Finite Group Databasebeta

← #13159025

Subgroups — #13159025

Order 1728.

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

normal subgroups only

This group has 928 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
1C4×#6308C211yesmaximal, characteristic
2C32×C4×Dic6C211yesmaximal, characteristic
3C2×#88799C211yesmaximal, characteristic
4C3×#63630C211yesmaximal, characteristic
5#11722054C211yesmaximal, characteristic
6#11722041C211yesmaximal, characteristic
7#11722038C211yesmaximal, characteristic
8#10592729—31nomaximal
9#10590383—34nomaximal
10C2×#6308C2211yescharacteristic
11C2×C32×Dic6C2211yescharacteristic
12C33×C42C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14#88857—21no—
15C4×C3⋊(C3⋊Dic3)—21no—
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4—21no—
20C32×C4×Dic3—21no—
21#88799C414yescomplemented
22C32×C4×Q8—31no—
23C2×#3756—31no—
24C4×#634—31no—
25C4×#634—34no—
26C4×#634—64no—
27C3×C4×Dic6D314yes—
28C2×#3719—34no—
29#63662—31no—
30#63637—31no—
31#63635—31no—
32#63630D311yescharacteristic
33C3×#803—34no—
34#63573—34no—
35#63567—34no—
36#63566—34no—
37C2×C33×C4D411yescharacteristic
38C2×C33×C4D411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)—21no—
41C2×C3⋊(C3⋊Dic3)—21no—
42#6308—21no—
43#6308C2×C412yes—
44C2×C32×Dic3—21no—
45C2×C32×Dic3—21no—
46C32×Dic6—21no—
47C32×Dic6C2×C412yes—
48C3×C3⋊(C3⋊C8)—21no—
49C3×C3⋊(C3⋊C8)C2×C412yes—
50#5030—94no—
51#4716—91no—
52C2×C32×Q8—31no—
53C2×#634—31no—
54C2×#634—34no—
55C2×#634—64no—
56C2×C3×Dic6C2×D314yes—
57C32×C4⋊C4—31no—
58C32×C4⋊C4—61no—
59C32×C42—24no—
60C32×C42—61no—
61C32×C42C2×D311yescharacteristic
62C32×C42C2×D314yes—
63#3756—34no—
64C4⋊(C3⋊Dic3)—31no—
65C4⋊(C3⋊Dic3)—34no—
66C4⋊(C3⋊Dic3)—64no—
67#3750—68no—
68#3750—61no—
69#3750—64no—
70C4×C3⋊Dic3—68no—
71C4×C3⋊Dic3—61no—
72C4×C3⋊Dic3—64no—
73C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C3×C4⋊Dic3C2×D314yes—
76C3×Dic3⋊C4—24no—
77C3×C4×Dic3—24no—
78C2×C3×C3⋊C8—34no—
79C2×C3×C3⋊C8—34no—
80#3719—316no—
81C22×C33C2×D411yescharacteristic
82C33×C4—21no—
83C33×C4—21no—
84C33×C4C22×C411yescharacteristic
85C33×C4#4111yescharacteristic
86C3⋊(C3⋊Dic3)—22no—
87C3⋊(C3⋊Dic3)—41no—
88C32×Dic3—22no—
89C32×Dic3—41no—
90C3×C4×Q8—34no—
91C2×#216—94no—
92C2×Dic12—91no—
93C4×Dic6—94no—
94C4×Dic6—91no—
95C4×Dic6—184no—
96C4×Dic6C3⋊D311yescharacteristic
97C3×#97—91no—
98#835—94no—
99#817—91no—
100#816—91no—
101#816—91no—
102#810—94no—
103#808—94no—
104#803—34no—
105C32×Q8—32no—
106C32×Q8—61no—
107C2×C32×C4—24no—
108C2×C32×C4—24no—
109C2×C32×C4—24no—
110C2×C32×C4—61no—
111C2×C32×C4—61no—
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—61no—
120C2×C3⋊Dic3—64no—
121C2×C3⋊Dic3—68no—
122C2×C3⋊Dic3—61no—
123C2×C3⋊Dic3—64no—
124#634—32no—
125#634—38no—
126#634—68no—
127#634—68no—
128#634—64no—
129#634—61no—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—24no—
132C3×Dic6—24no—
133C3×Dic6C4×D318yes—
134C3⋊(C3⋊C8)—21no—
135C3⋊(C3⋊C8)C4×D312yes—
136C3×C3⋊C8—38no—
137C3×C3⋊C8—64no—
138#441—271noSylow
139C2×C33C4×D411yescharacteristic
140C2×C33#13711yescharacteristic
141C2×C33#13711yescharacteristic
142C2×C3×Q8—34no—
143C2×Dic6—94no—
144C2×Dic6—91no—
145C2×Dic6—184no—
146C2×Dic6C2×C3⋊D311yescharacteristic
147C2×C3×C8—91no—
148C2×C3×C8—91no—
149C3×C4⋊C4—34no—
150C3×C4⋊C4—64no—
151C3×C42—24no—
152C3×C42—64no—
153C3×C42D3214yes—
154C3×C42C2×C3⋊D311yescharacteristic
155#216—916no—
156C4⋊Dic3—94no—
157C4⋊Dic3—91no—
158C4⋊Dic3—184no—
159C4⋊Dic3C2×C3⋊D311yescharacteristic
160Dic3⋊C4—21no—
161Dic3⋊C4—188no—
162Dic3⋊C4—184no—
163Dic3⋊C4—181no—
164C4×Dic3—21no—
165C4×Dic3—188no—
166C4×Dic3—184no—
167C4×Dic3—181no—
168C2×C3⋊C8—34no—
