Finite Group Databasebeta

← #29667868

Subgroups — #29667868

Order 15616.

784 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456960C211yesmaximal, characteristic, complemented
2#21456744C211yesmaximal, characteristic
3#21456379C211yesmaximal, characteristic
4C2×#16840158C211yesmaximal, characteristic, complemented
5#21456278C211yesmaximal, characteristic
6#21445453C212yesmaximal, complemented
7#21445442C212yesmaximal, complemented
8#21445417C212yesmaximal, complemented
9#21445407C212yesmaximal, complemented
10M16×#92418C211yesmaximal, characteristic, complemented
11C61×#3165C211yesmaximal, characteristic, complemented
12C2×#14519810—22no—
13#16840312C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15#16840158—22no—
16#16840158C2212yes—
17#16840158C2212yes—
18#16840158C2211yescharacteristic
19#16840158C2212yes—
20#16840156—22no—
21#16840142C2212yes—
22#16840136C2211yescharacteristic
23#16840136C2212yes—
24#16840135C2212yes—
25#16840112C2212yes—
26#16840112C2212yes—
27#16840112C2211yescharacteristic
28#16840110C2211yescharacteristic
29#16840090—22no—
30#16840047—22no—
31#16839993—22no—
32#16839964—22no—
33#16839901—22no—
34C4×#11758201—22no—
35#16839815C2211yescharacteristic
36C2×#14519708—22no—
37C2×C4×#92418C2211yescharacteristic
38#16839483C2212yes—
39C8×#92418C2212yes—
40C61×#533C2211yescharacteristic
41C2×C4⋊Q8×C61C2211yescharacteristic
42C61×#501—22no—
43C61×#500C2212yes—
44C61×#498C2212yes—
45C61×#428—22no—
46C4×M16×C61C2211yescharacteristic
47C2×#11758222—21no—
48C22×Dic122—21no—
49#14519810—48no—
50C2×Dic244—24no—
51C2×#11758190—24no—
52C2×#11758201—22no—
53C2×#11758201—24no—
54C22×#92418D412yes—
55#14519777—21no—
56#14519777—21no—
57#14519777—21no—
58#14519777C2312yes—
59#14519777C2311yescharacteristic
60#14519774C2311yescharacteristic
61#14519765—21no—
62#14519751—21no—
63#14519751—21no—
64C4×D244—21no—
65C4×D244—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68#14519708—22no—
69#14519708—24no—
70#14519708—24no—
71#14519708—24no—
72C4⋊#92418D412yes—
73#14519701—24no—
74#14519701—22no—
75#14519701—24no—
76#14519697—22no—
77C22⋊#92418D412yes—
78C4×#92418D412yes—
79C4×#92418C2311yescharacteristic
80C4×#92418C2311yescharacteristic
81C4×#92418C2312yes—
82C22×Q8×C61—21no—
83C2×M16×C61D412yes—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61C2312yes—
88C4⋊Q8×C61C2311yescharacteristic
89C4⋊Q8×C61C2311yescharacteristic
90#125×C61—21no—
91C22⋊Q8×C61—21no—
92C22⋊Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95#117×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C42×C61C2311yescharacteristic
99#107×C61—24no—
100#106×C61—24no—
101#97×C61D412yes—
102C4×C8×C61C2312yes—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—22no—
106C2×Q8×C61—41no—
107C2×Q8×C61—82no—
108C22×C4×C61—41no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C2×C22⋊D61—41no—
113C22×Dic61—41no—
114#11758222—44no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244—21no—
118C2×C4×D61—41no—
119C2×Dic122—22no—
120C2×Dic122—21no—
121C2×Dic122—44no—
122C2×Dic122—41no—
123M16×C61SD1618yescomplemented
124C2×C8×C61C2×D414yes—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—41no—
132C4⋊C4×C61—41no—
133C4⋊C4×C61—41no—
134C4⋊C4×C61—41no—
