Finite Group Databasebeta

← #27151687

Subgroups — #27151687

Order 12416.

418 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261301C211yesmaximal, characteristic
2#20261216C211yesmaximal, characteristic, complemented
3#20261062C211yesmaximal, characteristic, complemented
4#20261060C211yesmaximal, characteristic, complemented
5#20261057C211yesmaximal, characteristic, complemented
6#20261056C211yesmaximal, characteristic, complemented
7#20261053C211yesmaximal, characteristic, complemented
8#20261052C211yesmaximal, characteristic, complemented
9#20261050C211yesmaximal, characteristic, complemented
10#20260776C211yesmaximal, characteristic, complemented
11#20260772C211yesmaximal, characteristic, complemented
12#20260738C211yesmaximal, characteristic, complemented
13#102×D97C211yesmaximal, characteristic, complemented
14#20260700C211yesmaximal, characteristic, complemented
15#480×C97C211yesmaximal, characteristic, complemented
16C2×#13095511C2211yescharacteristic
17C2×D4×D97C2211yescharacteristic
18C2×#13095494C2211yescharacteristic
19C2×C8×D97C2211yescharacteristic
20#15690806—21no—
21#15690796—21no—
22#15690785C2211yescharacteristic
23#15690784C2211yescharacteristic
24#15690781C2211yescharacteristic
25#15690780—21no—
26#15690778—21no—
27#15690775—21no—
28#15690772—21no—
29C4×C22⋊D97—21no—
30C4×C22⋊D97—21no—
31#15690769C2211yescharacteristic, complemented
32#15690768C2211yescharacteristic
33#15690768C2211yescharacteristic
34#15690768C2211yescharacteristic, complemented
35C4⋊C4×D97C2211yescharacteristic
36C4⋊C4×D97C2211yescharacteristic, complemented
37#15690744C2211yescharacteristic, complemented
38#15690743C2211yescharacteristic, complemented
39#15690742C2211yescharacteristic, complemented
40#15690741C2211yescharacteristic, complemented
41#15690741C2211yescharacteristic, complemented
42#15690741C2211yescharacteristic, complemented
43#15690732C2211yescharacteristic
44#15690732C2211yescharacteristic
45#15690731C2211yescharacteristic
46#15690731C2211yescharacteristic
47#15690730C2211yescharacteristic
48#15690730C2211yescharacteristic
49C22⋊#11715895C2211yescharacteristic, complemented
50#15690722C2211yescharacteristic, complemented
51#15690722C2211yescharacteristic, complemented
52#121×C97C2211yescharacteristic
53C2×C4⋊C4×C97C2211yescharacteristic, complemented
54#107×C97C2211yescharacteristic, complemented
55#107×C97C2211yescharacteristic, complemented
56#102×C97C2211yescharacteristic
57#102×C97C2211yescharacteristic
58C22⋊C8×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60#13095511—22no—
61#13095511—22no—
62C2×Dic194D411yescharacteristic
63C2×D388—21no—
64C2×D388D411yescharacteristic
65C2×D388C2311yescharacteristic
66C2×C22⋊D97—21no—
67C2×C22⋊D97—21no—
68C2×C22⋊D97—21no—
69C2×C22⋊D97D411yescharacteristic
70C2×C22⋊D97D411yescharacteristic
71D4×D97—22no—
72D4×D97—22no—
73C2×C4×D97—21no—
74C2×C4×D97—21no—
75C2×C4×D97C2311yescharacteristic
76C2×C4×D97C2311yescharacteristic
77#13095494—22no—
78C2×#11715895C2311yescharacteristic
79C2×#11715895C2311yescharacteristic
80C8×D97—22no—
81C4⋊Dic97C2311yescharacteristic
82C4⋊Dic97C2311yescharacteristic
83C4⋊Dic97C2311yescharacteristic
84#13095490—21no—
85#13095490—21no—
86C22⋊Dic97—21no—
87C22⋊Dic97—21no—
88#13095488—21no—
89#13095488—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488—21no—
93C4×Dic97—21no—
