Finite Group Databasebeta

← #13147281

Subgroups — #13147281

Order 1728.

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

normal subgroups only

This group has 457 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
1C22×Q8⋊H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#88622C211yesmaximal, characteristic, complemented
3#11720765C211yesmaximal, characteristic, complemented
4#10596288C311yesmaximal, characteristic, complemented
5#5817⋊C3—31nomaximal
6C2×Q8⋊H(3,𝔽3)—21no—
7C2×Q8⋊H(3,𝔽3)C2211yescharacteristic
8#88622—21no—
9#3757⋊C3—41nomaximal
10C22×C32×Q8C2×C311yescharacteristic
11C2×#3832C2×C311yescharacteristic, complemented
12C22×C3×SL(2,𝔽3)—21no—
13C22×C3×SL(2,𝔽3)D311yescharacteristic, complemented
14C2×C3⋊(#41⋊C3)—31no—
15#64156—31no—
16#64155—31no—
17Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
18#64139—31no—
19C4×#638—31no—
20Dic3×SL(2,𝔽3)—31no—
21C23×H(3,𝔽3)—41no—
22C2×(C3⋊Dic3)⋊C3—41no—
23C2×(C3⋊Dic3)⋊C3—41no—
24Q8⋊H(3,𝔽3)D411yescharacteristic, complemented
25#5817—31no—
26#5817—91no—
27#548⋊C3—91no—
28C2×C32×Q8—21no—
29C2×C32×Q8—31no—
30C2×C32×Q8—61no—
31C2×C32×Q8C22×C311yescharacteristic
32C22×C32×C4—31no—
33C2×#638—31no—
34#3832—21no—
35#3832—61no—
36C2×C4⋊(C3⋊D3)—31no—
37C2×C4×C3⋊D3—31no—
38C2×C3×SL(2,𝔽3)—22no—
39C2×C3×SL(2,𝔽3)—21no—
40C2×C3×SL(2,𝔽3)—21no—
41C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
42C3⋊(#41⋊C3)—61no—
43#3757A411yescharacteristic
44#3752—32no—
45#3752—31no—
46C4⋊(C3⋊Dic3)—31no—
47#3750—31no—
48C4×C3⋊Dic3—31no—
49C3×#217—121no—
50C22×H(3,𝔽3)—41no—
51C22×H(3,𝔽3)—42no—
52C22×H(3,𝔽3)—81no—
53C22×H(3,𝔽3)—81no—
54(C3⋊Dic3)⋊C3—81no—
55(C3⋊Dic3)⋊C3—81no—
56C22×C3×Q8—31no—
57C22×C3×Q8C3×D311yescharacteristic
58C2×#239—31no—
59C2×#239—91no—
60C2×#41⋊C3—91no—
61C22×SL(2,𝔽3)—31no—
62C22×SL(2,𝔽3)—61no—
63#947—93no—
64#947—91no—
65#946—93no—
66#946—91no—
67Q8×Dic3—31no—
68Q8×Dic3—91no—
69#930—93no—
70#930—91no—
71C4×#67—91no—
72C4×#67—93no—
73C4×SL(2,𝔽3)—91no—
74C23×C32—41no—
75C23×C32—81no—
76C23×C32C2×A411yescharacteristic
77C22×C3⋊D3—31no—
78C32×Q8—31no—
79C32×Q8—61no—
80C32×Q8—61no—
81C32×Q8C3×D411yescharacteristic
82C2×C32×C4—31no—
83C2×C32×C4—31no—
84C2×C32×C4—61no—
85C2×C32×C4—61no—
86#638—62no—
87C2×C3⋊Dic3—31no—
88C2×C3⋊Dic3C2×A411yescharacteristic
89C2×C3⋊Dic3C2×A411yescharacteristic
90C4⋊(C3⋊D3)—61no—
91C4⋊(C3⋊D3)—61no—
92C4×C3⋊D3—61no—
93C4×C3⋊D3—61no—
94C2×C3×Dic3—121no—
95C2×C3×Dic3—121no—
96C3×SL(2,𝔽3)—21no—
97C3×SL(2,𝔽3)#6711yescharacteristic, complemented
98#548—91noSylow
99C2×H(3,𝔽3)—41no—
