Finite Group Databasebeta

← #27151918

Subgroups — #27151918

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#20261350C211yesmaximal, characteristic, complemented
2#20261207C211yesmaximal, characteristic, complemented
3#20261079C211yesmaximal, characteristic, complemented
4#20261057C211yesmaximal, characteristic, complemented
5#20261024C211yesmaximal, characteristic, complemented
6#20261013C211yesmaximal, characteristic, complemented
7#20260949C211yesmaximal, characteristic, complemented
8#20260948C211yesmaximal, characteristic, complemented
9#20260887C211yesmaximal, characteristic, complemented
10#20260867C211yesmaximal, characteristic
11#20260837C211yesmaximal, characteristic, complemented
12C4×#13095499C211yesmaximal, characteristic, complemented
13C4×#13095496C211yesmaximal, characteristic, complemented
14#20260797C211yesmaximal, characteristic, complemented
15#442×C97C211yesmaximal, characteristic, complemented
16C2×D4×D97C2211yescharacteristic
17C2×#13095500C2211yescharacteristic
18C2×#13095499C2211yescharacteristic
19C2×#13095496C2211yescharacteristic
20#15690809C2211yescharacteristic, complemented
21#15690803—21no—
22#15690797C2211yescharacteristic, complemented
23#15690796—21no—
24#15690793—21no—
25#15690790C2211yescharacteristic
26#15690789—21no—
27#15690781C2211yescharacteristic
28C4×Dic194C2211yescharacteristic, complemented
29C4×D388C2211yescharacteristic
30D4×Dic97C2211yescharacteristic
31#15690772C2211yescharacteristic
32C4×C22⋊D97—21no—
33#15690770—21no—
34#15690768C2211yescharacteristic
35#15690767C2211yescharacteristic
36C2×#13095488—21no—
37C22⋊C4×D97—21no—
38#15690744C2211yescharacteristic, complemented
39#15690743C2211yescharacteristic, complemented
40#15690741C2211yescharacteristic, complemented
41#15690739C2211yescharacteristic, complemented
42#15690739C2211yescharacteristic, complemented
43#15690735C2211yescharacteristic, complemented
44#15690733C2211yescharacteristic, complemented
45#15690732C2211yescharacteristic
46#15690731C2211yescharacteristic
47#15690731C2211yescharacteristic, complemented
48#15690730C2211yescharacteristic
49#15690722C2211yescharacteristic
50#15690722C2211yescharacteristic
51C4×#11715895C2211yescharacteristic, complemented
52C2×D8×C97C2211yescharacteristic
53C4×D4×C97C2211yescharacteristic
54C4×D4×C97C2211yescharacteristic, complemented
55#106×C97C2211yescharacteristic, complemented
56#102×C97C2211yescharacteristic
57#102×C97C2211yescharacteristic, complemented
58#97×C97C2211yescharacteristic, complemented
59C23×D97—21no—
60C2×Dic194—21no—
61C2×Dic194C2311yescharacteristic
62C2×D388C2311yescharacteristic
63C2×C22⋊D97—21no—
64C2×C22⋊D97—21no—
65D4×D97—22no—
66D4×D97—22no—
67C22×Dic97—21no—
68C2×C4×D97—21no—
69C2×C4×D97—21no—
70C2×C4×D97C2311yescharacteristic
71#13095500—22no—
72#13095499—22no—
73#13095496—22no—
74C2×#11715895C2311yescharacteristic
75C2×#11715895C2311yescharacteristic
76C4⋊Dic97—21no—
77C4⋊Dic97C2311yescharacteristic
78C4⋊Dic97C2311yescharacteristic
79#13095490—21no—
80#13095490—21no—
81#13095490D411yescharacteristic
82#13095490D411yescharacteristic
83C22⋊Dic97—21no—
