Finite Group Databasebeta

← #29834491

Subgroups — #29834491

Order 15616.

738 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
1C2×#16840200C211yesmaximal, characteristic, complemented
2C2×M32×D61C211yesmaximal, characteristic, complemented
3#21455401C211yesmaximal, characteristic, complemented
4C2×#16839999C211yesmaximal, characteristic, complemented
5#21447933C212yesmaximal, complemented
6#21447923C212yesmaximal, complemented
7#21447916C212yesmaximal, complemented
8#21447906C212yesmaximal, complemented
9#21447534C211yesmaximal, characteristic, complemented
10#21447529C211yesmaximal, characteristic, complemented
11C61×#2159C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C22×D488C2211yescharacteristic
14C22×C8×D61C2211yescharacteristic
15#16840274—21no—
16C2×#14519752—21no—
17#16840200—24no—
18C2×#14519722C2212yescomplemented
19C2×D976C2212yescomplemented
20C2×#14519717C2211yescharacteristic
21C2×#14519716C2212yescomplemented
22M32×D61—24no—
23C2×C16×D61C2212yescomplemented
24#16840004C2211yescharacteristic, complemented
25#16840004C2211yescharacteristic, complemented
26#16839999—21no—
27#16839999—21no—
28#16839999C2212yes—
29#16839999C2212yes—
30#16839999C2211yescharacteristic, complemented
31#16839999C2211yescharacteristic, complemented
32#16839992—21no—
33#16839992—21no—
34#16839756—21no—
35C2×#14519697—21no—
36#16839565C2212yescomplemented
37#16839562C2212yescomplemented
38#16839562C2212yescomplemented
39#16839562C2212yescomplemented
40C2×M32×C61C2211yescharacteristic
41C61×#427C2211yescharacteristic
42C2×#107×C61C2211yescharacteristic
43C61×#367C2212yescomplemented
44C61×#366C2212yescomplemented
45C2×#11758222—21no—
46C22×D244—21no—
47C22×D244—41no—
48C22×C4×D61D411yescharacteristic
49#14519799—22no—
50#14519799—22no—
51C2×Dic244C2311yescharacteristic
52C2×#11758190—21no—
53C2×D488—22no—
54C2×D488—22no—
55C2×D488—21no—
56C2×D488C2312yes—
57C2×D488C2311yescharacteristic
58C22×#92418D411yescharacteristic
59C2×C8×D61—22no—
60C2×C8×D61—22no—
61C2×C8×D61D411yescharacteristic
62C2×C8×D61D411yescharacteristic
63C2×C8×D61C2311yescharacteristic
64C2×C8×D61C2311yescharacteristic
65#14519761—41no—
66#14519757—21no—
67#14519757—21no—
68#14519752—21no—
69#14519752—21no—
70#14519752—22no—
71#14519752—22no—
72#14519752—41no—
73#14519752—41no—
74#14519747—41no—
75C4×D244—41no—
76C4×D244—41no—
77C2×#11758198—41no—
78#14519722—24no—
79D976—24no—
80#14519717—22no—
81#14519716—24no—
82C2×#11758185C2312yes—
83C16×D61—24no—
84#14519700—21no—
85#14519700—21no—
86#14519697—21no—
87#14519697—21no—
88#14519697—22no—
89#14519697—22no—
90C22×C8×C61C2311yescharacteristic
91#117×C61—21no—
92C2×C4⋊C4×C61—21no—
93M32×C61D414yescomplemented
94C2×C16×C61C2312yes—
95#107×C61—21no—
96#107×C61C2312yes—
97#107×C61C2311yescharacteristic
98#107×C61C2311yescharacteristic
99#106×C61—21no—
100C23×D61—41no—
101C23×D61D811yescharacteristic
102C22×C4×C61—41no—
