Finite Group Databasebeta

← #13139249

Subgroups — #13139249

Order 1728.

341 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 57 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
1D8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2#11720851C211yesmaximal, characteristic, complemented
3#11720566C211yesmaximal, characteristic, complemented
4#63654⋊C3C211yesmaximal, characteristic, complemented
5#11720255C211yesmaximal, characteristic, complemented
6#11720249C211yesmaximal, characteristic, complemented
7((C3⋊D3)⋊C3)⋊C16C211yesmaximal, characteristic, complemented
8#10591029—31nomaximal
9D3×#827—31nomaximal
10D4×(C3⋊D3)⋊C3—21no—
11((C3⋊D3)⋊C3)⋊Q8—21no—
12D8×H(3,𝔽3)C2211yescharacteristic
13#88530C2211yescharacteristic, complemented
14#3753⋊C3—21no—
15#3745⋊C3C2211yescharacteristic
16C8×(C3⋊D3)⋊C3C2211yescharacteristic
17#3746⋊C3C2211yescharacteristic, complemented
18#88439—21no—
19#88386C2211yescharacteristic, complemented
20(C3⋊(C3⋊C16))⋊C3C2211yescharacteristic, complemented
21D8×C3⋊D3D311yescharacteristic
22C3×D3×D8—31no—
23#63802—31no—
24D3×Dic12—31no—
25#63654D311yescharacteristic, complemented
26C3×#827—31no—
27C3×#827—31no—
28C3×#827—31no—
29C3×#827—61no—
30#63549—31no—
31#63549—31no—
32#63548—31no—
33#63547—31no—
34#63542—31no—
35D3×C3⋊C16—31no—
36C22×(C3⋊D3)⋊C3—41no—
37D4×H(3,𝔽3)—21no—
38#6229—21no—
39#638⋊C3—41no—
40C4⋊((C3⋊D3)⋊C3)—21no—
41C4×(C3⋊D3)⋊C3D411yescharacteristic
42#634⋊C3—21no—
43((C3⋊D3)⋊C3)⋊C4—41no—
44#6196—41no—
45C8×H(3,𝔽3)C2311yescharacteristic
46(C3⋊(C3⋊C8))⋊C3D411yescharacteristic
47C2×#827—91no—
48D3×SD32—91no—
49D3×SD32—91no—
50D4×C3⋊D3—21no—
51C3×D3×D4—61no—
52#3797—61no—
53D3×Dic6—61no—
54C32×D8—21no—
55C32×D8C2×D311yescharacteristic
56C32×D8C2×D311yescharacteristic
57#3753—21no—
58#3746C2×D311yescharacteristic, complemented
59#3745C2×D311yescharacteristic
60C8×C3⋊D3C2×D311yescharacteristic
61C3×#213—61no—
62C3×Dic12—31no—
63C3×Dic12—31no—
64C3×Dic12—31no—
65C3×Dic12—61no—
66C3×D24—31no—
67C3×D3×C8—31no—
68#3719—61no—
69#3718—61no—
70C3⋊(C3⋊C16)C2×D311yescharacteristic, complemented
71C3×C3⋊C16—31no—
72C3×C3⋊C16—31no—
73C3×C3⋊C16—31no—
74C3×C3⋊C16—61no—
75C22×H(3,𝔽3)—41no—
76C2×(C3⋊D3)⋊C3—41no—
77C2×(C3⋊D3)⋊C3—82no—
78C2×(C3⋊D3)⋊C3D811yescharacteristic
79C4×H(3,𝔽3)C2×D411yescharacteristic
80#1084—41no—
81(C3⋊Dic3)⋊C3—41no—
82(C3⋊Dic3)⋊C3D811yescharacteristic
83C2×C3×D8—91no—
84D3×Q16—91no—
85D3×Q16—91no—
86D3×D8—31no—
87D3×D8—91no—
88C2×Dic12—91no—
89C3×SD32—91no—
90C3×SD32—91no—
91#828—91no—
92#828—91no—
93#827—31no—
94#827—31no—
95#827—31no—
96#827—61no—
97#827—91no—
