Finite Group Databasebeta

← #72222619

Subgroups — #72222619

Order 4608.

675 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#10707078C211yesmaximal, characteristic, complemented
2#72222533C211yesmaximal, characteristic, complemented
3#72222526C211yesmaximal, characteristic, complemented
4#72222519—31nomaximal
5#72222515C211yesmaximal, characteristic
6#72222501C211yesmaximal, characteristic, complemented
7#72222500—31nomaximal
8C3×#10682523C211yesmaximal, characteristic, complemented
9#72222460C211yesmaximal, characteristic, complemented
10#11800601C411yescharacteristic
11C3×#69085C2211yescharacteristic
12#11797813C2211yescharacteristic
13C3×#69076—21no—
14#11777343—21no—
15#11773077C411yescharacteristic
16#11772984C411yescharacteristic
17#11772967C411yescharacteristic
18C3×#67519C2211yescharacteristic
19C3×#67519C2211yescharacteristic
20C3×#67514C2211yescharacteristic
21#11772179C2211yescharacteristic
22C3×#67443—21no—
23#11771606—21no—
24C32×#1313C2211yescharacteristic
25#10715814—31no—
26#10715812—31no—
27#10707078D311yescharacteristic, complemented
28#10707073—31no—
29#10707073—31no—
30#10707073—61no—
31#10682858—31no—
32#10682857—31no—
33#10682659—31no—
34#10682657—31no—
35#10682656—31no—
36#10682655—31no—
37#10682523D311yescharacteristic, complemented
38C3×#12068—31no—
39C3×#7333—31no—
40C2×#63626C2×C411yescharacteristic
41C4×#3745—21no—
42C8×C4⋊(C3⋊D3)—41no—
43C3×#4904C2×C411yescharacteristic
44C3×#4901—41no—
45C3×#4700—21no—
46C32×#346D411yescharacteristic
47C32×#346C2311yescharacteristic
48C32×#322D411yescharacteristic
49#10589375—21no—
50#10589371C2×C411yescharacteristic
51#10589325—41no—
52#10589323—21no—
53C3×#4510—21no—
54C3×#4508—41no—
55C3×#4506C2×C411yescharacteristic
56C3×#4499—21no—
57C3×#4499C2×C411yescharacteristic
58C3×#4499C2×C411yescharacteristic
59C3×#4463—21no—
60#10589074C812yescomplemented
61#10589071C812yescomplemented
62AGL(1,ℤ/32)—91noSylow
63C3×#2179—31no—
64C3×#2171—61no—
65#71509—31no—
66#69373—31no—
67#69085C2×D311yescharacteristic
68#69078—31no—
69#69078—31no—
70#69078—61no—
71#69076—21no—
72#68077—62no—
73#68077—61no—
74#68077—61no—
75C3×#1398—31no—
76C3×#1398—31no—
77C3×#1369—31no—
78C3×#1336—61no—
79C3×#1313—21no—
80C3×#1313C2×D311yescharacteristic
81C3×#1313C2×D311yescharacteristic
82#67828—31no—
83#67606—31no—
84#67604—31no—
85#67601—31no—
86#67583—31no—
87#67582—31no—
88#67519C2×D311yescharacteristic
89#67519C2×D311yescharacteristic
90#67516—62no—
91#67516—61no—
92#67516—61no—
93#67514C2×D311yescharacteristic
94#67513—31no—
95#67513—31no—
96#67513—61no—
97#67443—21no—
98C2×#3745—21no—
99C2×C8×C3⋊D3—81no—
100C4×C4⋊(C3⋊D3)—41no—
