Finite Group Databasebeta

← #27151920

Subgroups — #27151920

Order 12416.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261356C211yesmaximal, characteristic, complemented
2#20261076C211yesmaximal, characteristic, complemented
3#20261059C211yesmaximal, characteristic, complemented
4#20261025C211yesmaximal, characteristic, complemented
5#20261015C211yesmaximal, characteristic, complemented
6#20260950C211yesmaximal, characteristic, complemented
7#20260950C211yesmaximal, characteristic, complemented
8#20260886C211yesmaximal, characteristic, complemented
9#20260869C211yesmaximal, characteristic
10#20260845C211yesmaximal, characteristic, complemented
11#20260835C211yesmaximal, characteristic, complemented
12C4×#13095499C211yesmaximal, characteristic, complemented
13C4×#13095496C211yesmaximal, characteristic, complemented
14#20260796C211yesmaximal, characteristic, complemented
15#518×C97C211yesmaximal, characteristic, complemented
16C2×#13095499C2211yescharacteristic
17C2×#13095499C2211yescharacteristic
18C2×#13095496C2211yescharacteristic
19C2×#13095496C2211yescharacteristic
20#15690810C2211yescharacteristic, complemented
21#15690805—21no—
22#15690798C2211yescharacteristic, complemented
23#15690796—21no—
24#15690792—21no—
25#15690789C2211yescharacteristic
26#15690788—21no—
27#15690784C2211yescharacteristic
28C4×Dic194C2211yescharacteristic, complemented
29C4×D388C2211yescharacteristic, complemented
30#15690743C2211yescharacteristic, complemented
31#15690743C2211yescharacteristic, complemented
32#15690741C2211yescharacteristic, complemented
33#15690741C2211yescharacteristic, complemented
34C4⋊#11715895C2211yescharacteristic, complemented
35C4⋊#11715895C2211yescharacteristic, complemented
36#15690733C2211yescharacteristic, complemented
37#15690733C2211yescharacteristic, complemented
38#15690731C2211yescharacteristic
39#15690731C2211yescharacteristic
40#15690731C2211yescharacteristic
41#15690731C2211yescharacteristic
42#15690730C2211yescharacteristic, complemented
43#15690730C2211yescharacteristic, complemented
44C22⋊#11715895C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46#15690721C2211yescharacteristic, complemented
47C4×#11715895C2211yescharacteristic, complemented
48C22×D4×C97C2211yescharacteristic
49C4×D4×C97—21no—
50C4×D4×C97—21no—
51C4×D4×C97C2211yescharacteristic
52C4×D4×C97C2211yescharacteristic
53C4×D4×C97C2211yescharacteristic
54C4×D4×C97C2211yescharacteristic
55#117×C97C2211yescharacteristic
56#117×C97C2211yescharacteristic
57C2×C22⋊C4×C97—21no—
58C2×C22⋊C4×C97—21no—
59C2×Dic194—21no—
60C2×Dic194C2311yescharacteristic
61C2×D388C2311yescharacteristic
62C2×C22⋊D97—21no—
63C2×C4×D97—21no—
64#13095499—22no—
65#13095499—22no—
66#13095496—22no—
67#13095496—22no—
68C2×#11715895C2311yescharacteristic
69C2×#11715895C2311yescharacteristic
70C2×#11715895C2311yescharacteristic
71C2×#11715895C2311yescharacteristic
72C4⋊Dic97—21no—
73C4⋊Dic97C2311yescharacteristic
74C4⋊Dic97C2311yescharacteristic
75#13095490—21no—
76#13095490—21no—
77C22⋊Dic97—21no—
78#13095488—21no—
79#13095488—21no—
80#13095488—21no—
81#13095488—21no—
82C4×Dic97—21no—
83C24×C97—21no—
84C2×D4×C97—22no—
85C2×D4×C97—22no—
86C2×D4×C97—21no—
87C2×D4×C97C2311yescharacteristic
