Finite Group Databasebeta

← #13161479

Subgroups — #13161479

Order 1728.

654 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C3×C42×C3⋊D3C211yesmaximal, characteristic
2C3×C4⋊(C3⋊(C3⋊C8))C212yesmaximal, complemented
3#11722072C212yesmaximal
4#11722071C212yesmaximal, complemented
5#10590263—32nomaximal
6#10590260—31nomaximal
7C2×C3×C4×C3⋊D3C411yescharacteristic
8C2×C3×C4×C3⋊D3C411yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C33×C42C2211yescharacteristic
11C3×C4×C3⋊Dic3C411yescharacteristic
12C3×C4×C3⋊Dic3C411yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2214yes—
15C42×C3⋊D3D311yescharacteristic
16C3×C42×D3—32no—
17C3×C42×D3—61no—
18C4⋊(C3⋊(C3⋊C8))D312yescomplemented
19C3×C4⋊(C3⋊C8)—32no—
20C3×C4⋊(C3⋊C8)—32no—
21C3×C4⋊(C3⋊C8)—62no—
22#63557—32no—
23(C2×D3)⋊(C3⋊C8)—34no—
24#63555—32no—
25#63554—34no—
26C22×C3×C3⋊D3C2×C411yescharacteristic
27C2×C33×C4C2×C411yescharacteristic
28C2×C33×C4C2×C411yescharacteristic
29C2×C33×C4C2311yescharacteristic
30C2×C3×C3⋊Dic3C2×C411yescharacteristic
31C2×C3×C3⋊Dic3C2×C411yescharacteristic
32C2×C3×C3⋊Dic3C2×C411yescharacteristic
33C3×C4×C3⋊D3—22no—
34C3×C4×C3⋊D3—22no—
35C3×C4×C3⋊D3—22no—
36C3×C3⋊(C3⋊C8)—24no—
37#4926—91no—
38#4835—92no—
39C2×C4×C3⋊D3Dic311yescharacteristic
40C2×C4×C3⋊D3Dic311yescharacteristic
41C2×C4×C3⋊D3C2×D311yescharacteristic
42C2×C3×C4×D3—32no—
43C2×C3×C4×D3—32no—
44C2×C3×C4×D3—32no—
45C2×C3×C4×D3—61no—
46C2×C3×C4×D3—61no—
47C2×C3×C4×D3—61no—
48C32×C42—22no—
49C32×C42—21no—
50C32×C42—41no—
51C32×C42C2×D312yes—
52C32×C42C2×D311yescharacteristic
53C4×C3⋊Dic3Dic311yescharacteristic
54C4×C3⋊Dic3Dic311yescharacteristic
55C4×C3⋊Dic3C2×D311yescharacteristic
56C2×C3⋊(C3⋊C8)C2×D314yes—
57C3×C4×Dic3—32no—
58C3×C4×Dic3—32no—
59C3×C4×Dic3—32no—
60C3×C4×Dic3—61no—
61C3×C4×Dic3—61no—
62C3×C4×Dic3—61no—
63C2×C3×C3⋊C8—34no—
64C2×C3×C3⋊C8—34no—
65C2×C3×C3⋊C8—64no—
66C22×C33C22×C411yescharacteristic
67C2×C3×C3⋊D3—21no—
68C2×C3×C3⋊D3—21no—
69C2×C3×C3⋊D3M1612yes—
70C33×C4—21no—
71C33×C4M1612yes—
72C33×C4#4112yes—
73C3×C3⋊Dic3—21no—
74C3×C3⋊Dic3—21no—
75C3×C3⋊Dic3—21no—
76C2×C3×C42—91no—
77C42×D3—32no—
78C42×D3—61no—
79C3×C4⋊C8—92no—
80#830—92no—
81(C2×D3)⋊C8—94no—
82#815—94no—
83C4⋊(C3⋊C8)—32no—
84C4⋊(C3⋊C8)—32no—
85C4⋊(C3⋊C8)—62no—
86#803—92no—
87C22×C3⋊D3C2×Dic311yescharacteristic
88C22×C3×D3—32no—
89C22×C3×D3—61no—
90C2×C32×C4—22no—
91C2×C32×C4—22no—
92C2×C32×C4—21no—
93C2×C32×C4—21no—
94C2×C32×C4—22no—
95C2×C32×C4—21no—
96C2×C32×C4—41no—
97C2×C32×C4—41no—
98C2×C32×C4—41no—
99C2×C32×C4C4×D312yes—
100C2×C32×C4C4×D312yes—
101C2×C32×C4C2×Dic311yescharacteristic
102C2×C32×C4C2×Dic311yescharacteristic
103C2×C32×C4C22×D311yescharacteristic
104C2×C32×C4C22×D312yes—
105C2×C3⋊Dic3C2×Dic311yescharacteristic
106C2×C3⋊Dic3C2×Dic311yescharacteristic
