Finite Group Databasebeta

← C2×#11723107 #13173414

Subgroups — C2×#11723107 #13173414

Order 1728.

984 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 984 conjugacy classes of subgroups — too many to draw. Only its 109 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
2C2×C4⋊(C3⋊#157)C211yesmaximal, characteristic, complemented
3C2×#88988C211yesmaximal, characteristic, complemented
4#11723107C214yesmaximal, direct product factor
5C2×#63782—31nomaximal
6C23×C3×C3⋊D3—21no—
7C22×C3⋊#157—21no—
8C2×C33×D4C411yescharacteristic, complemented
9C2×C3×#638—21no—
10C3×D4×C3⋊D3—21no—
11C3×D4×C3⋊D3—21no—
12C3×D4×C3⋊D3C412yescomplemented
13C3×D4×C3⋊D3C2212yes—
14C2×C3×C4⋊(C3⋊D3)C411yescharacteristic, complemented
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3⋊#622—21no—
17C4⋊(C3⋊#157)—21no—
18C4⋊(C3⋊#157)C2212yescomplemented
19#88988—21no—
20#88988C2212yescomplemented
21C2×D4×C3⋊D3D311yescharacteristic
22C2×C3×D3×D4—62no—
23C2×C4⋊#157—31no—
24C2×#3787—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—42no—
31C22×C3×C3⋊D3—42no—
32C22×C3×C3⋊D3D411yescharacteristic
33C2×C3⋊#157—22no—
34C2×C3⋊#157—42no—
35C33×D4—21no—
36C33×D4C2×C412yescomplemented
37C2×C33×C4C2×C411yescharacteristic
38C3×#638—22no—
39C3×#638—22no—
40C2×C3×C3⋊Dic3D411yescharacteristic
41C3×C4⋊(C3⋊D3)—21no—
42C3×C4⋊(C3⋊D3)C2×C412yescomplemented
43C3×C4×C3⋊D3C2×C411yescharacteristic
44C3×C4×C3⋊D3D411yescharacteristic
45C3×C4×C3⋊D3D411yescharacteristic
46C3×C4×C3⋊D3C2311yescharacteristic
47C3⋊#622—22no—
48C2×#832—91nomaximal
49C23×C3⋊D3—21no—
50C23×C3×D3—64no—
51C22×#157—61no—
52C2×C32×D4—22no—
53C2×C32×D4—42no—
54C2×C32×D4Dic311yescharacteristic, complemented
55C2×#638—21no—
56D4×C3⋊D3—21no—
57D4×C3⋊D3—21no—
58D4×C3⋊D3Dic312yescomplemented
59D4×C3⋊D3C2×D312yes—
60C2×C4⋊(C3⋊D3)Dic311yescharacteristic, complemented
61C2×C4×C3⋊D3C2×D311yescharacteristic
62C2×C3×#67—64no—
63C3×D3×D4—64no—
64C3×D3×D4—64no—
65C3×D3×D4—64no—
66C3×D3×D4—64no—
67C2×C3×D12—62no—
68C2×C3×C4×D3—62no—
69C2×#622—61no—
70C4⋊#157—32no—
71C4⋊#157—61no—
72#3787—32no—
73#3787—61no—
74C22×C33—22no—
75C22×C33—42no—
76C22×C33C22⋊C411yescharacteristic
77C2×C3×C3⋊D3—22no—
78C2×C3×C3⋊D3—42no—
79C2×C3×C3⋊D3—44no—
80C2×C3×C3⋊D3—42no—
81C2×C3×C3⋊D3—42no—
82C2×C3×C3⋊D3C22⋊C411yescharacteristic
83C2×C3×C3⋊D3D812yes—
84C2×C3×C3⋊D3SD1612yes—
85C2×C3×C3⋊D3C2×D411yescharacteristic
86C3⋊#157—42no—
87C33×C4C22⋊C411yescharacteristic
