Finite Group Databasebeta

← #11923061

Subgroups — #11923061

Order 1152.

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

normal subgroups only

This group has 724 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
1C3×A4×#41C211yesmaximal, characteristic
2C2×#64259C211yesmaximal, characteristic, complemented
3#10593650C212yesmaximal, complemented
4#10593648C212yesmaximal, complemented
5#10593643C211yesmaximal, characteristic, complemented
6#85512—33nomaximal
7#84084—31nomaximal
8C3×Q8×A4C2211yescharacteristic
9C3×D4×A4—21no—
10C3×D4×A4C2211yescharacteristic
11C2×C3×C4×A4—21no—
12C2×C3×C4×A4C2211yescharacteristic
13C2×C3⋊SL(2,ℤ/4)—21no—
14#64259—21no—
15#64259C2212yes—
16#64160—41nomaximal
17C3⋊(A4⋊C8)C2212yes—
18C22×C3×#41D311yescharacteristic
19A4×#41D313yes—
20C2×A4⋊Q8—33no—
21C2×#951—31no—
22C2×#951—31no—
23C2×#924—31no—
24#5416—36no—
25#5414—36no—
26#5409—33no—
27#5239—32no—
28#5239—32no—
29#5232—31no—
30#5232—31no—
31#5228—32no—
32#5218—32no—
33#5207—31no—
34C22×C3×A4—21no—
35C22×C3×A4D411yescharacteristic
36C32×#41—41no—
37C2×#634—41no—
38C3×C4×A4—21no—
39C3×C4×A4D411yescharacteristic
40C3×C4×A4C2311yescharacteristic
41C3⋊SL(2,ℤ/4)—22no—
42#3756—42no—
43#3754—42no—
44#3748—41no—
45#3007—91noSylow
46C2×C3×#41—31no—
47C2×C3×#41—31no—
48C2×C3×#41—31no—
49C2×C3×#41—31no—
50C2×C3×#41—61no—
51C2×C3×#41—61no—
52C22×C3×Q8C2×D311yescharacteristic
53C22×C3×D4—21no—
54C22×C3×D4C2×D311yescharacteristic
55C23×C3×C4—21no—
56C23×C3×C4C2×D311yescharacteristic
57C22×Dic6—31no—
58Q8×A4C2×D313yes—
59D4×A4—23no—
60D4×A4C2×D313yes—
61C2×C4×A4—23no—
62C2×C4×A4C2×D313yes—
63C2×SL(2,ℤ/4)—63no—
64A4⋊Q8—36no—
65A4⋊Q8—63no—
66C2×#217—61no—
67#951—62no—
68#951—62no—
69#951—61no—
70#951—61no—
71#951—61no—
72#951—61no—
73#951—123no—
74C2×#216—32no—
75C2×#216—32no—
76C2×#214—32no—
77C2×#214—32no—
78C2×C4⋊Dic3—31no—
79#924—32no—
80#924—32no—
81#924—61no—
82#924—61no—
83C2×Dic3⋊C4—61no—
84C2×#208—31no—
85C2×#208—31no—
86A4⋊C8—36no—
87#835—32no—
88#835—32no—
89#832—32no—
90#832—32no—
91#830—32no—
92#830—32no—
93C2×C3×A4—21no—
94C2×C3×A4—41no—
95C2×C3×A4C2×D411yescharacteristic
96C32×Q8—41no—
97C32×D4—41no—
98C32×D4—81no—
99C2×C32×C4—41no—
100C2×C32×C4—81no—
101C2×C3⋊Dic3—81no—
102#634—42no—
103#634—81no—
104C3⋊(C3⋊C8)—42no—
105C22×#41C3⋊D311yescharacteristic
106C2×#137—91no—
107C2×#137—91no—
108C2×C22⋊Q8—91no—
109#452—92no—
110#452—92no—
111#451—92no—
112#450—92no—
113#417—91no—
114#417—91no—
115#407—91no—
116C24×C3—21no—
117C24×C3#6711yescharacteristic
118C22×A4—23no—
119C22×A4#6713yes—
120C3×#41—32no—
121C3×#41—32no—
122C3×#41—31no—
123C3×#41—31no—
124C3×#41—31no—
125C3×#41—43no—
126C3×#41—61no—
127C3×#41—61no—
128C3×#41—61no—
129C3×#41—62no—
130C3×#41—62no—
131C3×#41S411yescharacteristic
132C2×C3×Q8—31no—
133C2×C3×Q8—31no—
134C2×C3×Q8—61no—
135C2×C3×D4—31no—
136C2×C3×D4—31no—
137C2×C3×D4—61no—
138C2×C3×D4—61no—
139C2×C3×D4—61no—
140C2×C3×D4—61no—
141C2×C3×D4—61no—
142C22×C3×C4—21no—
143C22×C3×C4—31no—
144C22×C3×C4—31no—
145C22×C3×C4—31no—
146C22×C3×C4—31no—
147C22×C3×C4—61no—
148C22×C3×C4—61no—
149C22×C3×C4—61no—
150C22×C3×C4—61no—
151C22×C3×C4#6711yescharacteristic
