Finite Group Databasebeta

← #13158893

Subgroups — #13158893

Order 1728.

836 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 176 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#11723911C211yesmaximal, characteristic, complemented
2C32×C4×D12C211yesmaximal, characteristic
3C3×C4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
4#11722057C211yesmaximal, characteristic, complemented
5#11722054C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic
7#11722032C211yesmaximal, characteristic
8#10592708—31nomaximal
9#10590271—34nomaximal
10C2×C32×D12C2211yescharacteristic
11C2×C32×C4×D3—21no—
12C33×C42C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15#88857—21no—
16#88857—21no—
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×(C2×D3)⋊C4—21no—
20C32×C4⋊Dic3C2211yescharacteristic
21C32×C4×D4—31no—
22#64079—31no—
23#64079—34no—
24#64079—64no—
25C3×C4×D12D314yes—
26#63659—31no—
27#63659—31no—
28#63635—31no—
29#63634—31no—
30C4⋊(C3⋊(C3⋊C8))D311yescharacteristic, complemented
31C3×C4⋊(C3⋊C8)—34no—
32#63575—34no—
33#63573—34no—
34#63563—34no—
35#63563—34no—
36C22×C32×D3—21no—
37C2×C33×C4D411yescharacteristic
38C2×C33×C4D411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)—21no—
42C2×C32×Dic3—21no—
43C32×D12—21no—
44C32×D12Q812yes—
45C32×C4×D3—42no—
46C3×C3⋊(C3⋊C8)—21no—
47C3×C3⋊(C3⋊C8)—21no—
48#5009—94no—
49#4847—91no—
50C2×C32×D4—31no—
51C22×C32×C4—61no—
52C2×C3×D12C2×D314yes—
53C2×C3×C4×D3—24no—
54C32×C4⋊C4—31no—
55C32×C22⋊C4—61no—
56C32×C42—24no—
57C32×C42C2×D311yescharacteristic
58C32×C42C2×D314yes—
59C4⋊(C3⋊Dic3)—31no—
60C4⋊(C3⋊Dic3)—31no—
61C4⋊(C3⋊Dic3)—34no—
62C4⋊(C3⋊Dic3)—34no—
63C4⋊(C3⋊Dic3)—64no—
64C4⋊(C3⋊Dic3)—64no—
65#3750—68no—
66#3750—68no—
67#3750—61no—
68#3750—61no—
69#3750—64no—
70#3750—64no—
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
73C3×(C2×D3)⋊C4—24no—
74C3×C4⋊Dic3C2×D314yes—
75C2×C3×C3⋊C8—34no—
76C2×C3×C3⋊C8—34no—
77C22×C33C2×D411yescharacteristic
78C2×C32×D3—22no—
79C2×C32×D3—41no—
80C2×C32×D3—41no—
81C33×C4—21no—
82C33×C4—21no—
83C33×C4C2×Q811yescharacteristic
84C33×C4#4111yescharacteristic
85C3⋊(C3⋊Dic3)—41no—
86C3⋊(C3⋊Dic3)—41no—
87C32×Dic3—41no—
88C3×C4×D4—34no—
89C4×D12C3⋊D311yescharacteristic
90#870—94no—
91#870—91no—
92#870—184no—
93C3×C4⋊C8—91no—
94#832—94no—
95#832—94no—
96#821—91no—
97#821—91no—
98#818—91no—
99#817—91no—
100#808—94no—
101#807—94no—
102C4⋊(C3⋊C8)—34no—
103C23×C32—61no—
104C22×C3×D3—24no—
105C32×D4—32no—
106C32×D4—61no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4—24no—
