Finite Group Databasebeta

← #13159301

Subgroups — #13159301

Order 1728.

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

normal subgroups only

This group has 756 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
1C32×#888C211yesmaximal, characteristic
2#11723443C211yesmaximal, characteristic
3#11722180C211yesmaximal, characteristic
4C3×#63637C211yesmaximal, characteristic
5#11722041C211yesmaximal, characteristic
6#11722039C211yesmaximal, characteristic
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
8#10592792—31nomaximal
9#10590560—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×Dic6—21no—
13C33×C4⋊C4C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×Dic3⋊C4—21no—
18C32×C4×Dic3C2211yescharacteristic
19Dic3⋊(C3⋊Dic3)—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21C32×C4⋊Q8—31no—
22#64078—31no—
23C3×#888D314yes—
24#63866—34no—
25#63637—31no—
26#63637—31no—
27#63637—31no—
28#63637—34no—
29#63637—64no—
30#63637D311yescharacteristic
31C4×C3⋊(C3⋊C8)—31no—
32C3×#810—34no—
33#63567—34no—
34#63565—34no—
35Dic3×C3⋊C8—34no—
36C2×C33×C4—21no—
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)—21no—
39#6308—21no—
40#6308—21no—
41C2×C3×C3⋊Dic3—21no—
42C3×#634—21no—
43C3×#634—21no—
44C2×C32×Dic3—21no—
45C2×C32×Dic3D411yescharacteristic
46C2×C32×Dic3D411yescharacteristic
47C32×Dic6—42no—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C3×C3⋊(C3⋊C8)—21no—
50#5093—94no—
51#4792—91no—
52C2×C32×Q8—61no—
53C2×#634—31no—
54C2×#634—34no—
55C2×#634—64no—
56C2×#634C2×D311yescharacteristic
57C2×C3×Dic6—24no—
58C2×C3×Dic6—34no—
59C32×C4⋊C4—24no—
60C32×C4⋊C4—31no—
61C32×C4⋊C4—61no—
62C32×C4⋊C4C2×D314yes—
63C32×C4⋊C4C2×D311yescharacteristic
64C32×C42—31no—
65C4⋊(C3⋊Dic3)—61no—
66C4⋊(C3⋊Dic3)—61no—
67C2×C3⋊(C3⋊C8)—31no—
68C2×C3⋊(C3⋊C8)—34no—
69C2×C3⋊(C3⋊C8)—64no—
70C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
71C3×C4⋊Dic3C2×D314yes—
72C3×Dic3⋊C4—24no—
73C3×C4×Dic3C2×D314yes—
74C2×C3×C3⋊C8—34no—
75Dic3⋊Dic3—64no—
76Dic3⋊Dic3—64no—
77C22×C33C2×D411yescharacteristic
78C33×C4—41no—
79C33×C4#4111yescharacteristic
80C33×C4#4111yescharacteristic
81C3⋊(C3⋊Dic3)—41no—
82C3×C3⋊Dic3—41no—
83C32×Dic3—41no—
84C32×Dic3SD1612yes—
85C32×Dic3Q1612yes—
86C3×C4⋊Q8—34no—
87#944—91no—
88#888C3⋊D311yescharacteristic
89#869—94no—
90C3×#103—91no—
91#835—91no—
92#816—91no—
93C8×Dic3—91no—
94#810—34no—
95#810—94no—
96#810—94no—
97#810—94no—
98#810—91no—
99#810—184no—
100C4×C3⋊C8—94no—
101C32×Q8—122no—
102C2×C32×C4—21no—
103C2×C32×C4—24no—
104C2×C32×C4—24no—
105C2×C32×C4—31no—
106C2×C32×C4—31no—
107C2×C32×C4—44no—
108C2×C32×C4—61no—
109C2×C32×C4C22×D314yes—
110C2×C32×C4C22×D311yescharacteristic
111C2×C3⋊Dic3—21no—
112C2×C3⋊Dic3—61no—
113C2×C3⋊Dic3—64no—
114C2×C3⋊Dic3—124no—
115#634—21no—
116#634—21no—
117#634—68no—
118#634—61no—
119#634—61no—
120#634—64no—
121#634—64no—
122#634—124no—
123C2×C3×Dic3—24no—
124C2×C3×Dic3—64no—
125C2×C3×Dic3#6714yes—
126C2×C3×Dic3#6714yes—
127C3×Dic6—48no—
128C3×Dic6—64no—
129C3×Dic6—64no—
130C3⋊(C3⋊C8)—21no—
131C3⋊(C3⋊C8)—61no—
132C3⋊(C3⋊C8)—64no—
133C3⋊(C3⋊C8)—124no—
134C3×C3⋊C8—64no—
135#487—271noSylow
136C2×C33#12411yescharacteristic
137C2×C33C2×SD1611yescharacteristic
138C2×C33C2×Q1611yescharacteristic
139C2×C3×Q8—64no—
140C2×C3×Q8—91no—
141C2×Dic6—21no—
142C2×Dic6—34no—
143C2×Dic6—94no—
144C2×Dic6—91no—
145C2×Dic6—184no—
146C2×C3×C8—91no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4—34no—
149C3×C4⋊C4—64no—
150C3×C4⋊C4D3214yes—
151C3×C4⋊C4C2×C3⋊D311yescharacteristic
152C3×C42—34no—
153C4⋊Dic3—184no—
154C4⋊Dic3—184no—
155C4⋊Dic3C2×C3⋊D311yescharacteristic
156Dic3⋊C4—21no—
