Finite Group Databasebeta

← #10592666⋊C3 #13142606

Subgroups — #10592666⋊C3 #13142606

Order 1728.

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

normal subgroups only

This group has 491 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×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64139⋊C3C211yesmaximal, characteristic
3#63657⋊C3C211yesmaximal, characteristic, complemented
4#63636⋊C3C212yesmaximal, complemented
5#63634⋊C3C212yesmaximal, complemented
6#10592666C311yesmaximal, characteristic, complemented
7C3×#4967—31nomaximal
8C22×C4×H(3,𝔽3)—21no—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×#638⋊C3—21no—
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C4⋊C4×H(3,𝔽3)—21no—
13C4⋊C4×H(3,𝔽3)C2212yes—
14#3752⋊C3—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C2×C32×C4⋊C4—21no—
18C2×C32×C4⋊C4D311yescharacteristic, complemented
19C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
20#64139C2×C311yescharacteristic
21C3×#930—31no—
22#63657C2×C311yescharacteristic, complemented
23#63636C2×C312yescomplemented
24#63634C2×C312yescomplemented
25C3×#830—31no—
26C3×#809—32no—
27C3×#807—32no—
28C23×H(3,𝔽3)D411yescharacteristic
29C22×(C3⋊D3)⋊C3—21no—
30C2×C4×H(3,𝔽3)—22no—
31C2×C4×H(3,𝔽3)—22no—
32C2×C4×H(3,𝔽3)—42no—
33C2×C4×H(3,𝔽3)D411yescharacteristic
34C2×C4×H(3,𝔽3)C2311yescharacteristic
35#638⋊C3—42no—
36C2×(C3⋊Dic3)⋊C3—21no—
37C4⋊((C3⋊D3)⋊C3)—22no—
38(C3⋊(C3⋊C8))⋊C3—22no—
39C3×#480—91no—
40#4967—31no—
41#4967—91no—
42C22×C32×C4—21no—
43C22×C32×C4—22no—
44C22×C32×C4—21no—
45C22×C32×C4—21no—
46C22×C32×C4C2×D311yescharacteristic
47C22×C32×C4C22×C311yescharacteristic
48C2×#638—21no—
49C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
50C2×C3×#67—61no—
51C2×C3×D12—31no—
52C32×C4⋊C4—22no—
53C32×C4⋊C4—22no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4—21no—
56C32×C4⋊C4C2×D312yes—
57C32×C4⋊C4C22×C312yes—
58#3752—21no—
59C4⋊(C3⋊Dic3)C22×C311yescharacteristic
60C2×C3⋊(C3⋊C8)C22×C312yes—
61C3×(C2×D3)⋊C4—61no—
62C3×C4⋊Dic3—31no—
63C2×C3×C3⋊C8—32no—
64C22×H(3,𝔽3)—21no—
65C22×H(3,𝔽3)—21no—
66C22×H(3,𝔽3)D812yes—
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C2×(C3⋊D3)⋊C3—41no—
69C2×(C3⋊D3)⋊C3—41no—
70C2×(C3⋊D3)⋊C3—41no—
71C4×H(3,𝔽3)—21no—
72C4×H(3,𝔽3)—42no—
73C4×H(3,𝔽3)#4112yes—
74(C3⋊Dic3)⋊C3—41no—
75C3×#121—91no—
76C2×C3×C4⋊C4—31no—
77C2×C3×C4⋊C4—31no—
78C2×C3×C4⋊C4—61no—
79C2×C3×C4⋊C4C3×D311yescharacteristic
80#930—31no—
81#930—91no—
82C3×#107—92no—
83C3×#102—92no—
84C3×C22⋊C8—91no—
85#830—31no—
86#830—91no—
87#809—32no—
88#809—92no—
89#807—32no—
90#807—92no—
91C23×C32—21no—
92C23×C32#6711yescharacteristic
93C23×C32C3×D411yescharacteristic
94C22×C3⋊D3—21no—
95C22×C3×D3—61no—
96C2×C32×C4—24no—
97C2×C32×C4—22no—
98C2×C32×C4—22no—
99C2×C32×C4—24no—
100C2×C32×C4—22no—
