Finite Group Databasebeta

← #10589341⋊C3 #13135353

Subgroups — #10589341⋊C3 #13135353

Order 1728.

375 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 50 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#64146⋊C3C211yesmaximal, characteristic
2H(3,𝔽3)×#100C211yesmaximal, characteristic, complemented
3#63661⋊C3C211yesmaximal, characteristic, complemented
4#10589341C311yesmaximal, characteristic, complemented
5C3×#4475—31nomaximal
6C23×(C3⋊D3)⋊C3C411yescharacteristic, complemented
7C2×D4×H(3,𝔽3)C2211yescharacteristic
8C2×#638⋊C3—21no—
9M16×H(3,𝔽3)—21no—
10#3757⋊C3—21no—
11#3757⋊C3C411yescharacteristic, complemented
12#3752⋊C3—21no—
13#64146C2×C311yescharacteristic
14C3×#937—31no—
15C32×#100—21no—
16C32×#100D311yescharacteristic, complemented
17C32×#100C2×C311yescharacteristic, complemented
18#63661C2×C311yescharacteristic, complemented
19C3×#834—31no—
20C23×H(3,𝔽3)C2×C411yescharacteristic
21C23×H(3,𝔽3)D411yescharacteristic
22C22×(C3⋊D3)⋊C3—21no—
23C22×(C3⋊D3)⋊C3—42no—
24C22×(C3⋊D3)⋊C3—41no—
25D4×H(3,𝔽3)—41no—
26C2×C4×H(3,𝔽3)D411yescharacteristic
27#638⋊C3—42no—
28C2×(C3⋊Dic3)⋊C3—21no—
29C2×(C3⋊Dic3)⋊C3—41no—
30C8×H(3,𝔽3)—41no—
31C3×AΣL(1,𝔽16)—91no—
32#4475—31no—
33#4475—91no—
34C23×C3⋊D3C3×C411yescharacteristic, complemented
35C23×C3×D3—31no—
36C2×C32×D4—21no—
37C2×C32×D4C2×D311yescharacteristic
38C2×C32×D4C22×C311yescharacteristic
39C2×#638—21no—
40C2×C3×#67—61no—
41C32×M16—22no—
42C32×M16—21no—
43C32×M16—21no—
44#3757—21no—
45#3757C3×C411yescharacteristic, complemented
46#3752—21no—
47C3×#217—31no—
48C3×#217—61no—
49C3×(C2×D3)⋊C4—61no—
50C22×H(3,𝔽3)—21no—
51C22×H(3,𝔽3)—21no—
52C22×H(3,𝔽3)—41no—
53C22×H(3,𝔽3)—41no—
54C22×H(3,𝔽3)C22⋊C411yescharacteristic
55C2×(C3⋊D3)⋊C3—22no—
56C2×(C3⋊D3)⋊C3—41no—
57C2×(C3⋊D3)⋊C3—42no—
58C2×(C3⋊D3)⋊C3—41no—
59C2×(C3⋊D3)⋊C3—42no—
60C4×H(3,𝔽3)—21no—
61(C3⋊Dic3)⋊C3—41no—
62(C3⋊Dic3)⋊C3—81no—
63C3×#120—91no—
64#937—31no—
65#937—91no—
66C3×#100—31no—
67C3×#100—31no—
68C3×#100—61no—
69C3×#100C3×D311yescharacteristic
70C3×#99—91no—
71#834—31no—
72#834—91no—
73C23×C32—21no—
74C23×C32—21no—
75C23×C32C4×D311yescharacteristic
76C23×C32D1211yescharacteristic
77C23×C32C2×C3×C411yescharacteristic
78C23×C32C3×D411yescharacteristic
79C22×C3⋊D3—21no—
80C22×C3⋊D3—42no—
81C22×C3⋊D3—41no—
82C22×C3×D3—61no—
83C22×C3×D3—122no—
84C22×C3×D3—121no—
85C32×D4—41no—
86C32×D4—41no—
87C32×D4—42no—
88C2×C32×C4—21no—
89C2×C32×C4#6711yescharacteristic
90C2×C32×C4C3×D411yescharacteristic
91#638—42no—
92C2×C3⋊Dic3—21no—
93C2×C3⋊Dic3—41no—
94C3×#67—122no—
95C2×C3×Dic3—61no—