103C22×C4×C61C2×D411yescharacteristic
104C2×C22⋊D61—41no—
105C22×Dic61D811yescharacteristic
106#11758222—42no—
107#11758222—42no—
108C2×D244—22no—
109C2×D244—21no—
110C2×D244—42no—
111C2×D244—42no—
112C2×D244—42no—
113C2×D244—42no—
114C2×D244—41no—
115C2×D244—41no—
116C2×D244—41no—
117C2×D244—84no—
118C2×C4×D61—22no—
119C2×C4×D61—22no—
120C2×C4×D61—41no—
121C2×C4×D61D811yescharacteristic
122C2×C4×D61D811yescharacteristic
123C2×C4×D61C2×D411yescharacteristic
124C2×C4×D61C2×D411yescharacteristic
125C2×Dic122—21no—
126C2×C8×C61C2×D411yescharacteristic
127C2×C8×C61C2×D412yes—
128C2×C8×C61#4112yes—
129C2×C8×C61C2411yescharacteristic
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—41no—
134C22⋊C4×C61—41no—
135C42×C61—41no—
136#11758198—41no—
137#11758198—41no—
138#11758198—42no—
139#11758198—42no—
140#11758198—41no—
141#11758198—41no—
142C4⋊Dic61—41no—
143C4⋊Dic61—41no—
144C2×#92418—22no—
145C2×#92418C2×D411yescharacteristic
146C2×#92418C2×D411yescharacteristic
147Dic244—21no—
148Dic244—21no—
149D488—21no—
150D488—22no—
151D488—22no—
152D488—21no—
153D488—42no—
154#11758190—42no—
155C8×D61—22no—
156C8×D61—22no—
157C8×D61—21no—
158C8×D61—21no—
159C8×D61—21no—
160C8×D61—21no—
161C16×C61C2×D414yescomplemented
162#11758185—22no—
163C23×C61C2×D811yescharacteristic
164C22×D61—22no—
165C22×D61—22no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—42no—
169C22×D61—82no—
170C22×D61—82no—
171C22×D61—82no—
172C22×D61C2×D811yescharacteristic
173C22×D61C2×D811yescharacteristic
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—42no—
177C2×C4×C61—82no—
178C2×C4×C61#12112yes—
179C2×C4×C61C2×D811yescharacteristic
180C2×C4×C61#13512yes—
181C2×C4×C61C22×D411yescharacteristic
182C22⋊D61—82no—
183C2×Dic61—22no—
184C2×Dic61—41no—
185C2×Dic61C2×D811yescharacteristic
186C2×Dic61C2×D811yescharacteristic
187D244—41no—
188D244—42no—
189D244—42no—
190D244—41no—
191D244—82no—
192D244—84no—
193D244—82no—
194D244—84no—
195D244—82no—
196C4×D61—21no—
197C4×D61—21no—
198C4×D61—21no—
199C4×D61—21no—
200C4×D61—22no—
201C4×D61—22no—
202C4×D61—82no—
203Dic122—41no—
204Dic122—41no—
205C8×C61#12112yes—
206C8×C61C22×D411yescharacteristic
207C8×C61C2×#4111yescharacteristic
208#92418—21no—
209#92418—21no—
210#33605—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—21no—
213C22×C61#46612yes—
214C22×C61C22×D811yescharacteristic
215C2×D61—21no—
216C2×D61—21no—
217C2×D61—21no—
218C2×D61—21no—
219C2×D61—21no—
220C2×D61—44no—
221C2×D61—82no—
222C2×D61—81no—
223C2×D61—81no—
224C2×D61—82no—
225C2×D61—81no—
226C2×D61—82no—
227C2×D61—164no—
228C2×D61#50912yescomplemented
229C4×C61—82no—
230C4×C61—81no—
231C4×C61—81no—
232C4×C61#46612yes—
233C4×C61C2×#12111yescharacteristic
234C4×C61C2×#13511yescharacteristic
235Dic61—21no—
236Dic61—21no—
237Dic61—81no—
238C2×#509—611no—
239C22×M32—611no—
240#3050—611no—
241C2×#466—611no—
242#2220—612no—
243#2218—612no—
244#2217—612no—
