Finite Group Databasebeta

← #29661227

Subgroups — #29661227

Order 15616.

1152 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840504C211yesmaximal, characteristic, complemented
2#21456802C212yesmaximal, complemented
3#21445123C214yesmaximal, complemented
4#21445107C214yesmaximal, complemented
5#21444443C211yesmaximal, characteristic, complemented
6#21444416C212yesmaximal, complemented
7C61×#3153C211yesmaximal, characteristic, complemented
8#16840504—24no—
9#16840504—22no—
10#16840504C2214yes—
11C2×#14519766—21no—
12C2×#14519765C2212yes—
13C2×#14519763—22no—
14C2×#14519747—22no—
15C2×#14519745—21no—
16#16840149C2214yescomplemented
17#16840124C2214yescomplemented
18#16839977—24no—
19#16839900—22no—
20#16839895—22no—
21#16839819C2212yescomplemented
22#16839502C2214yescomplemented
23#16839501C2218yescomplemented
24C2×#137×C61—21no—
25C2×AGL(1,ℤ/8)×C61—21no—
26C61×#532C2211yescharacteristic, complemented
27C2×#124×C61C2211yescharacteristic
28C61×#497C2212yescomplemented
29C61×#496C2212yescomplemented
30C61×#495—22no—
31C4×M16×C61C2211yescharacteristic, complemented
32C24×D61—21no—
33C22×Dic122D412yes—
34C22×D244—21no—
35C22×D244D412yes—
36C22×C22⋊D61—22no—
37C2×#11758200—24no—
38C2×#11758199—24no—
39#14519766—24no—
40#14519766—24no—
41#14519765—22no—
42#14519765—22no—
43#14519765D414yes—
44#14519765C2314yes—
45#14519763—28no—
46#14519763—28no—
47#14519747—28no—
48#14519747—28no—
49#14519745—24no—
50#14519745—24no—
51C2×C22⋊Dic61—21no—
52C2×#11758198—22no—
53C2×#11758198—22no—
54C2×#11758198—22no—
55C2×C4×Dic61—21no—
56C4⋊#92418—22no—
57#14519700—24no—
58#14519697—24no—
59C22⋊#92418—22no—
60#14519688C2314yescomplemented
61C2×#41×C61—21no—
62C22×Q8×C61D411yescharacteristic
63C22×D4×C61D411yescharacteristic
64#137×C61—44no—
65AGL(1,ℤ/8)×C61—44no—
66C2×Q16×C61—22no—
67C2×SD16×C61—22no—
68C2×SD16×C61—22no—
69C2×D8×C61—22no—
70C2×M16×C61—21no—
71C4⋊Q8×C61C2311yescharacteristic, complemented
72#127×C61C2311yescharacteristic, complemented
73#124×C61—22no—
74#124×C61—21no—
75#124×C61D412yes—
76#124×C61C2312yes—
77#121×C61—22no—
78C4×D4×C61—22no—
79C2×C22⋊C4×C61—22no—
80C2×C42×C61C2311yescharacteristic
81#97×C61C2312yescomplemented
82C4×C8×C61—21no—
83C24×C61—21no—
84C23×D61—22no—
85C23×D61—24no—
86C23×D61—84no—
87C23×D61—82no—
88#41×C61—42no—
89#41×C61—42no—
90C2×Q8×C61—21no—
91C2×Q8×C61—21no—
92C2×Q8×C61—42no—
93C2×Q8×C61C2×D412yes—
94C2×D4×C61—21no—
95C2×D4×C61—22no—
96C2×D4×C61—21no—
97C2×D4×C61—42no—
98C2×D4×C61—41no—
99C2×D4×C61C2×D412yes—
100C22×C4×C61—21no—
101C22×C4×C61—41no—
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D412yes—
104C2×C22⋊D61—28no—
105C2×C22⋊D61—84no—
106C22×Dic61—22no—
107C2×D244—24no—
108C2×D244—22no—
109C2×D244—82no—
110C2×D244—82no—
111C2×D244C2×D414yes—
112C2×Dic122—22no—
113C2×Dic122—82no—
114C2×Dic122C2×D414yes—
115Q16×C61—44no—
116SD16×C61—44no—
117SD16×C61—44no—
118D8×C61—44no—
119M16×C61—24no—
120C2×C8×C61—22no—
121C4⋊C4×C61—22no—
122C22⋊C4×C61—24no—
123C22⋊C4×C61—24no—
124C22⋊C4×C61—42no—
125C42×C61C2×D412yes—
126C42×C61C2×D411yescharacteristic
127C42×C61C2411yescharacteristic
128C22⋊Dic61—24no—
129#11758200—84no—
130#11758199—84no—
131#11758198—28no—
