Finite Group Databasebeta

← #13166359

Subgroups — #13166359

Order 1728.

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

normal subgroups only

This group has 952 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×#64283C211yesmaximal, characteristic, complemented
4#11722729C214yesmaximal, complemented
5#10593825—31nomaximal
6C22×D3×C3⋊D3C412yes—
7C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×D12C411yescharacteristic
10C2×#6291C2212yes—
11C3⋊D3×D12—21no—
12C3⋊D3×D12—21no—
13C3⋊D3×D12—21no—
14C3⋊D3×D12—21no—
15#88993C2212yescomplemented
16#88989—21no—
17#88988—21no—
18C3×#3792C2212yescomplemented
19C3×#3788—21no—
20C3×#3787—21no—
21C2×D4×C3⋊D3—31no—
22C2×D3×D12—62no—
23#64283—31no—
24#64283D311yescharacteristic, complemented
25#63788—34no—
26C22×C3⋊(C3⋊D3)C2×C412yes—
27C22×C3×C3⋊D3C2×C411yescharacteristic
28C22×C32×D3C2×C412yes—
29C2×D3×C3⋊D3—22no—
30C2×D3×C3⋊D3—22no—
31C2×D3×C3⋊D3—44no—
32C2×C33×C4C2×C411yescharacteristic
33C4⋊(C3⋊(C3⋊D3))—21no—
34C4⋊(C3⋊(C3⋊D3))—21no—
35C2×C3×C3⋊Dic3C2311yescharacteristic
36C3×C4×C3⋊D3D411yescharacteristic
37C3×C4×C3⋊D3D411yescharacteristic
38C3×C4×C3⋊D3D411yescharacteristic
39C3×C4×C3⋊D3D411yescharacteristic
40C32×D12—21no—
41C32×D12—21no—
42#6291—22no—
43#6291—22no—
44C3⋊#622—22no—
45C3×#622—22no—
46C2×#823—91nomaximal
47C23×C3⋊D3—32no—
48C22×D32—64no—
49C2×C32×D4—31no—
50C2×#638—32no—
51D4×C3⋊D3—61no—
52D4×C3⋊D3—61no—
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—
66#3792—32no—
67#3792C2×D312yescomplemented
68#3788—21no—
69#3788—61no—
70#3787—21no—
71#3787—61no—
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⋊D3C22⋊C411yescharacteristic
78C2×C32×D3—22no—
79C2×C32×D3—42no—
80D3×C3⋊D3—44no—
81D3×C3⋊D3—44no—
82C33×C4C22⋊C411yescharacteristic
83C33×C4C22⋊C411yescharacteristic
84C3×C3⋊Dic3C2×D411yescharacteristic
85C3×C3⋊Dic3C2×D411yescharacteristic
86C2×D3×D4—182no—
87C22×D12—91no—
88C2×C3×M16—91no—
89C2×#208—91no—
90#823—94no—
91C23×C32—32no—
92C22×C3⋊D3—32no—
93C22×C3⋊D3—64no—
94C22×C3⋊D3—62no—
95C22×C3⋊D3—62no—
96C22×C3⋊D3—124no—
97C22×C3⋊D3—124no—
98C22×C3⋊D3C4×D311yescharacteristic
99C22×C3×D3—24no—
100C22×C3×D3—62no—
101C2×D32—68no—
102C2×D32—68no—
103C2×D32—1216no—
104C32×D4—61no—
105C32×D4—61no—
106C2×C32×C4—22no—
107C2×C32×C4—42no—
108C2×C32×C4C4×D311yescharacteristic
109#638—62no—
110#638—62no—
111C2×C3⋊Dic3C22×D311yescharacteristic
112C4⋊(C3⋊D3)—61no—
113C4⋊(C3⋊D3)—61no—
114C4⋊(C3⋊D3)—64no—
115C4⋊(C3⋊D3)—64no—
116C4⋊(C3⋊D3)—124no—
117C4⋊(C3⋊D3)—124no—
118C4×C3⋊D3D1211yescharacteristic
119C4×C3⋊D3D1211yescharacteristic
120C4×C3⋊D3#6711yescharacteristic
