Finite Group Databasebeta

← #10593650⋊C3 #13146902

Subgroups — #10593650⋊C3 #13146902

Order 1728.

308 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 38 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
1Q8×C22⋊H(3,𝔽3)C211yesmaximal, characteristic
2#64259⋊C3C211yesmaximal, characteristic
3(C3⋊(A4⋊C8))⋊C3C211yesmaximal, characteristic
4#10593650C311yesmaximal, characteristic, complemented
5C3×#5228—31nomaximal
6C4×C22⋊H(3,𝔽3)—21no—
7C4×C22⋊H(3,𝔽3)C2211yescharacteristic
8(C3⋊SL(2,ℤ/4))⋊C3—21no—
9#3756⋊C3—41nomaximal
10C22×C32×Q8D311yescharacteristic
11C3×Q8×A4—21no—
12C3×Q8×A4C2×C311yescharacteristic
13#64259C2×C311yescharacteristic
14C2×C3×#216—31no—
15C2×C3×#216—31no—
16C3×#924—31no—
17C3⋊(A4⋊C8)C2×C311yescharacteristic
18C3×#835—31no—
19C3×#835—31no—
20C3×#830—31no—
21C2×C22⋊H(3,𝔽3)D411yescharacteristic
22Q8×H(3,𝔽3)—41no—
23#634⋊C3—41no—
24(C3⋊(C3⋊C8))⋊C3—41no—
25#5416—91no—
26C3×#451—91no—
27#5228—31no—
28C2×C32×Q8—31no—
29C2×C32×Q8—31no—
30C2×C32×Q8—61no—
31C22×C32×C4—21no—
32C22×C32×C4C2×D311yescharacteristic
33C2×C3×Dic6—31no—
34C3×C4×A4—22no—
35C3×C4×A4—21no—
36C3×C4×A4—21no—
37C3×C4×A4C22×C311yescharacteristic
38C3⋊SL(2,ℤ/4)—21no—
39#3756—41no—
40C3×#217—61no—
41C3×#216—62no—
42C3×#216—61no—
43C3×#216—61no—
44C3×C4⋊Dic3—31no—
45C3×Dic3⋊C4—61no—
46C2×C3×C3⋊C8—31no—
47C2×C3×C3⋊C8—31no—
48C22⋊H(3,𝔽3)Q1611yescharacteristic, complemented
49C4×H(3,𝔽3)—41no—
50C4×H(3,𝔽3)—81no—
51(C3⋊Dic3)⋊C3—81no—
52C22×C3×Q8—31no—
53C22×C3×Q8C3×D311yescharacteristic
54Q8×A4—31no—
55Q8×A4—61no—
56A4⋊Q8—91no—
57C2×C3×Q16—91no—
58C2×C3×Q16—91no—
59C3×C22⋊Q8—91no—
60C2×#216—31no—
61C2×#216—31no—
62#924—31no—
63A4⋊C8—91no—
64C3×#103—91no—
65C3×#103—91no—
66C3×C22⋊C8—91no—
67#835—31no—
68#835—31no—
69#830—31no—
70C23×C32#6711yescharacteristic
71C2×C3×A4—21no—
72C2×C3×A4C3×D411yescharacteristic
73C32×Q8—32no—
74C32×Q8—31no—
75C32×Q8—41no—
76C32×Q8—61no—
77C32×Q8—81no—
78C32×Q8S411yescharacteristic
79C2×C32×C4—31no—
80C2×C32×C4—31no—
81C2×C32×C4—61no—
82C2×C32×C4—61no—
83#634—41no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C3×Dic6—61no—
87C3×Dic6—61no—
88C3⋊(C3⋊C8)—41no—
89C3×C3⋊C8—61no—
90C3×C3⋊C8—61no—
91#451—91noSylow
92C2×H(3,𝔽3)—41no—
93C2×C3×Q8—31no—
94C2×C3×Q8—31no—
95C2×C3×Q8—31no—
96C2×C3×Q8—31no—
97C2×C3×Q8—61no—
98C2×C3×Q8—61no—
99C2×C3×Q8—61no—
100C2×C3×Q8—61no—
101C2×C3×Q8—62no—
102C2×C3×Q8—91no—
103C22×C3×C4—21no—
104C22×C3×C4—31no—
105C22×C3×C4—61no—
106C22×C3×C4C2×C3×D311yescharacteristic
107C2×Dic6—31no—
108C4×A4—31no—
109C4×A4—61no—
110C4×A4—62no—
111C4×A4—61no—
112SL(2,ℤ/4)—181no—
113C3×Q16—182no—
114C3×Q16—181no—
115C3×Q16—181no—
116C2×C3×C8—91no—
117C2×C3×C8—91no—
118C3×C4⋊C4—91no—
119C3×C4⋊C4—181no—
120C3×C22⋊C4—181no—
121#217—61no—
122#216—62no—
123#216—61no—
124#216—61no—
125#216—361no—
126C4⋊Dic3—31no—
127Dic3⋊C4—61no—
128C2×C3⋊C8—31no—
129C2×C3⋊C8—31no—
130C22×C32—31no—
131C22×C32—31no—
132C22×C32#21611yescharacteristic, complemented
133C3×A4—21no—
134C3×A4C3×Q1611yescharacteristic, complemented
135C32×C4—21no—
136C32×C4—31no—
137C32×C4—41no—
138C32×C4—61no—
139C32×C4—81no—
140C32×C4—82no—
141C32×C4—81no—
142C32×C4C2×S411yescharacteristic
