Finite Group Databasebeta

← #13159405

Subgroups — #13159405

Order 1728.

828 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 186 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×#887C211yesmaximal, characteristic
2#11723399C211yesmaximal, characteristic
3C2×#88799C211yesmaximal, characteristic
4#11722180C211yesmaximal, characteristic
5C3×#63634C211yesmaximal, characteristic
6#11722037C211yesmaximal, characteristic
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
8C4×#3756—31nomaximal
9#10590481—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×Dic3⋊C4—21no—
17C32×C4×Dic3—21no—
18C32×C4×Dic3C2211yescharacteristic
19#88806—21no—
20Dic3×C3⋊Dic3—21no—
21#88799C414yescomplemented
22C32×C4×Q8—31no—
23C2×#3756—31no—
24C4×#634—31no—
25C3×#887D314yes—
26#63844—34no—
27C2×#3719—34no—
28#63662—31no—
29#63637—31no—
30#63637—34no—
31#63637—64no—
32#63634D311yescharacteristic
33C4×C3⋊(C3⋊C8)—31no—
34C3×#807—34no—
35#63565—34no—
36Dic3×C3⋊C8—34no—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40#6308—21no—
41#6308C2×C412yes—
42C2×C3×C3⋊Dic3—21no—
43C2×C32×Dic3—21no—
44C2×C32×Dic3D411yescharacteristic
45C2×C32×Dic3D411yescharacteristic
46C32×Dic6—21no—
47C32×Dic6C2×C412yes—
48C3⋊(C3⋊(C3⋊C8))—21no—
49C3×C3⋊(C3⋊C8)C2×C412yes—
50C4×#216—94no—
51#4873—91no—
52C2×C32×Q8—31no—
53C2×#634—31no—
54C2×#634—34no—
55C2×#634—64no—
56C2×C3×Dic6C2×D314yes—
57C32×C4⋊C4—24no—
58C32×C4⋊C4—61no—
59C32×C4⋊C4C2×D314yes—
60C32×C4⋊C4C2×D311yescharacteristic
61C32×C42—31no—
62C32×C42—61no—
63#3756—34no—
64C4⋊(C3⋊Dic3)C2×D311yescharacteristic
65#3750—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⋊Dic3—34no—
72C3×Dic3⋊C4—24no—
73C3×C4×Dic3—24no—
74C3×C4×Dic3C2×D314yes—
75C2×C3×C3⋊C8—34no—
76#3724—64no—
77Dic32—64no—
78#3719—316no—
79C22×C33C2×D411yescharacteristic
80C33×C4—41no—
81C33×C4C22×C411yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—22no—
84C3×C3⋊Dic3—41no—
85C32×Dic3—22no—
86C32×Dic3—21no—
87C32×Dic3Q1612yescomplemented
88C3×C4×Q8—34no—
89C2×#216—94no—
90C2×Dic12—91no—
91#887—91no—
92#887C3⋊D311yescharacteristic
93C4×Dic6—94no—
94C3×#107—91no—
95#835—94no—
96C8×Dic3—91no—
97#810—91no—
98#810—94no—
99#810—91no—
100#810—184no—
101#807—34no—
102C4×C3⋊C8—94no—
103C32×Q8—32no—
104C32×Q8—61no—
105C2×C32×C4—21no—
106C2×C32×C4—24no—
107C2×C32×C4—24no—
108C2×C32×C4—31no—
109C2×C32×C4—31no—
110C2×C32×C4—44no—
111C2×C32×C4—61no—
112C2×C32×C4C22×D314yes—
113C2×C32×C4C22×D311yescharacteristic
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—61no—
116C2×C3⋊Dic3—64no—
117C2×C3⋊Dic3—124no—
118#634—32no—
119#634—38no—
120#634—68no—
121#634—61no—
122#634—64no—
123#634—124no—
124C2×C3×Dic3—24no—
125C2×C3×Dic3—64no—
126C2×C3×Dic3#6714yes—
127C2×C3×Dic3#6714yes—
128C3×Dic6—24no—
129C3×Dic6C4×D318yes—
130C3⋊(C3⋊C8)—61no—
131C3⋊(C3⋊C8)—64no—
132C3⋊(C3⋊C8)—124no—
133C3⋊(C3⋊C8)C4×D312yes—
134C3×C3⋊C8—38no—
135C4×Q16—271noSylow
136C2×C33C4×D411yescharacteristic
137C2×C33C2×Q1611yescharacteristic
138C2×C33#13511yescharacteristic
139C2×C3×Q8—34no—
140C2×Dic6—94no—
141C2×Dic6—91no—
142C2×Dic6—184no—
143C2×Dic6C2×C3⋊D311yescharacteristic
144C2×C3×C8—91no—
145C3×C4⋊C4—24no—
146C3×C4⋊C4—64no—
147C3×C4⋊C4—91no—
148C3×C4⋊C4D3214yes—
149C3×C4⋊C4C2×C3⋊D311yescharacteristic
150C3×C42—34no—
151C3×C42—64no—
152#216—916no—
153C4⋊Dic3—34no—
154Dic3⋊C4—21no—
155Dic3⋊C4—184no—
156Dic3⋊C4—181no—
157C4×Dic3—21no—
158C4×Dic3—184no—
159C4×Dic3—181no—
160C4×Dic3C2×C3⋊D311yescharacteristic
161C2×C3⋊C8—34no—
162C2×C3⋊C8—94no—
163C2×C3⋊C8—91no—
164C2×C3⋊C8—184no—