152C22×C3×C4C22×D311yescharacteristic
153C22×Dic3—61no—
154C22×Dic3—61no—
155C2×Dic6—32no—
156C2×Dic6—61no—
157C2×Dic6—61no—
158C2×Dic6—61no—
159C2×Dic6—61no—
160C2×Dic6—61no—
161C2×Dic6—123no—
162C4×A4—23no—
163C4×A4#6713yes—
164C4×A4C22×D313yes—
165SL(2,ℤ/4)—66no—
166#217—62no—
167#217—62no—
168#216—64no—
169#216—62no—
170#216—62no—
171#216—126no—
172#214—64no—
173#214—62no—
174#214—62no—
175#214—126no—
176C4⋊Dic3—32no—
177C4⋊Dic3—61no—
178Dic3⋊C4—62no—
179Dic3⋊C4—62no—
180#208—61no—
181#208—61no—
182#208—61no—
183#208—61no—
184#208—123no—
185C2×C3⋊C8—32no—
186C2×C3⋊C8—32no—
187C22×C32—41no—
188C22×C32—81no—
189C3×A4#13711yescharacteristic, complemented
190C32×C4—41no—
191C32×C4—41no—
192C32×C4—81no—
193C3⋊Dic3—82no—
194C2×#41—31no—
195C2×#41—31no—
196C2×#41—31no—
197C2×#41—31no—
198C2×#41—61no—
199C2×#41—61no—
200C22×Q8—91no—
201C22×Q8C2×C3⋊D311yescharacteristic
202C22×D4—21no—
203C22×D4C2×C3⋊D311yescharacteristic
204C23×C4—21no—
205C23×C4C2×C3⋊D311yescharacteristic
206#137—182no—
207#137—182no—
208#137—181no—
209#137—181no—
210#137—181no—
211#137—181no—
212C2×Q16—92no—
213C2×Q16—92no—
214C2×SD16—92no—
215C2×SD16—92no—
216C2×M16—91no—
217C2×M16—91no—
218C22⋊Q8—92no—
219C22⋊Q8—92no—
220C22⋊Q8—181no—
221C22⋊Q8—181no—
222C2×C4⋊C4—91no—
223C2×C4⋊C4—181no—
224C2×C22⋊C4—181no—
225#103—92no—
226#103—92no—
227#102—92no—
228#102—92no—
229C22⋊C8—92no—
230C22⋊C8—92no—
231C23×C3—21no—
232C23×C3—31no—
233C23×C3—31no—
234C23×C3—41no—
235C23×C3—61no—
236C23×C3—61no—
237C23×C3—61no—
238C23×C3—121no—
239C23×C3C2×#6711yescharacteristic
240C2×A4—23no—
241C2×A4—43no—
242C2×A4C2×#6713yes—
243C3×Q8—32no—
244C3×Q8—31no—
245C3×Q8—43no—
246C3×Q8—61no—
247C3×Q8C2×S411yescharacteristic
248C3×D4—21no—
249C3×D4—32no—
250C3×D4—31no—
251C3×D4—43no—
252C3×D4—61no—
253C3×D4—61no—
254C3×D4—61no—
255C3×D4—62no—
256C3×D4—61no—
257C3×D4—83no—
258C3×D4C2×S411yescharacteristic
259C2×C3×C4—21no—
260C2×C3×C4—31no—
261C2×C3×C4—31no—
262C2×C3×C4—31no—
263C2×C3×C4—31no—
264C2×C3×C4—31no—
265C2×C3×C4—31no—
266C2×C3×C4—31no—
267C2×C3×C4—43no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—61no—
272C2×C3×C4—62no—
273C2×C3×C4—61no—
274C2×C3×C4—61no—
275C2×C3×C4—83no—
276C2×C3×C4C2×S411yescharacteristic
277C2×Dic3—62no—
278C2×Dic3—62no—
279C2×Dic3—121no—
280C2×Dic3—122no—
281C2×Dic3—121no—
282C2×Dic3—243no—
283Dic6—62no—
284Dic6—62no—
285Dic6—126no—
286Dic6—121no—
287Dic6—121no—
288Dic6—243no—
289C3⋊C8—62no—
290C3⋊C8—62no—
291C3⋊C8—126no—
292C2×C32—41no—
293C2×C32—81no—
294C2×C32—161no—
295C24—21no—
296C24#63811yescharacteristic
297#41—32no—
298#41—32no—
299#41—31no—
300#41—31no—
301#41—31no—
302#41—61no—
303#41—61no—
304#41—61no—
305#41—62no—
306#41—62no—
307#41C3⋊S411yescharacteristic
308C2×Q8—31no—
309C2×Q8—31no—
310C2×Q8—61no—
311C2×Q8—92no—
312C2×Q8—181no—
313C2×Q8—181no—
314C2×Q8—181no—
315C2×Q8—181no—
316C2×Q8—181no—
317C2×D4—31no—
318C2×D4—31no—
319C2×D4—61no—
320C2×D4—61no—
321C2×D4—61no—
322C2×D4—61no—
323C2×D4—61no—
324C22×C4—21no—
325C22×C4—31no—
326C22×C4—31no—
327C22×C4—31no—
328C22×C4—31no—
329C22×C4—61no—
330C22×C4—61no—
331C22×C4—61no—
332C22×C4—61no—
333C22×C4—181no—
334C22×C4—181no—
