Finite Group Databasebeta

← #13147218

Subgroups — #13147218

Order 1728.

413 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 48 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×#88623C211yesmaximal, characteristic
2C4×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
3#11720765C211yesmaximal, characteristic, complemented
4#10596043C311yesmaximal, characteristic, complemented
5#5572⋊C3—31nomaximal
6C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
7#88623C412yescomplemented
8#3752⋊C3—41nomaximal
9C2×Q8×C3⋊D3C2×C311yescharacteristic
10C2×D3×SL(2,𝔽3)—31no—
11C32×C4×Q8C2×C311yescharacteristic, complemented
12Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
13#64101—31no—
14#64096—31no—
15#64081—31no—
16C4×#634—31no—
17C3×C4×SL(2,𝔽3)—21no—
18C3×C4×SL(2,𝔽3)D311yescharacteristic, complemented
19Dic3×SL(2,𝔽3)—31no—
20C22×(C3⋊D3)⋊C3—41no—
21C2×C4×H(3,𝔽3)—41no—
22C2×(C3⋊Dic3)⋊C3—41no—
23Q8⋊H(3,𝔽3)C2×C411yescharacteristic
24#5572—31no—
25#5572—91no—
26#519⋊C3—91no—
27C2×C32×Q8C22×C311yescharacteristic
28Q8×C3⋊D3—32no—
29Q8×C3⋊D3C3×C412yescomplemented
30C2×C4×C3⋊D3—31no—
31C2×#634—31no—
32C2×C3×SL(2,𝔽3)—21no—
33C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
34D3×SL(2,𝔽3)—32no—
35C32×C4⋊C4—31no—
36C32×C42—31no—
37#3752—31no—
38#3752A411yescharacteristic
39C4⋊(C3⋊Dic3)—31no—
40#3750—31no—
41#3750—31no—
42C4×C3⋊Dic3—32no—
43C4×C3⋊Dic3—31no—
44C3×(C2×D3)⋊C4—121no—
45C22×H(3,𝔽3)—41no—
46C2×(C3⋊D3)⋊C3—42no—
47C2×(C3⋊D3)⋊C3—81no—
48C2×(C3⋊D3)⋊C3—81no—
49C4×H(3,𝔽3)—81no—
50(C3⋊Dic3)⋊C3—81no—
51C2×D3×Q8—31no—
52C2×D3×Q8—91no—
53C22×SL(2,𝔽3)—91no—
54C3×C4×Q8—31no—
55C3×C4×Q8C3×D311yescharacteristic
56Q8×Dic3—31no—
57Q8×Dic3—91no—
58#892—93no—
59#892—91no—
60#887—93no—
61#887—91no—
62#872—91no—
63#872—93no—
64C4×Dic6—93no—
65C4×Dic6—91no—
66C4×SL(2,𝔽3)—31no—
67C4×SL(2,𝔽3)—61no—
68C4×SL(2,𝔽3)—91no—
69C22×C3⋊D3C2×A411yescharacteristic
70C22×C3×D3—121no—
71C32×Q8—31no—
72C32×Q8C2×C3×C411yescharacteristic
73C2×C32×C4—31no—
74C2×C32×C4—31no—
75C2×C32×C4—41no—
76C2×C32×C4—81no—
77C2×C32×C4C2×A411yescharacteristic
78C2×C3⋊Dic3—31no—
79C2×C3⋊Dic3—31no—
80C2×C3⋊Dic3C2×A411yescharacteristic
81C4×C3⋊D3—32no—
82C4×C3⋊D3—32no—
83#634—32no—
84#634—32no—
85C2×C3×Dic3—121no—
86C3×SL(2,𝔽3)—21no—
87C3×SL(2,𝔽3)C4×D311yescharacteristic
88#519—91noSylow
89C2×H(3,𝔽3)—41no—
90C2×H(3,𝔽3)—41no—
91C2×H(3,𝔽3)—41no—
92(C3⋊D3)⋊C3—82no—
93C2×C3×Q8—31no—
94C2×C3×Q8C2×C3×D311yescharacteristic
95D3×Q8—32no—
96D3×Q8—92no—
97D3×Q8—96no—
98D3×Q8—92no—
99C2×C4×D3—91no—
100C2×C4×D3—93no—
101C2×Dic6—93no—
102C2×Dic6—91no—
103C2×SL(2,𝔽3)—31no—
104C2×SL(2,𝔽3)—61no—
105C2×SL(2,𝔽3)—92no—
106C3×C4⋊C4—31no—
107C3×C4⋊C4—33no—
108C3×C22⋊C4—361no—
109C3×C42—33no—
110C3×C42—31no—
111(C2×D3)⋊C4—31no—
112(C2×D3)⋊C4—91no—
113(C2×D3)⋊C4—91no—
114(C2×D3)⋊C4—93no—
115C4⋊Dic3—93no—
116C4⋊Dic3—91no—
117Dic3⋊C4—91no—
118Dic3⋊C4—93no—
119Dic3⋊C4—93no—
120Dic3⋊C4—91no—
121C4×Dic3—91no—
122C4×Dic3—96no—
123C4×Dic3—92no—
124C4×Dic3—93no—
125C22×C32—41no—
126C22×C32—81no—
127C22×C32C22×A411yescharacteristic
128C2×C3⋊D3—21no—
129C2×C3⋊D3—21no—
130C2×C3⋊D3C4×A412yes—
131C2×C3×D3—122no—
132C2×C3×D3—241no—
133C2×C3×D3—241no—
134C32×C4—21no—
135C32×C4—31no—
136C32×C4—31no—
137C32×C4—61no—
138C32×C4—81no—
139C32×C4—82no—
140C3⋊Dic3—21no—
141C3⋊Dic3—32no—
142C3⋊Dic3—61no—
143C3×Dic3—241no—
144C22×Q8—91no—
