Finite Group Databasebeta

← #27151935

Subgroups — #27151935

Order 12416.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261340C211yesmaximal, characteristic, complemented
2#20261053C211yesmaximal, characteristic, complemented
3#20261048C211yesmaximal, characteristic, complemented
4#20261042C211yesmaximal, characteristic, complemented
5#20261035C211yesmaximal, characteristic, complemented
6#20260935C211yesmaximal, characteristic, complemented
7#20260930C211yesmaximal, characteristic, complemented
8#20260880C211yesmaximal, characteristic, complemented
9#20260880C211yesmaximal, characteristic, complemented
10#20260845C211yesmaximal, characteristic, complemented
11#20260833C211yesmaximal, characteristic, complemented
12C4×#13095498C211yesmaximal, characteristic, complemented
13C4×#13095496C211yesmaximal, characteristic, complemented
14#20260796C211yesmaximal, characteristic, complemented
15#520×C97C211yesmaximal, characteristic, complemented
16C2×#13095498C2211yescharacteristic
17C2×#13095498C2211yescharacteristic
18C2×#13095496C2211yescharacteristic
19C2×#13095496C2211yescharacteristic
20#15690807C2211yescharacteristic
21#15690802C2211yescharacteristic
22#15690795—21no—
23#15690795—21no—
24#15690784C2211yescharacteristic
25#15690784C2211yescharacteristic
26#15690783—21no—
27#15690783—21no—
28C4×D388C2211yescharacteristic
29C4×D388C2211yescharacteristic
30#15690743C2211yescharacteristic
31#15690743C2211yescharacteristic
32#15690741C2211yescharacteristic
33#15690741C2211yescharacteristic
34C4⋊#11715895C2211yescharacteristic
35C4⋊#11715895C2211yescharacteristic
36#15690734C2211yescharacteristic
37#15690734C2211yescharacteristic
38#15690731C2211yescharacteristic
39#15690731C2211yescharacteristic
40#15690730C2211yescharacteristic
41#15690730C2211yescharacteristic
42#15690730C2211yescharacteristic
43#15690730C2211yescharacteristic
44C22⋊#11715895C2211yescharacteristic
45C22⋊#11715895C2211yescharacteristic
46#15690721C2211yescharacteristic
47C4×#11715895C2211yescharacteristic
48C2×#41×C97C2211yescharacteristic
49C4×Q8×C97C2211yescharacteristic
50C4×Q8×C97C2211yescharacteristic
51C4×D4×C97—21no—
52C4×D4×C97—21no—
53C4×D4×C97C2211yescharacteristic
54C4×D4×C97C2211yescharacteristic
55#117×C97—21no—
56#117×C97C2211yescharacteristic
57C2×C4⋊C4×C97—21no—
58C2×C4⋊C4×C97C2211yescharacteristic
59C2×D388—21no—
60C2×D388—21no—
61C2×D388C2311yescharacteristic
62C2×D388C2311yescharacteristic
63C2×C22⋊D97—21no—
64C2×C22⋊D97—21no—
65C2×C4×D97—21no—
66C2×C4×D97—21no—
67#13095498—22no—
68#13095498—22no—
69#13095496—22no—
70#13095496—22no—
71C2×#11715895C2311yescharacteristic
72C2×#11715895C2311yescharacteristic
73C2×#11715895C2311yescharacteristic
74C2×#11715895C2311yescharacteristic
75C4⋊Dic97C2311yescharacteristic
76C4⋊Dic97C2311yescharacteristic
77#13095490—21no—
78#13095490—21no—
79#13095488—21no—
80#13095488—21no—
81#13095488—21no—
82#13095488—21no—
83#41×C97—22no—
84#41×C97—22no—
85C2×Q8×C97C2311yescharacteristic
86C2×D4×C97—21no—
87C2×D4×C97C2311yescharacteristic
88C22×C4×C97—21no—