88C33×C4C22×C411yescharacteristic
89C3×C3⋊Dic3C22⋊C411yescharacteristic
90C3×C3⋊Dic3C2×D411yescharacteristic
91C22×C3×D4—91no—
92C2×D3×D4—62no—
93C2×C4⋊Dic3—91no—
94C22×C3⋊C8—91no—
95#832—94no—
96C23×C32—21no—
97C23×C32—24no—
98C23×C32—44no—
99C22×C3⋊D3—21no—
100C22×C3⋊D3—22no—
101C22×C3⋊D3—22no—
102C22×C3⋊D3—42no—
103C22×C3⋊D3—42no—
104C22×C3⋊D3#6711yescharacteristic
105C22×C3×D3—62no—
106C22×C3×D3—64no—
107C22×C3×D3—68no—
108C22×C3×D3—68no—
109C22×C3×D3—128no—
110C22×C3×D3—128no—
111C2×#157—62no—
112C2×#157—122no—
113C32×D4—21no—
114C32×D4—24no—
115C32×D4—24no—
116C32×D4—44no—
117C32×D4—44no—
118C32×D4C2×Dic312yescomplemented
119C2×C32×C4—22no—
120C2×C32×C4—42no—
121C2×C32×C4C2×Dic311yescharacteristic
122#638—22no—
123#638—22no—
124C2×C3⋊Dic3#6711yescharacteristic
125C4⋊(C3⋊D3)—21no—
126C4⋊(C3⋊D3)C2×Dic312yescomplemented
127C4×C3⋊D3C2×Dic311yescharacteristic
128C4×C3⋊D3#6711yescharacteristic
129C4×C3⋊D3#6711yescharacteristic
130C4×C3⋊D3C22×D311yescharacteristic
131C3×#67—68no—
132C3×#67—68no—
133C2×C3×Dic3—62no—
134C3×D12—64no—
135C3×D12—64no—
136C3×C4×D3—62no—
137C3×C4×D3—62no—
138C3×C4×D3—62no—
139C3×C4×D3—62no—
140#622—62no—
141C2×#102—271noSylow
142C2×C33—42no—
143C2×C33#10212yescomplemented
144C2×C33C2×C22⋊C411yescharacteristic
145C3×C3⋊D3—42no—
146C3×C3⋊D3#10212yescomplemented
147C3×C3⋊D3C2×D811yescharacteristic
148C3×C3⋊D3C2×SD1611yescharacteristic
149C24×C3—181no—
150C23×D3—64no—
151C2×C3×D4—22no—
152C2×C3×D4—42no—
153C2×C3×D4—92no—
154C2×C3×D4—92no—
155C2×C3×D4—91no—
156C2×C3×D4—181no—
157C2×C3×D4—181no—
158C2×C3×D4—181no—
159C2×C3×D4#15711yescharacteristic, complemented
160C22×C3×C4—91no—
161C2×#67—64no—
162C22×Dic3—181no—
163D3×D4—64no—
164D3×D4—64no—
165D3×D4—64no—
166D3×D4—64no—
167C2×D12—62no—
168C2×C4×D3—62no—
169C4⋊Dic3—92no—
170C4⋊Dic3—181no—
171C2×C3⋊C8—92no—
172C2×C3⋊C8—181no—
173C2×C3⋊C8—181no—
174C22×C32—22no—
175C22×C32—28no—
176C22×C32—22no—
177C22×C32—48no—
178C22×C32—42no—
179C22×C32—42no—
180C22×C32—48no—
181C22×C32—88no—
182C22×C32#21711yescharacteristic
183C2×C3⋊D3—22no—
184C2×C3⋊D3—42no—
185C2×C3⋊D3—44no—
186C2×C3⋊D3—42no—
187C2×C3⋊D3—42no—
188C2×C3⋊D3#21312yes—
189C2×C3⋊D3#21412yes—
190C2×C3⋊D3#21711yescharacteristic
191C2×C3⋊D3C2×#6711yescharacteristic
192C2×C3×D3—62no—
193C2×C3×D3—64no—
194C2×C3×D3—64no—
195C2×C3×D3—62no—
196C2×C3×D3—68no—
197C2×C3×D3—1216no—
198C2×C3×D3—128no—
199C2×C3×D3—128no—
200C2×C3×D3—128no—
