Finite Group Databasebeta

← #27154820

Subgroups — #27154820

Order 12416.

380 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261470C211yesmaximal, characteristic, complemented
2#20261125C211yesmaximal, characteristic, complemented
3#20261122C211yesmaximal, characteristic, complemented
4#20261117C211yesmaximal, characteristic, complemented
5#20260957C211yesmaximal, characteristic, complemented
6#20260954C211yesmaximal, characteristic, complemented
7#20260950C211yesmaximal, characteristic, complemented
8#20260947C211yesmaximal, characteristic, complemented
9#20260930C211yesmaximal, characteristic, complemented
10#20260927C211yesmaximal, characteristic, complemented
11#20260866C211yesmaximal, characteristic, complemented
12#20260835C211yesmaximal, characteristic, complemented
13#20260833C211yesmaximal, characteristic, complemented
14C4×#13095496C211yesmaximal, characteristic, complemented
15#559×C97C211yesmaximal, characteristic, complemented
16C2×#13095501C2211yescharacteristic
17C2×#13095499C2211yescharacteristic
18C2×#13095499C2211yescharacteristic
19C2×#13095498C2211yescharacteristic
20C2×#13095498C2211yescharacteristic
21C2×#13095496C2211yescharacteristic
22C2×#13095496C2211yescharacteristic
23C2×#13095496C2211yescharacteristic
24C2×#13095496C2211yescharacteristic
25#15690807C2211yescharacteristic
26#15690799—21no—
27#15690798C2211yescharacteristic
28#15690798C2211yescharacteristic
29#15690795—21no—
30#15690795—21no—
31C4×Dic194C2211yescharacteristic
32C4×D388C2211yescharacteristic
33C4×D388C2211yescharacteristic
34#15690772—21no—
35#15690743C2211yescharacteristic
36C4⋊#11715895C2211yescharacteristic
37C4⋊#11715895C2211yescharacteristic
38C4⋊#11715895C2211yescharacteristic
39#15690734C2211yescharacteristic
40#15690733C2211yescharacteristic
41#15690731C2211yescharacteristic
42#15690731C2211yescharacteristic
43#15690730C2211yescharacteristic
44#15690730C2211yescharacteristic
45#15690721C2211yescharacteristic
46#15690721C2211yescharacteristic
47C4×#11715895C2211yescharacteristic
48#127×C97C2211yescharacteristic
49#124×C97—21no—
50#124×C97C2211yescharacteristic
51#124×C97C2211yescharacteristic
52#123×C97—21no—
53#123×C97—21no—
54C4×Q8×C97C2211yescharacteristic
55C4×Q8×C97C2211yescharacteristic
56C4×D4×C97—21no—
57C4×D4×C97C2211yescharacteristic
58C4×D4×C97C2211yescharacteristic
59C2×Dic194C2311yescharacteristic
60C2×D388—21no—
61C2×D388—21no—
62C2×D388C2311yescharacteristic
63C2×D388C2311yescharacteristic
64C2×C4×D97—21no—
65C2×C4×D97—21no—
66#13095501—41no—
67#13095499—22no—
68#13095499—41no—
69#13095498—22no—
70#13095498—41no—
71#13095496—22no—
72#13095496—41no—
73#13095496—41no—
74#13095496—41no—
75C2×#11715895C2311yescharacteristic
76C2×#11715895C2311yescharacteristic
77C2×#11715895C2311yescharacteristic
78C2×#11715895C2311yescharacteristic
79C4⋊Dic97C2311yescharacteristic
80#13095490—21no—
81#13095488—21no—
82#13095488—21no—
83#13095488—21no—
84#13095488—21no—
85#13095488—21no—
86#13095488—21no—
87C4×Dic97—21no—
88C2×Q8×C97D411yescharacteristic
89C2×Q8×C97C2311yescharacteristic