121C4×C3⋊D3#6711yescharacteristic
122C2×C3×Dic3—62no—
123C3×D12—24no—
124C3×D12—24no—
125C3×C4×D3—62no—
126C3×C4×D3—62no—
127C3×C4×D3—62no—
128C3×C4×D3—62no—
129#626—68no—
130#626—68no—
131#622—22no—
132#622—62no—
133C2×#100—271noSylow
134C2×C33—21no—
135C2×C33C2×C22⋊C411yescharacteristic
136C3⋊(C3⋊D3)—42no—
137C3×C3⋊D3—21no—
138C3×C3⋊D3#10012yescomplemented
139C32×D3—42no—
140C23×D3—92no—
141C23×D3—184no—
142C2×C3×D4—62no—
143C22×C3×C4—91no—
144C2×#67—184no—
145D3×D4—184no—
146D3×D4—184no—
147D3×D4—184no—
148D3×D4—184no—
149C2×D12—92no—
150C2×D12—91no—
151C2×D12—182no—
152C2×D12—181no—
153C2×D12—181no—
154C2×D12—181no—
155C2×D12—181no—
156C2×D12—362no—
157C2×D12#15711yescharacteristic
158C2×C4×D3—62no—
159C3×M16—92no—
160C3×M16—181no—
161C2×C3×C8—181no—
162#208—92no—
163#208—181no—
164C2×C3⋊C8—181no—
165C22×C32—22no—
166C22×C32—42no—
167C22×C32—62no—
168C22×C32—122no—
169C22×C32C2×C4×D311yescharacteristic
170C2×C3⋊D3—22no—
171C2×C3⋊D3—62no—
172C2×C3⋊D3—68no—
173C2×C3⋊D3—128no—
174C2×C3⋊D3—124no—
175C2×C3⋊D3—124no—
176C2×C3⋊D3—122no—
177C2×C3⋊D3—128no—
178C2×C3⋊D3—248no—
179C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
180C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
181C2×C3×D3—28no—
182C2×C3×D3—48no—
183C2×C3×D3—62no—
184C2×C3×D3—62no—
185C2×C3×D3—68no—
186D32—1216no—
187D32—1216no—
188C32×C4—22no—
189C32×C4—22no—
190C32×C4—42no—
191C32×C4—42no—
192C32×C4(C2×D3)⋊C411yescharacteristic
193C32×C4(C2×D3)⋊C411yescharacteristic
194C3⋊Dic3C2×D1211yescharacteristic
195C3⋊Dic3C2×#6711yescharacteristic
196C3×Dic3—62no—
197C3×Dic3—62no—
198C22×D4—271no—
199C2×M16—91no—
200C2×M16—271no—
201#100—274no—
202C33C2×#10011yescharacteristic, complemented, Sylow
203C23×C3—64no—
204C23×C3—91no—
205C22×D3—62no—
206C22×D3—92no—
207C22×D3—184no—
208C22×D3—182no—
209C22×D3—182no—
210C22×D3—188no—
211C22×D3—188no—
212C22×D3—364no—
213C22×D3—364no—
214C22×D3—3616no—
215C22×D3C2×#15712yes—
216C3×D4—64no—
217C3×D4—64no—
218C2×C3×C4—22no—
219C2×C3×C4—42no—
220C2×C3×C4—91no—
221C2×C3×C4—91no—
222C2×C3×C4—91no—
223C2×C3×C4—91no—
224C2×C3×C4—91no—
225C2×C3×C4C2×#15711yescharacteristic
226#67—188no—
227#67—188no—
228C2×Dic3—62no—
229D12—21no—
230D12—21no—
231D12—184no—
232D12—184no—
233D12—182no—
234D12—182no—
235D12—181no—
236D12—181no—
237D12—364no—
238D12—364no—
239C4×D3—62no—
240C4×D3—62no—
241C4×D3—62no—
242C4×D3—62no—
243C3×C8—182no—
244C3⋊C8—182no—
245C2×C32—21no—
246C2×C32—24no—
247C2×C32—22no—
248C2×C32—44no—
249C2×C32—42no—
250C2×C32—122no—
251C2×C32C2×(C2×D3)⋊C411yescharacteristic
252C3⋊D3—21no—
253C3⋊D3—122no—
254C3⋊D3—128no—
255C3⋊D3—248no—
