Finite Group Databasebeta

← #29834697

Subgroups — #29834697

Order 15616.

556 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456352C211yesmaximal, characteristic, complemented
2#21454666C211yesmaximal, characteristic
3#21447978C211yesmaximal, characteristic, complemented
4#21447977C211yesmaximal, characteristic, complemented
5#21447976C211yesmaximal, characteristic, complemented
6#21447975C211yesmaximal, characteristic, complemented
7#21447965C211yesmaximal, characteristic, complemented
8#21447954C211yesmaximal, characteristic, complemented
9#21447953C211yesmaximal, characteristic, complemented
10#21447606C211yesmaximal, characteristic, complemented
11#21447602C211yesmaximal, characteristic, complemented
12#21447576C211yesmaximal, characteristic, complemented
13#21447533C211yesmaximal, characteristic
14#357×D61C211yesmaximal, characteristic
15C61×#2226C211yesmaximal, characteristic, complemented
16C2×D8×D61C2211yescharacteristic
17#16840389—21no—
18#16840234—21no—
19C2×#14519716C2211yescharacteristic
20C2×C16×D61C2211yescharacteristic
21#16840170—21no—
22#16840168—21no—
23#16840166C2211yescharacteristic, complemented
24#16840164C2211yescharacteristic, complemented
25#16840163C2211yescharacteristic, complemented
26#16839872—21no—
27#16839871—21no—
28C4×D488C2211yescharacteristic
29D8×Dic61C2211yescharacteristic
30#16839844C2211yescharacteristic
31#16839803—21no—
32#16839799C2211yescharacteristic
33#16839799C2211yescharacteristic
34#107×D61C2211yescharacteristic
35#16839765—21no—
36#16839716C2211yescharacteristic
37#16839590C2211yescharacteristic, complemented
38#16839589C2211yescharacteristic, complemented
39#16839588C2211yescharacteristic, complemented
40#16839587C2211yescharacteristic, complemented
41#16839587C2211yescharacteristic, complemented
42#16839587C2211yescharacteristic, complemented
43C4⋊#11758185C2211yescharacteristic, complemented
44#16839580C2211yescharacteristic, complemented
45#16839580C2211yescharacteristic, complemented
46#16839564C2211yescharacteristic
47#16839564C2211yescharacteristic
48#16839563C2211yescharacteristic
49#16839563C2211yescharacteristic
50#16839562C2211yescharacteristic
51#16839562C2211yescharacteristic
52C61×#500C2211yescharacteristic, complemented
53C4×D8×C61C2211yescharacteristic
54C61×#367C2211yescharacteristic, complemented
55C61×#367C2211yescharacteristic, complemented
56C61×C4⋊C16C2211yescharacteristic, complemented
57C61×#357C2211yescharacteristic
58C61×#357C2211yescharacteristic
59C2×D4×D61—21no—
60C2×D488C2311yescharacteristic
61C2×#11758199—21no—
62D8×D61—22no—
63D8×D61Q814yescomplemented
64C2×#11758189D411yescharacteristic
65C2×C8×D61C2311yescharacteristic
66#14519776—21no—
67#14519775—41no—
68#14519769—21no—
69#14519768—21no—
70#14519767—41no—
71#14519755—41no—
72#14519753—41no—
73C4×D244—21no—
74D4×Dic61—21no—
75#14519739—21no—
76C4×C22⋊D61—41no—
77#14519737—41no—
78#14519735—21no—
79#14519735—21no—
80#14519734D411yescharacteristic
81C4⋊C4×D61—21no—
82C2×#11758198—41no—
83C22⋊C4×D61—41no—
84#14519716—22no—
85C2×#11758185C2311yescharacteristic
86C2×#11758185C2311yescharacteristic
87C16×D61—22no—
88#14519711C2311yescharacteristic
89#14519711C2311yescharacteristic
90#14519711C2311yescharacteristic
91#14519710—21no—
92#14519710—21no—
93#14519709—21no—
94#14519708—21no—
95#14519708—21no—
96#14519708—21no—
97#14519699—21no—
98#14519698—21no—
99#14519697—21no—
100#14519688D411yescharacteristic
101#14519687D411yescharacteristic
102C8×Dic61C2311yescharacteristic
103C2×D8×C61C2311yescharacteristic
104C4⋊Q8×C61—21no—
105C4×D4×C61—21no—
106C2×C16×C61C2311yescharacteristic
107C2×C16×C61C2311yescharacteristic
108#107×C61—21no—
109#107×C61C2311yescharacteristic
