Finite Group Databasebeta

← #29575261

Subgroups — #29575261

Order 15616.

690 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840109C211yesmaximal, characteristic
2#21448883C211yesmaximal, characteristic
3#21446771C211yesmaximal, characteristic, complemented
4#21446741C211yesmaximal, characteristic, complemented
5C2×#16839505C211yesmaximal, characteristic
6#21443659C211yesmaximal, characteristic, complemented
7C61×#1279C211yesmaximal, characteristic, complemented
8C2×#14519777C2211yescharacteristic
9C2×#14519774C2211yescharacteristic
10C2×C4×D244—21no—
11C2×#14519739—21no—
12#16840109—22no—
13#16840109C414yescomplemented
14C2×C4⋊#92418C2211yescharacteristic
15C2×#14519702—21no—
16C2×#14519702—21no—
17C2×#14519699—22no—
18#16839664—21no—
19#16839663—21no—
20#16839640—22no—
21C4×C4⋊Dic61C2211yescharacteristic
22#16839524—21no—
23#16839524—21no—
24#16839505—22no—
25C2×C4⋊C8×C61C2211yescharacteristic
26C2×C4×C8×C61C2211yescharacteristic
27C4×C4⋊C4×C61C2211yescharacteristic
28C22×Dic122—21no—
29C22×D244—21no—
30C22×D244—21no—
31C22×D244D412yes—
32C22×C4×D61—22no—
33C22×#92418—21no—
34#14519777—21no—
35#14519777—21no—
36#14519777—21no—
37#14519777C2×C412yescomplemented
38#14519774—21no—
39#14519774—21no—
40#14519774C2×C412yes—
41#14519774D412yes—
42C4×D244—42no—
43C4×D244—42no—
44#14519739—42no—
45#14519739—42no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61—21no—
48C2×C4⋊Dic61D411yescharacteristic
49C2×C4⋊Dic61D411yescharacteristic
50C2×#11758198—22no—
51C2×#11758198—22no—
52C2×#11758198—22no—
53C2×C4×Dic61—21no—
54C4⋊#92418—21no—
55C4⋊#92418—21no—
56#14519702—42no—
57#14519702—42no—
58#14519699—28no—
59#14519684—21no—
60C22×C8×C61—21no—
61C22×C8×C61—21no—
62C2×C4⋊C4×C61D411yescharacteristic
63C2×C4⋊C4×C61D411yescharacteristic
64C2×C42×C61—21no—
65C2×C42×C61C2311yescharacteristic
66C4⋊C8×C61—21no—
67C4⋊C8×C61—21no—
68C4×C8×C61—21no—
69C4×C8×C61C2×C412yescomplemented
70#95×C61—21no—
71C23×D61—22no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61C2×D411yescharacteristic
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—41no—
80C2×D244—22no—
81C2×D244—22no—
82C2×D244—22no—
83C2×D244—22no—
84C2×D244—22no—
85C2×D244—41no—
86C2×D244—41no—
87C2×D244—41no—
88C2×D244—41no—
89C2×D244—41no—
90C2×D244—41no—
91C2×D244—41no—
92C2×D244—41no—
93C2×D244—42no—
94C2×D244—42no—
95C2×D244C22⋊C414yes—
96C2×C4×D61—44no—
97C2×C4×D61—44no—
98C2×C4×D61—82no—
99C2×Dic122—22no—
100C2×Dic122—41no—
101C2×Dic122—41no—
102C2×Dic122—41no—
103C2×Dic122—41no—
104C2×Dic122—41no—
105C2×C8×C61—22no—
106C2×C8×C61—41no—
107C2×C8×C61—41no—
108C2×C8×C61—41no—
109C2×C8×C61—41no—
110C2×C8×C61—41no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61SD1612yes—
115C42×C61—41no—
116C42×C61—41no—
117C42×C61C22×C411yescharacteristic
118C42×C61C2×D411yescharacteristic
119C42×C61#4111yescharacteristic
120C42×C61#4111yescharacteristic
121#11758198—44no—
122#11758198—44no—
123#11758198—82no—
124C4⋊Dic61—21no—
125C4⋊Dic61—22no—
126C4⋊Dic61—22no—
127C4⋊Dic61—21no—
128C4⋊Dic61—21no—
129C4⋊Dic61—41no—
130C4⋊Dic61—41no—
131C4⋊Dic61D812yes—
132C4×Dic61—41no—
133C4×Dic61—41no—
134C2×#92418—41no—
135C2×#92418—41no—
136C2×#92418—41no—
137C23×C61#12011yescharacteristic
138C22×D61—24no—
139C22×D61—42no—
140C22×D61—42no—
141C22×D61—42no—
142C22×D61—42no—
143C22×D61—42no—
144C22×D61—42no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—22no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61C2×C22⋊C411yescharacteristic
154C2×C4×C61C4×D411yescharacteristic
