Finite Group Databasebeta

← #13173429

Subgroups — #13173429

Order 1728.

688 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 89 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×C3×Q8×C3⋊D3C211yesmaximal, characteristic
2#11724514C211yesmaximal, characteristic
3#11724511C211yesmaximal, characteristic
4#11723109C214yesmaximal, complemented
5#10593829—31nomaximal
6C2×C33×Q8C411yescharacteristic, complemented
7C3×Q8×C3⋊D3—21no—
8C3×Q8×C3⋊D3—21no—
9C3×Q8×C3⋊D3C412yescomplemented
10C3×Q8×C3⋊D3C2212yes—
11C2×C3×C4×C3⋊D3—21no—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634—21no—
14C2×C3×#634C411yescharacteristic, complemented
15#88994—21no—
16#88993—21no—
17C4⋊(C3⋊#157)C2212yes—
18C4×C3⋊#157—21no—
19#88989—21no—
20#88988C2212yes—
21C2×Q8×C3⋊D3D311yescharacteristic
22C2×C3×D3×Q8—62no—
23#64286—31no—
24#64283—31no—
25#63784—34no—
26C22×C3×C3⋊D3D411yescharacteristic
27C2×C3⋊#157—22no—
28C33×Q8—21no—
29C33×Q8C2×C412yes—
30C2×C33×C4—21no—
31C2×C33×C4C2×C411yescharacteristic
32C2×C3×C3⋊Dic3—21no—
33C2×C3×C3⋊Dic3D411yescharacteristic
34C3×C4×C3⋊D3—22no—
35C3×C4×C3⋊D3—22no—
36C3×C4×C3⋊D3C2×C411yescharacteristic
37C3×C4×C3⋊D3D411yescharacteristic
38C3×C4×C3⋊D3D411yescharacteristic
39C3×C4×C3⋊D3C2311yescharacteristic
40C3×#634—21no—
41C3×#634—22no—
42C3×#634—22no—
43C3×#634C2×C412yes—
44C3⋊#622—22no—
45#5225—91nomaximal
46C2×C32×Q8—22no—
47C2×C32×Q8—42no—
48C2×C32×Q8Dic311yescharacteristic, complemented
49Q8×C3⋊D3—21no—
50Q8×C3⋊D3—21no—
51Q8×C3⋊D3Dic312yescomplemented
52Q8×C3⋊D3C2×D312yes—
53C2×C4×C3⋊D3—21no—
54C2×C4×C3⋊D3C2×D311yescharacteristic
55C2×#634—21no—
56C2×#634Dic311yescharacteristic, complemented
57C3×D3×Q8—64no—
58C3×D3×Q8—64no—
59C3×D3×Q8—64no—
60C3×D3×Q8—64no—
61C2×C3×C4×D3—62no—
62C2×C3×C4×D3—64no—
63C2×C3×Dic6—62no—
64C2×C3×Dic6—64no—
65C22⋊#157—61no—
66#3792—61no—
67C4⋊#157—32no—
68C4×#157—61no—
69#3788—61no—
70#3787—32no—
71C22×C33C22⋊C411yescharacteristic
72C2×C3×C3⋊D3—22no—
73C2×C3×C3⋊D3C22⋊C411yescharacteristic
74C2×C3×C3⋊D3C2×D411yescharacteristic
75C3⋊#157—42no—
76C33×C4—22no—
77C33×C4C22⋊C411yescharacteristic
78C33×C4C22×C411yescharacteristic
79C3×C3⋊Dic3—22no—
80C3×C3⋊Dic3C22⋊C411yescharacteristic
81C3×C3⋊Dic3C2×D411yescharacteristic
82C22×C3×Q8—91no—
83C2×D3×Q8—62no—
84#926—91no—
85C2×#208—91no—
86#835—94no—
87C22×C3⋊D3#6711yescharacteristic
88C22×C3×D3—62no—
89C2×#157—62no—
90C32×Q8—21no—
91C32×Q8—24no—
92C32×Q8—24no—
93C32×Q8—44no—
94C32×Q8—44no—
95C32×Q8C2×Dic312yes—
96C2×C32×C4—21no—
97C2×C32×C4—24no—
98C2×C32×C4—22no—
