Finite Group Databasebeta

← #29835222

Subgroups — #29835222

Order 15616.

544 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456462C211yesmaximal, characteristic
2#21454687C211yesmaximal, characteristic, complemented
3#21448011C211yesmaximal, characteristic, complemented
4#21448007C211yesmaximal, characteristic, complemented
5#21447985C211yesmaximal, characteristic, complemented
6#21447981C211yesmaximal, characteristic, complemented
7#21447857C211yesmaximal, characteristic, complemented
8#21447857C211yesmaximal, characteristic, complemented
9#21447736C211yesmaximal, characteristic, complemented
10#21447686C211yesmaximal, characteristic, complemented
11#21447684C211yesmaximal, characteristic, complemented
12SD32×Dic61C211yesmaximal, characteristic, complemented
13#21447668C211yesmaximal, characteristic, complemented
14#21447623C211yesmaximal, characteristic, complemented
15C61×#2185C211yesmaximal, characteristic, complemented
16#16840365—21no—
17#16840240—21no—
18C2×#14519720C2211yescharacteristic
19C2×#14519720C2211yescharacteristic
20#16840163C2211yescharacteristic
21#16840137C2211yescharacteristic
22#16840125—21no—
23#16840013—21no—
24#16839998C2211yescharacteristic
25#16839871—21no—
26#16839869—21no—
27#16839869—21no—
28Q16×Dic61C2211yescharacteristic
29Q16×Dic61C2211yescharacteristic
30Q16×Dic61C2211yescharacteristic
31Q16×Dic61C2211yescharacteristic
32D8×Dic61C2211yescharacteristic
33#16839844C2211yescharacteristic
34#16839800C2211yescharacteristic
35#16839771—21no—
36#16839717C2211yescharacteristic
37#16839590C2211yescharacteristic
38#16839589C2211yescharacteristic
39#16839588C2211yescharacteristic
40#16839587C2211yescharacteristic
41#16839580C2211yescharacteristic
42#16839580C2211yescharacteristic
43#16839566C2211yescharacteristic
44#16839566C2211yescharacteristic
45#16839564C2211yescharacteristic
46#16839564C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839563C2211yescharacteristic
49C22⋊#11758185C2211yescharacteristic
50#16839537C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C22×Q16×C61C2211yescharacteristic
53C2×SD32×C61C2211yescharacteristic
54C2×SD32×C61C2211yescharacteristic
55C61×#466C2211yescharacteristic
56C61×#358C2211yescharacteristic
57C61×#358C2211yescharacteristic
58C61×C22⋊C16C2211yescharacteristic
59C2×D488C2311yescharacteristic
60C2×#11758199—21no—
61C2×#11758194D411yescharacteristic
62#14519773—21no—
63#14519768—21no—
64#14519761—41no—
65#14519759—41no—
66#14519751—21no—
67#14519749—41no—
68#14519748—41no—
69Q8×Dic61—21no—
70Q8×Dic61—21no—
71Q8×Dic61—21no—
72Q8×Dic61—21no—
73Q8×Dic61—41no—
74Q8×Dic61—41no—
75D4×Dic61—21no—
76#14519739—21no—
77#14519736—21no—
78#14519736—41no—
79#14519736—41no—
80#14519736D411yescharacteristic
81#14519720—22no—
82#14519720—22no—
83C2×#11758185C2311yescharacteristic
84C2×#11758185C2311yescharacteristic
85#14519711C2311yescharacteristic
86#14519711C2311yescharacteristic
87#14519711C2311yescharacteristic