155C2×C4×C61C4×D411yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61#12311yescharacteristic
159C2×C4×C61C2×D811yescharacteristic
160C2×C4×C61C2×SD1611yescharacteristic
161C2×C4×C61#13511yescharacteristic
162C2×C4×C61#13511yescharacteristic
163C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C61#13711yescharacteristic
167C2×Dic61—22no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—42no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—81no—
175C2×Dic61—81no—
176D244—44no—
177D244—42no—
178D244—42no—
179D244—42no—
180D244—42no—
181D244—42no—
182D244—42no—
183D244—42no—
184D244—42no—
185D244—82no—
186C4×D61—88no—
187Dic122—42no—
188Dic122—42no—
189Dic122—81no—
190Dic122—81no—
191C8×C61—42no—
192C8×C61—81no—
193#92418—81no—
194#9057—611nomaximal, Sylow
195C22×C61#38611yescharacteristic
196C22×C61#44711yescharacteristic
197C22×C61#44911yescharacteristic
198C22×C61#45011yescharacteristic
199C22×C61#45211yescharacteristic
200C22×C61#45511yescharacteristic
201C22×C61#45511yescharacteristic
202C2×D61—44no—
203C2×D61—44no—
204C2×D61—44no—
205C2×D61—82no—
206C2×D61—82no—
207C2×D61—82no—
208C2×D61—82no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—42no—
212C4×C61—81no—
213C4×C61C4×D811yescharacteristic
214C4×C61C4×SD1611yescharacteristic
215C4×C61#44011yescharacteristic
216C4×C61#44211yescharacteristic
217C4×C61#45911yescharacteristic
218C4×C61#46111yescharacteristic
219C4×C61#46311yescharacteristic
220C4×C61#46311yescharacteristic
221Dic61—42no—
222Dic61—82no—
223Dic61—81no—
224C2×#486—611no—
225#2366—611no—
226#1988—611no—
227#1981—611no—
228C2×#331—611no—
229#1279—611no—
230#1279D6111yescharacteristic, complemented
231C2×C61#163911yescharacteristic
232C2×C61#164011yescharacteristic
233C2×C61#164111yescharacteristic
234C2×C61#164211yescharacteristic
235C2×C61#187711yescharacteristic
236C2×C61#187911yescharacteristic
237C2×C61#188511yescharacteristic
238D61—84no—
239C2×C4⋊Q8—611no—
240C2×#127—611no—
241C2×C4×D4—1221no—
242C2×C4×D4—1221no—
243#486—614no—
244#486—1222no—
245C2×C4⋊C8—611no—
246C2×C4⋊C8C2×D6111yescharacteristic
247C2×#103—1221no—
248C2×#103—1221no—
249C2×#102—1222no—
250C2×C4×C8C2×D6111yescharacteristic
251#390—1221no—
252#390—1221no—
253#386—1222no—
254C4×C4⋊C4—611no—
255C4×C4⋊C4C2×D6111yescharacteristic
256#340—1221no—
257#340—1221no—
258#331—1222no—
259C61#905711yescharacteristic, complemented, Sylow
260C22×Q8—1221no—
261C22×D4—612no—
262C22×D4—1221no—
263C22×D4—1221no—
264C23×C4—1222no—
265C22×C8—21no—
266C22×C8—21no—
267C22×C8—1221no—
268C4⋊Q8—612no—
269C4⋊Q8—1221no—
270C4⋊Q8—1221no—
271C4⋊Q8—1221no—
272#127—612no—
273#127—612no—
274#127—1221no—
275#127—1221no—
276C4×D4—2442no—
277C4×D4—2442no—
278C4×D4—2442no—
279C4×D4—2442no—
280C2×C4⋊C4—611no—
281C2×C4⋊C4—611no—
282C2×C4⋊C4—1221no—
283C2×C4⋊C4—1221no—
284C2×C4⋊C4D24411yescharacteristic
285C2×C4⋊C4D24411yescharacteristic
286C2×C22⋊C4—1222no—
287C2×C22⋊C4—1222no—
288C2×C22⋊C4—1222no—
289C2×C42—21no—
290C2×C42—1221no—
291C2×C42C22×D6111yescharacteristic
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—1221no—
295C4⋊C8—1221no—
296#103—2442no—
297#103—2442no—
298#102—1228no—
299C4×C8—21no—
300C4×C8C4×D6112yescomplemented
301#95—21no—
302#95—1221no—
303C24—1222no—
304C2×Q8—1222no—
305C2×Q8—2441no—
306C2×Q8—2441no—
307C2×Q8—2441no—
308C2×Q8—2441no—
309C2×Q8—2441no—
310C2×D4—614no—
311C2×D4—1222no—
312C2×D4—1222no—
313C2×D4—1222no—
314C2×D4—1222no—
315C2×D4—1222no—
316C2×D4—2442no—
317C2×D4—2442no—
318C2×D4—2441no—
319C2×D4—2441no—
320C2×D4—2441no—
321C2×D4—2441no—
322C2×D4—2441no—
323C2×D4—2441no—
324C2×D4—2441no—
