Finite Group Databasebeta

← #13166364

Subgroups — #13166364

Order 1728.

960 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 960 conjugacy classes of subgroups — too many to draw. Only its 78 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⋊D3×D12C211yesmaximal, characteristic
2#11724511C211yesmaximal, characteristic, complemented
3C3×#64286C211yesmaximal, characteristic, complemented
4#11722706C214yesmaximal, complemented
5#10593814—31nomaximal
6C22×D3×C3⋊D3—21no—
7C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic, complemented
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×D12C411yescharacteristic, complemented
10C2×#6291—21no—
11C3⋊D3×D12—21no—
12C3⋊D3×D12—21no—
13C3⋊D3×D12C412yescomplemented
14C3⋊D3×D12C2212yes—
15#88993—21no—
16#88989—21no—
17#88988C2212yescomplemented
18C3×C22⋊#157—21no—
19C3×C4⋊#157C2212yescomplemented
20C3×C4×#157—21no—
21C2×D4×C3⋊D3—31no—
22C2×D3×D12—62no—
23#64286D311yescharacteristic, complemented
24#64283—31no—
25#63782—34no—
26C22×C3⋊(C3⋊D3)—21no—
27C22×C3×C3⋊D3D411yescharacteristic
28C22×C32×D3—21no—
29C2×D3×C3⋊D3—22no—
30C2×D3×C3⋊D3—22no—
31C2×D3×C3⋊D3—44no—
32C2×C3×#157—22no—
33C2×C33×C4C2×C411yescharacteristic
34C4⋊(C3⋊(C3⋊D3))—21no—
35C4⋊(C3⋊(C3⋊D3))C2×C412yes—
36C2×C3×C3⋊Dic3D411yescharacteristic
37C3×C4×C3⋊D3C2×C411yescharacteristic
38C3×C4×C3⋊D3D411yescharacteristic
39C3×C4×C3⋊D3D411yescharacteristic
40C3×C4×C3⋊D3C2311yescharacteristic
41C32×D12—21no—
42C32×D12C2×C412yes—
43#6291—22no—
44#6291—22no—
45C3⋊#622—22no—
46#5001—91nomaximal
47C23×C3⋊D3—61no—
48C22×D32—64no—
49C2×C32×D4—31no—
50C2×#638—61no—
51D4×C3⋊D3—32no—
52D4×C3⋊D3—32no—
53D4×C3⋊D3—61no—
54D4×C3⋊D3—61no—
55C2×C4⋊(C3⋊D3)—31no—
56C2×C4⋊(C3⋊D3)—62no—
57C2×C4⋊(C3⋊D3)—122no—
58C2×C4×C3⋊D3C2×D311yescharacteristic
59C2×C3×D12—22no—
60C2×C3×C4×D3—62no—
61C2×#626—64no—
62D3×D12—64no—
63D3×D12—64no—
64D3×D12—64no—
65D3×D12—64no—
66C22⋊#157—21no—
67#3792—61no—
68C4⋊#157C2×D312yescomplemented
69C4×#157—21no—
70#3788—61no—
71#3787—32no—
72C22×C33C22⋊C411yescharacteristic
73C2×C3⋊(C3⋊D3)—22no—
74C2×C3⋊(C3⋊D3)—42no—
75C2×C3×C3⋊D3—22no—
76C2×C3×C3⋊D3C22⋊C411yescharacteristic
77C2×C3×C3⋊D3C2×D411yescharacteristic
78C2×C32×D3—22no—
79C2×C32×D3—42no—
80D3×C3⋊D3—44no—
81D3×C3⋊D3—44no—
82C3×#157—42no—
83C33×C4C22⋊C411yescharacteristic
84C33×C4C22×C411yescharacteristic
85C3×C3⋊Dic3C22⋊C411yescharacteristic
86C3×C3⋊Dic3C2×D411yescharacteristic
87C2×D3×D4—182no—
88C22×D12—91no—
89C3×#117—91no—
90C2×#208—91no—
91#809—94no—
92C23×C32—61no—
93C22×C3⋊D3—61no—
94C22×C3⋊D3—62no—
95C22×C3⋊D3—62no—
96C22×C3⋊D3—64no—
97C22×C3⋊D3—124no—
98C22×C3⋊D3—124no—
99C22×C3⋊D3#6711yescharacteristic
100C22×C3×D3—24no—
101C22×C3×D3—62no—
102C2×D32—68no—
103C2×D32—68no—
104C2×D32—1216no—
105C2×#157—22no—