107C2×C3⋊Dic3C2×Dic311yescharacteristic
108C4×C3⋊D3—22no—
109C4×C3⋊D3—22no—
110C4×C3⋊D3—22no—
111C2×C3×Dic3—32no—
112C2×C3×Dic3—32no—
113C2×C3×Dic3—32no—
114C2×C3×Dic3—61no—
115C2×C3×Dic3—61no—
116C2×C3×Dic3—61no—
117C3×C4×D3—64no—
118C3×C4×D3—64no—
119C3×C4×D3—64no—
120C3×C4×D3—64no—
121C3×C4×D3—64no—
122C3×C4×D3—62no—
123C3×C4×D3—62no—
124C3⋊(C3⋊C8)—24no—
125C3×C3⋊C8—64no—
126C3×C3⋊C8—64no—
127C3×C3⋊C8—124no—
128#432—271noSylow
129C2×C33#11711yescharacteristic
130C2×C33C2×M1611yescharacteristic
131C2×C33C2×M1611yescharacteristic
132C3×C3⋊D3—22no—
133C22×C3×C4—91no—
134C22×C3×C4—91no—
135C22×C3×C4—91no—
136C2×C4×D3—32no—
137C2×C4×D3—32no—
138C2×C4×D3—32no—
139C2×C4×D3—61no—
140C2×C4×D3—61no—
141C2×C4×D3—61no—
142C2×C3×C8—94no—
143C3×C42—22no—
144C3×C42—21no—
145C3×C42—41no—
146C3×C42—91no—
147C3×C42—91no—
148C3×C42—91no—
149C3×C42D3212yes—
150C3×C42D3211yescharacteristic
151C4×Dic3—32no—
152C4×Dic3—32no—
153C4×Dic3—32no—
154C4×Dic3—61no—
155C4×Dic3—61no—
156C4×Dic3—61no—
157C2×C3⋊C8—34no—
158C2×C3⋊C8—34no—
159C2×C3⋊C8—64no—
160C22×C32—22no—
161C22×C32—21no—
162C22×C32—41no—
163C22×C32C2×C4×D312yes—
164C22×C32C22×Dic311yescharacteristic
165C2×C3⋊D3—21no—
166C2×C3⋊D3—21no—
167C2×C3⋊D3#20812yes—
168C2×C3×D3—34no—
169C2×C3×D3—62no—
170C2×C3×D3—62no—
171C2×C3×D3—62no—
172C2×C3×D3—62no—
173C2×C3×D3—62no—
174C32×C4—21no—
175C32×C4—22no—
176C32×C4—22no—
177C32×C4—24no—
178C32×C4—22no—
179C32×C4—22no—
180C32×C4—22no—
181C32×C4—22no—
182C32×C4—42no—
183C32×C4—42no—
184C32×C4—42no—
185C32×C4—42no—
186C32×C4ΓL(1,𝔽25)14yes—
187C32×C4#20812yes—
188C32×C4#23512yes—
189C32×C4#23712yes—
190C32×C4#23912yes—
191C3⋊Dic3—21no—
192C3⋊Dic3—21no—
193C3⋊Dic3—21no—
194C3×Dic3—62no—
195C3×Dic3—62no—
196C3×Dic3—62no—
197C3×Dic3—62no—
198C3×Dic3—62no—
199C3×Dic3—62no—
200C2×C42—91no—
201C4⋊C8—92no—
202C22⋊C8—272no—
203#97—272no—
204C33#43211yescharacteristic, complemented, Sylow
205C23×C3—91no—
206C22×D3—32no—
207C22×D3—61no—
208C2×C3×C4—22no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—21no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—91no—
218C2×C3×C4—91no—
219C2×C3×C4—91no—
220C2×C3×C4—182no—
221C2×C3×C4—182no—
222C2×C3×C4—182no—
223C2×C3×C4D3×Dic312yes—
224C2×C3×C4D3×Dic312yes—
225C2×C3×C4#62411yescharacteristic
226C2×C3×C4#62411yescharacteristic
227C2×C3×C4C2×D3212yes—
228C2×C3×C4C2×D3211yescharacteristic
229C2×Dic3—32no—
230C2×Dic3—32no—
231C2×Dic3—32no—
232C2×Dic3—61no—
233C2×Dic3—61no—
234C2×Dic3—61no—
235C4×D3—64no—
236C4×D3—64no—
237C4×D3—64no—
238C4×D3—64no—
239C4×D3—64no—
240C4×D3—62no—
241C4×D3—62no—
242C3×C8—184no—
243C3⋊C8—64no—
244C3⋊C8—64no—
245C3⋊C8—124no—
246C2×C32—22no—
247C2×C32—21no—
248C2×C32—22no—
249C2×C32—21no—
250C2×C32—22no—
251C2×C32—21no—
252C2×C32—41no—
