Finite Group Databasebeta

← #29663847

Subgroups — #29663847

Order 15616.

656 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456378C211yesmaximal, characteristic
2#21455626C211yesmaximal, characteristic, complemented
3#21455293C211yesmaximal, characteristic
4C2×C8×Dic122C211yesmaximal, characteristic, complemented
5#21445250C212yesmaximal, complemented
6#21445236C212yesmaximal, complemented
7#21445202C212yesmaximal, complemented
8#21445168C212yesmaximal, complemented
9#21444510C211yesmaximal, characteristic, complemented
10#21444497C211yesmaximal, characteristic
11C61×#1566C211yesmaximal, characteristic, complemented
12C2×#14519810—21no—
13#16840319C2211yescharacteristic
14#16840312C2211yescharacteristic
15C2×C4×Dic122C2211yescharacteristic
16#16840160C2211yescharacteristic
17#16840135C2211yescharacteristic
18#16840113C2211yescharacteristic
19#16840109C2211yescharacteristic
20#16840070—22no—
21#16840058C2212yes—
22#16840048C2212yes—
23#16839991—21no—
24#16839991—21no—
25#16839985—22no—
26#16839973C2212yes—
27#16839963C2212yes—
28C8×Dic122—21no—
29C8×Dic122—21no—
30C8×Dic122C2212yes—
31C8×Dic122C2212yes—
32C4×#11758190—21no—
33C4×#11758190—21no—
34#16839804—21no—
35C2×C4⋊#92418C2211yescharacteristic
36C2×#14519706—21no—
37#16839772—21no—
38#16839764—21no—
39C2×C8×Dic61—21no—
40#16839510C2212yes—
41#16839509C2212yes—
42#16839507C2212yes—
43#16839505C2212yes—
44C61×#423C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C2×C4×C8×C61C2211yescharacteristic
47C61×#334C2212yes—
48C61×#334C2212yes—
49C2×#11758222—21no—
50C22×Dic122D411yescharacteristic
51#14519810—44no—
52C2×Dic244—22no—
53C2×#11758190—22no—
54C2×#11758190—21no—
55C2×#11758190—21no—
56C22×#92418—21no—
57#14519777C2311yescharacteristic
58#14519777C2311yescharacteristic
59#14519777C2311yescharacteristic
60#14519774C2311yescharacteristic
61#14519769—21no—
62#14519765—21no—
63#14519756—21no—
64#14519756—21no—
65#14519751—21no—
66#14519751—21no—
67C4×Dic122—21no—
68C4×Dic122—21no—
69C4×Dic122—21no—
70C4×Dic122—21no—
71C4×Dic122D412yes—
72C4×Dic122C2312yes—
73C4×D244—21no—
74C4×D244—21no—
75#14519736—21no—
76C2×C4⋊Dic61D411yescharacteristic
77C2×#11758196—21no—
78C2×C4×Dic61—21no—
79#14519711—22no—
80#14519709—21no—
81#14519709—21no—
82#14519709—22no—
83C4⋊#92418—21no—
84C4⋊#92418C2312yes—
85#14519706—22no—
86#14519706—22no—
87#14519702—22no—
88#14519702—22no—
89#14519702—21no—
90#14519702—21no—
91#14519702—22no—
92#14519699—21no—
93#14519699—21no—
94#14519699—22no—
95C8×Dic61—22no—
96C8×Dic61—22no—
97C2×M16×C61—21no—
98C2×M16×C61—21no—
99C22×C8×C61D411yescharacteristic
100C22×C8×C61D411yescharacteristic
101C2×C42×C61C2311yescharacteristic
102C4⋊C8×C61—21no—
103C4⋊C8×C61—21no—
104C4⋊C8×C61C2312yes—
105C4⋊C8×C61C2312yes—
106C4×C8×C61D411yescharacteristic
107C4×C8×C61D411yescharacteristic
108C4×C8×C61C2311yescharacteristic
109C4×C8×C61C2311yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61#4111yescharacteristic
113C2×C22⋊D61—41no—
114C22×Dic61—21no—
115C22×Dic61—21no—
116#11758222—42no—
117#11758222—42no—
118C2×D244—21no—
119C2×D244—21no—
120C2×D244—21no—
121C2×C4×D61—41no—
122C2×Dic122—21no—
123C2×Dic122—22no—
124C2×Dic122—21no—
125C2×Dic122—21no—
126C2×Dic122—21no—
127C2×Dic122SD1614yescomplemented
128C2×Dic122C2×D412yes—
129M16×C61—42no—
130M16×C61—42no—
131C2×C8×C61—22no—
132C2×C8×C61—22no—
133C2×C8×C61—22no—
134C2×C8×C61—41no—
135C2×C8×C61C2×D411yescharacteristic
136C2×C8×C61C2×D411yescharacteristic
137C2×C8×C61C2×D411yescharacteristic
138C2×C8×C61C2×D411yescharacteristic
139C42×C61C2×D411yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61#4111yescharacteristic
