Finite Group Databasebeta

← #29835188

Subgroups — #29835188

Order 15616.

556 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456474C211yesmaximal, characteristic, complemented
2#21454666C211yesmaximal, characteristic
3#21448029C211yesmaximal, characteristic, complemented
4#21448026C211yesmaximal, characteristic, complemented
5#21447979C211yesmaximal, characteristic, complemented
6#21447974C211yesmaximal, characteristic, complemented
7#21447863C211yesmaximal, characteristic, complemented
8#21447862C211yesmaximal, characteristic, complemented
9#21447749C211yesmaximal, characteristic
10#21447746C211yesmaximal, characteristic
11#21447685C211yesmaximal, characteristic, complemented
12C4×#14519720C211yesmaximal, characteristic, complemented
13C4×#14519721C211yesmaximal, characteristic, complemented
14#21447627C211yesmaximal, characteristic, complemented
15C61×#2175C211yesmaximal, characteristic, complemented
16C2×D8×D61C2211yescharacteristic
17#16840370—21no—
18#16840234—21no—
19C2×#14519722C2211yescharacteristic
20C2×#14519721C2211yescharacteristic
21C2×#14519720C2211yescharacteristic
22#16840163C2211yescharacteristic, complemented
23#16840140C2211yescharacteristic, complemented
24#16840129—21no—
25#16840019—21no—
26#16840004C2211yescharacteristic
27#16840000C2211yescharacteristic
28#16839872—21no—
29#16839871—21no—
30#16839867—21no—
31C4×Dic244C2211yescharacteristic, complemented
32C4×D488C2211yescharacteristic
33D8×Dic61C2211yescharacteristic
34#16839844C2211yescharacteristic
35#16839799C2211yescharacteristic
36#16839765—21no—
37#16839716C2211yescharacteristic
38#16839589C2211yescharacteristic, complemented
39#16839589C2211yescharacteristic, complemented
40#16839588C2211yescharacteristic, complemented
41#16839580C2211yescharacteristic, complemented
42#16839580C2211yescharacteristic, complemented
43#16839567C2211yescharacteristic, complemented
44#16839566C2211yescharacteristic, complemented
45#16839565C2211yescharacteristic, complemented
46#16839564C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839562C2211yescharacteristic
49#16839541C2211yescharacteristic
50#16839541C2211yescharacteristic
51C4×#11758185C2211yescharacteristic, complemented
52C2×SD32×C61C2211yescharacteristic
53C4×Q16×C61C2211yescharacteristic, complemented
54C4×D8×C61C2211yescharacteristic
55C61×#366C2211yescharacteristic, complemented
56C61×#358C2211yescharacteristic, complemented
57C61×#357C2211yescharacteristic
58C61×#346C2211yescharacteristic, complemented
59C2×D4×D61—21no—
60C2×Dic244C2311yescharacteristic
61C2×#11758202—21no—
62C2×D488C2311yescharacteristic
63C2×#11758199—21no—
64D8×D61—22no—
65D8×D61—22no—
66C2×#11758189D411yescharacteristic
67C2×C8×D61C2311yescharacteristic
68#14519775—21no—
69#14519772—41no—
70#14519771—41no—
71#14519769—41no—
72#14519768—21no—
73#14519767—41no—
74#14519757—21no—
75#14519753—21no—
76C4×Dic122—21no—
77C4×D244—21no—
78D4×Dic61—21no—
79#14519739—21no—
80C4×C22⋊D61—41no—
81#14519737—41no—
82#14519735—21no—
83#14519734D411yescharacteristic
84C2×#11758198—41no—
85C22⋊C4×D61—41no—
86#14519722—22no—
87#14519721—22no—
88#14519720—22no—
