Finite Group Databasebeta

← #11803088

Subgroups — #11803088

Order 1152.

1349 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64090C211yesmaximal, characteristic
2C3×#4950C212yesmaximal, complemented
3#10591711C212yesmaximal
4#10591705C211yesmaximal, characteristic, complemented
5C2×#63579C211yesmaximal, characteristic, complemented
6#10591325C211yesmaximal, characteristic, complemented
7#10591266C212yesmaximal, complemented
8#10591263C212yesmaximal, complemented
9#10591230C211yesmaximal, characteristic, complemented
10#10591229C211yesmaximal, characteristic, complemented
11#10591219C211yesmaximal, characteristic, complemented
12#74383—32nomaximal
13C22×#638C2211yescharacteristic
14C23×C3⋊Dic3C2211yescharacteristic
15C22×C4×C3⋊D3C2211yescharacteristic
16C23×C3×Dic3C2212yes—
17C2×C32×C22⋊C4C2211yescharacteristic
18C2×#3752C2211yescharacteristic
19C2×#3750C2211yescharacteristic
20C2×C4×C3⋊Dic3C2211yescharacteristic
21#64090C418yescomplemented
22C2×C3×#217C2212yes—
23C2×C3×C4⋊Dic3C2212yes—
24C2×C3×C4⋊Dic3C2212yes—
25C2×#3722C2212yes—
26C2×#3722C2212yes—
27C2×Dic32C2212yes—
28C3×C3⋊#95C2212yes—
29C3×C3⋊#95C2212yes—
30#63579C2212yes—
31#63579C2212yes—
32#63579C2212yes—
33#63579C2211yescharacteristic
34#63579C2211yescharacteristic
35#63579C2211yescharacteristic
36#63579C2211yescharacteristic
37C2×#67⋊C4—32no—
38C2×#67⋊C4—61no—
39C3×#385—32no—
40#5213—32no—
41C2×C3⋊#95—32no—
42#4990—32no—
43#4987—32no—
44#4978—32no—
45#4977—32no—
46#4958—32no—
47#4950D312yescomplemented
48Dic3×C22⋊C4—32no—
49#4670—32no—
50#4664—32no—
51#4655—32no—
52#4651—32no—
53C24×C32C2311yescharacteristic
54C23×C3⋊D3C2311yescharacteristic
55C22×C32×C4C2311yescharacteristic
56C22×C32×C4C2311yescharacteristic
57C2×#638—24no—
58C2×#638C2×C414yes—
59C22×C3⋊Dic3—42no—
60C22×C3⋊Dic3C2×C414yes—
61C22×C3⋊Dic3C2311yescharacteristic
62C22×C3⋊Dic3C2311yescharacteristic
63C22×C3⋊Dic3C2311yescharacteristic
64C2×C4×C3⋊D3—42no—
65C2×C4×C3⋊D3C2×C414yescomplemented
66C22×C3×Dic3—24no—
67C22×C3×Dic3—42no—
68C22×C3×Dic3—42no—
69C22×C3×Dic3C2312yes—
70C22×C3×Dic3C2312yes—
71C22×C3×Dic3C2312yes—
72C22×C3×Dic3C2312yes—
73C32×C22⋊C4C2×C414yes—
74#3752C2×C414yescomplemented
75#3750C2×C414yescomplemented
76C4×C3⋊Dic3C2×C414yescomplemented
77C3×#217—24no—
78C3×C4⋊Dic3—42no—
79C3×C4⋊Dic3—42no—
80#3722—42no—
81#3722—42no—
82Dic32—42no—
83#2318—91noSylow
84C23×C3×C4—32no—
85C22×#67—32no—
86C22×#67—61no—
87C23×Dic3—32no—
88C23×Dic3—61no—
89C23×Dic3C2×D312yes—
90C22×C4×D3—32no—
91C22×C4×D3—61no—
92C2×C3×C4⋊C4—32no—
93C2×C3×C4⋊C4—32no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C22⋊C4—32no—
96C2×C3×C22⋊C4C2×D312yes—
97C2×#217C2×D312yes—
98C2×(C2×D3)⋊C4—32no—
99C2×(C2×D3)⋊C4—32no—
100C2×(C2×D3)⋊C4—32no—
101C2×(C2×D3)⋊C4—32no—
102C2×(C2×D3)⋊C4—32no—
103C2×(C2×D3)⋊C4—61no—