245#2215—612no—
246#2159D6111yescharacteristic, complemented
247#2140—611no—
248#2139—611no—
249C2×C61—22no—
250C2×C61C2×#46611yescharacteristic
251C2×C61C2×#50911yescharacteristic
252C2×C61#341411yescharacteristic
253D61—22no—
254D61—41no—
255D61—161no—
256D61—162no—
257C2×#135—611no—
258C22×D8—611no—
259C23×C8—611no—
260#525—1221no—
261C2×#121—1221no—
262#509—1224no—
263C2×SD32—612no—
264C2×D16—612no—
265C2×M32—611no—
266C2×M32—612no—
267C2×M32—1224no—
268C2×M32C2×D6111yescharacteristic
269C22×C16—612no—
270#467—611no—
271#467—611no—
272#466—612no—
273#466—612no—
274#466—611no—
275#466—611no—
276#466—1221no—
277#466—1221no—
278#465—1221no—
279#465—1221no—
280#427C2×D6111yescharacteristic
281C2×#107C2×D6111yescharacteristic
282#416—1221no—
283C2×#102—1221no—
284#367C2×D6112yescomplemented
285#366C2×D6112yescomplemented
286#358—612no—
287#357—612no—
288#357—612no—
289#357—612no—
290C61#3360511yescharacteristic, complemented, Sylow
291C2×#41—1221no—
292C22×D4—1221no—
293C22×D4—2441no—
294C23×C4—611no—
295#135—1222no—
296#135—1222no—
297C2×Q16—611no—
298C2×SD16—1221no—
299C2×D8—612no—
300C2×D8—611no—
301C2×D8—1222no—
302C2×D8—1221no—
303C2×D8—1222no—
304C22×C8—611no—
305C22×C8—611no—
306C22×C8—611no—
307C22×C8—611no—
308C22×C8—611no—
309C22×C8—1222no—
310C22×C8—1222no—
311C22×C8C22×D6111yescharacteristic
312#123—2441no—
313C22⋊Q8—1221no—
314C22⋊Q8—1221no—
315#121—1222no—
316#121—1222no—
317#121—1221no—
318#121—1221no—
319#121—2441no—
320#121—2441no—
321#120—2441no—
322C4×D4—2441no—
323C4×D4—2441no—
324#117—21no—
325C2×C4⋊C4—21no—
326C2×C22⋊C4—2441no—
327SD32—1224no—
328D16—1224no—
329M32—1222no—
330M32—1224no—
331M32D24414yescomplemented
332C2×C16—612no—
333C2×C16—1224no—
334C2×C16C22×D6112yes—
335#107—21no—
336#107C22×D6111yescharacteristic
337#107C22×D6111yescharacteristic
338#107C22×D6112yes—
339#106—21no—
340#103—1221no—
341#103—1221no—
342#102—1222no—
343#102—1222no—
344#102—1221no—
345#102—1221no—
346C24—611no—
347C24—2441no—
348#41—2442no—
349#41—2442no—
350C2×Q8—1221no—
351C2×D4—1222no—
352C2×D4—1221no—
353C2×D4—4884no—
354C2×D4—2441no—
355C2×D4—2441no—
356C2×D4—2442no—
357C2×D4—2442no—
358C2×D4—2441no—
359C2×D4—2442no—
360C2×D4—2441no—
361C2×D4—2442no—
362C22×C4—41no—
363C22×C4—611no—
364C22×C4—611no—
365C22×C4—611no—
366C22×C4—611no—
367C22×C4—611no—
368C22×C4—1222no—
369C22×C4—1222no—
370C22×C4—2441no—
371C22×C4D4×D6111yescharacteristic
372Q16—1221no—
373Q16—1221no—
374SD16—2442no—
375D8—1221no—
376D8—1222no—
377D8—1222no—
378D8—1221no—
379D8—2442no—
380C2×C8—611no—
381C2×C8—611no—
382C2×C8—1222no—
383C2×C8—1221no—
384C2×C8—1221no—
385C2×C8—1221no—
386C2×C8—1221no—
387C2×C8—1222no—
388C2×C8—1222no—
389C2×C8C2×D24412yes—
390C2×C8D4×D6111yescharacteristic
391C2×C8#1175822612yes—
392C2×C8C23×D6111yescharacteristic
393C4⋊C4—22no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—2441no—
398C4⋊C4—2441no—
399C22⋊C4—41no—
400C22⋊C4—2442no—