201#157—122no—
202C32×C4—22no—
203C32×C4—22no—
204C32×C4—42no—
205C32×C4—42no—
206C32×C4#21711yescharacteristic
207C32×C4C22×Dic311yescharacteristic
208C3⋊Dic3#21711yescharacteristic
209C3⋊Dic3C2×#6711yescharacteristic
210C3×Dic3—62no—
211C3×Dic3—62no—
212C22×D4—91no—
213C22×C8—271no—
214C2×C4⋊C4—271no—
215#102—274no—
216C33C2×#10211yescharacteristic, complemented, Sylow
217C23×C3—24no—
218C23×C3—21no—
219C23×C3—44no—
220C23×C3—91no—
221C23×C3—182no—
222C23×C3—182no—
223C23×C3—181no—
224C23×C3—362no—
225C23×C3—362no—
226C22×D3—62no—
227C22×D3—64no—
228C22×D3—68no—
229C22×D3—68no—
230C22×D3—128no—
231C22×D3—128no—
232C3×D4—24no—
233C3×D4—21no—
234C3×D4—24no—
235C3×D4—44no—
236C3×D4—44no—
237C3×D4—92no—
238C3×D4—182no—
239C3×D4—182no—
240C3×D4—181no—
241C3×D4C2×#15712yescomplemented
242C2×C3×C4—22no—
243C2×C3×C4—42no—
244C2×C3×C4—91no—
245C2×C3×C4—91no—
246C2×C3×C4—91no—
247C2×C3×C4—91no—
248C2×C3×C4—91no—
249C2×C3×C4C2×#15711yescharacteristic
250#67—68no—
251#67—68no—
252C2×Dic3—62no—
253C2×Dic3—182no—
254C2×Dic3—362no—
255D12—64no—
256D12—64no—
257C4×D3—62no—
258C4×D3—62no—
259C4×D3—62no—
260C4×D3—62no—
261C3⋊C8—182no—
262C2×C32—22no—
263C2×C32—24no—
264C2×C32—44no—
265C2×C32—42no—
266C2×C32—42no—
267C2×C32—48no—
268C2×C32—88no—
269C2×C32#83212yescomplemented
270C2×C32C2×#21711yescharacteristic
271C3⋊D3—42no—
272C3⋊D3#83212yescomplemented
273C3⋊D3C2×#21311yescharacteristic
274C3⋊D3C2×#21411yescharacteristic
275C3×D3—64no—
276C3×D3—62no—
277C3×D3—62no—
278C3×D3—128no—
279C24—181no—
280C2×D4—92no—
281C2×D4—92no—
282C2×D4—91no—
283C2×D4—181no—
284C2×D4—181no—
285C2×D4—181no—
286C2×D4C3⋊#15711yescharacteristic, complemented
287C22×C4—91no—
288C22×C4—541no—
289C2×C8—272no—
290C2×C8—541no—
291C2×C8—541no—
292C4⋊C4—272no—
293C4⋊C4—541no—
294C22×C3—28no—
295C22×C3—22no—
296C22×C3—22no—
297C22×C3—48no—
298C22×C3—48no—
299C22×C3—42no—
300C22×C3—42no—
301C22×C3—88no—
302C22×C3—91no—
303C22×C3—91no—
304C22×C3—92no—
305C22×C3—92no—
306C22×C3—182no—
307C22×C3—362no—
308C22×C3—362no—
309C22×C3—364no—
310C22×C3—362no—
311C22×C3C22⋊#15711yescharacteristic
312C2×D3—62no—
313C2×D3—62no—
314C2×D3—64no—
315C2×D3—64no—
316C2×D3—68no—
317C2×D3—1216no—
318C2×D3—128no—
319C2×D3—128no—
320C2×D3—128no—
321C3×C4—22no—
322C3×C4—22no—
323C3×C4—42no—
324C3×C4—42no—
325C3×C4—91no—
326C3×C4—91no—
327C3×C4C22⋊#15711yescharacteristic
328C3×C4C22×#15711yescharacteristic
329Dic3—62no—
