Finite Group Databasebeta

← #13161802

Subgroups — #13161802

Order 1728.

1186 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1186 conjugacy classes of subgroups — too many to draw. Only its 199 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
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic
2C2×C3×#3748C212yesmaximal, complemented
3#11722072C214yesmaximal, complemented
4#10591074—31nomaximal
5#10591069—32nomaximal
6C23×C3×C3⋊D3C411yescharacteristic
7C22×C33×C4C2211yescharacteristic
8C22×C3×C3⋊Dic3C411yescharacteristic
9C2×C3×C4×C3⋊D3—24no—
10C2×C3×C4×C3⋊D3C412yes—
11C2×C3×C4×C3⋊D3C2212yes—
12C3×#3748—24no—
13C2×C3×C3⋊(C3⋊C8)C2214yescomplemented
14C22×C4×C3⋊D3D311yescharacteristic
15C22×C3×C4×D3—32no—
16C22×C3×C4×D3—61no—
17C2×#3748D312yescomplemented
18C2×C3×#208—32no—
19C2×C3×#208—32no—
20C2×C3×#208—62no—
21#63557—34no—
22(C2×D3)⋊(C3⋊C8)—38no—
23C23×C33C2×C411yescharacteristic
24C22×C3×C3⋊D3—22no—
25C22×C3×C3⋊D3—24no—
26C22×C3×C3⋊D3C2×C412yes—
27C2×C33×C4C2×C411yescharacteristic
28C2×C33×C4D414yes—
29C2×C33×C4C2311yescharacteristic
30C2×C3×C3⋊Dic3—22no—
31C2×C3×C3⋊Dic3C2×C412yes—
32C3×C4×C3⋊D3—24no—
33C3×C4×C3⋊D3—24no—
34C3×C3⋊(C3⋊C8)—24no—
35#5207—91no—
36#5126—92no—
37C23×C3⋊D3Dic311yescharacteristic
38C23×C3×D3—32no—
39C23×C3×D3—61no—
40C22×C32×C4—22no—
41C22×C32×C4—21no—
42C22×C32×C4—41no—
43C22×C32×C4C2×D312yes—
44C22×C32×C4C2×D311yescharacteristic
45C22×C3⋊Dic3Dic311yescharacteristic
46C2×C4×C3⋊D3—24no—
47C2×C4×C3⋊D3Dic312yes—
48C2×C4×C3⋊D3C2×D312yes—
49C22×C3×Dic3—32no—
50C22×C3×Dic3—61no—
51C2×C3×C4×D3—34no—
52C2×C3×C4×D3—34no—
53C2×C3×C4×D3—62no—
54C2×C3×C4×D3—62no—
55C2×C3×C4×D3—68no—
56C2×C3×C4×D3—64no—
57C2×C3×C4×D3—64no—
58#3748—24no—
59C2×C3⋊(C3⋊C8)C2×D314yescomplemented
60C3×#208—68no—
61C3×#208—64no—
62C3×#208—64no—
63C2×C3×C3⋊C8—34no—
64C2×C3×C3⋊C8—34no—
65C2×C3×C3⋊C8—64no—
66C22×C33—22no—
67C22×C33C22⋊C412yes—
68C22×C33C22×C411yescharacteristic
69C2×C3×C3⋊D3—28no—
70C2×C3×C3⋊D3—22no—
71C2×C3×C3⋊D3—22no—
72C2×C3×C3⋊D3M1614yescomplemented
73C33×C4C22⋊C412yes—
74C33×C4C2×D412yes—
75C3×C3⋊Dic3—22no—
76C23×C3×C4—91no—
77C22×C4×D3—32no—
78C22×C4×D3—61no—
79C2×C3×M16—92no—
80C2×#208—32no—
81C2×#208—32no—
82C2×#208—62no—
83#830—94no—
84(C2×D3)⋊C8—98no—
85C23×C32—22no—
86C23×C32—21no—
87C23×C32—41no—
88C23×C32C4×D312yes—
89C23×C32C2×Dic311yescharacteristic
90C22×C3⋊D3—22no—
91C22×C3⋊D3—24no—
92C22×C3⋊D3C2×Dic312yes—
93C22×C3×D3—34no—
94C22×C3×D3—62no—
95C22×C3×D3—68no—
96C22×C3×D3—68no—
97C22×C3×D3—64no—
98C22×C3×D3—64no—
99C2×C32×C4—22no—
100C2×C32×C4—21no—
101C2×C32×C4—24no—
102C2×C32×C4—24no—
103C2×C32×C4—24no—
104C2×C32×C4—22no—
105C2×C32×C4—21no—
106C2×C32×C4—41no—
107C2×C32×C4—44no—
108C2×C32×C4—41no—
109C2×C32×C4C4×D312yes—
110C2×C32×C4D1214yes—
111C2×C32×C4C2×Dic311yescharacteristic
112C2×C32×C4#6714yes—
113C2×C32×C4#6714yes—
114C2×C32×C4C22×D312yes—