100C2×H(3,𝔽3)—41no—
101C2×H(3,𝔽3)—41no—
102C2×H(3,𝔽3)—82no—
103C2×C3×Q8—21no—
104C2×C3×Q8—33no—
105C2×C3×Q8—31no—
106C2×C3×Q8—31no—
107C2×C3×Q8—61no—
108C2×C3×Q8—63no—
109C2×C3×Q8—61no—
110C2×C3×Q8C2×C3×D311yescharacteristic
111C22×C3×C4—33no—
112C22×C3×C4—31no—
113C2×#67—93no—
114C2×#67—91no—
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—181no—
124C2×SL(2,𝔽3)—31no—
125C2×SL(2,𝔽3)—61no—
126C2×SL(2,𝔽3)—62no—
127C2×SL(2,𝔽3)—61no—
128C3×C22⋊C4—361no—
129#217—31no—
130#217—91no—
131(C2×D3)⋊C4—91no—
132(C2×D3)⋊C4—93no—
133(C2×D3)⋊C4—96no—
134(C2×D3)⋊C4—92no—
135C4⋊Dic3—93no—
136C4⋊Dic3—91no—
137Dic3⋊C4—91no—
138Dic3⋊C4—93no—
139C4×Dic3—91no—
140C4×Dic3—93no—
141C22×C32—21no—
142C22×C32—21no—
143C22×C32—41no—
144C22×C32—42no—
145C22×C32—81no—
146C22×C32—81no—
147C22×C32—82no—
148C22×C32—82no—
149C22×C32—81no—
150C22×C32—82no—
151C22×C32#41⋊C312yescomplemented
152C22×C32C22×A411yescharacteristic
153C2×C3⋊D3—61no—
154C2×C3⋊D3—61no—
155C2×C3⋊D3—61no—
156C32×C4—31no—
157C32×C4—31no—
158C32×C4—61no—
159C3⋊Dic3—21no—
160C3⋊Dic3—21no—
161C3⋊Dic3—61no—
162C3×Dic3—241no—
163C3×Dic3—241no—
164C2×#41—91no—
165C22×Q8(C3⋊D3)⋊C311yescharacteristic
166#124—271no—
167C22⋊Q8—271no—
168#121—271no—
169C4×Q8—91no—
170C4×D4—271no—
171H(3,𝔽3)—41noSylow
172C23×C3—31no—
173C23×C3—121no—
174C23×C3—241no—
175C23×C3D3×A411yescharacteristic
176C22×D3—93no—
177C22×D3—91no—
178C3×Q8—31no—
179C3×Q8—33no—
180C3×Q8—31no—
181C3×Q8—61no—
182C3×Q8—61no—
183C3×Q8—63no—
184C3×Q8—63no—
185C3×Q8C3×#6711yescharacteristic
186C2×C3×C4—33no—
187C2×C3×C4—31no—
188C2×C3×C4—33no—
189C2×C3×C4—31no—
190C2×C3×C4—61no—
191C2×C3×C4—61no—
192C2×C3×C4—63no—
193C2×C3×C4—63no—
194C2×C3×C4—361no—
195C2×C3×C4—361no—
196#67—186no—
197#67—182no—
198C2×Dic3—31no—
199C2×Dic3—31no—
200C2×Dic3—91no—
201C2×Dic3—91no—
202C2×Dic3—93no—
203C2×Dic3—91no—
204D12—181no—
205D12—181no—
206D12—183no—
207D12—183no—
208C4×D3—181no—
209C4×D3—181no—
210C4×D3—183no—
211C4×D3—183no—
212SL(2,𝔽3)—31no—
213SL(2,𝔽3)—61no—
214C2×C32—22no—
215C2×C32—41no—
216C2×C32—41no—
217C2×C32—41no—
218C2×C32—82no—
219C2×C32—84no—
220C2×C32—81no—
221C2×C32—81no—
222C2×C32—81no—
223C2×C32D4×A411yescharacteristic
224C2×C32C2×#41⋊C311yescharacteristic
225C2×C322−1+4⋊C311yescharacteristic
226C3⋊D3—121no—
227#41—181no—
228#41—541no—
229C2×Q8—21no—
230C2×Q8—31no—
231C2×Q8—61no—
232C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
233C2×D4—271no—
234C2×D4—271no—
235C22×C4—31no—
236C22×C4—271no—
237C4⋊C4—271no—
238C4⋊C4—271no—
239C22⋊C4—91no—
240C22⋊C4—271no—
