Finite Group Databasebeta

← #11921997

Subgroups — #11921997

Order 1152.

1048 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 83 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×C4⋊(C3⋊S4)C211yesmaximal, characteristic, complemented
2C2×C3×C4⋊S4C211yesmaximal, characteristic, complemented
3C2×#64207C214yesmaximal, complemented
4C4×Dic3×A4C211yesmaximal, characteristic, complemented
5#85289—31nomaximal
6#82700—31nomaximal
7C22×C3⋊S4C2212yes—
8C22×C3×S4C2212yes—
9C2×C3×C4×A4C2211yescharacteristic
10C2×Dic3×A4C2212yes—
11C4⋊(C3⋊S4)—22no—
12C3×C4⋊S4—22no—
13#64207—28no—
14#63911—41nomaximal
15C2×C4⋊S4—34no—
16C2×C4⋊S4—31no—
17C2×C4⋊S4—61no—
18C2×C4⋊S4D311yescharacteristic, complemented
19C2×C3×#121—31no—
20C2×#940—31no—
21C2×#939—34no—
22C2×#930—31no—
23C22×C4×Dic3D311yescharacteristic, complemented
24C42×A4—31no—
25#5221—31no—
26#4988—31no—
27#4975—31no—
28#4957—34no—
29C22×C3×A4C2311yescharacteristic
30C2×C3⋊S4—22no—
31C2×C3⋊S4—24no—
32C2×C3×S4—24no—
33C2×C3×S4—22no—
34C2×C4⋊(C3⋊D3)—41no—
35C2×C3×D12—41no—
36C3×C4×A4D412yescomplemented
37C2×#626—44no—
38Dic3×A4D414yescomplemented
39C3×C4×Dic3—41no—
40#2856—91noSylow
41C22×S4—32no—
42C22×S4—62no—
43C22×S4C2×D312yes—
44C22×C3×D4—31no—
45C22×C3×D4—32no—
46C23×C3×C4C2×D311yescharacteristic
47C22×#67—34no—
48C22×#67—32no—
49C23×Dic3C2×D312yes—
50C22×D12—31no—
51C2×C4×A4—21no—
52C2×C4×A4—32no—
53C2×C4×A4C2×D311yescharacteristic
54C4⋊S4—22no—
55C4⋊S4—64no—
56C4⋊S4—68no—
57C4⋊S4—62no—
58C3×#121—62no—
59C3×#121—62no—
60C2×C3×C4⋊C4—31no—
61C2×C3×C22⋊C4—32no—
62C2×#217—34no—
63#940—61no—
64#940—62no—
65#940—61no—
66#939—68no—
67#939—68no—
68#930—62no—
69#930—62no—
70C2×(C2×D3)⋊C4—32no—
71C2×(C2×D3)⋊C4—34no—
72C2×C4⋊Dic3—31no—
73C2×Dic3⋊C4—34no—
74C2×C4×Dic3—31no—
75C2×C4×Dic3—32no—
76C2×C4×Dic3—31no—
77#874—121no—
78C22×C3⋊D3—42no—
79C22×C3×D3—42no—
80C2×C3×A4C2×D412yes—
81C2×C3×A4C2×D411yescharacteristic
82C3⋊S4—42no—
83C3×S4—42no—
84C2×C32×C4—41no—
85C4⋊(C3⋊D3)—82no—
86C2×C3×Dic3—42no—
87C3×D12—82no—
88#626—88no—
89C2×#127—91no—
90C2×#121—31no—
91C2×#121—94no—
92C2×#121—91no—
93C22×C42—31no—
94#390—94no—
95#390—91no—
96#390—91no—
97#384—91no—
98C24×C3—32no—
99C24×C3C22×D311yescharacteristic
100C23×D3—32no—
101C22×A4—21no—
102C22×A4C22×D311yescharacteristic
103C2×S4—24no—
104C2×S4—22no—
105C2×S4—64no—
106C2×S4—64no—
107C2×S4—62no—
108C2×S4—124no—
109C2×C3×D4—61no—
110C2×C3×D4—62no—
111C2×C3×D4—62no—
112C2×C3×D4—64no—
113C2×C3×D4—62no—
114C2×C3×D4—64no—
115C2×C3×D4—62no—
116C2×C3×D4—61no—
117C22×C3×C4—31no—
118C22×C3×C4—31no—
119C22×C3×C4—32no—
120C22×C3×C4—32no—
121C22×C3×C4D1212yescomplemented
122C2×#67—68no—
123C2×#67—64no—
124C2×#67—68no—
125C2×#67—62no—
126C2×#67—64no—
127C2×#67—62no—
128C2×#67—64no—
129C2×#67—64no—
130C22×Dic3—32no—
131C22×Dic3—32no—
132C22×Dic3—34no—
133C22×Dic3—32no—
134C22×Dic3D1214yescomplemented
135C2×D12—41no—
136C2×D12—62no—
137C2×D12—61no—
138C2×D12—62no—
139C2×D12—61no—
