Finite Group Databasebeta

← #72222624

Subgroups — #72222624

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
2C3×#10682658C211yesmaximal, characteristic, complemented
3#72222526C211yesmaximal, characteristic, complemented
4#72222524—31nomaximal
5C3×#10715810C211yesmaximal, characteristic
6#72222504—31nomaximal
7#72222501C211yesmaximal, characteristic, complemented
8#72222493C211yesmaximal, characteristic, complemented
9#72222464C211yesmaximal, characteristic, complemented
10C3×#71458C411yescharacteristic
11C3×#69085C2211yescharacteristic
12#11797611—21no—
13#11777340—21no—
14C3×#67579C411yescharacteristic
15C3×#67825C411yescharacteristic
16C3×#67580C411yescharacteristic
17C3×#67519C2211yescharacteristic
18C3×#67519C2211yescharacteristic
19C3×#67514C2211yescharacteristic
20#11772179C2211yescharacteristic
21#11771667—21no—
22#11771603—21no—
23#11771319C2211yescharacteristic
24C32×#2177C2211yescharacteristic
25#10715810D311yescharacteristic
26#10707077—31no—
27#10707076—31no—
28#10707072—31no—
29#10707072—31no—
30#10707072—61no—
31#10682858—31no—
32#10682857—31no—
33#10682658D311yescharacteristic, complemented
34#10682656—31no—
35#10682655—31no—
36#10682524—31no—
37#10682522—31no—
38C3×#13658—31no—
39C3×#7390—31no—
40C2×C32×D16C2×C411yescharacteristic
41C32×C4×D8—21no—
42D4×C3⋊(C3⋊C8)—41no—
43C3×#4904C2×C411yescharacteristic
44C3×#4700—21no—
45#10590622—41no—
46C32×#367C2×C411yescharacteristic
47C32×#357—21no—
48C32×#346C2311yescharacteristic
49D4⋊(C3⋊(C3⋊C8))—41no—
50#10589327D411yescharacteristic
51#10589323—21no—
52#10589321D411yescharacteristic
53C3×#4520C812yescomplemented
54C3×#4519C812yescomplemented
55C3×#4510—21no—
56C3×#4506C2×C411yescharacteristic
57C3×#4499—21no—
58C3×#4499C2×C411yescharacteristic
59C3×#4499C2×C411yescharacteristic
60C3×#4463—21no—
61#10589111—41no—
62AGL(1,ℤ/32)—91noSylow
63C3×#2263—31no—
64C3×#2179—31no—
65C3×#2177—21no—
66C3×#2177C2×D311yescharacteristic
67C3×#2177C2×D311yescharacteristic
68#71458Dic311yescharacteristic
69#70924—61no—
70#70878—61no—
71#70878—61no—
72#70878—62no—
73#69301—61no—
74#69085C2×D311yescharacteristic
75C3×#1426—31no—
76C3×#1419—31no—
77C3×#1418—31no—
78C3×#1398—31no—
79C3×#1398—31no—
80C3×#1369—31no—
81#67825Dic311yescharacteristic
82#67767—61no—
83#67762—61no—
84#67762—61no—
85#67762—62no—
86#67580Dic311yescharacteristic
87#67579Dic311yescharacteristic
88#67569—61no—
89#67519C2×D311yescharacteristic
90#67519C2×D311yescharacteristic
91#67514C2×D311yescharacteristic
92#67513—31no—
93#67513—31no—
94#67513—61no—
95#67511—31no—
96#67511—31no—
97#67511—61no—
98C2×C32×D8—21no—
99C32×C4×D4—41no—
