Finite Group Databasebeta

← #27153638

Subgroups — #27153638

Order 12416.

398 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261431C211yesmaximal, characteristic, complemented
2#20261089C211yesmaximal, characteristic, complemented
3#20261073C211yesmaximal, characteristic, complemented
4#20261051C211yesmaximal, characteristic, complemented
5#20261042C211yesmaximal, characteristic, complemented
6#20260969C211yesmaximal, characteristic, complemented
7#20260966C211yesmaximal, characteristic, complemented
8#20260947C211yesmaximal, characteristic, complemented
9#20260880C211yesmaximal, characteristic, complemented
10#20260874C211yesmaximal, characteristic, complemented
11C4×#13095498C211yesmaximal, characteristic, complemented
12D4×#11715895C211yesmaximal, characteristic, complemented
13#20260731C211yesmaximal, characteristic, complemented
14C2×#15690734C211yesmaximal, characteristic, complemented
15#548×C97C211yesmaximal, characteristic, complemented
16C2×#13095498C2211yescharacteristic
17C2×#13095498C2211yescharacteristic
18C2×#13095496C2211yescharacteristic
19C22×#11715895C2211yescharacteristic
20C22×#11715895C2211yescharacteristic
21#15690805—21no—
22#15690802C2211yescharacteristic
23#15690792—21no—
24#15690784C2211yescharacteristic
25#15690784C2211yescharacteristic
26#15690782—21no—
27C4×D388C2211yescharacteristic
28#15690770—21no—
29#15690769C2211yescharacteristic
30C2×C4⋊Dic97C2211yescharacteristic
31#15690743C2211yescharacteristic
32#15690741C2211yescharacteristic
33#15690741C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic
35#15690734C2211yescharacteristic
36#15690734C2211yescharacteristic
37#15690734C2211yescharacteristic
38#15690734C2211yescharacteristic
39#15690734C2211yescharacteristic
40#15690734C2211yescharacteristic
41#15690731C2211yescharacteristic
42#15690731C2211yescharacteristic
43#15690731C2211yescharacteristic
44#15690730C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46C22⋊#11715895C2211yescharacteristic
47C4×#11715895C2211yescharacteristic
48C2×#41×C97C2211yescharacteristic
49C22×Q8×C97C2211yescharacteristic
50#124×C97—21no—
51#124×C97C2211yescharacteristic
52C22⋊Q8×C97—21no—
53C22⋊Q8×C97C2211yescharacteristic
54#121×C97—21no—
55#121×C97C2211yescharacteristic
56C4×Q8×C97C2211yescharacteristic
57C4×D4×C97—21no—
58C4×D4×C97C2211yescharacteristic
59C2×D388C2311yescharacteristic
60C2×C22⋊D97—21no—
61C2×C22⋊D97—21no—
62C22×Dic97—21no—
63C2×C4×D97—21no—
64#13095498—22no—
65#13095498—41no—
66#13095496—41no—
67C2×#11715895—41no—
68C2×#11715895—41no—
69C2×#11715895C2311yescharacteristic
70C2×#11715895C2311yescharacteristic
71C2×#11715895C2311yescharacteristic
72C2×#11715895C2311yescharacteristic
73C4⋊Dic97—21no—
74C4⋊Dic97C2311yescharacteristic
75C4⋊Dic97C2311yescharacteristic
76C4⋊Dic97C2311yescharacteristic
77#13095490—21no—
78#13095490—21no—
79C22⋊Dic97—21no—
80#13095488—21no—
81#13095488—21no—
82#13095488—21no—
83C4×Dic97—21no—
84#41×C97—22no—
85#41×C97—41no—
86C2×Q8×C97—21no—
87C2×Q8×C97—22no—
88C2×Q8×C97—41no—
89C2×Q8×C97C2311yescharacteristic
90C2×Q8×C97C2311yescharacteristic
91C2×D4×C97—21no—
92C2×D4×C97—21no—
