Finite Group Databasebeta

← #13147293

Subgroups — #13147293

Order 1728.

443 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 443 conjugacy classes of subgroups — too many to draw. Only its 44 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×#41⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#88622C211yesmaximal, characteristic, complemented
3#11720765C211yesmaximal, characteristic, complemented
4#10596305C311yesmaximal, characteristic, complemented
5#5834⋊C3—31nomaximal
6#41⋊H(3,𝔽3)—21no—
7C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
8#88622—21no—
9#3750⋊C3—41nomaximal
10C2×C32×#41C2×C311yescharacteristic
11C2×#3832C2×C311yescharacteristic
12C2×C3×#41⋊C3—21no—
13C2×C3×#41⋊C3D311yescharacteristic, complemented
14C2×C3⋊(#41⋊C3)—31no—
15Q8×C3⋊Dic3C2×C311yescharacteristic
16#64149—31no—
17#64147—31no—
18#64139—31no—
19C4×#638—31no—
20Dic3×SL(2,𝔽3)—31no—
21C2×C4×H(3,𝔽3)—41no—
22C2×(C3⋊Dic3)⋊C3—41no—
23C2×(C3⋊Dic3)⋊C3—41no—
24Q8⋊H(3,𝔽3)D411yescharacteristic, complemented
25#5834—31no—
26#5834—91no—
27C4⋊(#41⋊C3)—91no—
28C32×#41—21no—
29C32×#41—61no—
30C2×C32×Q8C22×C311yescharacteristic
31C2×C32×D4—31no—
32C22×C32×C4—31no—
33C2×#638—32no—
34C2×#638—31no—
35#3832—21no—
36#3832—61no—
37C2×C4⋊(C3⋊D3)—31no—
38C2×C4×C3⋊D3—31no—
39C3×#41⋊C3—22no—
40C3×#41⋊C3—21no—
41C2×C3×SL(2,𝔽3)—21no—
42C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
43C3⋊(#41⋊C3)—61no—
44#3757—31no—
45#3752—31no—
46C4⋊(C3⋊Dic3)—31no—
47#3750A411yescharacteristic
48C4×C3⋊Dic3—31no—
49C3×Dic3⋊C4—121no—
50C22×H(3,𝔽3)—41no—
51C4×H(3,𝔽3)—81no—
52(C3⋊Dic3)⋊C3—42no—
53(C3⋊Dic3)⋊C3—81no—
54C2×C3×#41—31no—
55C2×C3×#41C3×D311yescharacteristic
56C2×#239—31no—
57C2×#239—91no—
58C2×#41⋊C3—31no—
59C2×#41⋊C3—61no—
60C2×#41⋊C3—91no—
61Q8×Dic3—31no—
62Q8×Dic3—91no—
63#940—93no—
64#940—91no—
65#938—93no—
66#938—91no—
67#930—93no—
68#930—91no—
69C4×#67—91no—
70C4×#67—93no—
71C4×SL(2,𝔽3)—91no—
72C23×C32—31no—
73C22×C3⋊D3—31no—
74C32×Q8—31no—
75C32×Q8C3×D411yescharacteristic
76C32×D4—61no—
77C32×D4—61no—
78C2×C32×C4—31no—
79C2×C32×C4—41no—
80C2×C32×C4—61no—
81C2×C32×C4—61no—
82C2×C32×C4—81no—
83C2×C32×C4C2×A411yescharacteristic
84#638—62no—
85#638—122no—
86C2×C3⋊Dic3—31no—
87C2×C3⋊Dic3C2×A411yescharacteristic
88C2×C3⋊Dic3C2×A411yescharacteristic
89C4⋊(C3⋊D3)—61no—
90C4⋊(C3⋊D3)—61no—
91C4×C3⋊D3—61no—
92C4×C3⋊D3—61no—
93C2×C3×Dic3—121no—
94C2×C3×Dic3—121no—
95C3×SL(2,𝔽3)—21no—
