Finite Group Databasebeta

← #11923048

Subgroups — #11923048

Order 1152.

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

normal subgroups only

This group has 720 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×C3×Q8×A4C211yesmaximal, characteristic, complemented
2#10596575C211yesmaximal, characteristic
3#10593651C212yesmaximal, complemented
4#10593650C212yesmaximal, complemented
5#10593643C211yesmaximal, characteristic, complemented
6#85499—33nomaximal
7#84059—31nomaximal
8C3×Q8×A4—21no—
9C3×Q8×A4C2212yes—
10C2×C3×C4×A4—21no—
11C2×C3×C4×A4C2211yescharacteristic
12#64269—21no—
13C4⋊(C3⋊S4)C2211yescharacteristic
14C4×C3⋊S4—21no—
15#64259C2211yescharacteristic
16#64151—41nomaximal
17C3⋊(A4⋊C8)C2212yes—
18C23×C3×Q8D311yescharacteristic, complemented
19C2×Q8×A4D313yescomplemented
20#5912—33no—
21C2×#942—31no—
22C2×#942—31no—
23#5791—31no—
24#5417—36no—
25#5416—36no—
26#5409—33no—
27#5228—32no—
28#5228—32no—
29#5227—32no—
30#5227—32no—
31#5225—31no—
32#5225—31no—
33#5207—31no—
34C22×C3×A4D411yescharacteristic
35C2×C3⋊S4—21no—
36C2×C32×Q8—41no—
37#3828—41no—
38C3×C4×A4—22no—
39C3×C4×A4D411yescharacteristic
40C3×C4×A4C2311yescharacteristic
41C3⋊SL(2,ℤ/4)—21no—
42#3756—42no—
43#3755—42no—
44#3748—41no—
45#3004—91noSylow
46C22×C3×Q8—21no—
47C22×C3×Q8—31no—
48C22×C3×Q8—31no—
49C22×C3×Q8—32no—
50C22×C3×Q8—61no—
51C22×C3×Q8—61no—
52C22×C3×Q8C2×D312yes—
53C23×C3×C4—21no—
54C23×C3×C4C2×D311yescharacteristic
55C2×#235—31no—
56Q8×A4—23no—
57Q8×A4C2×D316yes—
58C2×C4×A4—23no—
59C2×C4×A4C2×D313yes—
60GL(2,ℤ/4)—63no—
61C4⋊S4—33no—
62C4×S4—63no—
63A4⋊Q8—33no—
64#953—61no—
65C2×#216—32no—
66C2×#216—32no—
67#942—62no—
68#942—62no—
69#942—62no—
70#942—61no—
71#942—61no—
72#942—123no—
73C2×#215—32no—
74C2×#215—32no—
75#930—31no—
76#930—31no—
77#929—61no—
78C4×#67—61no—
79C4×#67—61no—
80#926—31no—
81#924—31no—
82#924—31no—
83C2×#208—31no—
84C2×#208—31no—
85A4⋊C8—36no—
86#835—32no—
87#835—32no—
88#835—32no—
89#835—32no—
90#830—32no—
91#830—32no—
92C2×C3×A4—21no—
93C2×C3×A4C2×D411yescharacteristic
94C3⋊S4—41no—
95C32×Q8—42no—
96C32×Q8—81no—
97C2×C32×C4—41no—
98C2×C32×C4—81no—
99#638—81no—
100C4⋊(C3⋊D3)—41no—
101C4×C3⋊D3—81no—
102#634—41no—
103C3⋊(C3⋊C8)—42no—
104C23×Q8C3⋊D311yescharacteristic, complemented
105C2×#137—91no—
106C2×#137—91no—
107#525—91no—
108#451—92no—
109#451—92no—
110#448—92no—
111#448—92no—
112#419—91no—
113#419—91no—
114#407—91no—
115C24×C3#6711yescharacteristic
116C22×A4#6713yes—
117C2×S4—63no—
118C2×C3×Q8—32no—
119C2×C3×Q8—32no—
120C2×C3×Q8—32no—
121C2×C3×Q8—32no—
122C2×C3×Q8—32no—
123C2×C3×Q8—31no—
124C2×C3×Q8—43no—
125C2×C3×Q8—61no—
126C2×C3×Q8—61no—
127C2×C3×Q8—61no—
128C2×C3×Q8—62no—
129C2×C3×Q8—61no—
130C2×C3×Q8—61no—
131C2×C3×Q8—62no—
132C2×C3×Q8—64no—
133C2×C3×Q8S411yescharacteristic, complemented
134C22×C3×C4—22no—
135C22×C3×C4—31no—
136C22×C3×C4—31no—
137C22×C3×C4—31no—
138C22×C3×C4—31no—
139C22×C3×C4—61no—
140C22×C3×C4—62no—
141C22×C3×C4—61no—
142C22×C3×C4#6711yescharacteristic
143C22×C3×C4C22×D311yescharacteristic
