Finite Group Databasebeta

← #11921921

Subgroups — #11921921

Order 1152.

756 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 71 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
2C2×C3×A4⋊Q8C211yesmaximal, characteristic, complemented
3C4×Dic3×A4C211yesmaximal, characteristic, complemented
4#10593242C214yesmaximal, complemented
5#85290—31nomaximal
6#82809—31nomaximal
7C22×C3⋊S4C2212yes—
8C2×C3×C4×A4C2211yescharacteristic
9C2×Dic3×A4C2212yes—
10C4⋊(C3⋊S4)—22no—
11C2×C3×SL(2,ℤ/4)C2212yes—
12C3×A4⋊Q8—22no—
13#63864—41nomaximal
14C2×C4⋊S4—31no—
15C2×C4⋊S4—61no—
16C2×A4⋊Q8D311yescharacteristic, complemented
17C2×C3×C22⋊Q8—31no—
18C2×#947—31no—
19C2×#930—31no—
20C22×C4×Dic3D311yescharacteristic, complemented
21C42×A4—31no—
22(C2×S4)⋊C4—34no—
23#5229—31no—
24#4990—34no—
25#4988—31no—
26#4977—31no—
27#4958—34no—
28C22×C3×A4C2311yescharacteristic
29C2×C3⋊S4—22no—
30C2×C3⋊S4—24no—
31C2×C4⋊(C3⋊D3)—41no—
32C2×C3×Dic6—41no—
33C3×C4×A4D412yes—
34Dic3×A4—22no—
35C3×SL(2,ℤ/4)—22no—
36C3×C4×Dic3—41no—
37#3722—44no—
38#2850—91noSylow
39C22×S4—32no—
40C22×S4—62no—
41C22×C3×Q8—31no—
42C23×C3×C4C2×D311yescharacteristic
43C22×#67—32no—
44C23×Dic3C2×D312yes—
45C22×D12—31no—
46C2×C4×A4—21no—
47C2×C4×A4—32no—
48C2×C4×A4C2×D311yescharacteristic
49C2×SL(2,ℤ/4)C2×D312yes—
50C4⋊S4—64no—
51C4⋊S4—62no—
52A4⋊Q8—22no—
53C3×C22⋊Q8—62no—
54C3×C22⋊Q8—62no—
55C2×C3×C4⋊C4—31no—
56C2×C3×C4⋊C4—32no—
57C2×C3×C22⋊C4—32no—
58#947—61no—
59#947—62no—
60#947—61no—
61#930—62no—
62#930—62no—
63C2×(C2×D3)⋊C4—34no—
64C2×(C2×D3)⋊C4—34no—
65C2×(C2×D3)⋊C4—32no—
66C2×C4⋊Dic3—31no—
67C2×C4×Dic3—31no—
68C2×C4×Dic3—31no—
69C2×C4×Dic3—32no—
70#875—121no—
71C3⋊#95—34no—
72C3⋊#95—34no—
73C22×C3⋊D3—42no—
74C2×C3×A4C2×D411yescharacteristic
75C2×C3×A4#4112yes—
76C3⋊S4—42no—
77C2×C32×C4—41no—
78C4⋊(C3⋊D3)—82no—
79C2×C3×Dic3—42no—
80C2×C3×Dic3—42no—
81C3×Dic6—82no—
82C2×#124—91no—
83C2×C22⋊Q8—31no—
84C2×#121—91no—
85C22×C42—31no—
86#391—91no—
87#390—91no—
88#388—94no—
89#386—94no—
90#382—91no—
91C24×C3C22×D311yescharacteristic
92C23×D3—32no—
93C22×A4—21no—
94C22×A4C22×D311yescharacteristic
95C2×S4—64no—
96C2×S4—64no—
97C2×S4—62no—
98C2×S4—124no—
99C2×C3×Q8—61no—
100C2×C3×Q8—62no—
101C2×C3×Q8—62no—
102C2×C3×Q8—61no—
103C22×C3×C4—31no—
104C22×C3×C4—31no—
105C22×C3×C4—32no—
106C22×C3×C4—32no—
107C22×C3×C4—32no—
108C22×C3×C4D1212yes—
109C2×#67—62no—
110C2×#67—64no—
111C2×#67—62no—
112C2×#67—64no—
113C22×Dic3—22no—
114C22×Dic3—32no—
115C22×Dic3—32no—
116C22×Dic3—32no—
117C22×Dic3—62no—
118C2×D12—62no—
119C2×D12—61no—
120C2×D12—62no—
121C2×D12—61no—