253C2×C32—41no—
254C2×C32—41no—
255C2×C32#89212yes—
256C2×C32C2×ΓL(1,𝔽25)12yes—
257C2×C32D3×M1612yes—
258C2×C32C2×#20811yescharacteristic
259C2×C32C2×#20811yescharacteristic
260C2×C32#92611yescharacteristic
261C3⋊D3—22no—
262C3×D3—64no—
263C3×D3—64no—
264C22×C4—91no—
265C22×C4—91no—
266C22×C4—91no—
267C2×C8—94no—
268C42—91no—
269C42—91no—
270C42—91no—
271C42#111311yescharacteristic
272C22×C3—21no—
273C22×C3—22no—
274C22×C3—41no—
275C22×C3—92no—
276C22×C3—181no—
277C22×C3—181no—
278C22×C3C2×D3×Dic312yes—
279C22×C3C2×#62411yescharacteristic
280C2×D3—34no—
281C2×D3—62no—
282C2×D3—62no—
283C2×D3—62no—
284C2×D3—62no—
285C2×D3—62no—
286C3×C4—22no—
287C3×C4—22no—
288C3×C4—21no—
289C3×C4—22no—
290C3×C4—22no—
291C3×C4—22no—
292C3×C4—22no—
293C3×C4—24no—
294C3×C4—42no—
295C3×C4—42no—
296C3×C4—42no—
297C3×C4—42no—
298C3×C4—181no—
299C3×C4—181no—
300C3×C4—181no—
301C3×C4#371114yes—
302C3×C4#371212yes—
303C3×C4#379512yes—
304C3×C4#379612yes—
305C3×C4#379912yes—
306Dic3—62no—
307Dic3—62no—
308Dic3—62no—
309Dic3—62no—
310Dic3—62no—
311Dic3—62no—
312C32—21no—
313C32—22no—
314C32—41no—
315C32#483512yescomplemented
316C32#492611yescharacteristic, complemented
317C23—91no—
318C2×C4—91no—
319C2×C4—91no—
320C2×C4—91no—
321C2×C4—182no—
322C2×C4—182no—
323C2×C4—182no—
324C2×C4#629411yescharacteristic
325C2×C4#629411yescharacteristic
326C2×C4C2×#111311yescharacteristic
327C8—184no—
328C2×C3—21no—
329C2×C3—22no—
330C2×C3—21no—
331C2×C3—22no—
332C2×C3—21no—
333C2×C3—22no—
334C2×C3—41no—
335C2×C3—41no—
336C2×C3—41no—
337C2×C3—182no—
338C2×C3D3×#20812yes—
339C2×C3#6381811yescharacteristic
340C2×C3C2×#371112yes—
341C2×C3C2×#371211yescharacteristic
342C2×C3#6384112yes—
343C2×C3#6384911yescharacteristic
344D3—64no—
345D3—64no—
346C22—92no—
347C22—181no—
348C22—181no—
349C22C2×#629411yescharacteristic
350C4—21no—
351C4—181no—
352C4—181no—
353C4—181no—
354C4#8880912yes—
355C4#8904112yes—
356C3—21no—
357C3—22no—
358C3—41no—
359C3#1059026011yescharacteristic, complemented
360C3#1059026312yescomplemented
361C2—182no—
362C2#1172365611yescharacteristic
363C2C2×#8880911yescharacteristic
364C2#1172366711yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×#1113
commutator, socleC22×C33C22×C4
FittingC33×C42C22
prime radicalC22×C3×C3⋊D3C2×C4
cosocleC2×C33×C4C23
hypercenterC42#1113
hypocenterC33#432
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×C4⋊(C3⋊(C3⋊C8))C2semidirect
4. #11722071C2semidirect
18. C4⋊(C3⋊(C3⋊C8))D3semidirect
204. C33#432semidirect
315. C32#4835semidirect
316. C32#4926semidirect
359. C3#10590260semidirect
360. C3#10590263semidirect

Maximal subgroups

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

subgroupindex
C3×C42×C3⋊D32
C3×C4⋊(C3⋊(C3⋊C8))2
#117220722
#117220712
#105902633
#105902603