101C2×C32×C4—22no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4—44no—
105C2×C32×C4—42no—
106C2×C32×C4—42no—
107C2×C32×C4#6711yescharacteristic
108C2×C32×C4C3×D411yescharacteristic
109C2×C32×C4C22×D311yescharacteristic
110C2×C32×C4C23×C311yescharacteristic
111#638—42no—
112C2×C3⋊Dic3—21no—
113C4⋊(C3⋊D3)—22no—
114C3×#67—122no—
115C2×C3×Dic3—61no—
116C3×D12—62no—
117C3⋊(C3⋊C8)—22no—
118C3×C3⋊C8—62no—
119#480—91noSylow
120C2×H(3,𝔽3)—22no—
121C2×H(3,𝔽3)#12311yescharacteristic
122C2×H(3,𝔽3)C2×D811yescharacteristic
123C2×H(3,𝔽3)#13711yescharacteristic
124(C3⋊D3)⋊C3—81no—
125C2×C3×D4—91no—
126C2×C3×D4—181no—
127C22×C3×C4—21no—
128C22×C3×C4—31no—
129C22×C3×C4—31no—
130C22×C3×C4—61no—
131C22×C3×C4—61no—
132C22×C3×C4—62no—
133C22×C3×C4—61no—
134C22×C3×C4C2×C3×D311yescharacteristic
135C2×#67—61no—
136C2×#67—181no—
137C2×D12—31no—
138C2×D12—91no—
139C2×C3×C8—92no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—32no—
142C3×C4⋊C4—32no—
143C3×C4⋊C4—61no—
144C3×C4⋊C4—61no—
145C3×C4⋊C4—62no—
146C3×C4⋊C4—62no—
147C3×C4⋊C4—91no—
148C3×C4⋊C4C2×C3×D312yes—
149C3×C22⋊C4—181no—
150(C2×D3)⋊C4—61no—
151(C2×D3)⋊C4—181no—
152C4⋊Dic3—31no—
153C4⋊Dic3—91no—
154C2×C3⋊C8—32no—
155C2×C3⋊C8—92no—
156C22×C32—22no—
157C22×C32—22no—
158C22×C32—21no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32—21no—
162C22×C32—22no—
163C22×C32—21no—
164C22×C32#21312yes—
165C22×C32C3×D812yes—
166C22×C32C2×#6711yescharacteristic
167C22×C32C2×C3×D411yescharacteristic
168C2×C3⋊D3—41no—
169C2×C3⋊D3—41no—
170C2×C3⋊D3—41no—
171C2×C3×D3—121no—
172C2×C3×D3—121no—
173C2×C3×D3—121no—
174C32×C4—22no—
175C32×C4—21no—
176C32×C4—21no—
177C32×C4—22no—
178C32×C4—44no—
179C32×C4—42no—
180C32×C4—42no—
181C32×C4#23512yes—
182C32×C4C3×#4112yes—
183C3⋊Dic3—41no—
184C3×Dic3—121no—
185#121—91no—
186C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
187#107—92no—
188#102—92no—
189C22⋊C8—91no—
190H(3,𝔽3)#48011yescharacteristic, complemented, Sylow
191C23×C3—31no—
192C23×C3—31no—
193C23×C3—61no—
194C23×C3—181no—
195C23×C3C3×#6711yescharacteristic
196C22×D3—61no—
197C22×D3—181no—
198C3×D4—182no—
199C3×D4—362no—
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—42no—
207C2×C3×C4—62no—
208C2×C3×C4—62no—
209C2×C3×C4—64no—
210C2×C3×C4—62no—
211C2×C3×C4—62no—
212C2×C3×C4—64no—
213C2×C3×C4—61no—
214C2×C3×C4—61no—
215C2×C3×C4—122no—
216C2×C3×C4—122no—
217C2×C3×C4—124no—
218C2×C3×C4—181no—
219C2×C3×C4C3×#6711yescharacteristic
220C2×C3×C4C22×C3×D311yescharacteristic
221#67—122no—
222#67—362no—
223C2×Dic3—61no—
224C2×Dic3—181no—
225D12—62no—
226D12—182no—
227C3×C8—182no—
228C3⋊C8—62no—
229C3⋊C8—182no—
230C2×C32—24no—
231C2×C32—22no—
232C2×C32—22no—
233C2×C32—21no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32#92911yescharacteristic
