Finite Group Databasebeta

← #72225901

Subgroups — #72225901

Order 1536.

868 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10713448C211yesmaximal, characteristic, complemented
2#10713436C211yesmaximal, characteristic, complemented
3#10711613C211yesmaximal, characteristic, complemented
4#10704187C211yesmaximal, characteristic, complemented
5#10695160C211yesmaximal, characteristic, complemented
6#10695026C211yesmaximal, characteristic, complemented
7C3×#33717C211yesmaximal, characteristic
8#141758—31nomaximal, Sylow
9C2×C3×#509—22no—
10C3×#3147C2211yescharacteristic
11C2×C3×#493C2211yescharacteristic
12C3×#2250—22no—
13C3×#2248C412yescomplemented
14C3×#2246C412yescomplemented
15C3×C4×M32C2211yescharacteristic
16C2×#4566—22no—
17C2×#4563—22no—
18#71802—22no—
19#71764—22no—
20C2×#4513—22no—
21#69792C2211yescharacteristic
22#69764C2211yescharacteristic
23C4×#812C2211yescharacteristic
24C2×#4455C2211yescharacteristic
25#67511—22no—
26#33717D311yescharacteristic
27#13213—31no—
28#13210—31no—
29#12903—31no—
30#11404—31no—
31#9862—31no—
32#9837—31no—
33C2×C3×#135—21no—
34C2×C3×#135—21no—
35C22×C3×D8—21no—
36C22×C3×D8—21no—
37C3×#532—21no—
38C2×C3×#127—21no—
39C3×#509—44no—
40C3×#509—44no—
41C2×C3×SD32—24no—
42C2×C3×D16—24no—
43C2×C3×M32D412yes—
44C3×#495—21no—
45C3×#495—21no—
46C3×#494C2×C411yescharacteristic
47C3×#493—21no—
48C3×#493—21no—
49C3×#493—21no—
50C3×#493C2×C411yescharacteristic
51C3×#493C2×C412yes—
52C3×#492—21no—
53C2×C3×C4×C8C2311yescharacteristic
54C2×#832—44no—
55C2×#819—21no—
56C2×#819—21no—
57C2×#817—21no—
58C2×#817—21no—
59C2×#812D412yes—
60#4935—21no—
61#4924—21no—
62C2×C4×C3⋊C8D412yes—
63C3×#346D412yes—
64C3×C4×C16C2×C412yes—
65#4566—44no—
66#4563—44no—
67D4⋊(C3⋊C8)—44no—
68#4513—44no—
69#4461—21no—
70C4×C3⋊C16—21no—
71#4455—21no—
72#4455—21no—
73C2×#509—22no—
74#3147C2×D311yescharacteristic
75C2×#493C2×D311yescharacteristic
76#2250—22no—
77#2248Dic312yescomplemented
78#2246Dic312yescomplemented
79C2×#365—62no—
80C2×#361—62no—
81C2×#360—62no—
82C4×M32—31no—
83C4×M32C2×D311yescharacteristic
84#1957—62no—
85#1848—31no—
86#1844—31no—
87#1492—62no—
88C2×#322—31no—
89#1313—62no—
90C2×C3×#41—41no—
91C22×C3×D4—41no—
92C22×C3×D4—42no—
93C3×#135—42no—
94C3×#135—42no—
95C3×#135—42no—
96C3×#135—42no—
97C2×C3×Q16—21no—
98C2×C3×Q16—21no—
99C2×C3×SD16—41no—
100C2×C3×SD16—41no—
101C2×C3×D8—21no—
102C2×C3×D8—22no—
103C2×C3×D8—22no—
104C2×C3×D8—21no—
105C2×C3×D8—41no—
106C2×C3×D8—41no—
107C2×C3×D8—42no—
108C2×C3×D8—42no—
109C2×C3×D8—42no—
110C2×C3×D8—42no—
111C22×C3×C8C2×D411yescharacteristic
112C22×C3×C8C2×D411yescharacteristic
113C3×C4⋊Q8—21no—
114C3×#127—22no—
115C3×#127—21no—
