Finite Group Databasebeta

← #72252814

Subgroups — #72252814

Order 5184.

895 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 895 conjugacy classes of subgroups — too many to draw. Only its 64 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×C22⋊A4C311yesmaximal, characteristic
2#13180074C311yesmaximal, characteristic, complemented
3#13180073C312yesmaximal, complemented
4#13180071C311yesmaximal, characteristic, complemented
5#13180069C312yesmaximal, complemented
6#13180067C311yesmaximal, characteristic, complemented
7#13180065C312yesmaximal, complemented
8C22×#88881C311yesmaximal, characteristic, complemented
9C22×#88880C312yesmaximal, complemented
10#13049240—41nomaximal
11#13049233—42nomaximal
12C2×#88881—31no—
13C2×#88880—32no—
14C3×C22⋊(C22⋊A4)C3211yescharacteristic
15C3×C22⋊(C22⋊A4)C3211yescharacteristic
16C22×C3×C22⋊A4C3211yescharacteristic
17C24×C3×A4C3211yescharacteristic
18A4×C22⋊A4—33no—
19C22⋊(C22⋊(C22⋊C9))C3212yescomplemented
20C22×C22⋊(C22⋊C9)C3212yescomplemented
21C22×C22⋊(C22⋊C9)C3212yescomplemented
22C22×C22⋊(C22⋊C9)C3211yescharacteristic, complemented
23C24×C22⋊C9C3212yescomplemented
24C32×C22⋊A4—41no—
25C3×A42—42no—
26C3×A42—121no—
27#88907—41no—
28C9⋊(C22⋊A4)—42no—
29C9×C22⋊A4—42no—
30C22×#1094—42no—
31C22×C9⋊A4—42no—
32C22×C9⋊A4—42no—
33#88881—42no—
34#88881—41no—
35#88881—42no—
36#88881—42no—
37#88881—42no—
38#88881—122no—
39#88881A411yescharacteristic, complemented
40#88880—42no—
41#88880—42no—
42#88880—42no—
43#88880—121no—
44#88880A412yescomplemented
45(C22⋊A4)⋊C9—42no—
46(C22⋊A4)⋊C9—41no—
47(C22⋊A4)⋊C9—42no—
48(C22⋊A4)⋊C9—42no—
49(C22⋊A4)⋊C9—122no—
50A4×3−1+2—161no—
51A4×3−1+2—162no—
52C2×C3×C22⋊A4—31no—
53C23×C3×A4—32no—
54C23×C3×A4—91no—
55C2×C22⋊(C22⋊C9)—32no—
56C2×C22⋊(C22⋊C9)—32no—
57C2×C22⋊(C22⋊C9)—31no—
58C23×C22⋊C9—32no—
59C23×C22⋊C9—32no—
60C23×C22⋊C9—92no—
61C2×#1094—122no—
62C2×C9⋊A4—122no—
63C2×C9⋊A4—122no—
64C26×C3C3311yescharacteristic
65C22⋊(C22⋊A4)—31no—
66C22⋊(C22⋊A4)3−1+213yescomplemented
67C22×C22⋊A4—31no—
68C24×A4—31no—
69C24×C32—41no—
70C3×C22⋊A4—42no—
71C3×C22⋊A4—41no—
72C3×C22⋊A4—42no—
73C3×C22⋊A4—41no—
74C3×C22⋊A4—121no—
75C3×C22⋊A4—121no—
76C3×C22⋊A4—122no—
77C3×C22⋊A4—122no—
78C3×C22⋊A4—123no—
79C3×C22⋊A4—361no—
80C3×C22⋊A4—361no—
81C3×C22⋊A4C3×A411yescharacteristic
82C22×C3×A4—31no—
83C22×C3×A4—31no—
84C22×C3×A4—42no—
85C22×C3×A4—91no—
86C22×C3×A4—92no—
87C22×C3×A4—121no—
88C22×C3×A4C3×A412yes—
89A42—126no—
90A42—129no—
91C24×C9—42no—
92C22⋊(C22⋊C9)—42no—
93C22⋊(C22⋊C9)—42no—
94C22⋊(C22⋊C9)—42no—
95C22⋊(C22⋊C9)—122no—
96C22⋊(C22⋊C9)—122no—
97C22⋊(C22⋊C9)—122no—
98C22⋊(C22⋊C9)—362no—
99C22⋊(C22⋊C9)C3×A412yescomplemented
100C22⋊(C22⋊C9)C3×A411yescharacteristic, complemented
101C22⋊(C22⋊C9)C3×A412yescomplemented
102C22×C22⋊C9—32no—
103C22×C22⋊C9—32no—
104C22×C22⋊C9—42no—
105C22×C22⋊C9—42no—
106C22×C22⋊C9—42no—
107C22×C22⋊C9—42no—
108C22×C22⋊C9—42no—
109C22×C22⋊C9—92no—
110C22×C22⋊C9—92no—
111C22×C22⋊C9—92no—
112C22×C22⋊C9—122no—
113C22×C22⋊C9—121no—
114C22×C22⋊C9—122no—
115C22×C22⋊C9C3×A412yescomplemented
116C22×C22⋊C9C3×A412yescomplemented
117C32×A4—162no—