325C2×D4—2441no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—1221no—
329C22×C4—1221no—
330C22×C4—4882no—
331C22×C4—2444no—
332C22×C4—2444no—
333C22×C4—2441no—
334C22×C4C2×D24411yescharacteristic
335C22×C4D4×D6111yescharacteristic
336C22×C4D4×D6111yescharacteristic
337C2×C8—22no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—2441no—
344C2×C8—2441no—
345C2×C8—2441no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—612no—
350C4⋊C4—1221no—
351C4⋊C4—1222no—
352C4⋊C4—1222no—
353C4⋊C4—1221no—
354C4⋊C4—1221no—
355C4⋊C4—2441no—
356C4⋊C4—2441no—
357C4⋊C4#1175819012yes—
358C22⋊C4—4882no—
359C22⋊C4—2444no—
360C22⋊C4—2444no—
361C42—41no—
362C42—41no—
363C42—2441no—
364C42—2441no—
365C42C2×C4×D6111yescharacteristic
366C42#1175822211yescharacteristic
367C42D4×D6111yescharacteristic
368C42#1175822611yescharacteristic
369C23—1224no—
370C23—2442no—
371C23—2442no—
372C23—2442no—
373C23—2442no—
374C23—2442no—
375C23—2442no—
376C23#1451974711yescharacteristic
377Q8—4881no—
378Q8—4881no—
379Q8—2442no—
380Q8—2442no—
381D4—4882no—
382D4—2444no—
383D4—2442no—
384D4—2442no—
385D4—2442no—
386D4—2442no—
387D4—2442no—
388D4—2442no—
389D4—2442no—
390D4—2442no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—1222no—
400C2×C4—4888no—
401C2×C4—4881no—
402C2×C4—4881no—
403C2×C4—2441no—
404C2×C4—2442no—
405C2×C4—2441no—
406C2×C4—2441no—
407C2×C4—2441no—
408C2×C4—2441no—
409C2×C4C22⋊C4×D6111yescharacteristic
410C2×C4#1451973911yescharacteristic
411C2×C4C4×D24411yescharacteristic
412C2×C4#1451975011yescharacteristic
413C2×C4#1451975211yescharacteristic
414C2×C4#1451976211yescharacteristic
415C2×C4D8×D6111yescharacteristic
416C2×C4C2×#1175819011yescharacteristic
417C2×C4#1451979911yescharacteristic
418C2×C4#1451980011yescharacteristic
419C2×C4#1451980411yescharacteristic
420C2×C4#1451980411yescharacteristic
421C2×C4#1451980511yescharacteristic
422C2×C4#1451981011yescharacteristic
423C8—42no—
424C8—81no—
425C8—4881no—
426C22—4882no—
427C22—4882no—
428C22—4882no—
429C22—4882no—
430C22—2444no—
431C22—2444no—
432C22—2444no—
433C22#1683964011yescharacteristic
434C22#1683989711yescharacteristic
435C22#1683991011yescharacteristic
436C22#1683991511yescharacteristic
437C22#1683992611yescharacteristic
438C22#1683994111yescharacteristic
439C22#1683994111yescharacteristic
440C4—21no—
441C4—21no—
442C4—42no—
443C4—81no—
444C4—4882no—
445C4—4881no—
446C4—2442no—
447C4#1683984411yescharacteristic
448C4C4×#1175819011yescharacteristic
449C4#1683986411yescharacteristic
450C4#1683987311yescharacteristic
451C4#1683995611yescharacteristic
452C4#1683996811yescharacteristic
453C4#1683997811yescharacteristic
454C4#1683998011yescharacteristic
455C2—4884no—
456C2#2144500111yescharacteristic
457C2#2144500811yescharacteristic
458C2#2144501511yescharacteristic
459C2#2144502211yescharacteristic
460C2#2144615211yescharacteristic
461C2#2144616811yescharacteristic
462C2#2144620611yescharacteristic
subgroupG/H
centerC23#14519747
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1279C2
socleC23×C61#120
prime radicalC2×#14519774C22
cosocleC2×C42×C61C23
hypercenter#1279D61
hypocenterC61#9057
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
3. #21446771C2semidirect
4. #21446741C2semidirect
6. #21443659C2semidirect
7. C61×#1279C2semidirect
13. #16840109C4semidirect
37. #14519777C2×C4semidirect
69. C4×C8×C61C2×C4semidirect
230. #1279D61semidirect
259. C61#9057semidirect
300. C4×C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401092
#214488832
#214467712
#214467412
C2×#168395052
#214436592
C61×#12792
#905761