Finite Group Databasebeta

← #10592700⋊C3 #13142695

Subgroups — #10592700⋊C3 #13142695

Order 1728.

741 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 741 conjugacy classes of subgroups — too many to draw. Only its 148 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
1C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
3C2×#3748⋊C3C211yesmaximal, characteristic, complemented
4#63636⋊C3C214yesmaximal, complemented
5#10592700C311yesmaximal, characteristic, complemented
6C3×#5001—31nomaximal
7C23×(C3⋊D3)⋊C3—21no—
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×C4⋊((C3⋊D3)⋊C3)—21no—
10C2×C4⋊((C3⋊D3)⋊C3)—22no—
11C2×C4⋊((C3⋊D3)⋊C3)C414yescomplemented
12C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
13C4⋊C4×H(3,𝔽3)C2212yescomplemented
14C22⋊C4×H(3,𝔽3)—21no—
15C42×H(3,𝔽3)—21no—
16#3748⋊C3—22no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
18C22×C4⋊(C3⋊D3)C2×C311yescharacteristic
19C22×C3×D12—31no—
20C32×#117—21no—
21C32×#117D311yescharacteristic, complemented
22C32×#117C2×C311yescharacteristic, complemented
23C2×#3748C2×C311yescharacteristic, complemented
24C2×C3×#208—31no—
25#63636C2×C314yescomplemented
26C3×#809—34no—
27C23×H(3,𝔽3)D411yescharacteristic
28C22×(C3⋊D3)⋊C3—24no—
29C22×(C3⋊D3)⋊C3—22no—
30C22×(C3⋊D3)⋊C3—44no—
31C2×C4×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)C2×C412yes—
33C2×C4×H(3,𝔽3)D411yescharacteristic
34C2×C4×H(3,𝔽3)D412yes—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36C4⋊((C3⋊D3)⋊C3)—24no—
37C4⋊((C3⋊D3)⋊C3)—22no—
38C4⋊((C3⋊D3)⋊C3)C2×C414yes—
39(C3⋊(C3⋊C8))⋊C3—22no—
40C3×#418—91no—
41#5001—31no—
42#5001—91no—
43C23×C3⋊D3—21no—
44C23×C3×D3—61no—
45C22×C32×C4—21no—
46C22×C32×C4C2×D311yescharacteristic
47C22×C32×C4C22×C311yescharacteristic
48C2×C4⋊(C3⋊D3)—21no—
49C2×C4⋊(C3⋊D3)—22no—
50C2×C4⋊(C3⋊D3)C3×C414yescomplemented
51C2×C4⋊(C3⋊D3)C22×C312yes—
52C2×C3×D12—34no—
53C2×C3×D12—32no—
54C2×C3×D12—62no—
55C2×C3×D12—61no—
56C32×C4⋊C4—22no—
57C32×C4⋊C4C2×D312yescomplemented
58C32×C4⋊C4C22×C312yescomplemented
59C32×C22⋊C4—22no—
60C32×C22⋊C4—21no—
61C32×C22⋊C4—21no—
62C32×C42—22no—
63C32×C42—21no—
64C32×C42—21no—
65#3748—22no—
66C2×C3⋊(C3⋊C8)C22×C312yescomplemented
67C3×#208—62no—
68C2×C3×C3⋊C8—32no—
69C22×H(3,𝔽3)—22no—
70C22×H(3,𝔽3)C22⋊C412yes—
71C22×H(3,𝔽3)C2×D411yescharacteristic
72C2×(C3⋊D3)⋊C3—24no—
73C2×(C3⋊D3)⋊C3—48no—
74C2×(C3⋊D3)⋊C3—44no—
75C4×H(3,𝔽3)—42no—
76C4×H(3,𝔽3)C22⋊C412yes—
77C4×H(3,𝔽3)C22×C411yescharacteristic
78C4×H(3,𝔽3)C2×D411yescharacteristic
79C22×C3×D4—91no—
80C22×D12—31no—
81C22×D12—91no—
82C2×C3×M16—91no—
83C3×#117—31no—
84C3×#117—31no—
85C3×#117—61no—
86C3×#117C3×D311yescharacteristic
87C2×#208—31no—
88C2×#208—91no—
89C3×#102—94no—
90#809—34no—
91#809—94no—
92C23×C32—21no—
93C23×C32#6711yescharacteristic
94C23×C32C3×D411yescharacteristic
95C22×C3⋊D3—24no—
96C22×C3⋊D3—22no—
97C22×C3⋊D3—44no—
98C22×C3×D3—64no—
99C22×C3×D3—62no—
100C22×C3×D3—124no—
101C2×C32×C4—24no—
102C2×C32×C4—22no—
103C2×C32×C4—22no—
104C2×C32×C4—22no—
105C2×C32×C4—22no—
106C2×C32×C4—21no—
107C2×C32×C4—21no—
108C2×C32×C4C4×D312yes—