106C32×D4—32no—
107C32×D4—61no—
108C2×C32×C4—22no—
109C2×C32×C4—42no—
110C2×C32×C4C4×D311yescharacteristic
111#638—62no—
112#638—62no—
113C2×C3⋊Dic3D1211yescharacteristic
114C4⋊(C3⋊D3)—32no—
115C4⋊(C3⋊D3)—64no—
116C4⋊(C3⋊D3)—61no—
117C4⋊(C3⋊D3)—64no—
118C4⋊(C3⋊D3)—124no—
119C4⋊(C3⋊D3)—124no—
120C4×C3⋊D3C4×D311yescharacteristic
121C4×C3⋊D3D1211yescharacteristic
122C4×C3⋊D3#6711yescharacteristic
123C4×C3⋊D3C22×D311yescharacteristic
124C2×C3×Dic3—62no—
125C3×D12—24no—
126C3×D12—24no—
127C3×C4×D3—62no—
128C3×C4×D3—62no—
129C3×C4×D3—62no—
130C3×C4×D3—62no—
131#626—68no—
132#626—68no—
133#622—62no—
134#418—271noSylow
135C2×C33—21no—
136C2×C33C2×C22⋊C411yescharacteristic
137C3⋊(C3⋊D3)—42no—
138C3×C3⋊D3—21no—
139C3×C3⋊D3AGL(1,ℤ/8)12yes—
140C32×D3—42no—
141C23×D3—181no—
142C23×D3—184no—
143C2×C3×D4—62no—
144C22×C3×C4—91no—
145C2×#67—184no—
146D3×D4—184no—
147D3×D4—184no—
148D3×D4—184no—
149D3×D4—184no—
150C2×D12—91no—
151C2×D12—92no—
152C2×D12—92no—
153C2×D12—182no—
154C2×D12—181no—
155C2×D12—181no—
156C2×D12—181no—
157C2×D12—362no—
158C2×D12#15711yescharacteristic, complemented
159C2×C4×D3—62no—
160C3×C4⋊C4—92no—
161C3×C22⋊C4—181no—
162C3×C42—181no—
163#208—181no—
164#208—181no—
165C2×C3⋊C8—92no—
166C22×C32—22no—
167C22×C32—42no—
168C22×C32—62no—
169C22×C32—122no—
170C22×C32(C2×D3)⋊C411yescharacteristic
171C2×C3⋊D3—22no—
172C2×C3⋊D3—62no—
173C2×C3⋊D3—68no—
174C2×C3⋊D3—128no—
175C2×C3⋊D3—124no—
176C2×C3⋊D3—122no—
177C2×C3⋊D3—124no—
178C2×C3⋊D3—128no—
179C2×C3⋊D3—248no—
180C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
181C2×C3⋊D3C2×#6711yescharacteristic
182C2×C3×D3—28no—
183C2×C3×D3—48no—
184C2×C3×D3—62no—
185C2×C3×D3—62no—
186C2×C3×D3—68no—
187D32—1216no—
188D32—1216no—
189#157—42no—
190C32×C4—22no—
191C32×C4—22no—
192C32×C4—42no—
193C32×C4—42no—
194C32×C4(C2×D3)⋊C411yescharacteristic
195C32×C4C2×C4×D311yescharacteristic
196C3⋊Dic3(C2×D3)⋊C411yescharacteristic
197C3⋊Dic3C2×D1211yescharacteristic
198C3×Dic3—62no—
199C3×Dic3—62no—
200C22×D4—271no—
201C2×M16—271no—
202#117—91no—
203#102—274no—
204C33#41811yescharacteristic, complemented, Sylow
205C23×C3—64no—
206C23×C3—91no—
207C22×D3—21no—
208C22×D3—62no—
209C22×D3—182no—
210C22×D3—182no—
211C22×D3—188no—
212C22×D3—188no—
213C22×D3—184no—
214C22×D3—181no—
215C22×D3—364no—
216C22×D3—364no—
217C22×D3—3616no—
218C3×D4—64no—
219C3×D4—64no—
220C2×C3×C4—22no—
221C2×C3×C4—42no—
222C2×C3×C4—91no—
223C2×C3×C4—91no—
224C2×C3×C4—91no—
225C2×C3×C4—91no—
226C2×C3×C4—91no—
227C2×C3×C4—182no—
228C2×C3×C4C2×#15711yescharacteristic
229#67—188no—
230#67—188no—
231C2×Dic3—62no—
232D12—21no—
233D12—92no—
234D12—184no—
235D12—184no—
236D12—182no—
237D12—182no—
238D12—181no—
239D12—364no—
240D12—364no—
241D12C2×#15712yes—
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—122no—