110C2×C32×C4—61no—
111C2×C32×C4—122no—
112C2×C32×C4D1211yescharacteristic
113C2×C32×C4D1211yescharacteristic
114C2×C32×C4#6714yes—
115C2×C32×C4#6714yes—
116C2×C32×C4C22×D314yes—
117C2×C32×C4C22×D311yescharacteristic
118C2×C3⋊Dic3—68no—
119C2×C3⋊Dic3—68no—
120C2×C3⋊Dic3—61no—
121C2×C3⋊Dic3—61no—
122C2×C3⋊Dic3—64no—
123C2×C3⋊Dic3—64no—
124C2×C3×Dic3—24no—
125C3×D12—24no—
126C3×D12Dic618yes—
127C3×C4×D3—48no—
128C3⋊(C3⋊C8)—21no—
129C3⋊(C3⋊C8)—21no—
130C3×C3⋊C8—64no—
131C3×C3⋊C8—64no—
132#478—271noSylow
133C2×C33C22⋊Q811yescharacteristic
134C2×C33#13511yescharacteristic
135C2×C33AGL(1,ℤ/8)11yescharacteristic
136C32×D3—42no—
137C2×C3×D4—34no—
138C22×C3×C4—64no—
139C2×D12C2×C3⋊D311yescharacteristic
140C2×C4×D3—21no—
141C2×C3×C8—91no—
142C2×C3×C8—91no—
143C3×C4⋊C4—34no—
144C3×C22⋊C4—64no—
145C3×C42—24no—
146C3×C42D3214yes—
147C3×C42C2×C3⋊D311yescharacteristic
148(C2×D3)⋊C4—21no—
149C4⋊Dic3—94no—
150C4⋊Dic3—94no—
151C4⋊Dic3—91no—
152C4⋊Dic3—91no—
153C4⋊Dic3—184no—
154C4⋊Dic3—184no—
155C4⋊Dic3C2×C3⋊D311yescharacteristic
156Dic3⋊C4—188no—
157Dic3⋊C4—188no—
158Dic3⋊C4—184no—
159Dic3⋊C4—184no—
160Dic3⋊C4—181no—
161Dic3⋊C4—181no—
162C2×C3⋊C8—34no—
163C2×C3⋊C8—34no—
164C22×C32—24no—
165C22×C32—62no—
166C22×C32—121no—
167C22×C32—121no—
168C22×C32C2×D1211yescharacteristic
169C22×C32C2×#6714yes—
170C2×C3×D3—28no—
171C2×C3×D3—44no—
172C2×C3×D3—44no—
173C32×C4—21no—
174C32×C4—21no—
175C32×C4—24no—
176C32×C4—24no—
177C32×C4—24no—
178C32×C4—24no—
179C32×C4—44no—
180C32×C4—44no—
181C32×C4—121no—
182C32×C4C2×Dic614yes—
183C32×C4#23514yes—
184C32×C4#23711yescharacteristic
185C32×C4D3×Q811yescharacteristic
186C3⋊Dic3—128no—
187C3⋊Dic3—128no—
188C3⋊Dic3—121no—
189C3⋊Dic3—121no—
190C3⋊Dic3—124no—
191C3⋊Dic3—124no—
192C3×Dic3—44no—
193#125—271no—
194C4×D4—31no—
195#107—271no—
196#106—271no—
197C4⋊C8—91no—
198#102—271no—
199#102—271no—
200C33#47811yescharacteristic, complemented, Sylow
201C23×C3—64no—
202C22×D3—21no—
203C3×D4—38no—
204C3×D4—64no—
205C2×C3×C4—24no—
206C2×C3×C4—24no—
207C2×C3×C4—24no—
208C2×C3×C4—64no—
209C2×C3×C4—128no—
210C2×C3×C4#62614yes—
211C2×C3×C4#62614yes—
212C2×C3×C4#63811yescharacteristic
213C2×C3×C4#63811yescharacteristic
214C2×C3×C4C2×D3214yes—
215C2×C3×C4C22×C3⋊D311yescharacteristic
216C2×Dic3—21no—
217C2×Dic3—188no—
218C2×Dic3—188no—
219C2×Dic3—184no—
220C2×Dic3—184no—
221C2×Dic3—181no—
222C2×Dic3—181no—
223D12—21no—
224D12#63412yes—
225C4×D3—42no—
226C3×C8—181no—
227C3×C8—181no—
228C3⋊C8—64no—
229C3⋊C8—64no—
230C2×C32—24no—
231C2×C32—24no—
232C2×C32—24no—
233C2×C32—122no—
