8—61no—
127C2×C3⋊C8—181no—
128C22×C32—22no—
129C22×C32—22no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32—21no—
134C22×C32—21no—
135C22×C32—21no—
136C22×C32—21no—
137C22×C32(C2×D3)⋊C411yescharacteristic
138C22×C32#21411yescharacteristic
139C22×C32#21611yescharacteristic
140C22×C32C3×C22⋊C411yescharacteristic
141C22×C32C3×SD1611yescharacteristic
142C22×C32C3×Q1611yescharacteristic
143C32×C4—21no—
144C32×C4—21no—
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—41no—
148C32×C4—41no—
149C32×C4—44no—
150C32×C4—42no—
151C32×C4—42no—
152C3⋊Dic3—41no—
153C3⋊Dic3—41no—
154C3×Dic3—121no—
155C3×Dic3—121no—
156C22⋊Q8—91no—
157C22⋊C8—91no—
158#95(C3⋊D3)⋊C311yescharacteristic
159H(3,𝔽3)#32811yescharacteristic, complemented, Sylow
160C23×C3—31no—
161C23×C3—31no—
162C23×C3—61no—
163C23×C3C3×#6711yescharacteristic
164C3×Q8—361no—
165C2×C3×C4—22no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—62no—
174C2×C3×C4—62no—
175C2×C3×C4—64no—
176C2×C3×C4—61no—
177C2×C3×C4—61no—
178C2×C3×C4—121no—
179C2×C3×C4—121no—
180C2×C3×C4—121no—
181C2×C3×C4—121no—
182C2×C3×C4—122no—
183C2×C3×C4—121no—
184C2×C3×C4—121no—
185C2×C3×C4—122no—
186C2×C3×C4—122no—
187C2×C3×C4—181no—
188C2×C3×C4—181no—
189C2×C3×C4C3×C4×D311yescharacteristic
190C2×C3×C4C3×D1211yescharacteristic
191C2×Dic3—61no—
192C2×Dic3—61no—
193C2×Dic3—181no—
194C2×Dic3—181no—
195Dic6—121no—
196Dic6—361no—
197C3×C8—361no—
198C3⋊C8—121no—
199C3⋊C8—361no—
200C2×C32—22no—
201C2×C32—22no—
202C2×C32—21no—
203C2×C32—21no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32#80511yescharacteristic
210C2×C32#80611yescharacteristic
211C2×C32#81011yescharacteristic
212C2×C32C3×#9911yescharacteristic
213C2×C32C3×#10311yescharacteristic
214C2×C32C3×#10411yescharacteristic
215C2×Q8—91no—
216C22×C4—21no—
217C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
218C2×C8—181no—
219C4⋊C4—91no—
220C4⋊C4—181no—
221C22⋊C4—181no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—31no—
225C22×C3—31no—
226C22×C3—31no—
227C22×C3—31no—
228C22×C3—31no—
229C22×C3—31no—
230C22×C3—61no—
231C22×C3—61no—
232C22×C3—61no—
233C22×C3—61no—
234C22×C3—62no—
235C22×C3—62no—
236C22×C3—61no—
237C22×C3—61no—
238C22×C3—61no—
239C22×C3C3×(C2×D3)⋊C411yescharacteristic
240C22×C3C3×#21411yescharacteristic
241C22×C3C3×#21611yescharacteristic
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—42no—
245C3×C4—61no—
246C3×C4—61no—
247C3×C4—61no—
248C3×C4—61no—
249C3×C4—121no—
250C3×C4—121no—
251C3×C4—122no—
252C3×C4—122no—
253C3×C4—124no—
254C3×C4—361no—
255C3×C4—361no—
256Dic3—121no—
257Dic3—121no—
258Dic3—361no—
259Dic3—361no—
260C32—21no—
261C32#445211yescharacteristic, complemented
262C32C3×#32811yescharacteristic, complemented
263C23#638⋊C311yescharacteristic
264Q8—361no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—181no—
270C2×C4—181no—
271C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
272C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
273C8—361no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3—31no—
277C2×C3—31no—
278C2×C3—31no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—31no—
282C2×C3—61no—
283C2×C3—61no—
284C2×C3—61no—
285C2×C3—61no—
286C2×C3—62no—
287C2×C3—62no—
288C2×C3—61no—
289C2×C3—61no—
290C2×C3—61no—
291C2×C3C3×#80511yescharacteristic
292C2×C3C3×#80611yescharacteristic
293C2×C3C3×#81011yescharacteristic
294C22—21no—
295C22—21no—
296C22#3752⋊C311yescharacteristic
297C22#3754⋊C311yescharacteristic
298C22#3756⋊C311yescharacteristic
299C4—21no—
300C4—21no—
301C4—42no—
302C4—361no—
303C4—361no—
304C3—31no—
305C3—31no—
306C3—61no—
307C3C3×#445211yescharacteristic
308C2—21no—
309C2—21no—
310C2#63632⋊C311yescharacteristic
311C2#63633⋊C311yescharacteristic
312C2#63637⋊C311yescharacteristic
subgroupG/H
centerC22#3752⋊C3
commutatorC2×C32×C4C2×C3×C4
FrattiniC22×C3×C4C2×C3×D3
FittingH(3,𝔽3)×#95C2
socleC22×C3C3×(C2×D3)⋊C4
prime radicalC23×H(3,𝔽3)D4
cosocleC22×C32×C4C22×C3
hypercenter#95(C3⋊D3)⋊C3
hypocenterC32C3×#328
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #95

Complements

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

NH ≅ G/Ntype
4. #10589318C3semidirect
8. C2×#634⋊C3C4semidirect
10. C4⋊((C3⋊Dic3)⋊C3)C4semidirect
40. C2×#634C3×C4semidirect
43. C4⋊(C3⋊Dic3)C3×C4semidirect
159. H(3,𝔽3)#328semidirect
261. C32#4452semidirect
262. C32C3×#328semidirect

Maximal subgroups

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

subgroupindex
#64133⋊C32
H(3,𝔽3)×#952
#63657⋊C32
#105893183
C3×#44523