Finite Group Databasebeta

← #13138735

Subgroups — #13138735

Order 1728.

377 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 65 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
1#11721779C211yesmaximal, characteristic, complemented
2C2×#3745⋊C3C211yesmaximal, characteristic, complemented
3#63639⋊C3C211yesmaximal, characteristic, complemented
4#11720255C212yesmaximal, complemented
5#11720253C212yesmaximal, complemented
6#10590634—31nomaximal
7#10590630—31nomaximal
8#88740—21no—
9C2×C4⋊((C3⋊D3)⋊C3)—21no—
10C2×C8×H(3,𝔽3)C2211yescharacteristic
11#88530C2211yescharacteristic, complemented
12#88529C2211yescharacteristic, complemented
13#88528—21no—
14#3745⋊C3—21no—
15#3745⋊C3C2212yes—
16(C3⋊(C3⋊C16))⋊C3C2212yescomplemented
17C2×#3745D311yescharacteristic, complemented
18C3×#904—31no—
19C3×#904—31no—
20C3×#904—61no—
21C2×C3×D24—31no—
22#63639D311yescharacteristic, complemented
23C3×#812—31no—
24#63549—32no—
25#63547—32no—
26#63546—32no—
27#63545—32no—
28C22×(C3⋊D3)⋊C3—41no—
29C2×C4×H(3,𝔽3)D411yescharacteristic
30#6235—41no—
31C4⋊#272—21no—
32C4×#272—41no—
33#6229—21no—
34C4⋊((C3⋊D3)⋊C3)—22no—
35C4⋊((C3⋊D3)⋊C3)—41no—
36C8×H(3,𝔽3)D411yescharacteristic
37C8×H(3,𝔽3)C2311yescharacteristic
38#5193—91no—
39#5147—91no—
40#4908—91no—
41C2×C4⋊(C3⋊D3)—21no—
42C3×#235—62no—
43C3×#235—61no—
44C3×#235—61no—
45C2×C3×D12—61no—
46C2×C32×C8—21no—
47C2×C32×C8C2×D311yescharacteristic
48C2×C32×C8C2×D311yescharacteristic
49#3745—21no—
50#3745C2×D312yes—
51C3×Dic12—31no—
52C3×Dic12—31no—
53C3×Dic12—61no—
54C3×D24—32no—
55C3×D24—31no—
56C3×D24—31no—
57C3×D24—61no—
58C3×D24—61no—
59C3×#204—62no—
60C3×#204—61no—
61C3×#204—61no—
62C3⋊(C3⋊C16)C2×D312yescomplemented
63C3×C3⋊C16—32no—
64C22×H(3,𝔽3)D811yescharacteristic
65C2×#272—41no—
66C2×(C3⋊D3)⋊C3—42no—
67C2×(C3⋊D3)⋊C3—82no—
68C4×H(3,𝔽3)D811yescharacteristic
69C4×H(3,𝔽3)C2×D411yescharacteristic
70#1084—41no—
71C3×#135—91no—
72C2×C3×D8—91no—
73#904—31no—
74#904—31no—
75#904—61no—
76C2×D24—31no—
77C2×D24—91no—
78C3×M32—91no—
79#828—92no—
80#827—92no—
81#826—92no—
82#826—92no—
83#812—31no—
84#812—91no—
85#800—92no—
86D48—92no—
87C22×C3⋊D3—41no—
88C22×C3×D3—121no—
89C2×C32×C4—21no—
90C2×C32×C4D1211yescharacteristic
91C2×C32×C4#6711yescharacteristic
92C4⋊(C3⋊D3)—22no—
93C4⋊(C3⋊D3)—41no—
94C3×#67—122no—
95C3×#67—121no—
96C3×#67—121no—
97C3×D12—62no—
98C3×D12—62no—
99C3×D12—61no—
100C3×D12—61no—
101C3×D12—121no—
102C3×C4×D3—122no—
103C3×C4×D3—121no—
104C3×C4×D3—121no—
105C3×Dic6—62no—
106C3×Dic6—61no—
107C3×Dic6—61no—
108C32×C8—21no—
109C32×C8—21no—
110C32×C8D1211yescharacteristic
111C32×C8#6711yescharacteristic
112C32×C8C22×D311yescharacteristic
113C32×C8C22×D311yescharacteristic
114#509—271noSylow
115C2×H(3,𝔽3)—21no—
116C2×H(3,𝔽3)C2×D811yescharacteristic
117#272—81no—
118(C3⋊D3)⋊C3—82no—
119C3×#41—181no—
120C2×C3×D4—181no—
121#235—61no—
122#235—61no—
123#235—62no—
124C2×D12—61no—
125C2×D12—181no—
126C3×Q16—91no—
127C3×SD16—181no—
128C3×D8—92no—
129C3×D8—91no—
130C3×D8—181no—
131C2×C3×C8—31no—
132C2×C3×C8—31no—
133C2×C3×C8—61no—
134C2×C3×C8D3211yescharacteristic
135Dic12—31no—
136Dic12—31no—
137Dic12—61no—
138D24—31no—
139D24—31no—
140D24—32no—
141D24—61no—
142D24—61no—
143D24—92no—
144D24—181no—
145#204—61no—
146#204—61no—
147#204—62no—
148C3×C16—92no—
149C3⋊C16—32no—
150C3⋊C16—92no—
151C22×C32—21no—
152C22×C32D2411yescharacteristic
153C22×C32#21311yescharacteristic
154C2×C3⋊D3—42no—
155C2×C3⋊D3—82no—
156C2×C3×D3—122no—
157C2×C3×D3—122no—
158C2×C3×D3—121no—
159C2×C3×D3—121no—
160C2×C3×D3—242no—