165Dic12—94no—
166C22×C32—24no—
167C22×C32D3×D411yescharacteristic
168C22×C32C2×#6714yes—
169C32×C4—24no—
170C32×C4—24no—
171C32×C4—32no—
172C32×C4—41no—
173C32×C4—44no—
174C32×C4—62no—
175C32×C4—61no—
176C32×C4—84no—
177C32×C4C2×C4×D314yes—
178C32×C4C2×C4×D311yescharacteristic
179C32×C4#23514yes—
180C32×C4#23911yescharacteristic
181C3⋊Dic3—41no—
182C3⋊Dic3—62no—
183C3⋊Dic3—68no—
184C3⋊Dic3—128no—
185C3×Dic3—28no—
186C3×Dic3—24no—
187C3×Dic3—124no—
188C3×Dic3#21618yescomplemented
189C2×Q16—271no—
190C4×Q8—31no—
191C4×Q8—271no—
192#107—91no—
193#103—271no—
194#103—271no—
195C4×C8—271no—
196C33C4×Q1611yescharacteristic, complemented, Sylow
197C3×Q8—38no—
198C3×Q8—64no—
199C2×C3×C4—24no—
200C2×C3×C4—21no—
201C2×C3×C4—24no—
202C2×C3×C4—34no—
203C2×C3×C4—34no—
204C2×C3×C4—44no—
205C2×C3×C4—64no—
206C2×C3×C4—181no—
207C2×C3×C4C2×D3214yes—
208C2×C3×C4C22×C3⋊D311yescharacteristic
209C2×Dic3—21no—
210C2×Dic3—64no—
211C2×Dic3—184no—
212C2×Dic3—181no—
213C2×Dic3—364no—
214C2×Dic3#63811yescharacteristic
215C2×Dic3#63811yescharacteristic
216Dic6—21no—
217Dic6—98no—
218Dic6—92no—
219Dic6—188no—
220Dic6—184no—
221Dic6—181no—
222Dic6—364no—
223Dic6C4×C3⋊D312yes—
224C3×C8—92no—
225C3⋊C8—38no—
226C3⋊C8—184no—
227C3⋊C8—181no—
228C3⋊C8—364no—
229C2×C32—24no—
230C2×C32—24no—
231C2×C32—24no—
232C2×C32#89311yescharacteristic
233C2×C32#91211yescharacteristic
234C2×C32D3×Q1611yescharacteristic
235C2×C32C4×#6714yes—
236C2×C32C2×#21614yes—
237C2×C32#95014yes—
238C2×Q8—31no—
239C2×Q8—271no—
240Q16—274no—
241C2×C8—91no—
242C2×C8—271no—
243C4⋊C4—61no—
244C4⋊C4—91no—
245C4⋊C4—541no—
246C4⋊C4D3×C3⋊D311yescharacteristic
247C42—31no—
248C42—61no—
249C42—541no—
250C22×C3—24no—
251C22×C3D3×#6714yes—
252C22×C3C2×#63811yescharacteristic
253C3×C4—24no—
254C3×C4—24no—
255C3×C4—38no—
256C3×C4—41no—
257C3×C4—44no—
258C3×C4—68no—
259C3×C4—64no—
260C3×C4—84no—
261C3×C4—361no—
262C3×C4#379914yes—
263C3×C4C4×D3214yes—
264C3×C4C2×C4×C3⋊D311yescharacteristic
265C3×C4#382811yescharacteristic
266Dic3—22no—
267Dic3—21no—
268Dic3—124no—
269Dic3—182no—
270Dic3—188no—
271Dic3—368no—
272Dic3#375612yescomplemented
273C32—24no—
274C32#487311yescharacteristic, complemented
275C32C4×#21614yescomplemented
276Q8—32no—
277Q8—61no—
278Q8—272no—
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—92no—
288C8—541no—
289C2×C3—24no—
290C2×C3—24no—
291C2×C3—24no—
292C2×C3#6390414yes—
293C2×C3#6393314yes—
294C2×C3D3×#21614yes—
295C2×C3C4×#63811yescharacteristic
296C2×C3C2×#375611yescharacteristic
297C2×C3#6415911yescharacteristic
298C22D3×#63811yescharacteristic
299C4—32no—
300C4—41no—
301C4—61no—
302C4—62no—
303C4—361no—
304C4—542no—
305C4#8902211yescharacteristic
306C4C4×D3×C3⋊D311yescharacteristic
307C3—24no—
308C3#1059048114yescomplemented
309C3C4×#375611yescharacteristic, complemented
310C2#1172351011yescharacteristic
311C2#1172355611yescharacteristic
312C2D3×#375611yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socle, prime radicalC22×C33C2×D4
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33C4×Q16
LCSG ▷ 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
21. #88799C4semidirect
87. C32×Dic3Q16semidirect
188. C3×Dic3#216semidirect
196. C33C4×Q16semidirect
272. Dic3#3756semidirect
274. C32#4873semidirect
275. C32C4×#216semidirect
308. C3#10590481semidirect
309. C3C4×#3756semidirect

Maximal subgroups

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

subgroupindex
C32×#8872
#117233992
C2×#887992
#117221802
C3×#636342
#117220372
Dic3×C3⋊(C3⋊C8)2
C4×#37563
#105904813