Finite Group Databasebeta

← #29663520

Subgroups — #29663520

Order 15616.

652 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457136C211yesmaximal, characteristic
2#21456746C211yesmaximal, characteristic
3#21456365C211yesmaximal, characteristic, complemented
4#21454565C211yesmaximal, characteristic
5#21445333C212yesmaximal, complemented
6#21445319C212yesmaximal, complemented
7#21445209C212yesmaximal, complemented
8#21445195C212yesmaximal, complemented
9#21444474C211yesmaximal, characteristic, complemented
10C2×#16839509C211yesmaximal, characteristic, complemented
11C61×#1565C211yesmaximal, characteristic, complemented
12C2×#14519785—21no—
13#16840319C2211yescharacteristic
14#16840312C2211yescharacteristic
15C2×#14519777C2211yescharacteristic
16#16840171—22no—
17#16840167C2212yes—
18#16840160C2212yes—
19#16840135—21no—
20#16840113—21no—
21#16840053C2212yes—
22#16840048—21no—
23#16839991—22no—
24#16839968C2212yes—
25#16839968C2212yes—
26#16839968C2212yes—
27#16839963—21no—
28#16839903—22no—
29#16839841C2212yes—
30#16839835—22no—
31C8×D244C2212yes—
32#16839804—21no—
33#16839804—21no—
34C2×C4⋊#92418C2211yescharacteristic
35#16839730—21no—
36#16839510C2211yescharacteristic
37#16839510C2211yescharacteristic
38#16839509C2212yes—
39#16839509C2211yescharacteristic
40#16839509C2212yes—
41#16839509C2211yescharacteristic
42C61×#431C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C4×M16×C61C2211yescharacteristic
45C61×#335C2212yes—
46C61×#334C2212yes—
47C2×#11758222—21no—
48C22×Dic122—21no—
49C2×#11758190—22no—
50C2×#11758190—22no—
51C2×#11758190—22no—
52#14519785—44no—
53C2×#11758189—22no—
54C22×#92418—21no—
55C2×C8×D61—22no—
56#14519777—21no—
57#14519777—21no—
58#14519777—21no—
59#14519777D411yescharacteristic
60#14519777C2312yes—
61#14519777C2311yescharacteristic
62#14519777C2311yescharacteristic
63#14519774D411yescharacteristic
64#14519769—21no—
65#14519765—21no—
66#14519756—21no—
67#14519756—21no—
68#14519751—21no—
69#14519751D412yes—
70C4×Dic122—21no—
71C4×Dic122—21no—
72C4×D244—21no—
73C4×D244C2312yes—
74#14519736—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77#14519711—22no—
78#14519711—22no—
79#14519709—22no—
80#14519709—22no—
81#14519709—22no—
82C4⋊#92418—21no—
83C4⋊#92418C2311yescharacteristic
84C4⋊#92418C2311yescharacteristic
85#14519702—22no—
86#14519702—22no—
87#14519702—22no—
88#14519699—22no—
89#14519689—22no—
90#14519689—22no—
91C2×M16×C61D411yescharacteristic
92C2×M16×C61D411yescharacteristic
93C22×C8×C61—21no—
94C2×C42×C61C2311yescharacteristic
95C4⋊C8×C61—21no—
96C4⋊C8×C61C2311yescharacteristic
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2312yes—
99C22⋊C8×C61—21no—
100#97×C61D412yes—
101C4×C8×C61—21no—
102C4×C8×C61C2312yes—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61#4111yescharacteristic
106C2×C22⋊D61—22no—
107C22×Dic61—41no—
108#11758222—42no—
109#11758222—42no—
110C2×D244—21no—
111C2×D244—21no—
112C2×D244C2×D412yes—
113C2×C4×D61—22no—
114C2×Dic122—21no—
115C2×Dic122—22no—
116C2×Dic122—21no—
117C2×Dic122—21no—
118C2×Dic122—41no—
119C2×Dic122—81no—
120C2×Dic122—81no—
121M16×C61—22no—
122M16×C61SD1614yescomplemented
123C2×C8×C61—21no—
124C2×C8×C61—21no—
125C2×C8×C61—22no—
126C2×C8×C61—81no—
127C2×C8×C61C2×D412yes—
128C2×C8×C61C2×D412yes—
129C42×C61C2×D411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C42×C61#4111yescharacteristic
132C42×C61C2411yescharacteristic
133C22⋊Dic61—41no—
