Finite Group Databasebeta

← C2×#11724261 #13180995

Subgroups — C2×#11724261 #13180995

Order 1728.

1240 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1240 conjugacy classes of subgroups — too many to draw. Only its 194 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×Dic3×C3⋊D3C211yesmaximal, characteristic
2C2×#88994C211yesmaximal, characteristic
3C2×C3×C22⋊#157C211yesmaximal, characteristic
4C2×Dic3⋊#157C212yesmaximal, complemented
5#11724261C218yesmaximal, direct product factor
6C2×Dic3×#157C212yesmaximal, complemented
7C2×#64286—31nomaximal
8C23×C3×C3⋊D3C2211yescharacteristic
9C22×C3⋊#157C2212yes—
10C22×C3×#157C2212yes—
11C22×C3⋊(C3⋊Dic3)C411yescharacteristic, complemented
12C22×C32×Dic3C411yescharacteristic, complemented
13C2×Dic3×C3⋊D3C414yescomplemented
14C2×Dic3×C3⋊D3C411yescharacteristic, complemented
15C2×Dic3×C3⋊D3C411yescharacteristic, complemented
16C2×Dic3×C3⋊D3C2211yescharacteristic
17C2×Dic3×C3⋊D3C2211yescharacteristic
18C2×Dic3×C3⋊D3C2214yes—
19#88994C2214yes—
20C3×C22⋊#157C2214yes—
21Dic3⋊#157C2218yescomplemented
22Dic3×#157C2218yescomplemented
23C22×C4×C3⋊D3—31no—
24C22×D3×Dic3—62no—
25C2×C22⋊#157—31no—
26C2×C22⋊#157D311yescharacteristic
27#64286—38no—
28C2×C4⋊#157—32no—
29C2×C4×#157—32no—
30C23×C33C2×C411yescharacteristic
31C22×C3×C3⋊D3—24no—
32C22×C3×C3⋊D3C2×C411yescharacteristic
33C22×C3×C3⋊D3C2×C412yes—
34C22×C3×C3⋊D3C2312yes—
35C22×C3×C3⋊D3C2311yescharacteristic
36C2×C3⋊#157—24no—
37C2×C3⋊#157C2314yes—
38C2×C3×#157—24no—
39C2×C3×#157C2314yes—
40C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
41C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
42C2×C3⋊(C3⋊Dic3)C2×C414yescomplemented
43C2×C32×Dic3C2×C411yescharacteristic
44C2×C32×Dic3C2×C411yescharacteristic
45C2×C32×Dic3C2×C414yescomplemented
46Dic3×C3⋊D3C2×C412yes—
47Dic3×C3⋊D3C2×C412yes—
48Dic3×C3⋊D3C2×C414yescomplemented
49Dic3×C3⋊D3C2×C414yescomplemented
50Dic3×C3⋊D3C2312yes—
51Dic3×C3⋊D3C2312yes—
52C2×#877—91nomaximal
53C23×C3⋊D3C2×D311yescharacteristic
54C23×C3×D3—62no—
55C22×#157—32no—
56C22×#157C2×D312yes—
57C22×C32×C4—31no—
58C22×C3⋊Dic3—31no—
59C22×C3⋊Dic3—62no—
60C22×C3⋊Dic3—122no—
61C2×C4×C3⋊D3—34no—
62C2×C4×C3⋊D3—31no—
63C2×C4×C3⋊D3—31no—
64C2×C4×C3⋊D3—31no—
65C2×C4×C3⋊D3—31no—
66C2×C4×C3⋊D3—34no—
67C22×C3×Dic3—22no—
68C2×D3×Dic3—68no—
69C2×D3×Dic3—62no—
70C2×D3×Dic3—62no—
71C2×D3×Dic3—62no—
72C2×D3×Dic3—62no—
73C2×D3×Dic3—68no—
74C22⋊#157—34no—