104C2×C4⋊Dic3C2×D312yes—
105C2×C4⋊Dic3C2×D312yes—
106C2×Dic3⋊C4—32no—
107C2×Dic3⋊C4—61no—
108C2×C4×Dic3—32no—
109C2×C4×Dic3—32no—
110C2×C4×Dic3—32no—
111C2×C4×Dic3—61no—
112#67⋊C4—316no—
113#67⋊C4—68no—
114C3×#95—32no—
115C3×#95—32no—
116C3⋊#95—32no—
117C3⋊#95—32no—
118C3⋊#95—32no—
119C3⋊#95—32no—
120C3⋊#95—32no—
121C3⋊#95—32no—
122C3⋊#95—32no—
123C3⋊#95—32no—
124C3⋊#95—32no—
125C3⋊#95—32no—
126C3⋊#95C2×D312yes—
127C3⋊#95C2×D312yes—
128C23×C32—21no—
129C23×C32—22no—
130C23×C32—21no—
131C23×C32C22×C412yes—
132C23×C32#4114yes—
133C23×C32C2411yescharacteristic
134C22×C3⋊D3—22no—
135C22×C3⋊D3—21no—
136C22×C3⋊D3—22no—
137C22×C3⋊D3—21no—
138C22×C3⋊D3C22×C412yes—
139C2×C32×C4—21no—
140C2×C32×C4—22no—
141C2×C32×C4—21no—
142C2×C32×C4C22×C412yes—
143C2×C32×C4C22×C412yes—
144#638—28no—
145C2×C3⋊Dic3—22no—
146C2×C3⋊Dic3—22no—
147C2×C3⋊Dic3—22no—
148C2×C3⋊Dic3—44no—
149C2×C3⋊Dic3C22×C412yes—
150C2×C3⋊Dic3C22×C412yes—
151C2×C3⋊Dic3C22×C412yes—
152C4×C3⋊D3—44no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—22no—
160C2×C3×Dic3—22no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3—22no—
164C2×C3×Dic3—22no—
165C2×C3×Dic3—48no—
166C2×C4×D4—91no—
167#388—92no—
168#387—92no—
169#386—92no—
170#385—32no—
171#383—91no—
172#382—91no—
173#381—91no—
174#380—91no—
175C4×C22⋊C4—91no—
176C2×#95—91no—
177C24×C3—21no—
178C24×C3C22×D312yes—
179C23×D3—32no—
180C23×D3—61no—
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—32no—
184C22×C3×C4—32no—
185C22×C3×C4—32no—
186C22×C3×C4—32no—
187C22×C3×C4—64no—
188C22×C3×C4—122no—
189C22×C3×C4—122no—
190C22×C3×C4C22×D312yes—
191C22×C3×C4C22×D312yes—
192C2×#67—38no—
193C2×#67—64no—
194C2×#67—68no—
195C2×#67—64no—
196C2×#67—64no—
197C22×Dic3—24no—
198C22×Dic3—32no—
199C22×Dic3—32no—
200C22×Dic3—32no—
201C22×Dic3—38no—
202C22×Dic3—42no—
203C22×Dic3—42no—
204C22×Dic3—61no—
205C22×Dic3—61no—
206C22×Dic3—64no—
207C22×Dic3—61no—
208C22×Dic3—122no—
209C22×Dic3—122no—
210C22×Dic3—124no—
211C22×Dic3C22×D312yes—
212C22×Dic3C22×D312yes—
213C22×Dic3C22×D312yes—
214C22×Dic3C22×D312yes—
215C2×C4×D3—38no—
216C2×C4×D3—64no—
217C2×C4×D3—122no—
218C2×C4×D3—122no—
219C2×C4×D3—124no—
220C3×C4⋊C4—122no—
221C3×C4⋊C4—122no—
222C3×C22⋊C4—24no—
223C3×C22⋊C4—64no—
224C3×C22⋊C4C4×D318yes—
225#217—24no—
226(C2×D3)⋊C4—38no—
227(C2×D3)⋊C4—64no—
228(C2×D3)⋊C4—122no—
229(C2×D3)⋊C4—122no—
230(C2×D3)⋊C4—122no—
231(C2×D3)⋊C4—122no—
232C4⋊Dic3—42no—
233C4⋊Dic3—42no—
234Dic3⋊C4—38no—
235Dic3⋊C4—64no—
236C4×Dic3—38no—
237C4×Dic3—64no—
238C4×Dic3—122no—
239C4×Dic3—122no—
240C22×C32—22no—
241C22×C32—22no—
242C22×C32—24no—
243C22×C32—24no—
244C22×C32#11714yes—
245C22×C32C23×C411yescharacteristic
246C22×C32C2×#4112yes—
247C22×C32H(5,𝔽2)11yescharacteristic
248C22×C32H(5,𝔽2)12yes—
249C22×C322−1+411yescharacteristic