134#11758198—22no—
135#11758198—22no—
136C4⋊Dic61—21no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61—22no—
140C4⋊Dic61—21no—
141C4⋊Dic61—22no—
142C4⋊Dic61—21no—
143C4⋊Dic61—41no—
144C4⋊Dic61—41no—
145C4⋊Dic61C2×D412yes—
146#11758196—41no—
147#11758196—41no—
148C4×Dic61—41no—
149C2×#92418—21no—
150C2×#92418—21no—
151C2×#92418—42no—
152#11758190—82no—
153#11758190—82no—
154#11758190—82no—
155#11758189—44no—
156C8×D61—44no—
157C23×C61#12111yescharacteristic
158C22×D61—22no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—41no—
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61C2×SD1612yes—
166C2×C4×C61#13712yes—
167C2×C4×C61C22×D411yescharacteristic
168C2×C4×C61C22×D411yescharacteristic
169C2×C4×C61C2×#4111yescharacteristic
170C22⋊D61—44no—
171C2×Dic61—22no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174C2×Dic61—42no—
175C2×Dic61—82no—
176D244—22no—
177D244—22no—
178D244—42no—
179D244—81no—
180C4×D61—42no—
181C4×D61—42no—
182Dic122—42no—
183Dic122—41no—
184Dic122—42no—
185Dic122—42no—
186Dic122—41no—
187Dic122—82no—
188Dic122—81no—
189C8×C61—22no—
190C8×C61—41no—
191C8×C61—42no—
192C8×C61—41no—
193C8×C61C2×SD1614yes—
194#92418—41no—
195#92418—41no—
196#19232—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61#46112yes—
200C22×C61C2×#12111yescharacteristic
201C2×D61—42no—
202C2×D61—42no—
203C2×D61—42no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61#46112yes—
208C4×C61C22×SD1611yescharacteristic
209C4×C61C2×#13711yescharacteristic
210C4×C61#57611yescharacteristic
211C4×C61#57711yescharacteristic
212Dic61—42no—
213Dic61—82no—
214Dic61—81no—
215Dic61—81no—
216#3246—611no—
217#3206—611no—
218#3165—611no—
219#2935—611no—
220#1693—612no—
221#1691—612no—
222#1673—612no—
223#1671—612no—
224#1565D6111yescharacteristic, complemented
225#1542—611no—
226C2×#332—611no—
227C2×C61—22no—
228C2×C61C2×#46111yescharacteristic
229C2×C61#306211yescharacteristic
230C2×C61#306911yescharacteristic
231D61—82no—
232C2×#131—1221no—
233#533—611no—
234#532—611no—
235C2×C4⋊Q8—611no—
236#501—1222no—
237#500—612no—
238#498—612no—
239#492—1221no—
240#487—1221no—
241#476—612no—
242#475—1221no—
243#465—1222no—
244#461—612no—
245#461—612no—
246#461—612no—
247#460—1221no—
248#448—1222no—
249#436—612no—
250#435—1222no—
251C8×D4—612no—
252#431C2×D6111yescharacteristic
253#428—1221no—
254#428—1221no—
255C2×C4⋊C8—611no—
256C2×C4⋊C8C2×D6111yescharacteristic
257#408—1221no—
258C4×M16C2×D6111yescharacteristic
259#335C2×D6112yes—
260#334C2×D6112yes—
261#333—611no—
262#333—611no—
263#332—612no—
264#332—612no—
265#332—611no—
266#332—611no—
267C61#1923211yescharacteristic, complemented, Sylow
268C2×#41—1221no—
269C22×Q8—1221no—
270C2×SD16—1222no—
271C2×SD16—1222no—
272C2×SD16—1222no—
273#131—2444no—
274C2×M16—1222no—
275C2×M16D24411yescharacteristic
276C2×M16D24411yescharacteristic
277C22×C8—21no—
278C22×C8—1221no—
279C22×C8—1222no—
280C4⋊Q8—611no—
281C4⋊Q8—612no—
282C4⋊Q8—611no—
283C4⋊Q8—611no—
284C4⋊Q8—1221no—
285C4⋊Q8—1221no—
286C4⋊Q8—1221no—
287#127—611no—
288#125—1221no—
289#124—1221no—
290C22⋊Q8—1221no—
291C22⋊Q8—1221no—
292#121—612no—
293#121—1221no—
294C4×Q8—1221no—
295C4×Q8—1221no—
296C4×D4—612no—
297C4×D4—1221no—
298#117—1221no—
299C2×C4⋊C4—1221no—
300C2×C4⋊C4—1221no—
301C2×C42C22×D6111yescharacteristic
302#107—1222no—
303#107—1222no—
304#106—1222no—
305#106—1222no—
306#106—1222no—
307C4⋊C8—21no—
308C4⋊C8—611no—
309C4⋊C8—611no—
