Finite Group Databasebeta

← #13173411

Subgroups — #13173411

Order 1728.

744 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3830C211yesmaximal, characteristic
2#11724514C211yesmaximal, characteristic, complemented
3#11724511C211yesmaximal, characteristic, complemented
4#11723108C214yesmaximal, complemented
5#10593816—31nomaximal
6C2×C33×D4C411yescharacteristic, complemented
7C2×C3×#638—21no—
8C22×C3×C3⋊Dic3—21no—
9C3×#3830—21no—
10C3×#3830—21no—
11C3×#3830C412yescomplemented
12C3×#3830C2212yes—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C3×#634C411yescharacteristic, complemented
15#88994—21no—
16#88993—21no—
17C4⋊(C3⋊#157)—21no—
18C4×C3⋊#157C2212yescomplemented
19#88989C2212yescomplemented
20#88988—21no—
21C2×#3830D311yescharacteristic
22C2×C3×#237—62no—
23#64286—31no—
24#64283—31no—
25#63783—34no—
26C23×C33—21no—
27C22×C3×C3⋊D3D411yescharacteristic
28C2×C3⋊#157—22no—
29C33×D4—21no—
30C33×D4C2×C412yes—
31C2×C33×C4C2×C411yescharacteristic
32C3×#638—22no—
33C3×#638—22no—
34C2×C3×C3⋊Dic3—21no—
35C2×C3×C3⋊Dic3—22no—
36C2×C3×C3⋊Dic3—22no—
37C2×C3×C3⋊Dic3D411yescharacteristic
38C3×C4×C3⋊D3C2×C411yescharacteristic
39C3×C4×C3⋊D3D411yescharacteristic
40C3×C4×C3⋊D3D411yescharacteristic
41C3×C4×C3⋊D3C2311yescharacteristic
42C3×#634—21no—
43C3×#634C2×C412yes—
44C3⋊#622—22no—
45#5236—91nomaximal
46C2×C32×D4—22no—
47C2×C32×D4—42no—
48C2×C32×D4Dic311yescharacteristic, complemented
49C2×#638—21no—
50C22×C3⋊Dic3—21no—
51#3830—21no—
52#3830—21no—
53#3830Dic312yescomplemented
54#3830C2×D312yes—
55C2×C4×C3⋊D3C2×D311yescharacteristic
56C2×#634Dic311yescharacteristic, complemented
57C2×C3×#67—64no—
58C22×C3×Dic3—64no—
59C3×#237—64no—
60C3×#237—64no—
61C3×#237—64no—
62C3×#237—64no—
63C2×C3×C4×D3—62no—
64C2×C3×Dic6—62no—
65C22⋊#157—61no—
66#3792—61no—
67C4⋊#157—61no—
68C4×#157—32no—
69#3788—32no—
70#3787—61no—
71C22×C33—22no—
72C22×C33—42no—
73C22×C33C22⋊C411yescharacteristic
74C2×C3×C3⋊D3—41no—
75C2×C3×C3⋊D3C22⋊C411yescharacteristic
76C2×C3×C3⋊D3C2×D411yescharacteristic
77C3⋊#157—42no—
78C33×C4C22⋊C411yescharacteristic
79C33×C4C22×C411yescharacteristic
80C3×C3⋊Dic3—22no—
81C3×C3⋊Dic3C22⋊C411yescharacteristic
82C3×C3⋊Dic3C2×D411yescharacteristic
83C2×C3×#41—91no—
84C2×#237—62no—
85#926—91no—
86C2×#208—91no—
87#837—94no—
88C23×C32—21no—
89C23×C32—24no—
90C23×C32—44no—
91C22×C3⋊D3#6711yescharacteristic
92C22×C3×D3—62no—
93C2×#157—62no—
94C32×D4—21no—
95C32×D4—24no—
96C32×D4—24no—
97C32×D4—44no—
98C32×D4—44no—
99C32×D4C2×Dic312yes—
100C2×C32×C4—22no—
101C2×C32×C4—42no—
102C2×C32×C4C2×Dic311yescharacteristic
103#638—22no—
104#638—22no—
105C2×C3⋊Dic3—21no—
106C2×C3⋊Dic3—22no—
