Finite Group Databasebeta

← C2×#11724187 #13180779

Subgroups — C2×#11724187 #13180779

Order 1728.

1654 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 1654 conjugacy classes of subgroups — too many to draw. Only its 125 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×C3⋊(C3⋊S4)C211yesmaximal, characteristic, complemented
2C22×C32×S4C211yesmaximal, characteristic, complemented
3C2×A4×C3⋊Dic3C211yesmaximal, characteristic, complemented
4#11724187C214yesmaximal, direct product factor
5C2×#64207—34nomaximal
6C2×#64148—31nomaximal
7C22×C32×A4C2211yescharacteristic
8C2×C3⋊(C3⋊S4)—22no—
9C2×C3⋊(C3⋊S4)C2212yescomplemented
10C2×C32×S4—22no—
11C2×C32×S4C2212yescomplemented
12C2×#6291—41nomaximal
13A4×C3⋊Dic3C2212yescomplemented
14C22×C3⋊S4—34no—
15C22×C3⋊S4—64no—
16C22×C3×S4D314yescomplemented
17C22×C32×D4—31no—
18C22×#638—31no—
19C22×#638—31no—
20C23×C3⋊Dic3D311yescharacteristic, complemented
21C2×Dic3×A4—34no—
22#64207—316no—
23C2×#3757—31no—
24#64148—34no—
25#64148—34no—
26C2×#3752—31no—
27C2×#3750—31no—
28C22×C3⋊(C3⋊D3)—41no—
29C22×C32×D3—41no—
30C2×C32×A4D412yescomplemented
31C2×C32×A4C2311yescharacteristic
32C3⋊(C3⋊S4)—22no—
33C32×S4—22no—
34C2×C3×C3⋊Dic3—41no—
35#6291—44no—
36C2×C4⋊S4—91no—
37C2×#939—94no—
38C24×C32—31no—
39C24×C32C2×D311yescharacteristic
40C23×C3⋊D3—31no—
41C22×C3×A4—24no—
42C22×C3×A4C2×D314yes—
43C2×C3⋊S4—38no—
44C2×C3⋊S4—68no—
45C2×C3⋊S4—616no—
46C2×C3⋊S4—68no—
47C2×C3×S4—28no—
48C2×C3×S4C2×D318yescomplemented
49C2×C32×D4—32no—
50C2×C32×D4—32no—
51C2×C32×D4—62no—
52C2×C32×D4—62no—
53C22×C32×C4—31no—
54C2×#638—34no—
55C2×#638—32no—
56C2×#638—32no—
57C2×#638—61no—
58C2×#638—62no—
59C2×#638—62no—
60C2×#638—62no—
61C2×#638—61no—
62C22×C3⋊Dic3—31no—
63C22×C3⋊Dic3—31no—
64C22×C3⋊Dic3—32no—
65C22×C3⋊Dic3—31no—
66C22×C3⋊Dic3C2×D312yescomplemented
67C2×#626—124no—
68Dic3×A4—38no—
69#3757—34no—
70#3752—34no—
71#3750—34no—
72C22×C33—41no—
73C2×C3⋊(C3⋊D3)—42no—
74C2×C3⋊(C3⋊D3)—82no—
75C2×C32×D3—42no—
76C2×C32×D3—82no—
77C32×A4C2×D411yescharacteristic, complemented
78C3×C3⋊Dic3—42no—
79C22×S4—91no—
80C22×S4—184no—
81C22×S4C3⋊D311yescharacteristic, complemented
82C22×C3×D4—34no—
83C22×#67—94no—
84C22×#67—94no—
85C23×Dic3—34no—
86C2×C4×A4—91no—
87C4⋊S4—94no—
88C2×#217—94no—
89#939—916no—
90#939—916no—
91C2×(C2×D3)⋊C4—94no—
92C2×Dic3⋊C4—94no—
93C23×C32—31no—
94C23×C32—31no—