234C2×C32#89711yescharacteristic
235C2×C32#90411yescharacteristic
236C2×C32#90811yescharacteristic
237C2×C32#92414yes—
238C2×C32#93214yes—
239C2×C32#95014yes—
240C3×D3—48no—
241C2×D4—31no—
242C22×C4—61no—
243C2×C8—91no—
244C2×C8—91no—
245C4⋊C4—31no—
246C4⋊C4—271no—
247C4⋊C4—271no—
248C4⋊C4—541no—
249C4⋊C4—541no—
250C22⋊C4—61no—
251C42D3×C3⋊D311yescharacteristic
252C22×C3—24no—
253C22×C3—68no—
254C22×C3—124no—
255C22×C3—124no—
256C22×C3C2×#62614yes—
257C22×C3C2×#63811yescharacteristic
258C2×D3—22no—
259C2×D3—41no—
260C2×D3—41no—
261C3×C4—24no—
262C3×C4—24no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—24no—
266C3×C4—24no—
267C3×C4—44no—
268C3×C4—44no—
269C3×C4—124no—
270C3×C4D3×Dic614yes—
271C3×C4#379514yes—
272C3×C4C2×#63411yescharacteristic
273C3×C4#382811yescharacteristic
274Dic3—41no—
275Dic3—368no—
276Dic3—368no—
277Dic3—364no—
278Dic3—364no—
279Dic3—361no—
280Dic3—361no—
281C32—24no—
282C32#484711yescharacteristic, complemented
283C32#500914yescomplemented
284C23—61no—
285D4—32no—
286D4—61no—
287C2×C4—61no—
288C2×C4—122no—
289C2×C4—541no—
290C2×C4—541no—
291C2×C4#629111yescharacteristic
292C2×C4#629111yescharacteristic
293C2×C4C2×D3×C3⋊D311yescharacteristic
294C8—181no—
295C8—181no—
296C2×C3—24no—
297C2×C3—24no—
298C2×C3—24no—
299C2×C3—128no—
300C2×C3#6382914yes—
301C2×C3#6383014yes—
302C2×C3#6385714yes—
303C2×C3#6413311yescharacteristic
304C2×C3#6414111yescharacteristic
305C2×C3#6415911yescharacteristic
306D3—42no—
307C22—62no—
308C22—121no—
309C22—121no—
310C22C2×#629111yescharacteristic
311C4—21no—
312C4—21no—
313C4—121no—
314C4—1081no—
315C4—1081no—
316C4D3×#63411yescharacteristic
317C4#8902111yescharacteristic
318C3—24no—
319C3#1059027114yescomplemented
320C3#1059270811yescharacteristic, complemented
321C2—122no—
322C2#1172337711yescharacteristic
323C2#1172337811yescharacteristic
324C2#1172342611yescharacteristic
subgroupG/H
centerC22C2×#6291
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypercenterC42D3×C3⋊D3
hypocenterC33#478
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11723911C2semidirect
3. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
4. #11722057C2semidirect
5. #11722054C2semidirect
30. C4⋊(C3⋊(C3⋊C8))D3semidirect
200. C33#478semidirect
282. C32#4847semidirect
283. C32#5009semidirect
319. C3#10590271semidirect
320. C3#10592708semidirect

Maximal subgroups

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

subgroupindex
#117239112
C32×C4×D122
C3×C4⋊(C3⋊(C3⋊C8))2
#117220572
#117220542
#117220322
#105927083
#105902713