Finite Group Databasebeta

← #10592730⋊C3 #13142795

Subgroups — #10592730⋊C3 #13142795

Order 1728.

448 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C4×Q8×H(3,𝔽3)C211yesmaximal, characteristic
2C2×#3756⋊C3C212yesmaximal
3#64078⋊C3C211yesmaximal, characteristic
4#63637⋊C3C212yesmaximal
5C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic
6#10592730C311yesmaximal, characteristic, complemented
7C3×#5031—31nomaximal
8C2×Q8×H(3,𝔽3)C2211yescharacteristic
9C2×#634⋊C3C2212yes—
10C4⋊C4×H(3,𝔽3)—21no—
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12C42×H(3,𝔽3)—21no—
13C42×H(3,𝔽3)C2211yescharacteristic
14#3756⋊C3—24no—
15C4⋊((C3⋊Dic3)⋊C3)—21no—
16C4⋊((C3⋊Dic3)⋊C3)—21no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18C32×C4×Q8—21no—
19C32×C4×Q8D311yescharacteristic
20C32×C4×Q8C2×C311yescharacteristic
21C2×#3756C2×C312yes—
22#64078C2×C311yescharacteristic
23C2×C3×#216—32no—
24C3×#869—31no—
25#63637C2×C312yes—
26C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic
27C3×#810—32no—
28C3×C4⋊(C3⋊C8)—31no—
29Q8×H(3,𝔽3)—21no—
30Q8×H(3,𝔽3)D412yes—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)D411yescharacteristic
34C2×C4×H(3,𝔽3)D411yescharacteristic
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36C2×(C3⋊Dic3)⋊C3—22no—
37#634⋊C3—22no—
38#634⋊C3—22no—
39(C3⋊(C3⋊C8))⋊C3—22no—
40C3×#462—91no—
41#5031—31no—
42#5031—91no—
43C2×C32×Q8—21no—
44C2×C32×Q8C2×D311yescharacteristic
45C2×C32×Q8C22×C311yescharacteristic
46C2×#634C22×C312yes—
47C2×C3×Dic6—32no—
48C32×C4⋊C4—22no—
49C32×C4⋊C4—21no—
50C32×C4⋊C4—21no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4C2×D311yescharacteristic
53C32×C4⋊C4C22×C311yescharacteristic
54C32×C42—22no—
55C32×C42—21no—
56C32×C42—21no—
57C32×C42—21no—
58C32×C42C2×D311yescharacteristic
59C32×C42C22×C311yescharacteristic
60#3756—24no—
61C4⋊(C3⋊Dic3)—21no—
62C4⋊(C3⋊Dic3)—21no—
63C2×C3⋊(C3⋊C8)C22×C312yes—
64C3×#216—64no—
65C3×C4⋊Dic3—61no—
66C3×C4⋊Dic3—61no—
67C2×C3×C3⋊C8—32no—
68C22×H(3,𝔽3)C2×D411yescharacteristic
69C4×H(3,𝔽3)—22no—
70C4×H(3,𝔽3)—21no—
71C4×H(3,𝔽3)—41no—
72C4×H(3,𝔽3)Q1612yescomplemented
73C4×H(3,𝔽3)C2×D411yescharacteristic
74C4×H(3,𝔽3)#4111yescharacteristic
75(C3⋊Dic3)⋊C3—42no—
76C2×C3×Q16—92no—
77C3×C4⋊Q8—91no—
78C3×C4×Q8—31no—
79C3×C4×Q8—31no—
80C3×C4×Q8—61no—
81C3×C4×Q8C3×D311yescharacteristic
82C2×#216—32no—
83C2×#216—92no—
84#869—31no—
85#869—91no—
86C3×C4⋊C8—91no—
87C3×#103—92no—
88#810—32no—
89#810—92no—
90C4⋊(C3⋊C8)—31no—
91C4⋊(C3⋊C8)—91no—
92C32×Q8—22no—
93C32×Q8—21no—
94C32×Q8—21no—
95C32×Q8—22no—
96C32×Q8D1212yes—
97C32×Q8C3×D412yes—