94C4×Dic97—21no—
95C4×Dic97—21no—
96C4×Dic97—21no—
97C2×D4×C97—21no—
98C2×D4×C97C2311yescharacteristic
99C22×C4×C97—21no—
100C22×C4×C97C2311yescharacteristic
101C2×C8×C97C2311yescharacteristic
102C2×C8×C97C2311yescharacteristic
103C4⋊C4×C97—21no—
104C4⋊C4×C97C2311yescharacteristic
105C4⋊C4×C97C2311yescharacteristic
106C4⋊C4×C97C2311yescharacteristic
107C22⋊C4×C97—21no—
108C23×C97—21no—
109C23×C97C2×D411yescharacteristic
110C22×D97—21no—
111C22×D97—42no—
112C22×D97—81no—
113C22×D97—81no—
114C22×D97C2×D411yescharacteristic
115C22×D97C2×D411yescharacteristic
116D4×C97—21no—
117D4×C97—21no—
118D4×C97—81no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—81no—
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2411yescharacteristic
127C22⋊D97—22no—
128C22⋊D97—22no—
129C22⋊D97—42no—
130C22⋊D97—81no—
131C22⋊D97—81no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97C2×D411yescharacteristic
136C2×Dic97C2×D411yescharacteristic
137D388—22no—
138D388—21no—
139D388—21no—
140D388—81no—
141C4×D97—21no—
142C4×D97—21no—
143C4×D97—81no—
144C4×D97—81no—
145C4×D97#4112yes—
146C4×D97#4112yes—
147Dic194—22no—
148C8×C97—21no—
149C8×C97—21no—
150#11715895—21no—
151#11715895—21no—
152C22×C97—21no—
153C22×C97—21no—
154C22×C97—21no—
155C22×C97—41no—
156C22×C97—41no—
157C22×C97—41no—
158C22×C97C22×D411yescharacteristic
159C2×D97—21no—
160C2×D97—21no—
161C2×D97—21no—
162C2×D97—21no—
163C2×D97—21no—
164C2×D97—41no—
165C2×D97—41no—
166C2×D97—41no—
167C2×D97—84no—
168C2×D97#12312yes—
169C4×C97—21no—
170C4×C97—41no—
171C4×C97—41no—
172C4×C97—41no—
173C4×C97C2×#4111yescharacteristic
174C4×C97C2×#4111yescharacteristic
175Dic97—21no—
176Dic97—41no—
177Dic97—41no—
178Dic97—41no—
179Dic97#12312yes—
180C2×C97—41no—
181C2×C97—81no—
182C2×C97C2×#12311yescharacteristic
183C2×C97#57611yescharacteristic
184C2×C97#57711yescharacteristic
185D97—22no—
186D97—41no—
187D97—81no—
188#3096—971nomaximal, Sylow
189C97#309611yescharacteristic, complemented, Sylow
190#535—971no—
191#520—971no—
192#482—971no—
193#482—971no—
194#482—971no—
195#482—971no—
196#481—971no—
197#481—971no—
198#480—971no—
199#480D9711yescharacteristic, complemented
200#428—971no—
201#427—971no—
202#418—971no—
203C2×#102—971no—
204#408—971no—
205C2×#41—971no—
206C22×D4—971no—
207C2×M16—971no—
208C22×C8—971no—
209#127—1941no—
210#124—1941no—
211#121—971no—
212#121—971no—
213#121—971no—
214#121C2×D9711yescharacteristic
215#120—1941no—
216#120—1941no—
217C4×Q8—1941no—
218C4×D4—1941no—
219C4×D4—1941no—
220C4×D4—1941no—
221#117—971no—
222#117—971no—
223#117—971no—
224#117—971no—
225C2×C4⋊C4—971no—
226C2×C4⋊C4—971no—
227C2×C4⋊C4C2×D9711yescharacteristic, complemented
228#107—971no—
229#107—971no—
230#107C2×D9711yescharacteristic, complemented
231#107C2×D9711yescharacteristic, complemented
232#106—971no—
233#106—971no—
234#106—971no—
235#106—971no—
236#102—971no—
237#102—971no—
238#102—971no—
239#102—971no—
240#102—971no—
241#102—971no—
242#102C2×D9711yescharacteristic
243#102C2×D9711yescharacteristic
244C22⋊C8—971no—
245C22⋊C8—971no—
246C22⋊C8—971no—
247C22⋊C8C2×D9711yescharacteristic, complemented
248C24—1941no—
249#41—1942no—
250#41—1942no—
251C2×Q8—971no—
252C2×D4—21no—
253C2×D4—971no—
254C2×D4—971no—
255C2×D4—971no—
256C2×D4—971no—
257C2×D4—1941no—
258C2×D4—1942no—
259C2×D4—1942no—
260C2×D4—1941no—
261C2×D4—1941no—
