Finite Group Databasebeta

← #72222612

Subgroups — #72222612

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
1#72222543C211yesmaximal, characteristic, complemented
2#72222533C211yesmaximal, characteristic, complemented
3C3×#10682654C211yesmaximal, characteristic, complemented
4#72222517—31nomaximal
5#72222515C211yesmaximal, characteristic
6C3×#10682856C211yesmaximal, characteristic, complemented
7#72222495—31nomaximal
8#72222493C211yesmaximal, characteristic, complemented
9#72222453C211yesmaximal, characteristic, complemented
10#11800601C411yescharacteristic
11C3×#69085C2211yescharacteristic
12#11797813C2211yescharacteristic
13#11797611—21no—
14#11777337—21no—
15#11773077C411yescharacteristic
16#11772984C411yescharacteristic
17#11772967C411yescharacteristic
18C3×#67519C2211yescharacteristic
19C3×#67519C2211yescharacteristic
20C3×#67514C2211yescharacteristic
21#11771667—21no—
22#11771600—21no—
23#11771319C2211yescharacteristic
24C32×#1368C2211yescharacteristic
25#10715814—31no—
26#10715812—31no—
27#10707077—31no—
28#10707076—31no—
29#10707071—31no—
30#10707071—31no—
31#10707071—61no—
32#10682856D311yescharacteristic, complemented
33#10682659—31no—
34#10682657—31no—
35#10682654D311yescharacteristic, complemented
36#10682524—31no—
37#10682522—31no—
38C3×#7457—31no—
39C3×#7389—31no—
40C2×#63626C2×C411yescharacteristic
41#10592533—41no—
42C4×#3745—21no—
43C3×#4904C2×C411yescharacteristic
44C3×#4700—21no—
45#10590622—41no—
46C32×#346C2311yescharacteristic
47C32×#334—21no—
48#10589375—21no—
49#10589371C2×C411yescharacteristic
50#10589349—41no—
51#10589327D411yescharacteristic
52#10589321D411yescharacteristic
53C3×#4510—21no—
54C3×#4506C2×C411yescharacteristic
55C3×#4499—21no—
56C3×#4499C2×C411yescharacteristic
57C3×#4499C2×C411yescharacteristic
58C3×#4463—21no—
59#10589111—41no—
60#10589074C812yescomplemented
61#10589071C812yescomplemented
62AGL(1,ℤ/32)—91noSylow
63C3×#2179—31no—
64#71509—31no—
65#70924—61no—
66#69373—31no—
67#69301—61no—
68#69085C2×D311yescharacteristic
69#69078—31no—
70#69078—31no—
71#69078—61no—
72#68518—61no—
73#68518—61no—
74#68518—62no—
75C3×#1398—31no—
76C3×#1398—31no—
77C3×#1369—31no—
78C3×#1368—21no—
79C3×#1368C2×D311yescharacteristic
80C3×#1368C2×D311yescharacteristic
81#67828—31no—
82#67767—61no—
83#67606—31no—
84#67604—31no—
85#67601—31no—
86#67583—31no—
87#67582—31no—
88#67569—61no—
89#67519C2×D311yescharacteristic
90#67519C2×D311yescharacteristic
91#67514C2×D311yescharacteristic
92#67511—31no—
93#67511—31no—
94#67511—61no—
95#67465—61no—
96#67465—61no—
97#67465—62no—
98C2×#3745—21no—
99C2×C8×C3⋊D3—81no—
100C4×C4⋊(C3⋊D3)—41no—
101C3×#904—21no—
102C3×#901—41no—
103C2×C32×C16C22×C411yescharacteristic
104C2×C32×C16#4111yescharacteristic
105C32×C4⋊C8—41no—
106C32×C4×C8C2×D411yescharacteristic
107#63648—41no—
108#63647—81no—
109#63645—21no—
110C2×C3⋊(C3⋊C16)—21no—
111C4×C3⋊(C3⋊C8)—21no—
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
120#63543—81no—
121D3×C3⋊C16—81no—
122C3×C3⋊C32—21no—
123C3×C3⋊C32—41no—
124C3×C3⋊C32C2×C812yescomplemented
125#13658—91no—
126#12068—91no—
127#12067—91no—
128#7457—31no—
129#7390—91no—
130#7389—31no—
131#7333—91no—
132C3×#508—31no—
133C3×#443—61no—
134#5140—121no—
135#4907—121no—
136#4904C4×D311yescharacteristic