140C2×D12—121no—
141C2×D12—124no—
142C2×D12—241no—
143C4×A4—22no—
144C4×A4—34no—
145C4×A4#6712yescomplemented
146C3×C4⋊C4—62no—
147C3×C22⋊C4—64no—
148C3×C42—121no—
149#217—68no—
150(C2×D3)⋊C4—68no—
151(C2×D3)⋊C4—64no—
152C4⋊Dic3—62no—
153Dic3⋊C4—68no—
154C4×Dic3—31no—
155C4×Dic3—32no—
156C4×Dic3—61no—
157C4×Dic3S411yescharacteristic, complemented
158C22×C32—41no—
159C2×C3⋊D3—82no—
160C2×C3⋊D3—84no—
161C2×C3×D3—84no—
162C2×C3×D3—82no—
163C3×A4#12711yescharacteristic, complemented
164C32×C4—42no—
165C3×Dic3—44no—
166C22×D4—31no—
167C22×D4—32no—
168C22×D4—91no—
169C22×D4—94no—
170C22×D4—92no—
171C23×C4—32no—
172C23×C4D3211yescharacteristic
173#127—181no—
174#127—182no—
175#127—181no—
176#121—62no—
177#121—62no—
178#121—188no—
179#121—182no—
180#121—188no—
181#121—182no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—94no—
184C2×C4⋊C4—91no—
185C2×C22⋊C4—32no—
186C2×C22⋊C4—92no—
187C2×C22⋊C4—94no—
188C2×C22⋊C4—94no—
189C2×C42—91no—
190C2×C42—92no—
191C2×C42—91no—
192C23×C3—31no—
193C23×C3—32no—
194C23×C3—31no—
195C23×C3—64no—
196C23×C3—62no—
197C23×C3—64no—
198C23×C3—62no—
199C23×C3—62no—
200C23×C3C2×D1212yes—
201C23×C3C2×D1211yescharacteristic
202C22×D3—42no—
203C22×D3—62no—
204C22×D3—64no—
205C22×D3—62no—
206C22×D3—64no—
207C22×D3—62no—
208C22×D3—122no—
209C22×D3—242no—
210C2×A4—22no—
211C2×A4—21no—
212C2×A4D3×D412yes—
213C2×A4C2×#6711yescharacteristic
214S4—42no—
215S4—124no—
216S4—122no—
217C3×D4—122no—
218C3×D4—122no—
219C3×D4—124no—
220C3×D4—122no—
221C3×D4—122no—
222C2×C3×C4—31no—
223C2×C3×C4—32no—
224C2×C3×C4—32no—
225C2×C3×C4—32no—
226C2×C3×C4—41no—
227C2×C3×C4—61no—
228C2×C3×C4—64no—
229C2×C3×C4—62no—
230C2×C3×C4—81no—
231C2×C3×C4—122no—
232C2×C3×C4C2×S411yescharacteristic
233#67—128no—
234#67—124no—
235#67—122no—
236#67—122no—
237#67—128no—
238C2×Dic3—32no—
239C2×Dic3—34no—
240C2×Dic3—34no—
241C2×Dic3—34no—
242C2×Dic3—62no—
243C2×Dic3—64no—
244C2×Dic3—62no—
245C2×Dic3C2×S412yes—
246D12—82no—
247D12—122no—
248D12—122no—
249D12—242no—
250D12—244no—
251D12—248no—
252C2×C32—42no—
253C2×C32—41no—
254C3⋊D3—162no—
255C3×D3—162no—
256C24—32no—
257C24—92no—
258C24C2×D3211yescharacteristic
259C2×D4—62no—
260C2×D4—61no—
261C2×D4—62no—
262C2×D4—62no—
263C2×D4—64no—
264C2×D4—64no—
265C2×D4—62no—
266C2×D4—61no—
267C2×D4—184no—
268C2×D4—182no—
269C2×D4—181no—
270C2×D4—182no—
271C2×D4—188no—
272C2×D4—184no—
273C2×D4—188no—
274C2×D4—182no—
275C2×D4—184no—
276C2×D4—184no—
277C2×D4—182no—
278C2×D4—181no—
279C22×C4—31no—
280C22×C4—31no—
281C22×C4—32no—
282C22×C4—32no—
283C22×C4—34no—
284C22×C4—92no—
285C22×C4—92no—
286C22×C4—94no—
287C22×C4—92no—
288C22×C4#62612yescomplemented
289C4⋊C4—62no—
290C4⋊C4—182no—
291C4⋊C4—188no—
292C22⋊C4—64no—
293C22⋊C4—184no—
294C22⋊C4—188no—
295C22⋊C4—188no—
296C42—31no—
297C42—92no—
298C42—91no—
299C42—181no—
300C22×C3—32no—
301C22×C3—31no—
302C22×C3—32no—
303C22×C3—32no—
304C22×C3—31no—
305C22×C3—31no—
306C22×C3—41no—
307C22×C3—61no—