145C4×Q8—91no—
146C4×Q8—271no—
147C4×Q8—271no—
148C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
149#117—271no—
150#117—271no—
151H(3,𝔽3)—41noSylow
152C23×C3—361no—
153C22×D3—31no—
154C22×D3—91no—
155C3×Q8—31no—
156C3×Q8—33no—
157C3×Q8—31no—
158C3×Q8C3×C4×D311yescharacteristic
159C2×C3×C4—33no—
160C2×C3×C4—31no—
161C2×C3×C4—33no—
162C2×C3×C4—31no—
163C2×C3×C4—31no—
164C2×C3×C4—121no—
165C2×C3×C4—241no—
166C2×C3×C4—361no—
167C2×C3×C4D3×A411yescharacteristic
168C2×Dic3—31no—
169C2×Dic3—91no—
170C2×Dic3—93no—
171C2×Dic3—91no—
172C2×Dic3—93no—
173C2×Dic3—91no—
174C4×D3—92no—
175C4×D3—92no—
176C4×D3—96no—
177C4×D3—96no—
178Dic6—96no—
179Dic6—96no—
180Dic6—92no—
181Dic6—92no—
182SL(2,𝔽3)—31no—
183SL(2,𝔽3)—61no—
184C2×C32—41no—
185C2×C32—41no—
186C2×C32—41no—
187C2×C32—81no—
188C2×C32—81no—
189C2×C32—81no—
190C2×C32C2×C4×A411yescharacteristic
191C2×C322−1+4⋊C311yescharacteristic
192C2×C322−1+4⋊C311yescharacteristic
193C3⋊D3—22no—
194C3×D3—242no—
195C2×Q8—92no—
196C2×Q8—271no—
197C2×Q8—272no—
198C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
199C22×C4—271no—
200C4⋊C4—31no—
201C4⋊C4—271no—
202C4⋊C4—271no—
203C4⋊C4—271no—
204C22⋊C4—91no—
205C22⋊C4—271no—
206C42—31no—
207C42—271no—
208C42—272no—
209C22×C3—31no—
210C22×C3—121no—
211C22×C3—241no—
212C22×C3—362no—
213C22×C3—721no—
214C22×C3—721no—
215C22×C3C2×D3×A411yescharacteristic
216C2×D3—32no—
217C2×D3—61no—
218C2×D3—61no—
219C2×D3—92no—
220C2×D3—181no—
221C2×D3—181no—
222C3×C4—21no—
223C3×C4—33no—
224C3×C4—33no—
225C3×C4—31no—
226C3×C4—31no—
227C3×C4—61no—
228C3×C4—63no—
229C3×C4—61no—
230C3×C4—241no—
231C3×C4—242no—
232C3×C4—721no—
233Dic3—61no—
234Dic3—96no—
235Dic3—92no—
236Dic3—181no—
237Dic3—183no—
238Dic3—181no—
239C32—41no—
240C32—81no—
241C32#519⋊C311yescharacteristic, complemented
242C23—91no—
243Q8—31no—
244Q8—272no—
245Q8—272no—
246Q8C4×(C3⋊D3)⋊C311yescharacteristic
247C2×C4—31no—
248C2×C4—31no—
249C2×C4—91no—
250C2×C4—272no—
251C2×C4—272no—
252C2×C4—271no—
253C2×C4—271no—
254C2×C4(C3⋊D3)⋊A411yescharacteristic
255C2×C3—31no—
256C2×C3—31no—
257C2×C3—31no—
258C2×C3—121no—
259C2×C3—121no—
260C2×C3—121no—
261C2×C3—241no—
262C2×C3—241no—
263C2×C3—241no—
264C2×C3—722no—
265C2×C3C4×D3×A411yescharacteristic
266C2×C3#1007⋊C311yescharacteristic
267C2×C3#1010⋊C311yescharacteristic
268D3—62no—
269D3—182no—
270C22—92no—
271C22—181no—
272C22—181no—
273C22C2×(C3⋊D3)⋊A411yescharacteristic
274C4—21no—
275C4—31no—
276C4—31no—
277C4—61no—
278C4—181no—
279C4—272no—
280C4—541no—
281C3—31no—
282C3—121no—
283C3—241no—
284C3#5572⋊C311yescharacteristic
285C2—182no—
286C2C4×(C3⋊D3)⋊A411yescharacteristic
287C2#1172243211yescharacteristic
288C2#1172243811yescharacteristic
subgroupG/H
centerC22C2×(C3⋊D3)⋊A4
commutator, hypocenterC32×Q8C2×C3×C4
Frattini, socleC22×C3C2×D3×A4
FittingC32×C4×Q8C2×C3
prime radicalC2×#88623C2
cosocleC2×C32×Q8C22×C3
hypercenterC2×C4(C3⋊D3)⋊A4
LCSG ▷ C32×Q8
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C4×Q8⋊H(3,𝔽3)C2semidirect
3. #11720765C2semidirect
4. #10596043C3semidirect
7. #88623C4semidirect
11. C32×C4×Q8C2×C3semidirect
12. Q8×C3⋊Dic3C2×C3semidirect
18. C3×C4×SL(2,𝔽3)D3semidirect
29. Q8×C3⋊D3C3×C4semidirect
148. C4×Q8(C3⋊D3)⋊C3semidirect
241. C32#519⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#886232
C4×Q8⋊H(3,𝔽3)2
#117207652
#105960433
#5572⋊C33
#3752⋊C34