98#827—91no—
99#827—91no—
100C2×C3⋊C16—91no—
101#800—91no—
102#800—91no—
103D3×C16—91no—
104D3×C16—91no—
105C22×C3⋊D3—41no—
106C22×C3×D3—121no—
107C32×D4—21no—
108C32×D4—21no—
109C32×D4—41no—
110#638—41no—
111C4⋊(C3⋊D3)—21no—
112C4×C3⋊D3#6711yescharacteristic
113#634—21no—
114C3×#67—121no—
115C3×D12—61no—
116C3×C4×D3—31no—
117C3×Dic6—61no—
118C3×Dic6—61no—
119C3×Dic6—61no—
120C3×Dic6—121no—
121#627—121no—
122#627—121no—
123#624—121no—
124D3×Dic3—121no—
125C32×C8—21no—
126C32×C8C22×D311yescharacteristic
127C32×C8C22×D311yescharacteristic
128C3⋊(C3⋊C8)#6711yescharacteristic
129C3×C3⋊C8—31no—
130C2×SD32—271noSylow
131C2×H(3,𝔽3)—81no—
132C2×H(3,𝔽3)C2×D811yescharacteristic
133(C3⋊D3)⋊C3—81no—
134(C3⋊D3)⋊C3SD3212yescomplemented
135C2×C3×D4—181no—
136D3×Q8—181no—
137D3×Q8—181no—
138D3×D4—61no—
139D3×D4—181no—
140C2×Dic6—181no—
141C3×Q16—91no—
142C3×Q16—91no—
143C3×D8—31no—
144C3×D8—31no—
145C3×D8—61no—
146C3×D8—91no—
147C3×D8—181no—
148C3×D8D3211yescharacteristic
149C2×C3×C8—91no—
150#216—181no—
151#216—181no—
152#213—61no—
153#213—181no—
154Dic12—31no—
155Dic12—31no—
156Dic12—31no—
157Dic12—61no—
158Dic12—91no—
159Dic12—181no—
160D24—31no—
161D24—91no—
162D3×C8—31no—
163D3×C8—91no—
164C3×C16—91no—
165C3×C16—91no—
166C3⋊C16—31no—
167C3⋊C16—31no—
168C3⋊C16—31no—
169C3⋊C16—61no—
170C3⋊C16—91no—
171C22×C32—41no—
172C22×C32—41no—
173C22×C32—81no—
174C2×C3⋊D3—41no—
175C2×C3⋊D3—82no—
176C2×C3⋊D3#21311yescharacteristic
177C2×C3×D3—31no—
178C2×C3×D3—121no—
179C2×C3×D3—242no—
180C32×C4—21no—
181C32×C4D3×D411yescharacteristic
182C32×C4C2×#6711yescharacteristic
183C3⋊Dic3—41no—
184C3⋊Dic3#21311yescharacteristic
185C3×Dic3—31no—
186C3×Dic3—121no—
187C3×Dic3—121no—
188C3×Dic3—121no—
189C3×Dic3—241no—
190C2×Q16—271no—
191C2×D8—91no—
192SD32—91no—
193SD32—91no—
194SD32—271no—
195SD32—271no—
196C2×C16—271no—
197H(3,𝔽3)C2×SD3211yescharacteristic, complemented, Sylow
198C23×C3—361no—
199C22×D3—121no—
200C22×D3—361no—
201C3×Q8—181no—
202C3×Q8—181no—
203C3×D4—21no—
204C3×D4—61no—
205C3×D4—61no—
206C3×D4—121no—
207C3×D4—181no—
208C3×D4—361no—
209C2×C3×C4—91no—
210#67—121no—
211#67—361no—
212C2×Dic3—361no—
213D12—61no—
214D12—181no—
215C4×D3—31no—
216C4×D3—91no—
217C4×D3—361no—
218C4×D3—361no—
219Dic6—61no—
220Dic6—61no—
221Dic6—61no—
222Dic6—121no—
223Dic6—181no—
224Dic6—361no—
225Dic6—361no—
226Dic6—361no—
227C3×C8—31no—
228C3×C8—31no—
229C3×C8—61no—
230C3×C8—91no—
231C3×C8C2×D3211yescharacteristic
232C3⋊C8—31no—
233C3⋊C8—91no—
234C2×C32—21no—
235C2×C32—81no—
236C2×C32—81no—
237C2×C32—161no—
238C2×C32D3×D811yescharacteristic