88#14519709—21no—
89#14519708—21no—
90#14519701—21no—
91#14519701—21no—
92#14519701—21no—
93#14519701—21no—
94#14519699—21no—
95#14519698—21no—
96#14519697—21no—
97#14519687D411yescharacteristic
98#14519687D411yescharacteristic
99C8×Dic61C2311yescharacteristic
100C8×Dic61C2311yescharacteristic
101C22×Q8×C61—21no—
102C2×Q16×C61—22no—
103C2×Q16×C61—22no—
104C2×Q16×C61—21no—
105C2×Q16×C61C2311yescharacteristic
106C2×Q16×C61C2311yescharacteristic
107C2×D8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109#121×C61—21no—
110SD32×C61—22no—
111SD32×C61—22no—
112C2×C16×C61C2311yescharacteristic
113C2×C16×C61C2311yescharacteristic
114#107×C61C2311yescharacteristic
115#102×C61—21no—
116C2×Q8×C61—24no—
117C2×Q8×C61—21no—
118C2×Q8×C61—21no—
119C2×Q8×C61—42no—
120C2×Q8×C61—41no—
121C2×D4×C61—21no—
122C2×D4×C61—41no—
123C22×C4×C61—41no—
124C22×C4×C61C2×D411yescharacteristic
125C2×C22⋊D61—41no—
126C2×C22⋊D61—41no—
127C2×C22⋊D61—41no—
128C22×Dic61—41no—
129C2×D244—21no—
130C2×C4×D61—41no—
131Q16×C61—24no—
132Q16×C61—21no—
133Q16×C61—21no—
134Q16×C61#4112yes—
135Q16×C61#4112yes—
136D8×C61—21no—
137D8×C61—21no—
138C2×C8×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61C2×D411yescharacteristic
141C2×C8×C61C2411yescharacteristic
142C4⋊C4×C61—21no—
143C22⋊C4×C61—41no—
144C22⋊Dic61—41no—
145C22⋊Dic61—41no—
146C22⋊Dic61D811yescharacteristic
147C22⋊Dic61D811yescharacteristic
148#11758199—42no—
149#11758198—41no—
150#11758198—41no—
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61—41no—
155C4⋊Dic61—41no—
156C4⋊Dic61—41no—
157C4⋊Dic61—41no—
158C4⋊Dic61—41no—
159C4⋊Dic61D811yescharacteristic
160C4⋊Dic61D811yescharacteristic
161#11758196—41no—
162C4×Dic61—41no—
163C4×Dic61—41no—
164C4×Dic61—41no—
165C4×Dic61—41no—
166C4×Dic61—41no—
167C4×Dic61C2×D411yescharacteristic
168C4×Dic61C2×D411yescharacteristic
169#11758194—41no—
170C2×#92418C2×D411yescharacteristic
171C2×#92418C2×D411yescharacteristic
172D488—21no—
173D488—21no—
174C16×C61—21no—
175C16×C61—21no—
176#11758185—21no—
177#11758185—21no—
178C23×C61—41no—
179C23×C61C2×D811yescharacteristic
180C22×D61—41no—
181Q8×C61—22no—
182Q8×C61—22no—
183Q8×C61—44no—
184Q8×C61—41no—
185Q8×C61—41no—
186D4×C61—41no—
187D4×C61—41no—
188D4×C61—161no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—44no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61C2×D811yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C22⋊D61—161no—
198C22⋊D61—161no—
199C22⋊D61—161no—
200C2×Dic61—41no—
201C2×Dic61—41no—
202C2×Dic61—41no—
203C2×Dic61—161no—
204C2×Dic61C2×D811yescharacteristic
205C2×Dic61C2×D811yescharacteristic
206C2×Dic61C2×D811yescharacteristic
207C2×Dic61C2×D811yescharacteristic
208D244—41no—
209D244—41no—
210C4×D61—161no—
211C8×C61—21no—
212C8×C61C2×#4111yescharacteristic
213C8×C61C2×#4111yescharacteristic
214#92418—21no—
215#92418—21no—
216#33686—611nomaximal, Sylow
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—81no—
221C22×C61—81no—