100C22×C3⋊(C3⋊C8)—81no—
101C3×#904—21no—
102C3×#901—41no—
103C32×D16—21no—
104C32×D16C2×C812yes—
105C2×C32×C16C22×C411yescharacteristic
106C2×C32×C16#4111yescharacteristic
107C32×#107—21no—
108C32×#102—41no—
109C32×C4×C8C2×D411yescharacteristic
110#63657—81no—
111C2×C3⋊(C3⋊C16)—21no—
112C4⋊(C3⋊(C3⋊C8))—41no—
113C4×C3⋊(C3⋊C8)—21no—
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—31no—
126#12068—91no—
127#12067—91no—
128#7457—91no—
129#7390—31no—
130#7389—91no—
131#7333—91no—
132C3×#508—31no—
133C2×C3×D16—21no—
134C2×C3×D16C4×D311yescharacteristic
135C2×C3×D16C2×Dic311yescharacteristic
136C3×#443—61no—
137C3×C4×D8—22no—
138C3×C4×D8—21no—
139C3×C4×D8—21no—
140#5140—121no—
141D4×C3⋊C8—122no—
142D4×C3⋊C8—121no—
143D4×C3⋊C8—121no—
144#4907—121no—
145#4904C2×Dic311yescharacteristic
146#4700—21no—
147C3×SD64—32no—
148C3×D32—32no—
149C3×M64—31no—
150C3×M64—31no—
151C3×M64—61no—
152C3×#368—31no—
153C3×#367—21no—
154C3×#367C4×D311yescharacteristic
155C3×#367C2×Dic311yescharacteristic
156C3×#364—61no—
157C3×#359—61no—
158C3×#357—22no—
159C3×#357—21no—
160C3×#357—21no—
161C3×#346—21no—
162C3×#346C22×D311yescharacteristic
163C3×#346C22×D311yescharacteristic
164#4561—121no—
165D4⋊(C3⋊C8)—122no—
166D4⋊(C3⋊C8)—121no—
167D4⋊(C3⋊C8)—121no—
168#4520C3⋊C812yescomplemented
169#4519C3⋊C812yescomplemented
170#4515—121no—
171#4510—21no—
172#4506C2×Dic311yescharacteristic
173#4499—21no—
174#4499C2×Dic311yescharacteristic
175#4499C2×Dic311yescharacteristic
176#4463—21no—
177#4461—31no—
178#4461—31no—
179#4461—61no—
180#4457—62no—
181#4457—61no—
182#4457—61no—
183#4455—31no—
184#4455—31no—
185#4455—61no—
186C2×C32×D4—41no—
187C22×C32×C4—81no—
188C3×#235—41no—
189C32×D8—22no—
190C32×D8—41no—
191C2×C32×C8C4×D411yescharacteristic
192C2×C32×C8AGL(1,ℤ/8)11yescharacteristic
193C32×C4⋊C4—41no—
194C32×C22⋊C4—81no—
195C32×C42AGL(1,ℤ/8)11yescharacteristic
196C2×C3⋊(C3⋊C8)—41no—
197C2×C3⋊(C3⋊C8)—81no—
198C2×C3⋊(C3⋊C8)—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—31no—
211#2179—31no—
212#2177D3211yescharacteristic
213#2171—181no—
214#1580—181no—
215#1426—31no—
216#1419—31no—
217#1418—31no—
218#1398—31no—
219#1398—31no—
220#1369—31no—
221#1368—91no—
222#1336—181no—
223#1333—181no—
224#1313—91no—
225C3×#135—61no—
226C2×C3×D8—22no—
227C2×C3×D8—21no—
228C2×C3×D8—21no—
229C3×#131—121no—
230C3×C4×D4—42no—
231C3×C4×D4—41no—
232C3×C4×D4—41no—
233C22×C3⋊C8—242no—
234C22×C3⋊C8—241no—
235C22×C3⋊C8—241no—
236#904—21no—
237#901—41no—
238C3×Q32—31no—
239C3×SD32—61no—
240C3×D16—22no—
241C3×D16—21no—
242C3×D16—21no—
243C3×D16—22no—
244C3×D16—31no—
245C3×D16D3×C812yes—
