Finite Group Databasebeta

← #29667889

Subgroups — #29667889

Order 15616.

772 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456965C211yesmaximal, characteristic, complemented
2#21456949C211yesmaximal, characteristic
3#21456378C211yesmaximal, characteristic
4#21456365C211yesmaximal, characteristic, complemented
5#21456278C211yesmaximal, characteristic
6C2×#16840113C211yesmaximal, characteristic, complemented
7#21445452C212yesmaximal, complemented
8#21445441C212yesmaximal, complemented
9#21445420C212yesmaximal, complemented
10#21445416C212yesmaximal, complemented
11C8×M16×C61C211yesmaximal, characteristic, complemented
12C2×#14519810—22no—
13#16840319C2211yescharacteristic
14#16840312C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16C2×#14519777C2211yescharacteristic
17#16840171—22no—
18#16840167C2212yes—
19#16840160C2212yes—
20#16840160C2212yes—
21#16840160C2212yes—
22#16840160C2211yescharacteristic
23#16840156—22no—
24#16840142C2212yes—
25#16840135C2212yes—
26#16840135C2211yescharacteristic
27#16840113—22no—
28#16840113C2212yes—
29#16840113C2212yes—
30#16840113C2211yescharacteristic
31#16840113C2212yes—
32#16840109C2211yescharacteristic
33#16840109C2212yes—
34#16840093—22no—
35#16840081—22no—
36#16840053—22no—
37#16840048—22no—
38#16839991—22no—
39C4×#11758190—22no—
40#16839804—22no—
41C2×#14519702—22no—
42C61×#431C2211yescharacteristic
43C4×M16×C61C2211yescharacteristic
44C2×C4×C8×C61C2211yescharacteristic
45C61×#322C2212yes—
46C82×C61C2212yes—
47C2×#11758222—21no—
48C22×Dic122—21no—
49C22×Dic122—21no—
50#14519810—48no—
51C2×Dic244—24no—
52C2×#11758190—22no—
53C2×#11758190—24no—
54#14519777—21no—
55#14519777—21no—
56#14519777—21no—
57#14519777—21no—
58#14519777—21no—
59#14519777—21no—
60#14519777C2312yes—
61#14519777C2312yes—
62#14519777C2311yescharacteristic
63#14519777C2311yescharacteristic
64#14519777C2311yescharacteristic
65#14519774C2311yescharacteristic
66#14519769—21no—
67#14519765—21no—
68#14519756—21no—
69#14519756—21no—
70#14519751—21no—
71#14519751—21no—
72C4×Dic122—21no—
73C4×Dic122—21no—
74C4×D244—21no—
75C4×D244—21no—
76#14519736—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81#14519711—24no—
82#14519709—22no—
83#14519709—24no—
84#14519709—24no—
85#14519709—24no—
86#14519702—22no—
87#14519702—24no—
88#14519702—24no—
89#14519702—24no—
90#14519699—22no—
91#14519699—24no—
92C2×M16×C61D412yes—
93C22×C8×C61D412yes—
94C2×C42×C61C2311yescharacteristic
95C4⋊C8×C61D412yes—
96C22⋊C8×C61D412yes—
97#97×C61D412yes—
98C4×C8×C61D412yes—
99C4×C8×C61C2311yescharacteristic
100C4×C8×C61C2312yes—
101C4×C8×C61C2312yes—
102C4×C8×C61C2311yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C2×C22⋊D61—41no—
107C22×Dic61—41no—
108C22×Dic61—41no—
109#11758222—44no—
110C2×D244—21no—
111C2×D244—21no—
112C2×D244—21no—
113C2×C4×D61—41no—
114C2×Dic122—22no—
115C2×Dic122—22no—
116C2×Dic122—21no—
117C2×Dic122—21no—
118C2×Dic122—21no—
119C2×Dic122—44no—
120C2×Dic122—44no—
121C2×Dic122—41no—
122C2×Dic122—41no—
123M16×C61SD1618yescomplemented
124C2×C8×C61—42no—
125C2×C8×C61C2×D412yes—
126C2×C8×C61C2×D412yes—
127C2×C8×C61C2×D414yes—
128C2×C8×C61C2×D414yes—
129C42×C61C2×D411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C42×C61C2×D411yescharacteristic
132C42×C61C2411yescharacteristic
133C22⋊Dic61—41no—
134#11758198—41no—
