Finite Group Databasebeta

← #13159399

Subgroups — #13159399

Order 1728.

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

normal subgroups only

This group has 804 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
2Dic3×#634C211yesmaximal, characteristic
3C2×#88798C211yesmaximal, characteristic
4#11722177C211yesmaximal, characteristic
5C3×#63637C211yesmaximal, characteristic
6#11722038C211yesmaximal, characteristic
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
8C4×#3756—31nomaximal
9#10590476—34nomaximal
10C2×C3×#634C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C33×C4⋊C4C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×Dic3⋊C4—21no—
17C32×C4×Dic3—21no—
18C32×C4×Dic3C2211yescharacteristic
19#88807—21no—
20Dic3×C3⋊Dic3—21no—
21#88798C414yescomplemented
22C32×C4×Q8—31no—
23C2×#3756—31no—
24C4×#634—31no—
25C3×#887D314yes—
26Dic3×Dic6—34no—
27C2×#3718—34no—
28#63662—31no—
29#63637D311yescharacteristic
30#63634—31no—
31#63634—34no—
32#63634—64no—
33C4×C3⋊(C3⋊C8)—31no—
34C3×#810—34no—
35#63566—34no—
36Dic3×C3⋊C8—34no—
37C2×C33×C4—21no—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)—21no—
40C2×C3×C3⋊Dic3—21no—
41C3×#634—21no—
42C3×#634C2×C412yes—
43C2×C32×Dic3—21no—
44C2×C32×Dic3D411yescharacteristic
45C2×C32×Dic3D411yescharacteristic
46C32×Dic6—21no—
47C32×Dic6C2×C412yes—
48C3⋊(C3⋊(C3⋊C8))C2×C412yes—
49C3×C3⋊(C3⋊C8)—21no—
50C4×#216—94no—
51#4790—91no—
52C2×C32×Q8—31no—
53C2×#634C2×D311yescharacteristic
54C2×C3×Dic6—34no—
55C2×C3×Dic6C2×D314yes—
56C32×C4⋊C4—24no—
57C32×C4⋊C4—61no—
58C32×C4⋊C4C2×D314yes—
59C32×C4⋊C4C2×D311yescharacteristic
60C32×C42—31no—
61C32×C42—61no—
62#3756—34no—
63C4⋊(C3⋊Dic3)—31no—
64C4⋊(C3⋊Dic3)—34no—
65C4⋊(C3⋊Dic3)—64no—
66#3750—61no—
67C4×C3⋊Dic3—61no—
68C2×C3⋊(C3⋊C8)—31no—
69C2×C3⋊(C3⋊C8)—34no—
70C2×C3⋊(C3⋊C8)—64no—
71C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
72C3×Dic3⋊C4—24no—
73C3×C4×Dic3—24no—
74C3×C4×Dic3C2×D314yes—
75C2×C3×C3⋊C8—34no—
76Dic3⋊Dic3—64no—
77Dic32—64no—
78#3718—316no—
79C22×C33C2×D411yescharacteristic
80C33×C4—41no—
81C33×C4C22×C411yescharacteristic
82C33×C4#4111yescharacteristic
83C3⋊(C3⋊Dic3)—41no—
84C3×C3⋊Dic3—22no—
85C32×Dic3—22no—
86C32×Dic3—21no—
87C32×Dic3Q1612yescomplemented
88C3×C4×Q8—34no—
89Q8×Dic3—91no—
90C2×#216—94no—
91C2×#216—91no—
92#887C3⋊D311yescharacteristic
93C4×Dic6—94no—
94C3×#103—91no—
95#835—94no—
96#816—91no—
97C8×Dic3—91no—
98#810—34no—
99#807—94no—
100#807—91no—
101#807—184no—
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—21no—
119#634C4×D312yes—
120C2×C3×Dic3—24no—
121C2×C3×Dic3—64no—
122C2×C3×Dic3#6714yes—
123C2×C3×Dic3#6714yes—
124C3×Dic6—24no—
125C3×Dic6—38no—
126C3×Dic6—64no—
127C3×Dic6C4×D318yes—
128C3⋊(C3⋊C8)—21no—
129C3⋊(C3⋊C8)—32no—
130C3⋊(C3⋊C8)—38no—
131C3⋊(C3⋊C8)—68no—
132C3×C3⋊C8—64no—
133C4×Q16—271noSylow
134C2×C33C4×D411yescharacteristic
135C2×C33C2×Q1611yescharacteristic
136C2×C33#13511yescharacteristic
137C2×C3×Q8—34no—
138C2×C3×Q8—91no—
139C2×Dic6—34no—
140C2×Dic6C2×C3⋊D311yescharacteristic
141C2×C3×C8—91no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—64no—
144C3×C4⋊C4D3214yes—
145C3×C4⋊C4C2×C3⋊D311yescharacteristic
146C3×C42—34no—
147C3×C42—64no—
