Finite Group Databasebeta

← #10592434⋊C3 #13141619

Subgroups — #10592434⋊C3 #13141619

Order 1728.

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

normal subgroups only

This group has 449 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
1#64139⋊C3C211yesmaximal, characteristic
2C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×C22⋊C8C211yesmaximal, characteristic, complemented
4#63648⋊C3C212yesmaximal, complemented
5#63644⋊C3C212yesmaximal, complemented
6#10592434C311yesmaximal, characteristic, complemented
7C3×#4735—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#638⋊C3—21no—
10C22×(C3⋊Dic3)⋊C3—21no—
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2212yes—
13#3752⋊C3—21no—
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2212yes—
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#64139C2×C311yescharacteristic
18C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
19C3×#930—31no—
20C2×C3×C4⋊Dic3—31no—
21C32×C22⋊C8—21no—
22C32×C22⋊C8D311yescharacteristic, complemented
23C32×C22⋊C8C2×C311yescharacteristic, complemented
24#63648C2×C312yescomplemented
25#63644C2×C312yescomplemented
26C3×#821—32no—
27C3×#817—32no—
28C23×H(3,𝔽3)D411yescharacteristic
29C22×(C3⋊D3)⋊C3—21no—
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33#638⋊C3—42no—
34C2×(C3⋊Dic3)⋊C3—22no—
35C2×(C3⋊Dic3)⋊C3—21no—
36C2×(C3⋊Dic3)⋊C3—42no—
37C4⋊((C3⋊D3)⋊C3)—22no—
38C8×H(3,𝔽3)—22no—
39C3×#481—91no—
40#4735—31no—
41#4735—91no—
42C22×C32×C4—21no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C32×C4C22×C311yescharacteristic
45C2×#638—21no—
46C22×C3⋊Dic3—21no—
47C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
48C2×C3×#67—61no—
49C22×C3×Dic3—61no—
50C2×C3×D12—31no—
51C2×C32×C8—22no—
52C2×C32×C8C2×D312yes—
53C2×C32×C8C22×C312yes—
54#3752—21no—
55C4⋊(C3⋊Dic3)—21no—
56C4⋊(C3⋊Dic3)C22×C312yes—
57C4⋊(C3⋊Dic3)C22×C311yescharacteristic
58C3×(C2×D3)⋊C4—61no—
59C3×C4⋊Dic3—32no—
60C3×C4⋊Dic3—31no—
61C3×C4⋊Dic3—61no—
62C22×H(3,𝔽3)—21no—
63C22×H(3,𝔽3)—21no—
64C22×H(3,𝔽3)SD1612yes—
65C22×H(3,𝔽3)C2×D411yescharacteristic
66C2×(C3⋊D3)⋊C3—41no—
67C2×(C3⋊D3)⋊C3—41no—
68C2×(C3⋊D3)⋊C3—41no—
69C4×H(3,𝔽3)—21no—
70C4×H(3,𝔽3)#4112yes—
71(C3⋊Dic3)⋊C3—41no—
72(C3⋊Dic3)⋊C3—42no—
73C3×#121—91no—
74C2×C3×C4⋊C4—91no—
75#930—31no—
76#930—91no—
77C2×C4⋊Dic3—31no—
78C2×C4⋊Dic3—91no—
79C3×#106—92no—
80C3×#102—92no—
81C3×C22⋊C8—31no—
82C3×C22⋊C8—31no—
83C3×C22⋊C8—61no—
84C3×C22⋊C8C3×D311yescharacteristic
85#821—32no—
86#821—92no—
87#817—32no—
88#817—92no—
89C23×C32—21no—
90C23×C32D1211yescharacteristic
91C23×C32C3×D411yescharacteristic
92C22×C3⋊D3—21no—
93C22×C3×D3—61no—
94C2×C32×C4—22no—
95C2×C32×C4—22no—
96C2×C32×C4—21no—
97C2×C32×C4—21no—
98C2×C32×C4—42no—
99C2×C32×C4D1211yescharacteristic
100C2×C32×C4C3×D411yescharacteristic
101C2×C32×C4C22×D311yescharacteristic