169C2×C3⋊C8—34no—
170Dic12—94no—
171C22×C32—24no—
172C22×C32C2×D1211yescharacteristic
173C22×C32C2×#6714yes—
174C32×C4—28no—
175C32×C4—21no—
176C32×C4—21no—
177C32×C4—24no—
178C32×C4—24no—
179C32×C4—24no—
180C32×C4—24no—
181C32×C4—44no—
182C32×C4—62no—
183C32×C4—121no—
184C32×C4C2×C4×D314yes—
185C32×C4C2×C4×D311yescharacteristic
186C32×C4#23514yes—
187C32×C4#23511yescharacteristic
188C3⋊Dic3—616no—
189C3⋊Dic3—62no—
190C3⋊Dic3—68no—
191C3⋊Dic3—128no—
192C3⋊Dic3—121no—
193C3⋊Dic3—124no—
194C3×Dic3—28no—
195C3×Dic3—44no—
196C2×Q16—271no—
197C4×Q8—31no—
198C4×Q8—271no—
199#106—271no—
200#103—271no—
201#103—271no—
202#97—91no—
203C33#44111yescharacteristic, complemented, Sylow
204C3×Q8—38no—
205C3×Q8—64no—
206C2×C3×C4—24no—
207C2×C3×C4—24no—
208C2×C3×C4—24no—
209C2×C3×C4—64no—
210C2×C3×C4—64no—
211C2×C3×C4#62614yes—
212C2×C3×C4#62614yes—
213C2×C3×C4#63811yescharacteristic
214C2×C3×C4#63811yescharacteristic
215C2×C3×C4C2×D3214yes—
216C2×C3×C4C22×C3⋊D311yescharacteristic
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—188no—
220C2×Dic3—184no—
221C2×Dic3—181no—
222C2×Dic3—188no—
223C2×Dic3—184no—
224C2×Dic3—181no—
225Dic6—21no—
226Dic6—98no—
227Dic6—92no—
228Dic6—188no—
229Dic6—188no—
230Dic6—184no—
231Dic6—181no—
232Dic6C4×C3⋊D312yes—
233C3×C8—92no—
234C3×C8—181no—
235C3⋊C8—38no—
236C3⋊C8—64no—
237C2×C32—24no—
238C2×C32—24no—
239C2×C32—24no—
240C2×C32C4×D1211yescharacteristic
241C2×C32#90911yescharacteristic
242C2×C32#90911yescharacteristic
243C2×C32C4×#6714yes—
244C2×C32#94214yes—
245C2×C32#95114yes—
246C2×Q8—31no—
247C2×Q8—271no—
248Q16—274no—
249C2×C8—91no—
250C2×C8—91no—
251C4⋊C4—31no—
252C4⋊C4—61no—
253C4⋊C4—271no—
254C4⋊C4—541no—
255C42—61no—
256C42—541no—
257C42D3×C3⋊D311yescharacteristic
258C22×C3—24no—
259C22×C3C2×#62614yes—
260C22×C3C2×#63811yescharacteristic
261C3×C4—28no—
262C3×C4—24no—
263C3×C4—24no—
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—24no—
267C3×C4—24no—
268C3×C4—44no—
269C3×C4—68no—
270C3×C4—124no—
271C3×C4#379814yes—
272C3×C4C4×D3214yes—
273C3×C4C2×C4×C3⋊D311yescharacteristic
274C3×C4#382811yescharacteristic
275Dic3—22no—
276Dic3—41no—
277Dic3—1816no—
278Dic3—182no—
279Dic3—188no—
280Dic3—368no—
281Dic3—364no—
282Dic3—361no—
283C32—24no—
284C32#471611yescharacteristic, complemented
285C32#503014yescomplemented
286Q8—32no—
287Q8—61no—
288Q8—272no—
289Q8—541no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—541no—
293C2×C4—541no—
294C2×C4#629111yescharacteristic
295C2×C4#629111yescharacteristic
296C2×C4C2×D3×C3⋊D311yescharacteristic
297C8—92no—
298C8—181no—
299C2×C3—24no—
300C2×C3—24no—
301C2×C3—24no—
302C2×C3#6383114yes—
303C2×C3#6383314yes—
304C2×C3C4×#62614yes—
305C2×C3C4×#63811yescharacteristic
306C2×C3#6415111yescharacteristic
307C2×C3#6416011yescharacteristic
308C22C2×#629111yescharacteristic
309C4—21no—
310C4—21no—
311C4—62no—
312C4—121no—
313C4—542no—
314C4—1081no—
315C4#8902011yescharacteristic
316C4C4×D3×C3⋊D311yescharacteristic
317C3—24no—
318C3#1059038314yescomplemented
319C3#1059272911yescharacteristic, complemented
320C2#1172337911yescharacteristic
321C2#1172338111yescharacteristic
322C2C4×#629111yescharacteristic
subgroupG/H
centerC22C2×#6291
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#441
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
21. #88799C4semidirect
203. C33#441semidirect
284. C32#4716semidirect
285. C32#5030semidirect
318. C3#10590383semidirect
319. C3#10592729semidirect

Maximal subgroups

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

subgroupindex
C4×#63082
C32×C4×Dic62
C2×#887992
C3×#636302
#117220542
#117220412
#117220382
#105927293
#105903833