98C2×C32×C4—22no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4—22no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4#6711yescharacteristic
108C2×C32×C4#6711yescharacteristic
109C2×C32×C4C3×D411yescharacteristic
110C2×C32×C4C3×D411yescharacteristic
111C2×C32×C4C22×D311yescharacteristic
112C2×C32×C4C23×C311yescharacteristic
113C2×C3⋊Dic3—22no—
114#634—22no—
115#634—22no—
116C2×C3×Dic3—62no—
117C3×Dic6—62no—
118C3×Dic6—62no—
119C3⋊(C3⋊C8)—22no—
120C3×C3⋊C8—62no—
121#462—91noSylow
122C2×H(3,𝔽3)#12111yescharacteristic
123C2×H(3,𝔽3)C2×Q1611yescharacteristic
124C2×H(3,𝔽3)#13711yescharacteristic
125C2×C3×Q8—31no—
126C2×C3×Q8—31no—
127C2×C3×Q8—61no—
128C2×C3×Q8—92no—
129C2×C3×Q8C2×C3×D311yescharacteristic
130C2×Dic6—32no—
131C2×Dic6—92no—
132C3×Q16—184no—
133C2×C3×C8—92no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—31no—
136C3×C4⋊C4—31no—
137C3×C4⋊C4—61no—
138C3×C4⋊C4—61no—
139C3×C4⋊C4—62no—
140C3×C4⋊C4—61no—
141C3×C4⋊C4—181no—
142C3×C4⋊C4—181no—
143C3×C4⋊C4C2×C3×D311yescharacteristic
144C3×C42—21no—
145C3×C42—31no—
146C3×C42—31no—
147C3×C42—61no—
148C3×C42—61no—
149C3×C42—62no—
150C3×C42—61no—
151C3×C42C2×C3×D311yescharacteristic
152#216—64no—
153#216—184no—
154C4⋊Dic3—61no—
155C4⋊Dic3—61no—
156C4⋊Dic3—181no—
157C4⋊Dic3—181no—
158C2×C3⋊C8—32no—
159C2×C3⋊C8—92no—
160C22×C32—21no—
161C22×C32C2×#6711yescharacteristic
162C22×C32C2×C3×D411yescharacteristic
163C32×C4—24no—
164C32×C4—22no—
165C32×C4—21no—
166C32×C4—22no—
167C32×C4—21no—
168C32×C4—22no—
169C32×C4—21no—
170C32×C4—21no—
171C32×C4—41no—
172C32×C4—42no—
173C32×C4—41no—
174C32×C4—41no—
175C32×C4#21612yescomplemented
176C32×C4C3×Q1612yescomplemented
177C32×C4C2×D1211yescharacteristic
178C32×C4#23511yescharacteristic
179C32×C4C2×C3×D411yescharacteristic
180C32×C4C3×#4111yescharacteristic
181C3⋊Dic3—42no—
182C3×Dic3—122no—
183C2×Q16—92no—
184C4⋊Q8—91no—
185C4×Q8(C3⋊D3)⋊C311yescharacteristic
186C4⋊C8—91no—
187#103—92no—
188H(3,𝔽3)#46211yescharacteristic, complemented, Sylow
189C3×Q8—21no—
190C3×Q8—32no—
191C3×Q8—32no—
192C3×Q8—61no—
193C3×Q8—61no—
194C3×Q8—62no—
195C3×Q8—62no—
196C3×Q8—182no—
197C3×Q8—182no—
198C3×Q8C3×D1212yes—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—61no—
208C2×C3×C4—61no—
209C2×C3×C4—62no—
210C2×C3×C4—61no—
211C2×C3×C4—61no—
212C2×C3×C4—62no—
213C2×C3×C4—61no—
214C2×C3×C4—61no—
215C2×C3×C4—61no—
216C2×C3×C4—182no—
217C2×C3×C4C3×#6711yescharacteristic
218C2×C3×C4C3×#6711yescharacteristic
219C2×C3×C4C22×C3×D311yescharacteristic
220C2×Dic3—62no—
221C2×Dic3—182no—
222Dic6—62no—
223Dic6—62no—
224Dic6—182no—
225Dic6—182no—
226C3×C8—182no—
227C3⋊C8—62no—
228C3⋊C8—182no—
