Finite Group Databasebeta

← #27154357

Subgroups — #27154357

Order 12416.

354 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261449C211yesmaximal, characteristic
2#20261443C211yesmaximal, characteristic, complemented
3#20261139C211yesmaximal, characteristic, complemented
4#20261074C211yesmaximal, characteristic, complemented
5#20261073C211yesmaximal, characteristic, complemented
6#20261047C211yesmaximal, characteristic, complemented
7#20261046C211yesmaximal, characteristic, complemented
8#20261028C211yesmaximal, characteristic, complemented
9#20261023C211yesmaximal, characteristic, complemented
10#20260978C211yesmaximal, characteristic, complemented
11SD16×Dic97C211yesmaximal, characteristic, complemented
12#20260811C211yesmaximal, characteristic, complemented
13#20260777C211yesmaximal, characteristic, complemented
14#20260776C211yesmaximal, characteristic, complemented
15#468×C97C211yesmaximal, characteristic, complemented
16C2×#13095494C2211yescharacteristic
17#15690809C2211yescharacteristic
18#15690809C2211yescharacteristic
19#15690800—21no—
20#15690797—21no—
21#15690796—21no—
22#15690790—21no—
23#15690787—21no—
24#15690786C2211yescharacteristic
25#15690782—21no—
26#15690781C2211yescharacteristic
27Q8×Dic97C2211yescharacteristic
28D4×Dic97C2211yescharacteristic
29C4×C22⋊D97C2211yescharacteristic
30#15690770—21no—
31#15690769C2211yescharacteristic
32#15690768C2211yescharacteristic
33#15690766—21no—
34C2×C4⋊Dic97C2211yescharacteristic
35C4⋊C4×D97C2211yescharacteristic
36#15690744C2211yescharacteristic
37#15690744C2211yescharacteristic
38#15690743C2211yescharacteristic
39#15690743C2211yescharacteristic
40#15690743C2211yescharacteristic
41#15690742C2211yescharacteristic
42#15690742C2211yescharacteristic
43#15690741C2211yescharacteristic
44#15690739C2211yescharacteristic
45#15690734C2211yescharacteristic
46#15690734C2211yescharacteristic
47#15690731C2211yescharacteristic
48#15690731C2211yescharacteristic
49C22⋊#11715895C2211yescharacteristic
50#15690722C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C2×SD16×C97C2211yescharacteristic
53C2×M16×C97C2211yescharacteristic
54C22⋊Q8×C97C2211yescharacteristic
55#121×C97C2211yescharacteristic
56#107×C97C2211yescharacteristic
57#103×C97C2211yescharacteristic
58#102×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×Dic194—21no—
61C2×C22⋊D97—21no—
62C2×C22⋊D97D411yescharacteristic
63C22×Dic97—21no—
64C22×Dic97—21no—
65C2×C4×D97—21no—
66C2×C4×D97C2311yescharacteristic
67#13095494—22no—
68C2×#11715895C2311yescharacteristic
69C2×#11715895C2311yescharacteristic
70C4⋊Dic97—21no—
71C4⋊Dic97—21no—
72C4⋊Dic97C2311yescharacteristic
73C4⋊Dic97C2311yescharacteristic
74C4⋊Dic97C2311yescharacteristic
75C4⋊Dic97C2311yescharacteristic
76#13095490—21no—
77#13095490—21no—
78#13095490—21no—
79#13095490—21no—
80#13095490D411yescharacteristic
81C22⋊Dic97—21no—
82C22⋊Dic97—21no—
83C22⋊Dic97—21no—
