Finite Group Databasebeta

← #13147049

Subgroups — #13147049

Order 1728.

414 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 43 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
1D4×C22⋊H(3,𝔽3)C211yesmaximal, characteristic
2#11722380C211yesmaximal, characteristic, complemented
3#11720736C211yesmaximal, characteristic, complemented
4C3×#5414—33nomaximal
5C3×#5218—31nomaximal
6C22×C22⋊H(3,𝔽3)—21no—
7C4×C22⋊H(3,𝔽3)C2211yescharacteristic
8#88613—21no—
9#88546—41nomaximal
10C22×C32×D4D311yescharacteristic
11C3×D4×A4D313yes—
12C3×A4⋊Q8—33no—
13C2×C3×#214—31no—
14C2×C3×#214—31no—
15C3×#924—31no—
16C3×A4⋊C8—33no—
17C3×#832—31no—
18C3×#832—31no—
19C3×#830—31no—
20C2×C22⋊H(3,𝔽3)—41no—
21C2×C22⋊H(3,𝔽3)D411yescharacteristic
22D4×H(3,𝔽3)—41no—
23#6229—41no—
24#6180—41no—
25#5414—33no—
26C3×#450—91no—
27#5218—31no—
28C24×C32—21no—
29C22×C3×A4—23no—
30C2×C32×D4—31no—
31C2×C32×D4—31no—
32C2×C32×D4—61no—
33C22×C32×C4C2×D311yescharacteristic
34C2×C3×Dic6—31no—
35C3×C4×A4C2×D313yes—
36C3×SL(2,ℤ/4)—63no—
37C3×#217—61no—
38C3×#214—62no—
39C3×#214—61no—
40C3×#214—61no—
41C3×#214—123no—
42C3×C4⋊Dic3—31no—
43C3×Dic3⋊C4—61no—
44C2×C3×C3⋊C8—31no—
45C2×C3×C3⋊C8—31no—
46C22×H(3,𝔽3)—81no—
47C22⋊H(3,𝔽3)SD1611yescharacteristic, complemented
48C4×H(3,𝔽3)—41no—
49#1084—81no—
50C22×C3×D4—31no—
51C22×C3×D4C3⋊D311yescharacteristic
52D4×A4—33no—
53A4⋊Q8—33no—
54C2×C3×SD16—91no—
55C2×C3×SD16—91no—
56C3×C22⋊Q8—91no—
57C2×#214—31no—
58C2×#214—31no—
59#924—31no—
60A4⋊C8—33no—
61C3×#102—91no—
62C3×#102—91no—
63C3×C22⋊C8—91no—
64#832—31no—
65#832—31no—
66#830—31no—
67C23×C32—41no—
68C23×C32—61no—
69C23×C32—61no—
70C23×C32—121no—
71C23×C32#6711yescharacteristic
72C2×C3×A4—43no—
73C2×C3×A4#6713yes—
74C32×D4—32no—
75C32×D4—31no—
76C32×D4—43no—
77C32×D4—61no—
78C32×D4S411yescharacteristic
79C2×C32×C4—31no—
80C2×C32×C4—31no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—61no—
83C3×Dic6—61no—
84C3×Dic6—61no—
85C3×Dic6—123no—
86C3×C3⋊C8—61no—
87C3×C3⋊C8—61no—
88C3×C3⋊C8—123no—
89#450—91noSylow
90C2×H(3,𝔽3)—41no—
91C2×H(3,𝔽3)—161no—
92C24×C3—21no—
93C24×C3—61no—
94C22×A4—63no—
95C2×C3×Q8—91no—
96C2×C3×D4—31no—
97C2×C3×D4—31no—
98C2×C3×D4—31no—
99C2×C3×D4—31no—
100C2×C3×D4—61no—
101C2×C3×D4—61no—
102C2×C3×D4—61no—
103C2×C3×D4—61no—
104C2×C3×D4—62no—
105C22×C3×C4—31no—
106C22×C3×C4C2×C3⋊D311yescharacteristic
107C2×Dic6—31no—
108C4×A4—33no—
109SL(2,ℤ/4)—63no—
110C3×SD16—182no—
111C3×SD16—181no—
112C3×SD16—181no—
113C2×C3×C8—91no—
114C2×C3×C8—91no—
115C3×C4⋊C4—91no—
116C3×C4⋊C4—181no—
117C3×C22⋊C4—181no—
118#217—61no—
119#214—62no—
120#214—61no—
121#214—61no—
122#214—123no—
123C4⋊Dic3—31no—
124Dic3⋊C4—61no—
125C2×C3⋊C8—31no—
126C2×C3⋊C8—31no—
127C22×C32—21no—
128C22×C32—31no—
129C22×C32—31no—
130C22×C32—61no—
131C22×C32—83no—
132C22×C32—121no—
133C22×C32—121no—
134C22×C32—121no—
135C22×C32—121no—
136C22×C32#21411yescharacteristic, complemented
137C3×A4#21413yescomplemented
138C32×C4—31no—
139C32×C4—43no—
140C32×C4C2×S411yescharacteristic
141C3×Dic3—121no—
142C3×Dic3—121no—
143C3×Dic3—243no—
144C22×D4#27211yescharacteristic
145C2×SD16—91no—
146C2×SD16—91no—
147C22⋊Q8—91no—
148#102—91no—
149#102—91no—
150C22⋊C8—91no—
151H(3,𝔽3)—41noSylow
152C23×C3—31no—
153C23×C3—41no—
154C23×C3—61no—
155C23×C3—61no—
156C23×C3—61no—
157C23×C3—61no—
158C23×C3—62no—
159C23×C3—62no—
160C23×C3—121no—
