Finite Group Databasebeta

← #27151921

Subgroups — #27151921

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#20261358C211yesmaximal, characteristic, complemented
2#20261210C211yesmaximal, characteristic, complemented
3#20261073C211yesmaximal, characteristic, complemented
4#20261060C211yesmaximal, characteristic, complemented
5#20261023C211yesmaximal, characteristic, complemented
6#20261013C211yesmaximal, characteristic, complemented
7#20260952C211yesmaximal, characteristic, complemented
8#20260952C211yesmaximal, characteristic, complemented
9#20260869C211yesmaximal, characteristic
10#20260844C211yesmaximal, characteristic, complemented
11#20260836C211yesmaximal, characteristic, complemented
12SD16×Dic97C211yesmaximal, characteristic, complemented
13#20260815C211yesmaximal, characteristic, complemented
14#20260793C211yesmaximal, characteristic, complemented
15#450×C97C211yesmaximal, characteristic, complemented
16C2×#13095496C2211yescharacteristic
17C2×#13095496C2211yescharacteristic
18#15690809C2211yescharacteristic, complemented
19#15690805—21no—
20#15690799C2211yescharacteristic, complemented
21#15690796—21no—
22#15690792—21no—
23#15690790—21no—
24#15690784C2211yescharacteristic
25Q8×Dic97C2211yescharacteristic, complemented
26D4×Dic97—21no—
27D4×Dic97—21no—
28D4×Dic97C2211yescharacteristic
29D4×Dic97C2211yescharacteristic
30D4×Dic97C2211yescharacteristic
31D4×Dic97C2211yescharacteristic
32#15690772C2211yescharacteristic, complemented
33#15690769C2211yescharacteristic
34#15690769C2211yescharacteristic
35C2×C22⋊Dic97—21no—
36C2×C22⋊Dic97—21no—
37#15690744C2211yescharacteristic, complemented
38#15690743C2211yescharacteristic, complemented
39#15690742C2211yescharacteristic, complemented
40#15690741C2211yescharacteristic, complemented
41#15690739C2211yescharacteristic, complemented
42#15690739C2211yescharacteristic, complemented
43#15690734C2211yescharacteristic, complemented
44#15690734C2211yescharacteristic, complemented
45#15690732C2211yescharacteristic, complemented
46#15690732C2211yescharacteristic, complemented
47#15690731C2211yescharacteristic
48#15690731C2211yescharacteristic
49C22⋊#11715895C2211yescharacteristic
50#15690720C2211yescharacteristic, complemented
51C8×Dic97C2211yescharacteristic, complemented
52C22×D4×C97C2211yescharacteristic
53C2×SD16×C97C2211yescharacteristic
54C2×SD16×C97C2211yescharacteristic
55C22⋊Q8×C97C2211yescharacteristic
56#102×C97C2211yescharacteristic
57#102×C97C2211yescharacteristic
58C22⋊C8×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×D388C2311yescharacteristic
61C2×C22⋊D97—21no—
62C22×Dic97—21no—
63C22×Dic97—21no—
64C22×Dic97—21no—
65C22×Dic97—21no—
66C2×C4×D97—21no—
67#13095496—22no—
68#13095496—22no—
69C2×#11715895C2311yescharacteristic
70C2×#11715895C2311yescharacteristic
71C4⋊Dic97—21no—
72C4⋊Dic97D411yescharacteristic
73C4⋊Dic97D411yescharacteristic
74C4⋊Dic97C2311yescharacteristic
75C4⋊Dic97C2311yescharacteristic