96C3×SL(2,𝔽3)#6711yescharacteristic, complemented
97#550—91noSylow
98C2×H(3,𝔽3)—41no—
99C2×H(3,𝔽3)—41no—
100C2×H(3,𝔽3)—41no—
101C3×#41—21no—
102C3×#41—61no—
103C3×#41—63no—
104C3×#41—61no—
105C2×C3×Q8—31no—
106C2×C3×Q8C2×C3×D311yescharacteristic
107C2×C3×D4—31no—
108C2×C3×D4—33no—
109C22×C3×C4—33no—
110C22×C3×C4—31no—
111C2×#67—91no—
112C2×#67—93no—
113C2×#67—96no—
114C2×#67—92no—
115#239—61no—
116#239—181no—
117#239—183no—
118#239—181no—
119C2×D12—91no—
120C2×D12—93no—
121C2×C4×D3—91no—
122C2×C4×D3—93no—
123#41⋊C3—61no—
124#41⋊C3—62no—
125#41⋊C3—181no—
126C2×SL(2,𝔽3)—31no—
127C2×SL(2,𝔽3)—61no—
128C3×C4⋊C4—361no—
129#217—91no—
130#217—93no—
131(C2×D3)⋊C4—93no—
132(C2×D3)⋊C4—91no—
133C4⋊Dic3—93no—
134C4⋊Dic3—91no—
135Dic3⋊C4—31no—
136Dic3⋊C4—91no—
137C4×Dic3—91no—
138C4×Dic3—93no—
139C22×C32—41no—
140C22×C32—61no—
141C22×C32—61no—
142C22×C32—61no—
143C22×C32—81no—
144C22×C32C22×A411yescharacteristic
145C2×C3⋊D3—61no—
146C2×C3⋊D3—61no—
147C2×C3⋊D3—61no—
148C32×C4—21no—
149C32×C4—31no—
150C32×C4—31no—
151C32×C4—81no—
152C32×C4—82no—
153C3⋊Dic3—21no—
154C3⋊Dic3—61no—
155C3⋊Dic3#41⋊C312yescomplemented
156C3×Dic3—122no—
157C3×Dic3—241no—
158C2×#41—91no—
159C2×#41(C3⋊D3)⋊C311yescharacteristic
160#127—271no—
161#121—271no—
162#121—271no—
163C4×Q8—91no—
164C4×D4—271no—
165H(3,𝔽3)—41noSylow
166C23×C3—33no—
167C23×C3—31no—
168C22×D3—93no—
169C22×D3—91no—
170C3×Q8—31no—
171C3×Q8—33no—
172C3×Q8—31no—
173C3×Q8C3×#6711yescharacteristic
174C3×D4—61no—
175C3×D4—61no—
176C3×D4—63no—
177C3×D4—63no—
178C2×C3×C4—33no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—61no—
182C2×C3×C4—61no—
183C2×C3×C4—63no—
184C2×C3×C4—63no—
185C2×C3×C4—121no—
186C2×C3×C4—241no—
187C2×C3×C4—361no—
188C2×C3×C4—361no—
189C2×C3×C4D3×A411yescharacteristic
190#67—182no—
191#67—186no—
192#67—366no—
193#67—362no—
194C2×Dic3—31no—
195C2×Dic3—31no—
196C2×Dic3—91no—
197C2×Dic3—91no—
198C2×Dic3—93no—
199C2×Dic3—91no—
200D12—181no—
201D12—181no—
202D12—183no—
203D12—183no—
204C4×D3—181no—
205C4×D3—181no—
206C4×D3—183no—
207C4×D3—183no—
208SL(2,𝔽3)—31no—
209SL(2,𝔽3)—61no—
210C2×C32—41no—
211C2×C32—41no—
212C2×C32—41no—
213C2×C32—81no—
214C2×C32—81no—
215C2×C32—81no—
216C2×C32—121no—
217C2×C32D4×A411yescharacteristic
218C2×C32C2×#41⋊C311yescharacteristic
219C2×C32#99411yescharacteristic
220C3⋊D3—121no—
221#41—21no—
222#41—61no—
223#41—181no—
224#41—541no—
225C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