161C23×C3—121no—
162C23×C3—121no—
163C23×C3—122no—
164C23×C3#63811yescharacteristic
165C2×A4—33no—
166C2×A4—123no—
167C3×Q8—181no—
168C3×Q8—181no—
169C3×D4—32no—
170C3×D4—31no—
171C3×D4—32no—
172C3×D4—31no—
173C3×D4—31no—
174C3×D4—61no—
175C3×D4—62no—
176C3×D4—61no—
177C3×D4—61no—
178C3×D4—62no—
179C3×D4—123no—
180C3×D4C3⋊S411yescharacteristic
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
184C2×C3×C4—31no—
185C2×C3×C4—61no—
186C2×C3×C4—61no—
187C2×C3×C4—181no—
188C2×C3×C4—181no—
189C2×Dic3—61no—
190C2×Dic3—61no—
191Dic6—61no—
192Dic6—61no—
193Dic6—123no—
194C3×C8—181no—
195C3×C8—181no—
196C3⋊C8—61no—
197C3⋊C8—61no—
198C3⋊C8—123no—
199C2×C32—31no—
200C2×C32—31no—
201C2×C32—41no—
202C2×C32—43no—
203C2×C32—121no—
204C2×C32—163no—
205C2×C32GL(2,ℤ/4)11yescharacteristic
206C24—21no—
207C2×Q8—91no—
208C2×D4—31no—
209C2×D4—31no—
210C2×D4—61no—
211C22×C4C2×#27211yescharacteristic
212SD16—182no—
213SD16—181no—
214SD16—181no—
215C2×C8—91no—
216C2×C8—91no—
217C4⋊C4—91no—
218C4⋊C4—181no—
219C22⋊C4—181no—
220C22×C3—21no—
221C22×C3—31no—
222C22×C3—31no—
223C22×C3—31no—
224C22×C3—31no—
225C22×C3—31no—
226C22×C3—61no—
227C22×C3—61no—
228C22×C3—61no—
229C22×C3—62no—
230C22×C3—61no—
231C22×C3—61no—
232C22×C3—121no—
233C22×C3—121no—
234C22×C3—121no—
235C22×C3—121no—
236C22×C3—121no—
237C22×C3—121no—
238C22×C3—121no—
239C22×C3—121no—
240C22×C3—122no—
241C22×C3—122no—
242C22×C3—122no—
243C22×C3—122no—
244C22×C3—243no—
245C22×C3#375411yescharacteristic
246A4—33no—
247C3×C4—31no—
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—61no—
251C3×C4—123no—
252C3×C4—361no—
253C3×C4—361no—
254C3×C4C2×C3⋊S411yescharacteristic
255Dic3—121no—
256Dic3—121no—
257Dic3—243no—
258C32—43no—
259C32#541411yescharacteristic, complemented
260C23—41no—
261C23—61no—
262C23—61no—
263C23—121no—
264C23#623511yescharacteristic
265Q8—181no—
266Q8—181no—
267D4—32no—
268D4—31no—
269D4—61no—
270D4#625811yescharacteristic
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—181no—
274C2×C4—181no—
275C8—181no—
276C8—181no—
277C2×C3—31no—
278C2×C3—31no—
279C2×C3—31no—
280C2×C3—31no—
281C2×C3—31no—
282C2×C3—41no—
283C2×C3—61no—
284C2×C3—61no—
285C2×C3—121no—
286C2×C3—122no—
287C2×C3—121no—
288C2×C3—121no—
289C2×C3—123no—
290C2×C3—483no—
291C2×C3#6426911yescharacteristic
292C22—21no—
293C22—31no—
294C22—31no—
295C22—61no—
296C22—121no—
297C22—121no—
298C22—121no—
299C22—121no—
300C22#8854611yescharacteristic, complemented
301C4—31no—
302C4—361no—
303C4—361no—
304C4C2×#625811yescharacteristic
305C3—31no—
306C3—123no—
307C3#1059364811yescharacteristic
308C2—31no—
309C2—31no—
310C2—41no—
311C2—121no—
312C2#1172239811yescharacteristic
subgroupG/H
centerC2×C3#64269
commutator, cosocleC4×C22⋊H(3,𝔽3)C22
FrattiniC3×C4C2×C3⋊S4
FittingC22×C32×D4D3
socleC23×C3#638
prime radicalD4×C22⋊H(3,𝔽3)C2
hypercenterC3×D4C3⋊S4
hypocenterC22⋊H(3,𝔽3)SD16
LCSG ▷ C4×C22⋊H(3,𝔽3) ▷ C2×C22⋊H(3,𝔽3) ▷ C22⋊H(3,𝔽3)
UCSC1 ◁ C2×C3 ◁ C3×C4 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
2. #11722380C2semidirect
3. #11720736C2semidirect
47. C22⋊H(3,𝔽3)SD16semidirect
136. C22×C32#214semidirect
137. C3×A4#214semidirect
259. C32#5414semidirect
300. C22#88546semidirect

Maximal subgroups

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

subgroupindex
D4×C22⋊H(3,𝔽3)2
#117223802
#117207362
C3×#54143
C3×#52183
#885464