107C2×C3⋊Dic3—22no—
108C2×C3⋊Dic3#6711yescharacteristic
109C4×C3⋊D3C2×Dic311yescharacteristic
110C4×C3⋊D3#6711yescharacteristic
111C4×C3⋊D3#6711yescharacteristic
112C4×C3⋊D3C22×D311yescharacteristic
113#634—21no—
114#634C2×Dic312yes—
115C3×#67—68no—
116C3×#67—68no—
117C2×C3×Dic3—62no—
118C2×C3×Dic3—64no—
119C2×C3×Dic3—68no—
120C2×C3×Dic3—68no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C3×C4×D3—62no—
124C3×C4×D3—62no—
125C3×Dic6—64no—
126C3×Dic6—64no—
127#622—62no—
128#421—271noSylow
129C2×C33—21no—
130C2×C33—42no—
131C2×C33C2×C22⋊C411yescharacteristic
132C3×C3⋊D3—21no—
133C3×C3⋊D3—21no—
134C3×#41—92no—
135C3×#41—92no—
136C3×#41—181no—
137C3×#41—181no—
138C2×C3×Q8—91no—
139C2×C3×D4—22no—
140C2×C3×D4—42no—
141C2×C3×D4—181no—
142C2×C3×D4#15711yescharacteristic, complemented
143C22×C3×C4—91no—
144C22×C3×C4—181no—
145C2×#67—64no—
146C22×Dic3—64no—
147#237—64no—
148#237—64no—
149#237—64no—
150#237—64no—
151C2×C4×D3—62no—
152C2×Dic6—62no—
153#217—181no—
154C4⋊Dic3—181no—
155C4×Dic3—92no—
156#208—92no—
157#208—181no—
158C2×C3⋊C8—181no—
159C22×C32—22no—
160C22×C32—28no—
161C22×C32—22no—
162C22×C32—48no—
163C22×C32—42no—
164C22×C32—42no—
165C22×C32—48no—
166C22×C32—88no—
167C22×C32#21711yescharacteristic
168C2×C3⋊D3—41no—
169C2×C3⋊D3#21711yescharacteristic
170C2×C3⋊D3C2×#6711yescharacteristic
171C2×C3×D3—62no—
172C2×C3×D3—62no—
173C2×C3×D3—124no—
174#157—122no—
175C32×C4—22no—
176C32×C4—22no—
177C32×C4—42no—
178C32×C4—42no—
179C32×C4#21711yescharacteristic
180C32×C4C22×Dic311yescharacteristic
181C3⋊Dic3—22no—
182C3⋊Dic3#21711yescharacteristic
183C3⋊Dic3C2×#6711yescharacteristic
184C3×Dic3—62no—
185C3×Dic3—62no—
186C3×Dic3—68no—
187C2×#41—91no—
188C2×M16—271no—
189#117—271no—
190#104—274no—
191C33#42111yescharacteristic, complemented, Sylow
192C23×C3—24no—
193C23×C3—21no—
194C23×C3—44no—
195C23×C3—91no—
196C22×D3—62no—
197C3×Q8—92no—
198C3×Q8—181no—
199C3×D4—24no—
200C3×D4—21no—
201C3×D4—24no—
202C3×D4—44no—
203C3×D4—44no—
204C3×D4—182no—
205C3×D4—182no—
206C3×D4C2×#15712yes—
207C2×C3×C4—22no—
208C2×C3×C4—42no—
209C2×C3×C4—91no—
210C2×C3×C4—91no—
211C2×C3×C4—91no—
212C2×C3×C4—91no—
213C2×C3×C4—91no—
214C2×C3×C4—182no—
215C2×C3×C4—182no—
216C2×C3×C4—181no—
217C2×C3×C4C2×#15711yescharacteristic
218#67—68no—
219#67—68no—
220C2×Dic3—62no—
221C2×Dic3—64no—
222C2×Dic3—68no—
223C2×Dic3—68no—
224C2×Dic3—182no—
225C4×D3—62no—
226C4×D3—62no—
227C4×D3—62no—
228C4×D3—62no—
229Dic6—64no—
230Dic6—64no—
231C3⋊C8—182no—
232C2×C32—21no—
233C2×C32—24no—
234C2×C32—22no—
235C2×C32—44no—
236C2×C32—42no—
237C2×C32—42no—
238C2×C32—48no—
239C2×C32—88no—
240C2×C32C2×#21711yescharacteristic
