Finite Group Databasebeta

← #11806882

Subgroups — #11806882

Order 1152.

2496 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×D4C211yesmaximal, characteristic, complemented
2C2×#64149C211yesmaximal, characteristic
3C2×C4×#638C212yesmaximal, complemented
4C2×#64102C211yesmaximal, characteristic, complemented
5#10592920C211yesmaximal, characteristic, complemented
6#10592912C212yesmaximal, complemented
7C4⋊C4×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#10592656C212yesmaximal, complemented
9#10592655C212yesmaximal, complemented
10#10592638C211yesmaximal, characteristic, complemented
11C4×#3750C211yesmaximal, characteristic, complemented
12#82343—34nomaximal
13C22×C32×D4C2211yescharacteristic
14C23×C32×C4C2212yes—
15C22×#638C2212yes—
16C22×#638C2212yes—
17C22×C4⋊(C3⋊D3)C2211yescharacteristic
18C22×C4×C3⋊D3C2212yes—
19C32×C4×D4—24no—
20C2×C32×C4⋊C4C2211yescharacteristic
21C2×C32×C22⋊C4C2212yes—
22C2×C32×C42C2211yescharacteristic
23C2×#3757C2212yes—
24C2×#3757C2212yes—
25#64149C418yescomplemented
26C4×#638—28no—
27C2×#3752C2212yes—
28C2×#3752C2212yes—
29C2×#3752C2212yes—
30C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
31C2×#3750C2211yescharacteristic
32C2×#3750C2211yescharacteristic
33C2×C4×C3⋊Dic3C2211yescharacteristic
34C2×C4×C3⋊Dic3C2212yes—
35C2×C4×C3⋊Dic3C2211yescharacteristic
36#64102—24no—
37#63658C2211yescharacteristic
38#63658C2211yescharacteristic
39#63658C2212yes—
40C2×C3×C4×D4D314yescomplemented
41C2×#940—34no—
42C2×C4×#67—38no—
43C2×#893—34no—
44#5221—34no—
45#5213—38no—
46Dic3×C4⋊C4—34no—
47#4957—38no—
48#4956—38no—
49#4939—34no—
50C4×Dic3⋊C4—34no—
51C24×C32C2312yes—
52C23×C3⋊D3C2312yes—
53C2×C32×D4—22no—
54C2×C32×D4—22no—
55C2×C32×D4C2×C414yescomplemented
56C22×C32×C4—24no—
57C22×C32×C4—24no—
58C22×C32×C4—42no—
59C22×C32×C4C2311yescharacteristic
60C22×C32×C4C2312yes—
61C22×C32×C4C2311yescharacteristic
62C22×C32×C4C2311yescharacteristic
63C2×#638—24no—
64C2×#638—24no—
65C2×#638—42no—
66C2×#638—42no—
67C2×#638C2×C418yescomplemented
68C2×#638C2×C418yescomplemented
69C22×C3⋊Dic3C2311yescharacteristic
70C22×C3⋊Dic3C2312yes—
71C22×C3⋊Dic3C2311yescharacteristic
72C22×C3⋊Dic3C2311yescharacteristic
73C22×C3⋊Dic3C2311yescharacteristic
74C2×C4⋊(C3⋊D3)—22no—
75C2×C4⋊(C3⋊D3)—22no—
76C2×C4⋊(C3⋊D3)C2×C414yescomplemented
77C2×C4×C3⋊D3—24no—
78C2×C4×C3⋊D3—24no—
79C2×C4×C3⋊D3—42no—
80C32×C4⋊C4D414yescomplemented
81C32×C22⋊C4—24no—
82C32×C42—22no—
83#3757—24no—
84#3757—42no—
85#3752—24no—
86#3752—24no—
87#3752—42no—
88C4⋊(C3⋊Dic3)—22no—
89#3750—22no—
90#3750D414yescomplemented
91C4×C3⋊Dic3—22no—
92C4×C3⋊Dic3—24no—
93C4×C3⋊Dic3C2×C414yescomplemented
94#2366—91noSylow
95C22×C3×D4C2×D314yes—
96C23×C3×C4C2×D318yes—
97C22×#67—38no—
98C22×#67—38no—
99C22×D12—34no—
100C22×C4×D3—38no—
101C3×C4×D4—216no—
102C2×C3×C4⋊C4C2×D314yes—
103C2×C3×C22⋊C4C2×D318yes—
104C2×C3×C42C2×D314yes—
105C2×#217—38no—
106C2×#217—38no—
107#940—332no—
108C4×#67—632no—
109C2×(C2×D3)⋊C4—38no—