96C2×C3×Dic3—121no—
97C32×C8—42no—
98C32×C8—41no—
99C32×C8—41no—
100AΣL(1,𝔽16)—91noSylow
101C2×H(3,𝔽3)—21no—
102C2×H(3,𝔽3)—41no—
103C2×H(3,𝔽3)—41no—
104C2×H(3,𝔽3)#9911yescharacteristic
105(C3⋊D3)⋊C3—42no—
106C24×C3—91no—
107C23×D3—31no—
108C23×D3—91no—
109C2×C3×D4—31no—
110C2×C3×D4—31no—
111C2×C3×D4—61no—
112C2×C3×D4—181no—
113C2×C3×D4C2×C3×D311yescharacteristic
114C2×#67—61no—
115C2×#67—181no—
116C3×M16—21no—
117C3×M16—61no—
118C3×M16—61no—
119C3×M16—62no—
120C3×C22⋊C4—91no—
121C3×C22⋊C4—181no—
122C3×C22⋊C4—181no—
123#217—31no—
124#217—61no—
125#217—91no—
126#217—181no—
127(C2×D3)⋊C4—61no—
128(C2×D3)⋊C4—181no—
129C22×C32—21no—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32—21no—
134C22×C32—41no—
135C22×C32—41no—
136C22×C32—41no—
137C22×C32—41no—
138C22×C32—41no—
139C22×C32—41no—
140C22×C32—81no—
141C22×C32—81no—
142C22×C32(C2×D3)⋊C411yescharacteristic
143C22×C32C3×C22⋊C411yescharacteristic
144C2×C3⋊D3—22no—
145C2×C3⋊D3—41no—
146C2×C3⋊D3—42no—
147C2×C3⋊D3—41no—
148C2×C3⋊D3—42no—
149C2×C3×D3—62no—
150C2×C3×D3—121no—
151C2×C3×D3—122no—
152C2×C3×D3—121no—
153C2×C3×D3—122no—
154C32×C4—22no—
155C32×C4—21no—
156C32×C4—21no—
157C3⋊Dic3—41no—
158C3⋊Dic3—81no—
159C3×Dic3—121no—
160C3×Dic3—241no—
161#120—91no—
162#100(C3⋊D3)⋊C311yescharacteristic, complemented
163#99—91no—
164H(3,𝔽3)AΣL(1,𝔽16)11yescharacteristic, complemented, Sylow
165C23×C3—31no—
166C23×C3—31no—
167C23×C3—31no—
168C23×C3—31no—
169C23×C3—61no—
170C23×C3—61no—
171C23×C3—181no—
172C23×C3—361no—
173C23×C3—362no—
174C23×C3C3×C4×D311yescharacteristic
175C23×C3C3×D1211yescharacteristic
176C22×D3—61no—
177C22×D3—122no—
178C22×D3—121no—
179C22×D3—181no—
180C22×D3—362no—
181C22×D3—361no—
182C3×D4—41no—
183C3×D4—121no—
184C3×D4—121no—
185C3×D4—122no—
186C3×D4—362no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—61no—
190C2×C3×C4—181no—
191C2×C3×C4—361no—
192C2×C3×C4C3×#6711yescharacteristic
193#67—122no—
194#67—362no—
195C2×Dic3—61no—
196C2×Dic3—121no—
197C2×Dic3—181no—
198C2×Dic3—361no—
199C3×C8—41no—
200C3×C8—121no—
201C3×C8—121no—
202C3×C8—122no—
203C2×C32—21no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32—81no—
212C2×C32—81no—
213C2×C32#80611yescharacteristic
214C2×C32C3×#9911yescharacteristic
215C3⋊D3—42no—
216C3×D3—122no—
217C24—91no—
218C2×D4—181no—
219C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
220M16—21no—
221C22⋊C4—91no—
222C22⋊C4—181no—
223C22⋊C4—181no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—31no—
227C22×C3—31no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—61no—