76C4⋊Dic97C2311yescharacteristic
77#13095490—21no—
78C22⋊Dic97—21no—
79C22⋊Dic97—21no—
80C22⋊Dic97—21no—
81C22⋊Dic97—21no—
82C22⋊Dic97—21no—
83C22⋊Dic97D411yescharacteristic
84C22⋊Dic97D411yescharacteristic
85#13095488—21no—
86#13095488—21no—
87#13095488—21no—
88#13095488—21no—
89C4×Dic97—21no—
90C4×Dic97—21no—
91C4×Dic97C2311yescharacteristic
92C4×Dic97C2311yescharacteristic
93C24×C97—21no—
94C2×Q8×C97C2311yescharacteristic
95C2×D4×C97—22no—
96C2×D4×C97—22no—
97C2×D4×C97—21no—
98C2×D4×C97C2311yescharacteristic
99C2×D4×C97C2311yescharacteristic
100C22×C4×C97C2311yescharacteristic
101SD16×C97—22no—
102SD16×C97—22no—
103C2×C8×C97C2311yescharacteristic
104C2×C8×C97C2311yescharacteristic
105C4⋊C4×C97—21no—
106C4⋊C4×C97C2311yescharacteristic
107C22⋊C4×C97—21no—
108C23×C97—24no—
109C23×C97—21no—
110C23×C97—21no—
111C23×C97—81no—
112C23×C97—81no—
113C23×C97C2×D411yescharacteristic
114C22×D97—21no—
115Q8×C97—21no—
116Q8×C97—21no—
117D4×C97—24no—
118D4×C97—21no—
119D4×C97—21no—
120D4×C97#4112yes—
121D4×C97#4112yes—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2411yescharacteristic
128C22⋊D97—81no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—81no—
133C2×Dic97—81no—
134C2×Dic97—81no—
135C2×Dic97—81no—
136C2×Dic97C2×D411yescharacteristic
137C2×Dic97C2×D411yescharacteristic
138C2×Dic97C2×D411yescharacteristic
139C2×Dic97C2×D411yescharacteristic
140D388—21no—
141D388—21no—
142C4×D97—81no—
143Dic194—81no—
144C8×C97—21no—
145C8×C97—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—41no—
164C4×C97—41no—
165C4×C97C2×#4111yescharacteristic
166C4×C97C2×#4111yescharacteristic
167Dic97—21no—
168Dic97—21no—
169Dic97—21no—
170Dic97—21no—
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#518—971no—
185#485—971no—
186#482—971no—
187#476—971no—
188#474—971no—
189#463—971no—
190#463—971no—
191#450D9711yescharacteristic, complemented
192#447—971no—
193#442—971no—
194#440—971no—
195C4×SD16—971no—
196C4×D8—971no—
197#433—971no—
198C22×D4C2×D9711yescharacteristic
199C2×SD16C2×D9711yescharacteristic
200C2×SD16C2×D9711yescharacteristic
201C2×D8—971no—
202C2×D8—971no—
203C4⋊Q8—971no—
204#126—1941no—
205#124—971no—
206#124—1941no—
207#123—1941no—
208C22⋊Q8—1941no—
209C22⋊Q8C2×D9711yescharacteristic
210#121—971no—
211C4×Q8—971no—
212C4×D4—971no—
213C4×D4—971no—
214C4×D4—971no—
215C4×D4—971no—
216C4×D4—971no—
217C4×D4—1941no—
218C4×D4—1941no—
219#117—971no—
220#117—971no—
221C2×C22⋊C4—1941no—
222C2×C22⋊C4—1941no—
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#102C2×D9711yescharacteristic
236#102C2×D9711yescharacteristic
237C22⋊C8—971no—
238C22⋊C8C2×D9711yescharacteristic
239#97—971no—
240C4×C8—971no—
241C24—21no—
242C2×Q8—1941no—
243C2×Q8C22×D9711yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—21no—
247C2×D4—971no—
248C2×D4—1941no—
249C2×D4C22×D9711yescharacteristic
250C2×D4C22×D9711yescharacteristic
251C22×C4—1941no—
252C22×C4—1941no—
253C22×C4—1941no—
254C22×C4—1941no—
255C22×C4—1941no—