95C23×C32—32no—
96C23×C32—32no—
97C23×C32—62no—
98C23×C32—61no—
99C23×C32—62no—
100C23×C32—61no—
101C23×C32D1212yescomplemented
102C23×C32C22×D311yescharacteristic
103C22×C3⋊D3—32no—
104C22×C3⋊D3—62no—
105C22×C3⋊D3—62no—
106C22×C3⋊D3—61no—
107C22×C3⋊D3—61no—
108C22×C3⋊D3—124no—
109C22×C3⋊D3—244no—
110C22×C3×D3—44no—
111C2×C3×A4—28no—
112C2×C3×A4—24no—
113C2×C3×A4#6718yescomplemented
114C2×C3×A4C22×D314yes—
115C3⋊S4—616no—
116C3⋊S4—68no—
117C3×S4—28no—
118C32×D4—64no—
119C32×D4—62no—
120C32×D4—62no—
121C2×C32×C4—32no—
122C2×C32×C4—62no—
123#638—64no—
124#638—64no—
125#638—62no—
126#638—62no—
127#638—64no—
128C2×C3⋊Dic3—31no—
129C2×C3⋊Dic3—32no—
130C2×C3⋊Dic3—32no—
131C2×C3⋊Dic3—32no—
132C2×C3⋊Dic3—32no—
133C2×C3⋊Dic3—61no—
134C2×C3⋊Dic3—62no—
135C2×C3⋊Dic3S411yescharacteristic, complemented
136C2×C3×Dic3—124no—
137#626—1216no—
138C2×#121—271noSylow
139C2×C33—42no—
140C2×C33—41no—
141C3⋊(C3⋊D3)—82no—
142C32×D3—82no—
143C24×C3—34no—
144C24×C3D3214yes—
145C23×D3—94no—
146C22×A4—24no—
147C22×A4C2×C3⋊D311yescharacteristic
148C2×S4—22no—
149C2×S4—92no—
150C2×S4—188no—
151C2×S4—1816no—
152C2×S4—182no—
153C2×S4C2×C3⋊D312yescomplemented
154C2×C3×D4—38no—
155C2×C3×D4—38no—
156C2×C3×D4—68no—
157C2×C3×D4—68no—
158C22×C3×C4—34no—
159C2×#67—916no—
160C2×#67—98no—
161C2×#67—98no—
162C2×#67—184no—
163C2×#67—188no—
164C2×#67—184no—
165C2×#67—188no—
166C2×#67—188no—
167C22×Dic3—38no—
168C22×Dic3—94no—
169C22×Dic3—94no—
170C22×Dic3—98no—
171C22×Dic3—94no—
172C2×D12—361no—
173C4×A4—92no—
174#217—916no—
175(C2×D3)⋊C4—916no—
176Dic3⋊C4—916no—
177C22×C32—32no—
178C22×C32—31no—
179C22×C32—32no—
180C22×C32—32no—
181C22×C32—31no—
182C22×C32—31no—
183C22×C32—44no—
184C22×C32—61no—
185C22×C32—62no—
186C22×C32—62no—
187C22×C32—62no—
188C22×C32—84no—
189C22×C32—122no—
190C22×C32—122no—
191C22×C32C2×D1211yescharacteristic, complemented
192C22×C32C2×S411yescharacteristic
193C2×C3⋊D3—62no—
194C2×C3⋊D3—62no—
195C2×C3⋊D3—62no—
196C2×C3⋊D3—128no—
197C2×C3⋊D3—122no—
198C2×C3⋊D3—122no—
199C2×C3⋊D3—248no—
200C2×C3⋊D3—248no—
201C2×C3⋊D3—2416no—
202C2×C3×D3—48no—
203C2×C3×D3—88no—
204C3×A4—24no—
205C3×A4C2×#6714yescomplemented
206C32×C4—62no—
207C3⋊Dic3—32no—
208C3⋊Dic3—62no—
209C3⋊Dic3C2×S412yescomplemented
210C3×Dic3—128no—
211C22×D4—31no—
212C22×D4—271no—
213C22×D4—271no—
214C23×C4—91no—
215#121—274no—
216#121—274no—
217C2×C4⋊C4—271no—