148#216—916no—
149#216—94no—
150C4⋊Dic3—94no—
151C4⋊Dic3—91no—
152C4⋊Dic3—184no—
153C4⋊Dic3—181no—
154Dic3⋊C4—21no—
155Dic3⋊C4—184no—
156C4×Dic3—21no—
157C4×Dic3—184no—
158C4×Dic3—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—41no—
171C32×C4—44no—
172C32×C4—62no—
173C32×C4—61no—
174C32×C4—84no—
175C32×C4C2×C4×D314yes—
176C32×C4C2×C4×D311yescharacteristic
177C32×C4#23514yes—
178C32×C4#23711yescharacteristic
179C3⋊Dic3—22no—
180C3⋊Dic3—121no—
181C3⋊Dic3—124no—
182C3⋊Dic3—244no—
183C3×Dic3—28no—
184C3×Dic3—24no—
185C3×Dic3—68no—
186C3×Dic3#21618yescomplemented
187C2×Q16—271no—
188C4×Q8—31no—
189C4×Q8—271no—
190#107—271no—
191#103—91no—
192#103—271no—
193C4×C8—271no—
194C33C4×Q1611yescharacteristic, complemented, Sylow
195C3×Q8—38no—
196C3×Q8—64no—
197C3×Q8—92no—
198C3×Q8—181no—
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—38no—
218Dic6—64no—
219Dic6C4×C3⋊D312yes—
220C3×C8—181no—
221C3⋊C8—64no—
222C3⋊C8—98no—
223C3⋊C8—92no—
224C3⋊C8—188no—
225C2×C32—24no—
226C2×C32—24no—
227C2×C32—24no—
228C2×C32#67⋊C411yescharacteristic
229C2×C32#91611yescharacteristic
230C2×C32D3×Q1611yescharacteristic
231C2×C32C4×#6714yes—
232C2×C32C2×#21614yes—
233C2×C32#95014yes—
234C2×Q8—31no—
235C2×Q8—91no—
236Q16—274no—
237C2×C8—91no—
238C2×C8—271no—
239C4⋊C4—61no—
240C4⋊C4—271no—
241C4⋊C4—541no—
242C4⋊C4D3×C3⋊D311yescharacteristic
243C42—31no—
244C42—61no—
245C42—541no—
246C22×C3—24no—
247C22×C3D3×#6714yes—
248C22×C3C2×#63811yescharacteristic
249C3×C4—24no—
250C3×C4—24no—
251C3×C4—38no—
252C3×C4—41no—
253C3×C4—44no—
254C3×C4—68no—
255C3×C4—64no—
256C3×C4—84no—
257C3×C4—182no—
258C3×C4#379514yes—
259C3×C4C4×D3214yes—
260C3×C4C2×C4×C3⋊D311yescharacteristic
261C3×C4#382811yescharacteristic
262Dic3—22no—
263Dic3—21no—
264Dic3—68no—
265Dic3—361no—
266Dic3—364no—
267Dic3—724no—
268Dic3#375612yescomplemented
269C32—24no—
270C32#479011yescharacteristic, complemented
271C32C4×#21614yescomplemented
272Q8—32no—
273Q8—61no—
274Q8—92no—
275Q8—181no—
276C2×C4—21no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—61no—
280C2×C4—181no—
281C2×C4—541no—
282C2×C4C2×D3×C3⋊D311yescharacteristic
283C8—181no—
284C8—272no—
285C2×C3—24no—
286C2×C3—24no—
287C2×C3—24no—
288C2×C3#6387914yes—
289C2×C3#6393514yes—
290C2×C3D3×#21614yes—
291C2×C3C4×#63811yescharacteristic
292C2×C3C2×#375611yescharacteristic
293C2×C3#6415911yescharacteristic
294C22D3×#63811yescharacteristic
295C4—32no—
296C4—41no—
297C4—61no—
298C4—62no—
299C4—182no—
300C4—1081no—
301C4#8902111yescharacteristic
302C4C4×D3×C3⋊D311yescharacteristic
303C3—24no—
304C3#1059047614yescomplemented
305C3C4×#375611yescharacteristic, complemented
306C2#1172350611yescharacteristic
307C2#1172355811yescharacteristic
308C2D3×#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. #88798C4semidirect
87. C32×Dic3Q16semidirect
186. C3×Dic3#216semidirect
194. C33C4×Q16semidirect
268. Dic3#3756semidirect
270. C32#4790semidirect
271. C32C4×#216semidirect
304. C3#10590476semidirect
305. C3C4×#3756semidirect

Maximal subgroups

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

subgroupindex
C32×#8872
Dic3×#6342
C2×#887982
#117221772
C3×#636372
#117220382
Dic3×C3⋊(C3⋊C8)2
C4×#37563
#105904763