144C2×#67—61no—
145C2×#67—61no—
146C2×#67—61no—
147#235—61no—
148#235—61no—
149#235—61no—
150#235—61no—
151#235—123no—
152C2×D12—31no—
153C2×C4×D3—61no—
154C2×Dic6—31no—
155C4×A4—26no—
156C4×A4#6713yes—
157C4×A4C22×D313yes—
158SL(2,ℤ/4)—63no—
159#217—61no—
160#217—61no—
161#217—61no—
162#216—64no—
163#216—62no—
164#216—62no—
165#216—126no—
166#215—64no—
167#215—62no—
168#215—62no—
169#215—126no—
170(C2×D3)⋊C4—61no—
171(C2×D3)⋊C4—61no—
172C4⋊Dic3—31no—
173C4⋊Dic3—31no—
174Dic3⋊C4—61no—
175Dic3⋊C4—61no—
176C4×Dic3—61no—
177#208—61no—
178#208—61no—
179#208—61no—
180#208—61no—
181#208—123no—
182C2×C3⋊C8—32no—
183C2×C3⋊C8—32no—
184C22×C32—41no—
185C2×C3⋊D3—81no—
186C3×A4#13711yescharacteristic, complemented
187C32×C4—41no—
188C32×C4—41no—
189C32×C4—82no—
190C3⋊Dic3—81no—
191C2×#41—91no—
192C22×Q8—21no—
193C22×Q8—31no—
194C22×Q8—31no—
195C22×Q8—32no—
196C22×Q8—61no—
197C22×Q8—61no—
198C22×Q8C2×C3⋊D312yes—
199C23×C4—21no—
200C23×C4C2×C3⋊D311yescharacteristic
201#137—182no—
202#137—182no—
203#137—182no—
204#137—181no—
205#137—181no—
206C2×Q16—92no—
207C2×Q16—92no—
208C2×SD16—92no—
209C2×SD16—92no—
210C2×M16—91no—
211C2×M16—91no—
212#123—181no—
213C22⋊Q8—91no—
214C22⋊Q8—91no—
215#121—91no—
216#121—91no—
217#120—181no—
218C4×D4—181no—
219C4×D4—181no—
220#117—91no—
221#103—92no—
222#103—92no—
223#103—92no—
224#103—92no—
225C22⋊C8—92no—
226C22⋊C8—92no—
227C23×C3—21no—
228C23×C3—31no—
229C23×C3—31no—
230C23×C3—61no—
231C23×C3C2×#6711yescharacteristic
232C22×D3—61no—
233C2×A4—23no—
234C2×A4C2×#6713yes—
235S4—123no—
236C3×Q8—21no—
237C3×Q8—34no—
238C3×Q8—32no—
239C3×Q8—46no—
240C3×Q8—62no—
241C3×Q8—62no—
242C3×Q8—62no—
243C3×Q8—61no—
244C3×Q8—83no—
245C3×Q8C2×S412yes—
246C2×C3×C4—21no—
247C2×C3×C4—31no—
248C2×C3×C4—31no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—43no—
255C2×C3×C4—62no—
256C2×C3×C4—62no—
257C2×C3×C4—61no—
258C2×C3×C4—62no—
259C2×C3×C4—61no—
260C2×C3×C4—62no—
261C2×C3×C4—83no—
262C2×C3×C4C2×S411yescharacteristic
263#67—121no—
264#67—121no—
265#67—122no—
266#67—121no—
267#67—121no—
268#67—243no—
269C2×Dic3—61no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272D12—61no—
273D12—61no—
274D12—123no—
275C4×D3—121no—
276C4×D3—121no—
277C4×D3—243no—
278Dic6—61no—
279Dic6—61no—
280Dic6—123no—
281C3⋊C8—62no—
282C3⋊C8—62no—
283C3⋊C8—126no—
284C2×C32—41no—
285C2×C32—81no—
286C3⋊D3—161no—
287C24#63811yescharacteristic
288#41—181no—
289#41—181no—
290#41—181no—
291#41—181no—
292C2×Q8—32no—
293C2×Q8—32no—
294C2×Q8—32no—
295C2×Q8—32no—
296C2×Q8—32no—
297C2×Q8—31no—
298C2×Q8—61no—
299C2×Q8—61no—
300C2×Q8—61no—
301C2×Q8—62no—
302C2×Q8—61no—
303C2×Q8—61no—
304C2×Q8—62no—
305C2×Q8—64no—
306C2×Q8—91no—
307C2×Q8C3⋊S411yescharacteristic, complemented
308C2×D4—91no—
309C2×D4—181no—
310C2×D4—181no—
311C2×D4—181no—
312C22×C4—22no—
313C22×C4—31no—
314C22×C4—31no—
315C22×C4—31no—
316C22×C4—31no—
317C22×C4—61no—
318C22×C4—62no—
319C22×C4—61no—
320C22×C4—181no—
321C22×C4#63811yescharacteristic
322C22×C4C22×C3⋊D311yescharacteristic
