Finite Group Databasebeta

← #10593651⋊C3 #13146903

Subgroups — #10593651⋊C3 #13146903

Order 1728.

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

normal subgroups only

This group has 334 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, complemented
2C4⋊((C3⋊S4)⋊C3)C211yesmaximal, characteristic
3(C3⋊(A4⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4#10593651C311yesmaximal, characteristic, complemented
5C3×#5227—31nomaximal
6C2×(C3⋊S4)⋊C3—21no—
7C4×C22⋊H(3,𝔽3)—21no—
8C4×C22⋊H(3,𝔽3)C2211yescharacteristic
9#3755⋊C3—41nomaximal
10C22×C32×Q8D311yescharacteristic, complemented
11C3×Q8×A4—21no—
12C3×Q8×A4C2×C311yescharacteristic, complemented
13C4⋊(C3⋊S4)C2×C311yescharacteristic
14C2×C3×#215—31no—
15C2×C3×#215—31no—
16C3×#930—31no—
17C3⋊(A4⋊C8)C2×C311yescharacteristic, complemented
18C3×#835—31no—
19C3×#835—31no—
20C3×#830—31no—
21C2×C22⋊H(3,𝔽3)D411yescharacteristic
22(C3⋊S4)⋊C3—41no—
23Q8×H(3,𝔽3)—41no—
24C4⋊((C3⋊D3)⋊C3)—41no—
25(C3⋊(C3⋊C8))⋊C3—41no—
26#5417—91no—
27C3×#448—91no—
28#5227—31no—
29C2×C3⋊S4—21no—
30C2×C32×Q8—31no—
31C2×C32×Q8—31no—
32C2×C32×Q8—61no—
33C22×C32×C4—21no—
34C22×C32×C4C2×D311yescharacteristic
35C2×C3×#67—61no—
36C2×C3×D12—31no—
37C3×C4×A4—22no—
38C3×C4×A4—21no—
39C3×C4×A4—21no—
40C3×C4×A4C22×C311yescharacteristic
41#3755—41no—
42C3×#215—62no—
43C3×#215—61no—
44C3×#215—61no—
45C3×(C2×D3)⋊C4—61no—
46C3×C4⋊Dic3—31no—
47C2×C3×C3⋊C8—31no—
48C2×C3×C3⋊C8—31no—
49C2×(C3⋊D3)⋊C3—81no—
50C22⋊H(3,𝔽3)SD1611yescharacteristic, complemented
51C4×H(3,𝔽3)—41no—
52C4×H(3,𝔽3)—81no—
53C22×C3×Q8—31no—
54C22×C3×Q8C3×D311yescharacteristic
55Q8×A4—31no—
56Q8×A4—61no—
57C4⋊S4—91no—
58C2×C3×SD16—91no—
59C2×C3×SD16—91no—
60C3×#121—91no—
61C2×#215—31no—
62C2×#215—31no—
63#930—31no—
64A4⋊C8—91no—
65C3×#103—91no—
66C3×#103—91no—
67C3×C22⋊C8—91no—
68#835—31no—
69#835—31no—
70#830—31no—
71C23×C32#6711yescharacteristic
72C22×C3×D3—61no—
73C2×C3×A4—21no—
74C2×C3×A4C3×D411yescharacteristic
75C3⋊S4—41no—
76C32×Q8—32no—
77C32×Q8—31no—
78C32×Q8—41no—
79C32×Q8—61no—
80C32×Q8—81no—
81C32×Q8S411yescharacteristic, complemented
82C2×C32×C4—31no—
83C2×C32×C4—31no—
84C2×C32×C4—61no—
85C2×C32×C4—61no—
86C4⋊(C3⋊D3)—41no—
87C3×#67—121no—
88C3×#67—121no—
89C2×C3×Dic3—61no—
90C3×D12—61no—
91C3×D12—61no—
92C3⋊(C3⋊C8)—41no—
93C3×C3⋊C8—61no—
94C3×C3⋊C8—61no—
95#448—91noSylow
96C2×H(3,𝔽3)—41no—
97(C3⋊D3)⋊C3—161no—
98C2×S4—181no—
99C2×C3×Q8—31no—
100C2×C3×Q8—31no—
101C2×C3×Q8—31no—
102C2×C3×Q8—31no—
103C2×C3×Q8—61no—
104C2×C3×Q8—61no—
105C2×C3×Q8—61no—
106C2×C3×Q8—61no—
107C2×C3×Q8—62no—
108C2×C3×D4—91no—
109C2×C3×D4—181no—
110C22×C3×C4—21no—
111C22×C3×C4—31no—
112C22×C3×C4—61no—
113C22×C3×C4C2×C3×D311yescharacteristic
114C2×#67—61no—
115C2×D12—31no—
116C4×A4—31no—
117C4×A4—61no—
118C4×A4—62no—
119C4×A4—61no—
120C3×SD16—182no—
121C3×SD16—181no—
122C3×SD16—181no—
123C2×C3×C8—91no—
124C2×C3×C8—91no—
125C3×C4⋊C4—91no—
126C3×C22⋊C4—181no—
127#215—62no—
128#215—61no—
129#215—61no—
130#215—361no—
131(C2×D3)⋊C4—61no—
132C4⋊Dic3—31no—
133C2×C3⋊C8—31no—
134C2×C3⋊C8—31no—
135C22×C32—31no—
136C22×C32—31no—
137C22×C32#21511yescharacteristic, complemented
138C2×C3⋊D3—81no—
139C2×C3×D3—121no—
140C2×C3×D3—121no—
141C2×C3×D3—121no—
142C3×A4—21no—
143C3×A4C3×SD1611yescharacteristic, complemented
144C32×C4—21no—
145C32×C4—31no—
146C32×C4—41no—
147C32×C4—61no—
148C32×C4—81no—
149C32×C4—82no—
150C32×C4—81no—
151C32×C4C2×S411yescharacteristic
152C3×Dic3—121no—
153C22×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
154C2×SD16—91no—
155C2×SD16—91no—