90C2×D4×C97—21no—
91C2×D4×C97—21no—
92C2×D4×C97C2311yescharacteristic
93C2×D4×C97C2311yescharacteristic
94C22×C4×C97—21no—
95C22×C4×C97—21no—
96C4⋊C4×C97—21no—
97C4⋊C4×C97D411yescharacteristic
98C4⋊C4×C97D411yescharacteristic
99C4⋊C4×C97D411yescharacteristic
100C4⋊C4×C97C2311yescharacteristic
101C22⋊C4×C97—21no—
102C22⋊C4×C97—21no—
103C22⋊C4×C97—21no—
104C22⋊C4×C97—21no—
105C22⋊C4×C97—21no—
106C22⋊C4×C97—21no—
107C42×C97—21no—
108C42×C97C2311yescharacteristic
109C42×C97C2311yescharacteristic
110C42×C97C2311yescharacteristic
111C23×C97—21no—
112C23×C97—21no—
113C22×D97—21no—
114C22×D97—21no—
115Q8×C97—21no—
116Q8×C97—21no—
117Q8×C97—41no—
118D4×C97—21no—
119D4×C97—21no—
120D4×C97—21no—
121D4×C97—21no—
122D4×C97—42no—
123D4×C97—81no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
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×C97C2×D411yescharacteristic
134C2×C4×C97C2411yescharacteristic
135C2×Dic97—21no—
136C2×Dic97—21no—
137D388—21no—
138D388—21no—
139D388—21no—
140D388—21no—
141D388—42no—
142D388—81no—
143C4×D97—81no—
144C4×D97—81no—
145Dic194—21no—
146Dic194—21no—
147#11715895—21no—
148#11715895—21no—
149#11715895—21no—
150#11715895#4112yes—
151C22×C97—41no—
152C22×C97—41no—
153C22×C97—41no—
154C22×C97—41no—
155C22×C97—41no—
156C22×C97—41no—
157C22×C97C22×D411yescharacteristic
158C2×D97—41no—
159C2×D97—41no—
160C2×D97—41no—
161C2×D97—41no—
162C2×D97—41no—
163C2×D97—41no—
164C4×C97—21no—
165C4×C97—21no—
166C4×C97—21no—
167C4×C97—21no—
168C4×C97—21no—
169C4×C97—21no—
170C4×C97—41no—
171C4×C97—41no—
172C4×C97C2×#4111yescharacteristic
173C4×C97H(5,𝔽2)11yescharacteristic
174Dic97—41no—
175Dic97—41no—
176C2×C97—81no—
177C2×C97—81no—
178C2×C97#55011yescharacteristic
179C2×C97#57511yescharacteristic
180C2×C97#57611yescharacteristic
181D97—81no—
182D97—81no—
183#3403—971nomaximal, Sylow
184C97#340311yescharacteristic, complemented, Sylow
185#559—971no—
186#559D9711yescharacteristic, complemented
187#496—971no—
188#496—971no—
189#495—971no—
190#464—971no—
191#464—971no—
192#463—971no—
193#463—971no—
194#459—971no—
195#459—971no—
196#446—971no—
197#440—971no—
198#440—971no—
199C4×D8—971no—
200C2×Q16—971no—
201C2×SD16—971no—
202C2×SD16—971no—
203C2×SD16—971no—
204C2×SD16—971no—
205C2×D8—971no—
206C2×D8—971no—
207C2×D8—971no—
208C2×D8—971no—
209#127—971no—
210#127C2×D9711yescharacteristic
211#124—21no—
212#124—971no—
213#124—971no—
214#124—1941no—
215#124C2×D9711yescharacteristic
216#124C2×D9711yescharacteristic
217#123—21no—
218#123—21no—
219#123—1941no—
220#123—1941no—
221C4×Q8—971no—
222C4×Q8C2×D9711yescharacteristic
223C4×Q8C2×D9711yescharacteristic
224C4×D4—21no—
225C4×D4—971no—
226C4×D4—971no—
227C4×D4—1941no—
228C4×D4C2×D9711yescharacteristic
229C4×D4C2×D9711yescharacteristic
230#107—971no—
231C4⋊C8—971no—
232C4⋊C8—971no—
233C4⋊C8—971no—
234#103—971no—
235#103—971no—
236#102—971no—
237#102—971no—
238#102—971no—
239#102—971no—
240#97—971no—
241#97—971no—
242C4×C8—971no—
243C2×Q8—971no—
244C2×Q8C22⋊D9711yescharacteristic
245C2×Q8C22×D9711yescharacteristic
