Finite Group Databasebeta

← #29832931

Subgroups — #29832931

Order 15616.

1124 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840575C211yesmaximal, characteristic
2C2×#16840180C211yesmaximal, characteristic, complemented
3#21454416C211yesmaximal, characteristic, complemented
4#21447537C212yesmaximal, complemented
5#21447532C211yesmaximal, characteristic, complemented
6#21447530C212yesmaximal, complemented
7#21447525C212yesmaximal, complemented
8#21447524C211yesmaximal, characteristic, complemented
9C2×#16839566C211yesmaximal, characteristic, complemented
10#358×D61C212yesmaximal, complemented
11C61×#2141C211yesmaximal, characteristic, complemented
12C2×#14519825—21no—
13#16840575—24no—
14#16840575C418yescomplemented
15C2×#14519809—22no—
16C2×#14519808—21no—
17C2×#14519800C2212yes—
18C2×#14519799C2211yescharacteristic
19C2×Q16×D61C2212yes—
20C2×#14519783C2211yescharacteristic
21#16840242—21no—
22#16840180—24no—
23C2×#14519716C2212yes—
24C22×#11758185C2211yescharacteristic
25C2×C16×D61C2212yes—
26#16839805—21no—
27#16839801—22no—
28#16839799C2212yes—
29C2×#14519711C2211yescharacteristic
30#107×D61C2212yes—
31#16839567C2212yes—
32#16839566C2212yes—
33#16839566C2212yes—
34#16839565C2211yescharacteristic
35#16839565C2211yescharacteristic
36#16839564C2212yes—
37#16839562C2211yescharacteristic
38#16839562C2211yescharacteristic
39C22×Q16×C61C2211yescharacteristic
40C2×M32×C61C2211yescharacteristic
41C61×#427C2211yescharacteristic
42C61×#358C2212yes—
43C61×#358C2212yes—
44#14519825—28no—
45#14519825—44no—
46C2×#11758226—22no—
47C2×#11758226—41no—
48C2×#11758222—21no—
49C2×#11758222—42no—
50C2×#11758222D411yescharacteristic
51C2×Q8×D61—22no—
52C2×Q8×D61—41no—
53#14519809—216no—
54#14519808—28no—
55#14519800—24no—
56#14519800C2×C418yescomplemented
57#14519799—22no—
58#14519799C2×C414yescomplemented
59C2×Dic244C2311yescharacteristic
60C2×#11758202—22no—
61Q16×D61—24no—
62Q16×D61C2×C418yescomplemented
63C2×#11758190—21no—
64C2×#11758201—22no—
65C2×D488C2311yescharacteristic
66#14519783C2×C414yes—
67#14519783D414yes—
68C2×#11758194D411yescharacteristic
69C2×#11758189D412yes—
70C2×C8×D61C2312yes—
71C4×Dic122—41no—
72C4×D244—41no—
73#14519737—42no—
74#14519735—22no—
75C2×C4⋊Dic61—21no—
76C4⋊C4×D61—22no—
77#14519716—24no—
78C2×#11758185—22no—
79C2×#11758185C2311yescharacteristic
80C2×#11758185C2311yescharacteristic
81C16×D61—24no—
82#14519711—21no—
83#14519711C2312yes—
84#14519710—22no—
85#14519708—22no—
86C22×Q8×C61—21no—
87C2×Q16×C61—21no—
88C2×Q16×C61—22no—
89C2×Q16×C61C2×C414yescomplemented
90C2×Q16×C61C2312yes—
91C22×C8×C61C2311yescharacteristic
92#117×C61—21no—
93M32×C61—22no—
94C2×C16×C61C2312yes—
95#107×C61C2311yescharacteristic
96#107×C61C2311yescharacteristic
97#106×C61—21no—
98C2×Q8×C61—24no—
99C2×Q8×C61—22no—
100C2×Q8×C61—41no—
101C2×Q8×C61—42no—
102C22×C4×C61—41no—
103C22×C4×C61C2×D411yescharacteristic
104C2×C22⋊D61—41no—
105C2×C22⋊D61D812yes—
106C22×Dic61—41no—
107#11758226—28no—
108#11758226—48no—
109#11758226—44no—
110Q8×D61—28no—
111Q8×D61—44no—
112Q8×D61—48no—
113#11758222—24no—
114#11758222—42no—
115#11758222—416no—
116#11758222C22⋊C414yes—
117#11758222SD1614yes—
118C2×D244—21no—
119C2×D244—42no—
120C2×D244D811yescharacteristic
121C2×C4×D61—42no—
122C2×C4×D61—41no—
123C2×C4×D61—42no—
124C2×C4×D61—42no—
125C2×C4×D61C2×D412yes—
126C2×Dic122—21no—
127C2×Dic122—42no—
128C2×Dic122D811yescharacteristic
129Q16×C61—24no—
130Q16×C61—22no—
131Q16×C61C22×C414yes—
132C2×C8×C61C22×C412yes—