116C3×#127—41no—
117C3×#127—42no—
118C3×#124—41no—
119C3×#121—42no—
120C3×C4×D4—42no—
121C2×C3×C42C2×D411yescharacteristic
122C2×C4⋊Dic3—41no—
123C2×C4⋊Dic3—42no—
124C2×#208—41no—
125C22×C3⋊C8—22no—
126C3×SD32—48no—
127C3×D16—48no—
128C3×M32D814yes—
129C3×M32SD1614yes—
130C2×C3×C16C22⋊C414yes—
131C3×C4×C8C22×C411yescharacteristic
132C3×C4×C8C2×D411yescharacteristic
133C3×C4×C8#4111yescharacteristic
134C3×C4×C8#4111yescharacteristic
135#832—88no—
136#819—42no—
137#819—42no—
138#817—42no—
139#817—42no—
140#812—22no—
141#812—22no—
142C2×C3⋊C16—22no—
143C4⋊(C3⋊C8)—42no—
144C4×C3⋊C8—22no—
145C4×C3⋊C8—22no—
146C2×#135—21no—
147C2×#135—21no—
148C22×D8—21no—
149C22×D8—21no—
150#532—21no—
151C2×#127—21no—
152#509—44no—
153#509—44no—
154C2×SD32—24no—
155C2×D16—24no—
156C2×M32—32no—
157C2×M32#6712yes—
158#495—21no—
159#495—21no—
160#494C2×Dic311yescharacteristic
161#493—21no—
162#493—21no—
163#493—21no—
164#493C2×Dic312yes—
165#493C2×Dic311yescharacteristic
166#492—21no—
167C2×#108—61no—
168C2×#108—61no—
169C2×#106—61no—
170C2×#106—61no—
171#423—61no—
172C2×#102—124no—
173C2×C4×C8—32no—
174C2×C4×C8C22×D311yescharacteristic
175#384—61no—
176#365—124no—
177#361—124no—
178#360—124no—
179#346—61no—
180#346#6712yes—
181C4×C16—61no—
182C4×C16C2×Dic312yes—
183D4⋊C8—124no—
184#322—61no—
185#322—61no—
186C24×C3—81no—
187C3×#41—81no—
188C3×#41—81no—
189C3×#41—81no—
190C3×#41—81no—
191C2×C3×Q8—41no—
192C2×C3×D4—41no—
193C2×C3×D4—42no—
194C2×C3×D4—42no—
195C2×C3×D4—44no—
196C2×C3×D4—84no—
197C2×C3×D4—84no—
198C2×C3×D4—81no—
199C2×C3×D4—82no—
200C2×C3×D4—81no—
201C2×C3×D4—81no—
202C2×C3×D4—81no—
203C2×C3×D4—81no—
204C2×C3×D4—81no—
205C22×C3×C4—81no—
206C22×C3×C4#12711yescharacteristic
207C22×C3×C4AGL(1,ℤ/8)12yes—
208C22×Dic3—81no—
209C3×Q16—42no—
210C3×Q16—42no—
211C3×SD16—82no—
212C3×SD16—82no—
213C3×D8—42no—
214C3×D8—44no—
215C3×D8—44no—
216C3×D8—42no—
217C3×D8—82no—
218C3×D8—82no—
219C2×C3×C8C2×C22⋊C411yescharacteristic
220C2×C3×C8C4×D411yescharacteristic
221C2×C3×C8#12111yescharacteristic
222C2×C3×C8#12411yescharacteristic
223C2×C3×C8C2×D812yes—
224C2×C3×C8C2×SD1612yes—
225C2×C3×C8AGL(1,ℤ/8)12yes—
226C2×C3×C8AGL(1,ℤ/8)12yes—
227C3×C4⋊C4—42no—
228C3×C22⋊C4—82no—
229C3×C42C4×D411yescharacteristic
230C3×C42#12111yescharacteristic
231C3×C42AGL(1,ℤ/8)12yes—
232C4⋊Dic3—84no—
233C4⋊Dic3—81no—
234C4⋊Dic3—81no—
235#208—81no—
236#208—81no—
237C2×C3⋊C8—44no—
238C2×C3⋊C8—42no—
239C2×C3⋊C8—81no—
240C3×C16#10218yescomplemented
241C3⋊C16—42no—
242C2×#41—41no—
243C22×D4—42no—
244C22×D4—41no—
245#135—42no—
246#135—42no—
247#135—42no—
248#135—42no—
249C2×Q16—21no—
250C2×Q16—21no—
251C2×SD16—41no—
252C2×SD16—41no—