75C22⋊#157C2×D314yes—
76C4⋊#157—38no—
77C4×#157—38no—
78C22×C33—22no—
79C22×C33C22×C412yes—
80C22×C33C22×C411yescharacteristic
81C2×C3×C3⋊D3—22no—
82C2×C3×C3⋊D3—24no—
83C2×C3×C3⋊D3—24no—
84C2×C3×C3⋊D3C22×C412yes—
85C2×C3×C3⋊D3C22×C411yescharacteristic
86C2×C3×C3⋊D3#4114yes—
87C2×C3×C3⋊D3C2411yescharacteristic
88C3⋊#157—24no—
89C3×#157—24no—
90C3⋊(C3⋊Dic3)C22×C412yes—
91C3⋊(C3⋊Dic3)C22×C412yes—
92C32×Dic3C22×C412yes—
93C32×Dic3C22×C412yes—
94C23×Dic3—91no—
95C22×C4×D3—182no—
96C2×C3×C22⋊C4—91no—
97C2×#217—91no—
98C2×Dic3⋊C4—92no—
99C2×C4×Dic3—92no—
100#877—98no—
101C23×C32—22no—
102C23×C32—42no—
103C23×C32C4×D311yescharacteristic
104C22×C3⋊D3—24no—
105C22×C3⋊D3C4×D311yescharacteristic
106C22×C3⋊D3C4×D312yes—
107C22×C3⋊D3C22×D312yes—
108C22×C3⋊D3C22×D311yescharacteristic
109C22×C3×D3—64no—
110C22×C3×D3—62no—
111C22×C3×D3—62no—
112C22×C3×D3—64no—
113C22×C3×D3—616no—
114C2×#157—24no—
115C2×#157—34no—
116C2×#157—64no—
117C2×#157C22×D314yes—
118C2×C32×C4—31no—
119C2×C32×C4—31no—
120C2×C32×C4—34no—
121C2×C3⋊Dic3—31no—
122C2×C3⋊Dic3—31no—
123C2×C3⋊Dic3—34no—
124C2×C3⋊Dic3—62no—
125C2×C3⋊Dic3—62no—
126C2×C3⋊Dic3—68no—
127C2×C3⋊Dic3—122no—
128C2×C3⋊Dic3—122no—
129C2×C3⋊Dic3—128no—
130C4×C3⋊D3—32no—
131C4×C3⋊D3—32no—
132C4×C3⋊D3—34no—
133C4×C3⋊D3—34no—
134C4×C3⋊D3—32no—
135C4×C3⋊D3—32no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—28no—
139D3×Dic3—64no—
140D3×Dic3—64no—
141D3×Dic3—68no—
142D3×Dic3—68no—
143D3×Dic3—64no—
144D3×Dic3—64no—
145C2×#117—271noSylow
146C2×C33—22no—
147C2×C33#11712yescomplemented
148C2×C33C23×C411yescharacteristic
149C3×C3⋊D3—22no—
150C3×C3⋊D3#11712yescomplemented
151C3×C3⋊D3C2×#4112yes—
152C24×C3—91no—
153C23×D3—62no—
154C22×C3×C4—62no—
155C22×C3×C4—92no—
156C22×Dic3—91no—
157C22×Dic3—91no—
158C22×Dic3—91no—
159C22×Dic3—91no—
160C22×Dic3—91no—
161C22×Dic3—94no—
162C22×Dic3—94no—
163C22×Dic3—92no—
164C22×Dic3—182no—
165C22×Dic3—362no—
166C22×Dic3#15711yescharacteristic, complemented
167C2×C4×D3—182no—
168C2×C4×D3—182no—
169C2×C4×D3—188no—
170C2×C4×D3—188no—
171C2×C4×D3—182no—
172C2×C4×D3—182no—
173C3×C22⋊C4—94no—
174#217—94no—
175Dic3⋊C4—98no—
176C4×Dic3—98no—
177C22×C32—22no—
178C22×C32—28no—
179C22×C32—24no—
180C22×C32—22no—
181C22×C32—48no—
182C22×C32—44no—
183C22×C32—42no—
184C22×C32C2×C4×D312yes—
185C22×C32C2×C4×D311yescharacteristic
186C2×C3⋊D3—22no—
187C2×C3⋊D3—24no—
188C2×C3⋊D3—24no—
189C2×C3⋊D3C2×C4×D312yes—
190C2×C3⋊D3C2×C4×D311yescharacteristic
