Finite Group Databasebeta

← #13173415

Subgroups — #13173415

Order 1728.

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

normal subgroups only

This group has 960 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×D4×C3⋊D3C211yesmaximal, characteristic
2#11724514C211yesmaximal, characteristic, complemented
3#11724511C211yesmaximal, characteristic, complemented
4#11723107C214yesmaximal, complemented
5#10593814—31nomaximal
6C23×C3×C3⋊D3—21no—
7C2×C33×D4C411yescharacteristic, complemented
8C2×C3×#638—21no—
9C3×D4×C3⋊D3—21no—
10C3×D4×C3⋊D3—21no—
11C3×D4×C3⋊D3C412yescomplemented
12C3×D4×C3⋊D3C2212yes—
13C2×C3×C4⋊(C3⋊D3)C411yescharacteristic, complemented
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15#88994—21no—
16#88993—21no—
17C4⋊(C3⋊#157)C2212yescomplemented
18C4×C3⋊#157—21no—
19#88989—21no—
20#88988C2212yescomplemented
21C2×D4×C3⋊D3D311yescharacteristic
22C2×C3×D3×D4—62no—
23#64286—31no—
24#64283—31no—
25#63782—34no—
26C23×C33—21no—
27C22×C3×C3⋊D3—21no—
28C22×C3×C3⋊D3—22no—
29C22×C3×C3⋊D3—22no—
30C22×C3×C3⋊D3—44no—
31C22×C3×C3⋊D3D411yescharacteristic
32C2×C3⋊#157—22no—
33C33×D4—21no—
34C33×D4C2×C412yes—
35C2×C33×C4C2×C411yescharacteristic
36C3×#638—22no—
37C3×#638—22no—
38C2×C3×C3⋊Dic3D411yescharacteristic
39C3×C4⋊(C3⋊D3)—21no—
40C3×C4⋊(C3⋊D3)C2×C412yes—
41C3×C4×C3⋊D3C2×C411yescharacteristic
42C3×C4×C3⋊D3D411yescharacteristic
43C3×C4×C3⋊D3D411yescharacteristic
44C3×C4×C3⋊D3C2311yescharacteristic
45C3⋊#622—22no—
46#5215—91nomaximal
47C23×C3⋊D3—21no—
48C23×C3×D3—64no—
49C2×C32×D4—22no—
50C2×C32×D4—42no—
51C2×C32×D4Dic311yescharacteristic, complemented
52C2×#638—21no—
53D4×C3⋊D3—21no—
54D4×C3⋊D3—21no—
55D4×C3⋊D3Dic312yescomplemented
56D4×C3⋊D3C2×D312yes—
57C2×C4⋊(C3⋊D3)Dic311yescharacteristic, complemented
58C2×C4×C3⋊D3C2×D311yescharacteristic
59C2×C3×#67—64no—
60C3×D3×D4—64no—
61C3×D3×D4—64no—
62C3×D3×D4—64no—
63C3×D3×D4—64no—
64C2×C3×D12—62no—
65C2×C3×C4×D3—62no—
66C22⋊#157—61no—
67#3792—61no—
68C4⋊#157—32no—
69C4×#157—61no—
70#3788—61no—
71#3787—32no—
72C22×C33—22no—
73C22×C33—42no—
74C22×C33C22⋊C411yescharacteristic
75C2×C3×C3⋊D3—22no—
76C2×C3×C3⋊D3—22no—
77C2×C3×C3⋊D3—44no—
78C2×C3×C3⋊D3—42no—
79C2×C3×C3⋊D3—44no—
80C2×C3×C3⋊D3C22⋊C411yescharacteristic
81C2×C3×C3⋊D3C2×D411yescharacteristic
82C3⋊#157—42no—
83C33×C4C22⋊C411yescharacteristic
84C33×C4C22×C411yescharacteristic
85C3×C3⋊Dic3C22⋊C411yescharacteristic
86C3×C3⋊Dic3C2×D411yescharacteristic
87C22×C3×D4—91no—
88C2×D3×D4—62no—
89#926—91no—
90C2×#208—91no—
91#832—94no—
92C23×C32—21no—
93C23×C32—24no—
94C23×C32—44no—
95C22×C3⋊D3—21no—
96C22×C3⋊D3—22no—
97C22×C3⋊D3—22no—
98C22×C3⋊D3—44no—
99C22×C3⋊D3#6711yescharacteristic
100C22×C3×D3—62no—
101C22×C3×D3—64no—
102C22×C3×D3—68no—
103C22×C3×D3—68no—
104C22×C3×D3—1216no—
105C2×#157—62no—
106C32×D4—21no—
107C32×D4—24no—
108C32×D4—24no—
109C32×D4—44no—