229C2×C32—21no—
230C2×C32—21no—
231C2×C32—21no—
232C2×C32#93011yescharacteristic
233C2×C32C2×#21611yescharacteristic
234C2×C32#95111yescharacteristic
235C2×C32C3×#12111yescharacteristic
236C2×C32C2×C3×Q1611yescharacteristic
237C2×C32C3×#13711yescharacteristic
238C2×Q8—92no—
239C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
240Q16—184no—
241C2×C8—92no—
242C4⋊C4—21no—
243C4⋊C4—181no—
244C4⋊C4—181no—
245C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
246C42—21no—
247C42C2×(C3⋊D3)⋊C311yescharacteristic
248C22×C3—31no—
249C22×C3—31no—
250C22×C3—61no—
251C22×C3C2×C3×#6711yescharacteristic
252C3×C4—22no—
253C3×C4—21no—
254C3×C4—31no—
255C3×C4—31no—
256C3×C4—31no—
257C3×C4—31no—
258C3×C4—32no—
259C3×C4—32no—
260C3×C4—41no—
261C3×C4—61no—
262C3×C4—61no—
263C3×C4—62no—
264C3×C4—62no—
265C3×C4—64no—
266C3×C4—61no—
267C3×C4—61no—
268C3×C4—62no—
269C3×C4—121no—
270C3×C4—121no—
271C3×C4—121no—
272C3×C4—122no—
273C3×C4—362no—
274C3×C4C3×#21612yes—
275C3×C4C2×C3×D1211yescharacteristic
276C3×C4C3×#23511yescharacteristic
277Dic3—122no—
278Dic3—362no—
279C32—21no—
280C32#503111yescharacteristic, complemented
281C32C3×#46211yescharacteristic, complemented
282Q8—21no—
283Q8—182no—
284Q8—182no—
285Q8C4⋊((C3⋊D3)⋊C3)12yes—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—182no—
289C2×C4#638⋊C311yescharacteristic
290C2×C4#638⋊C311yescharacteristic
291C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
292C8—182no—
293C2×C3—31no—
294C2×C3—31no—
295C2×C3—31no—
296C2×C3—31no—
297C2×C3—31no—
298C2×C3—31no—
299C2×C3—61no—
300C2×C3—61no—
301C2×C3—61no—
302C2×C3C3×#93011yescharacteristic
303C2×C3C2×C3×#21611yescharacteristic
304C2×C3C3×#95111yescharacteristic
305C22C2×#638⋊C311yescharacteristic
306C4—22no—
307C4—21no—
308C4—41no—
309C4—362no—
310C4#3756⋊C312yescomplemented
311C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
312C4#3828⋊C311yescharacteristic
313C3—31no—
314C3—31no—
315C3—61no—
316C3C3×#503111yescharacteristic
317C2#64139⋊C311yescharacteristic
318C2C2×#3756⋊C311yescharacteristic
319C2#64160⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC4×Q8×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#462
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
6. #10592730C3semidirect
72. C4×H(3,𝔽3)Q16semidirect
175. C32×C4#216semidirect
176. C32×C4C3×Q16semidirect
188. H(3,𝔽3)#462semidirect
280. C32#5031semidirect
281. C32C3×#462semidirect
310. C4#3756⋊C3semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×H(3,𝔽3)2
C2×#3756⋊C32
#64078⋊C32
#63637⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105927303
C3×#50313