156#121—91no—
157#103—91no—
158#103—91no—
159C22⋊C8—91no—
160H(3,𝔽3)—41noSylow
161C23×C3—31no—
162C23×C3—181no—
163C23×C3C3×#6711yescharacteristic
164C22×D3—61no—
165C2×A4—31no—
166C2×A4—61no—
167S4—361no—
168C3×Q8—32no—
169C3×Q8—32no—
170C3×Q8—31no—
171C3×Q8—31no—
172C3×Q8—31no—
173C3×Q8—61no—
174C3×Q8—62no—
175C3×Q8—61no—
176C3×Q8—61no—
177C3×Q8—62no—
178C3×Q8—121no—
179C3×Q8—241no—
180C3×Q8C3×S411yescharacteristic
181C3×D4—181no—
182C3×D4—181no—
183C3×D4—361no—
184C3×D4—361no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—61no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—61no—
193C2×C3×C4—61no—
194C2×C3×C4—62no—
195C2×C3×C4—62no—
196C2×C3×C4—61no—
197C2×C3×C4—181no—
198#67—121no—
199#67—121no—
200C2×Dic3—61no—
201D12—61no—
202D12—61no—
203D12—361no—
204C3×C8—181no—
205C3×C8—181no—
206C3⋊C8—61no—
207C3⋊C8—61no—
208C3⋊C8—361no—
209C2×C32—31no—
210C2×C32—31no—
211C2×C32—41no—
212C2×C32—81no—
213C2×C32GL(2,ℤ/4)11yescharacteristic
214C3⋊D3—161no—
215C3×D3—241no—
216C2×Q8—31no—
217C2×Q8—31no—
218C2×Q8—61no—
219C2×D4—91no—
220C2×D4—181no—
221C22×C4—21no—
222C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
223SD16—182no—
224SD16—181no—
225SD16—181no—
226C2×C8—91no—
227C2×C8—91no—
228C4⋊C4—91no—
229C22⋊C4—181no—
230C22×C3—31no—
231C22×C3—31no—
232C22×C3—31no—
233C22×C3—31no—
234C22×C3—31no—
235C22×C3—61no—
236C22×C3—61no—
237C22×C3—361no—
238C22×C3—361no—
239C22×C3—361no—
240C22×C3C3×#21511yescharacteristic
241C2×D3—121no—
242C2×D3—121no—
243C2×D3—121no—
244C2×D3—721no—
245A4—31no—
246A4—61no—
247C3×C4—21no—
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—61no—
252C3×C4—61no—
253C3×C4—62no—
254C3×C4—61no—
255C3×C4—61no—
256C3×C4—121no—
257C3×C4—241no—
258C3×C4—242no—
259C3×C4—241no—
260C3×C4—361no—
261C3×C4C2×C3×S411yescharacteristic
262Dic3—121no—
263C32—41no—
264C32—81no—
265C32#541711yescharacteristic, complemented
266C23—181no—
267C23#638⋊C311yescharacteristic
268Q8—32no—
269Q8—31no—
270Q8—61no—
271Q8(C3⋊S4)⋊C311yescharacteristic, complemented
272D4—181no—
273D4—181no—
274D4—361no—
275D4—361no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—61no—
279C2×C4—61no—
280C2×C4—181no—
281C8—181no—
282C8—181no—
283C2×C3—31no—
284C2×C3—31no—
285C2×C3—31no—
286C2×C3—31no—
287C2×C3—31no—
288C2×C3—61no—
289C2×C3—61no—
290C2×C3—121no—
291C2×C3—241no—
292C2×C3—721no—
293C2×C3C3×GL(2,ℤ/4)11yescharacteristic
294D3—241no—
295D3—1441no—
296C22—31no—
297C22—31no—
298C22—361no—
299C22—361no—
300C22—361no—
301C22#3755⋊C311yescharacteristic, complemented
302C4—21no—
303C4—31no—
304C4—61no—
305C4—361no—
306C4C2×(C3⋊S4)⋊C311yescharacteristic
307C3—31no—
308C3—121no—
309C3—241no—
310C3C3×#541711yescharacteristic
311C2—31no—
312C2—31no—
313C2—721no—
314C2#64269⋊C311yescharacteristic
subgroupG/H
centerC2#64269⋊C3
commutator, cosocleC3×C4×A4C22×C3
FrattiniC3×C4C2×C3×S4
FittingC22×C32×Q8D3
socleC23×C3C3×#67
prime radicalC4⋊((C3⋊S4)⋊C3)C2
hypercenterQ8(C3⋊S4)⋊C3
hypocenterC3×A4C3×SD16
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
1. Q8×C22⋊H(3,𝔽3)C2semidirect
3. (C3⋊(A4⋊C8))⋊C3C2semidirect
4. #10593651C3semidirect
10. C22×C32×Q8D3semidirect
12. C3×Q8×A4C2×C3semidirect
17. C3⋊(A4⋊C8)C2×C3semidirect
50. C22⋊H(3,𝔽3)SD16semidirect
81. C32×Q8S4semidirect
137. C22×C32#215semidirect
143. C3×A4C3×SD16semidirect
153. C22×Q8(C3⋊D3)⋊C3semidirect
265. C32#5417semidirect
271. Q8(C3⋊S4)⋊C3semidirect
301. C22#3755⋊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
C4⋊((C3⋊S4)⋊C3)2
(C3⋊(A4⋊C8))⋊C32
#105936513
C3×#52273
#3755⋊C34