101C3×#904—21no—
102C3×#901—41no—
103C2×C32×C16—21no—
104C2×C32×C16C22×C411yescharacteristic
105C2×C32×C16#4111yescharacteristic
106C32×C4×C8—21no—
107C32×C4×C8C2×D411yescharacteristic
108#63648—41no—
109#63647—81no—
110#63645—21no—
111C4⋊(C3⋊(C3⋊C8))—41no—
112#63626—21no—
113#63626C2×C812yes—
114C3×#819—21no—
115C3×#812—41no—
116C3×#805—41no—
117C3×Dic24C2×C811yescharacteristic, complemented
118C3×#800—21no—
119C3×D48C2×C811yescharacteristic, complemented
120C3×#798—81no—
121C3×D3×C16—81no—
122C3×C3⋊C32—21no—
123C3×C3⋊C32—41no—
124C3×C3⋊C32C2×C812yescomplemented
125#13658—91no—
126#12068—31no—
127#12067—91no—
128#7457—91no—
129#7390—91no—
130#7389—91no—
131#7333—31no—
132C3×#508—31no—
133C3×#504—121no—
134C3×#443—61no—
135#4904C4×D311yescharacteristic
136C2×D48—31no—
137C2×D48—31no—
138C2×D48—61no—
139#4901—41no—
140#4700—21no—
141C4×D24—62no—
142C4×D24—61no—
143C4×D24—61no—
144C8×D12—122no—
145C8×D12—121no—
146C8×D12—121no—
147C3×M64—31no—
148C3×M64—31no—
149C3×M64—61no—
150C3×#368—31no—
151C3×#364—61no—
152C3×#359—61no—
153C3×#350—121no—
154C3×#346—21no—
155C3×#346—21no—
156C3×#346D1211yescharacteristic
157C3×#346#6711yescharacteristic
158C3×#346C22×D311yescharacteristic
159C3×#346C22×D311yescharacteristic
160C3×#322—21no—
161C3×#322D1211yescharacteristic
162C3×#322#6711yescharacteristic
163#4521—32no—
164#4519—32no—
165#4510—21no—
166#4509—62no—
167#4509—61no—
168#4509—61no—
169#4508—41no—
170#4506C4×D311yescharacteristic
171#4505—31no—
172#4505—31no—
173#4505—61no—
174#4499—21no—
175#4499C4×D311yescharacteristic
176#4499C4×D311yescharacteristic
177#4463—21no—
178#4459—122no—
179#4459—121no—
180#4459—121no—
181#4457—62no—
182#4457—61no—
183#4457—61no—
184#4449—32no—
185D96—32no—
186C2×C4⋊(C3⋊D3)—41no—
187C2×C4×C3⋊D3—81no—
188C3×#235—41no—
189C2×C32×C8—41no—
190C2×C32×C8C4×D411yescharacteristic
191C2×C32×C8AGL(1,ℤ/8)11yescharacteristic
192C32×C42AGL(1,ℤ/8)11yescharacteristic
193#3752—81no—
194C4⋊(C3⋊Dic3)—41no—
195C2×C3⋊(C3⋊C8)—81no—
196#3745—22no—
197#3745—41no—
198C8×C3⋊D3—162no—
199C3×#208—41no—
200C3×Dic12—21no—
201C3×D24—21no—
202C3×#204—41no—
203C3×ΓL(1,𝔽25)—81no—
204C3×D3×C8—81no—
205C32×C16—21no—
206C32×C16—41no—
207C32×C16C22×C811yescharacteristic
208C32×C16#13111yescharacteristic
209C3×C3⋊C16—81no—
210#2263—91no—
211#2179—31no—
212#2177—91no—
213#2171—61no—
214#1580—181no—
215#1426—91no—
216#1419—91no—
217#1418—91no—
218#1398—31no—
219#1398—31no—
220#1369—31no—
221#1368—91no—
222#1336—61no—
223#1333—181no—
224#1313D3211yescharacteristic
225C3×#135—61no—
226C3×#131—121no—
227#904—21no—
228C2×D24—62no—
229C2×D24—61no—
230C2×D24—61no—
231#901—41no—
232C2×D3×C8—242no—
