Finite Group Databasebeta

← #11807755

Subgroups — #11807755

Order 1152.

1365 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64147C211yesmaximal, characteristic
2C3×#5220C212yesmaximal, complemented
3#10591711C212yesmaximal, complemented
4#10591704C212yesmaximal, complemented
5#10591703C212yesmaximal, complemented
6#10591693C212yesmaximal, complemented
7#10591355C211yesmaximal, characteristic, complemented
8#10591271C212yesmaximal, complemented
9#10591219C211yesmaximal, characteristic, complemented
10#74883—32nomaximal
11C22×C32×D4C2211yescharacteristic
12C22×#638C2212yes—
13C22×C4×C3⋊D3C2211yescharacteristic
14C23×C3×Dic3C2212yes—
15C2×#3757C2212yes—
16#64147C418yescomplemented
17C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×#217C2212yes—
19C2×C3×#217C2212yes—
20C2×C3×#217C2212yes—
21C2×#3724C2212yes—
22C2×#3724C2212yes—
23C2×#3722C2211yescharacteristic
24C2×#3722C2211yescharacteristic
25C2×#3722C2212yes—
26C2×Dic32C2212yes—
27C3×C3⋊#95C2212yes—
28C3×C3⋊#95C2212yes—
29#63579C2212yes—
30#63579C2212yes—
31#63579C2211yescharacteristic
32#63579C2211yescharacteristic
33C2×#938—32no—
34C2×#938—61no—
35C3×#386—32no—
36#5220D312yescomplemented
37#5213—32no—
38#5210—32no—
39#4958—32no—
40#4957—32no—
41#4957—32no—
42#4946—32no—
43#4945—32no—
44#4944—32no—
45#4667—32no—
46#4664—32no—
47#4654—32no—
48#4653—32no—
49C24×C32C2312yes—
50C23×C3⋊D3C2311yescharacteristic
51C2×C32×D4—42no—
52C2×C32×D4C2×C414yes—
53C22×C32×C4C2311yescharacteristic
54C2×#638—42no—
55C2×#638—42no—
56C2×#638C2×C418yescomplemented
57C22×C3⋊Dic3C2311yescharacteristic
58C22×C3⋊Dic3C2312yes—
59C2×C4×C3⋊D3—42no—
60C2×C4×C3⋊D3C2×C414yescomplemented
61C22×C3×Dic3—42no—
62C22×C3×Dic3—42no—
63C22×C3×Dic3—42no—
64C22×C3×Dic3C2312yes—
65C22×C3×Dic3C2312yes—
66C22×C3×Dic3C2312yes—
67C22×C3×Dic3C2312yes—
68#3757C2×C418yescomplemented
69C4⋊(C3⋊Dic3)C2×C414yescomplemented
70C3×#217—42no—
71C3×#217—42no—
72C3×#217—42no—
73#3724—42no—
74#3724—42no—
75#3722—41no—
76#3722—42no—
77#3722—41no—
78Dic32—42no—
79#2377—91noSylow
80C22×C3×D4—21no—
81C22×C3×D4C2×D312yes—
82C23×C3×C4—32no—
83C22×#67—32no—
84C22×#67—32no—
85C22×#67—62no—
86C23×Dic3C2×D312yes—
87C22×C4×D3—32no—
88C22×C4×D3—61no—
89C2×C3×C22⋊C4—32no—
90C2×C3×C22⋊C4—32no—
91C2×C3×C22⋊C4—32no—
92C2×#217—32no—
93C2×#217—32no—
94C2×#217—62no—
95C2×#217C2×D312yes—
96C2×#217C2×D312yes—
97C2×#217C2×D312yes—
98#938—316no—
99#938—68no—
100C2×(C2×D3)⋊C4—32no—
101C2×(C2×D3)⋊C4—32no—
102C2×(C2×D3)⋊C4—32no—
103C2×(C2×D3)⋊C4—32no—
104C2×C4⋊Dic3—32no—
105C2×C4⋊Dic3—61no—
106C2×Dic3⋊C4—32no—
107C2×Dic3⋊C4—32no—
108C2×Dic3⋊C4—32no—
109C2×Dic3⋊C4—32no—
110C2×C4×Dic3—32no—
111C2×C4×Dic3—32no—
112C3×#95—32no—
113C3×#95—32no—
