Finite Group Databasebeta

← #27153619

Subgroups — #27153619

Order 12416.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261420C211yesmaximal, characteristic, complemented
2#20261396C211yesmaximal, characteristic
3#20261091C211yesmaximal, characteristic, complemented
4#20261074C211yesmaximal, characteristic, complemented
5#20261048C211yesmaximal, characteristic, complemented
6#20261040C211yesmaximal, characteristic, complemented
7#20260975C211yesmaximal, characteristic, complemented
8#20260964C211yesmaximal, characteristic, complemented
9#20260960C211yesmaximal, characteristic, complemented
10#20260900C211yesmaximal, characteristic, complemented
11#20260891C211yesmaximal, characteristic, complemented
12#20260827C211yesmaximal, characteristic, complemented
13C8×C22⋊D97C211yesmaximal, characteristic, complemented
14#20260730C211yesmaximal, characteristic, complemented
15C2×#103×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic
17C2×#13095501C2211yescharacteristic
18C2×#13095501C2211yescharacteristic
19C2×#13095500C2211yescharacteristic
20C2×C8×D97C2211yescharacteristic
21#15690805—21no—
22#15690801C2211yescharacteristic
23#15690800—21no—
24#15690799—21no—
25#15690799C2211yescharacteristic
26#15690793—21no—
27#15690790C2211yescharacteristic
28#15690789C2211yescharacteristic
29#15690786—21no—
30#15690786C2211yescharacteristic
31#15690784—21no—
32#15690784C2211yescharacteristic
33Q8×Dic97C2211yescharacteristic
34#15690775C2211yescharacteristic
35C4×C22⋊D97—21no—
36C4×C22⋊D97C2211yescharacteristic
37#15690769—21no—
38#15690768C2211yescharacteristic
39#15690743C2211yescharacteristic
40#15690743C2211yescharacteristic
41#15690742C2211yescharacteristic
42#15690739C2211yescharacteristic
43#15690735C2211yescharacteristic
44#15690734C2211yescharacteristic
45#15690734C2211yescharacteristic
46#15690732C2211yescharacteristic
47#15690730C2211yescharacteristic
48#15690730C2211yescharacteristic
49C22⋊#11715895C2211yescharacteristic
50#15690722C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C22×Q8×C97C2211yescharacteristic
53C22×C8×C97C2211yescharacteristic
54C2×C4⋊C4×C97C2211yescharacteristic
55#103×C97C2211yescharacteristic
56#103×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58#103×C97C2211yescharacteristic
59#13095512—22no—
60#13095512—41no—
61C2×Dic194C2311yescharacteristic
62C2×D388—21no—
63C2×D388C2311yescharacteristic
64C2×C22⋊D97—21no—
65C2×C22⋊D97D411yescharacteristic
66C2×C4×D97—21no—
67C2×C4×D97C2311yescharacteristic
68#13095501—22no—
69#13095501—41no—
70#13095500—41no—
71C2×#11715895C2311yescharacteristic
72C2×#11715895C2311yescharacteristic
73C8×D97—41no—
74C4⋊Dic97—21no—
75C4⋊Dic97—21no—
76C4⋊Dic97—21no—
77C4⋊Dic97C2311yescharacteristic
78C4⋊Dic97C2311yescharacteristic
79#13095490—21no—
80#13095490—21no—
81#13095490—21no—
82#13095490—21no—
83#13095490D411yescharacteristic
84C22⋊Dic97—21no—
85C22⋊Dic97D411yescharacteristic
86#13095488—21no—
87#13095488—21no—
88#13095488—21no—
89#13095488—21no—
90#13095488—21no—
91#13095488—21no—
92#13095488D411yescharacteristic
93C4×Dic97—21no—
94C4×Dic97—21no—
95C4×Dic97—21no—
96C4×Dic97C2311yescharacteristic
97C2×Q8×C97—21no—
98C2×Q8×C97—22no—
99C2×Q8×C97—41no—
100C2×Q8×C97C2311yescharacteristic
101C2×Q8×C97C2311yescharacteristic
102C22×C4×C97—21no—
103C22×C4×C97—21no—
104C22×C4×C97C2311yescharacteristic
105C2×C8×C97—41no—
106C2×C8×C97C2311yescharacteristic
107C2×C8×C97C2311yescharacteristic
108C4⋊C4×C97—21no—
109C4⋊C4×C97C2311yescharacteristic
110C4⋊C4×C97C2311yescharacteristic
111C23×C97C2×D411yescharacteristic
112C22×D97—21no—
113C22×D97C2×D411yescharacteristic
114Q8×C97—21no—
115Q8×C97—21no—
116Q8×C97—21no—
117Q8×C97—42no—
118Q8×C97#4112yes—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—42no—
126C2×C4×C97—42no—
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2411yescharacteristic
129C22⋊D97—22no—
130C22⋊D97—42no—
131C2×Dic97—21no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134C2×Dic97C2×D411yescharacteristic
135C2×Dic97C2×D411yescharacteristic
136C2×Dic97C2×D411yescharacteristic
137D388—21no—
138D388—21no—
139D388—42no—
140C4×D97—21no—
141C4×D97—21no—
142C4×D97—42no—
143Dic194—21no—
144Dic194—21no—
145C8×C97—21no—
146C8×C97—21no—
147#11715895—21no—
148#11715895—21no—
149C22×C97—21no—
150C22×C97—21no—
151C22×C97#13512yes—
152C22×C97C22×D411yescharacteristic
153C2×D97—21no—
154C2×D97—21no—
155C2×D97—21no—
156C2×D97—41no—