253C2×D8—22no—
254C2×D8—21no—
255C2×D8—22no—
256C2×D8—21no—
257C2×D8—42no—
258C2×D8—42no—
259C2×D8—42no—
260C2×D8—42no—
261C2×D8—41no—
262C2×D8—41no—
263C2×M16—121no—
264C22×C8—62no—
265C22×C8D3×D411yescharacteristic
266C22×C8C2×#6711yescharacteristic
267C4⋊Q8—21no—
268#127—22no—
269#127—21no—
270#127—42no—
271#127—41no—
272#124—41no—
273#121—42no—
274C4×D4—42no—
275C2×C4⋊C4—122no—
276C2×C4⋊C4—121no—
277C2×C42D3×D411yescharacteristic
278SD32—48no—
279D16—48no—
280M32—62no—
281M32—62no—
282M32#21314yes—
283M32#21414yes—
284C2×C16—62no—
285C2×C16#21714yes—
286#108—122no—
287#108—122no—
288#106—122no—
289#106—122no—
290C4⋊C8—122no—
291#102—248no—
292C4×C8—62no—
293C4×C8—62no—
294C4×C8#23711yescharacteristic
295C4×C8#23711yescharacteristic
296C4×C8C22×Dic311yescharacteristic
297C4×C8C2×#6711yescharacteristic
298C23×C3—81no—
299C23×C3—82no—
300C23×C3—162no—
301C23×C3—162no—
302C23×C3—161no—
303C23×C3—161no—
304C23×C3#49611yescharacteristic
305C3×Q8—81no—
306C3×Q8—81no—
307C3×D4—84no—
308C3×D4—88no—
309C3×D4—82no—
310C3×D4—81no—
311C3×D4—82no—
312C3×D4—81no—
313C3×D4—164no—
314C3×D4—161no—
315C3×D4—161no—
316C3×D4—161no—
317C3×D4—161no—
318C2×C3×C4—24no—
319C2×C3×C4—81no—
320C2×C3×C4—161no—
321C2×C3×C4—161no—
322C2×C3×C4#39011yescharacteristic
323C2×C3×C4#44212yes—
324C2×C3×C4#46912yes—
325C2×C3×C4#49211yescharacteristic
326C2×C3×C4#49311yescharacteristic
327C2×C3×C4#49611yescharacteristic
328C2×C3×C4#49611yescharacteristic
329C2×C3×C4#49611yescharacteristic
330C2×Dic3—162no—
331C2×Dic3—161no—
332C3×C8C2×#10212yes—
333C3×C8#44212yes—
334C3×C8#46912yes—
335C3×C8#48612yes—
336C3⋊C8—82no—
337C3⋊C8—161no—
338C24—81no—
339#41—81no—
340#41—81no—
341#41—81no—
342#41—81no—
343C2×Q8—41no—
344C2×D4—42no—
345C2×D4—44no—
346C2×D4—41no—
347C2×D4—42no—
348C2×D4—84no—
349C2×D4—84no—
350C2×D4—81no—
351C2×D4—81no—
352C2×D4—81no—
353C2×D4—81no—
354C2×D4—82no—
355C2×D4—81no—
356C2×D4—81no—
357C22×C4—81no—
358C22×C4—241no—
359C22×C4#91112yes—
360C22×C4#94011yescharacteristic
361Q16—42no—
362Q16—42no—
363SD16—82no—
364SD16—82no—
365D8—44no—
366D8—42no—
367D8—42no—
368D8—44no—
369D8—82no—
370D8—82no—
371M16—241no—
372M16—241no—
373C2×C8—124no—
374C2×C8—122no—
375C2×C8—241no—
376C2×C8#91112yes—
377C2×C8#91112yes—
378C2×C8C2×#21312yes—
379C2×C8C2×#21412yes—
380C2×C8D4×Dic311yescharacteristic
381C2×C8#93611yescharacteristic
382C2×C8#93811yescharacteristic
383C2×C8C2×#21711yescharacteristic
384C4⋊C4—42no—
385C4⋊C4—244no—
386C4⋊C4—241no—
387C4⋊C4—241no—
388C22⋊C4—82no—
389C42#91112yes—
390C42D4×Dic311yescharacteristic
391C42#93811yescharacteristic
392C16—122no—
393C16#83218yescomplemented
394C22×C3—22no—
