Finite Group Databasebeta

← #72258792

Subgroups — #72258792

Order 1536.

742 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10813660C211yesmaximal, characteristic
2#10713326C211yesmaximal, characteristic, complemented
3#10711285C211yesmaximal, characteristic
4C2×#67441C211yesmaximal, characteristic
5#10682358C211yesmaximal, characteristic, complemented
6#10682126C211yesmaximal, characteristic, complemented
7C3×#8813C211yesmaximal, characteristic, complemented
8#136361—31nomaximal, Sylow
9#79116—21no—
10#79106—21no—
11C2×#4700C2211yescharacteristic
12C3×#1844—22no—
13C3×#1844C2211yescharacteristic
14C3×#1844C2211yescharacteristic
15C2×C3×C82C2211yescharacteristic
16#70592—21no—
17#70572—21no—
18C2×#4488—21no—
19C2×#4487—21no—
20#69779C2211yescharacteristic
21C2×#4463C2211yescharacteristic
22C2×#4463C2211yescharacteristic
23#69592—22no—
24#67721—21no—
25#67720—21no—
26#67441C414yescomplemented
27#22689—31no—
28#13186—31no—
29#12823—31no—
30C2×#1335—31no—
31#8813D311yescharacteristic, complemented
32#7268—31no—
33#7179—31no—
34C2×#904D411yescharacteristic
35C2×#901—21no—
36#5504—22no—
37C2×C3×#106—21no—
38C2×C3×#106—21no—
39C2×C3×#106—21no—
40C2×C3×#106—21no—
41C2×C3×#106—44no—
42C2×C3×C4×C8—21no—
43C2×C3×C4×C8C2311yescharacteristic
44C3×#384—21no—
45C3×#384D412yes—
46C2×#819D411yescharacteristic
47C2×#819D411yescharacteristic
48#5106D411yescharacteristic
49#5101—21no—
50C2×#812—21no—
51C2×#812—21no—
52#4966—22no—
53#4964—22no—
54C2×#805—21no—
55#4869—41no—
56#4861—42no—
57#4860—41no—
58D3×#106—41no—
59#4853—41no—
60#4852—42no—
61#4700—21no—
62#4700—21no—
63#4700C2×C412yes—
64#4700Q812yes—
65C3×C82—21no—
66C3×C82C2×C412yescomplemented
67#4530—22no—
68#4523—21no—
69#4488—42no—
70#4487—42no—
71#4463—21no—
72#4463—21no—
73#4463—21no—
74#4463C2×C412yescomplemented
75C2×#443—31no—
76#2952—61no—
77#2912—61no—
78C2×#365—61no—
79C2×#364—31no—
80C2×#364—31no—
81C2×#360—61no—
82#1951—61no—
83#1859—62no—
84#1844—22no—
85#1844C2×D311yescharacteristic
86#1844C2×D311yescharacteristic
87#1815—61no—
88#1772—31no—
89C2×C82C2×D311yescharacteristic
90#1386—61no—
91#1383—61no—
92#1335—34no—
93C2×#235—21no—
94C22×C3×C8—22no—
95C22×C3×C8C2×D411yescharacteristic
96C22×C3×C8C2×D411yescharacteristic
97C2×C3×C4⋊C4—22no—
98C2×C3×C4⋊C4—42no—
99C2×C3×C4⋊C4—42no—
100C2×C3×C4⋊C4—41no—
101C2×C3×C42C2×D411yescharacteristic
102C4×#67—82no—
103#926—21no—
104C2×#208—21no—
105C2×#208—21no—
106C2×Dic12SD1611yescharacteristic
107#904—21no—
108#904—21no—
109#904C4⋊C412yes—
110#904SD1612yes—
111C2×D24SD1611yescharacteristic
112C2×#204—21no—
113#901—22no—
114#901—22no—
115#901—41no—
116#901—41no—
117C2×ΓL(1,𝔽25)—41no—
118C2×D3×C8—41no—
119#893—42no—
120#892—42no—
121D3×C4⋊C4—42no—
122#887—42no—
123C3×#106—41no—
124C3×#106—41no—
125C3×#106—41no—