157C2×D97—41no—
158C2×D97—41no—
159C4×C97—22no—
160C4×C97—21no—
161C4×C97—41no—
162C4×C97—41no—
163C4×C97—41no—
164C4×C97C2×#4111yescharacteristic
165C4×C97H(5,𝔽2)11yescharacteristic
166Dic97—21no—
167Dic97—21no—
168Dic97—21no—
169Dic97—41no—
170Dic97—41no—
171Dic97—41no—
172C2×C97—22no—
173C2×C97#54611yescharacteristic
174C2×C97C2×#13511yescharacteristic
175C2×C97#57811yescharacteristic
176D97—41no—
177D97—81no—
178#3300—971nomaximal, Sylow
179C97#330011yescharacteristic, complemented, Sylow
180#552—971no—
181#548—971no—
182#488—971no—
183#485—971no—
184#480—971no—
185#479—971no—
186#467—971no—
187#465—971no—
188#464—971no—
189#452—971no—
190#451—971no—
191C4×Q16—971no—
192C8×D4—971no—
193#416—971no—
194C2×#103D9711yescharacteristic, complemented
195C2×#41—971no—
196C22×Q8C2×D9711yescharacteristic
197C2×Q16—971no—
198C2×Q16—971no—
199C2×SD16—971no—
200C22×C8—971no—
201C22×C8C2×D9711yescharacteristic
202#126—1941no—
203#125—971no—
204#125—1941no—
205#124—971no—
206#124—1941no—
207#123—1941no—
208C22⋊Q8—971no—
209C22⋊Q8—971no—
210C22⋊Q8—971no—
211C22⋊Q8—1941no—
212#121—971no—
213#121—1941no—
214C4×Q8—971no—
215C4×Q8—971no—
216C4×D4—971no—
217C4×D4—1941no—
218#117—971no—
219#117—1941no—
220C2×C4⋊C4C2×D9711yescharacteristic
221#107—971no—
222#107—971no—
223#106—971no—
224C4⋊C8—971no—
225#103—971no—
226#103—971no—
227#103—971no—
228#103C2×D9711yescharacteristic
229#103C2×D9711yescharacteristic
230#103C2×D9711yescharacteristic
231#103C2×D9711yescharacteristic
232#102—971no—
233#102—971no—
234#102—971no—
235C22⋊C8—971no—
236C22⋊C8—971no—
237C4×C8—971no—
238#41—3881no—
239#41—1942no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—971no—
244C2×Q8C22×D9711yescharacteristic
245C2×Q8C22×D9711yescharacteristic
246C2×D4—971no—
247C2×D4—971no—
248C2×D4—1941no—
249C2×D4—1941no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—971no—
253C22×C4—1941no—
254C22×C4C22×D9711yescharacteristic
255Q16—3881no—
256Q16—1942no—
257SD16—3881no—
258C2×C8—41no—
259C2×C8—971no—
260C2×C8—971no—
261C2×C8—3881no—
262C2×C8C22×D9711yescharacteristic
263C2×C8C22×D9711yescharacteristic
264C4⋊C4—21no—
265C4⋊C4—971no—
266C4⋊C4—971no—
267C4⋊C4—971no—
268C4⋊C4—1941no—
269C4⋊C4—1941no—
270C4⋊C4—1941no—
271C4⋊C4—1941no—
272C4⋊C4—1941no—
273C4⋊C4—1941no—
274C4⋊C4—1941no—
275C4⋊C4C22×D9711yescharacteristic
276C4⋊C4C22×D9711yescharacteristic
277C22⋊C4—971no—
278C22⋊C4—971no—
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—
289C42—1941no—
290C23—971no—
291C23—1941no—
292C23D4×D9711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—42no—
297Q8—1941no—
298Q8—1941no—
299Q8#1309551112yes—
300D4—3882no—
301D4—3882no—
302D4—1942no—
303D4—1941no—
304D4—1941no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—971no—
314C2×C4—971no—
315C2×C4—971no—
316C2×C4—3882no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4—1941no—
320C2×C4—1941no—
321C2×C4—1941no—
322C2×C4D4×D9711yescharacteristic
323C2×C4C23×D9711yescharacteristic
324C8—21no—
325C8—21no—
326C8—1941no—
327C8—1941no—
328C22—21no—
329C22—21no—
330C22—3881no—
331C22—3881no—
332C22—3881no—
333C22—1941no—
334C22—1941no—
335C22—1941no—
336C22#1569083112yes—
337C22C2×D4×D9711yescharacteristic
338C4—21no—
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—3881no—
344C4—3881no—
345C4—3881no—
346C4—1941no—
347C4—1941no—
348C4—1941no—
349C4C2×#1309551111yescharacteristic
350C4#1569085511yescharacteristic
351C2—22no—
352C2—7761no—
353C2—3881no—
354C2#2026137511yescharacteristic
355C2C2×#1569083111yescharacteristic
356C2#2026154811yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#103×C97C2
socleC22×C97C22×D4
prime radical#20261396C2
hypercenterC2×#103D97
hypocenterC97#3300
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. #20261420C2semidirect
3. #20261091C2semidirect
4. #20261074C2semidirect
5. #20261048C2semidirect
6. #20261040C2semidirect
7. #20260975C2semidirect
8. #20260964C2semidirect
9. #20260960C2semidirect
10. #20260900C2semidirect
11. #20260891C2semidirect
12. #20260827C2semidirect
13. C8×C22⋊D97C2semidirect
14. #20260730C2semidirect
15. C2×#103×C97C2semidirect
179. C97#3300semidirect
194. C2×#103D97semidirect

Maximal subgroups

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

subgroupindex
#202614202
#202613962
#202610912
#202610742
#202610482
#202610402
#202609752
#202609642
#202609602
#202609002
#202608912
#202608272
C8×C22⋊D972
#202607302
C2×#103×C972
#330097