250C2×C3⋊D3—22no—
251C2×C3⋊D3—22no—
252C2×C3⋊D3—22no—
253C2×C3⋊D3—42no—
254C2×C3⋊D3—42no—
255C32×C4—22no—
256C32×C4—22no—
257C3⋊Dic3—22no—
258C3⋊Dic3—22no—
259C3⋊Dic3—22no—
260C3×Dic3—42no—
261C3×Dic3—42no—
262C3×Dic3—42no—
263C3×Dic3—42no—
264C22×D4—91no—
265C23×C4—32no—
266C23×C4—91no—
267C23×C4—91no—
268C4×D4—98no—
269C2×C4⋊C4—32no—
270C2×C4⋊C4—32no—
271C2×C4⋊C4—91no—
272C2×C22⋊C4—32no—
273C2×C22⋊C4—91no—
274C2×C22⋊C4—92no—
275C2×C22⋊C4—92no—
276C2×C22⋊C4D3211yescharacteristic
277C2×C42—91no—
278C2×C42—92no—
279#95—32no—
280#95—32no—
281#95—91no—
282#95—91no—
283#95—92no—
284#95—91no—
285#95—91no—
286#95—92no—
287#95—92no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3—22no—
291C23×C3—22no—
292C23×C3—21no—
293C23×C3—24no—
294C23×C3—22no—
295C23×C3—42no—
296C23×C3—41no—
297C23×C3—41no—
298C23×C3C2×C4×D314yes—
299C23×C3#23714yes—
300C23×C3#23914yes—
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—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—42no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
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—62no—
333C2×C3×C4—62no—
334C2×C3×C4—62no—
335C2×C3×C4—128no—
336C2×C3×C4C2×C4×D314yes—
337C2×C3×C4C2×C4×D314yes—
338#67—616no—
339#67—616no—
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—22no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—34no—
353C2×Dic3—34no—
354C2×Dic3—34no—
355C2×Dic3—48no—
356C2×Dic3—62no—
357C2×Dic3—62no—
358C2×Dic3—62no—
359C2×Dic3—64no—
360C2×Dic3—64no—
361C2×Dic3—62no—
362C2×Dic3—62no—
363C2×Dic3—62no—
364C2×Dic3—62no—
365C2×Dic3—62no—
366C2×Dic3—62no—
367C2×Dic3—122no—
368C2×Dic3—128no—
369C2×Dic3—128no—
370C4×D3—128no—
371C4×D3—128no—
372C2×C32—24no—
373C2×C32C2×#11711yescharacteristic
374C2×C32#51811yescharacteristic
375C2×C32#52011yescharacteristic
376C2×C32#53812yes—
377C2×C32#53912yes—
378C3⋊D3—42no—
379C24—91no—
380C24C2×D3211yescharacteristic
381C2×D4—94no—
382C2×D4—184no—
383C22×C4—32no—
384C22×C4—32no—
385C22×C4—32no—
386C22×C4—32no—
387C22×C4—64no—
388C22×C4—91no—
389C22×C4—91no—
390C22×C4—91no—
391C22×C4—94no—
392C22×C4—94no—
393C22×C4—122no—
394C22×C4—122no—
395C22×C4—362no—
396C22×C4—362no—
397C22×C4C2×D3211yescharacteristic
398C22×C4C2×D3211yescharacteristic
399C4⋊C4—94no—
400C4⋊C4—122no—
401C4⋊C4—122no—
402C22⋊C4—64no—
403C22⋊C4—94no—
404C22⋊C4—362no—
405C22⋊C4—362no—
406C22⋊C4#62414yes—
407C42—94no—
408C42—362no—
409C22×C3—24no—
410C22×C3—24no—
411C22×C3—24no—
412C22×C3—24no—
413C22×C3—24no—
414C22×C3—24no—
415C22×C3—22no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—22no—
419C22×C3—21no—
420C22×C3—24no—
421C22×C3—42no—
422C22×C3—44no—
423C22×C3—44no—
424C22×C3—42no—
425C22×C3#89218yes—
426C22×C3C22×C4×D312yes—
427C22×C3C2×#23712yes—
428C22×C3#100412yes—
429C22×C3#100412yes—
430C22×C3C2×#23912yes—