89C2×#11758185C2311yescharacteristic
90C2×#11758185C2311yescharacteristic
91#14519711C2311yescharacteristic
92#14519711C2311yescharacteristic
93#14519708—21no—
94#14519708—21no—
95#14519702—21no—
96#14519701—21no—
97#14519700—21no—
98#14519699—21no—
99#14519698—21no—
100#14519697—21no—
101#14519688D411yescharacteristic
102#14519687D411yescharacteristic
103C8×Dic61C2311yescharacteristic
104C2×Q16×C61C2311yescharacteristic
105C2×D8×C61C2311yescharacteristic
106C4×Q8×C61—21no—
107C4×D4×C61—21no—
108SD32×C61—22no—
109C2×C16×C61C2311yescharacteristic
110C2×C16×C61C2311yescharacteristic
111#107×C61C2311yescharacteristic
112#107×C61C2311yescharacteristic
113#103×C61—21no—
114#102×C61—21no—
115C4×C8×C61C2311yescharacteristic
116C23×D61—41no—
117C2×Q8×C61—21no—
118C2×D4×C61—21no—
119C22×C4×C61—41no—
120C2×C22⋊D61—41no—
121C22×Dic61—41no—
122D4×D61—24no—
123D4×D61—42no—
124C2×D244—21no—
125C2×C4×D61—41no—
126C2×C4×D61—41no—
127C2×C4×D61C2×D411yescharacteristic
128C2×Dic122—21no—
129Q16×C61—21no—
130Q16×C61—21no—
131D8×C61—21no—
132D8×C61#4112yes—
133C2×C8×C61C2×D411yescharacteristic
134C2×C8×C61C2411yescharacteristic
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—41no—
138C22⋊C4×C61—41no—
139C42×C61—41no—
140C42×C61C2×D411yescharacteristic
141C22⋊Dic61—41no—
142#11758202—42no—
143#11758199—24no—
144#11758198—41no—
145#11758198—41no—
146#11758198—41no—
147#11758198—41no—
148#11758198—41no—
149#11758198D811yescharacteristic
150#11758198D811yescharacteristic
151C4⋊Dic61—21no—
152C4⋊Dic61—21no—
153C4⋊Dic61—41no—
154C4⋊Dic61—41no—
155#11758196—41no—
156#11758196—41no—
157#11758196—41no—
158#11758196—41no—
159#11758196—41no—
160#11758196D811yescharacteristic
161#11758196D811yescharacteristic
162C4×Dic61—41no—
163C4×Dic61—41no—
164C4×Dic61C2×D411yescharacteristic
165C2×#92418C2×D411yescharacteristic
166C2×#92418C2×D411yescharacteristic
167Dic244—21no—
168Dic244—21no—
169D488—21no—
170D488#4112yes—
171#11758189—41no—
172C8×D61—21no—
173C8×D61—21no—
174C16×C61—21no—
175C16×C61—21no—
176#11758185—21no—
177#11758185—21no—
178C23×C61—41no—
179C22×D61—44no—
180C22×D61—41no—
181C22×D61—161no—
182C22×D61—161no—
183C22×D61C2×D811yescharacteristic
184Q8×C61—41no—
185Q8×C61—41no—
186D4×C61—22no—
187D4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—161no—
192C2×C4×C61C2×D811yescharacteristic
193C2×C4×C61C2×D811yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C22⋊D61—44no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—161no—
200C2×Dic61C2×D811yescharacteristic
201C2×Dic61C2×D811yescharacteristic
202C2×Dic61C2×D811yescharacteristic
203D244—22no—
204D244—41no—
205C4×D61—21no—
206C4×D61—21no—
207C4×D61—161no—
208C4×D61—161no—
209Dic122—41no—
210Dic122—41no—
211C8×C61—21no—
212C8×C61C2×#4111yescharacteristic
213C8×C61C2×#4111yescharacteristic
214#92418—21no—
215#92418—21no—
216#33683—611nomaximal, Sylow
217C22×C61—42no—
218C22×C61—81no—
219C22×C61—81no—
220C22×C61C22×D811yescharacteristic
221C2×D61—21no—
222C2×D61—21no—
223C2×D61—21no—
224C2×D61—42no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61—164no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—81no—