110#107×C61C2311yescharacteristic
111#107×C61C2311yescharacteristic
112#102×C61—21no—
113C4×C8×C61C2311yescharacteristic
114C23×D61—41no—
115C2×Q8×C61—41no—
116C2×D4×C61—21no—
117C22×C4×C61—41no—
118C2×C22⋊D61—41no—
119C22×Dic61—41no—
120D4×D61—24no—
121D4×D61—42no—
122C2×D244—21no—
123C2×C4×D61—41no—
124C2×C4×D61—41no—
125C2×C4×D61—41no—
126C2×C4×D61C2×D411yescharacteristic
127C2×Dic122—41no—
128D8×C61—21no—
129D8×C61C2×Q812yes—
130C2×C8×C61C2×D411yescharacteristic
131C2×C8×C61C2411yescharacteristic
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—41no—
136C22⋊C4×C61—41no—
137C42×C61C2×D411yescharacteristic
138C22⋊Dic61—41no—
139#11758199—24no—
140#11758198—41no—
141#11758198—41no—
142#11758198—41no—
143#11758198—41no—
144#11758198D811yescharacteristic
145#11758198D811yescharacteristic
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—41no—
150#11758196—41no—
151#11758196—41no—
152#11758196—41no—
153#11758196—41no—
154#11758196—41no—
155#11758196D811yescharacteristic
156#11758196D811yescharacteristic
157C4×Dic61—41no—
158C4×Dic61—41no—
159C4×Dic61C2×D411yescharacteristic
160C2×#92418C2×D411yescharacteristic
161C2×#92418C2×D411yescharacteristic
162D488—21no—
163D488C2×Q812yes—
164#11758189—22no—
165C8×D61C2×Q812yes—
166C8×D61#4112yes—
167C16×C61—21no—
168C16×C61—21no—
169#11758185—21no—
170#11758185—21no—
171C23×C61—41no—
172C22×D61—41no—
173C22×D61—44no—
174C22×D61—161no—
175C22×D61—161no—
176C22×D61C2×D811yescharacteristic
177Q8×C61—161no—
178D4×C61—22no—
179D4×C61—41no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61—161no—
184C2×C4×C61C2×D811yescharacteristic
185C2×C4×C61C2×D811yescharacteristic
186C2×C4×C61C22×D411yescharacteristic
187C22⋊D61—44no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61—161no—
192C2×Dic61C2×D811yescharacteristic
193C2×Dic61C2×D811yescharacteristic
194C2×Dic61C2×D811yescharacteristic
195D244—22no—
196D244—41no—
197C4×D61—161no—
198C4×D61—161no—
199C4×D61—161no—
200C4×D61C22⋊Q812yes—
201C4×D61#13712yes—
202Dic122—161no—
203C8×C61—21no—
204C8×C61C22×Q811yescharacteristic
205C8×C61C2×#4111yescharacteristic
206#92418—21no—
207#92418C22⋊Q812yes—
208#33631—611nomaximal, Sylow
209C22×C61—42no—
210C22×C61—81no—
211C22×C61—81no—
212C22×C61C22×D811yescharacteristic
213C2×D61—21no—
214C2×D61—21no—
215C2×D61—42no—
216C2×D61—81no—
217C2×D61—81no—
218C2×D61—164no—
219C2×D61#47412yes—
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—81no—
223C4×C61—81no—
224C4×C61—81no—
225C4×C61C2×C22⋊Q811yescharacteristic
226C4×C61C2×#13711yescharacteristic
227Dic61—21no—
228Dic61—21no—
229Dic61—81no—
230Dic61—81no—
231Dic61—81no—
232Dic61#47412yes—
233#3164—611no—
234#2948—611no—
235#2228—611no—
236#2228—611no—
237#2228—611no—
238#2228—611no—
239#2226—611no—
240#2226D6111yescharacteristic, complemented
241#2224—611no—
242#2224—611no—
243#2159—611no—
244#2158—611no—
245#2151—611no—
246#2140—611no—
247C2×#357—611no—
248C2×C61—82no—
249C2×C61C2×#47411yescharacteristic
250C2×C61#341511yescharacteristic
251C2×C61#341611yescharacteristic
252D61—22no—
253D61—82no—
254C22×D8—611no—
255#544—1221no—
256#518—1221no—
257C2×M32—611no—
258C22×C16—611no—
259#501—1221no—
260#501—1221no—
261#500—611no—
262#500C2×D6111yescharacteristic, complemented
263#499—611no—
264#499—611no—
265#442—1221no—
266#442—1221no—
267C4×D8—611no—
268C4×D8—611no—
269C4×D8—611no—
270C4×D8C2×D6111yescharacteristic
271#428—1221no—
272#427—611no—
273#427—611no—
274C2×#107—611no—
275#418—1221no—
276#405—611no—
277#367—611no—
278#367—611no—
279#367C2×D6111yescharacteristic, complemented
280#367C2×D6111yescharacteristic, complemented
281#366—611no—
282#366—611no—
283#366—611no—
284#366—611no—
285C4⋊C16—611no—