233C2×D3×C8—241no—
234C2×D3×C8—241no—
235C4×D12—122no—
236C4×D12—121no—
237C4×D12—121no—
238C3×Q32—31no—
239C3×SD32—61no—
240C3×D16—31no—
241C3×M32—121no—
242C3×M32—241no—
243C2×C3×C16—21no—
244C2×C3×C16—21no—
245C2×C3×C16—21no—
246C2×C3×C16—21no—
247C2×C3×C16—41no—
248C2×C3×C16—241no—
249C2×C3×C16C2×C4×D311yescharacteristic
250C2×C3×C16C2×C4×D311yescharacteristic
251C2×C3×C16#23511yescharacteristic
252C2×C3×C16#23511yescharacteristic
253C3×#108—61no—
254C3×#104—121no—
255C3×C4×C8—22no—
256C3×C4×C8—21no—
257C3×C4×C8—21no—
258C3×C4×C8—21no—
259C3×C4×C8C2×D1211yescharacteristic
260C3×C4×C8C2×#6711yescharacteristic
261#821—122no—
262#821—121no—
263#821—121no—
264(C2×D3)⋊C8—242no—
265(C2×D3)⋊C8—241no—
266(C2×D3)⋊C8—241no—
267#819—21no—
268#818—62no—
269#818—61no—
270#818—61no—
271#812—41no—
272#805—41no—
273C4⋊(C3⋊C8)—122no—
274C4⋊(C3⋊C8)—121no—
275C4⋊(C3⋊C8)—121no—
276Dic24D3×C811yescharacteristic, complemented
277#800—21no—
278D48—32no—
279D48—32no—
280D48—62no—
281D48—62no—
282D48—61no—
283D48—61no—
284D48D3×C811yescharacteristic, complemented
285#798—81no—
286D3×C16—81no—
287C3×C32—32no—
288C3×C32—61no—
289C3×C32—121no—
290C3⋊C32—21no—
291C3⋊C32—41no—
292C3⋊C32D3×C812yescomplemented
293C22×C3⋊D3—81no—
294C2×C32×C4—21no—
295C2×C32×C4#44211yescharacteristic
296C2×C3⋊Dic3—81no—
297C4⋊(C3⋊D3)—42no—
298C4⋊(C3⋊D3)—81no—
299C4×C3⋊D3—161no—
300C4×C3⋊D3—161no—
301C3×#67—81no—
302C3×D12—41no—
303C3×C4×D3—81no—
304C3×Dic6—41no—
305C32×C8—21no—
306C32×C8—81no—
307C32×C8C8×D411yescharacteristic
308C32×C8#44411yescharacteristic
309C3⋊(C3⋊C8)—161no—
310C3×C3⋊C8—81no—
311#508—31no—
312C2×D16—91no—
313#504—121no—
314#443—61no—
315C4×D8—181no—
316C8×D4—361no—
317SD64—92no—
318D32—92no—
319M64—31no—
320M64—31no—
321M64—61no—
322#368—31no—
323#367—91no—
324#364—61no—
325#359—61no—
326#357—181no—
327#350—121no—
328#346#62611yescharacteristic
329#346C2×D3211yescharacteristic
330#334—181no—
331D4⋊C8—361no—
332#322#62611yescharacteristic
333C3×#41—121no—
334#235—41no—
335C2×D12—122no—
336C2×D12—121no—
337C2×D12—121no—
338C2×C4×D3—242no—
339C2×C4×D3—241no—
340C2×C4×D3—241no—
341C3×Q16—61no—
342C3×SD16—121no—
343C3×D8—61no—
344C3×M16—121no—
345C3×M16—241no—
346C2×C3×C8—21no—
347C2×C3×C8—21no—
348C2×C3×C8—42no—
349C2×C3×C8—41no—
350C2×C3×C8—41no—
351C2×C3×C8—241no—
352C2×C3×C8C4×D1211yescharacteristic
353C2×C3×C8#90811yescharacteristic
354C2×C3×C8C4×#6711yescharacteristic
355C2×C3×C8#93211yescharacteristic
356C3×C42—21no—
357C3×C42#90811yescharacteristic
358C3×C42#94911yescharacteristic
359(C2×D3)⋊C4—242no—
360(C2×D3)⋊C4—241no—
361(C2×D3)⋊C4—241no—
362C4⋊Dic3—122no—
363C4⋊Dic3—121no—
364C4⋊Dic3—121no—