226C2×D4—31no—
227C2×D4—271no—
228C2×D4—271no—
229C2×D4—272no—
230C22×C4—31no—
231C22×C4—271no—
232C4⋊C4—91no—
233C4⋊C4—271no—
234C22⋊C4—271no—
235C22⋊C4—271no—
236C42—271no—
237C22×C3—31no—
238C22×C3—63no—
239C22×C3—63no—
240C22×C3—63no—
241C22×C3—61no—
242C22×C3—61no—
243C22×C3—61no—
244C22×C3—121no—
245C22×C3—241no—
246C22×C3C2×D3×A411yescharacteristic
247C2×D3—183no—
248C2×D3—183no—
249C2×D3—183no—
250C2×D3—181no—
251C2×D3—181no—
252C2×D3—181no—
253C3×C4—21no—
254C3×C4—33no—
255C3×C4—33no—
256C3×C4—31no—
257C3×C4—31no—
258C3×C4—61no—
259C3×C4—241no—
260C3×C4—242no—
261C3×C4—362no—
262C3×C4—721no—
263Dic3—32no—
264Dic3—61no—
265Dic3—92no—
266Dic3—181no—
267Dic3—183no—
268Dic3—181no—
269C32—41no—
270C32—81no—
271C32C4⋊(#41⋊C3)11yescharacteristic, complemented
272C23—31no—
273C23—271no—
274Q8—31no—
275Q8#638⋊C311yescharacteristic
276D4—61no—
277D4—61no—
278D4—541no—
279D4—541no—
280D4—542no—
281D4—1082no—
282C2×C4—31no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—91no—
286C2×C4—91no—
287C2×C4—271no—
288C2×C4—541no—
289C2×C4—541no—
290C2×C4(C3⋊D3)⋊A411yescharacteristic
291C2×C3—31no—
292C2×C3—31no—
293C2×C3—31no—
294C2×C3—123no—
295C2×C3—121no—
296C2×C3—121no—
297C2×C3—121no—
298C2×C3—121no—
299C2×C3—241no—
300C2×C3—241no—
301C2×C3—241no—
302C2×C3C3⋊#99411yescharacteristic
303C2×C3D3×#41⋊C311yescharacteristic
304C2×C3A4×#6711yescharacteristic
305D3—363no—
306D3—361no—
307C22—61no—
308C22—61no—
309C22—61no—
310C22—541no—
311C22—541no—
312C22—541no—
313C22C2×(C3⋊D3)⋊A411yescharacteristic
314C4—21no—
315C4—31no—
316C4—31no—
317C4—92no—
318C4—181no—
319C4—541no—
320C3—31no—
321C3—121no—
322C3—241no—
323C3#5834⋊C311yescharacteristic
324C2—121no—
325C2—1081no—
326C2#1172243611yescharacteristic
327C2#1172243711yescharacteristic
328C2#638⋊A411yescharacteristic
subgroupG/H
centerC22C2×(C3⋊D3)⋊A4
commutator, cosocleC2×C32×Q8C22×C3
Frattini, socleC22×C3C2×D3×A4
FittingC2×C32×#41C2×C3
hypercenterC2×C4(C3⋊D3)⋊A4
hypocenterC32×Q8C3×D4
LCSG ▷ C2×C32×Q8 ▷ C32×Q8
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#41⋊H(3,𝔽3)C2semidirect
2. C2×#88622C2semidirect
3. #11720765C2semidirect
4. #10596305C3semidirect
13. C2×C3×#41⋊C3D3semidirect
24. Q8⋊H(3,𝔽3)D4semidirect
96. C3×SL(2,𝔽3)#67semidirect
155. C3⋊Dic3#41⋊C3semidirect
271. C32C4⋊(#41⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C2×#41⋊H(3,𝔽3)2
C2×#886222
#117207652
#105963053
#5834⋊C33
#3750⋊C34