246C3×D16C2×C3⋊C812yes—
247C3×M32—121no—
248C2×C3×C16—21no—
249C2×C3×C16—21no—
250C2×C3×C16C2×C4×D311yescharacteristic
251C2×C3×C16#23711yescharacteristic
252C2×C3×C16#23711yescharacteristic
253C2×C3×C16C22×Dic311yescharacteristic
254C3×#108—61no—
255C3×#107—22no—
256C3×#107—21no—
257C3×#107—21no—
258C3×#104—121no—
259C3×#102—42no—
260C3×#102—41no—
261C3×#102—41no—
262C3×C4×C8—21no—
263C3×C4×C8D3×D411yescharacteristic
264C3×C4×C8D3×D411yescharacteristic
265#830—242no—
266#830—241no—
267#830—241no—
268#819—21no—
269#812—41no—
270#812—241no—
271C2×C3⋊C16—61no—
272C2×C3⋊C16—61no—
273C2×C3⋊C16—121no—
274C2×C3⋊C16—241no—
275#805—41no—
276C4⋊(C3⋊C8)—122no—
277C4⋊(C3⋊C8)—121no—
278C4⋊(C3⋊C8)—121no—
279C4×C3⋊C8—62no—
280C4×C3⋊C8—61no—
281C4×C3⋊C8—61no—
282Dic24C2×C3⋊C811yescharacteristic, complemented
283#800—21no—
284D48C2×C3⋊C811yescharacteristic, complemented
285#798—241no—
286D3×C16—241no—
287C3×C32—32no—
288C3×C32—61no—
289C3×C32—121no—
290C3⋊C32—21no—
291C3⋊C32—41no—
292C3⋊C32C2×C3⋊C812yescomplemented
293C23×C32—81no—
294C32×D4—42no—
295C32×D4—81no—
296C2×C32×C4—21no—
297C2×C32×C4—81no—
298C2×C32×C4—161no—
299C2×C32×C4—161no—
300C2×C32×C4#44211yescharacteristic
301C3×#67—81no—
302C3×D12—41no—
303C3×C4×D3—81no—
304C3×Dic6—41no—
305C32×C8—21no—
306C32×C8C8×D411yescharacteristic
307C32×C8#44411yescharacteristic
308C3⋊(C3⋊C8)—81no—
309C3⋊(C3⋊C8)—161no—
310C3×C3⋊C8—81no—
311#508—31no—
312C2×D16D3×Dic311yescharacteristic
313#504—361no—
314#443—61no—
315C4×D8—21no—
316C8×D4—361no—
317SD64—32no—
318D32—32no—
319M64—31no—
320M64—31no—
321M64—61no—
322#368—31no—
323#367D3×Dic311yescharacteristic
324#364—61no—
325#359—61no—
326#357—21no—
327#350—361no—
328#346—91no—
329#346C2×D3211yescharacteristic
330#334—181no—
331D4⋊C8—361no—
332#322—91no—
333C3×#41—121no—
334C2×C3×D4—42no—
335C2×C3×D4—41no—
336C2×C3×D4—41no—
337C22×C3×C4—82no—
338C22×C3×C4—81no—
339C22×C3×C4—81no—
340#235—41no—
341C3×Q16—61no—
342C3×SD16—121no—
343C3×D8—24no—
344C3×D8—22no—
345C3×D8—22no—
346C3×D8—42no—
347C3×D8—41no—
348C3×D8—41no—
349C3×D8—61no—
350C3×M16—121no—
351C3×M16—241no—
352C2×C3×C8—21no—
353C2×C3×C8—21no—
354C2×C3×C8—241no—
355C2×C3×C8#67⋊C411yescharacteristic
356C2×C3×C8#91111yescharacteristic
357C2×C3×C8#91111yescharacteristic
358C2×C3×C8D4×Dic311yescharacteristic
359C3×C4⋊C4—42no—
360C3×C4⋊C4—41no—
361C3×C4⋊C4—41no—
362C3×C22⋊C4—82no—
363C3×C22⋊C4—81no—
364C3×C22⋊C4—81no—
365C3×C42—21no—
366C3×C42#91111yescharacteristic
367C3×C42#91411yescharacteristic
368#208—41no—
369C2×C3⋊C8—122no—
370C2×C3⋊C8—121no—
371C2×C3⋊C8—121no—
372C2×C3⋊C8—242no—
373C2×C3⋊C8—241no—
374C2×C3⋊C8—241no—