365#208—41no—
366C2×C3⋊C8—242no—
367C2×C3⋊C8—241no—
368C2×C3⋊C8—241no—
369Dic12—21no—
370D24—21no—
371D24—64no—
372D24—62no—
373D24—62no—
374D24—122no—
375D24—121no—
376D24—121no—
377#204—41no—
378ΓL(1,𝔽25)—81no—
379D3×C8—81no—
380D3×C8—484no—
381D3×C8—482no—
382D3×C8—482no—
383C3×C16—21no—
384C3×C16—21no—
385C3×C16—21no—
386C3×C16—21no—
387C3×C16—41no—
388C3×C16—41no—
389C3×C16—41no—
390C3×C16—81no—
391C3×C16—241no—
392C3×C16C2×D3×C811yescharacteristic
393C3×C16C2×D3×C811yescharacteristic
394C3×C16#90111yescharacteristic
395C3×C16#90111yescharacteristic
396C3⋊C16—81no—
397C22×C32AGL(1,ℤ/16)11yescharacteristic
398C2×C3⋊D3—82no—
399C2×C3⋊D3—161no—
400C2×C3⋊D3—161no—
401C2×C3×D3—81no—
402C32×C4—41no—
403C32×C4#158011yescharacteristic
404C32×C4AGL(1,ℤ/16)11yescharacteristic
405C3⋊Dic3—161no—
406C3×Dic3—81no—
407#135—61no—
408C2×D8—181no—
409#131—121no—
410C22×C8—721no—
411C4×D4—361no—
412Q32—31no—
413SD32—61no—
414D16—31no—
415D16—92no—
416D16—181no—
417M32—121no—
418M32—241no—
419C2×C16—21no—
420C2×C16—241no—
421C2×C16#379811yescharacteristic
422C2×C16C4×D3211yescharacteristic
423#108—61no—
424#107—181no—
425C4⋊C8—361no—
426#104—121no—
427#102—361no—
428C22⋊C8—721no—
429C4×C8—21no—
430C4×C8C2×#62611yescharacteristic
431C32—32no—
432C32—61no—
433C32—121no—
434C22×D3—242no—
435C22×D3—241no—
436C22×D3—241no—
437C3×Q8—121no—
438C3×D4—121no—
439C3×D4—241no—
440C2×C3×C4—22no—
441C2×C3×C4—21no—
442C2×C3×C4—21no—
443C2×C3×C4—21no—
444C2×C3×C4—241no—
445C2×C3×C4#471111yescharacteristic
446C2×C3×C4#501411yescharacteristic
447#67—81no—
448C2×Dic3—242no—
449C2×Dic3—241no—
450C2×Dic3—241no—
451D12—41no—
452D12—124no—
453D12—122no—
454D12—122no—
455D12—242no—
456D12—241no—
457D12—241no—
458C4×D3—81no—
459C4×D3—482no—
460C4×D3—481no—
461C4×D3—481no—
462C4×D3—482no—
463C4×D3—481no—
464C4×D3—481no—
465Dic6—41no—
466C3×C8—21no—
467C3×C8—21no—
468C3×C8—21no—
469C3×C8—21no—
470C3×C8—41no—
471C3×C8—82no—
472C3×C8—81no—
473C3×C8—81no—
474C3×C8—241no—
475C3×C8C8×D1211yescharacteristic
476C3×C8#471711yescharacteristic
477C3×C8C8×#6711yescharacteristic
478C3×C8#514511yescharacteristic
479C3⋊C8—81no—
480C3⋊C8—482no—
481C3⋊C8—481no—
482C3⋊C8—481no—
483C2×C32—21no—
484C2×C32#1206711yescharacteristic
485C3⋊D3—162no—
486C3×D3—161no—
487#41—121no—
488C2×D4—361no—
489C22×C4—721no—
490Q16—61no—
491SD16—121no—
492D8—61no—
493D8—182no—
494D8—361no—
495M16—121no—
496M16—241no—
497C2×C8—41no—
498C2×C8—241no—
499C2×C8—721no—
500C2×C8—1442no—
501C2×C8#6383011yescharacteristic
502C2×C8C4×#62611yescharacteristic
503C4⋊C4—361no—
504C22⋊C4—721no—
505C42#6382511yescharacteristic
506C16—21no—
507C16—41no—
508C16—241no—