89C22×C4×C97—21no—
90C22×C4×C97—21no—
91C22×C4×C97—21no—
92C22×C4×C97C2311yescharacteristic
93C4⋊C4×C97—21no—
94C4⋊C4×C97—21no—
95C4⋊C4×C97—21no—
96C4⋊C4×C97D411yescharacteristic
97C4⋊C4×C97D411yescharacteristic
98C4⋊C4×C97C2311yescharacteristic
99C4⋊C4×C97C2311yescharacteristic
100C22⋊C4×C97—21no—
101C22⋊C4×C97—21no—
102C22⋊C4×C97D411yescharacteristic
103C22⋊C4×C97D411yescharacteristic
104C42×C97—21no—
105C42×C97—21no—
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97—21no—
109C23×C97C2×D411yescharacteristic
110C22×D97—21no—
111C22×D97—21no—
112Q8×C97—21no—
113Q8×C97#4112yes—
114D4×C97—24no—
115D4×C97—21no—
116D4×C97#4112yes—
117C2×C4×C97—24no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—81no—
124C2×C4×C97—81no—
125C2×C4×C97—81no—
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2×D411yescharacteristic
130C2×C4×C97C2×D411yescharacteristic
131C2×C4×C97C2411yescharacteristic
132C22⋊D97—81no—
133C22⋊D97—81no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136D388—21no—
137D388—21no—
138D388—21no—
139D388—21no—
140D388—81no—
141D388—81no—
142C4×D97—81no—
143C4×D97—81no—
144#11715895—21no—
145#11715895—21no—
146#11715895—21no—
147#11715895—21no—
148C22×C97—21no—
149C22×C97—22no—
150C22×C97—21no—
151C22×C97—21no—
152C22×C97—41no—
153C22×C97—41no—
154C22×C97C22×D411yescharacteristic
155C2×D97—41no—
156C2×D97—41no—
157C2×D97—41no—
158C2×D97—41no—
159C2×D97—41no—
160C2×D97—41no—
161C4×C97—21no—
162C4×C97—22no—
163C4×C97—21no—
164C4×C97—21no—
165C4×C97—21no—
166C4×C97—21no—
167C4×C97—41no—
168C4×C97—41no—
169C4×C97C2×#4111yescharacteristic
170C4×C97C2×#4111yescharacteristic
171Dic97—41no—
172Dic97—41no—
173C2×C97—41no—
174C2×C97—42no—
175C2×C97#52511yescharacteristic
176C2×C97#57611yescharacteristic
177C2×C97#57711yescharacteristic
178D97—81no—
179D97—81no—
180#3119—971nomaximal, Sylow
181C97#311911yescharacteristic, complemented, Sylow
182#540—971no—
183#520D9711yescharacteristic, complemented
184#481—971no—
185#480—971no—
186#479—971no—
187#478—971no—
188#460—971no—
189#459—971no—
190#449—971no—
191#449—971no—
192#442—971no—
193#440—971no—
194C4×SD16—971no—
195C4×D8—971no—
196#433—971no—
197C2×#41C2×D9711yescharacteristic
198C2×SD16—971no—
199C2×SD16—971no—
200C2×D8—971no—
201C2×D8—971no—
202#127—971no—
203#125—971no—
204#123—1941no—
205#123—1941no—
206#121—971no—
207#121—971no—
208#121—1941no—
209#121—1941no—
210C4×Q8C2×D9711yescharacteristic
211C4×Q8C2×D9711yescharacteristic
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—971no—
215C4×D4—971no—
216C4×D4C2×D9711yescharacteristic
217C4×D4C2×D9711yescharacteristic
218#117—21no—
219#117C2×D9711yescharacteristic
220C2×C4⋊C4—21no—
221C2×C4⋊C4C2×D9711yescharacteristic
222#107—971no—
223#107—971no—
224#106—971no—
225#106—971no—
226C4⋊C8—971no—
227C4⋊C8—971no—
228#103—971no—
229#103—971no—
230#102—971no—
231#102—971no—
232#102—971no—
233#102—971no—
234#102—971no—
235#102—971no—
236C22⋊C8—971no—
237C22⋊C8—971no—
238#97—971no—
239C4×C8—971no—
240#41—22no—
241#41—22no—
242C2×Q8C22×D9711yescharacteristic