218C2×C22⋊C4—271no—
219C2×C22⋊C4—271no—
220C33—41noSylow
221C23×C3—34no—
222C23×C3—34no—
223C23×C3—38no—
224C23×C3—38no—
225C23×C3—64no—
226C23×C3—68no—
227C23×C3—68no—
228C23×C3—64no—
229C23×C3#62618yescomplemented
230C23×C3C2×D3214yes—
231C22×D3—41no—
232C22×D3—98no—
233C22×D3—188no—
234C22×D3—184no—
235C22×D3—188no—
236C22×D3—184no—
237C22×D3—361no—
238C22×D3—724no—
239C2×A4—28no—
240C2×A4—24no—
241C2×A4#63812yescomplemented
242C2×A4C22×C3⋊D311yescharacteristic
243S4—22no—
244S4—1816no—
245S4—182no—
246C3×D4—616no—
247C3×D4—68no—
248C3×D4—68no—
249C2×C3×C4—38no—
250C2×C3×C4—68no—
251C2×C3×C4—361no—
252#67—1816no—
253#67—1816no—
254#67—1816no—
255#67—188no—
256#67—188no—
257C2×Dic3—34no—
258C2×Dic3—98no—
259C2×Dic3—98no—
260C2×Dic3—94no—
261C2×Dic3—98no—
262C2×Dic3—98no—
263C2×Dic3—184no—
264C2×Dic3—188no—
265D12—364no—
266C2×C32—31no—
267C2×C32—31no—
268C2×C32—32no—
269C2×C32—44no—
270C2×C32—48no—
271C2×C32—84no—
272C2×C32—88no—
273C2×C32—122no—
274C2×C32C4⋊S412yescomplemented
275C2×C32C22×S411yescharacteristic
276C3⋊D3—122no—
277C3⋊D3—248no—
278C3⋊D3—2416no—
279C3×D3—88no—
280C24—31no—
281C24—271no—
282C24D3×C3⋊D311yescharacteristic
283C2×D4—32no—
284C2×D4—32no—
285C2×D4—62no—
286C2×D4—62no—
287C2×D4—274no—
288C2×D4—272no—
289C2×D4—272no—
290C2×D4—541no—
291C2×D4—542no—
292C2×D4—541no—
293C2×D4—542no—
294C2×D4—542no—
295C22×C4—31no—
296C22×C4—92no—
297C22×C4—271no—
298C22×C4—271no—
299C22×C4—272no—
300C22×C4—271no—
301C4⋊C4—274no—
302C22⋊C4—274no—
303C22⋊C4—274no—
304C22×C3—38no—
305C22×C3—34no—
306C22×C3—34no—
307C22×C3—34no—
308C22×C3—38no—
309C22×C3—38no—
310C22×C3—41no—
311C22×C3—64no—
312C22×C3—68no—
313C22×C3—68no—
314C22×C3—68no—
315C22×C3—84no—
316C22×C3—128no—
317C22×C3—128no—
318C22×C3C2×#62614yescomplemented
319C22×C3D3×S414yes—
320C2×D3—42no—
321C2×D3—82no—
322C2×D3—188no—
323C2×D3—188no—
324C2×D3—188no—
325C2×D3—362no—
326C2×D3—368no—
327C2×D3—368no—
328C2×D3—728no—
329C2×D3—722no—
330C2×D3—7216no—
331A4—24no—
332A4C2×#63811yescharacteristic, complemented
333C3×C4—68no—
334C3×C4—362no—
335Dic3—38no—
336Dic3—98no—
337Dic3—188no—
338C32—44no—
339C32—84no—
340C32C2×C4⋊S411yescharacteristic, complemented
341C23—31no—
342C23—31no—
343C23—32no—
344C23—32no—
345C23—62no—
346C23—61no—
347C23—62no—
348C23—61no—
349C23—272no—
350C23—542no—
351C23—541no—
352C23—542no—
353C23—541no—