133C2×C8×C61C2×D411yescharacteristic
134C2×C8×C61C2×D412yes—
135C2×C8×C61C2411yescharacteristic
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C22⋊C4×C61—41no—
139C42×C61—41no—
140#11758202—28no—
141#11758201—28no—
142#11758198—42no—
143C4⋊Dic61—22no—
144C4⋊Dic61—41no—
145#11758196—42no—
146C4×Dic61—42no—
147#11758194C22⋊C414yes—
148C2×#92418C2×D412yes—
149Dic244—21no—
150Dic244C22×C412yes—
151D488—21no—
152D488C22×C412yes—
153#11758190—24no—
154#11758189C22⋊C418yes—
155C8×D61C22×C414yes—
156C8×D61C2×D414yes—
157C16×C61—22no—
158#11758185—21no—
159#11758185—21no—
160C23×C61C2×D811yescharacteristic
161C22×D61—41no—
162C22×D61C2×D812yes—
163Q8×C61—24no—
164Q8×C61—42no—
165Q8×C61—44no—
166C2×C4×C61—42no—
167C2×C4×C61—44no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61C2×C22⋊C412yes—
171C2×C4×C61C2×D811yescharacteristic
172C2×C4×C61C2×SD1612yes—
173C2×C4×C61C22×D411yescharacteristic
174C22⋊D61—44no—
175C22⋊D61#10218yes—
176C2×Dic61—41no—
177C2×Dic61—42no—
178C2×Dic61—161no—
179C2×Dic61C2×D812yes—
180D244—22no—
181D244—48no—
182D244—41no—
183D244#10214yes—
184C4×D61—48no—
185C4×D61—44no—
186C4×D61—48no—
187C4×D61—162no—
188C4×D61C2×C22⋊C414yes—
189C4×D61C2×SD1614yes—
190Dic122—22no—
191Dic122—41no—
192Dic122—48no—
193Dic122#10214yes—
194C8×C61C2×C22⋊C412yes—
195C8×C61C23×C411yescharacteristic
196C8×C61C22×D411yescharacteristic
197#92418C2×C22⋊C414yes—
198#33382—611nomaximal, Sylow
199C22×C61—21no—
200C22×C61—21no—
201C22×C61C2×#10212yes—
202C22×C61C22×D811yescharacteristic
203C2×D61—22no—
204C2×D61—22no—
205C2×D61—42no—
206C2×D61—81no—
207C2×D61—81no—
208C2×D61C2×#10214yes—
209C4×C61—44no—
210C4×C61—81no—
211C4×C61—81no—
212C4×C61C2×#10212yes—
213C4×C61C22×C22⋊C411yescharacteristic
214C4×C61C22×SD1611yescharacteristic
215Dic61—42no—
216Dic61—82no—
217Dic61C2×#10214yes—
218C2×#578—611no—
219C2×#504—611no—
220#2913—611no—
221#2141—612no—
222#2141D6111yescharacteristic, complemented
223#2139—611no—
224#2139—612no—
225#2138—611no—
226#2138—612no—
227C2×#358—612no—
228C2×#358—611no—
229C2×C61—22no—
230C2×C61C22×#10211yescharacteristic
231C2×C61#341611yescharacteristic
232C2×C61#341711yescharacteristic
233D61—24no—
234D61—82no—
235C2×2−1+4—1221no—
236#578—618no—
237#578—1224no—
238C2×#137—1222no—
239C2×#137—1221no—
240C2×#135—612no—
241C2×#135—611no—
242C22×Q16—612no—
243C22×Q16C2×D6111yescharacteristic
244C2×#131—611no—
245#520—1221no—
246#504—1224no—
247C2×M32—612no—
248C2×M32C2×D6111yescharacteristic
249C22×C16—611no—
250C22×C16—612no—
251#428—1222no—
252#428—1221no—
253#427—612no—
254#427C2×D6111yescharacteristic
255C2×#107—611no—
256C2×#107—612no—
257#358—612no—
258#358—612no—
259#358—612no—
260#358—611no—
261#358—611no—
262#358C2×D6112yes—
263#358C2×D6112yes—
264#357—612no—
265#357—611no—
266#357—611no—
267C61#3338211yescharacteristic, complemented, Sylow
2682−1+4—1228no—
2692−1+4—2444no—
270C2×#41—611no—
271C2×#41—1222no—
272C2×#41—1221no—
273C2×#41—2442no—
274C2×#41—2441no—
275C22×Q8—21no—
276C22×Q8—1222no—
277C22×Q8—2441no—
278#137—12216no—
279#137—1228no—
280#135—618no—
281#135—614no—
282#135—1224no—
283#135—1222no—
284C2×Q16—21no—
285C2×Q16—22no—
286C2×Q16—618no—
287C2×Q16—611no—
288C2×Q16—1224no—
289C2×Q16—1222no—
290C2×Q16C4×D6114yescomplemented
291C2×Q16C22×D6112yes—
292C2×SD16—1222no—
293C2×SD16—1221no—
294C2×D8—611no—
295#131—614no—
296#131—614no—
297C2×M16—611no—
298C2×M16—612no—
299C22×C8—612no—
300C22×C8C22×D6111yescharacteristic
301C4×Q8—2441no—
302C4×D4—2441no—
303C4×D4—2442no—
304#117—21no—