109C2×C32×C4D1212yes—
110C2×C32×C4#6711yescharacteristic
111C2×C32×C4C2×C3×C412yes—
112C2×C32×C4C3×D412yes—
113C2×C32×C4C3×D411yescharacteristic
114C2×C32×C4C22×D311yescharacteristic
115C2×C32×C4C23×C311yescharacteristic
116C4⋊(C3⋊D3)—24no—
117C4⋊(C3⋊D3)—22no—
118C4⋊(C3⋊D3)C2×C3×C414yes—
119C3×D12—34no—
120C3×D12—64no—
121C3×D12—62no—
122C3⋊(C3⋊C8)—22no—
123C3×C3⋊C8—62no—
124#418—91noSylow
125C2×H(3,𝔽3)—22no—
126C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
127C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
128(C3⋊D3)⋊C3—44no—
129C24×C3—181no—
130C23×D3—61no—
131C23×D3—181no—
132C2×C3×D4—92no—
133C2×C3×D4—94no—
134C2×C3×D4—182no—
135C2×C3×D4—181no—
136C22×C3×C4—31no—
137C22×C3×C4—31no—
138C22×C3×C4—61no—
139C22×C3×C4C2×C3×D311yescharacteristic
140C2×D12—34no—
141C2×D12—32no—
142C2×D12—62no—
143C2×D12—61no—
144C2×D12—92no—
145C2×D12—94no—
146C2×D12—182no—
147C2×D12—181no—
148C3×M16—182no—
149C2×C3×C8—92no—
150C3×C4⋊C4—32no—
151C3×C4⋊C4—32no—
152C3×C4⋊C4—62no—
153C3×C4⋊C4C2×C3×D312yes—
154C3×C22⋊C4—21no—
155C3×C22⋊C4—61no—
156C3×C22⋊C4—61no—
157C3×C22⋊C4—62no—
158C3×C42—21no—
159C3×C42—61no—
160C3×C42—61no—
161C3×C42—62no—
162#208—62no—
163#208—182no—
164C2×C3⋊C8—32no—
165C2×C3⋊C8—92no—
166C22×C32—22no—
167C22×C32—22no—
168C22×C32—22no—
169C22×C32—21no—
170C22×C32—42no—
171C22×C32(C2×D3)⋊C412yes—
172C22×C32C3×C22⋊C412yes—
173C22×C32C2×#6711yescharacteristic
174C22×C32C2×C3×D411yescharacteristic
175C2×C3⋊D3—24no—
176C2×C3⋊D3—48no—
177C2×C3⋊D3—44no—
178C2×C3×D3—64no—
179C2×C3×D3—128no—
180C2×C3×D3—124no—
181C32×C4—22no—
182C32×C4—21no—
183C32×C4—21no—
184C32×C4—44no—
185C32×C4—42no—
186C32×C4—42no—
187C32×C4(C2×D3)⋊C412yes—
188C32×C4C3×C22⋊C412yes—
189C32×C4C2×C4×D311yescharacteristic
190C32×C4C2×D1211yescharacteristic
191C32×C4C22×C3×C411yescharacteristic
192C32×C4C2×C3×D411yescharacteristic
193C22×D4—91no—
194C2×M16—91no—
195#117(C3⋊D3)⋊C311yescharacteristic, complemented
196#102—94no—
197H(3,𝔽3)#41811yescharacteristic, complemented, Sylow
198C23×C3—31no—
199C23×C3—31no—
200C23×C3—61no—
201C23×C3—182no—
202C23×C3—184no—
203C23×C3—364no—
204C23×C3C3×#6711yescharacteristic
205C22×D3—64no—
206C22×D3—62no—
207C22×D3—124no—
208C22×D3—182no—
209C22×D3—184no—
210C22×D3—364no—
211C3×D4—94no—
212C3×D4—182no—
213C3×D4—184no—
214C2×C3×C4—22no—
215C2×C3×C4—32no—
216C2×C3×C4—32no—
217C2×C3×C4—32no—
218C2×C3×C4—32no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—62no—
224C2×C3×C4—62no—
225C2×C3×C4—64no—
226C2×C3×C4—62no—
227C2×C3×C4—62no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4C3×C4×D312yes—
231C2×C3×C4C3×D1212yes—
232C2×C3×C4C3×#6711yescharacteristic
233C2×C3×C4C22×C3×D311yescharacteristic
234D12—34no—
235D12—64no—
236D12—62no—
237D12—94no—
238D12—182no—
239D12—184no—
240C3×C8—182no—
241C3⋊C8—62no—
242C3⋊C8—182no—
243C2×C32—22no—
244C2×C32—22no—
245C2×C32—21no—
246C2×C32—22no—
247C2×C32—42no—
248C2×C32C2×(C2×D3)⋊C411yescharacteristic
249C2×C32#94912yes—
250C2×C32C2×C3×C22⋊C411yescharacteristic
251C2×C32C3×AGL(1,ℤ/8)12yes—
252C3⋊D3—44no—
253C3×D3—124no—