132#11758198—24no—
133#11758198—24no—
134#11758198—24no—
135#11758198—24no—
136C4×Dic61—24no—
137C2×#92418—24no—
138C23×C61—22no—
139C23×C61—44no—
140C23×C61—42no—
141C23×C61—41no—
142C23×C61#12011yescharacteristic
143C22×D61—24no—
144C22×D61—44no—
145C22×D61—44no—
146C22×D61—44no—
147C22×D61—84no—
148C22×D61—88no—
149C22×D61—816no—
150Q8×C61—42no—
151Q8×C61—41no—
152Q8×C61—41no—
153Q8×C61—81no—
154D4×C61—42no—
155D4×C61—44no—
156D4×C61—41no—
157D4×C61—41no—
158D4×C61—81no—
159D4×C61—82no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—42no—
163C2×C4×C61—42no—
164C2×C4×C61—41no—
165C2×C4×C61—82no—
166C2×C4×C61#12011yescharacteristic
167C2×C4×C61#12012yes—
168C2×C4×C61C22×D412yes—
169C2×C4×C61C22×D411yescharacteristic
170C22⋊D61—88no—
171C2×Dic61—24no—
172C2×Dic61—44no—
173D244—44no—
174D244—82no—
175D244—84no—
176Dic122—44no—
177Dic122—82no—
178C8×C61—24no—
179#92418—44no—
180#18992—611nomaximal, Sylow
181C22×C61—22no—
182C22×C61—42no—
183C22×C61—44no—
184C22×C61—84no—
185C22×C61—81no—
186C22×C61—82no—
187C22×C61#45312yes—
188C22×C61C2×#12011yescharacteristic
189C2×D61—44no—
190C2×D61—44no—
191C2×D61—44no—
192C2×D61—88no—
193C2×D61—816no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—22no—
197C4×C61—22no—
198C4×C61—42no—
199C4×C61—81no—
200Dic61—44no—
201C2×H(3,𝔽4)—611no—
202#3213—612no—
203#3153D6111yescharacteristic, complemented
204#1659—614no—
205#1655—614no—
206#1533—611no—
207#1527—612no—
208C2×C61—22no—
209C2×C61—82no—
210C2×C61—161no—
211C2×C61C2×#45311yescharacteristic
212C2×C61#301912yes—
213D61—84no—
214C2×#137—21no—
215C2×AGL(1,ℤ/8)—21no—
216H(3,𝔽4)—614no—
217H(3,𝔽4)—1224no—
218H(3,𝔽4)—1222no—
219#532C2×D6111yescharacteristic, complemented
220C2×#124—612no—
221C2×#124—1222no—
222C2×#124—1221no—
223C2×#124C2×D6111yescharacteristic
224C2×#120—1222no—
225C2×#120—1221no—
226#497C2×D6112yescomplemented
227#496—614no—
228#496C2×D6112yescomplemented
229#495—22no—
230#489—614no—
231#463—1224no—
232#448—1222no—
233#447—1222no—
234#432—612no—
235C4×M16C2×D6111yescharacteristic, complemented
236#329—614no—
237#329—618no—
238C61#1899211yescharacteristic, complemented, Sylow
239C25—1221no—
240C2×#41—21no—
241C22×Q8—612no—
242C22×Q8C22⋊D6111yescharacteristic
243C22×D4—612no—
244C22×D4—1221no—
245C22×D4—1222no—
246C22×D4C22⋊D6111yescharacteristic
247#137—44no—
248AGL(1,ℤ/8)—44no—
249C2×Q16—22no—
250C2×SD16—22no—
251C2×SD16—22no—
252C2×SD16—1224no—
253C2×D8—22no—
254C2×D8—1224no—
255C2×M16—21no—
256C4⋊Q8C22×D6111yescharacteristic, complemented
257#127C22×D6111yescharacteristic, complemented
258#124—22no—
259#124—21no—
260#124—614no—
261#124—614no—
262#124—1228no—
263#124—1228no—
264#124—1224no—
265#124—1224no—
266#124—1222no—
267#124—1222no—
268#124C22⋊D6112yes—
269#124C22×D6112yes—
270#121—22no—
271#120—1228no—
272#120—1228no—
273#120—1224no—
274#120—1224no—
275C4×D4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—1222no—
278C2×C22⋊C4—1221no—
279C2×C22⋊C4—1222no—
280C2×C22⋊C4—1222no—
281C2×C42—1221no—
282C2×C42C22×D6111yescharacteristic
283C4⋊C8—1222no—
284#103—1224no—
285#102—1224no—
286C22⋊C8—1222no—
287#97—614no—
288#97C22×D6112yescomplemented
289C4×C8—21no—
290C24—21no—
291C24—1224no—