88C2×D4×C97C2311yescharacteristic
89C22×C4×C97—21no—
90C22×C4×C97—21no—
91C22×C4×C97—21no—
92C22×C4×C97—21no—
93C22×C4×C97C2311yescharacteristic
94C4⋊C4×C97D411yescharacteristic
95C4⋊C4×C97D411yescharacteristic
96C4⋊C4×C97C2311yescharacteristic
97C4⋊C4×C97C2311yescharacteristic
98C22⋊C4×C97—21no—
99C22⋊C4×C97—21no—
100C22⋊C4×C97—21no—
101C22⋊C4×C97—21no—
102C22⋊C4×C97—21no—
103C22⋊C4×C97D411yescharacteristic
104C22⋊C4×C97D411yescharacteristic
105C42×C97—21no—
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97—24no—
109C23×C97—21no—
110C23×C97—21no—
111C23×C97—81no—
112C23×C97—81no—
113C23×C97C2×D411yescharacteristic
114C22×D97—21no—
115D4×C97—24no—
116D4×C97—21no—
117D4×C97—21no—
118D4×C97#4112yes—
119D4×C97#4112yes—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—81no—
125C2×C4×C97—81no—
126C2×C4×C97—81no—
127C2×C4×C97—81no—
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2×D411yescharacteristic
130C2×C4×C97C2×D411yescharacteristic
131C2×C4×C97C2×D411yescharacteristic
132C2×C4×C97C2×D411yescharacteristic
133C2×C4×C97C2411yescharacteristic
134C22⋊D97—81no—
135C2×Dic97—21no—
136C2×Dic97—21no—
137C2×Dic97—21no—
138D388—21no—
139D388—21no—
140C4×D97—81no—
141Dic194—21no—
142Dic194—21no—
143Dic194—81no—
144#11715895—21no—
145#11715895—21no—
146#11715895—21no—
147#11715895—21no—
148C22×C97—21no—
149C22×C97—22no—
150C22×C97—22no—
151C22×C97—21no—
152C22×C97—21no—
153C22×C97—41no—
154C22×C97—41no—
155C22×C97—41no—
156C22×C97—41no—
157C22×C97—84no—
158C22×C97C22×D411yescharacteristic
159C2×D97—41no—
160C2×D97—41no—
161C2×D97—41no—
162C4×C97—21no—
163C4×C97—21no—
164C4×C97—21no—
165C4×C97—21no—
166C4×C97—21no—
167C4×C97—41no—
168C4×C97—41no—
169C4×C97C2×#4111yescharacteristic
170C4×C97C2×#4111yescharacteristic
171Dic97—41no—
172Dic97—41no—
173Dic97—41no—
174C2×C97—41no—
175C2×C97—42no—
176C2×C97—42no—
177C2×C97#52511yescharacteristic
178C2×C97#57611yescharacteristic
179C2×C97#57711yescharacteristic
180D97—81no—
181#3118—971nomaximal, Sylow
182C97#311811yescharacteristic, complemented, Sylow
183#542—971no—
184#518D9711yescharacteristic, complemented
185#485—971no—
186#482—971no—
187#476—971no—
188#474—971no—
189#463—971no—
190#463—971no—
191#450—971no—
192#447—971no—
193#442—971no—
194#440—971no—
195C4×SD16—971no—
196C4×D8—971no—
197#433—971no—
198C22×D4C2×D9711yescharacteristic
199C2×SD16—971no—
200C2×SD16—971no—
201C2×D8—971no—
202C2×D8—971no—
203C4⋊Q8—971no—
204#126—1941no—
205#124—971no—
206#124—1941no—
207#123—1941no—
208C22⋊Q8—971no—
209C22⋊Q8—1941no—
210#121—971no—
211C4×Q8—971no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—971no—
215C4×D4C2×D9711yescharacteristic
216C4×D4C2×D9711yescharacteristic
217C4×D4C2×D9711yescharacteristic
218C4×D4C2×D9711yescharacteristic
219#117C2×D9711yescharacteristic
220#117C2×D9711yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223#107—971no—
224#107—971no—
225#106—971no—
226#106—971no—
227C4⋊C8—971no—
228C4⋊C8—971no—
229#103—971no—
230#103—971no—
231#102—971no—
232#102—971no—
233#102—971no—
234#102—971no—
235#102—971no—
236#102—971no—
237C22⋊C8—971no—
238C22⋊C8—971no—
239#97—971no—
240C4×C8—971no—
241C24—21no—
242C2×Q8—971no—
243C2×Q8—1941no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—971no—