122C2×D12—121no—
123C2×D12—241no—
124C2×Dic6—41no—
125C4×A4—22no—
126C4×A4—62no—
127C4×A4#6712yes—
128SL(2,ℤ/4)—22no—
129C3×C4⋊C4—62no—
130C3×C4⋊C4—64no—
131C3×C22⋊C4—62no—
132C3×C22⋊C4—62no—
133C3×C42—121no—
134(C2×D3)⋊C4—68no—
135(C2×D3)⋊C4—64no—
136(C2×D3)⋊C4—64no—
137(C2×D3)⋊C4—64no—
138(C2×D3)⋊C4—124no—
139C4⋊Dic3—62no—
140C4×Dic3—31no—
141C4×Dic3—32no—
142C4×Dic3—61no—
143C4×Dic3S411yescharacteristic, complemented
144C22×C32—41no—
145C2×C3⋊D3—82no—
146C2×C3⋊D3—84no—
147C3×A4#12411yescharacteristic, complemented
148C32×C4—42no—
149C3×Dic3—82no—
150C3×Dic3—82no—
151C22×Q8—31no—
152C22×D4—91no—
153C22×D4—92no—
154C23×C4—32no—
155C23×C4D3211yescharacteristic
156#124—181no—
157#124—182no—
158#124—181no—
159C22⋊Q8—62no—
160C22⋊Q8—62no—
161#121—182no—
162#121—182no—
163C2×C4⋊C4—31no—
164C2×C4⋊C4—32no—
165C2×C4⋊C4—91no—
166C2×C22⋊C4—32no—
167C2×C22⋊C4—92no—
168C2×C22⋊C4—94no—
169C2×C22⋊C4—94no—
170C2×C42—91no—
171C2×C42—92no—
172C2×C42—91no—
173#95—94no—
174#95—94no—
175C23×C3—31no—
176C23×C3—31no—
177C23×C3—32no—
178C23×C3C2×D1211yescharacteristic
179C23×C3#23512yes—
180C22×D3—62no—
181C22×D3—64no—
182C22×D3—62no—
183C22×D3—64no—
184C22×D3—62no—
185C22×D3—122no—
186C22×D3—242no—
187C2×A4—22no—
188C2×A4—21no—
189C2×A4#23912yes—
190C2×A4C2×#6711yescharacteristic
191S4—124no—
192S4—122no—
193C3×Q8—122no—
194C3×Q8—122no—
195C2×C3×C4—31no—
196C2×C3×C4—32no—
197C2×C3×C4—32no—
198C2×C3×C4—32no—
199C2×C3×C4—41no—
200C2×C3×C4—61no—
201C2×C3×C4—64no—
202C2×C3×C4—62no—
203C2×C3×C4—62no—
204C2×C3×C4—62no—
205C2×C3×C4—62no—
206C2×C3×C4—81no—
207C2×C3×C4—122no—
208C2×C3×C4C2×S411yescharacteristic
209#67—122no—
210#67—124no—
211#67—122no—
212C2×Dic3—34no—
213C2×Dic3—32no—
214C2×Dic3—42no—
215C2×Dic3—62no—
216C2×Dic3—62no—
217C2×Dic3—62no—
218C2×Dic3—62no—
219C2×Dic3—64no—
220C2×Dic3C2×S412yes—
221D12—122no—
222D12—122no—
223D12—242no—
224D12—244no—
225Dic6—82no—
226C2×C32—41no—
227C2×C32—42no—
228C3⋊D3—162no—
229C24—92no—
230C24C2×D3211yescharacteristic
231C2×Q8—62no—
232C2×Q8—61no—
233C2×Q8—62no—
234C2×Q8—61no—
235C2×D4—182no—
236C2×D4—181no—
237C2×D4—182no—
238C2×D4—182no—
239C2×D4—184no—
240C2×D4—184no—
241C2×D4—182no—
242C2×D4—181no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—32no—
246C22×C4—32no—
247C22×C4—32no—
248C22×C4—62no—
249C22×C4—92no—
250C22×C4—92no—
251C22×C4—92no—
252C22×C4—182no—
253C22×C4#62612yes—
254C4⋊C4—62no—
255C4⋊C4—64no—
256C4⋊C4—182no—
257C22⋊C4—62no—
258C22⋊C4—62no—
259C22⋊C4—184no—
260C22⋊C4—184no—
261C22⋊C4—184no—
262C22⋊C4—188no—
263C42—31no—
264C42—92no—