241C3⋊D3—21no—
242C3⋊D3—21no—
243C3×D3—122no—
244C3×D3—122no—
245#41—92no—
246#41—92no—
247#41—181no—
248#41—181no—
249C2×Q8—91no—
250C2×D4—181no—
251C2×D4C3⋊#15711yescharacteristic, complemented
252C22×C4—91no—
253C22×C4—181no—
254M16—272no—
255M16—541no—
256C2×C8—541no—
257C4⋊C4—541no—
258C22⋊C4—541no—
259C42—272no—
260C22×C3—28no—
261C22×C3—22no—
262C22×C3—22no—
263C22×C3—48no—
264C22×C3—42no—
265C22×C3—48no—
266C22×C3—42no—
267C22×C3—88no—
268C22×C3—91no—
269C22×C3—91no—
270C22×C3—361no—
271C22×C3C22⋊#15711yescharacteristic
272C2×D3—62no—
273C2×D3—62no—
274C2×D3—124no—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—42no—
278C3×C4—42no—
279C3×C4—91no—
280C3×C4—91no—
281C3×C4—182no—
282C3×C4C22⋊#15711yescharacteristic
283C3×C4C22×#15711yescharacteristic
284Dic3—62no—
285Dic3—62no—
286Dic3—68no—
287Dic3—362no—
288C32—22no—
289C32—42no—
290C32#523611yescharacteristic, complemented
291C23—21no—
292C23—91no—
293Q8—92no—
294Q8—181no—
295D4—21no—
296D4—182no—
297D4—182no—
298D4C2×C3⋊#15712yes—
299C2×C4—91no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—91no—
303C2×C4—91no—
304C2×C4—182no—
305C2×C4—182no—
306C2×C4—181no—
307C2×C4—542no—
308C2×C4C2×C3⋊#15711yescharacteristic
309C8—542no—
310C2×C3—24no—
311C2×C3—21no—
312C2×C3—22no—
313C2×C3—44no—
314C2×C3—48no—
315C2×C3—42no—
316C2×C3—42no—
317C2×C3—88no—
318C2×C3—181no—
319C2×C3—181no—
320C2×C3C2×C22⋊#15711yescharacteristic
321D3—122no—
322D3—122no—
323C22—22no—
324C22—42no—
325C22—91no—
326C22—91no—
327C22—361no—
328C22#8899411yescharacteristic
329C4—91no—
330C4—91no—
331C4—182no—
332C4—1082no—
333C4#8899411yescharacteristic
334C4C22×C3⋊#15711yescharacteristic
335C3—22no—
336C3—42no—
337C3#1059381611yescharacteristic, complemented
338C2—21no—
339C2—42no—
340C2—181no—
341C2—181no—
342C2C2×#8899411yescharacteristic
subgroupG/H
centerC2C2×#88994
commutatorC33×C4C22×C4
FrattiniC4C22×C3⋊#157
FittingC2×C33×D4C4
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×#3830C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×D4C3⋊#157
hypocenterC33#421
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. #11724514C2semidirect
3. #11724511C2semidirect
4. #11723108C2semidirect
6. C2×C33×D4C4semidirect
11. C3×#3830C4semidirect
14. C2×C3×#634C4semidirect
18. C4×C3⋊#157C22semidirect
19. #88989C22semidirect
48. C2×C32×D4Dic3semidirect
53. #3830Dic3semidirect
56. C2×#634Dic3semidirect
142. C2×C3×D4#157semidirect
191. C33#421semidirect
251. C2×D4C3⋊#157semidirect
290. C32#5236semidirect
337. C3#10593816semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
#117245142
#117245112
#117231082
#105938163
#52369