509C16D32×C811yescharacteristic
510C16#6380311yescharacteristic
511C22×C3—21no—
512C22×C3—241no—
513C22×C3#6909111yescharacteristic
514C22×C3#7092911yescharacteristic
515C2×D3—81no—
516C2×D3—244no—
517C2×D3—242no—
518C2×D3—242no—
519C2×D3—482no—
520C2×D3—481no—
521C2×D3—481no—
522C2×D3—482no—
523C2×D3—481no—
524C2×D3—481no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—42no—
528C3×C4—41no—
529C3×C4—41no—
530C3×C4—241no—
531C3×C4#6807711yescharacteristic
532C3×C4#6909111yescharacteristic
533C3×C4#7084011yescharacteristic
534C3×C4#7091811yescharacteristic
535Dic3—81no—
536Dic3—482no—
537Dic3—481no—
538Dic3—481no—
539C32AGL(1,ℤ/32)11yescharacteristic, complemented, Sylow
540C23—721no—
541Q8—121no—
542D4—121no—
543D4—241no—
544D4—362no—
545D4—721no—
546C2×C4—21no—
547C2×C4—241no—
548C2×C4—721no—
549C2×C4—1441no—
550C2×C4—1441no—
551C2×C4#1059034611yescharacteristic
552C8—21no—
553C8—81no—
554C8—241no—
555C8—1441no—
556C8C8×#62611yescharacteristic
557C8#1059092411yescharacteristic
558C2×C3—22no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—481no—
563C2×C3#1070707011yescharacteristic
564C2×C3#1070707311yescharacteristic
565D3—161no—
566D3—484no—
567D3—482no—
568D3—482no—
569C22—241no—
570C22—722no—
571C22—1441no—
572C22—1441no—
573C22#1179797211yescharacteristic
574C4—41no—
575C4—241no—
576C4—1441no—
577C4#1177735411yescharacteristic
578C4#1179797311yescharacteristic
579C3—21no—
580C3#7222250011yescharacteristic, complemented
581C3#7222251911yescharacteristic, complemented
582C2—21no—
583C2—481no—
584C2—1442no—
585C2#7222261811yescharacteristic
subgroupG/H
centerC2#72222618
commutatorC32×C16C22×C8
Frattini#346C2×D32
FittingC32×#1313C22
socleC2×C32#12067
prime radical#11800601C4
cosocleC32×#346C23
hypercenter#1313D32
hypocenterC32AGL(1,ℤ/32)
LCSG ▷ C32×C16 ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #1313

Complements

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

NH ≅ G/Ntype
1. C3×#10707078C2semidirect
2. #72222533C2semidirect
3. #72222526C2semidirect
6. #72222501C2semidirect
8. C3×#10682523C2semidirect
9. #72222460C2semidirect
27. #10707078D3semidirect
37. #10682523D3semidirect
60. #10589074C8semidirect
61. #10589071C8semidirect
117. C3×Dic24C2×C8semidirect
119. C3×D48C2×C8semidirect
124. C3×C3⋊C32C2×C8semidirect
276. Dic24D3×C8semidirect
284. D48D3×C8semidirect
292. C3⋊C32D3×C8semidirect
539. C32AGL(1,ℤ/32)semidirect
580. C3#72222500semidirect
581. C3#72222519semidirect

Maximal subgroups

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

subgroupindex
C3×#107070782
#722225332
#722225262
#722225193
#722225152
#722225012
#722225003
C3×#106825232
#722224602