401C22⋊C4—2442no—
402C22⋊C4—2441no—
403C22⋊C4—2441no—
404C22⋊C4—2441no—
405C22⋊C4—2441no—
406C42—41no—
407C16—1222no—
408C16C2×D24414yescomplemented
409C23—611no—
410C23—611no—
411C23—1222no—
412C23—1222no—
413C23—4882no—
414C23—4882no—
415C23—4882no—
416C23—2441no—
417C23—2441no—
418C23—2442no—
419C23D8×D6111yescharacteristic
420Q8—2441no—
421Q8—2441no—
422D4—4884no—
423D4—4882no—
424D4—4884no—
425D4—4882no—
426D4—4882no—
427D4—4882no—
428D4—2442no—
429D4—2441no—
430D4—2442no—
431D4—2441no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—82no—
436C2×C4—611no—
437C2×C4—611no—
438C2×C4—1222no—
439C2×C4—1222no—
440C2×C4—1222no—
441C2×C4—1221no—
442C2×C4—1221no—
443C2×C4—1221no—
444C2×C4—1221no—
445C2×C4—4882no—
446C2×C4—2441no—
447C2×C4#1451975212yes—
448C2×C4D8×D6111yescharacteristic
449C2×C4#1451980012yes—
450C2×C4C2×D4×D6111yescharacteristic
451C8—1221no—
452C8—1221no—
453C8#1451975212yes—
454C8C22×D24411yescharacteristic
455C8C2×#1175822611yescharacteristic
456C22—21no—
457C22—21no—
458C22—612no—
459C22—1221no—
460C22—1221no—
461C22—1221no—
462C22—1221no—
463C22—1221no—
464C22—9764no—
465C22—4881no—
466C22—4882no—
467C22—4881no—
468C22—4882no—
469C22—4882no—
470C22—4881no—
471C22—2444no—
472C22#1683999912yes—
473C22C2×D8×D6111yescharacteristic
474C4—81no—
475C4—82no—
476C4—81no—
477C4—1221no—
478C4—1221no—
479C4—4881no—
480C4#1683999912yes—
481C4C2×#1451975211yescharacteristic
482C4C2×#1451980011yescharacteristic
483C2—22no—
484C2—1222no—
485C2—9762no—
486C2—9761no—
487C2—2441no—
488C2C2×#1683999911yescharacteristic
489C2#509×D6111yescharacteristic
490C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2159C2
socleC22×C61C22×D8
hypercenter#2159D61
hypocenterC61#33605
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
1. C2×#16840200C2semidirect
2. C2×M32×D61C2semidirect
3. #21455401C2semidirect
4. C2×#16839999C2semidirect
5. #21447933C2semidirect
6. #21447923C2semidirect
7. #21447916C2semidirect
8. #21447906C2semidirect
9. #21447534C2semidirect
10. #21447529C2semidirect
11. C61×#2159C2semidirect
18. C2×#14519722C22semidirect
19. C2×D976C22semidirect
21. C2×#14519716C22semidirect
23. C2×C16×D61C22semidirect
24. #16840004C22semidirect
25. #16840004C22semidirect
30. #16839999C22semidirect
31. #16839999C22semidirect
36. #16839565C22semidirect
37. #16839562C22semidirect
38. #16839562C22semidirect
39. #16839562C22semidirect
43. C61×#367C22semidirect
44. C61×#366C22semidirect
93. M32×C61D4semidirect
161. C16×C61C2×D4semidirect
228. C2×D61#509semidirect
246. #2159D61semidirect
284. #367C2×D61semidirect
285. #366C2×D61semidirect
290. C61#33605semidirect
331. M32D244semidirect
408. C16C2×D244semidirect

Maximal subgroups

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

subgroupindex
C2×#168402002
C2×M32×D612
#214554012
C2×#168399992
#214479332
#214479232
#214479162
#214479062
#214475342
#214475292
C61×#21592
#3360561