114C3⋊#95—32no—
115C3⋊#95—32no—
116C3⋊#95—32no—
117C3⋊#95—32no—
118C3⋊#95—32no—
119C3⋊#95—32no—
120C3⋊#95C2×D312yes—
121C3⋊#95C2×D312yes—
122C23×C32—22no—
123C23×C32—22no—
124C23×C32—22no—
125C23×C32—22no—
126C23×C32—22no—
127C23×C32—22no—
128C23×C32C22×C414yes—
129C23×C32C2411yescharacteristic
130C22×C3⋊D3—22no—
131C22×C3⋊D3—21no—
132C22×C3⋊D3—22no—
133C22×C3⋊D3—21no—
134C22×C3⋊D3C22×C412yes—
135C32×D4—44no—
136C2×C32×C4—22no—
137C2×C32×C4C22×C412yes—
138#638—48no—
139C2×C3⋊Dic3—22no—
140C2×C3⋊Dic3—22no—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3C22×C414yes—
143C2×C3⋊Dic3C22×C412yes—
144C4×C3⋊D3—44no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3—22no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—48no—
158C2×#121—91no—
159#390—92no—
160#388—92no—
161#387—92no—
162#386—32no—
163#386—92no—
164#385—92no—
165#381—91no—
166#380—91no—
167C24×C3—22no—
168C24×C3C22×D312yes—
169C24×C3C22×D312yes—
170C23×D3—32no—
171C23×D3—61no—
172C2×C3×D4—24no—
173C2×C3×D4—42no—
174C2×C3×D4—42no—
175C2×C3×D4—44no—
176C2×C3×D4C4×D318yes—
177C22×C3×C4—21no—
178C22×C3×C4—32no—
179C22×C3×C4—32no—
180C22×C3×C4—32no—
181C22×C3×C4—32no—
182C22×C3×C4—122no—
183C22×C3×C4—122no—
184C22×C3×C4—122no—
185C22×C3×C4C22×D312yes—
186C2×#67—38no—
187C2×#67—38no—
188C2×#67—68no—
189C2×#67—122no—
190C2×#67—122no—
191C2×#67—122no—
192C2×#67—122no—
193C2×#67—124no—
194C2×#67—124no—
195C22×Dic3—32no—
196C22×Dic3—32no—
197C22×Dic3—32no—
198C22×Dic3—42no—
199C22×Dic3—42no—
200C22×Dic3—42no—
201C22×Dic3—62no—
202C22×Dic3—61no—
203C22×Dic3C22×D312yes—
204C22×Dic3C22×D312yes—
205C22×Dic3C22×D312yes—
206C22×Dic3C22×D312yes—
207C2×C4×D3—38no—
208C2×C4×D3—64no—
209C2×C4×D3—122no—
210C2×C4×D3—122no—
211C2×C4×D3—124no—
212C3×C22⋊C4—122no—
213C3×C22⋊C4—122no—
214C3×C22⋊C4—122no—
215#217—38no—
216#217—38no—
217#217—42no—
218#217—42no—
219#217—42no—
220#217—68no—
221(C2×D3)⋊C4—122no—
222(C2×D3)⋊C4—122no—
223(C2×D3)⋊C4—122no—
224(C2×D3)⋊C4—122no—
225C4⋊Dic3—38no—
226C4⋊Dic3—64no—
227Dic3⋊C4—122no—
228Dic3⋊C4—122no—
229Dic3⋊C4—122no—
230Dic3⋊C4—122no—
231C4×Dic3—122no—
232C4×Dic3—122no—
233C22×C32—24no—
234C22×C32—24no—
235C22×C32—24no—
236C22×C32—42no—
237C22×C32—42no—
238C22×C32—42no—
239C22×C32—42no—
240C22×C32C23×C411yescharacteristic
241C22×C32H(5,𝔽2)12yes—
242C22×C32H(5,𝔽2)11yescharacteristic
243C22×C32H(5,𝔽2)12yes—
244C22×C322−1+411yescharacteristic
245C2×C3⋊D3—22no—
246C2×C3⋊D3—22no—
247C2×C3⋊D3—22no—
248C2×C3⋊D3—42no—
249C2×C3⋊D3—42no—
250C32×C4—22no—
251C3⋊Dic3—22no—
252C3⋊Dic3—24no—
253C3×Dic3—42no—
254C3×Dic3—42no—
255C3×Dic3—42no—
256C3×Dic3—42no—
257C22×D4—92no—
258C22×D4D3211yescharacteristic
259C23×C4—32no—
260C23×C4—91no—
261#121—98no—