231C22×C3—61no—
232C22×C3—61no—
233C22×C3—61no—
234C22×C3—61no—
235C22×C3—121no—
236C22×C3—121no—
237C22×C3—121no—
238C22×C3—121no—
239C22×C3—121no—
240C22×C3—121no—
241C22×C3—182no—
242C22×C3—241no—
243C22×C3—241no—
244C22×C3—361no—
245C22×C3—361no—
246C22×C3—362no—
247C22×C3—362no—
248C22×C3C3×(C2×D3)⋊C411yescharacteristic
249C2×D3—62no—
250C2×D3—121no—
251C2×D3—122no—
252C2×D3—121no—
253C2×D3—122no—
254C2×D3—182no—
255C2×D3—361no—
256C2×D3—361no—
257C2×D3—362no—
258C2×D3—362no—
259C3×C4—21no—
260C3×C4—61no—
261C3×C4—61no—
262C3×C4—62no—
263C3×C4—361no—
264C3×C4—721no—
265Dic3—121no—
266Dic3—241no—
267Dic3—361no—
268Dic3—721no—
269C32—21no—
270C32#447511yescharacteristic, complemented
271C32C3×AΣL(1,𝔽16)11yescharacteristic, complemented
272C23—181no—
273C23—361no—
274C23—362no—
275C23C4×(C3⋊D3)⋊C311yescharacteristic
276C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
277D4—41no—
278D4—362no—
279C2×C4—181no—
280C2×C4—361no—
281C2×C4#638⋊C311yescharacteristic
282C8—41no—
283C2×C3—21no—
284C2×C3—31no—
285C2×C3—31no—
286C2×C3—41no—
287C2×C3—41no—
288C2×C3—61no—
289C2×C3—61no—
290C2×C3—61no—
291C2×C3—121no—
292C2×C3—121no—
293C2×C3—121no—
294C2×C3—121no—
295C2×C3—121no—
296C2×C3—241no—
297C2×C3—241no—
298C2×C3—362no—
299C2×C3C3×#80611yescharacteristic
300D3—122no—
301D3—362no—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—182no—
307C22—361no—
308C22—362no—
309C22—362no—
310C22—361no—
311C22#3752⋊C311yescharacteristic
312C4—21no—
313C4—361no—
314C4—721no—
315C3—31no—
316C3—31no—
317C3—61no—
318C3C3×#447511yescharacteristic
319C2—21no—
320C2—41no—
321C2—41no—
322C2—362no—
323C2#63633⋊C311yescharacteristic
subgroupG/H
centerC2#63633⋊C3
commutatorC23×C32C2×C3×C4
FrattiniC2×C3×D4C2×C3×D3
FittingH(3,𝔽3)×#100C2
socleC2×C3C3×#806
prime radical#64146⋊C3C2
cosocleC2×C32×D4C22×C3
hypercenter#100(C3⋊D3)⋊C3
hypocenterC32C3×AΣL(1,𝔽16)
LCSG ▷ C23×C32 ▷ C22×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #100

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#100C2semidirect
3. #63661⋊C3C2semidirect
4. #10589341C3semidirect
6. C23×(C3⋊D3)⋊C3C4semidirect
11. #3757⋊C3C4semidirect
16. C32×#100D3semidirect
17. C32×#100C2×C3semidirect
18. #63661C2×C3semidirect
34. C23×C3⋊D3C3×C4semidirect
45. #3757C3×C4semidirect
162. #100(C3⋊D3)⋊C3semidirect
164. H(3,𝔽3)AΣL(1,𝔽16)semidirect
270. C32#4475semidirect
271. C32C3×AΣL(1,𝔽16)semidirect

Maximal subgroups

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

subgroupindex
#64146⋊C32
H(3,𝔽3)×#1002
#63661⋊C32
#105893413
C3×#44753