135#11758198—41no—
136C4⋊Dic61—22no—
137C4⋊Dic61—21no—
138C4⋊Dic61—22no—
139C4⋊Dic61—22no—
140C4⋊Dic61—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143C4⋊Dic61—21no—
144C4⋊Dic61—22no—
145C4⋊Dic61—21no—
146C4⋊Dic61—41no—
147C4⋊Dic61—41no—
148C4⋊Dic61—41no—
149C4⋊Dic61—41no—
150#11758196—41no—
151#11758196—41no—
152C4×Dic61—41no—
153Dic244—48no—
154#11758190—44no—
155#11758190—48no—
156C23×C61#12711yescharacteristic
157C22×D61—41no—
158C2×C4×C61—22no—
159C2×C4×C61—22no—
160C2×C4×C61#12711yescharacteristic
161C2×C4×C61#12711yescharacteristic
162C2×C4×C61#12711yescharacteristic
163C2×C4×C61C2×SD1614yes—
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C22×D411yescharacteristic
166C2×C4×C61C22×D411yescharacteristic
167C22⋊D61—82no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61—41no—
173C2×Dic61—82no—
174C2×Dic61—82no—
175D244—42no—
176D244—42no—
177D244—42no—
178C4×D61—82no—
179Dic122—44no—
180Dic122—42no—
181Dic122—42no—
182Dic122—44no—
183Dic122—42no—
184Dic122—82no—
185Dic122—82no—
186C8×C61—22no—
187C8×C61—22no—
188C8×C61—24no—
189C8×C61C2×SD1618yes—
190#19624—611nomaximal, Sylow
191C22×C61—21no—
192C22×C61—21no—
193C22×C61#49212yes—
194C22×C61C2×#12711yescharacteristic
195C2×D61—81no—
196C2×D61—81no—
197C2×D61—81no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61#49212yes—
201C4×C61C22×SD1612yes—
202C4×C61#57512yes—
203C4×C61#57812yes—
204Dic61—82no—
205Dic61—81no—
206Dic61—82no—
207Dic61—81no—
208Dic61—81no—
209#3229—611no—
210#3227—611no—
211#3167—611no—
212#3165—611no—
213#3152—611no—
214C2×#487—611no—
215#1716—612no—
216#1714—612no—
217#1708—612no—
218#1707—612no—
219C8×M16D6111yescharacteristic, complemented
220C2×C61—22no—
221C2×C61C2×#49211yescharacteristic
222C2×C61#315311yescharacteristic
223C2×C61#315611yescharacteristic
224D61—161no—
225C2×#137—1222no—
226#533—611no—
227#532—611no—
228C2×C4⋊Q8—611no—
229C2×C4⋊Q8—611no—
230#501—1222no—
231#500—612no—
232#498—612no—
233#498—611no—
234#498—612no—
235#498—612no—
236#497—1222no—
237#494—612no—
238#492—612no—
239#492—611no—
240#487—612no—
241#487—612no—
242#487—611no—
243#487—612no—
244#487—1222no—
245#486—611no—
246#486—612no—
247#483—1222no—
248#481—1222no—
249#476—1222no—
250#475—1222no—
251#465—1222no—
252C4×SD16—1222no—
253#431C2×D6111yescharacteristic
254#428—1222no—
255C2×#103—1222no—
256C4×M16C2×D6111yescharacteristic
257C2×C4×C8C2×D6111yescharacteristic
258#322C2×D6112yes—
259C82C2×D6112yes—
260C61#1962411yescharacteristic, complemented, Sylow
261C2×#41—1221no—
262C22×Q8—1221no—
263C22×Q8—1221no—
264#137—2448no—
265C2×Q16—1224no—
266C2×SD16—1224no—
267C2×SD16—1222no—
268C2×M16D24412yes—
269C22×C8D24412yes—
270C4⋊Q8—612no—
271C4⋊Q8—612no—
272C4⋊Q8—611no—
273C4⋊Q8—611no—
274C4⋊Q8—611no—
275C4⋊Q8—1221no—
276C4⋊Q8—1221no—
277C4⋊Q8—1221no—
278C4⋊Q8—1221no—
279C4⋊Q8—1221no—
280C4⋊Q8—1221no—
281#127—611no—
282#125—1221no—
283#124—1221no—
284C22⋊Q8—1221no—
285C22⋊Q8—1221no—
286#121—1221no—
287#121—1221no—
288C4×Q8—1221no—
289C4×Q8—1221no—
290C4×D4—1221no—
291C4×D4—1221no—
292#117—1221no—
293C2×C4⋊C4—1221no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4—1221no—
296C2×C4⋊C4—1221no—
297C2×C42C22×D6111yescharacteristic
298#107—1224no—
299#106—1224no—
300#106—1222no—
301#106—1224no—
302#106—1224no—
303C4⋊C8D24412yes—
304#103—1224no—