354C23#629112yescomplemented
355C23C2×D3×C3⋊D311yescharacteristic
356D4—64no—
357D4—62no—
358D4—62no—
359D4—544no—
360D4—544no—
361D4—544no—
362D4—542no—
363D4—542no—
364C2×C4—32no—
365C2×C4—62no—
366C2×C4—91no—
367C2×C4—272no—
368C2×C4—272no—
369C2×C4—271no—
370C2×C4—272no—
371C2×C4—272no—
372C2×C4—541no—
373C2×C4—542no—
374C2×C3—34no—
375C2×C3—34no—
376C2×C3—38no—
377C2×C3—42no—
378C2×C3—41no—
379C2×C3—88no—
380C2×C3—84no—
381C2×C3—128no—
382C2×C3#6420718yescomplemented
383C2×C3C2×D3×S414yes—
384D3—82no—
385D3—368no—
386D3—722no—
387D3—7216no—
388C22—32no—
389C22—31no—
390C22—32no—
391C22—32no—
392C22—31no—
393C22—31no—
394C22—61no—
395C22—62no—
396C22—62no—
397C22—62no—
398C22—122no—
399C22—122no—
400C22—542no—
401C22—542no—
402C22—542no—
403C22—1082no—
404C22—1082no—
405C22C2×#629111yescharacteristic, complemented
406C22C3⋊D3×S411yescharacteristic
407C4—62no—
408C4—92no—
409C4—272no—
410C4—542no—
411C3—41no—
412C3—84no—
413C3C2×#6420714yescomplemented
414C2—31no—
415C2—31no—
416C2—32no—
417C2—122no—
418C2—1082no—
419C2#1172418712yesdirect product factor
420C2C2×C3⋊D3×S411yescharacteristic
subgroupG/H
center, hypercenterC22C3⋊D3×S4
commutator, cosocleC2×C32×A4C23
FrattiniC2C2×C3⋊D3×S4
Fitting, socleC24×C32C2×D3
hypocenterC32×A4C2×D4
LCSG ▷ C2×C32×A4 ▷ C32×A4
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C3⋊(C3⋊S4)C2semidirect
2. C22×C32×S4C2semidirect
3. C2×A4×C3⋊Dic3C2semidirect
4. #11724187C2direct (N × H)
9. C2×C3⋊(C3⋊S4)C22semidirect
11. C2×C32×S4C22semidirect
13. A4×C3⋊Dic3C22semidirect
16. C22×C3×S4D3semidirect
20. C23×C3⋊Dic3D3semidirect
30. C2×C32×A4D4semidirect
48. C2×C3×S4C2×D3semidirect
66. C22×C3⋊Dic3C2×D3semidirect
77. C32×A4C2×D4semidirect
81. C22×S4C3⋊D3semidirect
101. C23×C32D12semidirect
113. C2×C3×A4#67semidirect
135. C2×C3⋊Dic3S4semidirect
153. C2×S4C2×C3⋊D3semidirect
191. C22×C32C2×D12semidirect
205. C3×A4C2×#67semidirect
209. C3⋊Dic3C2×S4semidirect
229. C23×C3#626semidirect
241. C2×A4#638semidirect
274. C2×C32C4⋊S4semidirect
318. C22×C3C2×#626semidirect
332. A4C2×#638semidirect
340. C32C2×C4⋊S4semidirect
354. C23#6291semidirect
382. C2×C3#64207semidirect
405. C22C2×#6291semidirect
413. C3C2×#64207semidirect
419. C2#11724187direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊(C3⋊S4)2
C22×C32×S42
C2×A4×C3⋊Dic32
#117241872
C2×#642073
C2×#641483
C2×#62914