157Dic3⋊C4—181no—
158Dic3⋊C4—181no—
159C4×Dic3C2×C3⋊D311yescharacteristic
160C2×C3⋊C8—34no—
161C2×C3⋊C8—94no—
162C2×C3⋊C8—91no—
163C2×C3⋊C8—184no—
164C22×C32—24no—
165C22×C32D3×D411yescharacteristic
166C22×C32C2×#6714yes—
167C32×C4—24no—
168C32×C4—24no—
169C32×C4—32no—
170C32×C4—32no—
171C32×C4—41no—
172C32×C4—44no—
173C32×C4—84no—
174C32×C4—121no—
175C32×C4#23511yescharacteristic
176C32×C4#23711yescharacteristic
177C32×C4#23914yes—
178C32×C4#23914yes—
179C3⋊Dic3—41no—
180C3⋊Dic3—128no—
181C3⋊Dic3—121no—
182C3⋊Dic3—124no—
183C3×Dic3—44no—
184C3×Dic3—124no—
185C3×Dic3#21418yes—
186C3×Dic3#21618yes—
187C4⋊Q8—31no—
188C4⋊Q8—271no—
189#103—91no—
190#103—271no—
191#103—271no—
192#103—271no—
193C4×C8—271no—
194C33#48711yescharacteristic, complemented, Sylow
195C3×Q8—128no—
196C3×Q8—181no—
197C3×Q8—181no—
198C2×C3×C4—24no—
199C2×C3×C4—21no—
200C2×C3×C4—24no—
201C2×C3×C4—34no—
202C2×C3×C4—34no—
203C2×C3×C4—44no—
204C2×C3×C4—64no—
205C2×C3×C4—181no—
206C2×C3×C4C2×D3214yes—
207C2×C3×C4C22×C3⋊D311yescharacteristic
208C2×Dic3—21no—
209C2×Dic3—64no—
210C2×Dic3—184no—
211C2×Dic3—181no—
212C2×Dic3—364no—
213C2×Dic3#63811yescharacteristic
214C2×Dic3#63811yescharacteristic
215Dic6—42no—
216Dic6—64no—
217Dic6—64no—
218Dic6—188no—
219Dic6—184no—
220Dic6—184no—
221Dic6—181no—
222Dic6—181no—
223Dic6—364no—
224C3×C8—181no—
225C3⋊C8—64no—
226C3⋊C8—184no—
227C3⋊C8—181no—
228C3⋊C8—364no—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32—24no—
232C2×C32#88511yescharacteristic
233C2×C32D3×SD1611yescharacteristic
234C2×C32D3×Q1611yescharacteristic
235C2×C32C2×#21414yes—
236C2×C32C2×#21614yes—
237C2×C32#94714yes—
238C2×Q8—61no—
239C2×Q8—91no—
240C2×Q8—271no—
241C2×C8—91no—
242C2×C8—271no—
243C4⋊C4—31no—
244C4⋊C4—61no—
245C4⋊C4—541no—
246C4⋊C4—541no—
247C4⋊C4D3×C3⋊D311yescharacteristic
248C42—31no—
249C22×C3—24no—
250C22×C3D3×#6714yes—
251C22×C3C2×#63811yescharacteristic
252C3×C4—24no—
253C3×C4—24no—
254C3×C4—38no—
255C3×C4—38no—
256C3×C4—41no—
257C3×C4—44no—
258C3×C4—84no—
259C3×C4—124no—
260C3×C4—361no—
261C3×C4#379614yes—
262C3×C4#379914yes—
263C3×C4#383211yescharacteristic
264C3×C4#383211yescharacteristic
265Dic3—41no—
266Dic3—124no—
267Dic3—368no—
268Dic3—361no—
269Dic3—364no—
270Dic3#375412yes—
271Dic3#375612yes—
272C32—24no—
273C32#479211yescharacteristic, complemented
274C32#509314yescomplemented
275Q8—122no—
276Q8—181no—
277Q8—181no—
278Q8—541no—
279Q8—541no—
280C2×C4—21no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—61no—
284C2×C4—181no—
285C2×C4—541no—
286C2×C4C2×D3×C3⋊D311yescharacteristic
287C8—181no—
288C8—541no—
289C2×C3—24no—
290C2×C3—24no—
291C2×C3—24no—
292C2×C3#6390714yes—
293C2×C3D3×#21414yes—
294C2×C3D3×#21614yes—
295C2×C3C2×#375411yescharacteristic
296C2×C3C2×#375611yescharacteristic
297C2×C3#6415611yescharacteristic
298C22D3×#63811yescharacteristic
299C4—32no—
300C4—32no—
301C4—41no—
302C4—121no—
303C4—361no—
304C4—1081no—
305C4#8901711yescharacteristic
306C4#8901811yescharacteristic
307C3—24no—
308C3#1059056014yescomplemented
309C3#1059279211yescharacteristic, complemented
310C2#1172339211yescharacteristic
311C2D3×#375411yescharacteristic
312C2D3×#375611yescharacteristic
subgroupG/H
centerC22D3×#638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#487
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
194. C33#487semidirect
273. C32#4792semidirect
274. C32#5093semidirect
308. C3#10590560semidirect
309. C3#10592792semidirect

Maximal subgroups

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

subgroupindex
C32×#8882
#117234432
#117221802
C3×#636372
#117220412
#117220392
Dic3×C3⋊(C3⋊C8)2
#105927923
#105905603