222C22×C61—81no—
223C22×C61C22×D811yescharacteristic
224C2×D61—81no—
225C2×D61—81no—
226C2×D61—81no—
227C4×C61—21no—
228C4×C61—42no—
229C4×C61—42no—
230C4×C61—81no—
231C4×C61#52511yescharacteristic
232C4×C61#57511yescharacteristic
233Dic61—21no—
234Dic61—21no—
235Dic61—21no—
236Dic61—21no—
237Dic61—81no—
238Dic61—81no—
239Dic61—81no—
240#3176—611no—
241#2950—611no—
242#2234—611no—
243#2234—611no—
244#2229—611no—
245#2229—611no—
246#2207—611no—
247#2207—611no—
248#2185—611no—
249#2185D6111yescharacteristic, complemented
250#2175—611no—
251#2175—611no—
252C4×SD32—611no—
253C4×SD32—611no—
254#2164—611no—
255C2×C61—41no—
256C2×C61—161no—
257C2×C61#310011yescharacteristic
258C2×C61#341611yescharacteristic
259C2×C61#341611yescharacteristic
260D61—161no—
261C22×Q16C2×D6111yescharacteristic
262#540—1221no—
263#519—1221no—
264C2×SD32—611no—
265C2×SD32—611no—
266C2×SD32C2×D6111yescharacteristic
267C2×SD32C2×D6111yescharacteristic
268#499—611no—
269#493—611no—
270#490—1221no—
271#469—1221no—
272#466—611no—
273#466C2×D6111yescharacteristic
274#442—1221no—
275#441—1221no—
276#441—1221no—
277C4×Q16—611no—
278C4×Q16—611no—
279C4×Q16—611no—
280C4×Q16—611no—
281C4×D8—611no—
282C4×D8—611no—
283#427—611no—
284#419—1221no—
285#405—611no—
286#367—611no—
287#367—611no—
288#366—611no—
289#366—611no—
290C4⋊C16—611no—
291C4⋊C16—611no—
292#358—611no—
293#358—611no—
294#358C2×D6111yescharacteristic
295#358C2×D6111yescharacteristic
296#357—611no—
297#357—611no—
298#357—611no—
299#357—611no—
300C22⋊C16—611no—
301C22⋊C16C2×D6111yescharacteristic
302#346—611no—
303C4×C16—611no—
304C61#3368611yescharacteristic, complemented, Sylow
305C22×Q8—21no—
306C2×Q16—22no—
307C2×Q16—22no—
308C2×Q16—21no—
309C2×Q16C22×D6111yescharacteristic
310C2×Q16C22×D6111yescharacteristic
311C2×D8—611no—
312C2×D8—1221no—
313C2×D8C22×D6111yescharacteristic
314C2×M16—611no—
315C22×C8C22×D6111yescharacteristic
316#127—1221no—
317#125—1221no—
318#123—2441no—
319#123—2441no—
320#121—21no—
321#121—1221no—
322#121—2441no—
323#121—2441no—
324C4×Q8—1221no—
325C4×Q8—1221no—
326C4×Q8—1221no—
327C4×Q8—1221no—
328C4×Q8—2441no—
329C4×Q8—2441no—
330C4×D4—1221no—
331C4×D4—1221no—
332#117—611no—
333#117—1221no—
334#117—2441no—
335#117—2441no—
336SD32—22no—
337SD32—22no—
338SD32—1222no—
339SD32—1222no—
340C2×C16—611no—
341C2×C16—611no—
342C2×C16C22×D6111yescharacteristic
343C2×C16C22×D6111yescharacteristic
344#107—611no—
345#107—611no—
346#107—611no—
347#107C22×D6111yescharacteristic
348#106—1221no—
349#106—1221no—
350#103—1221no—
351#103—1221no—
352#103—1221no—
353#103—1221no—
354#102—21no—
355#102—1221no—
356#102—1221no—
357#102—1221no—
358#97—611no—
359#97—611no—
360C4×C8—611no—
361C4×C8—611no—
362C2×Q8—24no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—42no—
366C2×Q8—41no—
367C2×D4—21no—
368C2×D4—41no—
369C2×D4—1221no—
370C2×D4—2441no—
371C2×D4—2441no—
372C2×D4—2441no—
373C22×C4—41no—
374C22×C4—2441no—
375C22×C4—2441no—