99C2×C32×C4—44no—
100C2×C32×C4—42no—
101C2×C32×C4C2×Dic311yescharacteristic
102C2×C3⋊Dic3—21no—
103C2×C3⋊Dic3#6711yescharacteristic
104C4×C3⋊D3—22no—
105C4×C3⋊D3—22no—
106C4×C3⋊D3C2×Dic311yescharacteristic
107C4×C3⋊D3#6711yescharacteristic
108C4×C3⋊D3#6711yescharacteristic
109C4×C3⋊D3C22×D311yescharacteristic
110#634—21no—
111#634—22no—
112#634—22no—
113#634C2×Dic312yes—
114C2×C3×Dic3—62no—
115C2×C3×Dic3—64no—
116C3×C4×D3—62no—
117C3×C4×D3—62no—
118C3×C4×D3—62no—
119C3×C4×D3—62no—
120C3×C4×D3—68no—
121C3×C4×D3—68no—
122C3×Dic6—64no—
123C3×Dic6—68no—
124C3×Dic6—68no—
125C3×Dic6—64no—
126#622—62no—
127#419—271noSylow
128C2×C33—21no—
129C2×C33C2×C22⋊C411yescharacteristic
130C3×C3⋊D3—21no—
131C3×C3⋊D3#13712yes—
132C2×C3×Q8—22no—
133C2×C3×Q8—42no—
134C2×C3×Q8—92no—
135C2×C3×Q8—92no—
136C2×C3×Q8—91no—
137C2×C3×Q8—181no—
138C2×C3×Q8—181no—
139C2×C3×Q8—181no—
140C2×C3×Q8#15711yescharacteristic, complemented
141C22×C3×C4—91no—
142C22×C3×C4—181no—
143D3×Q8—64no—
144D3×Q8—64no—
145D3×Q8—64no—
146D3×Q8—64no—
147C2×C4×D3—62no—
148C2×C4×D3—64no—
149C2×Dic6—62no—
150C2×Dic6—64no—
151#217—181no—
152C4⋊Dic3—92no—
153C4×Dic3—181no—
154#208—181no—
155#208—181no—
156C2×C3⋊C8—92no—
157C22×C32—22no—
158C22×C32—42no—
159C22×C32#21711yescharacteristic
160C2×C3⋊D3—22no—
161C2×C3⋊D3#21711yescharacteristic
162C2×C3⋊D3C2×#6711yescharacteristic
163C2×C3×D3—62no—
164C2×C3×D3—62no—
165C2×C3×D3—68no—
166#157—122no—
167C32×C4—22no—
168C32×C4—28no—
169C32×C4—22no—
170C32×C4—22no—
171C32×C4—48no—
172C32×C4—42no—
173C32×C4—42no—
174C32×C4#21711yescharacteristic
175C32×C4C22×Dic311yescharacteristic
176C3⋊Dic3—22no—
177C3⋊Dic3#21711yescharacteristic
178C3⋊Dic3C2×#6711yescharacteristic
179C3×Dic3—62no—
180C3×Dic3—62no—
181C3×Dic3—68no—
182C22×Q8—91no—
183C2×M16—271no—
184#117—271no—
185#103—274no—
186C33#41911yescharacteristic, complemented, Sylow
187C23×C3—91no—
188C22×D3—62no—
189C3×Q8—24no—
190C3×Q8—21no—
191C3×Q8—24no—
192C3×Q8—44no—
193C3×Q8—44no—
194C3×Q8—92no—
195C3×Q8—182no—
196C3×Q8—182no—
197C3×Q8—181no—
198C3×Q8C2×#15712yes—
199C2×C3×C4—24no—
200C2×C3×C4—21no—
201C2×C3×C4—22no—
202C2×C3×C4—44no—
203C2×C3×C4—42no—
204C2×C3×C4—91no—
205C2×C3×C4—91no—
206C2×C3×C4—91no—
207C2×C3×C4—91no—
208C2×C3×C4—91no—
209C2×C3×C4—182no—
210C2×C3×C4—182no—
211C2×C3×C4—181no—
212C2×C3×C4C2×#15711yescharacteristic
213C2×Dic3—62no—
214C2×Dic3—64no—
215C2×Dic3—182no—
216C4×D3—62no—
217C4×D3—62no—
218C4×D3—62no—
219C4×D3—62no—
220C4×D3—68no—
221C4×D3—68no—
222Dic6—64no—
223Dic6—68no—
224Dic6—68no—