330Dic3—62no—
331Dic3—362no—
332C32—22no—
333C32—42no—
334C32C2×#83211yescharacteristic, complemented
335C23—21no—
336C23—91no—
337C23—182no—
338C23—182no—
339C23—181no—
340C23—362no—
341C23—362no—
342D4—21no—
343D4—92no—
344D4—182no—
345D4—182no—
346D4—181no—
347D4C2×C3⋊#15712yescomplemented
348C2×C4—91no—
349C2×C4—91no—
350C2×C4—91no—
351C2×C4—91no—
352C2×C4—91no—
353C2×C4—542no—
354C2×C4—1082no—
355C2×C4C2×C3⋊#15711yescharacteristic
356C8—542no—
357C2×C3—24no—
358C2×C3—22no—
359C2×C3—48no—
360C2×C3—42no—
361C2×C3—44no—
362C2×C3—42no—
363C2×C3—88no—
364C2×C3—91no—
365C2×C3—91no—
366C2×C3—92no—
367C2×C3—362no—
368C2×C3#6378212yescomplemented
369C2×C3C2×C22⋊#15711yescharacteristic
370D3—64no—
371D3—62no—
372D3—62no—
373D3—128no—
374C22—22no—
375C22—42no—
376C22—92no—
377C22—92no—
378C22—91no—
379C22—91no—
380C22—182no—
381C22—362no—
382C22—364no—
383C22—362no—
384C22—362no—
385C22#8899411yescharacteristic
386C4—91no—
387C4—91no—
388C4—1082no—
389C4#8899411yescharacteristic
390C4C22×C3⋊#15711yescharacteristic
391C3—22no—
392C3—42no—
393C3C2×#6378211yescharacteristic, complemented
394C2—42no—
395C2—91no—
396C2—92no—
397C2—91no—
398C2—362no—
399C2#1172310712yesdirect product factor
400C2C2×#8899411yescharacteristic
subgroupG/H
centerC22#88994
commutatorC33×C4C22×C4
FrattiniC4C22×C3⋊#157
FittingC2×C33×D4C4
socleC22×C33C22⋊C4
prime radicalC2×C3×D4×C3⋊D3C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×D4C3⋊#157
hypocenterC33C2×#102
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊(C3⋊#157)C2semidirect
3. C2×#88988C2semidirect
4. #11723107C2direct (N × H)
8. C2×C33×D4C4semidirect
12. C3×D4×C3⋊D3C4semidirect
14. C2×C3×C4⋊(C3⋊D3)C4semidirect
18. C4⋊(C3⋊#157)C22semidirect
20. #88988C22semidirect
36. C33×D4C2×C4semidirect
42. C3×C4⋊(C3⋊D3)C2×C4semidirect
54. C2×C32×D4Dic3semidirect
58. D4×C3⋊D3Dic3semidirect
60. C2×C4⋊(C3⋊D3)Dic3semidirect
118. C32×D4C2×Dic3semidirect
126. C4⋊(C3⋊D3)C2×Dic3semidirect
143. C2×C33#102semidirect
146. C3×C3⋊D3#102semidirect
159. C2×C3×D4#157semidirect
216. C33C2×#102semidirect
241. C3×D4C2×#157semidirect
269. C2×C32#832semidirect
272. C3⋊D3#832semidirect
286. C2×D4C3⋊#157semidirect
334. C32C2×#832semidirect
347. D4C2×C3⋊#157semidirect
368. C2×C3#63782semidirect
393. C3C2×#63782semidirect
399. C2#11723107direct (N × H)

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
C2×C4⋊(C3⋊#157)2
C2×#889882
#117231072
C2×#637823
C2×#8329