115C2×C32×C4C22×D311yescharacteristic
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3C2×Dic312yes—
118C4×C3⋊D3—24no—
119C4×C3⋊D3—24no—
120C2×C3×Dic3—34no—
121C2×C3×Dic3—62no—
122C2×C3×Dic3—64no—
123C2×C3×Dic3—64no—
124C3×C4×D3—68no—
125C3×C4×D3—68no—
126C3×C4×D3—68no—
127C3×C4×D3—64no—
128C3×C4×D3—64no—
129C3⋊(C3⋊C8)—24no—
130C3×C3⋊C8—68no—
131C3×C3⋊C8—64no—
132C3×C3⋊C8—64no—
133#407—271noSylow
134C2×C33—22no—
135C2×C33C2×C22⋊C411yescharacteristic
136C2×C33C2×M1612yes—
137C3×C3⋊D3—24no—
138C24×C3—91no—
139C23×D3—32no—
140C23×D3—61no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4—41no—
144C22×C3×C4—91no—
145C22×C3×C4—92no—
146C22×C3×C4—92no—
147C22×C3×C4—184no—
148C22×C3×C4D3212yes—
149C22×C3×C4D3211yescharacteristic
150C22×Dic3—32no—
151C22×Dic3—61no—
152C2×C4×D3—34no—
153C2×C4×D3—34no—
154C2×C4×D3—62no—
155C2×C4×D3—62no—
156C2×C4×D3—68no—
157C2×C4×D3—64no—
158C2×C4×D3—64no—
159C3×M16—184no—
160C2×C3×C8—94no—
161#208—68no—
162#208—64no—
163#208—64no—
164C2×C3⋊C8—34no—
165C2×C3⋊C8—34no—
166C2×C3⋊C8—64no—
167C22×C32—22no—
168C22×C32—24no—
169C22×C32—24no—
170C22×C32—24no—
171C22×C32—22no—
172C22×C32—22no—
173C22×C32—21no—
174C22×C32—42no—
175C22×C32—41no—
176C22×C32—44no—
177C22×C32—44no—
178C22×C32(C2×D3)⋊C414yes—
179C22×C32#21712yes—
180C22×C32C2×C4×D312yes—
181C22×C32C22×Dic311yescharacteristic
182C2×C3⋊D3—28no—
183C2×C3⋊D3—22no—
184C2×C3⋊D3—22no—
185C2×C3⋊D3#20814yescomplemented
186C2×C3×D3—38no—
187C2×C3×D3—64no—
188C2×C3×D3—64no—
189C2×C3×D3—616no—
190C2×C3×D3—616no—
191C2×C3×D3—64no—
192C2×C3×D3—64no—
193C2×C3×D3—64no—
194C32×C4—24no—
195C32×C4—22no—
196C32×C4—22no—
197C32×C4—22no—
198C32×C4—22no—
199C32×C4—42no—
200C32×C4—42no—
201C32×C4(C2×D3)⋊C414yes—
202C32×C4#21712yes—
203C32×C4C2×D1212yes—
204C32×C4C2×#6712yes—
205C32×C4C2×#6712yes—
206C3⋊Dic3—22no—
207C3×Dic3—64no—
208C3×Dic3—64no—
209C23×C4—91no—
210C2×M16—92no—
211C22⋊C8—274no—
212C33#40711yescharacteristic, complemented, Sylow
213C23×C3—21no—
214C23×C3—22no—
215C23×C3—41no—
216C23×C3—92no—
217C23×C3—184no—
218C23×C3—182no—
219C23×C3D3×Dic312yes—
220C23×C3#62411yescharacteristic
221C22×D3—34no—
222C22×D3—62no—
223C22×D3—68no—
224C22×D3—64no—
225C22×D3—68no—
226C22×D3—64no—
227C2×C3×C4—22no—
228C2×C3×C4—21no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—22no—
233C2×C3×C4—21no—
234C2×C3×C4—41no—
235C2×C3×C4—44no—
236C2×C3×C4—41no—
237C2×C3×C4—92no—
238C2×C3×C4—182no—
239C2×C3×C4—184no—
240C2×C3×C4—184no—
241C2×C3×C4D3×Dic312yes—
242C2×C3×C4#62411yescharacteristic
243C2×C3×C4#62514yes—
244C2×C3×C4#62614yes—
245C2×C3×C4#62614yes—
246C2×C3×C4C2×D3212yes—
247C2×C3×C4C2×D3211yescharacteristic
248C2×Dic3—34no—
249C2×Dic3—62no—
250C2×Dic3—64no—
251C2×Dic3—64no—
252C4×D3—68no—
253C4×D3—68no—
254C4×D3—68no—
255C4×D3—64no—
256C4×D3—64no—
257C3×C8—184no—
258C3⋊C8—68no—
259C3⋊C8—64no—
260C3⋊C8—64no—
261C2×C32—22no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32—24no—