118C32×A4—161no—
119C32×A4—481no—
120C22×3−1+2—161no—
121C22×3−1+2—162no—
122#1094—42no—
123#1094—162no—
124#1094—162no—
125#1094—481no—
126C9⋊A4—42no—
127C9⋊A4—42no—
128C9⋊A4—161no—
129C9⋊A4—162no—
130C9⋊A4—162no—
131C9⋊A4—162no—
132C9⋊A4—162no—
133C9⋊A4—162no—
134C9⋊A4—161no—
135C9⋊A4—162no—
136C9⋊A4—482no—
137C9×A4—161no—
138C9×A4—162no—
139C9×A4—162no—
140C9×A4—162no—
141C9×A4—162no—
142C9×A4—482no—
143C25×C3—32no—
144C25×C3—31no—
145C25×C3—91no—
146C25×C3—92no—
147C25×C3—271no—
148C2×C22⋊A4—91no—
149C23×A4—92no—
150C23×A4—93no—
151C3×3−1+2—641noSylow
152C23×C32—122no—
153C23×C32—361no—
154C2×C3×A4—32no—
155C2×C3×A4—91no—
156C2×C3×A4—122no—
157C2×C3×A4—361no—
158C23×C9—122no—
159C23×C9—122no—
160C23×C9—362no—
161C2×C22⋊C9—32no—
162C2×C22⋊C9—32no—
163C2×C22⋊C9—92no—
164C2×C22⋊C9—122no—
165C2×C22⋊C9—122no—
166C2×C22⋊C9—122no—
167C2×C22⋊C9—122no—
168C2×C22⋊C9—122no—
169C2×C22⋊C9—362no—
170C2×C22⋊C9—362no—
171C2×C22⋊C9—361no—
172C26C3×3−1+211yescharacteristic, complemented, Sylow
173C2×3−1+2—481no—
174C2×3−1+2—482no—
175C24×C3—31no—
176C24×C3—31no—
177C24×C3—32no—
178C24×C3—32no—
179C24×C3—91no—
180C24×C3—92no—
181C24×C3—92no—
182C24×C3—92no—
183C24×C3—92no—
184C24×C3—91no—
185C24×C3—91no—
186C24×C3—92no—
187C24×C3—121no—
188C24×C3—272no—
189C24×C3—271no—
190C24×C3—272no—
191C24×C3—271no—
192C24×C3—271no—
193C24×C3—272no—
194C24×C3—272no—
195C24×C3—271no—
196C24×C3—271no—
197C24×C3—274no—
198C24×C3—272no—
199C24×C3C32×A412yes—
200C24×C3C32×A411yescharacteristic
201C22⋊A4—31no—
202C22⋊A4—46no—
203C22⋊A4—43no—
204C22⋊A4—123no—
205C22⋊A4—126no—
206C22⋊A4—121no—
207C22⋊A4—123no—
208C22⋊A4—122no—
209C22⋊A4—363no—
210C22⋊A4—362no—
211C22⋊A4—363no—
212C22×A4—33no—
213C22×A4—32no—
214C22×A4—91no—
215C22×A4—93no—
216C22×A4—96no—
217C22×A4—123no—
218C22×A4—122no—
219C22×C32—42no—
220C22×C32—121no—
221C22×C32—121no—
222C22×C32—161no—
223C22×C32—361no—
224C22×C32—362no—
225C3×A4—42no—
226C3×A4—121no—
227C3×A4—162no—
228C3×A4—161no—
229C3×A4—162no—
230C3×A4—161no—
231C3×A4—482no—
232C3×A4—481no—
233C3×A4—481no—
234C3×A4—482no—
235C3×A4—483no—
236C3×A4—486no—
237C3×A4—489no—
238C3×A4—1441no—
239C3×A4—1441no—
240C3×A4A4211yescharacteristic
241C22×C9—42no—
242C22×C9—42no—
243C22×C9—122no—
244C22×C9—122no—
245C22×C9—162no—
246C22×C9—162no—
247C22×C9—161no—
248C22×C9—362no—
249C22×C9—362no—
250C22×C9—362no—
251C22⋊C9—42no—
252C22⋊C9—42no—
253C22⋊C9—42no—
254C22⋊C9—42no—
255C22⋊C9—42no—
256C22⋊C9—122no—
257C22⋊C9—122no—
258C22⋊C9—121no—
259C22⋊C9—162no—
260C22⋊C9—162no—
261C22⋊C9—162no—
262C22⋊C9—482no—
263C22⋊C9—482no—
264C22⋊C9—482no—
265C22⋊C9—1442no—
266C22⋊C9A4212yescomplemented
267C25—32no—
268C25—31no—
269C25—92no—
270C25—91no—
271C25—271no—
272C33—641no—
2733−1+2—161no—
2743−1+2—162no—
2753−1+2—642no—
2763−1+2—641no—
2773−1+2—642no—
2783−1+2—641no—
279C3×C9—642no—
280C3×C9—641no—
281C23×C3—32no—
282C23×C3—32no—
283C23×C3—32no—
284C23×C3—92no—
285C23×C3—91no—
286C23×C3—91no—
287C23×C3—92no—
288C23×C3—92no—
289C23×C3—91no—
290C23×C3—92no—
291C23×C3—91no—
292C23×C3—271no—
293C23×C3—272no—
294C23×C3—272no—
295C23×C3—271no—
296C23×C3—271no—