102C2×C32×C4C23×C311yescharacteristic
103#638—42no—
104C2×C3⋊Dic3—22no—
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3—42no—
107C4⋊(C3⋊D3)—22no—
108C3×#67—122no—
109C2×C3×Dic3—62no—
110C2×C3×Dic3—61no—
111C2×C3×Dic3—122no—
112C3×D12—62no—
113C32×C8—22no—
114C32×C8—22no—
115C32×C8—42no—
116#481—91noSylow
117C2×H(3,𝔽3)—22no—
118C2×H(3,𝔽3)#12311yescharacteristic
119C2×H(3,𝔽3)C2×SD1611yescharacteristic
120C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
121(C3⋊D3)⋊C3—81no—
122C2×C3×D4—91no—
123C2×C3×D4—181no—
124C22×C3×C4—31no—
125C22×C3×C4—31no—
126C22×C3×C4—61no—
127C22×C3×C4—181no—
128C22×C3×C4C2×C3×D311yescharacteristic
129C2×#67—61no—
130C2×#67—181no—
131C22×Dic3—61no—
132C22×Dic3—181no—
133C2×D12—31no—
134C2×D12—91no—
135C2×C3×C8—32no—
136C2×C3×C8—32no—
137C2×C3×C8—62no—
138C2×C3×C8C2×C3×D312yes—
139C3×C4⋊C4—92no—
140C3×C4⋊C4—91no—
141C3×C4⋊C4—181no—
142C3×C22⋊C4—181no—
143(C2×D3)⋊C4—61no—
144(C2×D3)⋊C4—181no—
145C4⋊Dic3—31no—
146C4⋊Dic3—32no—
147C4⋊Dic3—61no—
148C4⋊Dic3—91no—
149C4⋊Dic3—92no—
150C4⋊Dic3—181no—
151C22×C32—21no—
152C22×C32—21no—
153C22×C32—21no—
154C22×C32—21no—
155C22×C32—22no—
156C22×C32—21no—
157C22×C32—41no—
158C22×C32—41no—
159C22×C32#20412yes—
160C22×C32C3×SD1612yes—
161C22×C32C2×D1211yescharacteristic
162C22×C32C2×C3×D411yescharacteristic
163C2×C3⋊D3—41no—
164C2×C3⋊D3—41no—
165C2×C3⋊D3—41no—
166C2×C3×D3—121no—
167C2×C3×D3—121no—
168C2×C3×D3—121no—
169C32×C4—21no—
170C32×C4—21no—
171C32×C4—22no—
172C32×C4—41no—
173C32×C4#23712yes—
174C32×C4C3×#4112yes—
175C3⋊Dic3—41no—
176C3⋊Dic3—42no—
177C3×Dic3—121no—
178C3×Dic3—122no—
179#121—91no—
180C2×C4⋊C4—91no—
181#106—92no—
182#102—92no—
183C22⋊C8(C3⋊D3)⋊C311yescharacteristic, complemented
184H(3,𝔽3)#48111yescharacteristic, complemented, Sylow
185C23×C3—31no—
186C23×C3—31no—
187C23×C3—61no—
188C23×C3—181no—
189C23×C3C3×D1211yescharacteristic
190C22×D3—61no—
191C22×D3—181no—
192C3×D4—182no—
193C3×D4—362no—
194C2×C3×C4—22no—
195C2×C3×C4—31no—
196C2×C3×C4—31no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—62no—
200C2×C3×C4—62no—
201C2×C3×C4—61no—
202C2×C3×C4—61no—
203C2×C3×C4—122no—
204C2×C3×C4—181no—
205C2×C3×C4—182no—
206C2×C3×C4—362no—
207C2×C3×C4C3×D1211yescharacteristic
208C2×C3×C4C22×C3×D311yescharacteristic
209#67—122no—
210#67—362no—
211C2×Dic3—62no—
212C2×Dic3—61no—
213C2×Dic3—122no—
214C2×Dic3—182no—
215C2×Dic3—181no—
216C2×Dic3—362no—
217D12—62no—
218D12—182no—
219C3×C8—22no—
220C3×C8—62no—
221C3×C8—62no—
222C3×C8—122no—
223C2×C32—22no—
224C2×C32—22no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—42no—
229C2×C32#88611yescharacteristic
230C2×C32C2×#20411yescharacteristic
231C2×C32#90811yescharacteristic
232C2×C32C3×#12311yescharacteristic
233C2×C32C2×C3×SD1611yescharacteristic
234C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
235C3⋊D3—81no—