237C2×C32C2×#21311yescharacteristic
238C2×C32#94211yescharacteristic
239C2×C32C3×#12311yescharacteristic
240C2×C32C2×C3×D811yescharacteristic
241C2×C32C3×#13711yescharacteristic
242C3⋊D3—81no—
243C3×D3—241no—
244C2×D4—91no—
245C2×D4—181no—
246C22×C4—21no—
247C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
248C2×C8—92no—
249C4⋊C4—21no—
250C4⋊C4—91no—
251C4⋊C4C2×(C3⋊D3)⋊C312yes—
252C22⋊C4—181no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—32no—
258C22×C3—32no—
259C22×C3—61no—
260C22×C3—61no—
261C22×C3—61no—
262C22×C3—61no—
263C22×C3—62no—
264C22×C3—62no—
265C22×C3—61no—
266C22×C3—62no—
267C22×C3—361no—
268C22×C3—361no—
269C22×C3—361no—
270C22×C3C3×#21312yes—
271C22×C3C2×C3×#6711yescharacteristic
272C2×D3—121no—
273C2×D3—121no—
274C2×D3—121no—
275C2×D3—361no—
276C2×D3—361no—
277C2×D3—361no—
278C3×C4—21no—
279C3×C4—32no—
280C3×C4—32no—
281C3×C4—42no—
282C3×C4—61no—
283C3×C4—61no—
284C3×C4—62no—
285C3×C4—62no—
286C3×C4—122no—
287C3×C4—122no—
288C3×C4—124no—
289C3×C4—361no—
290C3×C4C3×#23512yes—
291Dic3—121no—
292Dic3—361no—
293C32—21no—
294C32#496711yescharacteristic, complemented
295C32C3×#48011yescharacteristic, complemented
296C23—181no—
297C23#638⋊C311yescharacteristic
298D4—182no—
299D4—362no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—42no—
303C2×C4—181no—
304C2×C4#638⋊C311yescharacteristic
305C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
306C8—182no—
307C2×C3—22no—
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—62no—
315C2×C3—62no—
316C2×C3—64no—
317C2×C3—61no—
318C2×C3—61no—
319C2×C3—61no—
320C2×C3—721no—
321C2×C3C3×#92911yescharacteristic
322C2×C3C2×C3×#21311yescharacteristic
323C2×C3C3×#94211yescharacteristic
324D3—241no—
325D3—721no—
326C22—21no—
327C22—21no—
328C22—361no—
329C22—361no—
330C22—361no—
331C22#3753⋊C312yes—
332C22C2×#638⋊C311yescharacteristic
333C4—21no—
334C4—42no—
335C4—361no—
336C4#3828⋊C312yes—
337C3—31no—
338C3—31no—
339C3—61no—
340C3C3×#496711yescharacteristic
341C2—22no—
342C2—721no—
343C2#64138⋊C311yescharacteristic
344C2C2×#3753⋊C311yescharacteristic
345C2#64151⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
prime radical#64139⋊C3C2
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C3×#480
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊C4×H(3,𝔽3)C2semidirect
3. #63657⋊C3C2semidirect
4. #63636⋊C3C2semidirect
5. #63634⋊C3C2semidirect
6. #10592666C3semidirect
18. C2×C32×C4⋊C4D3semidirect
19. C2×C32×C4⋊C4C2×C3semidirect
22. #63657C2×C3semidirect
23. #63636C2×C3semidirect
24. #63634C2×C3semidirect
186. C2×C4⋊C4(C3⋊D3)⋊C3semidirect
190. H(3,𝔽3)#480semidirect
294. C32#4967semidirect
295. C32C3×#480semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×H(3,𝔽3)2
#64139⋊C32
#63657⋊C32
#63636⋊C32
#63634⋊C32
#105926663
C3×#49673