137C2×D48—31no—
138C2×D48—31no—
139C2×D48—61no—
140#4834—122no—
141#4834—121no—
142#4834—121no—
143#4700—21no—
144C4×D24—62no—
145C4×D24—61no—
146C4×D24—61no—
147C3×M64—31no—
148C3×M64—31no—
149C3×M64—61no—
150C3×#368—31no—
151C3×#364—61no—
152C3×#359—61no—
153C3×#346—21no—
154C3×#346C22×D311yescharacteristic
155C3×#346C22×D311yescharacteristic
156C3×#334—22no—
157C3×#334—21no—
158C3×#334—21no—
159#4561—121no—
160#4521—32no—
161#4519—32no—
162#4515—121no—
163#4510—21no—
164#4509—62no—
165#4509—61no—
166#4509—61no—
167#4506C4×D311yescharacteristic
168#4505—31no—
169#4505—31no—
170#4505—61no—
171#4499—21no—
172#4499C4×D311yescharacteristic
173#4499C4×D311yescharacteristic
174#4483—122no—
175#4483—121no—
176#4483—121no—
177#4463—21no—
178#4461—31no—
179#4461—31no—
180#4461—61no—
181#4455—31no—
182#4455—31no—
183#4455—61no—
184#4449—32no—
185D96—32no—
186C2×C4⋊(C3⋊D3)—41no—
187C2×C4×C3⋊D3—81no—
188C3×#235—41no—
189C2×C32×C8—81no—
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)—41no—
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×C16C22×C811yescharacteristic
207C32×C16#13111yescharacteristic
208C3⋊(C3⋊C16)—41no—
209C3×C3⋊C16—81no—
210#2263—91no—
211#2179—31no—
212#2177—91no—
213#2171—181no—
214#1580—181no—
215#1426—91no—
216#1419—91no—
217#1418—91no—
218#1398—31no—
219#1398—31no—
220#1369—31no—
221#1368D3211yescharacteristic
222#1336—181no—
223#1333—181no—
224#1313—91no—
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—
242C2×C3×C16—21no—
243C2×C3×C16—21no—
244C2×C3×C16C2×C4×D311yescharacteristic
245C2×C3×C16C2×C4×D311yescharacteristic
246C2×C3×C16#23711yescharacteristic
247C2×C3×C16#23911yescharacteristic
248C3×#108—61no—
249C3×C4⋊C8—42no—
250C3×C4⋊C8—41no—
251C3×C4⋊C8—41no—
252C3×#104—121no—
253C3×C4×C8—21no—
254C3×C4×C8D3×D411yescharacteristic
255C3×C4×C8D3×D411yescharacteristic
256#821—122no—
257#821—121no—
258#821—121no—
259(C2×D3)⋊C8—242no—
260(C2×D3)⋊C8—241no—
261(C2×D3)⋊C8—241no—
262#819—21no—
263#818—62no—
264#818—61no—
265#818—61no—
266#812—41no—
267#812—241no—
268C2×C3⋊C16—61no—
269C2×C3⋊C16—61no—
270C2×C3⋊C16—121no—
271C2×C3⋊C16—241no—
272#805—41no—
273C4×C3⋊C8—62no—
274C4×C3⋊C8—61no—
275C4×C3⋊C8—61no—
276Dic24D3×C811yescharacteristic, complemented
277#800—21no—
278D48—32no—
279D48—32no—
280D48—62no—
281D48—62no—
282D48—61no—
283D48—61no—
284D48D3×C811yescharacteristic, complemented
285#798—241no—
286D3×C16—241no—
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—161no—
307C32×C8C8×D411yescharacteristic
308C32×C8#44411yescharacteristic
309C3⋊(C3⋊C8)—81no—
310C3×C3⋊C8—81no—
311#508—31no—
312C2×D16—91no—
313#504—361no—
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—361no—
328#346—91no—
329#346C2×D3211yescharacteristic
330#334—21no—
331D4⋊C8—361no—
332#322—91no—
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—82no—
349C2×C3×C8—81no—
350C2×C3×C8—81no—
351C2×C3×C8—241no—
352C2×C3×C8#67⋊C411yescharacteristic
353C2×C3×C8#89311yescharacteristic
354C2×C3×C8#91411yescharacteristic
355C2×C3×C8#91411yescharacteristic
356C3×C42—21no—
357C3×C42#91111yescharacteristic
358C3×C42#91411yescharacteristic
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—122no—
367C2×C3⋊C8—121no—
368C2×C3⋊C8—121no—
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—241no—