110C2×(C2×D3)⋊C4—38no—
111C2×(C2×D3)⋊C4—38no—
112C2×C4⋊Dic3—34no—
113C2×Dic3⋊C4—34no—
114C2×Dic3⋊C4—34no—
115C2×C4×Dic3—34no—
116C2×C4×Dic3—34no—
117C2×C4×Dic3—38no—
118#893—616no—
119C3⋊#95—34no—
120C3⋊#95—34no—
121C3⋊#95—38no—
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—22no—
132C22×C3⋊D3—22no—
133C22×C3⋊D3—22no—
134C22×C3⋊D3—22no—
135C22×C3⋊D3—22no—
136C22×C3⋊D3C22×C414yes—
137C32×D4—28no—
138C2×C32×C4—22no—
139C2×C32×C4—21no—
140C2×C32×C4—22no—
141C2×C32×C4—21no—
142C2×C32×C4—21no—
143C2×C32×C4—21no—
144C2×C32×C4—21no—
145C2×C32×C4—48no—
146C2×C32×C4C22×C412yes—
147C2×C32×C4C2×D412yes—
148C2×C32×C4C2×D414yes—
149C2×C32×C4C2×D412yes—
150C2×C32×C4#4112yes—
151#638—216no—
152#638—48no—
153C2×C3⋊Dic3—21no—
154C2×C3⋊Dic3—21no—
155C2×C3⋊Dic3—21no—
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—22no—
159C2×C3⋊Dic3—21no—
160C2×C3⋊Dic3—22no—
161C2×C3⋊Dic3—22no—
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3C22×C412yes—
164C2×C3⋊Dic3C22×C412yes—
165C2×C3⋊Dic3C2×D412yes—
166C2×C3⋊Dic3C2×D412yes—
167C2×C3⋊Dic3#4112yes—
168C4⋊(C3⋊D3)—28no—
169C4×C3⋊D3—48no—
170C2×#127—91no—
171C2×C4×D4—92no—
172C2×C4×D4—91no—
173C2×C4×D4C3⋊D311yescharacteristic, complemented
174#390—91no—
175#390—92no—
176#390—91no—
177#386—92no—
178#386—92no—
179C4×C4⋊C4—91no—
180C4×C4⋊C4—91no—
181C24×C3C22×D318yes—
182C23×D3—38no—
183C2×C3×D4—28no—
184C2×C3×D4—28no—
185C2×C3×D4C4×D3116yescomplemented
186C22×C3×C4—216no—
187C22×C3×C4—216no—
188C22×C3×C4—48no—
189C22×C3×C4C22×D314yes—
190C22×C3×C4C22×D314yes—
191C22×C3×C4C22×D318yes—
192C22×C3×C4C22×D314yes—
193C2×#67—332no—
194C2×#67—332no—
195C2×#67—616no—
196C2×#67—616no—
197C2×#67—128no—
198C2×#67—128no—
199C22×Dic3—34no—
200C22×Dic3—34no—
201C22×Dic3—34no—
202C22×Dic3—34no—
203C22×Dic3—38no—
204C2×D12—316no—
205C2×D12—68no—
206C2×D12—68no—
207C2×C4×D3—616no—
208C2×C4×D3—616no—
209C2×C4×D3—128no—
210C3×C4⋊C4#67116yescomplemented
211C3×C22⋊C4—216no—
212C3×C42—28no—
213#217—616no—
214#217—128no—
215(C2×D3)⋊C4—616no—
216(C2×D3)⋊C4—616no—
217(C2×D3)⋊C4—128no—
218C4⋊Dic3—68no—
219Dic3⋊C4—316no—
220Dic3⋊C4—68no—
221C4×Dic3—316no—
222C4×Dic3—616no—
223C4×Dic3—68no—
224C22×C32—24no—
225C22×C32—24no—
226C22×C32—24no—
227C22×C32—42no—
228C22×C32—42no—
229C22×C32—42no—
230C22×C32—42no—
231C22×C32C23×C411yescharacteristic
232C22×C32C22×D411yescharacteristic
233C22×C32C22×D411yescharacteristic
234C22×C32C2×#4111yescharacteristic
235C22×C32C2×#4111yescharacteristic
236C22×C32H(5,𝔽2)11yescharacteristic
237C22×C32H(5,𝔽2)11yescharacteristic
238C2×C3⋊D3—24no—
239C2×C3⋊D3—24no—
240C2×C3⋊D3—24no—
241C2×C3⋊D3—42no—
242C2×C3⋊D3—42no—
243C2×C3⋊D3—42no—
244C2×C3⋊D3—42no—
245C32×C4—24no—
246C32×C4—22no—
247C32×C4—41no—
248C32×C4C4×D414yescomplemented
249C3⋊Dic3—22no—
250C3⋊Dic3—22no—
251C3⋊Dic3—42no—
252C3⋊Dic3—41no—
253C3⋊Dic3C4×D414yescomplemented
254C22×D4—92no—
255C22×D4—92no—
256C22×D4—91no—
257C22×D4C2×C3⋊D311yescharacteristic
258C23×C4—92no—
259C23×C4C2×C3⋊D312yes—