126C3×#106—41no—
127C3×#106—41no—
128C3×#106—41no—
129C3×#106—41no—
130C3×#106—41no—
131C3×#106—88no—
132C3×C4×C8—22no—
133C3×C4×C8—22no—
134C3×C4×C8C22×C411yescharacteristic
135C3×C4×C8C2×Q811yescharacteristic
136C3×C4×C8#4111yescharacteristic
137C3×C4×C8#4111yescharacteristic
138#819—21no—
139#819—21no—
140#819—21no—
141#819C4⋊C412yes—
142#818SD1612yes—
143#817—21no—
144#815—41no—
145C8×Dic3—41no—
146#812—22no—
147#812—41no—
148#812—41no—
149#812—41no—
150C2×C3⋊C16—41no—
151C2×C3⋊C16—41no—
152#810—42no—
153#809—42no—
154#808—42no—
155#807—42no—
156#805—22no—
157#805—22no—
158C2×#135—31no—
159C2×#131—61no—
160#520—62no—
161C2×M32—61no—
162C2×M32—61no—
163#443—32no—
164#443—32no—
165#443—61no—
166#443—61no—
167#442—121no—
168#441—121no—
169C4×SD16—122no—
170C2×#108—31no—
171C2×#108—31no—
172#428—62no—
173#428—122no—
174#427—31no—
175#427—121no—
176C2×#106—21no—
177C2×#106—21no—
178C2×#106—21no—
179C2×#106—21no—
180C2×#106—44no—
181C2×#106—121no—
182C2×#104—61no—
183#417—62no—
184#405—61no—
185C2×C4×C8—21no—
186C2×C4×C8C22×D311yescharacteristic
187#384—21no—
188#384#6712yes—
189#365—122no—
190#364—32no—
191#364—61no—
192#364—61no—
193#364—61no—
194#360—122no—
195#343—62no—
196#339—61no—
197C82—21no—
198C82C4×D312yescomplemented
199C22×C3×C4—21no—
200C22×C3×C4—42no—
201C22×C3×C4—81no—
202C22×C3×C4#12011yescharacteristic
203C2×#67—41no—
204#235—22no—
205#235—22no—
206#235—41no—
207#235—41no—
208C2×D12—21no—
209C2×C4×D3—41no—
210C2×C4×D3—82no—
211C2×Dic6—21no—
212C2×C3×C8—24no—
213C2×C3×C8—28no—
214C2×C3×C8C2×C4⋊C411yescharacteristic
215C2×C3×C8C4×D411yescharacteristic
216C2×C3×C8C22⋊Q811yescharacteristic
217C2×C3×C8#12311yescharacteristic
218C2×C3×C8C2×SD1611yescharacteristic
219C2×C3×C8C2×SD1611yescharacteristic
220C2×C3×C8C2×SD1611yescharacteristic
221C2×C3×C8#13511yescharacteristic
222C2×C3×C8AGL(1,ℤ/8)11yescharacteristic
223C2×C3×C8AGL(1,ℤ/8)11yescharacteristic
224C2×C3×C8AGL(1,ℤ/8)11yescharacteristic
225C2×C3×C8#13711yescharacteristic
226C3×C4⋊C4—42no—
227C3×C4⋊C4—42no—
228C3×C4⋊C4—84no—
229C3×C4⋊C4—84no—
230C3×C4⋊C4—81no—
231C3×C4⋊C4—81no—
232C3×C42—21no—
233C3×C42C4×D411yescharacteristic
234C3×C42C22⋊Q811yescharacteristic
235#217—41no—
236(C2×D3)⋊C4—82no—
237C4⋊Dic3—22no—
238Dic3⋊C4—82no—
239C4×Dic3—41no—
240C4×Dic3—82no—
241#208—22no—
242#208—41no—
243#208—41no—
244#208—41no—
245C2×C3⋊C8—41no—
246C2×C3⋊C8—41no—
247Dic12—21no—
248Dic12#10612yescomplemented
249D24—21no—
250D24#10612yescomplemented
251#204—22no—
252#204—22no—
253ΓL(1,𝔽25)—42no—
254ΓL(1,𝔽25)—42no—
255D3×C8—42no—
256D3×C8—42no—
257C3⋊C16—42no—
258C3⋊C16—81no—
259C2×#41—61no—
260#135—32no—
261#135—32no—
262#135—61no—
263#135—61no—
264C2×Q16—31no—
265C2×SD16—61no—
266C2×D8—31no—
267#131—62no—
268#131—62no—
269#131—121no—
270#131—121no—