143C3⋊Dic3—81no—
144C3×Dic3—121no—
145C3×Dic3—121no—
146C22×Q8(C3⋊D3)⋊C311yescharacteristic
147C2×Q16—91no—
148C2×Q16—91no—
149C22⋊Q8—91no—
150#103—91no—
151#103—91no—
152C22⋊C8—91no—
153H(3,𝔽3)—41noSylow
154C23×C3—31no—
155C23×C3C3×#6711yescharacteristic
156C2×A4—31no—
157C2×A4—61no—
158C3×Q8—32no—
159C3×Q8—32no—
160C3×Q8—31no—
161C3×Q8—31no—
162C3×Q8—31no—
163C3×Q8—61no—
164C3×Q8—62no—
165C3×Q8—61no—
166C3×Q8—61no—
167C3×Q8—62no—
168C3×Q8—121no—
169C3×Q8—181no—
170C3×Q8—181no—
171C3×Q8—241no—
172C3×Q8C3×S411yescharacteristic
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—61no—
178C2×C3×C4—61no—
179C2×C3×C4—61no—
180C2×C3×C4—61no—
181C2×C3×C4—61no—
182C2×C3×C4—62no—
183C2×C3×C4—62no—
184C2×C3×C4—61no—
185C2×C3×C4—181no—
186C2×C3×C4—181no—
187C2×Dic3—61no—
188C2×Dic3—61no—
189Dic6—61no—
190Dic6—61no—
191Dic6—361no—
192C3×C8—181no—
193C3×C8—181no—
194C3⋊C8—61no—
195C3⋊C8—61no—
196C3⋊C8—361no—
197C2×C32—31no—
198C2×C32—31no—
199C2×C32—41no—
200C2×C32—81no—
201C2×C32GL(2,ℤ/4)11yescharacteristic
202C2×Q8—31no—
203C2×Q8—31no—
204C2×Q8—61no—
205C2×Q8—91no—
206C22×C4—21no—
207C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
208Q16—182no—
209Q16—181no—
210Q16—181no—
211C2×C8—91no—
212C2×C8—91no—
213C4⋊C4—91no—
214C4⋊C4—181no—
215C22⋊C4—181no—
216C22×C3—31no—
217C22×C3—31no—
218C22×C3—31no—
219C22×C3—31no—
220C22×C3—31no—
221C22×C3—61no—
222C22×C3—61no—
223C22×C3C3×#21611yescharacteristic
224A4—31no—
225A4—61no—
226C3×C4—21no—
227C3×C4—31no—
228C3×C4—31no—
229C3×C4—31no—
230C3×C4—61no—
231C3×C4—61no—
232C3×C4—62no—
233C3×C4—61no—
234C3×C4—61no—
235C3×C4—121no—
236C3×C4—241no—
237C3×C4—242no—
238C3×C4—241no—
239C3×C4—361no—
240C3×C4—361no—
241C3×C4C2×C3×S411yescharacteristic
242Dic3—121no—
243Dic3—121no—
244Dic3—721no—
245C32—41no—
246C32—81no—
247C32#541611yescharacteristic, complemented
248C23#638⋊C311yescharacteristic
249Q8—32no—
250Q8—31no—
251Q8—61no—
252Q8—181no—
253Q8—181no—
254Q8(C3⋊S4)⋊C311yescharacteristic
255C2×C4—31no—
256C2×C4—31no—
257C2×C4—61no—
258C2×C4—61no—
259C2×C4—181no—
260C2×C4—181no—
261C8—181no—
262C8—181no—
263C2×C3—31no—
264C2×C3—31no—
265C2×C3—31no—
266C2×C3—31no—
267C2×C3—31no—
268C2×C3—61no—
269C2×C3—61no—
270C2×C3—121no—
271C2×C3—241no—
272C2×C3C3×GL(2,ℤ/4)11yescharacteristic
273C22—31no—
274C22—31no—
275C22#3756⋊C311yescharacteristic, complemented
276C4—21no—
277C4—31no—
278C4—61no—
279C4—361no—
280C4—361no—
281C4C2×(C3⋊S4)⋊C311yescharacteristic
282C3—31no—
283C3—121no—
284C3—241no—
285C3C3×#541611yescharacteristic
286C2—31no—
287C2—31no—
288C2#64269⋊C311yescharacteristic
subgroupG/H
centerC2#64269⋊C3
commutator, cosocleC3×C4×A4C22×C3
FrattiniC3×C4C2×C3×S4
FittingC22×C32×Q8D3
socleC23×C3C3×#67
prime radicalC2×C22⋊H(3,𝔽3)D4
hypercenterQ8(C3⋊S4)⋊C3
hypocenterC3×A4C3×Q16
LCSG ▷ C3×C4×A4 ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
4. #10593650C3semidirect
48. C22⋊H(3,𝔽3)Q16semidirect
132. C22×C32#216semidirect
134. C3×A4C3×Q16semidirect
247. C32#5416semidirect
275. C22#3756⋊C3semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊H(3,𝔽3)2
#64259⋊C32
(C3⋊(A4⋊C8))⋊C32
#105936503
C3×#52283
#3756⋊C34