260#127—98no—
261C4×D4—24no—
262C4×D4—188no—
263C4×D4—184no—
264C2×C4⋊C4—91no—
265C2×C4⋊C4—91no—
266C2×C4⋊C4—91no—
267C2×C4⋊C4C2×C3⋊D311yescharacteristic
268C2×C22⋊C4—92no—
269C2×C22⋊C4—92no—
270C2×C22⋊C4—92no—
271C2×C22⋊C4—92no—
272C2×C22⋊C4—92no—
273C2×C22⋊C4C2×C3⋊D312yes—
274C2×C42—91no—
275C2×C42—92no—
276C2×C42—91no—
277C2×C42C2×C3⋊D311yescharacteristic
278#95—91no—
279#95—91no—
280#95—92no—
281C23×C3—28no—
282C23×C3—28no—
283C23×C3—28no—
284C23×C3—28no—
285C23×C3—28no—
286C23×C3—28no—
287C23×C3C2×C4×D3116yes—
288C23×C3C23×D314yes—
289C22×D3—316no—
290C22×D3—68no—
291C22×D3—68no—
292C22×D3—68no—
293C22×D3—68no—
294C22×D3—68no—
295C22×D3—68no—
296C3×D4—232no—
297C2×C3×C4—28no—
298C2×C3×C4—28no—
299C2×C3×C4—24no—
300C2×C3×C4—24no—
301C2×C3×C4—24no—
302C2×C3×C4—24no—
303C2×C3×C4—24no—
304C2×C3×C4—432no—
305C2×C3×C4C2×C4×D318yes—
306C2×C3×C4#23518yes—
307C2×C3×C4D3×D418yes—
308C2×C3×C4C2×#67116yes—
309C2×C3×C4C2×#6718yes—
310#67—664no—
311#67—1232no—
312C2×Dic3—38no—
313C2×Dic3—38no—
314C2×Dic3—38no—
315C2×Dic3—38no—
316C2×Dic3—38no—
317C2×Dic3—64no—
318C2×Dic3—64no—
319C2×Dic3—64no—
320C2×Dic3—64no—
321C2×Dic3—68no—
322C2×Dic3—68no—
323C2×Dic3—64no—
324C2×Dic3—68no—
325C2×Dic3—64no—
326C2×Dic3—64no—
327D12—632no—
328C4×D3—1232no—
329C2×C32—44no—
330C2×C32C2×C4×D411yescharacteristic
331C2×C32C2×C4×D411yescharacteristic
332C2×C32#51811yescharacteristic
333C2×C32D4211yescharacteristic
334C2×C32#55011yescharacteristic
335C2×C32#55011yescharacteristic
336C2×C32#55911yescharacteristic
337C3⋊D3—44no—
338C24—92no—
339C24C22×C3⋊D312yes—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—98no—
343C2×D4—98no—
344C2×D4—94no—
345C2×D4—184no—
346C2×D4—184no—
347C2×D4—182no—
348C2×D4—182no—
349C2×D4—362no—
350C2×D4—362no—
351C2×D4C4×C3⋊D314yescomplemented
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—42no—
355C22×C4—91no—
356C22×C4—91no—
357C22×C4—91no—
358C22×C4—92no—
359C22×C4—91no—
360C22×C4—184no—
361C22×C4—184no—
362C22×C4—362no—
363C22×C4C22×C3⋊D311yescharacteristic
364C22×C4C22×C3⋊D311yescharacteristic
365C22×C4C22×C3⋊D312yes—
366C22×C4C22×C3⋊D311yescharacteristic
367C4⋊C4—94no—
368C4⋊C4—182no—
369C4⋊C4—182no—
370C4⋊C4#63814yescomplemented
371C22⋊C4—24no—
372C22⋊C4—184no—
373C22⋊C4—184no—
374C22⋊C4—184no—
375C22⋊C4—362no—
376C22⋊C4—362no—
377C42—22no—
378C42—94no—
379C42—184no—
380C42—182no—
381C22×C3—216no—
382C22×C3—216no—
383C22×C3—216no—
384C22×C3—48no—
385C22×C3—48no—
386C22×C3—48no—
387C22×C3—48no—
388C22×C3C22×C4×D314yes—
389C22×C3C2×#23514yes—
390C22×C3C2×D3×D414yes—
391C22×C3#100414yes—
392C22×C3D3×#4114yes—
393C22×C3#100914yes—
394C22×C3C22×#6714yes—
395C2×D3—616no—
396C2×D3—616no—
397C2×D3—616no—
398C2×D3—128no—
399C2×D3—128no—
400C2×D3—128no—
401C2×D3—128no—
402C3×C4—216no—
403C3×C4—28no—
404C3×C4—44no—
405C3×C4C4×#67116yescomplemented
406Dic3—316no—
407Dic3—68no—
408Dic3—68no—
409Dic3—128no—
410Dic3—124no—
411C32#236611yescharacteristic, complemented, Sylow
412C23—22no—