239C2×C32C2×#21311yescharacteristic
240C3⋊D3—81no—
241C3⋊D3#82712yescomplemented
242C3×D3—32no—
243C3×D3—241no—
244C2×Q8—541no—
245C2×D4—181no—
246Q16—91no—
247Q16—91no—
248Q16—541no—
249D8—91no—
250D8—181no—
251D8((C3⋊D3)⋊C3)⋊C211yescharacteristic
252C2×C8—91no—
253C16—91no—
254C16—91no—
255C22×C3—41no—
256C22×C3—91no—
257C22×C3—121no—
258C22×C3—121no—
259C22×C3—241no—
260C22×C3—361no—
261C22×C3—722no—
262C2×D3—31no—
263C2×D3—91no—
264C2×D3—121no—
265C2×D3—242no—
266C2×D3—361no—
267C2×D3—722no—
268C3×C4—31no—
269C3×C4—31no—
270C3×C4—61no—
271C3×C4—91no—
272C3×C4—361no—
273C3×C4—361no—
274C3×C4D3×#6711yescharacteristic
275Dic3—31no—
276Dic3—91no—
277Dic3—121no—
278Dic3—121no—
279Dic3—121no—
280Dic3—241no—
281Dic3—361no—
282C32—21no—
283C32D3×SD3211yescharacteristic, complemented
284C32C2×#82711yescharacteristic, complemented
285C23—361no—
286Q8—181no—
287Q8—181no—
288Q8—1081no—
289D4—21no—
290D4—181no—
291D4—361no—
292C2×C4—91no—
293C2×C4—1081no—
294C8—91no—
295C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
296C2×C3—31no—
297C2×C3—31no—
298C2×C3—61no—
299C2×C3—81no—
300C2×C3—92no—
301C2×C3—241no—
302C2×C3—241no—
303C2×C3—481no—
304C2×C3—721no—
305C2×C3D3×#21311yescharacteristic
306D3—32no—
307D3—92no—
308D3—241no—
309D3—721no—
310C22—41no—
311C22—91no—
312C22—361no—
313C22—722no—
314C4—91no—
315C4—361no—
316C4—361no—
317C4#8867611yescharacteristic
318C3—31no—
319C3—31no—
320C3—61no—
321C3D3×#82711yescharacteristic
322C2—81no—
323C2—92no—
324C2—721no—
325C2#1172119411yescharacteristic
subgroupG/H
centerC2#11721194
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C2×D32
FittingD8×H(3,𝔽3)C22
socleC2×C3D3×#213
prime radicalD8×(C3⋊D3)⋊C3C2
hypercenterD8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×SD32
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C4 ◁ C8 ◁ D8

Complements

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

NH ≅ G/Ntype
2. #11720851C2semidirect
3. #11720566C2semidirect
4. #63654⋊C3C2semidirect
5. #11720255C2semidirect
6. #11720249C2semidirect
7. ((C3⋊D3)⋊C3)⋊C16C2semidirect
13. #88530C22semidirect
17. #3746⋊C3C22semidirect
19. #88386C22semidirect
20. (C3⋊(C3⋊C16))⋊C3C22semidirect
25. #63654D3semidirect
58. #3746C2×D3semidirect
70. C3⋊(C3⋊C16)C2×D3semidirect
134. (C3⋊D3)⋊C3SD32semidirect
197. H(3,𝔽3)C2×SD32semidirect
241. C3⋊D3#827semidirect
283. C32D3×SD32semidirect
284. C32C2×#827semidirect

Maximal subgroups

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

subgroupindex
D8×(C3⋊D3)⋊C32
#117208512
#117205662
#63654⋊C32
#117202552
#117202492
((C3⋊D3)⋊C3)⋊C162
#105910293
D3×#8273