265C42—91no—
266C42—181no—
267C22×C3—31no—
268C22×C3—32no—
269C22×C3—31no—
270C22×C3—31no—
271C22×C3—32no—
272C22×C3—32no—
273C22×C3—41no—
274C22×C3—61no—
275C22×C3—81no—
276C22×C3#87511yescharacteristic, complemented
277C22×C3C22×S411yescharacteristic
278C2×D3—122no—
279C2×D3—122no—
280C2×D3—124no—
281C2×D3—122no—
282C2×D3—124no—
283C2×D3—242no—
284C2×D3—244no—
285C2×D3—244no—
286C2×D3—484no—
287A4—21no—
288A4#94711yescharacteristic, complemented
289C3×C4—32no—
290C3×C4—42no—
291C3×C4—82no—
292C3×C4—122no—
293C3×C4—122no—
294C3×C4—242no—
295C3×C4C4⋊S412yes—
296Dic3—22no—
297Dic3—62no—
298Dic3—82no—
299Dic3—122no—
300C32—41noSylow
301C23—31no—
302C23—31no—
303C23—32no—
304C23—182no—
305C23—184no—
306C23—184no—
307C23—182no—
308C23—182no—
309C23#379912yes—
310C23C2×#62611yescharacteristic
311Q8—122no—
312Q8—122no—
313D4—362no—
314D4—364no—
315D4—362no—
316D4—362no—
317D4—362no—
318C2×C4—31no—
319C2×C4—32no—
320C2×C4—32no—
321C2×C4—32no—
322C2×C4—32no—
323C2×C4—61no—
324C2×C4—62no—
325C2×C4—62no—
326C2×C4—64no—
327C2×C4—62no—
328C2×C4—62no—
329C2×C4—94no—
330C2×C4—92no—
331C2×C4—182no—
332C2×C4—182no—
333C2×C4—182no—
334C2×C4—182no—
335C2×C4—184no—
336C2×C4D3×S411yescharacteristic
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—32no—
340C2×C3—42no—
341C2×C3—41no—
342C2×C3—82no—
343C2×C3—81no—
344C2×C3C2×C4⋊S411yescharacteristic
345C2×C3#591212yes—
346D3—242no—
347D3—482no—
348D3—484no—
349C22—32no—
350C22—31no—
351C22—32no—
352C22—32no—
353C22—31no—
354C22—31no—
355C22—61no—
356C22—362no—
357C22—362no—
358C22—364no—
359C22—364no—
360C22—362no—
361C22#6386411yescharacteristic, complemented
362C22C2×D3×S411yescharacteristic
363C4—32no—
364C4—62no—
365C4—122no—
366C4—122no—
367C4—182no—
368C4—362no—
369C4#6420712yes—
370C3—41no—
371C3—81no—
372C3#8529011yescharacteristic, complemented
373C2—31no—
374C2—31no—
375C2—32no—
376C2—722no—
377C2#1059642812yes—
378C2C2×#6420711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC22×C3×A4C23
FittingC23×C3×C4C2×D3
socleC24×C3C22×D3
prime radicalC2×C4⋊(C3⋊S4)C2
hypercenterC2×C4D3×S4
hypocenterC3×A4#124
LCSG ▷ C22×C3×A4 ▷ C3×A4
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×A4⋊Q8C2semidirect
3. C4×Dic3×A4C2semidirect
4. #10593242C2semidirect
16. C2×A4⋊Q8D3semidirect
20. C22×C4×Dic3D3semidirect
143. C4×Dic3S4semidirect
147. C3×A4#124semidirect
276. C22×C3#875semidirect
288. A4#947semidirect
361. C22#63864semidirect
372. C3#85290semidirect

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×A4⋊Q82
C4×Dic3×A42
#105932422
#852903
#828093
#638644