413C23—22no—
414C23—22no—
415C23—22no—
416C23—22no—
417C23—22no—
418C23—94no—
419C23—182no—
420C23—182no—
421C23—182no—
422C23—182no—
423C23—182no—
424C23—182no—
425C23C2×C4×C3⋊D314yes—
426C23C23×C3⋊D311yescharacteristic
427D4—28no—
428D4—1816no—
429D4—188no—
430D4—368no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—48no—
439C2×C4—92no—
440C2×C4—92no—
441C2×C4—92no—
442C2×C4—92no—
443C2×C4—92no—
444C2×C4—181no—
445C2×C4—181no—
446C2×C4—181no—
447C2×C4—181no—
448C2×C4—182no—
449C2×C4—182no—
450C2×C4—181no—
451C2×C4—182no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4—368no—
455C2×C4C2×C4×C3⋊D312yes—
456C2×C4#382812yes—
457C2×C4D4×C3⋊D312yes—
458C2×C4C2×#63814yes—
459C2×C4C2×#63812yes—
460C2×C3—416no—
461C2×C3C4×D3×D414yes—
462C2×C3#554514yes—
463C2×C3#555314yes—
464C2×C3#555914yes—
465C2×C3C2×C4×#6714yes—
466C2×C3D4×#6714yes—
467C2×C3#583414yes—
468D3—1216no—
469C22—24no—
470C22—24no—
471C22—24no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—184no—
477C22—184no—
478C22—184no—
479C22—362no—
480C22—362no—
481C22—362no—
482C22—362no—
483C22C22×C4×C3⋊D311yescharacteristic
484C22C2×#382811yescharacteristic
485C22C2×D4×C3⋊D311yescharacteristic
486C22#6436111yescharacteristic
487C22#41×C3⋊D311yescharacteristic
488C22#6436611yescharacteristic
489C22C22×#63811yescharacteristic
490C4—22no—
491C4—24no—
492C4—41no—
493C4—94no—
494C4—182no—
495C4—182no—
496C4—362no—
497C4—361no—
498C4C4×#63814yescomplemented
499C3#8234314yescomplemented
500C2—44no—
501C2—364no—
502C2C4×D4×C3⋊D311yescharacteristic
503C2#1059601611yescharacteristic
504C2#1059602411yescharacteristic
505C2#1059603011yescharacteristic
506C2C2×C4×#63811yescharacteristic
507C2D4×#63811yescharacteristic
508C2#1059630511yescharacteristic
subgroupG/H
center, FrattiniC23C23×C3⋊D3
commutatorC22×C32C23×C4
FittingC2×C32×C4×D4C2
socle, cosocleC23×C32C24
prime radicalC2×#64149C2
hypercenterC2×C4×D4C3⋊D3
hypocenterC32#2366
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C32×C4×D4C2semidirect
3. C2×C4×#638C2semidirect
4. C2×#64102C2semidirect
5. #10592920C2semidirect
6. #10592912C2semidirect
7. C4⋊C4×C3⋊Dic3C2semidirect
8. #10592656C2semidirect
9. #10592655C2semidirect
10. #10592638C2semidirect
11. C4×#3750C2semidirect
25. #64149C4semidirect
40. C2×C3×C4×D4D3semidirect
55. C2×C32×D4C2×C4semidirect
67. C2×#638C2×C4semidirect
68. C2×#638C2×C4semidirect
76. C2×C4⋊(C3⋊D3)C2×C4semidirect
80. C32×C4⋊C4D4semidirect
90. #3750D4semidirect
93. C4×C3⋊Dic3C2×C4semidirect
173. C2×C4×D4C3⋊D3semidirect
185. C2×C3×D4C4×D3semidirect
210. C3×C4⋊C4#67semidirect
248. C32×C4C4×D4semidirect
253. C3⋊Dic3C4×D4semidirect
351. C2×D4C4×C3⋊D3semidirect
370. C4⋊C4#638semidirect
405. C3×C4C4×#67semidirect
411. C32#2366semidirect
498. C4C4×#638semidirect
499. C3#82343semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×D42
C2×#641492
C2×C4×#6382
C2×#641022
#105929202
#105929122
C4⋊C4×C3⋊Dic32
#105926562
#105926552
#105926382
C4×#37502
#823433