84C22⋊Dic97—21no—
85C22⋊Dic97—21no—
86C22⋊Dic97—21no—
87#13095488—21no—
88#13095488—21no—
89#13095488—21no—
90#13095488—21no—
91#13095488D411yescharacteristic
92#13095488D411yescharacteristic
93C4×Dic97—21no—
94C4×Dic97—21no—
95C4×Dic97C2311yescharacteristic
96C2×D4×C97C2311yescharacteristic
97C2×D4×C97C2311yescharacteristic
98C22×C4×C97—21no—
99C22×C4×C97—21no—
100D8×C97—22no—
101C2×C8×C97C2311yescharacteristic
102C2×C8×C97C2311yescharacteristic
103C4⋊C4×C97C2311yescharacteristic
104C4⋊C4×C97C2311yescharacteristic
105C22⋊C4×C97—21no—
106C22⋊C4×C97—21no—
107C42×C97C2311yescharacteristic
108C23×C97—21no—
109C23×C97—21no—
110C22×D97—21no—
111C22×D97—24no—
112C22×D97—81no—
113C22×D97—81no—
114C22×D97C2×D411yescharacteristic
115D4×C97—21no—
116D4×C97—21no—
117D4×C97—21no—
118D4×C97#4112yes—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—81no—
122C2×C4×C97—81no—
123C2×C4×C97C2×D411yescharacteristic
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2411yescharacteristic
126C22⋊D97—24no—
127C22⋊D97—81no—
128C2×Dic97—21no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—81no—
132C2×Dic97C2×D411yescharacteristic
133C2×Dic97C2×D411yescharacteristic
134C2×Dic97C2×D411yescharacteristic
135D388—21no—
136D388#4112yes—
137C4×D97—21no—
138C4×D97—21no—
139C4×D97—81no—
140C4×D97—81no—
141Dic194—21no—
142Dic194—21no—
143Dic194—81no—
144C8×C97—21no—
145C8×C97—21no—
146#11715895—21no—
147#11715895—21no—
148C22×C97—22no—
149C22×C97—41no—
150C22×C97—41no—
151C22×C97—41no—
152C22×C97—41no—
153C22×C97—41no—
154C22×C97C22×D411yescharacteristic
155C2×D97—21no—
156C2×D97—22no—
157C2×D97—21no—
158C2×D97—21no—
159C2×D97—41no—
160C2×D97—41no—
161C2×D97—84no—
162C4×C97—21no—
163C4×C97—21no—
164C4×C97—41no—
165C4×C97—41no—
166C4×C97C2×#4111yescharacteristic
167C4×C97C2×#4111yescharacteristic
168Dic97—21no—
169Dic97—21no—
170Dic97—21no—
171Dic97—41no—
172Dic97—41no—
173Dic97—41no—
174C2×C97—42no—
175C2×C97—81no—
176C2×C97#52511yescharacteristic
177C2×C97#57611yescharacteristic
178C2×C97#57711yescharacteristic
179D97—41no—
180D97—42no—
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#450—971no—
192#447—971no—
193#442D9711yescharacteristic, complemented
194#440—971no—
195C4×SD16—971no—
196C4×D8—971no—
197#433—971no—
198C22×D4—971no—
199C2×SD16—971no—
200C2×SD16—971no—
201C2×D8—971no—
202C2×D8C2×D9711yescharacteristic
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—971no—
213C4×D4—971no—
214C4×D4—971no—
215C4×D4—1941no—
216C4×D4—1941no—
217C4×D4C2×D9711yescharacteristic
218C4×D4C2×D9711yescharacteristic, complemented
219#117—971no—
220#117—971no—
221C2×C22⋊C4—1941no—
222C2×C22⋊C4—1941no—
223#107—971no—
224#107—971no—
225#106—971no—
226#106C2×D9711yescharacteristic, complemented
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, complemented
237C22⋊C8—971no—
238C22⋊C8—971no—
239#97C2×D9711yescharacteristic, complemented
240C4×C8—971no—
241C24—1941no—
242C2×Q8—971no—
243C2×Q8—1941no—
244C2×D4—971no—
245C2×D4—1941no—
246C2×D4—1942no—
247C2×D4—1941no—
248C2×D4—1942no—
249C2×D4C22×D9711yescharacteristic
250C2×D4C22×D9711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—971no—