241C22⋊C4—272no—
242C42—271no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—32no—
246C22×C3—31no—
247C22×C3—61no—
248C22×C3—61no—
249C22×C3—122no—
250C22×C3—121no—
251C22×C3—241no—
252C22×C3—241no—
253C22×C3—242no—
254C22×C3—242no—
255C22×C3—242no—
256C22×C3—241no—
257C22×C3C3⋊(#41⋊C3)12yes—
258C22×C3C2×D3×A411yescharacteristic
259C2×D3—183no—
260C2×D3—183no—
261C2×D3—183no—
262C2×D3—181no—
263C2×D3—181no—
264C2×D3—181no—
265C3×C4—33no—
266C3×C4—33no—
267C3×C4—31no—
268C3×C4—31no—
269C3×C4—63no—
270C3×C4—61no—
271C3×C4—721no—
272C3×C4—721no—
273Dic3—61no—
274Dic3—61no—
275Dic3—181no—
276Dic3—181no—
277Dic3—183no—
278Dic3—181no—
279C32—41no—
280C32—81no—
281C32#548⋊C311yescharacteristic, complemented
282C23—271no—
283C23(C3⋊D3)⋊A411yescharacteristic
284Q8—31no—
285Q8—61no—
286Q8—61no—
287Q8#638⋊C311yescharacteristic
288D4—541no—
289D4—541no—
290D4—542no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—61no—
294C2×C4—61no—
295C2×C4—91no—
296C2×C4—91no—
297C2×C4—271no—
298C2×C4—541no—
299C2×C4—541no—
300C2×C3—22no—
301C2×C3—31no—
302C2×C3—31no—
303C2×C3—31no—
304C2×C3—62no—
305C2×C3—121no—
306C2×C3—121no—
307C2×C3—121no—
308C2×C3—242no—
309C2×C3—244no—
310C2×C3—241no—
311C2×C3—241no—
312C2×C3—241no—
313C2×C3C2×C3⋊(#41⋊C3)11yescharacteristic
314C2×C3#1007⋊C311yescharacteristic
315C2×C3A4×#6711yescharacteristic
316D3—363no—
317D3—361no—
318C22—21no—
319C22—21no—
320C22—541no—
321C22—541no—
322C22—541no—
323C22#8862212yescomplemented
324C22C2×(C3⋊D3)⋊A411yescharacteristic
325C4—31no—
326C4—31no—
327C4—61no—
328C4—181no—
329C4—181no—
330C4—541no—
331C3—31no—
332C3—121no—
333C3—241no—
334C3#5817⋊C311yescharacteristic
335C2—22no—
336C2—1081no—
337C2C2×#8862211yescharacteristic
338C2#1172243211yescharacteristic
339C2#638⋊A411yescharacteristic
subgroupG/H
centerC22C2×(C3⋊D3)⋊A4
commutator, cosocleC2×C32×Q8C22×C3
Frattini, socleC22×C3C2×D3×A4
FittingC22×C32×Q8C2×C3
hypercenterC23(C3⋊D3)⋊A4
hypocenterC32×Q8C3×D4
LCSG ▷ C2×C32×Q8 ▷ C32×Q8
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×Q8⋊H(3,𝔽3)C2semidirect
2. C2×#88622C2semidirect
3. #11720765C2semidirect
4. #10596288C3semidirect
11. C2×#3832C2×C3semidirect
13. C22×C3×SL(2,𝔽3)D3semidirect
17. Q8×C3⋊Dic3C2×C3semidirect
24. Q8⋊H(3,𝔽3)D4semidirect
97. C3×SL(2,𝔽3)#67semidirect
151. C22×C32#41⋊C3semidirect
281. C32#548⋊C3semidirect
323. C22#88622semidirect

Maximal subgroups

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

subgroupindex
C22×Q8⋊H(3,𝔽3)2
C2×#886222
#117207652
#105962883
#5817⋊C33
#3757⋊C34