286C4⋊C16—611no—
287C4⋊C16—611no—
288C4⋊C16C2×D6111yescharacteristic, complemented
289#357—611no—
290#357—611no—
291#357—611no—
292#357—611no—
293#357—611no—
294#357—611no—
295#357C2×D6111yescharacteristic
296#357C2×D6111yescharacteristic
297C61#3363111yescharacteristic, complemented, Sylow
298C22×D4—1221no—
299C2×D8—614no—
300C2×D8—611no—
301C2×D8—1222no—
302C2×D8—1221no—
303C2×D8C22×D6111yescharacteristic
304C2×M16—611no—
305C22×C8—611no—
306C4⋊Q8—21no—
307C4⋊Q8—1221no—
308C4⋊Q8—2441no—
309#125—1221no—
310#125—1221no—
311#125—2441no—
312C22⋊Q8—2441no—
313C22⋊Q8—2441no—
314C4×D4—21no—
315C4×D4—1221no—
316C4×D4—1221no—
317C4×D4—1221no—
318C4×D4—2441no—
319C4×D4—2441no—
320#117—611no—
321#117—1221no—
322#117—1221no—
323C2×C4⋊C4—1221no—
324C2×C22⋊C4—2441no—
325C2×C22⋊C4—2441no—
326M32—1222no—
327C2×C16—611no—
328C2×C16—611no—
329C2×C16—1222no—
330C2×C16C22×D6111yescharacteristic
331C2×C16C22×D6111yescharacteristic
332#107—21no—
333#107—611no—
334#107—611no—
335#107—611no—
336#107—1221no—
337#107—1221no—
338#107C22×D6111yescharacteristic
339#107C22×D6111yescharacteristic
340#107C22×D6111yescharacteristic
341#106—1221no—
342#106—1221no—
343#106—1221no—
344#106—1221no—
345#102—21no—
346#102—1221no—
347#102—1221no—
348#102—1221no—
349#97—611no—
350#97—611no—
351C4×C8—611no—
352C4×C8C22×D6111yescharacteristic
353C24—2441no—
354C2×Q8—41no—
355C2×Q8—2441no—
356C2×D4—21no—
357C2×D4—1224no—
358C2×D4—1221no—
359C2×D4—2442no—
360C2×D4—2441no—
361C22×C4—41no—
362C22×C4—611no—
363C22×C4—2441no—
364C22×C4—2441no—
365C22×C4—2441no—
366C22×C4—2441no—
367D8—21no—
368D8—612no—
369D8—1224no—
370D8—1221no—
371D8Q8×D6112yes—
372M16—1222no—
373C2×C8—611no—
374C2×C8—612no—
375C2×C8—612no—
376C2×C8—611no—
377C2×C8D4×D6111yescharacteristic
378C2×C8C23×D6111yescharacteristic
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—611no—
384C4⋊C4—611no—
385C4⋊C4—1221no—
386C4⋊C4—1221no—
387C4⋊C4—1221no—
388C4⋊C4—2441no—
389C4⋊C4—2441no—
390C4⋊C4—2441no—
391C4⋊C4—2441no—
392C4⋊C4—2441no—
393C4⋊C4—2441no—
394C22⋊C4—41no—
395C22⋊C4—611no—
396C22⋊C4—611no—
397C22⋊C4—2441no—
398C22⋊C4—2441no—
399C22⋊C4—2441no—
400C22⋊C4—2441no—
401C22⋊C4—2441no—
402C42—611no—
403C42—2441no—
404C42—2441no—
405C42D4×D6111yescharacteristic
406C16—21no—
407C16—21no—
408C16—1221no—
409C16—1221no—
410C23—41no—
411C23—611no—
412C23—9761no—
413C23—9761no—
414C23—2444no—
415C23—2441no—
416Q8—161no—
417Q8—9761no—
418D4—22no—
419D4—41no—
420D4—1222no—
421D4—2444no—
422D4—2441no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—161no—
427C2×C4—611no—
428C2×C4—612no—
429C2×C4—612no—
430C2×C4—611no—
431C2×C4—611no—
432C2×C4—9761no—
433C2×C4—9761no—
434C2×C4—9761no—
435C2×C4—9761no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4—2441no—
439C2×C4D8×D6111yescharacteristic
440C2×C4D8×D6111yescharacteristic
441C2×C4C2×D4×D6111yescharacteristic
442C8—21no—
443C8—612no—
444C8—1221no—
445C8C2×Q8×D6111yescharacteristic
446C8#41×D6111yescharacteristic
447C22—42no—
448C22—81no—
449C22—81no—
450C22—612no—
451C22—1221no—
452C22—1221no—
453C22—9764no—
454C22—4881no—
455C22—4881no—
456C22—2442no—
457C22C2×D8×D6111yescharacteristic
458C4—21no—
459C4—21no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—612no—
464C4—1221no—
465C4—1221no—
466C4—4881no—
467C4—4881no—
468C4—4881no—
469C4C22⋊Q8×D6111yescharacteristic
470C4#137×D6111yescharacteristic
471C2—82no—
472C2—1222no—
473C2—4882no—
474C2#474×D6111yescharacteristic
475C2#2145895911yescharacteristic
476C2#2145896111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2226C2
socleC22×C61C22×D8
prime radicalC2×D8×D61C22
hypercenter#2226D61
hypocenterC61#33631
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2226