93C2×D4×C97D411yescharacteristic
94C2×D4×C97C2311yescharacteristic
95C22×C4×C97—21no—
96C22×C4×C97—21no—
97C22×C4×C97C2311yescharacteristic
98C22×C4×C97C2311yescharacteristic
99C4⋊C4×C97—21no—
100C4⋊C4×C97—21no—
101C4⋊C4×C97D411yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C22⋊C4×C97—21no—
104C22⋊C4×C97—21no—
105C22⋊C4×C97—21no—
106C22⋊C4×C97—21no—
107C22⋊C4×C97D411yescharacteristic
108C22⋊C4×C97D411yescharacteristic
109C42×C97—21no—
110C42×C97C2311yescharacteristic
111C23×C97—21no—
112C23×C97C2×D411yescharacteristic
113C23×C97C2×D411yescharacteristic
114C22×D97—21no—
115Q8×C97—21no—
116Q8×C97—21no—
117Q8×C97—21no—
118Q8×C97—42no—
119Q8×C97#4112yes—
120D4×C97—22no—
121D4×C97—21no—
122D4×C97—21no—
123D4×C97—42no—
124D4×C97—42no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—21no—
128C2×C4×C97—21no—
129C2×C4×C97—21no—
130C2×C4×C97—21no—
131C2×C4×C97—21no—
132C2×C4×C97—42no—
133C2×C4×C97—42no—
134C2×C4×C97C2×D411yescharacteristic
135C2×C4×C97C2×D411yescharacteristic
136C2×C4×C97C2×D411yescharacteristic
137C2×C4×C97C2×D411yescharacteristic
138C2×C4×C97C2411yescharacteristic
139C22⋊D97—42no—
140C22⋊D97—81no—
141C2×Dic97—21no—
142C2×Dic97—21no—
143C2×Dic97—21no—
144C2×Dic97—81no—
145D388—21no—
146D388—21no—
147C4×D97—81no—
148#11715895—21no—
149#11715895—21no—
150#11715895—21no—
151#11715895—21no—
152C22×C97—21no—
153C22×C97—21no—
154C22×C97—21no—
155C22×C97—21no—
156C22×C97—21no—
157C22×C97—41no—
158C22×C97—41no—
159C22×C97—41no—
160C22×C97#13512yes—
161C22×C97C22×D411yescharacteristic
162C2×D97—41no—
163C2×D97—41no—
164C2×D97—41no—
165C4×C97—21no—
166C4×C97—21no—
167C4×C97—22no—
168C4×C97—21no—
169C4×C97—21no—
170C4×C97—41no—
171C4×C97—41no—
172C4×C97C2×#4111yescharacteristic
173C4×C97H(5,𝔽2)11yescharacteristic
174Dic97—41no—
175Dic97—41no—
176Dic97—41no—
177C2×C97—22no—
178C2×C97—41no—
179C2×C97—81no—
180C2×C97#54611yescharacteristic
181C2×C97C2×#13511yescharacteristic
182C2×C97#57711yescharacteristic
183D97—81no—
184#3301—971nomaximal, Sylow
185C97#330111yescharacteristic, complemented, Sylow
186#553—971no—
187#548D9711yescharacteristic, complemented
188#488—971no—
189#485—971no—
190#481—971no—
191#479—971no—
192#466—971no—
193#465—971no—
194#463—971no—
195#449—971no—
196#448—971no—
197C4×SD16—971no—
198C8×D4—971no—
199#416—971no—
200C2×#103—971no—
201C2×#41C2×D9711yescharacteristic
202C22×Q8C2×D9711yescharacteristic
203C2×SD16—971no—
204C2×SD16—971no—
205C2×D8—971no—
206C22×C8—971no—
207C22×C8—971no—
208#126—1941no—
209#125—971no—
210#124—21no—
211#124C2×D9711yescharacteristic
212#123—1941no—
213C22⋊Q8—21no—
214C22⋊Q8C2×D9711yescharacteristic
215#121—21no—
216#121—971no—
217#121—971no—
218#121—1941no—
219#121C2×D9711yescharacteristic
220C4×Q8C2×D9711yescharacteristic
221C4×D4—21no—
222C4×D4—971no—
223C4×D4—1941no—
224C4×D4C2×D9711yescharacteristic
225#117—971no—
226C2×C4⋊C4—971no—
227#107—971no—
228#106—971no—
229#106—971no—
230C4⋊C8—971no—
231#103—971no—
232#103—971no—
233#103—971no—
234#103—971no—
235#103—971no—
236#103—971no—
237#102—971no—
238#102—971no—
239#102—971no—
240#102—971no—
241C22⋊C8—971no—
242C22⋊C8—971no—