135C22⋊C4×C61—41no—
136C42×C61—41no—
137C42×C61C2×D411yescharacteristic
138C42×C61C2×D411yescharacteristic
139C42×C61C2×D411yescharacteristic
140C42×C61C2411yescharacteristic
141#11758201—44no—
142#11758201—84no—
143#11758198—41no—
144#11758198—41no—
145C4⋊Dic61—21no—
146C4⋊Dic61—22no—
147C4⋊Dic61—21no—
148C4⋊Dic61—22no—
149C4⋊Dic61—41no—
150C4⋊Dic61—41no—
151C2×#92418—24no—
152C2×#92418C2×D412yes—
153C2×#92418C2×D414yes—
154C2×#92418C2×D412yes—
155Dic244—48no—
156#11758190—48no—
157C23×C61#12711yescharacteristic
158C22×D61—41no—
159Q8×C61—42no—
160Q8×C61—44no—
161Q8×C61—42no—
162Q8×C61—82no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—42no—
168C2×C4×C61—82no—
169C2×C4×C61#12711yescharacteristic
170C2×C4×C61#12711yescharacteristic
171C2×C4×C61#12711yescharacteristic
172C2×C4×C61C2×SD1614yes—
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C2×C4×C61C22×D411yescharacteristic
176C22⋊D61—82no—
177C2×Dic61—42no—
178C2×Dic61—41no—
179C2×Dic61—82no—
180D244—42no—
181D244—42no—
182D244—42no—
183C4×D61—82no—
184Dic122—44no—
185Dic122—42no—
186Dic122—82no—
187C8×C61C2×SD1618yes—
188#92418—22no—
189#92418—24no—
190#92418#13714yescomplemented
191#19622—611nomaximal, Sylow
192C22×C61—21no—
193C22×C61—21no—
194C22×C61#49212yes—
195C22×C61C2×#12711yescharacteristic
196C2×D61—81no—
197C2×D61—81no—
198C2×D61—81no—
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—81no—
202C4×C61—82no—
203C4×C61—81no—
204C4×C61#49212yes—
205C4×C61C22×SD1612yes—
206C4×C61C2×#13712yes—
207C4×C61#57712yes—
208Dic61—82no—
209Dic61—81no—
210#3229—611no—
211#3206—611no—
212#3167—611no—
213#3165D6111yescharacteristic, complemented
214C2×#498—611no—
215#3152—611no—
216#1716—612no—
217#1714—612no—
218#1707—612no—
219#1705—612no—
220C8×M16—611no—
221C2×C61—22no—
222C2×C61C2×#49211yescharacteristic
223C2×C61#315211yescharacteristic
224C2×C61#315411yescharacteristic
225D61—161no—
226C2×#137—1222no—
227#533C2×D6111yescharacteristic
228#532—611no—
229C2×C4⋊Q8—611no—
230C2×C4⋊Q8C2×D6111yescharacteristic
231#501—22no—
232#500C2×D6112yes—
233#498—612no—
234#498—612no—
235#498—611no—
236#498—612no—
237#498—1222no—
238#498C2×D6112yes—
239#497—1222no—
240#494—612no—
241#492—611no—
242#492—612no—
243#492—612no—
244#487—611no—
245#487—612no—
246#487—612no—
247#486—611no—
248#483—1222no—
249#475—1222no—
250#465—1222no—
251#461—1222no—
252#448—1222no—
253C4×SD16—1222no—
254#431—611no—
255#428—22no—
256C2×#106—1222no—
257C4×M16C2×D6111yescharacteristic
258C2×C4×C8—611no—
259#322—612no—
260C82—612no—
261C61#1962211yescharacteristic, complemented, Sylow
262C2×#41—1221no—
263C22×Q8—21no—
264C22×Q8—1221no—
265#137—2448no—
266C2×Q16—1224no—
267C2×SD16—1224no—
268C2×SD16—1222no—
269C2×SD16—1224no—
270C2×M16C22⋊D6112yes—
271C22×C8—612no—
272C4⋊Q8—21no—
273C4⋊Q8—21no—
274C4⋊Q8—21no—
275C4⋊Q8—612no—
276C4⋊Q8—611no—
277C4⋊Q8—1221no—
278C4⋊Q8—1221no—
279C4⋊Q8—1221no—
280C4⋊Q8C22×D6112yes—
281C4⋊Q8C22×D6111yescharacteristic
282C4⋊Q8C22×D6111yescharacteristic
283#127—611no—
284#125—21no—
285#124—1221no—
286C22⋊Q8—21no—
287C22⋊Q8—21no—
288#121—1221no—
289#121—1221no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×D4—1221no—
293C4×D4—1221no—
294#117—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—1221no—