248C2×D4—1941no—
249C2×D4C22×D9711yescharacteristic
250C2×D4C22×D9711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—1941no—
256C22×C4C22×D9711yescharacteristic
257SD16—1942no—
258SD16—1942no—
259D8—1942no—
260D8—1942no—
261C2×C8—971no—
262C2×C8—971no—
263C2×C8—971no—
264C2×C8—971no—
265C4⋊C4—971no—
266C4⋊C4—971no—
267C4⋊C4—1941no—
268C4⋊C4—1941no—
269C4⋊C4—1941no—
270C4⋊C4C22⋊D9711yescharacteristic
271C4⋊C4C22⋊D9711yescharacteristic
272C4⋊C4C22×D9711yescharacteristic
273C4⋊C4C22×D9711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1941no—
280C22⋊C4—1941no—
281C22⋊C4—1941no—
282C22⋊C4—1941no—
283C22⋊C4—1941no—
284C22⋊C4C22⋊D9711yescharacteristic
285C22⋊C4C22⋊D9711yescharacteristic
286C42—21no—
287C42—1941no—
288C42C22×D9711yescharacteristic
289C42C22×D9711yescharacteristic
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—1941no—
296C23C2×C22⋊D9711yescharacteristic
297Q8—7761no—
298Q8—1941no—
299Q8—1941no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—7761no—
304D4—1941no—
305D4—1941no—
306D4#1309551112yes—
307D4#1309551112yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—7761no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4—1941no—
320C2×C4C2×C22⋊D9711yescharacteristic
321C2×C4C2×C22⋊D9711yescharacteristic
322C2×C4C2×C22⋊D9711yescharacteristic
323C2×C4C2×C22⋊D9711yescharacteristic
324C2×C4C2×C22⋊D9711yescharacteristic
325C2×C4C23×D9711yescharacteristic
326C8—1941no—
327C8—1941no—
328C8—1941no—
329C8—1941no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—84no—
340C22—3881no—
341C22—3881no—
342C22—3881no—
343C22C22×C22⋊D9711yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—41no—
350C4—41no—
351C4—3881no—
352C4—3881no—
353C4—3881no—
354C4C2×#1309551111yescharacteristic
355C4C2×#1309551111yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—7761no—
360C2#2026124611yescharacteristic
361C2#2026153811yescharacteristic
362C2#2026154611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#518×C97C2
socleC22×C97C22×D4
prime radical#20260869C2
hypercenter#518D97
hypocenterC97#3118
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #20261356C2semidirect
2. #20261076C2semidirect
3. #20261059C2semidirect
4. #20261025C2semidirect
5. #20261015C2semidirect
6. #20260950C2semidirect
7. #20260950C2semidirect
8. #20260886C2semidirect
10. #20260845C2semidirect
11. #20260835C2semidirect
12. C4×#13095499C2semidirect
13. C4×#13095496C2semidirect
14. #20260796C2semidirect
15. #518×C97C2semidirect
20. #15690810C22semidirect
22. #15690798C22semidirect
28. C4×Dic194C22semidirect
29. C4×D388C22semidirect
30. #15690743C22semidirect
31. #15690743C22semidirect
32. #15690741C22semidirect
33. #15690741C22semidirect
34. C4⋊#11715895C22semidirect
35. C4⋊#11715895C22semidirect
36. #15690733C22semidirect
37. #15690733C22semidirect
42. #15690730C22semidirect
43. #15690730C22semidirect
46. #15690721C22semidirect
47. C4×#11715895C22semidirect
182. C97#3118semidirect
184. #518D97semidirect

Maximal subgroups

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

subgroupindex
#202613562
#202610762
#202610592
#202610252
#202610152
#202609502
#202608862
#202608692
#202608452
#202608352
C4×#130954992
C4×#130954962
#202607962
#518×C972
#311897