395C22×C3—161no—
396C22×C3—162no—
397C22×C3—164no—
398C22×C3—162no—
399C22×C3—322no—
400C22×C3—321no—
401C22×C3—321no—
402C22×C3#171511yescharacteristic
403C22×C3#172111yescharacteristic
404C22×C3#197211yescharacteristic
405C3×C4—22no—
406C3×C4—161no—
407C3×C4#171511yescharacteristic
408C3×C4#172111yescharacteristic
409C3×C4#195711yescharacteristic
410C3×C4#197211yescharacteristic
411C3×C4AGL(1,ℤ/16)14yes—
412Dic3—321no—
413C23—81no—
414C23—82no—
415C23—162no—
416C23—162no—
417C23—161no—
418C23—161no—
419C23#515411yescharacteristic
420Q8—81no—
421Q8—81no—
422D4—82no—
423D4—81no—
424D4—88no—
425D4—84no—
426D4—82no—
427D4—81no—
428D4—164no—
429D4—161no—
430D4—161no—
431D4—161no—
432D4—161no—
433C2×C4—24no—
434C2×C4—81no—
435C2×C4—161no—
436C2×C4—161no—
437C2×C4—482no—
438C2×C4—481no—
439C2×C4#507311yescharacteristic
440C2×C4#507611yescharacteristic
441C2×C4#515212yes—
442C2×C4#515411yescharacteristic
443C2×C4#515411yescharacteristic
444C2×C4#515411yescharacteristic
445C2×C4#515912yes—
446C2×C4#522111yescharacteristic
447C8—242no—
448C8—481no—
449C8#507012yes—
450C8#515212yes—
451C8#515912yes—
452C8C2×#83212yes—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—321no—
456C2×C3—322no—
457C2×C3#983711yescharacteristic
458C2×C3#1324912yes—
459C22—22no—
460C22—164no—
461C22—162no—
462C22—161no—
463C22—162no—
464C22—322no—
465C22—321no—
466C22—321no—
467C22#7127711yescharacteristic
468C22#7128011yescharacteristic
469C22#7203911yescharacteristic
470C4—22no—
471C4—161no—
472C4—961no—
473C4#7127711yescharacteristic
474C4#7128011yescharacteristic
475C4#7151514yes—
476C4#7176411yescharacteristic
477C4#7203911yescharacteristic
478C3#14175811yescharacteristic, complemented, Sylow
479C2—21no—
480C2—21no—
481C2—322no—
482C2—321no—
483C2#1069502611yescharacteristic
484C2#1071363912yes—
subgroupG/H
centerC22#72039
commutatorC3×C4×C8C22×C4
FrattiniC2×C4×C8C22×D3
Fitting, prime radicalC3×#33717C2
socleC22×C3#1972
cosocleC2×C3×C4×C8C23
hypercenter#33717D3
hypocenterC3#141758
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #33717

Complements

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

NH ≅ G/Ntype
1. #10713448C2semidirect
2. #10713436C2semidirect
3. #10711613C2semidirect
4. #10704187C2semidirect
5. #10695160C2semidirect
6. #10695026C2semidirect
13. C3×#2248C4semidirect
14. C3×#2246C4semidirect
77. #2248Dic3semidirect
78. #2246Dic3semidirect
240. C3×C16#102semidirect
393. C16#832semidirect
478. C3#141758semidirect

Maximal subgroups

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

subgroupindex
#107134482
#107134362
#107116132
#107041872
#106951602
#106950262
C3×#337172
#1417583