231C4×C61—81no—
232C4×C61—81no—
233C4×C61#52511yescharacteristic
234C4×C61#57511yescharacteristic
235Dic61—21no—
236Dic61—21no—
237Dic61—21no—
238Dic61—81no—
239Dic61—81no—
240Dic61—81no—
241#3177—611no—
242#2948—611no—
243#2237—611no—
244#2237—611no—
245#2228—611no—
246#2228—611no—
247#2208—611no—
248#2208—611no—
249#2187—611no—
250#2187—611no—
251#2175—611no—
252#2175D6111yescharacteristic, complemented
253C4×SD32—611no—
254C4×SD32—611no—
255#2164—611no—
256C2×C61—82no—
257C2×C61#310011yescharacteristic
258C2×C61#341611yescharacteristic
259C2×C61#341611yescharacteristic
260D61—41no—
261D61—82no—
262C22×D8—611no—
263#541—1221no—
264#518—1221no—
265C2×SD32—611no—
266C2×SD32—611no—
267C2×SD32—611no—
268C2×SD32C2×D6111yescharacteristic
269#499—611no—
270#494—611no—
271#491—1221no—
272#470—1221no—
273#467—611no—
274#467—611no—
275#442—1221no—
276#442—1221no—
277#441—1221no—
278C4×Q16—611no—
279C4×Q16C2×D6111yescharacteristic, complemented
280C4×D8—611no—
281C4×D8—611no—
282C4×D8—611no—
283C4×D8C2×D6111yescharacteristic
284#427—611no—
285#418—1221no—
286#405—611no—
287#367—611no—
288#367—611no—
289#366—611no—
290#366C2×D6111yescharacteristic, complemented
291C4⋊C16—611no—
292C4⋊C16—611no—
293#358—611no—
294#358—611no—
295#358—611no—
296#358C2×D6111yescharacteristic, complemented
297#357—611no—
298#357—611no—
299#357—611no—
300#357C2×D6111yescharacteristic
301C22⋊C16—611no—
302C22⋊C16—611no—
303#346C2×D6111yescharacteristic, complemented
304C4×C16—611no—
305C61#3368311yescharacteristic, complemented, Sylow
306C22×D4—1221no—
307C2×Q16—611no—
308C2×Q16—1221no—
309C2×Q16C22×D6111yescharacteristic
310C2×D8—611no—
311C2×D8—1221no—
312C2×D8—1222no—
313C2×D8—1222no—
314C2×D8C22×D6111yescharacteristic
315C2×M16—611no—
316C22×C8—611no—
317C4⋊Q8—1221no—
318#126—2441no—
319#126—2441no—
320#125—1221no—
321#125—2441no—
322#125—2441no—
323C22⋊Q8—1221no—
324C22⋊Q8—1221no—
325C4×Q8—21no—
326C4×Q8—1221no—
327C4×D4—21no—
328C4×D4—1221no—
329C4×D4—1221no—
330C4×D4—1221no—
331C4×D4—2441no—
332C4×D4—2441no—
333#117—611no—
334#117—1221no—
335C2×C22⋊C4—2441no—
336C2×C22⋊C4—2441no—
337SD32—22no—
338SD32—1222no—
339SD32—1222no—
340SD32—1222no—
341C2×C16—611no—
342C2×C16—611no—
343C2×C16C22×D6111yescharacteristic
344C2×C16C22×D6111yescharacteristic
345#107—611no—
346#107—611no—
347#107C22×D6111yescharacteristic
348#107C22×D6111yescharacteristic
349#106—1221no—
350#106—1221no—
351#103—21no—
352#103—1221no—
353#103—1221no—
354#103—1221no—
355#102—21no—
356#102—1221no—
357#102—1221no—
358#102—1221no—
359#97—611no—
360#97—611no—
361C4×C8—611no—
362C4×C8C22×D6111yescharacteristic
363C24—2441no—
364C2×Q8—21no—
365C2×Q8—1221no—
366C2×D4—21no—
367C2×D4—1224no—
368C2×D4—1221no—
369C2×D4—2441no—
370C2×D4—2442no—
371C22×C4—41no—
372C22×C4—611no—
373C22×C4—2441no—
374C22×C4—2441no—
375C22×C4—2441no—
376Q16—21no—
377Q16—21no—
378Q16—1221no—
379Q16—1221no—
380Q16—2442no—
381D8—21no—
382D8—612no—
383D8—1224no—
384D8—1221no—
385D8#1175822212yes—
386M16—2441no—
387C2×C8—611no—
388C2×C8—611no—
389C2×C8—1221no—
390C2×C8—1221no—
391C2×C8D4×D6111yescharacteristic