191C2×C3⋊D3#23714yes—
192C2×C3⋊D3C23×D311yescharacteristic
193C2×C3×D3—62no—
194C2×C3×D3—64no—
195C2×C3×D3—68no—
196C2×C3×D3—62no—
197C2×C3×D3—616no—
198C2×C3×D3—616no—
199C2×C3×D3—68no—
200#157—24no—
201#157—64no—
202C32×C4—32no—
203C32×C4—32no—
204C3⋊Dic3—32no—
205C3⋊Dic3—32no—
206C3⋊Dic3—64no—
207C3⋊Dic3—64no—
208C3⋊Dic3—124no—
209C3⋊Dic3—124no—
210C3×Dic3—24no—
211C3×Dic3—24no—
212C23×C4—271no—
213#117—278no—
214C2×C4⋊C4—272no—
215C2×C22⋊C4—91no—
216C2×C22⋊C4—271no—
217C2×C42—272no—
218C33C2×#11711yescharacteristic, complemented, Sylow
219C23×C3—22no—
220C23×C3—42no—
221C23×C3—91no—
222C23×C3—92no—
223C23×C3—92no—
224C23×C3—91no—
225C23×C3—184no—
226C23×C3C2×#15711yescharacteristic
227C22×D3—64no—
228C22×D3—62no—
229C22×D3—62no—
230C22×D3—64no—
231C22×D3—616no—
232C2×C3×C4—68no—
233C2×C3×C4—62no—
234C2×C3×C4—62no—
235C2×C3×C4—94no—
236C2×C3×C4—184no—
237C2×Dic3—94no—
238C2×Dic3—91no—
239C2×Dic3—91no—
240C2×Dic3—92no—
241C2×Dic3—92no—
242C2×Dic3—92no—
243C2×Dic3—92no—
244C2×Dic3—94no—
245C2×Dic3—94no—
246C2×Dic3—94no—
247C2×Dic3—188no—
248C2×Dic3—182no—
249C2×Dic3—182no—
250C2×Dic3—184no—
251C2×Dic3—368no—
252C2×Dic3—362no—
253C2×Dic3—362no—
254C2×Dic3C2×#15711yescharacteristic
255C2×Dic3C2×#15711yescharacteristic
256C2×Dic3C2×#15714yescomplemented
257C4×D3—184no—
258C4×D3—184no—
259C4×D3—184no—
260C4×D3—184no—
261C4×D3—188no—
262C4×D3—188no—
263C2×C32—22no—
264C2×C32—28no—
265C2×C32—24no—
266C2×C32—22no—
267C2×C32—48no—
268C2×C32—44no—
269C2×C32—42no—
270C2×C32#87712yescomplemented
271C2×C32C22×C4×D311yescharacteristic
272C3⋊D3—22no—
273C3⋊D3#87712yescomplemented
274C3⋊D3C2×#23712yes—
275C3×D3—64no—
276C3×D3—64no—
277C3×D3—68no—
278C24—91no—
279C22×C4—31no—
280C22×C4—92no—
281C22×C4—271no—
282C22×C4—271no—
283C22×C4—274no—
284C22×C4—274no—
285C22×C4—271no—
286C22×C4—271no—
287C22×C4—271no—
288C22×C4—272no—
289C4⋊C4—278no—
290C22⋊C4—94no—
291C22⋊C4—274no—
292C42—278no—
293C22×C3—28no—
294C22×C3—22no—
295C22×C3—24no—
296C22×C3—22no—
297C22×C3—48no—
298C22×C3—44no—
299C22×C3—42no—
300C22×C3—94no—
301C22×C3—91no—
302C22×C3—91no—
303C22×C3—92no—
304C22×C3—184no—
305C22×C3—184no—
306C22×C3—182no—
307C22×C3C22×#15712yes—
308C22×C3C22×#15711yescharacteristic
309C2×D3—62no—
310C2×D3—64no—
311C2×D3—62no—
312C2×D3—68no—
313C2×D3—616no—
314C2×D3—616no—
315C2×D3—68no—
316C3×C4—64no—
317C3×C4—64no—
318C3×C4—184no—
319Dic3—92no—
320Dic3—92no—
321Dic3—184no—
322Dic3—184no—