243C4×C8—971no—
244#41—22no—
245#41—41no—
246C2×Q8—21no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8C22×D9711yescharacteristic
250C2×Q8C22×D9711yescharacteristic
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—971no—
254C2×D4—1941no—
255C2×D4—1941no—
256C2×D4C22⋊D9711yescharacteristic
257C2×D4C22×D9711yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—1941no—
261C22×C4—1941no—
262C22×C4C22×D9711yescharacteristic
263C22×C4C22×D9711yescharacteristic
264SD16—3881no—
265SD16—1942no—
266D8—3881no—
267C2×C8—971no—
268C2×C8—971no—
269C2×C8—971no—
270C2×C8—971no—
271C2×C8—3881no—
272C2×C8—3881no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—971no—
276C4⋊C4—971no—
277C4⋊C4—971no—
278C4⋊C4—1941no—
279C4⋊C4—1941no—
280C4⋊C4—1941no—
281C4⋊C4C22⋊D9711yescharacteristic
282C4⋊C4C22×D9711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—1941no—
288C22⋊C4—1941no—
289C22⋊C4—1941no—
290C22⋊C4—1941no—
291C22⋊C4C22⋊D9711yescharacteristic
292C22⋊C4C22⋊D9711yescharacteristic
293C42—21no—
294C42—1941no—
295C42C22×D9711yescharacteristic
296C23—21no—
297C23—1941no—
298C23C2×C22⋊D9711yescharacteristic
299C23C2×C22⋊D9711yescharacteristic
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—42no—
304Q8#1309551012yes—
305D4—22no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—42no—
310D4—7761no—
311D4—3882no—
312D4—1941no—
313D4—1941no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—7761no—
324C2×C4—7761no—
325C2×C4—1941no—
326C2×C4—1941no—
327C2×C4—1941no—
328C2×C4C2×C22⋊D9711yescharacteristic
329C2×C4C2×C22⋊D9711yescharacteristic
330C2×C4C2×C22⋊D9711yescharacteristic
331C2×C4C2×C22⋊D9711yescharacteristic
332C2×C4C23×D9711yescharacteristic
333C8—1941no—
334C8—1941no—
335C8—1941no—
336C8—1941no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—3881no—
346C22—3881no—
347C22—3881no—
348C22#1569083012yes—
349C22C22×C22⋊D9711yescharacteristic
350C4—21no—
351C4—21no—
352C4—22no—
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—3881no—
358C4—3881no—
359C4—3881no—
360C4C2×#1309551011yescharacteristic
361C4#1569085511yescharacteristic
362C2—22no—
363C2—41no—
364C2—81no—
365C2—7761no—
366C2#2026137911yescharacteristic
367C2C2×#1569083011yescharacteristic
368C2#2026154311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#548×C97C2
socleC22×C97C22×D4
hypercenter#548D97
hypocenterC97#3301
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #20261431C2semidirect
2. #20261089C2semidirect
3. #20261073C2semidirect
4. #20261051C2semidirect
5. #20261042C2semidirect
6. #20260969C2semidirect
7. #20260966C2semidirect
8. #20260947C2semidirect
9. #20260880C2semidirect
10. #20260874C2semidirect
11. C4×#13095498C2semidirect
12. D4×#11715895C2semidirect
13. #20260731C2semidirect
14. C2×#15690734C2semidirect
15. #548×C97C2semidirect
185. C97#3301semidirect
187. #548D97semidirect

Maximal subgroups

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

subgroupindex
#202614312
#202610892
#202610732
#202610512
#202610422
#202609692
#202609662
#202609472
#202608802
#202608742
C4×#130954982
D4×#117158952
#202607312
C2×#156907342
#548×C972
#330197