271C2×M16—61no—
272C2×M16—61no—
273C2×M16—121no—
274C22×C8—22no—
275C22×C8—121no—
276C22×C8D3×D411yescharacteristic
277C22×C8D3×D411yescharacteristic
278C4×Q8—122no—
279C4×D4—122no—
280C4×D4—242no—
281#117—61no—
282#117—122no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—41no—
287C2×C4⋊C4—122no—
288C2×C42C2×#6711yescharacteristic
289M32—62no—
290M32—121no—
291M32—121no—
292M32—121no—
293C2×C16—121no—
294C2×C16—121no—
295#108—32no—
296#108—61no—
297#108—61no—
298#108—61no—
299#107—32no—
300#107—122no—
301#106—41no—
302#106—41no—
303#106—41no—
304#106—41no—
305#106—41no—
306#106—41no—
307#106—41no—
308#106—41no—
309#106—61no—
310#106—88no—
311#106—122no—
312#104—62no—
313#104—62no—
314#103—122no—
315#102—122no—
316#97—121no—
317C4×C8—22no—
318C4×C8—22no—
319C4×C8—121no—
320C4×C8C2×C4×D311yescharacteristic
321C4×C8#23511yescharacteristic
322C4×C8D3×Q811yescharacteristic
323C4×C8#23911yescharacteristic
324C23×C3#45311yescharacteristic
325C22×D3—41no—
326C2×C3×C4—24no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—82no—
330C2×C3×C4—162no—
331C2×C3×C4—162no—
332C2×C3×C4—161no—
333C2×C3×C4#38511yescharacteristic
334C2×C3×C4#44811yescharacteristic
335C2×C3×C4#44911yescharacteristic
336C2×C3×C4#45011yescharacteristic
337C2×C3×C4#45011yescharacteristic
338C2×C3×C4#45611yescharacteristic
339#67—42no—
340#67—42no—
341C2×Dic3—41no—
342C2×Dic3—42no—
343D12—22no—
344D12—41no—
345C4×D3—42no—
346C4×D3—42no—
347C4×D3—322no—
348Dic6—22no—
349Dic6—41no—
350C3×C8—28no—
351C3×C8C2×#10611yescharacteristic
352C3×C8#42811yescharacteristic
353C3×C8C4×SD1611yescharacteristic
354C3×C8#44211yescharacteristic
355C3×C8#47611yescharacteristic
356C3×C8#47811yescharacteristic
357C3×C8#48111yescharacteristic
358C3×C8#48311yescharacteristic
359C3⋊C8—42no—
360C3⋊C8—81no—
361#41—62no—
362#41—62no—
363#41—121no—
364#41—121no—
365C2×Q8—61no—
366C2×D4—61no—
367C2×D4—121no—
368C22×C4—21no—
369C22×C4—42no—
370C22×C4—81no—
371C22×C4—121no—
372C22×C4—242no—
373C22×C4#93711yescharacteristic
374Q16—32no—
375Q16—61no—
376SD16—62no—
377SD16—62no—
378D8—32no—
379D8—61no—
380M16—62no—
381M16—121no—
382M16—122no—
383M16—122no—
384M16—121no—
385M16—121no—
386C2×C8—24no—
387C2×C8—28no—
388C2×C8—121no—
389C2×C8—122no—
390C2×C8—122no—
391C2×C8—121no—
392C2×C8D3×C4⋊C411yescharacteristic
393C2×C8#89311yescharacteristic
394C2×C8#89411yescharacteristic
395C2×C8#89611yescharacteristic
396C2×C8D3×SD1611yescharacteristic
397C2×C8D3×SD1611yescharacteristic
398C2×C8D3×SD1611yescharacteristic
399C2×C8#91411yescharacteristic
400C2×C8#91411yescharacteristic
401C2×C8#91411yescharacteristic
402C2×C8#91511yescharacteristic
403C2×C8#91611yescharacteristic
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—62no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C4⋊C4—81no—