254C24—181no—
255C2×D4—94no—
256C2×D4—92no—
257C2×D4—182no—
258C2×D4—181no—
259C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
260M16—182no—
261C2×C8—92no—
262C4⋊C4C2×(C3⋊D3)⋊C312yescomplemented
263C22⋊C4—21no—
264C42—21no—
265C22×C3—22no—
266C22×C3—31no—
267C22×C3—31no—
268C22×C3—32no—
269C22×C3—32no—
270C22×C3—62no—
271C22×C3—62no—
272C22×C3—61no—
273C22×C3—62no—
274C22×C3—122no—
275C22×C3—184no—
276C22×C3—364no—
277C22×C3—368no—
278C22×C3C3×(C2×D3)⋊C412yes—
279C22×C3C2×C3×#6711yescharacteristic
280C2×D3—64no—
281C2×D3—128no—
282C2×D3—124no—
283C2×D3—184no—
284C2×D3—364no—
285C2×D3—368no—
286C3×C4—31no—
287C3×C4—31no—
288C3×C4—31no—
289C3×C4—31no—
290C3×C4—32no—
291C3×C4—32no—
292C3×C4—42no—
293C3×C4—61no—
294C3×C4—61no—
295C3×C4—62no—
296C3×C4—122no—
297C3×C4—122no—
298C3×C4—124no—
299C3×C4C3×(C2×D3)⋊C412yes—
300C3×C4C2×C3×C4×D311yescharacteristic
301C3×C4C2×C3×D1211yescharacteristic
302C32—21no—
303C32#500111yescharacteristic, complemented
304C32C3×#41811yescharacteristic, complemented
305C23—182no—
306C23—184no—
307C23—364no—
308C23#638⋊C311yescharacteristic
309D4—94no—
310D4—182no—
311D4—184no—
312C2×C4—22no—
313C2×C4C4×(C3⋊D3)⋊C312yes—
314C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
315C2×C4#638⋊C311yescharacteristic
316C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
317C8—182no—
318C2×C3—22no—
319C2×C3—31no—
320C2×C3—31no—
321C2×C3—32no—
322C2×C3—32no—
323C2×C3—62no—
324C2×C3—62no—
325C2×C3—61no—
326C2×C3—62no—
327C2×C3—122no—
328C2×C3—364no—
329C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
330C2×C3C3×#94912yes—
331D3—124no—
332D3—364no—
333C22—22no—
334C22—184no—
335C22—368no—
336C22—364no—
337C22#3752⋊C312yes—
338C22C2×#638⋊C311yescharacteristic
339C4—42no—
340C4#3752⋊C312yes—
341C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
342C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
343C3—31no—
344C3—31no—
345C3—61no—
346C3C3×#500111yescharacteristic
347C2—22no—
348C2—364no—
349C2C2×#3752⋊C311yescharacteristic
350C2#64158⋊C312yes—
subgroupG/H
centerC22C2×#638⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#117C2
socleC22×C3C2×C3×#67
prime radicalC22×C4⋊((C3⋊D3)⋊C3)C2
cosocleC2×C32×C4C23×C3
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C3×#418
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. H(3,𝔽3)×#117C2semidirect
3. C2×#3748⋊C3C2semidirect
4. #63636⋊C3C2semidirect
5. #10592700C3semidirect
11. C2×C4⋊((C3⋊D3)⋊C3)C4semidirect
13. C4⋊C4×H(3,𝔽3)C22semidirect
17. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
21. C32×#117D3semidirect
22. C32×#117C2×C3semidirect
23. C2×#3748C2×C3semidirect
25. #63636C2×C3semidirect
50. C2×C4⋊(C3⋊D3)C3×C4semidirect
57. C32×C4⋊C4C2×D3semidirect
58. C32×C4⋊C4C22×C3semidirect
66. C2×C3⋊(C3⋊C8)C22×C3semidirect
195. #117(C3⋊D3)⋊C3semidirect
197. H(3,𝔽3)#418semidirect
262. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
303. C32#5001semidirect
304. C32C3×#418semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊((C3⋊D3)⋊C3)2
H(3,𝔽3)×#1172
C2×#3748⋊C32
#63636⋊C32
#105927003
C3×#50013