375C2×C3⋊C8—484no—
376C2×C3⋊C8—482no—
377C2×C3⋊C8—482no—
378Dic12—21no—
379D24—21no—
380#204—41no—
381ΓL(1,𝔽25)—81no—
382D3×C8—81no—
383C3×C16—21no—
384C3×C16—21no—
385C3×C16—21no—
386C3×C16—21no—
387C3×C16—41no—
388C3×C16—241no—
389C3×C16C2×D3×C811yescharacteristic
390C3×C16#90711yescharacteristic
391C3×C16C22×C3⋊C811yescharacteristic
392C3×C16#94811yescharacteristic
393C3⋊C16—81no—
394C3⋊C16—121no—
395C3⋊C16—121no—
396C3⋊C16—241no—
397C22×C32—82no—
398C22×C32—161no—
399C22×C32—161no—
400C22×C32AGL(1,ℤ/16)11yescharacteristic
401C2×C3×D3—81no—
402C32×C4—41no—
403C32×C4—161no—
404C32×C4#158011yescharacteristic
405C32×C4AGL(1,ℤ/16)11yescharacteristic
406C3×Dic3—81no—
407#135—61no—
408C2×D8—21no—
409#131—121no—
410C22×C8—721no—
411C4×D4—41no—
412Q32—31no—
413SD32—61no—
414D16—21no—
415D16—31no—
416D16D3×C3⋊C812yes—
417M32—121no—
418M32—721no—
419C2×C16—181no—
420C2×C16—721no—
421C2×C16C2×D3×Dic311yescharacteristic
422C2×C16#380511yescharacteristic
423#108—61no—
424#107—21no—
425C4⋊C8—361no—
426#104—121no—
427#102—41no—
428C22⋊C8—721no—
429C4×C8—181no—
430C4×C8#381111yescharacteristic
431C32—32no—
432C32—61no—
433C32—121no—
434C23×C3—82no—
435C23×C3—81no—
436C23×C3—81no—
437C3×Q8—121no—
438C3×D4—44no—
439C3×D4—42no—
440C3×D4—42no—
441C3×D4—82no—
442C3×D4—81no—
443C3×D4—81no—
444C3×D4—121no—
445C3×D4—241no—
446C2×C3×C4—22no—
447C2×C3×C4—21no—
448C2×C3×C4—21no—
449C2×C3×C4—21no—
450C2×C3×C4—82no—
451C2×C3×C4—81no—
452C2×C3×C4—81no—
453C2×C3×C4—162no—
454C2×C3×C4—161no—
455C2×C3×C4—161no—
456C2×C3×C4—162no—
457C2×C3×C4—161no—
458C2×C3×C4—161no—
459C2×C3×C4—241no—
460C2×C3×C4#478511yescharacteristic
461C2×C3×C4#515211yescharacteristic
462#67—81no—
463D12—41no—
464C4×D3—81no—
465Dic6—41no—
466C3×C8—21no—
467C3×C8—21no—
468C3×C8—21no—
469C3×C8—21no—
470C3×C8—41no—
471C3×C8—241no—
472C3×C8#67⋊C811yescharacteristic
473C3×C8#482311yescharacteristic
474C3×C8D4×C3⋊C811yescharacteristic
475C3×C8#519711yescharacteristic
476C3⋊C8—81no—
477C3⋊C8—242no—
478C3⋊C8—241no—
479C3⋊C8—241no—
480C3⋊C8—482no—
481C3⋊C8—481no—
482C3⋊C8—481no—
483C2×C32—21no—
484C2×C32—162no—
485C2×C32#1206711yescharacteristic
486C3×D3—161no—
487#41—121no—
488C2×D4—41no—
489C22×C4—81no—
490Q16—61no—
491SD16—121no—
492D8—22no—
493D8—41no—
494D8—61no—
495M16—121no—
496M16—241no—
497C2×C8—241no—
498C2×C8—361no—
499C2×C8—721no—
500C2×C8—1442no—
501C2×C8#6392711yescharacteristic
502C2×C8#67⋊Dic311yescharacteristic
503C4⋊C4—41no—
504C22⋊C4—81no—
505C42#6393711yescharacteristic
506C16—21no—
507C16—241no—
508C16—361no—
509C16C2×D3×C3⋊C811yescharacteristic
510C16#6381511yescharacteristic
511C22×C3—21no—