262C2×D4—1941no—
263C2×D4C22×D9711yescharacteristic
264C22×C4—21no—
265C22×C4—971no—
266C22×C4—971no—
267C22×C4—1941no—
268C22×C4—1941no—
269C22×C4C22×D9711yescharacteristic
270M16—1942no—
271C2×C8—971no—
272C2×C8—971no—
273C2×C8—1942no—
274C2×C8C22×D9711yescharacteristic
275C2×C8C22×D9711yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—971no—
278C4⋊C4—971no—
279C4⋊C4—971no—
280C4⋊C4—1941no—
281C4⋊C4—1941no—
282C4⋊C4C22×D9711yescharacteristic
283C4⋊C4C22×D9711yescharacteristic
284C4⋊C4C22×D9711yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—1941no—
287C22⋊C4—1941no—
288C22⋊C4—1941no—
289C22⋊C4—1941no—
290C22⋊C4—1941no—
291C22⋊C4—1941no—
292C22⋊C4—1941no—
293C42—1941no—
294C42—1941no—
295C42—1941no—
296C42—1941no—
297C23—21no—
298C23—7761no—
299C23—7761no—
300C23—971no—
301C23—971no—
302C23—3882no—
303C23—1941no—
304C23D4×D9711yescharacteristic
305Q8—1942no—
306D4—21no—
307D4—21no—
308D4—81no—
309D4—7761no—
310D4—7761no—
311D4—7761no—
312D4—3882no—
313D4—1942no—
314D4—1942no—
315D4—1942no—
316D4—1941no—
317D4—1941no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—81no—
324C2×C4—7761no—
325C2×C4—7761no—
326C2×C4—971no—
327C2×C4—972no—
328C2×C4—972no—
329C2×C4—971no—
330C2×C4—1941no—
331C2×C4—1941no—
332C2×C4—1941no—
333C2×C4—1941no—
334C2×C4—1941no—
335C2×C4D4×D9711yescharacteristic
336C2×C4C23×D9711yescharacteristic
337C8—21no—
338C8—21no—
339C8—1941no—
340C8—1941no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—7764no—
348C22—972no—
349C22—3881no—
350C22—3881no—
351C22—3881no—
352C22—1941no—
353C22—1941no—
354C22—1941no—
355C22—1941no—
356C22—1941no—
357C22C2×D4×D9711yescharacteristic
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—972no—
363C4—3881no—
364C4—3881no—
365C4—3881no—
366C4—1941no—
367C4#41×D9711yescharacteristic
368C4#41×D9711yescharacteristic
369C2—41no—
370C2—81no—
371C2—7761no—
372C2—3881no—
373C2—1942no—
374C2#123×D9711yescharacteristic
375C2#2026153611yescharacteristic
376C2#2026154411yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#480×C97C2
socleC22×C97C22×D4
prime radical#20261301C2
hypercenter#480D97
hypocenterC97#3096
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
2. #20261216C2semidirect
3. #20261062C2semidirect
4. #20261060C2semidirect
5. #20261057C2semidirect
6. #20261056C2semidirect
7. #20261053C2semidirect
8. #20261052C2semidirect
9. #20261050C2semidirect
10. #20260776C2semidirect
11. #20260772C2semidirect
12. #20260738C2semidirect
13. #102×D97C2semidirect
14. #20260700C2semidirect
15. #480×C97C2semidirect
31. #15690769C22semidirect
34. #15690768C22semidirect
36. C4⋊C4×D97C22semidirect
37. #15690744C22semidirect
38. #15690743C22semidirect
39. #15690742C22semidirect
40. #15690741C22semidirect
41. #15690741C22semidirect
42. #15690741C22semidirect
49. C22⋊#11715895C22semidirect
50. #15690722C22semidirect
51. #15690722C22semidirect
53. C2×C4⋊C4×C97C22semidirect
54. #107×C97C22semidirect
55. #107×C97C22semidirect
58. C22⋊C8×C97C22semidirect
189. C97#3096semidirect
199. #480D97semidirect
227. C2×C4⋊C4C2×D97semidirect
230. #107C2×D97semidirect
231. #107C2×D97semidirect
247. C22⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613012
#202612162
#202610622
#202610602
#202610572
#202610562
#202610532
#202610522
#202610502
#202607762
#202607722
#202607382
#102×D972
#202607002
#480×C972
#309697