236C3×D3—241no—
237C2×D4—91no—
238C2×D4—181no—
239C22×C4—181no—
240C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
241C2×C8C2×(C3⋊D3)⋊C312yes—
242C4⋊C4—92no—
243C4⋊C4—91no—
244C4⋊C4—181no—
245C22⋊C4—181no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—31no—
249C22×C3—31no—
250C22×C3—32no—
251C22×C3—32no—
252C22×C3—61no—
253C22×C3—61no—
254C22×C3—61no—
255C22×C3—61no—
256C22×C3—61no—
257C22×C3—62no—
258C22×C3—121no—
259C22×C3—121no—
260C22×C3—361no—
261C22×C3—361no—
262C22×C3—361no—
263C22×C3C3×#20412yes—
264C22×C3C2×C3×D1211yescharacteristic
265C2×D3—121no—
266C2×D3—121no—
267C2×D3—121no—
268C2×D3—361no—
269C2×D3—361no—
270C2×D3—361no—
271C3×C4—21no—
272C3×C4—32no—
273C3×C4—32no—
274C3×C4—61no—
275C3×C4—61no—
276C3×C4—62no—
277C3×C4—121no—
278C3×C4—362no—
279C3×C4—361no—
280C3×C4C3×#23712yes—
281Dic3—121no—
282Dic3—122no—
283Dic3—362no—
284Dic3—361no—
285C32—21no—
286C32#473511yescharacteristic, complemented
287C32C3×#48111yescharacteristic, complemented
288C23—181no—
289C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
290D4—182no—
291D4—362no—
292C2×C4—22no—
293C2×C4—181no—
294C2×C4—182no—
295C2×C4—362no—
296C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
297C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
298C8—22no—
299C2×C3—22no—
300C2×C3—31no—
301C2×C3—31no—
302C2×C3—31no—
303C2×C3—31no—
304C2×C3—31no—
305C2×C3—31no—
306C2×C3—62no—
307C2×C3—62no—
308C2×C3—61no—
309C2×C3—61no—
310C2×C3—61no—
311C2×C3—122no—
312C2×C3—721no—
313C2×C3C3×#88611yescharacteristic
314C2×C3C2×C3×#20411yescharacteristic
315C2×C3C3×#90811yescharacteristic
316D3—241no—
317D3—721no—
318C22—21no—
319C22—21no—
320C22—361no—
321C22—361no—
322C22—361no—
323C22#3744⋊C312yes—
324C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
325C4—21no—
326C4—361no—
327C4—362no—
328C4#3830⋊C312yes—
329C3—31no—
330C3—31no—
331C3—61no—
332C3C3×#473511yescharacteristic
333C2—22no—
334C2—721no—
335C2#64095⋊C311yescharacteristic
336C2C2×#3744⋊C311yescharacteristic
337C2#64117⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C22⋊C8C2
socleC22×C3C2×C3×D12
prime radical#64139⋊C3C2
hypercenterC22⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#481
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
3. H(3,𝔽3)×C22⋊C8C2semidirect
4. #63648⋊C3C2semidirect
5. #63644⋊C3C2semidirect
6. #10592434C3semidirect
18. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
22. C32×C22⋊C8D3semidirect
23. C32×C22⋊C8C2×C3semidirect
24. #63648C2×C3semidirect
25. #63644C2×C3semidirect
183. C22⋊C8(C3⋊D3)⋊C3semidirect
184. H(3,𝔽3)#481semidirect
286. C32#4735semidirect
287. C32C3×#481semidirect

Maximal subgroups

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

subgroupindex
#64139⋊C32
C2×C4⋊((C3⋊Dic3)⋊C3)2
H(3,𝔽3)×C22⋊C82
#63648⋊C32
#63644⋊C32
#105924343
C3×#47353