410C4⋊C4—81no—
411C4⋊C4—242no—
412C22⋊C4—121no—
413C22⋊C4—242no—
414C42—21no—
415C42—121no—
416C42—242no—
417C42C4×#6711yescharacteristic
418C42#94611yescharacteristic
419C16—122no—
420C16—241no—
421C22×C3—22no—
422C22×C3#162211yescharacteristic
423C22×C3#162911yescharacteristic
424C22×C3#185911yescharacteristic
425C2×D3—42no—
426C2×D3—82no—
427C3×C4—22no—
428C3×C4—22no—
429C3×C4—162no—
430C3×C4—321no—
431C3×C4#165111yescharacteristic
432C3×C4#165411yescharacteristic
433C3×C4#185311yescharacteristic
434C3×C4#186411yescharacteristic
435Dic3—42no—
436Dic3—82no—
437C23—121no—
438C23#507711yescharacteristic
439Q8—62no—
440Q8—121no—
441D4—62no—
442D4—122no—
443D4—122no—
444D4—121no—
445C2×C4—24no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—82no—
449C2×C4—121no—
450C2×C4—122no—
451C2×C4—122no—
452C2×C4—122no—
453C2×C4—162no—
454C2×C4—162no—
455C2×C4—161no—
456C2×C4—962no—
457C2×C4#498911yescharacteristic
458C2×C4#509511yescharacteristic
459C2×C4#516911yescharacteristic
460C2×C4#516911yescharacteristic
461C2×C4#517011yescharacteristic
462C2×C4#517211yescharacteristic
463C8—28no—
464C8—122no—
465C8—241no—
466C8#485211yescharacteristic
467C8D3×#10611yescharacteristic
468C8#486111yescharacteristic
469C8#486211yescharacteristic
470C8#486311yescharacteristic
471C8#486911yescharacteristic
472C8#487011yescharacteristic
473C8#487111yescharacteristic
474C2×C3—21no—
475C2×C3—21no—
476C2×C3#888611yescharacteristic
477C2×C3#1236412yes—
478D3—82no—
479C22—22no—
480C22—122no—
481C22—242no—
482C22#6959211yescharacteristic
483C22#7126411yescharacteristic
484C22#7126611yescharacteristic
485C4—22no—
486C4—22no—
487C4—122no—
488C4—162no—
489C4—242no—
490C4—321no—
491C4#7057911yescharacteristic
492C4#7058411yescharacteristic
493C4#7134211yescharacteristic
494C4#7134311yescharacteristic
495C3#13636111yescharacteristic, complemented, Sylow
496C2—21no—
497C2—21no—
498C2—242no—
499C2#1068937211yescharacteristic
500C2#1070894412yes—
subgroupG/H
centerC22#71266
commutatorC3×C4×C8C22×C4
FrattiniC2×C4×C8C22×D3
FittingC3×#8813C2
socleC22×C3#1629
prime radicalC2×#904D4
cosocleC2×C3×C4×C8C23
hypercenter#8813D3
hypocenterC3#136361
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ #8813

Complements

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

NH ≅ G/Ntype
2. #10713326C2semidirect
5. #10682358C2semidirect
6. #10682126C2semidirect
7. C3×#8813C2semidirect
26. #67441C4semidirect
31. #8813D3semidirect
66. C3×C82C2×C4semidirect
74. #4463C2×C4semidirect
198. C82C4×D3semidirect
248. Dic12#106semidirect
250. D24#106semidirect
495. C3#136361semidirect

Maximal subgroups

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

subgroupindex
#108136602
#107133262
#107112852
C2×#674412
#106823582
#106821262
C3×#88132
#1363613