253C2×C32C2×(C2×D3)⋊C411yescharacteristic
254C3⋊D3—21no—
255C3⋊D3—122no—
256C3⋊D3—128no—
257C3⋊D3—248no—
258C3⋊D3#94912yes—
259C3×D3—48no—
260C3×D3—64no—
261C3×D3—64no—
262C24—541no—
263C2×D4—31no—
264C2×D4—272no—
265C2×D4—272no—
266C2×D4—271no—
267C2×D4—541no—
268C2×D4—541no—
269C2×D4—541no—
270C22×C4—91no—
271M16—541no—
272M16—541no—
273C2×C8—272no—
274C4⋊C4—92no—
275C22⋊C4—181no—
276C42—181no—
277C22×C3—22no—
278C22×C3—42no—
279C22×C3—68no—
280C22×C3—91no—
281C22×C3—91no—
282C22×C3—128no—
283C22×C3—182no—
284C22×C3C22⋊#15711yescharacteristic
285C2×D3—22no—
286C2×D3—42no—
287C2×D3—62no—
288C2×D3—62no—
289C2×D3—68no—
290C2×D3—188no—
291C2×D3—182no—
292C2×D3—368no—
293C2×D3—364no—
294C2×D3—364no—
295C2×D3—3616no—
296C2×D3—3616no—
297C2×D3—362no—
298C2×D3—368no—
299C2×D3—728no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—42no—
303C3×C4—42no—
304C3×C4—91no—
305C3×C4—91no—
306C3×C4—362no—
307C3×C4C22⋊#15711yescharacteristic
308C3×C4C22×#15711yescharacteristic
309Dic3—62no—
310Dic3—62no—
311C32—22no—
312C32—42no—
313C32#500111yescharacteristic, complemented
314C23—61no—
315C23—91no—
316C23—542no—
317C23—542no—
318C23—541no—
319C23—1084no—
320D4—32no—
321D4—61no—
322D4—272no—
323D4—542no—
324D4—542no—
325D4—541no—
326C2×C4—91no—
327C2×C4—91no—
328C2×C4—91no—
329C2×C4—91no—
330C2×C4—91no—
331C2×C4—182no—
332C2×C4D3×#15711yescharacteristic
333C8—542no—
334C2×C3—24no—
335C2×C3—21no—
336C2×C3—22no—
337C2×C3—44no—
338C2×C3—42no—
339C2×C3—92no—
340C2×C3—128no—
341C2×C3—181no—
342C2×C3C2×C22⋊#15711yescharacteristic
343D3—42no—
344D3—64no—
345D3—64no—
346D3—368no—
347D3—362no—
348D3—728no—
349C22—62no—
350C22—91no—
351C22—91no—
352C22—122no—
353C22—182no—
354C22—542no—
355C22—1082no—
356C22—1084no—
357C22—1084no—
358C22(C2×D3)⋊#15711yescharacteristic
359C4—91no—
360C4—91no—
361C4—362no—
362C4(C2×D3)⋊#15711yescharacteristic
363C4C2×D3×#15711yescharacteristic
364C3—22no—
365C3—42no—
366C3#1059381411yescharacteristic, complemented
367C2—21no—
368C2—92no—
369C2—122no—
370C2—181no—
371C2—1082no—
372C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC2C2×(C2×D3)⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC2×C3⋊D3×D12C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33#418
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724511C2semidirect
3. C3×#64286C2semidirect
4. #11722706C2semidirect
7. C2×C4⋊(C3⋊(C3⋊D3))C4semidirect
9. C2×C32×D12C4semidirect
13. C3⋊D3×D12C4semidirect
17. #88988C22semidirect
19. C3×C4⋊#157C22semidirect
23. #64286D3semidirect
68. C4⋊#157C2×D3semidirect
158. C2×D12#157semidirect
204. C33#418semidirect
313. C32#5001semidirect
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⋊D3×D122
#117245112
C3×#642862
#117227062
#105938143
#50019