376C22×C4D4×D6111yescharacteristic
377Q16—24no—
378Q16—21no—
379Q16—21no—
380Q16#1175822412yes—
381Q16#1175822412yes—
382D8—21no—
383D8—21no—
384D8—1221no—
385D8—1221no—
386D8—2442no—
387M16—2441no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—611no—
391C2×C8—611no—
392C2×C8D4×D6111yescharacteristic
393C2×C8C23×D6111yescharacteristic
394C4⋊C4—21no—
395C4⋊C4—611no—
396C4⋊C4—611no—
397C4⋊C4—1221no—
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4—2441no—
401C4⋊C4—2441no—
402C4⋊C4—2441no—
403C4⋊C4—2441no—
404C4⋊C4—2441no—
405C4⋊C4—2441no—
406C22⋊C4—41no—
407C22⋊C4—611no—
408C22⋊C4—611no—
409C22⋊C4—2441no—
410C22⋊C4—2441no—
411C22⋊C4—2441no—
412C22⋊C4—2441no—
413C42—611no—
414C42—611no—
415C42—2441no—
416C42—2441no—
417C42—2441no—
418C42—2441no—
419C42—2441no—
420C16—21no—
421C16—21no—
422C16—1221no—
423C16—1221no—
424C23—41no—
425C23—2441no—
426C23D8×D6111yescharacteristic
427Q8—22no—
428Q8—22no—
429Q8—44no—
430Q8—41no—
431Q8—41no—
432D4—41no—
433D4—41no—
434D4—161no—
435D4—9761no—
436D4—9761no—
437D4—9761no—
438D4—2441no—
439D4—2441no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—44no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—611no—
447C2×C4—611no—
448C2×C4—611no—
449C2×C4—611no—
450C2×C4—9761no—
451C2×C4—9761no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4D8×D6111yescharacteristic
456C2×C4C2×D4×D6111yescharacteristic
457C8—21no—
458C8—1221no—
459C8—1221no—
460C8C2×#1175822411yescharacteristic
461C8C2×#1175822411yescharacteristic
462C22—21no—
463C22—21no—
464C22—21no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—4881no—
469C22—4881no—
470C22—4881no—
471C22C2×D8×D6111yescharacteristic
472C4—21no—
473C4—42no—
474C4—42no—
475C4—81no—
476C4—1221no—
477C4—1221no—
478C4—1221no—
479C4—1221no—
480C4—4881no—
481C4—4881no—
482C4—4881no—
483C4#1684026811yescharacteristic
484C4#1684055811yescharacteristic
485C2—41no—
486C2—161no—
487C2—9761no—
488C2#2145581411yescharacteristic
489C2#2145896111yescharacteristic
490C2#2145896411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2185C2
socleC22×C61C22×D8
prime radical#21456462C2
hypercenter#2185D61
hypocenterC61#33686
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2185

Complements

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

NH ≅ G/Ntype
2. #21454687C2semidirect
3. #21448011C2semidirect
4. #21448007C2semidirect
5. #21447985C2semidirect
6. #21447981C2semidirect
7. #21447857C2semidirect
8. #21447857C2semidirect
9. #21447736C2semidirect
10. #21447686C2semidirect
11. #21447684C2semidirect
12. SD32×Dic61C2semidirect
13. #21447668C2semidirect
14. #21447623C2semidirect
15. C61×#2185C2semidirect
249. #2185D61semidirect
304. C61#33686semidirect

Maximal subgroups

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

subgroupindex
#214564622
#214546872
#214480112
#214480072
#214479852
#214479812
#214478572
#214477362
#214476862
#214476842
SD32×Dic612
#214476682
#214476232
C61×#21852
#3368661