Finite Group Databasebeta

← #13161174

Subgroups — #13161174

Order 1728.

900 conjugacy classes of subgroups; only the 196 normal subgroups are shown.

normal subgroups only

This group has 900 conjugacy classes of subgroups — too many to draw. Only its 196 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
1C2×#88863C211yesmaximal, characteristic
2C32×Q8×Dic3C211yesmaximal, characteristic
3#11723399C211yesmaximal, characteristic
4C3×#63662C211yesmaximal, characteristic
5#11722056C211yesmaximal, characteristic
6#11722041C211yesmaximal, characteristic
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
8C4×#3756—31nomaximal
9#10591537—34nomaximal
10C2×C33×Q8C2211yescharacteristic
11C2×#6308C2211yescharacteristic
12#88863C414yescomplemented
13C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3—21no—
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4×Dic3—21no—
19C32×C4×Dic3C2211yescharacteristic
20#88806—21no—
21Dic3×C3⋊Dic3—21no—
22C32×C4×Q8—31no—
23C2×#3756—31no—
24C2×#3756—34no—
25C2×#3756—64no—
26C4×#634—31no—
27C3×Q8×Dic3D314yes—
28#63844—34no—
29#63662D311yescharacteristic
30#63637—31no—
31#63634—31no—
32C4×C3⋊(C3⋊C8)—31no—
33C3×#835—34no—
34#63576—34no—
35#63567—34no—
36Dic3×C3⋊C8—34no—
37C33×Q8—21no—
38C33×Q8C2×C412yes—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42#6308—21no—
43#6308C2×C412yes—
44C2×C3×C3⋊Dic3—21no—
45C2×C32×Dic3—21no—
46C2×C32×Dic3D411yescharacteristic
47C2×C32×Dic3D411yescharacteristic
48C3⋊(C3⋊(C3⋊C8))C2×C412yes—
49C3×C3⋊(C3⋊C8)—21no—
50C4×#216—94no—
51#4790—91no—
52C2×C32×Q8—24no—
53C2×C32×Q8C2×D314yes—
54C2×C32×Q8C2×D311yescharacteristic
55C2×#634—31no—
56C2×#634—34no—
57C2×#634—64no—
58C32×C4⋊C4—31no—
59C32×C4⋊C4—61no—
60C32×C42—31no—
61C32×C42—61no—
62#3756—34no—
63#3756—316no—
64#3756—616no—
65C4⋊(C3⋊Dic3)C2×D311yescharacteristic
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×C4⋊Dic3—24no—
73C3×C4⋊Dic3—34no—
74C3×C4⋊Dic3C2×D314yes—
75C3×C4×Dic3—24no—
76C3×C4×Dic3C2×D314yes—
77C2×C3×C3⋊C8—34no—
78#3724—64no—
79Dic32—64no—
80C22×C33C2×D411yescharacteristic
81C33×C4—22no—
82C33×C4C22×C411yescharacteristic
83C33×C4#4111yescharacteristic
84C3⋊(C3⋊Dic3)—22no—
85C3×C3⋊Dic3—41no—
86C32×Dic3—21no—
87C32×Dic3—41no—
88C32×Dic3Q1612yescomplemented
89C3×C4×Q8—34no—
90Q8×Dic3C3⋊D311yescharacteristic
91C2×#216—94no—
92C2×#216—91no—
93C2×#216—184no—
94#887—91no—
95C4×Dic6—94no—
96C3×#103—91no—
97#835—34no—
98#816—91no—
99C8×Dic3—91no—
100#810—94no—
101#807—94no—
102#807—91no—
103C4×C3⋊C8—94no—
104C32×Q8—21no—
105C32×Q8—28no—
106C32×Q8—24no—
107C32×Q8—44no—
108C32×Q8C4×D318yes—
109C32×Q8C4×D312yes—
110C2×C32×C4—21no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—31no—
114C2×C32×C4—31no—
115C2×C32×C4—44no—
116C2×C32×C4—61no—
117C2×C32×C4C22×D314yes—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—61no—
121C2×C3⋊Dic3—64no—
122C2×C3⋊Dic3—124no—
123#634—32no—
124#634—38no—
125#634—68no—
126#634—61no—
127#634—64no—
128#634—124no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3—64no—
131C2×C3×Dic3#6714yes—
132C2×C3×Dic3#6714yes—
133C3⋊(C3⋊C8)—21no—
134C3⋊(C3⋊C8)—32no—
135C3⋊(C3⋊C8)—38no—
136C3⋊(C3⋊C8)—68no—
137C3×C3⋊C8—64no—
138C4×Q16—271noSylow
139C2×C33C4×D411yescharacteristic
140C2×C33C2×Q1611yescharacteristic
141C2×C33#13511yescharacteristic
142C2×C3×Q8—24no—
143C2×C3×Q8D3214yes—
144C2×C3×Q8C2×C3⋊D311yescharacteristic
145C2×Dic6—94no—
146C2×Dic6—91no—
147C2×Dic6—184no—
148C2×C3×C8—91no—
149C3×C4⋊C4—34no—
150C3×C4⋊C4—64no—
151C3×C4⋊C4—91no—
152C3×C42—34no—
153C3×C42—64no—
154#216—916no—
155#216—94no—
156#216—1816no—
157C4⋊Dic3—21no—
158C4⋊Dic3—34no—
159C4⋊Dic3C2×C3⋊D311yescharacteristic
160Dic3⋊C4—184no—
161Dic3⋊C4—181no—
162C4×Dic3—21no—
163C4×Dic3—184no—
164C4×Dic3—181no—