84C22⋊Dic97D411yescharacteristic
85#13095488—21no—
86#13095488—21no—
87#13095488D411yescharacteristic
88C4×Dic97—21no—
89C4×Dic97—21no—
90C4×Dic97—21no—
91C4×Dic97C2311yescharacteristic
92C2×Q8×C97C2311yescharacteristic
93C2×D4×C97—21no—
94C2×D4×C97C2311yescharacteristic
95C22×C4×C97C2311yescharacteristic
96SD16×C97—22no—
97M16×C97—22no—
98C2×C8×C97C2311yescharacteristic
99C2×C8×C97C2311yescharacteristic
100C4⋊C4×C97—21no—
101C4⋊C4×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C22⋊C4×C97—21no—
104C22⋊C4×C97—21no—
105C23×C97—21no—
106C23×C97C2×D411yescharacteristic
107C22×D97C2×D411yescharacteristic
108Q8×C97—21no—
109Q8×C97—21no—
110D4×C97—21no—
111D4×C97—21no—
112D4×C97—81no—
113C2×C4×C97—21no—
114C2×C4×C97—21no—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97C2×D411yescharacteristic
119C2×C4×C97C2411yescharacteristic
120C22⋊D97—22no—
121C22⋊D97—81no—
122C2×Dic97—21no—
123C2×Dic97—21no—
124C2×Dic97—21no—
125C2×Dic97—21no—
126C2×Dic97—42no—
127C2×Dic97—81no—
128C2×Dic97C2×D411yescharacteristic
129C2×Dic97C2×D411yescharacteristic
130C2×Dic97C2×D411yescharacteristic
131C4×D97—21no—
132C4×D97—21no—
133C4×D97—81no—
134Dic194—81no—
135Dic194—81no—
136C8×C97—21no—
137C8×C97#4112yes—
138#11715895—21no—
139#11715895—21no—
140C22×C97—21no—
141C22×C97—21no—
142C22×C97—21no—
143C22×C97—41no—
144C22×C97—41no—
145C22×C97—41no—
146C22×C97C22×D411yescharacteristic
147C2×D97—21no—
148C2×D97—21no—
149C2×D97—21no—
150C4×C97—21no—
151C4×C97—41no—
152C4×C97—41no—
153C4×C97—41no—
154C4×C97C2×#4111yescharacteristic
155C4×C972−1+411yescharacteristic
156Dic97—21no—
157Dic97—21no—
158Dic97—41no—
159Dic97—41no—
160Dic97—41no—
161Dic97—41no—
162Dic97#13712yes—
163C2×C97—41no—
164C2×C97—81no—
165C2×C97#54711yescharacteristic
166C2×C97C2×#13711yescharacteristic
167C2×C97#57611yescharacteristic
168D97—41no—
169#3357—971nomaximal, Sylow
170C97#335711yescharacteristic, complemented, Sylow
171#555—971no—
172#554—971no—
173#500—971no—
174#485—971no—
175#485—971no—
176#480—971no—
177#480—971no—
178#477—971no—
179#476—971no—
180#468—971no—
181#468D9711yescharacteristic, complemented
182C4×SD16—971no—
183#436—971no—
184#428—971no—
185#428—971no—
186C2×SD16C2×D9711yescharacteristic
187C2×M16—971no—
188C2×M16C2×D9711yescharacteristic
189C4⋊Q8—971no—
190C4⋊Q8—971no—
191#125—1941no—
192#124—1941no—
193#124—1941no—
194C22⋊Q8—971no—
195C22⋊Q8—1941no—
196C22⋊Q8—1941no—
197C22⋊Q8C2×D9711yescharacteristic
198#121—971no—
199#121—1941no—
200#121C2×D9711yescharacteristic
201C4×Q8—971no—
202C4×D4—971no—
203C4×D4—971no—
204C4×D4—1941no—
205#117—971no—
206#117—971no—
207#117—1941no—
208C2×C4⋊C4—971no—
209C2×C4⋊C4—971no—
210#107—971no—
211#107—971no—
212#107—971no—
213#107—971no—
214#107—971no—
215#107C2×D9711yescharacteristic
216#106—971no—
217#106—971no—
218#106—971no—
219C4⋊C8—971no—
220#103—971no—
221#103—971no—
222#103C2×D9711yescharacteristic