262C2×C4⋊C4—91no—
263C2×C4⋊C4—92no—
264C2×C4⋊C4—92no—
265C2×C22⋊C4—32no—
266C2×C22⋊C4—32no—
267C2×C22⋊C4—32no—
268C2×C22⋊C4—92no—
269C2×C22⋊C4—91no—
270C2×C22⋊C4—91no—
271C2×C22⋊C4—92no—
272C2×C42—92no—
273#95—32no—
274#95—32no—
275#95—91no—
276#95—91no—
277#95—92no—
278#95—92no—
279C23×C3—22no—
280C23×C3—22no—
281C23×C3—22no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—21no—
292C23×C3—24no—
293C23×C3—42no—
294C23×C3—42no—
295C23×C3—42no—
296C23×C3—42no—
297C23×C3—42no—
298C23×C3—42no—
299C23×C3C2×C4×D314yes—
300C23×C3C2×C4×D314yes—
301C23×C3C23×D312yes—
302C22×D3—34no—
303C22×D3—62no—
304C22×D3—64no—
305C22×D3—62no—
306C22×D3—62no—
307C22×D3—62no—
308C22×D3—62no—
309C22×D3—62no—
310C22×D3—62no—
311C22×D3—122no—
312C22×D3—121no—
313C22×D3—121no—
314C3×D4—48no—
315C3×D4—48no—
316C2×C3×C4—24no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—62no—
321C2×C3×C4—62no—
322C2×C3×C4—62no—
323C2×C3×C4—62no—
324C2×C3×C4—62no—
325C2×C3×C4—62no—
326C2×C3×C4—62no—
327C2×C3×C4—62no—
328C2×C3×C4—62no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—62no—
332C2×C3×C4—128no—
333C2×C3×C4C2×C4×D314yes—
334#67—128no—
335#67—128no—
336#67—1216no—
337C2×Dic3—22no—
338C2×Dic3—22no—
339C2×Dic3—22no—
340C2×Dic3—22no—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—22no—
347C2×Dic3—22no—
348C2×Dic3—22no—
349C2×Dic3—34no—
350C2×Dic3—34no—
351C2×Dic3—34no—
352C2×Dic3—48no—
353C2×Dic3—64no—
354C2×Dic3—62no—
355C2×Dic3—62no—
356C2×Dic3—62no—
357C2×Dic3—62no—
358C2×Dic3—62no—
359C2×Dic3—62no—
360C2×Dic3—62no—
361C2×Dic3—122no—
362C2×Dic3—122no—
363C2×Dic3—122no—
364C4×D3—128no—
365C4×D3—128no—
366C2×C32—44no—
367C2×C32#51811yescharacteristic
368C2×C32#51811yescharacteristic
369C2×C32#52011yescharacteristic
370C2×C32#56012yes—
371C2×C32#56212yes—
372C3⋊D3—42no—
373C24—91no—
374C24C2×D3212yes—
375C2×D4—42no—
376C2×D4—98no—
377C2×D4—362no—
378C2×D4—362no—
379C2×D4#62414yes—
380C22×C4—32no—
381C22×C4—32no—
382C22×C4—32no—
383C22×C4—32no—
384C22×C4—92no—
385C22×C4—91no—
386C22×C4—94no—
387C22×C4—122no—
388C22×C4—122no—
389C22×C4—122no—
390C22×C4—362no—
391C22×C4C2×D3211yescharacteristic
392C4⋊C4—94no—
393C4⋊C4—362no—
394C4⋊C4—362no—
395C22⋊C4—98no—
396C22⋊C4—122no—
397C22⋊C4—122no—
398C22⋊C4—122no—
399C22⋊C4—361no—
400C22⋊C4—361no—
401C22⋊C4—362no—
402C42—362no—
403C22×C3—24no—
404C22×C3—24no—
405C22×C3—24no—
406C22×C3—24no—
407C22×C3—24no—
408C22×C3—24no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—22no—
412C22×C3—21no—
413C22×C3—22no—
414C22×C3—44no—
415C22×C3—44no—
416C22×C3—44no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3—82no—
426C22×C3—82no—
427C22×C3—82no—
428C22×C3—82no—
429C22×C3C22×C4×D312yes—
430C22×C3#100412yes—