254C22×C4—1941no—
255C22×C4—1941no—
256C22×C4—1941no—
257SD16—1942no—
258SD16—1942no—
259D8—22no—
260D8—1942no—
261C2×C8—971no—
262C2×C8—971no—
263C2×C8C22×D9711yescharacteristic
264C2×C8C22×D9711yescharacteristic
265C4⋊C4—971no—
266C4⋊C4—971no—
267C4⋊C4—971no—
268C4⋊C4—971no—
269C4⋊C4—1941no—
270C4⋊C4—1941no—
271C4⋊C4—1941no—
272C4⋊C4C22×D9711yescharacteristic
273C4⋊C4C22×D9711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—971no—
277C22⋊C4—971no—
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—1941no—
288C42—1941no—
289C42C22×D9711yescharacteristic
290C23—21no—
291C23—21no—
292C23—7761no—
293C23—7761no—
294C23—971no—
295C23—1944no—
296C23—1941no—
297Q8—7761no—
298Q8—1941no—
299Q8—1941no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—7761no—
304D4—972no—
305D4—1944no—
306D4—1941no—
307D4#1309551112yes—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—7761no—
313C2×C4—7761no—
314C2×C4—7761no—
315C2×C4—971no—
316C2×C4—971no—
317C2×C4—971no—
318C2×C4—1941no—
319C2×C4—1941no—
320C2×C4—1941no—
321C2×C4—1941no—
322C2×C4—1941no—
323C2×C4D4×D9711yescharacteristic
324C2×C4D4×D9711yescharacteristic
325C2×C4C23×D9711yescharacteristic
326C8—21no—
327C8—21no—
328C8—1941no—
329C8—1941no—
330C22—22no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—7764no—
337C22—3881no—
338C22—3881no—
339C22—1941no—
340C22—1942no—
341C22—1941no—
342C22—1941no—
343C22C2×D4×D9711yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—3881no—
349C4—3881no—
350C4—3881no—
351C4—1941no—
352C4—1941no—
353C4—1941no—
354C4#41×D9711yescharacteristic
355C4C2×#1309551111yescharacteristic
356C2—42no—
357C2—81no—
358C2—3881no—
359C2—3882no—
360C2#2026124411yescharacteristic
361C2#2026153911yescharacteristic
362C2#2026154511yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#442×C97C2
socleC22×C97C22×D4
prime radical#20260867C2
hypercenter#442D97
hypocenterC97#3118
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #20261350C2semidirect
2. #20261207C2semidirect
3. #20261079C2semidirect
4. #20261057C2semidirect
5. #20261024C2semidirect
6. #20261013C2semidirect
7. #20260949C2semidirect
8. #20260948C2semidirect
9. #20260887C2semidirect
11. #20260837C2semidirect
12. C4×#13095499C2semidirect
13. C4×#13095496C2semidirect
14. #20260797C2semidirect
15. #442×C97C2semidirect
20. #15690809C22semidirect
22. #15690797C22semidirect
28. C4×Dic194C22semidirect
38. #15690744C22semidirect
39. #15690743C22semidirect
40. #15690741C22semidirect
41. #15690739C22semidirect
42. #15690739C22semidirect
43. #15690735C22semidirect
44. #15690733C22semidirect
47. #15690731C22semidirect
51. C4×#11715895C22semidirect
54. C4×D4×C97C22semidirect
55. #106×C97C22semidirect
57. #102×C97C22semidirect
58. #97×C97C22semidirect
182. C97#3118semidirect
193. #442D97semidirect
218. C4×D4C2×D97semidirect
226. #106C2×D97semidirect
236. #102C2×D97semidirect
239. #97C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613502
#202612072
#202610792
#202610572
#202610242
#202610132
#202609492
#202609482
#202608872
#202608672
#202608372
C4×#130954992
C4×#130954962
#202607972
#442×C972
#311897