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—971no—
249C2×D4—971no—
250C2×D4—1941no—
251C2×D4—1941no—
252C2×D4C22×D9711yescharacteristic
253C2×D4C22×D9711yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—1941no—
257C22×C4—1941no—
258Q16—3881no—
259SD16—3881no—
260SD16—3881no—
261SD16—1942no—
262SD16—1942no—
263D8—3881no—
264D8—3881no—
265D8—3881no—
266D8—1942no—
267C2×C8—971no—
268C2×C8—971no—
269C2×C8—971no—
270C2×C8—971no—
271C4⋊C4—21no—
272C4⋊C4—971no—
273C4⋊C4—1941no—
274C4⋊C4C22⋊D9711yescharacteristic
275C4⋊C4C22⋊D9711yescharacteristic
276C4⋊C4C22⋊D9711yescharacteristic
277C4⋊C4C22×D9711yescharacteristic
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—1941no—
285C22⋊C4—1941no—
286C22⋊C4—1941no—
287C22⋊C4—1941no—
288C22⋊C4—1941no—
289C22⋊C4—1941no—
290C42—21no—
291C42—1941no—
292C42C22×D9711yescharacteristic
293C42C22×D9711yescharacteristic
294C42C22×D9711yescharacteristic
295C23—21no—
296C23—21no—
297C23—1941no—
298C23—1941no—
299Q8—21no—
300Q8—21no—
301Q8—41no—
302Q8—1941no—
303Q8—1941no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—81no—
310D4—7761no—
311D4—3882no—
312D4—1941no—
313D4—1941no—
314D4—1941no—
315D4—1941no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—7761no—
321C2×C4—7761no—
322C2×C4—1941no—
323C2×C4—1941no—
324C2×C4C2×C22⋊D9711yescharacteristic
325C2×C4C2×C22⋊D9711yescharacteristic
326C2×C4C2×C22⋊D9711yescharacteristic
327C2×C4C2×C22⋊D9711yescharacteristic
328C2×C4C2×C22⋊D9711yescharacteristic
329C2×C4C2×C22⋊D9711yescharacteristic
330C2×C4C23×D9711yescharacteristic
331C8—972no—
332C8—1941no—
333C8—1941no—
334C8—1941no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—3881no—
342C22—3881no—
343C22—3881no—
344C22—3881no—
345C22—3881no—
346C22—3881no—
347C22C22×C22⋊D9711yescharacteristic
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—41no—
355C4—41no—
356C4—3881no—
357C4—3881no—
358C4#41×D9711yescharacteristic
359C4#1569085611yescharacteristic
360C2—81no—
361C2—81no—
362C2—7761no—
363C2—7761no—
364C2#2026140411yescharacteristic
365C2#2026153311yescharacteristic
366C2#2026153811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#559×C97C2
socleC22×C97C22×D4
hypercenter#559D97
hypocenterC97#3403
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #559

Complements

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

NH ≅ G/Ntype
1. #20261470C2semidirect
2. #20261125C2semidirect
3. #20261122C2semidirect
4. #20261117C2semidirect
5. #20260957C2semidirect
6. #20260954C2semidirect
7. #20260950C2semidirect
8. #20260947C2semidirect
9. #20260930C2semidirect
10. #20260927C2semidirect
11. #20260866C2semidirect
12. #20260835C2semidirect
13. #20260833C2semidirect
14. C4×#13095496C2semidirect
15. #559×C97C2semidirect
184. C97#3403semidirect
186. #559D97semidirect

Maximal subgroups

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

subgroupindex
#202614702
#202611252
#202611222
#202611172
#202609572
#202609542
#202609502
#202609472
#202609302
#202609272
#202608662
#202608352
#202608332
C4×#130954962
#559×C972
#340397