292C24—1222no—
293C24—4884no—
294C24—4882no—
295#41—42no—
296#41—42no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—42no—
300C2×Q8—614no—
301C2×Q8—1222no—
302C2×Q8—4882no—
303C2×Q8C2×C22⋊D6112yes—
304C2×D4—22no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—42no—
308C2×D4—41no—
309C2×D4—614no—
310C2×D4—1224no—
311C2×D4—1228no—
312C2×D4—1222no—
313C2×D4—4882no—
314C2×D4—4882no—
315C2×D4—4884no—
316C2×D4C2×C22⋊D6112yes—
317C22×C4—21no—
318C22×C4—41no—
319C22×C4—1222no—
320C22×C4D4×D6112yes—
321C22×C4C2×C22⋊D6111yescharacteristic
322Q16—44no—
323SD16—44no—
324SD16—44no—
325SD16—4884no—
326D8—44no—
327D8—4884no—
328M16—24no—
329C2×C8—22no—
330C2×C8—1224no—
331C4⋊C4—22no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—42no—
335C22⋊C4—1228no—
336C22⋊C4—1224no—
337C22⋊C4—1224no—
338C22⋊C4—1224no—
339C22⋊C4—1224no—
340C22⋊C4—1224no—
341C42—1224no—
342C42D4×D6112yes—
343C42C2×C22⋊D6111yescharacteristic
344C42C23×D6111yescharacteristic
345C23—22no—
346C23—44no—
347C23—42no—
348C23—41no—
349C23—1224no—
350C23—4884no—
351C23—4888no—
352C23—48816no—
353C23—2444no—
354C23—2444no—
355C23—2444no—
356C23#1451974511yescharacteristic
357Q8—42no—
358Q8—41no—
359Q8—41no—
360Q8—81no—
361Q8—4882no—
362Q8—2444no—
363D4—42no—
364D4—41no—
365D4—44no—
366D4—41no—
367D4—81no—
368D4—82no—
369D4—4882no—
370D4—4884no—
371D4—4888no—
372D4—2444no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—82no—
379C2×C4—1224no—
380C2×C4—2444no—
381C2×C4#1451974511yescharacteristic
382C2×C4#1451974512yes—
383C2×C4C2×D4×D6112yes—
384C2×C4C22×C22⋊D6111yescharacteristic
385C8—24no—
386C8—2444no—
387C22—22no—
388C22—42no—
389C22—44no—
390C22—84no—
391C22—81no—
392C22—82no—
393C22—4888no—
394C22—48816no—
395C22—2444no—
396C22—2444no—
397C22—2444no—
398C22#1683992912yes—
399C22C2×#1451974511yescharacteristic
400C4—21no—
401C4—21no—
402C4—22no—
403C4—22no—
404C4—42no—
405C4—81no—
406C4—2444no—
407C2—22no—
408C2—82no—
409C2—161no—
410C2—4884no—
411C2C2×#1683992911yescharacteristic
412C2#2145517412yes—
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3153C2
socleC22×C61C2×#120
hypercenter#3153D61
hypocenterC61#18992
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3153

Complements

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

NH ≅ G/Ntype
1. C2×#16840504C2semidirect
2. #21456802C2semidirect
3. #21445123C2semidirect
4. #21445107C2semidirect
5. #21444443C2semidirect
6. #21444416C2semidirect
7. C61×#3153C2semidirect
16. #16840149C22semidirect
17. #16840124C22semidirect
21. #16839819C22semidirect
22. #16839502C22semidirect
23. #16839501C22semidirect
26. C61×#532C22semidirect
28. C61×#497C22semidirect
29. C61×#496C22semidirect
31. C4×M16×C61C22semidirect
60. #14519688C23semidirect
71. C4⋊Q8×C61C23semidirect
72. #127×C61C23semidirect
81. #97×C61C23semidirect
203. #3153D61semidirect
219. #532C2×D61semidirect
226. #497C2×D61semidirect
228. #496C2×D61semidirect
235. C4×M16C2×D61semidirect
238. C61#18992semidirect
256. C4⋊Q8C22×D61semidirect
257. #127C22×D61semidirect
288. #97C22×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168405042
#214568022
#214451232
#214451072
#214444432
#214444162
C61×#31532
#1899261