389C3×C16C2×D3×C811yescharacteristic
390C3×C16C2×D3×C811yescharacteristic
391C3×C16#90711yescharacteristic
392C3×C16#90711yescharacteristic
393C3⋊C16—81no—
394C3⋊C16—121no—
395C3⋊C16—121no—
396C3⋊C16—241no—
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—721no—
419C2×C16—181no—
420C2×C16—721no—
421C2×C16#379611yescharacteristic
422C2×C16C4×D3211yescharacteristic
423#108—61no—
424#107—181no—
425C4⋊C8—41no—
426#104—121no—
427#102—361no—
428C22⋊C8—721no—
429C4×C8—181no—
430C4×C8#381111yescharacteristic
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#478511yescharacteristic
446C2×C3×C4#486911yescharacteristic
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—162no—
472C3×C8—161no—
473C3×C8—161no—
474C3×C8—241no—
475C3×C8#67⋊C811yescharacteristic
476C3×C8#482311yescharacteristic
477C3×C8#483411yescharacteristic
478C3×C8#489411yescharacteristic
479C3⋊C8—81no—
480C3⋊C8—242no—
481C3⋊C8—241no—
482C3⋊C8—241no—
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—81no—
498C2×C8—241no—
499C2×C8—361no—
500C2×C8—1442no—
501C2×C8#6388111yescharacteristic
502C2×C8#6394111yescharacteristic
503C4⋊C4—361no—
504C22⋊C4—721no—
505C42#6393711yescharacteristic
506C16—21no—
507C16—241no—
508C16—361no—
509C16D32×C811yescharacteristic
510C16#6380511yescharacteristic
511C22×C3—21no—
512C22×C3—241no—
513C22×C3#6924911yescharacteristic
514C22×C3#6931011yescharacteristic
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#6815111yescharacteristic
532C3×C4#6851811yescharacteristic
533C3×C4#6925411yescharacteristic
534C3×C4#6931011yescharacteristic
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#1059046211yescharacteristic
552C8—21no—
553C8—161no—
554C8—241no—
555C8—721no—
556C8#1059090611yescharacteristic
557C8#1059097811yescharacteristic
558C2×C3—22no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—481no—
563C2×C3#1070706911yescharacteristic
564C2×C3#1070707111yescharacteristic
565D3—161no—
566D3—484no—
567D3—482no—
568D3—482no—
569C22—241no—
570C22—722no—
571C22—1441no—
572C22—1441no—
573C22#1179796711yescharacteristic
574C4—41no—
575C4—241no—
576C4—1441no—
577C4#1177734911yescharacteristic
578C4#1179798211yescharacteristic
579C3—21no—
580C3#7222249511yescharacteristic, complemented
581C3#7222251711yescharacteristic, complemented
582C2—21no—
583C2—481no—
584C2—1442no—
585C2#7222261111yescharacteristic
subgroupG/H
centerC2#72222611
commutatorC32×C16C22×C8
Frattini#346C2×D32
FittingC32×#1368C22
socleC2×C32#12067
prime radical#11800601C4
cosocleC32×#346C23
hypercenter#1368D32
hypocenterC32AGL(1,ℤ/32)
LCSG ▷ C32×C16 ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #346 ◁ #1368

Complements

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

NH ≅ G/Ntype
1. #72222543C2semidirect
2. #72222533C2semidirect
3. C3×#10682654C2semidirect
6. C3×#10682856C2semidirect
8. #72222493C2semidirect
9. #72222453C2semidirect
32. #10682856D3semidirect
35. #10682654D3semidirect
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#72222495semidirect
581. C3#72222517semidirect

Maximal subgroups

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

subgroupindex
#722225432
#722225332
C3×#106826542
#722225173
#722225152
C3×#106828562
#722224953
#722224932
#722224532