297C23×C3—271no—
298C23×C3—272no—
299C23×C3—272no—
300C23×C3—271no—
301C23×C3—272no—
302C23×C3—272no—
303C23×C3—272no—
304C23×C3—271no—
305C23×C3—272no—
306C23×C3—272no—
307C23×C3—271no—
308C23×C3—274no—
309C23×C3—274no—
310C23×C3—274no—
311C23×C3—274no—
312C23×C3—272no—
313C23×C3—272no—
314C23×C3—272no—
315C23×C3—362no—
316C23×C3—363no—
317C2×A4—92no—
318C2×A4—93no—
319C2×A4—362no—
320C2×A4—363no—
321C2×C32—122no—
322C2×C32—361no—
323C2×C32—481no—
324C2×C9—122no—
325C2×C9—122no—
326C2×C9—362no—
327C2×C9—481no—
328C2×C9—482no—
329C2×C9—482no—
330C24—31no—
331C24—31no—
332C24—32no—
333C24—32no—
334C24—91no—
335C24—92no—
336C24—92no—
337C24—92no—
338C24—92no—
339C24—91no—
340C24—91no—
341C24—92no—
342C24—272no—
343C24—271no—
344C24—272no—
345C24—271no—
346C24—271no—
347C24—272no—
348C24—272no—
349C24—271no—
350C24—271no—
351C24—272no—
352C24—274no—
353C24A4×3−1+212yescomplemented
354C24A4×3−1+211yescharacteristic, complemented
355C22×C3—31no—
356C22×C3—32no—
357C22×C3—32no—
358C22×C3—31no—
359C22×C3—92no—
360C22×C3—91no—
361C22×C3—92no—
362C22×C3—92no—
363C22×C3—92no—
364C22×C3—92no—
365C22×C3—91no—
366C22×C3—91no—
367C22×C3—123no—
368C22×C3—122no—
369C22×C3—271no—
370C22×C3—272no—
371C22×C3—271no—
372C22×C3—272no—
373C22×C3—271no—
374C22×C3—272no—
375C22×C3—274no—
376C22×C3—272no—
377C22×C3—272no—
378C22×C3—271no—
379C22×C3—271no—
380C22×C3—361no—
381C22×C3—363no—
382C22×C3—366no—
383C22×C3—481no—
384C22×C3C3×A4212yes—
385C22×C3C3×A4211yescharacteristic
386A4—31no—
387A4—123no—
388A4—122no—
389A4—166no—
390A4—163no—
391A4—483no—
392A4—486no—
393A4—483no—
394A4—482no—
395A4—481no—
396A4—1443no—
397A4—1442no—
398A4—1443no—
399C32—41no—
400C32—161no—
401C32—641no—
402C32—641no—
403C32—1923no—
404C9—42no—
405C9—162no—
406C9—161no—
407C9—162no—
408C9—642no—
409C23—32no—
410C23—32no—
411C23—32no—
412C23—92no—
413C23—91no—
414C23—91no—
415C23—92no—
416C23—92no—
417C23—91no—
418C23—92no—
419C23—91no—
420C23—271no—
421C23—272no—
422C23—272no—
423C23—271no—
424C23—271no—
425C23—271no—
426C23—272no—
427C23—272no—
428C23—271no—
429C23—272no—
430C23—272no—
431C23—272no—
432C23—271no—
433C23—272no—
434C23—272no—
435C23—271no—
436C23—274no—
437C23—274no—
438C23—274no—
439C23—272no—
440C23—272no—
441C23—274no—
442C23—272no—
443C2×C3—31no—
444C2×C3—32no—
445C2×C3—92no—
446C2×C3—91no—
447C2×C3—271no—
448C2×C3—362no—
449C2×C3—363no—
450C2×C3—1441no—
451C22—31no—
452C22—32no—
453C22—32no—
454C22—31no—
455C22—92no—
456C22—91no—
457C22—92no—
458C22—92no—
459C22—92no—
460C22—91no—
461C22—91no—
462C22—92no—
463C22—272no—
464C22—271no—
465C22—272no—
466C22—271no—
467C22—271no—
468C22—272no—
469C22—274no—
470C22—272no—
471C22—272no—
472C22—271no—
473C22—271no—
474C22#1304923312yescomplemented
475C22#1304924011yescharacteristic, complemented
476C3—121no—
477C3—481no—
478C3—643no—
479C3—1921no—
480C3A4311yescharacteristic
481C2—32no—
482C2—31no—
483C2—92no—
484C2—91no—
485C2—271no—
subgroupG/H
center, Frattini, hypercenterC3A43
commutator, Fitting, socle, cosocleC26×C3C33
prime radicalC3×A4×C22⋊A4C3
hypocenterC26C3×3−1+2
LCSG ▷ C26×C3 ▷ C26
UCSC1 ◁ C3