512C22×C3—84no—
513C22×C3—82no—
514C22×C3—82no—
515C22×C3—162no—
516C22×C3—161no—
517C22×C3—161no—
518C22×C3—162no—
519C22×C3—161no—
520C22×C3—161no—
521C22×C3—241no—
522C22×C3#6925411yescharacteristic
523C22×C3#7151511yescharacteristic
524C2×D3—81no—
525C3×C4—21no—
526C3×C4—21no—
527C3×C4—42no—
528C3×C4—41no—
529C3×C4—41no—
530C3×C4—162no—
531C3×C4—161no—
532C3×C4—161no—
533C3×C4—241no—
534C3×C4#6815111yescharacteristic
535C3×C4#6924911yescharacteristic
536C3×C4#7087811yescharacteristic
537C3×C4#7151511yescharacteristic
538Dic3—81no—
539C32AGL(1,ℤ/32)11yescharacteristic, complemented, Sylow
540C23—81no—
541Q8—121no—
542D4—42no—
543D4—81no—
544D4—121no—
545D4—241no—
546C2×C4—21no—
547C2×C4—81no—
548C2×C4—161no—
549C2×C4—161no—
550C2×C4—241no—
551C2×C4#1059143611yescharacteristic
552C8—21no—
553C8—241no—
554C8—721no—
555C8—1441no—
556C8#67⋊(C3⋊C8)11yescharacteristic
557C8#1059167611yescharacteristic
558C2×C3—22no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—164no—
563C2×C3—162no—
564C2×C3—162no—
565C2×C3—481no—
566C2×C3#1070706911yescharacteristic
567C2×C3#1070707211yescharacteristic
568D3—161no—
569C22—82no—
570C22—161no—
571C22—161no—
572C22—241no—
573C22#1179797911yescharacteristic
574C4—41no—
575C4—161no—
576C4—241no—
577C4#1177735211yescharacteristic
578C4#1179797011yescharacteristic
579C3—21no—
580C3#7222250411yescharacteristic, complemented
581C3#7222252411yescharacteristic, complemented
582C2—21no—
583C2—162no—
584C2—481no—
585C2#7222262111yescharacteristic
subgroupG/H
centerC2#72222621
commutatorC32×C16C22×C8
Frattini#346C2×D32
FittingC32×#2177C22
socleC2×C32#12067
prime radicalC3×#71458C4
cosocleC32×#346C23
hypercenter#2177D32
hypocenterC32AGL(1,ℤ/32)
LCSG ▷ C32×C16 ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #346 ◁ #2177

Complements

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

NH ≅ G/Ntype
1. #72222543C2semidirect
2. C3×#10682658C2semidirect
3. #72222526C2semidirect
7. #72222501C2semidirect
8. #72222493C2semidirect
9. #72222464C2semidirect
33. #10682658D3semidirect
53. C3×#4520C8semidirect
54. C3×#4519C8semidirect
117. C3×Dic24C2×C8semidirect
119. C3×D48C2×C8semidirect
124. C3×C3⋊C32C2×C8semidirect
168. #4520C3⋊C8semidirect
169. #4519C3⋊C8semidirect
282. Dic24C2×C3⋊C8semidirect
284. D48C2×C3⋊C8semidirect
292. C3⋊C32C2×C3⋊C8semidirect
539. C32AGL(1,ℤ/32)semidirect
580. C3#72222504semidirect
581. C3#72222524semidirect

Maximal subgroups

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

subgroupindex
#722225432
C3×#106826582
#722225262
#722225243
C3×#107158102
#722225043
#722225012
#722224932
#722224642