142C42×C61C2411yescharacteristic
143C22⋊Dic61—41no—
144#11758198—41no—
145#11758198—41no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61C2×D412yes—
154#11758196—22no—
155#11758196—22no—
156#11758196—41no—
157#11758196—41no—
158C4×Dic61—22no—
159C4×Dic61—22no—
160C4×Dic61—41no—
161C2×#92418—22no—
162C2×#92418—42no—
163Dic244—44no—
164#11758190—42no—
165#11758190—44no—
166#11758190—42no—
167C23×C61#12111yescharacteristic
168C22×D61—41no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—41no—
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61#12111yescharacteristic
175C2×C4×C61C2×SD1612yes—
176C2×C4×C61#13512yes—
177C2×C4×C61C22×D411yescharacteristic
178C2×C4×C61C22×D411yescharacteristic
179C2×C4×C61C2×#4111yescharacteristic
180C22⋊D61—82no—
181C2×Dic61—22no—
182C2×Dic61—24no—
183C2×Dic61—22no—
184C2×Dic61—42no—
185C2×Dic61—41no—
186C2×Dic61—41no—
187C2×Dic61—41no—
188D244—41no—
189D244—41no—
190D244—41no—
191D244—41no—
192D244—81no—
193C4×D61—82no—
194Dic122—22no—
195Dic122—24no—
196Dic122—41no—
197Dic122—41no—
198Dic122—41no—
199Dic122—41no—
200Dic122—81no—
201Dic122C2×SD1614yes—
202C8×C61—21no—
203C8×C61—21no—
204C8×C61—21no—
205C8×C61—42no—
206C8×C61—42no—
207C8×C61#13712yescomplemented
208#92418—42no—
209#19260—611nomaximal, Sylow
210C22×C61—21no—
211C22×C61—21no—
212C22×C61#46512yes—
213C22×C61C2×#12111yescharacteristic
214C2×D61—81no—
215C2×D61—81no—
216C2×D61—81no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61#46512yes—
221C4×C61C22×SD1611yescharacteristic
222C4×C61C2×#13511yescharacteristic
223C4×C61C2×#13711yescharacteristic
224C4×C61#57511yescharacteristic
225Dic61—24no—
226Dic61—42no—
227Dic61—81no—
228Dic61—81no—
229Dic61—81no—
230#3167—611no—
231#3078—611no—
232#3037—611no—
233C2×C8×Q8—611no—
234#1680—612no—
235#1678—612no—
236#1672—612no—
237#1666—612no—
238#1566D6111yescharacteristic, complemented
239#1550—611no—
240#1547—611no—
241C2×C61—22no—
242C2×C61#303711yescharacteristic
243C2×C61C2×#46511yescharacteristic
244C2×C61#306711yescharacteristic
245D61—161no—
246C2×#137—1221no—
247#533—611no—
248#532—611no—
249C2×C4×Q8—611no—
250#498—611no—
251#492—611no—
252#487—611no—
253#486—611no—
254#479—1222no—
255#477—612no—
256#475—612no—
257#465—1221no—
258#465—1221no—
259#464—1222no—
260#462—612no—
261#460—612no—
262C8×Q8—612no—
263C8×Q8—612no—
264C8×Q8—1221no—
265C8×Q8—1221no—
266C4×SD16—1221no—
267C4×SD16—1221no—
268#428—1221no—
269#423C2×D6111yescharacteristic
270#423C2×D6111yescharacteristic
271C2×C4⋊C8—611no—
272C2×C4⋊C8—1221no—
273#419—1221no—
274#417—1221no—
275C2×C4×C8—1221no—
276C2×C4×C8C2×D6111yescharacteristic
277#334C2×D6112yes—
278#334C2×D6112yes—
279#333—612no—
280#332—612no—
281#332—612no—
282#331—612no—
283C61#1926011yescharacteristic, complemented, Sylow
284C2×#41—1221no—
285C22×Q8—611no—
286#137—2444no—
287C2×Q16—1222no—
288C2×SD16—1222no—
289C2×SD16—1221no—
290C2×SD16—1221no—
291C2×M16—21no—
292C2×M16—21no—
293C22×C8—1221no—
294C22×C8D24411yescharacteristic
295C22×C8D24411yescharacteristic
296C4⋊Q8—611no—
297C4⋊Q8—611no—
298C4⋊Q8—611no—
299#127—611no—
300#125—1221no—
301#124—1221no—
302C22⋊Q8—1221no—
303C22⋊Q8—1221no—
304#121—1221no—
305#121—1221no—
306C4×Q8—612no—
307C4×Q8—612no—
308C4×Q8—1221no—
309C4×Q8—1221no—
310C4×Q8—1221no—
311C4×Q8—1221no—
312C4×D4—1221no—
313C4×D4—1221no—
314#117—1221no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—1221no—
317C2×C42—1221no—
318C2×C42C22×D6111yescharacteristic
319#107—1222no—
320#106—1222no—
321#106—1221no—
322#106—1221no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—612no—
326C4⋊C8—1222no—
327C4⋊C8—1222no—
328C4⋊C8—1221no—
329C4⋊C8C22×D6112yes—
330C4⋊C8C22×D6112yes—