392C2×C8C23×D6111yescharacteristic
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—41no—
396C4⋊C4—611no—
397C4⋊C4—611no—
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4—2441no—
401C4⋊C4—2441no—
402C4⋊C4—2441no—
403C4⋊C4—2441no—
404C4⋊C4—2441no—
405C4⋊C4—2441no—
406C4⋊C4—2441no—
407C22⋊C4—41no—
408C22⋊C4—611no—
409C22⋊C4—611no—
410C22⋊C4—2441no—
411C22⋊C4—2441no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C42—41no—
417C42—611no—
418C42—2441no—
419C42—2441no—
420C42D4×D6111yescharacteristic
421C16—21no—
422C16—21no—
423C16—1221no—
424C16—1221no—
425C23—41no—
426C23—611no—
427C23—9761no—
428C23—9761no—
429C23—2444no—
430C23—2441no—
431Q8—41no—
432Q8—41no—
433Q8—2441no—
434Q8—2441no—
435D4—22no—
436D4—41no—
437D4—1222no—
438D4—2444no—
439D4—2441no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—161no—
444C2×C4—611no—
445C2×C4—611no—
446C2×C4—611no—
447C2×C4—1221no—
448C2×C4—1221no—
449C2×C4—9761no—
450C2×C4—9761no—
451C2×C4—9761no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4D8×D6111yescharacteristic
456C2×C4D8×D6111yescharacteristic
457C2×C4C2×D4×D6111yescharacteristic
458C8—21no—
459C8—1221no—
460C8—1221no—
461C8#41×D6111yescharacteristic
462C8C2×#1175822211yescharacteristic
463C22—42no—
464C22—81no—
465C22—81no—
466C22—1221no—
467C22—1221no—
468C22—1221no—
469C22—9764no—
470C22—4881no—
471C22—4881no—
472C22—2442no—
473C22C2×D8×D6111yescharacteristic
474C4—21no—
475C4—21no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—1221no—
480C4—1221no—
481C4—1221no—
482C4—4881no—
483C4—4881no—
484C4—4881no—
485C4#1684027111yescharacteristic
486C4#1684055911yescharacteristic
487C2—82no—
488C2—4882no—
489C2—2441no—
490C2#2145581711yescharacteristic
491C2#2145896411yescharacteristic
492C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2175C2
socleC22×C61C22×D8
prime radicalC2×D8×D61C22
hypercenter#2175D61
hypocenterC61#33683
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
1. #21456474C2semidirect
3. #21448029C2semidirect
4. #21448026C2semidirect
5. #21447979C2semidirect
6. #21447974C2semidirect
7. #21447863C2semidirect
8. #21447862C2semidirect
11. #21447685C2semidirect
12. C4×#14519720C2semidirect
13. C4×#14519721C2semidirect
14. #21447627C2semidirect
15. C61×#2175C2semidirect
22. #16840163C22semidirect
23. #16840140C22semidirect
31. C4×Dic244C22semidirect
38. #16839589C22semidirect
39. #16839589C22semidirect
40. #16839588C22semidirect
41. #16839580C22semidirect
42. #16839580C22semidirect
43. #16839567C22semidirect
44. #16839566C22semidirect
45. #16839565C22semidirect
51. C4×#11758185C22semidirect
53. C4×Q16×C61C22semidirect
55. C61×#366C22semidirect
56. C61×#358C22semidirect
58. C61×#346C22semidirect
252. #2175D61semidirect
279. C4×Q16C2×D61semidirect
290. #366C2×D61semidirect
296. #358C2×D61semidirect
303. #346C2×D61semidirect
305. C61#33683semidirect

Maximal subgroups

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

subgroupindex
#214564742
#214546662
#214480292
#214480262
#214479792
#214479742
#214478632
#214478622
#214477492
#214477462
#214476852
C4×#145197202
C4×#145197212
#214476272
C61×#21752
#3368361