305#117—1222no—
306C2×C4⋊C4—1222no—
307C2×C4⋊C4—1221no—
308M32—22no—
309M32—1224no—
310C2×C16—611no—
311C2×C16—611no—
312C2×C16—1224no—
313C2×C16—1222no—
314C2×C16C22×D6112yes—
315#107—612no—
316#107—1222no—
317#107—1221no—
318#107C22×D6111yescharacteristic
319#107C22×D6111yescharacteristic
320#106—21no—
321#106—1222no—
322#41—614no—
323#41—614no—
324#41—1228no—
325#41—1224no—
326#41—24416no—
327#41—2444no—
328#41—2448no—
329#41—2442no—
330C2×Q8—22no—
331C2×Q8—24no—
332C2×Q8—41no—
333C2×Q8—42no—
334C2×Q8—611no—
335C2×Q8—1228no—
336C2×Q8—1221no—
337C2×Q8—2442no—
338C2×Q8—2444no—
339C2×Q8—2448no—
340C2×D4—612no—
341C2×D4—611no—
342C2×D4—1221no—
343C2×D4—2442no—
344C2×D4—2441no—
345C22×C4—41no—
346C22×C4—612no—
347C22×C4—2442no—
348C22×C4—2442no—
349C22×C4—2442no—
350C22×C4—2441no—
351C22×C4—2441no—
352C22×C4D4×D6111yescharacteristic
353Q16—22no—
354Q16—24no—
355Q16—612no—
356Q16—1228no—
357Q16—1221no—
358Q16C2×C4×D6114yes—
359SD16—1228no—
360SD16—1224no—
361D8—612no—
362D8—1221no—
363M16—614no—
364M16—618no—
365C2×C8—612no—
366C2×C8—614no—
367C2×C8—614no—
368C2×C8C2×C4×D6112yes—
369C2×C8D4×D6112yes—
370C2×C8D4×D6111yescharacteristic
371C2×C8C23×D6111yescharacteristic
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—1222no—
375C4⋊C4—2442no—
376C4⋊C4—2441no—
377C22⋊C4—41no—
378C22⋊C4—2442no—
379C42—41no—
380C42—2442no—
381C16—22no—
382C16—1221no—
383C16—1221no—
384C23—612no—
385C23—2441no—
386C23D8×D6111yescharacteristic
387Q8—24no—
388Q8—44no—
389Q8—42no—
390Q8—614no—
391Q8—1222no—
392Q8—2448no—
393Q8—2441no—
394D4—618no—
395D4—614no—
396D4—1222no—
397D4—2448no—
398D4—2444no—
399D4—2441no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—612no—
405C2×C4—614no—
406C2×C4—614no—
407C2×C4—9762no—
408C2×C4—9761no—
409C2×C4—2448no—
410C2×C4—2448no—
411C2×C4—2442no—
412C2×C4—2441no—
413C2×C4—2444no—
414C2×C4C22⋊C4×D6112yes—
415C2×C4D8×D6111yescharacteristic
416C2×C4SD16×D6112yes—
417C2×C4C2×D4×D6111yescharacteristic
418C8—614no—
419C8C22⋊C4×D6112yes—
420C8C22×C4×D6111yescharacteristic
421C8C2×D4×D6111yescharacteristic
422C22—21no—
423C22—21no—
424C22—614no—
425C22—1222no—
426C22—1222no—
427C22—4881no—
428C22—4881no—
429C22—2442no—
430C22#102×D6112yes—
431C22C2×D8×D6111yescharacteristic
432C4—44no—
433C4—81no—
434C4—81no—
435C4—614no—
436C4—4882no—
437C4—2442no—
438C4#102×D6112yes—
439C4C2×C22⋊C4×D6111yescharacteristic
440C4C2×SD16×D6111yescharacteristic
441C2—22no—
442C2—1224no—
443C2—4882no—
444C2C2×#102×D6111yescharacteristic
445C2#2145896411yescharacteristic
446C2#2145897111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2141C2
socleC22×C61C22×D8
prime radicalC2×#16840575C2
cosocleC2×C8×C61C24
hypercenter#2141D61
hypocenterC61#33382
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2141

Complements

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

NH ≅ G/Ntype
2. C2×#16840180C2semidirect
3. #21454416C2semidirect
4. #21447537C2semidirect
5. #21447532C2semidirect
6. #21447530C2semidirect
7. #21447525C2semidirect
8. #21447524C2semidirect
9. C2×#16839566C2semidirect
10. #358×D61C2semidirect
11. C61×#2141C2semidirect
14. #16840575C4semidirect
56. #14519800C2×C4semidirect
58. #14519799C2×C4semidirect
62. Q16×D61C2×C4semidirect
89. C2×Q16×C61C2×C4semidirect
222. #2141D61semidirect
267. C61#33382semidirect
290. C2×Q16C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168405752
C2×#168401802
#214544162
#214475372
#214475322
#214475302
#214475252
#214475242
C2×#168395662
#358×D612
C61×#21412
#3338261