331#103—1222no—
332#103—1222no—
333#103—1222no—
334#103—1221no—
335#103—1221no—
336#102—1222no—
337#102—1221no—
338#102—1221no—
339C4×C8—1222no—
340C4×C8—1222no—
341C4×C8D24411yescharacteristic
342C4×C8D24411yescharacteristic
343C4×C8C22×D6111yescharacteristic
344C4×C8C22×D6111yescharacteristic
345#41—2442no—
346#41—2442no—
347C2×Q8—612no—
348C2×Q8—614no—
349C2×Q8—1221no—
350C2×Q8—1222no—
351C2×Q8—1221no—
352C2×Q8—1221no—
353C2×Q8—1221no—
354C2×D4—1221no—
355C2×D4—1221no—
356C2×D4—1221no—
357C2×D4—2441no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4—2441no—
361C22×C4C2×D24411yescharacteristic
362C22×C4D4×D6111yescharacteristic
363C22×C4#1175822611yescharacteristic
364Q16—2444no—
365SD16—2442no—
366SD16—2444no—
367SD16—2442no—
368M16—42no—
369M16—42no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—41no—
374C2×C8—1222no—
375C2×C8—2442no—
376C2×C8C2×D24411yescharacteristic
377C2×C8C2×D24411yescharacteristic
378C2×C8C2×D24411yescharacteristic
379C2×C8C2×D24411yescharacteristic
380C4⋊C4—612no—
381C4⋊C4—1222no—
382C4⋊C4—1221no—
383C4⋊C4—1222no—
384C4⋊C4—1221no—
385C4⋊C4—1221no—
386C4⋊C4—1221no—
387C4⋊C4—1221no—
388C4⋊C4—1221no—
389C4⋊C4—1221no—
390C4⋊C4—2441no—
391C4⋊C4—2441no—
392C22⋊C4—2441no—
393C22⋊C4—2441no—
394C22⋊C4—2441no—
395C42—1222no—
396C42—1222no—
397C42—2441no—
398C42C2×D24411yescharacteristic
399C42D4×D6111yescharacteristic
400C42#1175822611yescharacteristic
401C42C23×D6111yescharacteristic
402C23—2441no—
403C23#1451975211yescharacteristic
404Q8—614no—
405Q8—1224no—
406Q8—1222no—
407Q8—4881no—
408Q8—2441no—
409Q8—2441no—
410Q8—2441no—
411Q8—2441no—
412D4—4881no—
413D4—4882no—
414D4—2441no—
415D4—2441no—
416D4—2441no—
417D4—2441no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—41no—
421C2×C4—1224no—
422C2×C4—1222no—
423C2×C4—1222no—
424C2×C4—4882no—
425C2×C4—2441no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4—2442no—
429C2×C4#1451975211yescharacteristic
430C2×C4#1451975211yescharacteristic
431C2×C4#1451975211yescharacteristic
432C2×C4SD16×D6112yes—
433C2×C4#1451979812yes—
434C2×C4C2×D4×D6111yescharacteristic
435C2×C4C22×D24411yescharacteristic
436C2×C4C2×#1175822611yescharacteristic
437C8—21no—
438C8—21no—
439C8—21no—
440C8—42no—
441C8—42no—
442C8—2442no—
443C8#1451981012yescomplemented
444C22—21no—
445C22—21no—
446C22—4881no—
447C22—4881no—
448C22—4881no—
449C22#1683999212yes—
450C22C2×#1451975211yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—1224no—
455C4—4881no—
456C4—4881no—
457C4—4881no—
458C4—2442no—
459C4#1683999212yes—
460C4C2×SD16×D6111yescharacteristic
461C4C2×#1451979811yescharacteristic
462C4C2×#1451981011yescharacteristic
463C4#1684056211yescharacteristic
464C2—22no—
465C2—9761no—
466C2#2145529311yescharacteristic
467C2C2×#1683999211yescharacteristic
468C2#2145554011yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1566C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#1566D61
hypocenterC61#19260
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1566

Complements

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

NH ≅ G/Ntype
2. #21455626C2semidirect
4. C2×C8×Dic122C2semidirect
5. #21445250C2semidirect
6. #21445236C2semidirect
7. #21445202C2semidirect
8. #21445168C2semidirect
9. #21444510C2semidirect
11. C61×#1566C2semidirect
127. C2×Dic122SD16semidirect
207. C8×C61#137semidirect
238. #1566D61semidirect
283. C61#19260semidirect
443. C8#14519810semidirect

Maximal subgroups

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

subgroupindex
#214563782
#214556262
#214552932
C2×C8×Dic1222
#214452502
#214452362
#214452022
#214451682
#214445102
#214444972
C61×#15662
#1926061