323Dic3—184no—
324Dic3—364no—
325Dic3—364no—
326Dic3C22×#15712yes—
327Dic3C22×#15712yes—
328C32—22no—
329C32—42no—
330C32C2×#87711yescharacteristic, complemented
331C23—91no—
332C23—92no—
333C23—92no—
334C23—91no—
335C23—184no—
336C23D3×#15711yescharacteristic
337C2×C4—34no—
338C2×C4—31no—
339C2×C4—31no—
340C2×C4—94no—
341C2×C4—184no—
342C2×C4—272no—
343C2×C4—272no—
344C2×C4—272no—
345C2×C4—272no—
346C2×C4—274no—
347C2×C4—274no—
348C2×C4—274no—
349C2×C4—271no—
350C2×C4—271no—
351C2×C4—274no—
352C2×C4—544no—
353C2×C3—28no—
354C2×C3—22no—
355C2×C3—22no—
356C2×C3—24no—
357C2×C3—48no—
358C2×C3—42no—
359C2×C3—44no—
360C2×C3—92no—
361C2×C3—92no—
362C2×C3—182no—
363C2×C3#6428612yescomplemented
364C2×C3C23×#15711yescharacteristic
365D3—64no—
366D3—64no—
367D3—68no—
368C22—22no—
369C22—91no—
370C22—91no—
371C22—94no—
372C22—92no—
373C22—184no—
374C22—184no—
375C22—182no—
376C22C2×D3×#15712yes—
377C22C2×D3×#15711yescharacteristic
378C4—32no—
379C4—32no—
380C4—184no—
381C4—272no—
382C4—272no—
383C4—544no—
384C3—22no—
385C3—42no—
386C3C2×#6428611yescharacteristic, complemented
387C2—22no—
388C2—92no—
389C2—92no—
390C2—182no—
391C2#1172426112yesdirect product factor
392C2C22×D3×#15711yescharacteristic
subgroupG/H
centerC22C2×D3×#157
commutatorC2×C33C23×C4
FrattiniC2C22×D3×#157
FittingC23×C33C2×C4
socleC22×C33C22×C4
prime radicalC23×C3×C3⋊D3C22
cosocleC2×C3×C3⋊D3C24
hypercenterC23D3×#157
hypocenterC33C2×#117
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
4. C2×Dic3⋊#157C2semidirect
5. #11724261C2direct (N × H)
6. C2×Dic3×#157C2semidirect
11. C22×C3⋊(C3⋊Dic3)C4semidirect
12. C22×C32×Dic3C4semidirect
13. C2×Dic3×C3⋊D3C4semidirect
14. C2×Dic3×C3⋊D3C4semidirect
15. C2×Dic3×C3⋊D3C4semidirect
21. Dic3⋊#157C22semidirect
22. Dic3×#157C22semidirect
42. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
45. C2×C32×Dic3C2×C4semidirect
48. Dic3×C3⋊D3C2×C4semidirect
49. Dic3×C3⋊D3C2×C4semidirect
147. C2×C33#117semidirect
150. C3×C3⋊D3#117semidirect
166. C22×Dic3#157semidirect
218. C33C2×#117semidirect
256. C2×Dic3C2×#157semidirect
270. C2×C32#877semidirect
273. C3⋊D3#877semidirect
330. C32C2×#877semidirect
363. C2×C3#64286semidirect
386. C3C2×#64286semidirect
391. C2#11724261direct (N × H)

Maximal subgroups

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

subgroupindex
C22×Dic3×C3⋊D32
C2×#889942
C2×C3×C22⋊#1572
C2×Dic3⋊#1572
#117242612
C2×Dic3×#1572
C2×#642863
C2×#8779