110C32×D4—44no—
111C32×D4C2×Dic312yes—
112C2×C32×C4—22no—
113C2×C32×C4—42no—
114C2×C32×C4C2×Dic311yescharacteristic
115#638—22no—
116#638—22no—
117C2×C3⋊Dic3#6711yescharacteristic
118C4⋊(C3⋊D3)—21no—
119C4⋊(C3⋊D3)C2×Dic312yes—
120C4×C3⋊D3C2×Dic311yescharacteristic
121C4×C3⋊D3#6711yescharacteristic
122C4×C3⋊D3#6711yescharacteristic
123C4×C3⋊D3C22×D311yescharacteristic
124C3×#67—68no—
125C3×#67—68no—
126C2×C3×Dic3—62no—
127C3×D12—64no—
128C3×D12—64no—
129C3×C4×D3—62no—
130C3×C4×D3—62no—
131C3×C4×D3—62no—
132C3×C4×D3—62no—
133#622—62no—
134#418—271noSylow
135C2×C33—21no—
136C2×C33—42no—
137C2×C33C2×C22⋊C411yescharacteristic
138C3×C3⋊D3—21no—
139C3×C3⋊D3—42no—
140C3×C3⋊D3AGL(1,ℤ/8)12yes—
141C24×C3—181no—
142C23×D3—64no—
143C2×C3×D4—22no—
144C2×C3×D4—42no—
145C2×C3×D4—92no—
146C2×C3×D4—92no—
147C2×C3×D4—91no—
148C2×C3×D4—181no—
149C2×C3×D4—181no—
150C2×C3×D4—181no—
151C2×C3×D4#15711yescharacteristic, complemented
152C22×C3×C4—91no—
153C2×#67—64no—
154D3×D4—64no—
155D3×D4—64no—
156D3×D4—64no—
157D3×D4—64no—
158C2×D12—62no—
159C2×C4×D3—62no—
160#217—181no—
161C4⋊Dic3—92no—
162C4×Dic3—181no—
163#208—181no—
164#208—181no—
165C2×C3⋊C8—92no—
166C22×C32—22no—
167C22×C32—28no—
168C22×C32—22no—
169C22×C32—48no—
170C22×C32—42no—
171C22×C32—42no—
172C22×C32—48no—
173C22×C32—88no—
174C22×C32#21711yescharacteristic
175C2×C3⋊D3—22no—
176C2×C3⋊D3—22no—
177C2×C3⋊D3—44no—
178C2×C3⋊D3—42no—
179C2×C3⋊D3—44no—
180C2×C3⋊D3#21711yescharacteristic
181C2×C3⋊D3C2×#6711yescharacteristic
182C2×C3×D3—62no—
183C2×C3×D3—62no—
184C2×C3×D3—68no—
185C2×C3×D3—68no—
186C2×C3×D3—1216no—
187C2×C3×D3—1216no—
188C2×C3×D3—128no—
189#157—122no—
190C32×C4—22no—
191C32×C4—22no—
192C32×C4—42no—
193C32×C4—42no—
194C32×C4#21711yescharacteristic
195C32×C4C22×Dic311yescharacteristic
196C3⋊Dic3#21711yescharacteristic
197C3⋊Dic3C2×#6711yescharacteristic
198C3×Dic3—62no—
199C3×Dic3—62no—
200C22×D4—91no—
201C2×M16—271no—
202#117—271no—
203#102—274no—
204C33#41811yescharacteristic, complemented, Sylow
205C23×C3—24no—
206C23×C3—21no—
207C23×C3—44no—
208C23×C3—91no—
209C23×C3—182no—
210C23×C3—182no—
211C23×C3—181no—
212C23×C3—364no—
213C22×D3—62no—
214C22×D3—64no—
215C22×D3—68no—
216C22×D3—68no—
217C22×D3—1216no—
218C3×D4—24no—
219C3×D4—21no—
220C3×D4—24no—
221C3×D4—44no—
222C3×D4—44no—
223C3×D4—92no—
224C3×D4—182no—
225C3×D4—182no—
226C3×D4—181no—
227C3×D4C2×#15712yes—
228C2×C3×C4—22no—
229C2×C3×C4—42no—
230C2×C3×C4—91no—
231C2×C3×C4—91no—
232C2×C3×C4—91no—
233C2×C3×C4—91no—
234C2×C3×C4—91no—
235C2×C3×C4C2×#15711yescharacteristic
236#67—68no—
237#67—68no—
238C2×Dic3—62no—
239C2×Dic3—182no—
240D12—64no—
241D12—64no—
242C4×D3—62no—
243C4×D3—62no—
244C4×D3—62no—
245C4×D3—62no—
246C3⋊C8—182no—
247C2×C32—21no—
248C2×C32—24no—
249C2×C32—22no—
250C2×C32—44no—
251C2×C32—42no—
252C2×C32—42no—
253C2×C32—48no—
254C2×C32—88no—