223#102—971no—
224#102—971no—
225#102C2×D9711yescharacteristic
226C22⋊C8—971no—
227C22⋊C8—971no—
228C4×C8—971no—
229C2×Q8—1941no—
230C2×Q8—1941no—
231C2×Q8C22×D9711yescharacteristic
232C2×D4—21no—
233C2×D4—971no—
234C2×D4—1941no—
235C2×D4C22×D9711yescharacteristic
236C22×C4—971no—
237C22×C4—1941no—
238C22×C4—1941no—
239C22×C4—1941no—
240C22×C4C22×D9711yescharacteristic
241SD16—22no—
242M16—22no—
243M16—1942no—
244C2×C8—971no—
245C2×C8—971no—
246C2×C8C22×D9711yescharacteristic
247C2×C8C22×D9711yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—971no—
250C4⋊C4—971no—
251C4⋊C4—971no—
252C4⋊C4—971no—
253C4⋊C4—971no—
254C4⋊C4—1941no—
255C4⋊C4—1941no—
256C4⋊C4—1941no—
257C4⋊C4—1941no—
258C4⋊C4—1941no—
259C4⋊C4—1941no—
260C4⋊C4C22×D9711yescharacteristic
261C4⋊C4C22×D9711yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—971no—
265C22⋊C4—971no—
266C22⋊C4—1941no—
267C22⋊C4—1941no—
268C22⋊C4—1941no—
269C22⋊C4—1941no—
270C22⋊C4—1941no—
271C42—971no—
272C42—1941no—
273C42—1941no—
274C42—1941no—
275C23—21no—
276C23—971no—
277C23D4×D9711yescharacteristic
278Q8—21no—
279Q8—21no—
280Q8—7761no—
281Q8—7761no—
282D4—21no—
283D4—21no—
284D4—81no—
285D4—7761no—
286D4—1942no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—7761no—
293C2×C4—7761no—
294C2×C4—971no—
295C2×C4—971no—
296C2×C4—971no—
297C2×C4—3882no—
298C2×C4—1941no—
299C2×C4—1941no—
300C2×C4—1941no—
301C2×C4—1941no—
302C2×C4—1941no—
303C2×C4—1941no—
304C2×C4D4×D9711yescharacteristic
305C2×C4C23×D9711yescharacteristic
306C8—21no—
307C8—1941no—
308C8—1941no—
309C8#1309551012yes—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—1941no—
317C22—1941no—
318C22—1941no—
319C22C2×D4×D9711yescharacteristic
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—972no—
325C4—3881no—
326C4—3881no—
327C4—3881no—
328C4—3881no—
329C4—1941no—
330C4—1941no—
331C4C2×#1309551011yescharacteristic
332C4#1569085711yescharacteristic
333C2—41no—
334C2—81no—
335C2—3881no—
336C2#2026138511yescharacteristic
337C2#137×D9711yescharacteristic
338C2#2026153911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#468×C97C2
socleC22×C97C22×D4
prime radical#20261449C2
hypercenter#468D97
hypocenterC97#3357
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
2. #20261443C2semidirect
3. #20261139C2semidirect
4. #20261074C2semidirect
5. #20261073C2semidirect
6. #20261047C2semidirect
7. #20261046C2semidirect
8. #20261028C2semidirect
9. #20261023C2semidirect
10. #20260978C2semidirect
11. SD16×Dic97C2semidirect
12. #20260811C2semidirect
13. #20260777C2semidirect
14. #20260776C2semidirect
15. #468×C97C2semidirect
170. C97#3357semidirect
181. #468D97semidirect

Maximal subgroups

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

subgroupindex
#202614492
#202614432
#202611392
#202610742
#202610732
#202610472
#202610462
#202610282
#202610232
#202609782
SD16×Dic972
#202608112
#202607772
#202607762
#468×C972
#335797