165C4×Dic3C2×C3⋊D311yescharacteristic
166C2×C3⋊C8—34no—
167C2×C3⋊C8—94no—
168C2×C3⋊C8—91no—
169C2×C3⋊C8—184no—
170C22×C32—24no—
171C22×C32D3×D411yescharacteristic
172C22×C32C2×#6714yes—
173C32×C4—22no—
174C32×C4—24no—
175C32×C4—24no—
176C32×C4—28no—
177C32×C4—32no—
178C32×C4—48no—
179C32×C4—61no—
180C32×C4—121no—
181C32×C4C2×C4×D314yes—
182C32×C4C2×C4×D311yescharacteristic
183C32×C4#23514yes—
184C32×C4#23711yescharacteristic
185C3⋊Dic3—41no—
186C3⋊Dic3—62no—
187C3⋊Dic3—68no—
188C3⋊Dic3—128no—
189C3×Dic3—24no—
190C3×Dic3—44no—
191C3×Dic3—124no—
192C3×Dic3#21618yescomplemented
193C2×Q16—271no—
194C4×Q8—31no—
195C4×Q8—271no—
196#107—271no—
197#103—91no—
198#103—271no—
199C4×C8—271no—
200C33C4×Q1611yescharacteristic, complemented, Sylow
201C3×Q8—24no—
202C3×Q8—21no—
203C3×Q8—28no—
204C3×Q8—44no—
205C3×Q8#62418yes—
206C3×Q8C4×C3⋊D312yes—
207C2×C3×C4—24no—
208C2×C3×C4—21no—
209C2×C3×C4—24no—
210C2×C3×C4—34no—
211C2×C3×C4—34no—
212C2×C3×C4—44no—
213C2×C3×C4—64no—
214C2×C3×C4—181no—
215C2×C3×C4C2×D3214yes—
216C2×C3×C4C22×C3⋊D311yescharacteristic
217C2×Dic3—21no—
218C2×Dic3—64no—
219C2×Dic3—184no—
220C2×Dic3—181no—
221C2×Dic3—364no—
222C2×Dic3#63811yescharacteristic
223C2×Dic3#63811yescharacteristic
224Dic6—98no—
225Dic6—92no—
226Dic6—188no—
227Dic6—184no—
228Dic6—181no—
229Dic6—364no—
230C3×C8—181no—
231C3⋊C8—64no—
232C3⋊C8—98no—
233C3⋊C8—92no—
234C3⋊C8—188no—
235C2×C32—24no—
236C2×C32—24no—
237C2×C32—24no—
238C2×C32#67⋊C411yescharacteristic
239C2×C32#91611yescharacteristic
240C2×C32D3×Q1611yescharacteristic
241C2×C32C4×#6714yes—
242C2×C32C2×#21614yes—
243C2×C32#95014yes—
244C2×Q8—271no—
245C2×Q8D3×C3⋊D311yescharacteristic
246Q16—274no—
247C2×C8—91no—
248C2×C8—271no—
249C4⋊C4—31no—
250C4⋊C4—61no—
251C4⋊C4—91no—
252C4⋊C4—541no—
253C42—31no—
254C42—61no—
255C42—541no—
256C22×C3—24no—
257C22×C3D3×#6714yes—
258C22×C3C2×#63811yescharacteristic
259C3×C4—22no—
260C3×C4—28no—
261C3×C4—24no—
262C3×C4—24no—
263C3×C4—38no—
264C3×C4—48no—
265C3×C4—64no—
266C3×C4—124no—
267C3×C4—361no—
268C3×C4#380414yes—
269C3×C4C2×#62414yes—
270C3×C4C2×C4×C3⋊D311yescharacteristic
271C3×C4#382811yescharacteristic
272Dic3—21no—
273Dic3—41no—
274Dic3—124no—
275Dic3—182no—
276Dic3—188no—
277Dic3—368no—
278Dic3#375612yescomplemented
279C32—24no—
280C32#479011yescharacteristic, complemented
281C32C4×#21614yescomplemented
282Q8—21no—
283Q8—272no—
284Q8—541no—
285Q8#628912yes—
286C2×C4—21no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—61no—
290C2×C4—181no—
291C2×C4—541no—
292C2×C4C2×D3×C3⋊D311yescharacteristic
293C8—181no—
294C8—272no—
295C2×C3—24no—
296C2×C3—24no—
297C2×C3—24no—
298C2×C3D3×#21614yes—
299C2×C3#6394714yes—
300C2×C3#6395214yes—
301C2×C3C4×#63811yescharacteristic
302C2×C3C2×#375611yescharacteristic
303C2×C3#6415911yescharacteristic
304C22D3×#63811yescharacteristic
305C4—22no—
306C4—32no—
307C4—61no—
308C4—121no—
309C4—361no—
310C4—542no—
311C4#8903111yescharacteristic
312C4C2×#628911yescharacteristic
313C3—24no—
314C3#1059153714yescomplemented
315C3C4×#375611yescharacteristic, complemented
316C2D3×#375611yescharacteristic
317C2#1172358011yescharacteristic
318C2#1172361811yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socle, prime radicalC22×C33C2×D4
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33C4×Q16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
12. #88863C4semidirect
88. C32×Dic3Q16semidirect
192. C3×Dic3#216semidirect
200. C33C4×Q16semidirect
278. Dic3#3756semidirect
280. C32#4790semidirect
281. C32C4×#216semidirect
314. C3#10591537semidirect
315. C3C4×#3756semidirect

Maximal subgroups

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

subgroupindex
C2×#888632
C32×Q8×Dic32
#117233992
C3×#636622
#117220562
#117220412
Dic3×C3⋊(C3⋊C8)2
C4×#37563
#105915373