335C22×C4#63811yescharacteristic
336C22×C4C22×C3⋊D311yescharacteristic
337Q16—184no—
338Q16—182no—
339Q16—182no—
340SD16—184no—
341SD16—182no—
342SD16—182no—
343M16—181no—
344M16—181no—
345M16—181no—
346M16—181no—
347C2×C8—92no—
348C2×C8—92no—
349C4⋊C4—92no—
350C4⋊C4—181no—
351C4⋊C4—182no—
352C4⋊C4—182no—
353C22⋊C4—182no—
354C22⋊C4—182no—
355C22×C3—21no—
356C22×C3—31no—
357C22×C3—31no—
358C22×C3—31no—
359C22×C3—43no—
360C22×C3—61no—
361C22×C3—61no—
362C22×C3—61no—
363C22×C3—61no—
364C22×C3—61no—
365C22×C3—83no—
366C22×C3—121no—
367C22×C3—121no—
368C22×C3—121no—
369C22×C3—121no—
370C22×C3#95111yescharacteristic, complemented
371C22×C3GL(2,ℤ/4)11yescharacteristic
372A4#95113yescomplemented
373C3×C4—21no—
374C3×C4—31no—
375C3×C4—31no—
376C3×C4—43no—
377C3×C4—43no—
378C3×C4—61no—
379C3×C4—83no—
380C3×C4GL(2,ℤ/4)11yescharacteristic
381C3×C4C22×S411yescharacteristic
382Dic3—122no—
383Dic3—122no—
384Dic3—246no—
385C32—41noSylow
386C23—21no—
387C23—31no—
388C23—31no—
389C23—41no—
390C23—61no—
391C23—61no—
392C23—61no—
393C23—121no—
394C23C2×#63811yescharacteristic
395Q8—32no—
396Q8—31no—
397Q8—61no—
398Q8—182no—
399Q8—182no—
400Q8—361no—
401Q8—361no—
402Q8C2×C3⋊S411yescharacteristic
403D4—21no—
404D4—32no—
405D4—31no—
406D4—61no—
407D4—61no—
408D4—61no—
409D4—62no—
410D4—61no—
411D4C2×C3⋊S411yescharacteristic
412C2×C4—21no—
413C2×C4—31no—
414C2×C4—31no—
415C2×C4—31no—
416C2×C4—31no—
417C2×C4—31no—
418C2×C4—31no—
419C2×C4—31no—
420C2×C4—61no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—61no—
424C2×C4—62no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—182no—
428C2×C4—182no—
429C2×C4—361no—
430C2×C4—362no—
431C2×C4—361no—
432C2×C4C2×C3⋊S411yescharacteristic
433C8—182no—
434C8—182no—
435C2×C3—21no—
436C2×C3—31no—
437C2×C3—31no—
438C2×C3—41no—
439C2×C3—43no—
440C2×C3—61no—
441C2×C3—83no—
442C2×C3—121no—
443C2×C3—163no—
444C2×C3C2×GL(2,ℤ/4)11yescharacteristic
445C22—21no—
446C22—31no—
447C22—31no—
448C22—31no—
449C22—61no—
450C22—61no—
451C22—61no—
452C22—61no—
453C22—61no—
454C22—121no—
455C22—121no—
456C22—121no—
457C22—121no—
458C22#6416011yescharacteristic, complemented
459C22#6426911yescharacteristic
460C4—21no—
461C4—31no—
462C4—31no—
463C4—61no—
464C4—362no—
465C4—362no—
466C4#6426911yescharacteristic
467C4C22×C3⋊S411yescharacteristic
468C3—43no—
469C3#8551211yescharacteristic, complemented
470C2—21no—
471C2—31no—
472C2—31no—
473C2—41no—
474C2—61no—
475C2—121no—
476C2C2×#6426911yescharacteristic
subgroupG/H
centerC2C2×#64269
commutator, cosocleC3×C4×A4C23
FrattiniC4C22×C3⋊S4
FittingC22×C3×#41D3
socleC23×C3C2×#67
prime radicalC3×A4×#41C2
hypercenter#41C3⋊S4
hypocenterC3×A4#137
LCSG ▷ C3×C4×A4 ▷ C2×C3×A4 ▷ C3×A4
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
2. C2×#64259C2semidirect
3. #10593650C2semidirect
4. #10593648C2semidirect
5. #10593643C2semidirect
189. C3×A4#137semidirect
370. C22×C3#951semidirect
372. A4#951semidirect
458. C22#64160semidirect
469. C3#85512semidirect

Maximal subgroups

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

subgroupindex
C3×A4×#412
C2×#642592
#105936502
#105936482
#105936432
#855123
#840843
#641604