265C2×C32—22no—
266C2×C32—22no—
267C2×C32—21no—
268C2×C32—41no—
269C2×C32—42no—
270C2×C32—44no—
271C2×C32—44no—
272C2×C32D3×M1614yes—
273C2×C32C2×#20812yes—
274C2×C32C2×(C2×D3)⋊C412yes—
275C2×C32C2×#21711yescharacteristic
276C3⋊D3—24no—
277C3×D3—68no—
278C3×D3—68no—
279C24—91no—
280C22×C4—91no—
281C22×C4—92no—
282C22×C4—92no—
283C22×C4—184no—
284C22×C4#111311yescharacteristic
285M16—184no—
286C2×C8—94no—
287C22×C3—24no—
288C22×C3—24no—
289C22×C3—22no—
290C22×C3—21no—
291C22×C3—22no—
292C22×C3—22no—
293C22×C3—24no—
294C22×C3—44no—
295C22×C3—44no—
296C22×C3—41no—
297C22×C3—42no—
298C22×C3—94no—
299C22×C3—188no—
300C22×C3—182no—
301C22×C3—182no—
302C22×C3(C2×D3)⋊Dic314yes—
303C22×C3#372212yes—
304C22×C3C2×D3×Dic312yes—
305C22×C3C2×#62411yescharacteristic
306C2×D3—38no—
307C2×D3—64no—
308C2×D3—64no—
309C2×D3—616no—
310C2×D3—616no—
311C2×D3—64no—
312C2×D3—64no—
313C2×D3—64no—
314C3×C4—22no—
315C3×C4—22no—
316C3×C4—22no—
317C3×C4—22no—
318C3×C4—24no—
319C3×C4—42no—
320C3×C4—42no—
321C3×C4—182no—
322C3×C4(C2×D3)⋊Dic314yes—
323C3×C4#372212yes—
324C3×C4C2×#62512yes—
325C3×C4C2×#62612yes—
326C3×C4C2×#62612yes—
327Dic3—64no—
328Dic3—64no—
329C32—22no—
330C32—21no—
331C32—41no—
332C32#512612yescomplemented
333C32#520711yescharacteristic, complemented
334C23—92no—
335C23—184no—
336C23—182no—
337C23#629411yescharacteristic
338C2×C4—92no—
339C2×C4—182no—
340C2×C4—184no—
341C2×C4—184no—
342C2×C4#629411yescharacteristic
343C2×C4#629514yes—
344C2×C4C2×#111311yescharacteristic
345C8—184no—
346C2×C3—24no—
347C2×C3—24no—
348C2×C3—22no—
349C2×C3—22no—
350C2×C3—24no—
351C2×C3—21no—
352C2×C3—22no—
353C2×C3—44no—
354C2×C3—44no—
355C2×C3—42no—
356C2×C3—41no—
357C2×C3—184no—
358C2×C3D3×#20814yes—
359C2×C3#6381812yes—
360C2×C3C2×(C2×D3)⋊Dic312yes—
361C2×C3C2×#372211yescharacteristic
362D3—68no—
363D3—68no—
364C22—22no—
365C22—94no—
366C22—188no—
367C22—182no—
368C22—182no—
369C22#8881412yes—
370C22C2×#629411yescharacteristic
371C4—182no—
372C4#8881412yes—
373C4C2×#629512yes—
374C3—21no—
375C3—22no—
376C3—41no—
377C3#1059106912yescomplemented
378C3#1059107411yescharacteristic, complemented
379C2—22no—
380C2—184no—
381C2#1172365612yes—
382C2C2×#8881411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×#1113
commutator, socleC22×C33C22×C4
FittingC22×C33×C4C22
prime radicalC23×C3×C3⋊D3C4
cosocleC2×C33×C4C23
hypercenterC22×C4#1113
hypocenterC33#407
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3748C2semidirect
3. #11722072C2semidirect
13. C2×C3×C3⋊(C3⋊C8)C22semidirect
17. C2×#3748D3semidirect
59. C2×C3⋊(C3⋊C8)C2×D3semidirect
72. C2×C3×C3⋊D3M16semidirect
185. C2×C3⋊D3#208semidirect
212. C33#407semidirect
332. C32#5126semidirect
333. C32#5207semidirect
377. C3#10591069semidirect
378. C3#10591074semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×C3⋊D32
C2×C3×#37482
#117220722
#105910743
#105910693