323Q16—184no—
324Q16—182no—
325Q16—182no—
326SD16—184no—
327SD16—182no—
328SD16—182no—
329M16—181no—
330M16—181no—
331M16—181no—
332M16—181no—
333C2×C8—92no—
334C2×C8—92no—
335C4⋊C4—91no—
336C4⋊C4—91no—
337C4⋊C4—181no—
338C4⋊C4—181no—
339C22⋊C4—181no—
340C22⋊C4—181no—
341C22⋊C4—181no—
342C22⋊C4—181no—
343C22⋊C4—181no—
344C42—181no—
345C22×C3—31no—
346C22×C3—31no—
347C22×C3—32no—
348C22×C3—31no—
349C22×C3—43no—
350C22×C3—61no—
351C22×C3—61no—
352C22×C3—61no—
353C22×C3#94211yescharacteristic, complemented
354C22×C3GL(2,ℤ/4)11yescharacteristic
355C2×D3—121no—
356C2×D3—121no—
357C2×D3—121no—
358C2×D3—243no—
359A4#94213yescomplemented
360C3×C4—22no—
361C3×C4—31no—
362C3×C4—31no—
363C3×C4—43no—
364C3×C4—43no—
365C3×C4—62no—
366C3×C4—86no—
367C3×C4GL(2,ℤ/4)11yescharacteristic
368C3×C4C22×S411yescharacteristic
369Dic3—121no—
370Dic3—121no—
371Dic3—121no—
372Dic3—243no—
373C32—41noSylow
374C23—21no—
375C23—31no—
376C23—31no—
377C23—61no—
378C23—181no—
379C23C2×#63811yescharacteristic
380Q8—21no—
381Q8—34no—
382Q8—32no—
383Q8—62no—
384Q8—62no—
385Q8—62no—
386Q8—61no—
387Q8—181no—
388Q8—181no—
389Q8C2×C3⋊S412yes—
390D4—181no—
391D4—181no—
392D4—361no—
393D4—361no—
394D4—362no—
395D4—361no—
396D4—361no—
397C2×C4—21no—
398C2×C4—31no—
399C2×C4—31no—
400C2×C4—31no—
401C2×C4—31no—
402C2×C4—31no—
403C2×C4—31no—
404C2×C4—31no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—61no—
408C2×C4—62no—
409C2×C4—61no—
410C2×C4—62no—
411C2×C4—181no—
412C2×C4—181no—
413C2×C4—181no—
414C2×C4—361no—
415C2×C4—361no—
416C2×C4C2×C3⋊S411yescharacteristic
417C8—182no—
418C8—182no—
419C2×C3—21no—
420C2×C3—31no—
421C2×C3—31no—
422C2×C3—43no—
423C2×C3—61no—
424C2×C3—83no—
425C2×C3C2×GL(2,ℤ/4)11yescharacteristic
426D3—241no—
427D3—483no—
428C22—31no—
429C22—31no—
430C22—32no—
431C22—31no—
432C22—61no—
433C22—61no—
434C22—61no—
435C22—361no—
436C22—361no—
437C22—361no—
438C22#6415111yescharacteristic, complemented
439C22#6426911yescharacteristic
440C4—22no—
441C4—31no—
442C4—31no—
443C4—62no—
444C4—361no—
445C4—361no—
446C4—361no—
447C4#6426911yescharacteristic
448C4C22×C3⋊S411yescharacteristic
449C3—43no—
450C3#8549911yescharacteristic, complemented
451C2—21no—
452C2—31no—
453C2—31no—
454C2—61no—
455C2—721no—
456C2C2×#6426911yescharacteristic
subgroupG/H
centerC2C2×#64269
commutator, cosocleC3×C4×A4C23
FrattiniC4C22×C3⋊S4
FittingC23×C3×Q8D3
socleC23×C3C2×#67
prime radical#10596575C2
hypercenterC2×Q8C3⋊S4
hypocenterC3×A4#137
LCSG ▷ C3×C4×A4 ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×Q8×A4C2semidirect
3. #10593651C2semidirect
4. #10593650C2semidirect
5. #10593643C2semidirect
18. C23×C3×Q8D3semidirect
19. C2×Q8×A4D3semidirect
104. C23×Q8C3⋊D3semidirect
133. C2×C3×Q8S4semidirect
186. C3×A4#137semidirect
307. C2×Q8C3⋊S4semidirect
353. C22×C3#942semidirect
359. A4#942semidirect
438. C22#64151semidirect
450. C3#85499semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×A42
#105965752
#105936512
#105936502
#105936432
#854993
#840593
#641514