256C3⋊D3#82312yescomplemented
257C3×D3—48no—
258C3×D3—64no—
259C3×D3—64no—
260C24—272no—
261C2×D4—31no—
262C2×D4—272no—
263C2×D4—271no—
264C2×D4—541no—
265C2×D4—541no—
266C2×D4—541no—
267C2×D4—541no—
268C22×C4—91no—
269M16—92no—
270M16—181no—
271M16—272no—
272M16—541no—
273C2×C8—181no—
274C2×C8—541no—
275C22×C3—22no—
276C22×C3—42no—
277C22×C3—68no—
278C22×C3—91no—
279C22×C3—91no—
280C22×C3—128no—
281C22×C3—182no—
282C22×C3C22×#15711yescharacteristic
283C2×D3—22no—
284C2×D3—42no—
285C2×D3—62no—
286C2×D3—62no—
287C2×D3—68no—
288C2×D3—188no—
289C2×D3—182no—
290C2×D3—368no—
291C2×D3—364no—
292C2×D3—364no—
293C2×D3—3616no—
294C2×D3—3616no—
295C2×D3—368no—
296C2×D3—362no—
297C2×D3—728no—
298C3×C4—22no—
299C3×C4—22no—
300C3×C4—42no—
301C3×C4—42no—
302C3×C4—91no—
303C3×C4—91no—
304C3×C4C22⋊#15711yescharacteristic
305C3×C4C22⋊#15711yescharacteristic
306Dic3—62no—
307Dic3—62no—
308C32—22no—
309C32—42no—
310C32C2×#82311yescharacteristic, complemented
311C23—32no—
312C23—91no—
313C23—272no—
314C23—542no—
315C23—542no—
316C23—1084no—
317D4—61no—
318D4—61no—
319D4—542no—
320D4—542no—
321D4—541no—
322D4—541no—
323C2×C4—91no—
324C2×C4—91no—
325C2×C4—91no—
326C2×C4—91no—
327C2×C4—91no—
328C2×C4D3×#15711yescharacteristic
329C8—182no—
330C8—542no—
331C2×C3—24no—
332C2×C3—21no—
333C2×C3—22no—
334C2×C3—44no—
335C2×C3—42no—
336C2×C3—92no—
337C2×C3—128no—
338C2×C3—181no—
339C2×C3C2×C22⋊#15711yescharacteristic
340D3—42no—
341D3—64no—
342D3—64no—
343D3—368no—
344D3—362no—
345D3—728no—
346C22—62no—
347C22—91no—
348C22—91no—
349C22—122no—
350C22—182no—
351C22—542no—
352C22—1082no—
353C22—1084no—
354C22—1084no—
355C22C2×D3×#15711yescharacteristic
356C4—91no—
357C4—91no—
358C4(C2×D3)⋊#15711yescharacteristic
359C4(C2×D3)⋊#15711yescharacteristic
360C3—22no—
361C3—42no—
362C3#1059382511yescharacteristic, complemented
363C2—21no—
364C2—92no—
365C2—122no—
366C2—181no—
367C2—1082no—
368C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC2C2×(C2×D3)⋊#157
commutatorC22×C33C22×C4
FrattiniC22C2×D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC2×C3⋊D3×D12C2
cosocleC2×C3×C3⋊Dic3C23
hypercenterC2×C4D3×#157
hypocenterC33C2×#100
LCSG ▷ C22×C33 ▷ 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×#64283C2semidirect
4. #11722729C2semidirect
15. #88993C22semidirect
18. C3×#3792C22semidirect
24. #64283D3semidirect
67. #3792C2×D3semidirect
138. C3×C3⋊D3#100semidirect
202. C33C2×#100semidirect
256. C3⋊D3#823semidirect
310. C32C2×#823semidirect
362. C3#10593825semidirect

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×#642832
#117227292
#105938253
C2×#8239