298C2×C4⋊C4—1221no—
299C2×C42C22×D6111yescharacteristic
300#107—24no—
301#106—24no—
302#106—1224no—
303#106—1224no—
304#106—1222no—
305#106—1224no—
306C4⋊C8—612no—
307#103—1222no—
308#103—1224no—
309#103—1224no—
310#102—1222no—
311C22⋊C8—612no—
312#97C22⋊D6112yes—
313C4×C8—612no—
314C4×C8—611no—
315C4×C8—612no—
316C4×C8—611no—
317C4×C8C22×D6112yes—
318#41—2444no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—22no—
322C2×Q8—41no—
323C2×Q8—82no—
324C2×Q8—1221no—
325C2×Q8—1222no—
326C2×Q8—2444no—
327C2×Q8—2441no—
328C2×D4—1221no—
329C2×D4—1221no—
330C2×D4—1221no—
331C2×D4—2441no—
332C22×C4—41no—
333C22×C4—2441no—
334C22×C4—2441no—
335C22×C4D4×D6111yescharacteristic
336C22×C4D4×D6111yescharacteristic
337C22×C4C2×C22⋊D6111yescharacteristic
338Q16—2448no—
339SD16—4884no—
340SD16—2444no—
341SD16—2448no—
342M16#1175820118yescomplemented
343C2×C8—612no—
344C2×C8—612no—
345C2×C8—614no—
346C2×C8—1224no—
347C2×C8C2×C22⋊D6114yes—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—22no—
353C4⋊C4—21no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—1221no—
359C4⋊C4—1222no—
360C4⋊C4—1221no—
361C4⋊C4—1222no—
362C4⋊C4—2441no—
363C4⋊C4—2441no—
364C22⋊C4—41no—
365C22⋊C4—2441no—
366C22⋊C4—2441no—
367C42—41no—
368C42D4×D6111yescharacteristic
369C42D4×D6111yescharacteristic
370C42C2×C22⋊D6111yescharacteristic
371C42C23×D6111yescharacteristic
372C23—2441no—
373C23#1451977311yescharacteristic
374Q8—42no—
375Q8—44no—
376Q8—42no—
377Q8—82no—
378Q8—4882no—
379Q8—2444no—
380Q8—2442no—
381D4—4882no—
382D4—2442no—
383D4—2442no—
384D4—2442no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—4882no—
392C2×C4—4882no—
393C2×C4—2441no—
394C2×C4—2442no—
395C2×C4#1451977311yescharacteristic
396C2×C4#1451977311yescharacteristic
397C2×C4#1451977311yescharacteristic
398C2×C4C2×#1175820114yes—
399C2×C4C2×D4×D6111yescharacteristic
400C2×C4C2×D4×D6111yescharacteristic
401C2×C4C22×C22⋊D6111yescharacteristic
402C8—614no—
403C8—1222no—
404C8—1224no—
405C8C2×#1175820118yes—
406C22—21no—
407C22—21no—
408C22—4881no—
409C22—4881no—
410C22—4881no—
411C22#1684013612yes—
412C22C2×#1451977311yescharacteristic
413C4—21no—
414C4—21no—
415C4—81no—
416C4—82no—
417C4—81no—
418C4—4881no—
419C4—4882no—
420C4#1684013612yes—
421C4C22×#1175820112yes—
422C4#137×D6112yes—
423C4#1684057212yes—
424C2—22no—
425C2—9761no—
426C2C2×#1684013611yescharacteristic
427C2#2145627411yescharacteristic
428C2#2145629311yescharacteristic
subgroupG/H
centerC22C2×#14519773
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3165C2
socleC22×C61C2×#127
prime radical#16840312C22
hypercenter#3165D61
hypocenterC61#19622
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
1. #21456960C2semidirect
4. C2×#16840158C2semidirect
6. #21445453C2semidirect
7. #21445442C2semidirect
8. #21445417C2semidirect
9. #21445407C2semidirect
10. M16×#92418C2semidirect
11. C61×#3165C2semidirect
123. M16×C61SD16semidirect
190. #92418#137semidirect
213. #3165D61semidirect
261. C61#19622semidirect
342. M16#11758201semidirect

Maximal subgroups

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

subgroupindex
#214569602
#214567442
#214563792
C2×#168401582
#214562782
#214454532
#214454422
#214454172
#214454072
M16×#924182
C61×#31652
#1962261