431C22×C3#100912yes—
432C22×C3#101012yes—
433C2×D3—64no—
434C2×D3—64no—
435C2×D3—64no—
436C2×D3—64no—
437C2×D3—122no—
438C2×D3—122no—
439C2×D3—124no—
440C2×D3—124no—
441C2×D3—122no—
442C2×D3—122no—
443C2×D3—122no—
444C2×D3—122no—
445C3×C4—24no—
446C3×C4—24no—
447C3×C4—42no—
448C3×C4—42no—
449C3×C4—122no—
450C3×C4—122no—
451C3×C4—122no—
452C3×C4—122no—
453Dic3—42no—
454Dic3—42no—
455Dic3—42no—
456Dic3—42no—
457Dic3—64no—
458Dic3—64no—
459Dic3—64no—
460Dic3—64no—
461Dic3—64no—
462Dic3—122no—
463C32#231811yescharacteristic, complemented, Sylow
464C23—21no—
465C23—21no—
466C23—22no—
467C23—92no—
468C23—182no—
469C23—181no—
470C23—181no—
471C23—182no—
472C23#379614yes—
473C23C2×#62412yes—
474C23C22×D3211yescharacteristic
475D4—188no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—62no—
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—92no—
492C2×C4—92no—
493C2×C4—92no—
494C2×C4—128no—
495C2×C4—182no—
496C2×C4—182no—
497C2×C4—182no—
498C2×C4—364no—
499C2×C4—364no—
500C2×C4C2×#62412yes—
501C2×C4C2×#62412yes—
502C2×C3—28no—
503C2×C3—21no—
504C2×C3—22no—
505C2×C3—22no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—44no—
509C2×C3C2×#89212yes—
510C2×C3#554512yes—
511C2×C3#554612yes—
512C2×C3#559312yes—
513C2×C3#564412yes—
514C2×C3#564512yes—
515C2×C3#566512yes—
516D3—124no—
517D3—124no—
518C22—24no—
519C22—24no—
520C22—22no—
521C22—22no—
522C22—182no—
523C22—182no—
524C22—182no—
525C22—362no—
526C22—362no—
527C22#6384914yes—
528C22C2×#379612yes—
529C22#6430911yescharacteristic
530C22#6431311yescharacteristic
531C22#6431412yes—
532C22C22×#62411yescharacteristic
533C4—22no—
534C4—22no—
535C4—122no—
536C4—122no—
537C4—122no—
538C4—122no—
539C4—182no—
540C4—182no—
541C4—182no—
542C3—21no—
543C3#7438312yescomplemented
544C2—24no—
545C2—362no—
546C2#1059417312yes—
547C2#1059417912yes—
548C2C2×#6384911yescharacteristic
549C2#1059521911yescharacteristic
550C2#1059522311yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C22⋊C4C22
socle, cosocleC23×C32C24
prime radicalC22×#638C22
hypercenterC2×C22⋊C4D32
hypocenterC32#2318
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#4950C2semidirect
4. #10591705C2semidirect
5. C2×#63579C2semidirect
6. #10591325C2semidirect
7. #10591266C2semidirect
8. #10591263C2semidirect
9. #10591230C2semidirect
10. #10591229C2semidirect
11. #10591219C2semidirect
21. #64090C4semidirect
47. #4950D3semidirect
65. C2×C4×C3⋊D3C2×C4semidirect
74. #3752C2×C4semidirect
75. #3750C2×C4semidirect
76. C4×C3⋊Dic3C2×C4semidirect
463. C32#2318semidirect
543. C3#74383semidirect

Maximal subgroups

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

subgroupindex
C2×#640902
C3×#49502
#105917112
#105917052
C2×#635792
#105913252
#105912662
#105912632
#105912302
#105912292
#105912192
#743833