Complements

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

NH ≅ G/Ntype
1. #21456352C2semidirect
3. #21447978C2semidirect
4. #21447977C2semidirect
5. #21447976C2semidirect
6. #21447975C2semidirect
7. #21447965C2semidirect
8. #21447954C2semidirect
9. #21447953C2semidirect
10. #21447606C2semidirect
11. #21447602C2semidirect
12. #21447576C2semidirect
15. C61×#2226C2semidirect
23. #16840166C22semidirect
24. #16840164C22semidirect
25. #16840163C22semidirect
37. #16839590C22semidirect
38. #16839589C22semidirect
39. #16839588C22semidirect
40. #16839587C22semidirect
41. #16839587C22semidirect
42. #16839587C22semidirect
43. C4⋊#11758185C22semidirect
44. #16839580C22semidirect
45. #16839580C22semidirect
52. C61×#500C22semidirect
54. C61×#367C22semidirect
55. C61×#367C22semidirect
56. C61×C4⋊C16C22semidirect
63. D8×D61Q8semidirect
240. #2226D61semidirect
262. #500C2×D61semidirect
279. #367C2×D61semidirect
280. #367C2×D61semidirect
288. C4⋊C16C2×D61semidirect
297. C61#33631semidirect

Maximal subgroups

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

subgroupindex
#214563522
#214546662
#214479782
#214479772
#214479762
#214479752
#214479652
#214479542
#214479532
#214476062
#214476022
#214475762
#214475332
#357×D612
C61×#22262
#3363161