431C22×C3#100412yes—
432C22×C3#100412yes—
433C22×C3#100412yes—
434C22×C3#100912yes—
435C22×C3#101012yes—
436C2×D3—64no—
437C2×D3—64no—
438C2×D3—64no—
439C2×D3—64no—
440C2×D3—122no—
441C2×D3—122no—
442C2×D3—124no—
443C2×D3—124no—
444C2×D3—122no—
445C2×D3—122no—
446C2×D3—122no—
447C2×D3—122no—
448C3×C4—24no—
449C3×C4—24no—
450C3×C4—122no—
451C3×C4—122no—
452C3×C4—122no—
453C3×C4—122no—
454Dic3—42no—
455Dic3—42no—
456Dic3—42no—
457Dic3—42no—
458Dic3—64no—
459Dic3—64no—
460Dic3—64no—
461Dic3—122no—
462Dic3—124no—
463C32#237711yescharacteristic, complemented, Sylow
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—92no—
471C23—181no—
472C23—181no—
473C23—182no—
474C23—182no—
475C23C2×#62414yes—
476C23C22×D3211yescharacteristic
477D4—44no—
478D4—368no—
479C2×C4—22no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—62no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—92no—
493C2×C4—94no—
494C2×C4—128no—
495C2×C4—182no—
496C2×C4—182no—
497C2×C4—182no—
498C2×C4—364no—
499C2×C4C2×#62412yes—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—22no—
504C2×C3—22no—
505C2×C3—44no—
506C2×C3—44no—
507C2×C3—84no—
508C2×C3#548612yes—
509C2×C3#554512yes—
510C2×C3#554612yes—
511C2×C3#559512yes—
512C2×C3#562012yes—
513C2×C3#562112yes—
514C2×C3#566712yes—
515D3—124no—
516D3—124no—
517C22—24no—
518C22—24no—
519C22—24no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—182no—
525C22—182no—
526C22—182no—
527C22—362no—
528C22—362no—
529C22#6430911yescharacteristic
530C22#6431311yescharacteristic
531C22#6431412yes—
532C22C22×#62411yescharacteristic
533C22#6433012yes—
534C4—22no—
535C4—122no—
536C4—122no—
537C4—122no—
538C4—122no—
539C4—184no—
540C4—182no—
541C3—21no—
542C3#7488312yescomplemented
543C2—44no—
544C2—362no—
545C2#1059521911yescharacteristic
546C2#1059522311yescharacteristic
547C2#1059525812yes—
548C2#1059526512yes—
549C2#1059551011yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC22×C32×D4C22
socle, cosocleC23×C32C24
prime radicalC2×#64147C2
hypercenterC22×D4D32
hypocenterC32#2377
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C3×#5220C2semidirect
3. #10591711C2semidirect
4. #10591704C2semidirect
5. #10591703C2semidirect
6. #10591693C2semidirect
7. #10591355C2semidirect
8. #10591271C2semidirect
9. #10591219C2semidirect
16. #64147C4semidirect
36. #5220D3semidirect
56. C2×#638C2×C4semidirect
60. C2×C4×C3⋊D3C2×C4semidirect
68. #3757C2×C4semidirect
69. C4⋊(C3⋊Dic3)C2×C4semidirect
463. C32#2377semidirect
542. C3#74883semidirect

Maximal subgroups

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

subgroupindex
C2×#641472
C3×#52202
#105917112
#105917042
#105917032
#105916932
#105913552
#105912712
#105912192
#748833