225Dic6—64no—
226C3⋊C8—182no—
227C2×C32—21no—
228C2×C32—24no—
229C2×C32—22no—
230C2×C32—44no—
231C2×C32—42no—
232C2×C32C2×#21711yescharacteristic
233C3⋊D3—21no—
234C3⋊D3#94212yes—
235C3×D3—64no—
236C3×D3—64no—
237C2×Q8—92no—
238C2×Q8—92no—
239C2×Q8—91no—
240C2×Q8—181no—
241C2×Q8—181no—
242C2×Q8—181no—
243C2×Q8C3⋊#15711yescharacteristic, complemented
244C22×C4—91no—
245C22×C4—181no—
246M16—541no—
247M16—541no—
248C2×C8—272no—
249C4⋊C4—272no—
250C22⋊C4—541no—
251C42—541no—
252C22×C3—22no—
253C22×C3—42no—
254C22×C3—91no—
255C22×C3—91no—
256C22×C3—182no—
257C22×C3C22⋊#15711yescharacteristic
258C2×D3—62no—
259C2×D3—62no—
260C2×D3—68no—
261C3×C4—28no—
262C3×C4—22no—
263C3×C4—22no—
264C3×C4—22no—
265C3×C4—48no—
266C3×C4—42no—
267C3×C4—42no—
268C3×C4—91no—
269C3×C4—91no—
270C3×C4—182no—
271C3×C4C22⋊#15711yescharacteristic
272C3×C4C22×#15711yescharacteristic
273Dic3—62no—
274Dic3—62no—
275Dic3—68no—
276Dic3—362no—
277C32—22no—
278C32—42no—
279C32#522511yescharacteristic, complemented
280C23—91no—
281Q8—21no—
282Q8—92no—
283Q8—182no—
284Q8—182no—
285Q8—181no—
286Q8C2×C3⋊#15712yes—
287C2×C4—21no—
288C2×C4—91no—
289C2×C4—91no—
290C2×C4—91no—
291C2×C4—91no—
292C2×C4—91no—
293C2×C4—182no—
294C2×C4—182no—
295C2×C4—181no—
296C2×C4—542no—
297C2×C4C2×C3⋊#15711yescharacteristic
298C8—542no—
299C2×C3—24no—
300C2×C3—21no—
301C2×C3—22no—
302C2×C3—44no—
303C2×C3—42no—
304C2×C3—92no—
305C2×C3—181no—
306C2×C3C2×C22⋊#15711yescharacteristic
307D3—64no—
308D3—64no—
309C22—91no—
310C22—91no—
311C22—182no—
312C22#8899411yescharacteristic
313C4—22no—
314C4—91no—
315C4—91no—
316C4—182no—
317C4—1082no—
318C4#8899411yescharacteristic
319C4C22×C3⋊#15711yescharacteristic
320C3—22no—
321C3—42no—
322C3#1059382911yescharacteristic, complemented
323C2—21no—
324C2—92no—
325C2—181no—
326C2C2×#8899411yescharacteristic
subgroupG/H
centerC2C2×#88994
commutatorC33×C4C22×C4
FrattiniC4C22×C3⋊#157
FittingC2×C33×Q8C4
socleC2×C33C2×C22⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC3×C4×C3⋊D3C23
hypercenterC2×Q8C3⋊#157
hypocenterC33#419
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
4. #11723109C2semidirect
6. C2×C33×Q8C4semidirect
9. C3×Q8×C3⋊D3C4semidirect
14. C2×C3×#634C4semidirect
48. C2×C32×Q8Dic3semidirect
51. Q8×C3⋊D3Dic3semidirect
56. C2×#634Dic3semidirect
140. C2×C3×Q8#157semidirect
186. C33#419semidirect
243. C2×Q8C3⋊#157semidirect
279. C32#5225semidirect
322. C3#10593829semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×C3⋊D32
#117245142
#117245112
#117231092
#105938293
#52259