308C22×C3—81no—
309C22×C3—122no—
310C22×C3—124no—
311C22×C3—122no—
312C22×C3—124no—
313C22×C3—122no—
314C22×C3#87411yescharacteristic, complemented
315C22×C3C22×S411yescharacteristic
316C2×D3—82no—
317C2×D3—84no—
318C2×D3—122no—
319C2×D3—122no—
320C2×D3—124no—
321C2×D3—122no—
322C2×D3—124no—
323C2×D3—244no—
324C2×D3—242no—
325C2×D3—244no—
326C2×D3—484no—
327A4—21no—
328A4#94011yescharacteristic, complemented
329C3×C4—32no—
330C3×C4—42no—
331C3×C4—82no—
332C3×C4—124no—
333C3×C4—122no—
334C3×C4C4⋊S412yescomplemented
335Dic3—34no—
336Dic3—122no—
337Dic3C4⋊S414yescomplemented
338C32—41noSylow
339C23—31no—
340C23—31no—
341C23—32no—
342C23—62no—
343C23—64no—
344C23—62no—
345C23—64no—
346C23—62no—
347C23—184no—
348C23—182no—
349C23—184no—
350C23—182no—
351C23—182no—
352C23D3×D1212yes—
353C23C2×#62611yescharacteristic
354D4—122no—
355D4—124no—
356D4—122no—
357D4—122no—
358D4—122no—
359D4—368no—
360D4—362no—
361D4—368no—
362D4—364no—
363D4—362no—
364D4—362no—
365D4—362no—
366C2×C4—32no—
367C2×C4—31no—
368C2×C4—32no—
369C2×C4—32no—
370C2×C4—32no—
371C2×C4—61no—
372C2×C4—62no—
373C2×C4—64no—
374C2×C4—94no—
375C2×C4—94no—
376C2×C4—92no—
377C2×C4—94no—
378C2×C4—182no—
379C2×C4—184no—
380C2×C4—182no—
381C2×C4D3×S411yescharacteristic
382C2×C3—31no—
383C2×C3—32no—
384C2×C3—31no—
385C2×C3—42no—
386C2×C3—41no—
387C2×C3—82no—
388C2×C3—81no—
389C2×C3—242no—
390C2×C3C2×C4⋊S412yes—
391C2×C3C2×C4⋊S411yescharacteristic
392D3—162no—
393D3—242no—
394D3—482no—
395D3—484no—
396C22—32no—
397C22—31no—
398C22—32no—
399C22—32no—
400C22—31no—
401C22—31no—
402C22—61no—
403C22—122no—
404C22—122no—
405C22—124no—
406C22—124no—
407C22—122no—
408C22—362no—
409C22—364no—
410C22—362no—
411C22—364no—
412C22—362no—
413C22#6391111yescharacteristic, complemented
414C22C2×D3×S411yescharacteristic
415C4—34no—
416C4—32no—
417C4—94no—
418C4—122no—
419C4—362no—
420C4#6420712yescomplemented
421C3—41no—
422C3—81no—
423C3#8528911yescharacteristic, complemented
424C2—31no—
425C2—32no—
426C2—31no—
427C2—242no—
428C2—722no—
429C2D3×C4⋊S412yes—
430C2C2×#6420711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC22×C3×A4C23
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
hypercenterC2×C4D3×S4
hypocenterC3×A4#127
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊(C3⋊S4)C2semidirect
2. C2×C3×C4⋊S4C2semidirect
3. C2×#64207C2semidirect
4. C4×Dic3×A4C2semidirect
18. C2×C4⋊S4D3semidirect
23. C22×C4×Dic3D3semidirect
36. C3×C4×A4D4semidirect
38. Dic3×A4D4semidirect
121. C22×C3×C4D12semidirect
134. C22×Dic3D12semidirect
145. C4×A4#67semidirect
157. C4×Dic3S4semidirect
163. C3×A4#127semidirect
288. C22×C4#626semidirect
314. C22×C3#874semidirect
328. A4#940semidirect
334. C3×C4C4⋊S4semidirect
337. Dic3C4⋊S4semidirect
413. C22#63911semidirect
420. C4#64207semidirect
423. C3#85289semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(C3⋊S4)2
C2×C3×C4⋊S42
C2×#642072
C4×Dic3×A42
#852893
#827003
#639114