255C2×C32C2×#21711yescharacteristic
256C3⋊D3—21no—
257C3⋊D3—42no—
258C3⋊D3#93212yes—
259C3×D3—64no—
260C3×D3—64no—
261C3×D3—128no—
262C24—181no—
263C2×D4—92no—
264C2×D4—92no—
265C2×D4—91no—
266C2×D4—181no—
267C2×D4—181no—
268C2×D4—181no—
269C2×D4C3⋊#15711yescharacteristic, complemented
270C22×C4—91no—
271M16—541no—
272M16—541no—
273C2×C8—272no—
274C4⋊C4—272no—
275C22⋊C4—541no—
276C42—541no—
277C22×C3—28no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—48no—
281C22×C3—48no—
282C22×C3—42no—
283C22×C3—42no—
284C22×C3—88no—
285C22×C3—91no—
286C22×C3—91no—
287C22×C3—182no—
288C22×C3—182no—
289C22×C3—362no—
290C22×C3—364no—
291C22×C3—364no—
292C22×C3C22⋊#15711yescharacteristic
293C2×D3—62no—
294C2×D3—62no—
295C2×D3—68no—
296C2×D3—68no—
297C2×D3—1216no—
298C2×D3—1216no—
299C2×D3—128no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—42no—
303C3×C4—42no—
304C3×C4—91no—
305C3×C4—91no—
306C3×C4C22⋊#15711yescharacteristic
307C3×C4C22×#15711yescharacteristic
308Dic3—62no—
309Dic3—62no—
310Dic3—362no—
311C32—22no—
312C32—42no—
313C32#521511yescharacteristic, complemented
314C23—21no—
315C23—91no—
316C23—182no—
317C23—182no—
318C23—181no—
319C23—364no—
320D4—21no—
321D4—92no—
322D4—182no—
323D4—182no—
324D4—181no—
325D4C2×C3⋊#15712yes—
326C2×C4—91no—
327C2×C4—91no—
328C2×C4—91no—
329C2×C4—91no—
330C2×C4—91no—
331C2×C4—542no—
332C2×C4C2×C3⋊#15711yescharacteristic
333C8—542no—
334C2×C3—24no—
335C2×C3—21no—
336C2×C3—22no—
337C2×C3—44no—
338C2×C3—48no—
339C2×C3—42no—
340C2×C3—42no—
341C2×C3—88no—
342C2×C3—92no—
343C2×C3—181no—
344C2×C3—362no—
345C2×C3C2×C22⋊#15711yescharacteristic
346D3—64no—
347D3—64no—
348D3—128no—
349C22—22no—
350C22—42no—
351C22—91no—
352C22—91no—
353C22—182no—
354C22—182no—
355C22—362no—
356C22—364no—
357C22—364no—
358C22#8899411yescharacteristic
359C4—91no—
360C4—91no—
361C4—1082no—
362C4#8899411yescharacteristic
363C4C22×C3⋊#15711yescharacteristic
364C3—22no—
365C3—42no—
366C3#1059381411yescharacteristic, complemented
367C2—21no—
368C2—42no—
369C2—92no—
370C2—181no—
371C2—362no—
372C2C2×#8899411yescharacteristic
subgroupG/H
centerC2C2×#88994
commutatorC33×C4C22×C4
FrattiniC4C22×C3⋊#157
FittingC2×C33×D4C4
socleC2×C33C2×C22⋊C4
prime radicalC2×C3×D4×C3⋊D3C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×D4C3⋊#157
hypocenterC33#418
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. #11723107C2semidirect
7. C2×C33×D4C4semidirect
11. C3×D4×C3⋊D3C4semidirect
13. C2×C3×C4⋊(C3⋊D3)C4semidirect
17. C4⋊(C3⋊#157)C22semidirect
20. #88988C22semidirect
51. C2×C32×D4Dic3semidirect
55. D4×C3⋊D3Dic3semidirect
57. C2×C4⋊(C3⋊D3)Dic3semidirect
151. C2×C3×D4#157semidirect
204. C33#418semidirect
269. C2×D4C3⋊#157semidirect
313. C32#5215semidirect
366. C3#10593814semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×C3⋊D32
#117245142
#117245112
#117231072
#105938143
#52159