161C32×C4—21no—
162C32×C4—21no—
163C32×C4D2411yescharacteristic
164C32×C4#21311yescharacteristic
165C32×C4C2×D1211yescharacteristic
166C32×C4C2×#6711yescharacteristic
167C3×Dic3—122no—
168C3×Dic3—121no—
169C3×Dic3—121no—
170#135—91no—
171C2×D8—91no—
172SD32—272no—
173D16—272no—
174M32—91no—
175H(3,𝔽3)#50911yescharacteristic, complemented, Sylow
176C23×C3—361no—
177C22×D3—121no—
178C22×D3—361no—
179C3×Q8—181no—
180C3×D4—182no—
181C3×D4—181no—
182C3×D4—361no—
183C3×D4—361no—
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—61no—
187C2×C3×C4—361no—
188C2×C3×C4#62611yescharacteristic
189#67—121no—
190#67—121no—
191#67—122no—
192D12—61no—
193D12—61no—
194D12—62no—
195D12—62no—
196D12—121no—
197D12—182no—
198D12—361no—
199C4×D3—121no—
200C4×D3—121no—
201C4×D3—122no—
202Dic6—61no—
203Dic6—61no—
204Dic6—62no—
205C3×C8—31no—
206C3×C8—31no—
207C3×C8—31no—
208C3×C8—31no—
209C3×C8—61no—
210C3×C8—61no—
211C3×C8#62611yescharacteristic
212C3×C8C2×D3211yescharacteristic
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—41no—
217C2×C32C2×D2411yescharacteristic
218C2×C32C2×#21311yescharacteristic
219C3⋊D3—82no—
220C3×D3—242no—
221C3×D3—241no—
222C3×D3—242no—
223C3×D3—241no—
224#41—181no—
225C2×D4—181no—
226Q16—91no—
227SD16—181no—
228D8—91no—
229D8—92no—
230D8—181no—
231C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
232C16—92no—
233C22×C3—31no—
234C22×C3—31no—
235C22×C3—61no—
236C22×C3—362no—
237C22×C3—361no—
238C22×C3—722no—
239C22×C3#371411yescharacteristic
240C2×D3—121no—
241C2×D3—121no—
242C2×D3—122no—
243C2×D3—122no—
244C2×D3—242no—
245C2×D3—362no—
246C2×D3—722no—
247C3×C4—31no—
248C3×C4—31no—
249C3×C4—31no—
250C3×C4—31no—
251C3×C4—61no—
252C3×C4—61no—
253C3×C4—361no—
254C3×C4#371411yescharacteristic
255C3×C4C2×#62611yescharacteristic
256Dic3—121no—
257Dic3—121no—
258Dic3—122no—
259C32—21no—
260C32#490811yescharacteristic, complemented
261C32#514711yescharacteristic, complemented
262C23—361no—
263Q8—181no—
264D4—182no—
265D4—181no—
266D4—361no—
267D4—361no—
268C2×C4—361no—
269C2×C4#620011yescharacteristic
270C8#620011yescharacteristic
271C8C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
272C2×C3—21no—
273C2×C3—31no—
274C2×C3—31no—
275C2×C3—61no—
276C2×C3—61no—
277C2×C3—61no—
278C2×C3—121no—
279C2×C3—722no—
280C2×C3—721no—
281C2×C3C2×#371411yescharacteristic
282D3—241no—
283D3—241no—
284D3—242no—
285D3—242no—
286D3—722no—
287C22—361no—
288C22—362no—
289C22—722no—
290C22#8843611yescharacteristic
291C4—361no—
292C4#8843611yescharacteristic
293C4C2×#620011yescharacteristic
294C3—31no—
295C3—31no—
296C3—61no—
297C3#1059063311yescharacteristic
298C2—21no—
299C2—722no—
300C2—721no—
301C2C2×#8843611yescharacteristic
subgroupG/H
centerC2C2×#88436
commutator, cosocleC8×H(3,𝔽3)C23
FrattiniC3×C8C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC2×C3C2×#3714
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#509
LCSG ▷ C8×H(3,𝔽3) ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. #11721779C2semidirect
2. C2×#3745⋊C3C2semidirect
3. #63639⋊C3C2semidirect
4. #11720255C2semidirect
5. #11720253C2semidirect
11. #88530C22semidirect
12. #88529C22semidirect
16. (C3⋊(C3⋊C16))⋊C3C22semidirect
17. C2×#3745D3semidirect
22. #63639D3semidirect
62. C3⋊(C3⋊C16)C2×D3semidirect
175. H(3,𝔽3)#509semidirect
260. C32#4908semidirect
261. C32#5147semidirect

Maximal subgroups

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

subgroupindex
#117217792
C2×#3745⋊C32
#63639⋊C32
#117202552
#117202532
#105906343
#105906303