Complements

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

NH ≅ G/Ntype
2. #13180074C3semidirect
3. #13180073C3semidirect
4. #13180071C3semidirect
5. #13180069C3semidirect
6. #13180067C3semidirect
7. #13180065C3semidirect
8. C22×#88881C3semidirect
9. C22×#88880C3semidirect
19. C22⋊(C22⋊(C22⋊C9))C32semidirect
20. C22×C22⋊(C22⋊C9)C32semidirect
21. C22×C22⋊(C22⋊C9)C32semidirect
22. C22×C22⋊(C22⋊C9)C32semidirect
23. C24×C22⋊C9C32semidirect
39. #88881A4semidirect
44. #88880A4semidirect
66. C22⋊(C22⋊A4)3−1+2semidirect
99. C22⋊(C22⋊C9)C3×A4semidirect
100. C22⋊(C22⋊C9)C3×A4semidirect
101. C22⋊(C22⋊C9)C3×A4semidirect
115. C22×C22⋊C9C3×A4semidirect
116. C22×C22⋊C9C3×A4semidirect
172. C26C3×3−1+2semidirect
266. C22⋊C9A42semidirect
353. C24A4×3−1+2semidirect
354. C24A4×3−1+2semidirect
474. C22#13049233semidirect
475. C22#13049240semidirect

Maximal subgroups

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

subgroupindex
C3×A4×C22⋊A43
#131800743
#131800733
#131800713
#131800693
#131800673
#131800653
C22×#888813
C22×#888803
#130492404
#130492334