310C4⋊C8—1221no—
311C4⋊C8C22×D6112yes—
312C4⋊C8C22×D6111yescharacteristic
313C4⋊C8C22×D6111yescharacteristic
314#103—1222no—
315#103—1222no—
316#103—1222no—
317#102—1222no—
318C22⋊C8—21no—
319C22⋊C8—1222no—
320C22⋊C8—1222no—
321#97D24412yes—
322C4×C8—21no—
323C4×C8C22×D6112yes—
324#41—2442no—
325#41—2442no—
326C2×Q8—1221no—
327C2×Q8—1221no—
328C2×Q8—1222no—
329C2×Q8—1221no—
330C2×Q8—4881no—
331C2×Q8—4881no—
332C2×Q8—2441no—
333C2×D4—612no—
334C2×D4—1222no—
335C2×D4—1221no—
336C2×D4—1221no—
337C22×C4—1222no—
338C22×C4—2441no—
339C22×C4C2×D24411yescharacteristic
340C22×C4D4×D6111yescharacteristic
341C22×C4#1175822611yescharacteristic
342SD16—4882no—
343SD16—4882no—
344SD16—4882no—
345M16—22no—
346M16—2444no—
347M16#1175819014yescomplemented
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—22no—
351C2×C8—81no—
352C2×C8—1221no—
353C2×C8—1221no—
354C2×C8—2442no—
355C2×C8—2444no—
356C2×C8C2×D24412yes—
357C2×C8C2×D24412yes—
358C4⋊C4—612no—
359C4⋊C4—1221no—
360C4⋊C4—1221no—
361C4⋊C4—1221no—
362C4⋊C4—1222no—
363C4⋊C4—1221no—
364C4⋊C4—1222no—
365C4⋊C4—1221no—
366C4⋊C4—2441no—
367C4⋊C4—2441no—
368C4⋊C4—2441no—
369C4⋊C4—2441no—
370C22⋊C4—1222no—
371C22⋊C4—1222no—
372C22⋊C4—2441no—
373C42—2441no—
374C42C2×D24411yescharacteristic
375C42D4×D6111yescharacteristic
376C42#1175822611yescharacteristic
377C42C23×D6111yescharacteristic
378C23—1222no—
379C23#1451975211yescharacteristic
380Q8—4881no—
381Q8—4882no—
382Q8—2441no—
383Q8—2442no—
384Q8—2442no—
385Q8—2441no—
386Q8—2442no—
387D4—1222no—
388D4—1222no—
389D4—4881no—
390D4—2444no—
391D4—2442no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—1222no—
396C2×C4—4882no—
397C2×C4—2441no—
398C2×C4—2441no—
399C2×C4—2442no—
400C2×C4—2442no—
401C2×C4—2442no—
402C2×C4#1451975211yescharacteristic
403C2×C4#1451975211yescharacteristic
404C2×C4#1451975211yescharacteristic
405C2×C4C2×#1175819012yes—
406C2×C4#1451981012yes—
407C2×C4C2×D4×D6111yescharacteristic
408C2×C4C22×D24411yescharacteristic
409C2×C4C2×#1175822611yescharacteristic
410C8—22no—
411C8—42no—
412C8—41no—
413C8—41no—
414C8—2441no—
415C8—2441no—
416C8C2×#1175819014yes—
417C22—21no—
418C22—21no—
419C22—2442no—
420C22—2442no—
421C22—2442no—
422C22#1683996812yes—
423C22C2×#1451975211yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—4882no—
428C4—4881no—
429C4—4881no—
430C4—2442no—
431C4#1683996812yes—
432C4C22×#1175819011yescharacteristic
433C4C2×#1451981011yescharacteristic
434C4#1684056611yescharacteristic
435C4#1684057111yescharacteristic
436C2—22no—
437C2—4882no—
438C2C2×#1683996811yescharacteristic
439C2#2145550011yescharacteristic
440C2#2145555711yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1565C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#1565D61
hypocenterC61#19232
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1565

Complements

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

NH ≅ G/Ntype
3. #21456365C2semidirect
5. #21445333C2semidirect
6. #21445319C2semidirect
7. #21445209C2semidirect
8. #21445195C2semidirect
9. #21444474C2semidirect
10. C2×#16839509C2semidirect
11. C61×#1565C2semidirect
122. M16×C61SD16semidirect
224. #1565D61semidirect
267. C61#19232semidirect
347. M16#11758190semidirect

Maximal subgroups

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

subgroupindex
#214571362
#214567462
#214563652
#214545652
#214453332
#214453192
#214452092
#214451952
#214444742
C2×#168395092
C61×#15652
#1923261