305#103—1224no—
306#103—1222no—
307#103—1224no—
308#102—1222no—
309#102—1224no—
310C22⋊C8D24412yes—
311#97D24412yes—
312C4×C8D24412yes—
313C4×C8C22×D6111yescharacteristic
314C4×C8C22×D6112yes—
315C4×C8C22×D6112yes—
316C4×C8C22×D6111yescharacteristic
317#41—2444no—
318C2×Q8—1221no—
319C2×Q8—1222no—
320C2×Q8—1221no—
321C2×Q8—1221no—
322C2×Q8—1222no—
323C2×Q8—2444no—
324C2×Q8—2444no—
325C2×Q8—2441no—
326C2×Q8—2441no—
327C2×D4—1221no—
328C2×D4—1221no—
329C2×D4—1221no—
330C2×D4—2441no—
331C22×C4—2441no—
332C22×C4—2441no—
333C22×C4—2441no—
334C22×C4C2×D24411yescharacteristic
335C22×C4C2×D24411yescharacteristic
336C22×C4C2×D24411yescharacteristic
337Q16—2448no—
338SD16—2444no—
339SD16—2448no—
340M16#1175819018yescomplemented
341C2×C8—42no—
342C2×C8C2×D24412yes—
343C2×C8C2×D24412yes—
344C2×C8C2×D24414yes—
345C2×C8C2×D24414yes—
346C4⋊C4—1222no—
347C4⋊C4—1221no—
348C4⋊C4—1222no—
349C4⋊C4—1222no—
350C4⋊C4—1221no—
351C4⋊C4—1221no—
352C4⋊C4—1221no—
353C4⋊C4—1221no—
354C4⋊C4—1222no—
355C4⋊C4—1221no—
356C4⋊C4—2441no—
357C4⋊C4—2441no—
358C4⋊C4—2441no—
359C4⋊C4—2441no—
360C4⋊C4—2441no—
361C4⋊C4—2441no—
362C22⋊C4—2441no—
363C22⋊C4—2441no—
364C22⋊C4—2441no—
365C42—2441no—
366C42C2×D24411yescharacteristic
367C42C2×D24411yescharacteristic
368C42C2×D24411yescharacteristic
369C42C23×D6111yescharacteristic
370C23—2441no—
371C23#1451977411yescharacteristic
372Q8—4882no—
373Q8—4882no—
374Q8—2444no—
375Q8—2442no—
376Q8—2442no—
377Q8—2444no—
378Q8—2442no—
379D4—4882no—
380D4—2442no—
381D4—2442no—
382D4—2442no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—4882no—
386C2×C4—4882no—
387C2×C4—4882no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2442no—
392C2×C4—2442no—
393C2×C4#1451977411yescharacteristic
394C2×C4#1451977411yescharacteristic
395C2×C4#1451977411yescharacteristic
396C2×C4C2×#1175819014yes—
397C2×C4C22×D24411yescharacteristic
398C2×C4C22×D24411yescharacteristic
399C2×C4C22×D24411yescharacteristic
400C8—22no—
401C8—22no—
402C8—24no—
403C8C2×#1175819018yes—
404C22—21no—
405C22—21no—
406C22—4881no—
407C22—4881no—
408C22—4881no—
409C22#1684013512yes—
410C22C2×#1451977411yescharacteristic
411C4—21no—
412C4—21no—
413C4—4881no—
414C4—4882no—
415C4—4882no—
416C4—4881no—
417C4—4881no—
418C4#1684013512yes—
419C4C22×#1175819012yes—
420C4#1684056212yes—
421C4#1684057912yes—
422C2—22no—
423C2—9761no—
424C2C2×#1684013511yescharacteristic
425C2#2145628511yescharacteristic
426C2#2145631311yescharacteristic
subgroupG/H
centerC22C2×#14519774
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC8×M16×C61C2
socleC22×C61C2×#127
prime radical#16840312C22
hypercenterC8×M16D61
hypocenterC61#19624
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C8×M16

Complements

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

NH ≅ G/Ntype
1. #21456965C2semidirect
4. #21456365C2semidirect
6. C2×#16840113C2semidirect
7. #21445452C2semidirect
8. #21445441C2semidirect
9. #21445420C2semidirect
10. #21445416C2semidirect
11. C8×M16×C61C2semidirect
123. M16×C61SD16semidirect
219. C8×M16D61semidirect
260. C61#19624semidirect
340. M16#11758190semidirect

Maximal subgroups

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

subgroupindex
#214569652
#214569492
#214563782
#214563652
#214562782
C2×#168401132
#214454522
#214454412
#214454202
#214454162
C8×M16×C612
#1962461