256C22×C4C22×D9711yescharacteristic
257SD16—22no—
258SD16—22no—
259D8—1942no—
260D8—1942no—
261C2×C8—971no—
262C2×C8—971no—
263C2×C8C22×D9711yescharacteristic
264C2×C8C22×D9711yescharacteristic
265C4⋊C4—21no—
266C4⋊C4—971no—
267C4⋊C4—971no—
268C4⋊C4—971no—
269C4⋊C4—971no—
270C4⋊C4—971no—
271C4⋊C4—1941no—
272C4⋊C4—1941no—
273C4⋊C4C22×D9711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—971no—
276C22⋊C4—971no—
277C22⋊C4—1941no—
278C22⋊C4—1941no—
279C22⋊C4—1941no—
280C22⋊C4—1941no—
281C22⋊C4—1941no—
282C22⋊C4—1941no—
283C22⋊C4—1941no—
284C22⋊C4—1941no—
285C22⋊C4—1941no—
286C42—971no—
287C42—971no—
288C42—1941no—
289C42—1941no—
290C23—24no—
291C23—21no—
292C23—21no—
293C23—81no—
294C23—81no—
295C23—1941no—
296C23D4×D9711yescharacteristic
297Q8—21no—
298Q8—21no—
299Q8—7761no—
300D4—24no—
301D4—21no—
302D4—21no—
303D4—7761no—
304D4—1941no—
305D4—1941no—
306D4#1309551012yes—
307D4#1309551012yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—7761no—
313C2×C4—7761no—
314C2×C4—7761no—
315C2×C4—7761no—
316C2×C4—7761no—
317C2×C4—971no—
318C2×C4—971no—
319C2×C4—971no—
320C2×C4—971no—
321C2×C4—1941no—
322C2×C4—1941no—
323C2×C4—1941no—
324C2×C4D4×D9711yescharacteristic
325C2×C4C23×D9711yescharacteristic
326C8—21no—
327C8—21no—
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—
343C22C2×D4×D9711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—3881no—
348C4—3881no—
349C4—3881no—
350C4—1941no—
351C4—1941no—
352C4—1941no—
353C4—1941no—
354C4C2×#1309551011yescharacteristic
355C4C2×#1309551011yescharacteristic
356C2—41no—
357C2—42no—
358C2—42no—
359C2—7761no—
360C2#2026124111yescharacteristic
361C2#2026153911yescharacteristic
362C2#2026154111yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#450×C97C2
socleC22×C97C22×D4
prime radical#20260869C2
hypercenter#450D97
hypocenterC97#3118
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #20261358C2semidirect
2. #20261210C2semidirect
3. #20261073C2semidirect
4. #20261060C2semidirect
5. #20261023C2semidirect
6. #20261013C2semidirect
7. #20260952C2semidirect
8. #20260952C2semidirect
10. #20260844C2semidirect
11. #20260836C2semidirect
12. SD16×Dic97C2semidirect
13. #20260815C2semidirect
14. #20260793C2semidirect
15. #450×C97C2semidirect
18. #15690809C22semidirect
20. #15690799C22semidirect
25. Q8×Dic97C22semidirect
32. #15690772C22semidirect
37. #15690744C22semidirect
38. #15690743C22semidirect
39. #15690742C22semidirect
40. #15690741C22semidirect
41. #15690739C22semidirect
42. #15690739C22semidirect
43. #15690734C22semidirect
44. #15690734C22semidirect
45. #15690732C22semidirect
46. #15690732C22semidirect
50. #15690720C22semidirect
51. C8×Dic97C22semidirect
182. C97#3118semidirect
191. #450D97semidirect

Maximal subgroups

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

subgroupindex
#202613582
#202612102
#202610732
#202610602
#202610232
#202610132
#202609522
#202608692
#202608442
#202608362
SD16×Dic972
#202608152
#202607932
#450×C972
#311897