243C2×D4—21no—
244C2×D4—971no—
245C2×D4—971no—
246C2×D4—1941no—
247C2×D4—1941no—
248C2×D4—1941no—
249C2×D4—1941no—
250C2×D4C22×D9711yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—1941no—
256C22×C4—1941no—
257C22×C4C22×D9711yescharacteristic
258SD16—1942no—
259SD16—1942no—
260D8—1942no—
261D8—1942no—
262C2×C8—971no—
263C2×C8—971no—
264C2×C8—971no—
265C2×C8—971no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—971no—
270C4⋊C4—971no—
271C4⋊C4—1941no—
272C4⋊C4—1941no—
273C4⋊C4C22⋊D9711yescharacteristic
274C4⋊C4C22⋊D9711yescharacteristic
275C4⋊C4C22×D9711yescharacteristic
276C4⋊C4C22×D9711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—1941no—
280C22⋊C4—1941no—
281C22⋊C4—1941no—
282C22⋊C4—1941no—
283C22⋊C4C22⋊D9711yescharacteristic
284C22⋊C4C22⋊D9711yescharacteristic
285C42—21no—
286C42—21no—
287C42C22×D9711yescharacteristic
288C42C22×D9711yescharacteristic
289C23—21no—
290C23—1941no—
291C23—1941no—
292C23C2×C22⋊D9711yescharacteristic
293Q8—21no—
294Q8#1309551112yes—
295D4—24no—
296D4—21no—
297D4—7761no—
298D4—7761no—
299D4—7761no—
300D4—7761no—
301D4—1941no—
302D4—1941no—
303D4—1941no—
304D4—1941no—
305D4#1309551112yes—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—7761no—
316C2×C4—7761no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4C2×C22⋊D9711yescharacteristic
320C2×C4C2×C22⋊D9711yescharacteristic
321C2×C4C2×C22⋊D9711yescharacteristic
322C2×C4C2×C22⋊D9711yescharacteristic
323C2×C4C2×C22⋊D9711yescharacteristic
324C2×C4C23×D9711yescharacteristic
325C8—1941no—
326C8—1941no—
327C8—1941no—
328C8—1941no—
329C22—21no—
330C22—22no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—3881no—
336C22—3881no—
337C22—3881no—
338C22—3881no—
339C22—3881no—
340C22—3881no—
341C22C22×C22⋊D9711yescharacteristic
342C4—21no—
343C4—22no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—41no—
349C4—41no—
350C4—3881no—
351C4—3881no—
352C4C2×#1309551111yescharacteristic
353C4C2×#1309551111yescharacteristic
354C2—41no—
355C2—42no—
356C2—7761no—
357C2—7761no—
358C2#2026124611yescharacteristic
359C2#2026153811yescharacteristic
360C2#2026154311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#520×C97C2
socleC22×C97C22×D4
hypercenter#520D97
hypocenterC97#3119
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #520

Complements

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

NH ≅ G/Ntype
1. #20261340C2semidirect
2. #20261053C2semidirect
3. #20261048C2semidirect
4. #20261042C2semidirect
5. #20261035C2semidirect
6. #20260935C2semidirect
7. #20260930C2semidirect
8. #20260880C2semidirect
9. #20260880C2semidirect
10. #20260845C2semidirect
11. #20260833C2semidirect
12. C4×#13095498C2semidirect
13. C4×#13095496C2semidirect
14. #20260796C2semidirect
15. #520×C97C2semidirect
181. C97#3119semidirect
183. #520D97semidirect

Maximal subgroups

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

subgroupindex
#202613402
#202610532
#202610482
#202610422
#202610352
#202609352
#202609302
#202608802
#202608452
#202608332
C4×#130954982
C4×#130954962
#202607962
#520×C972
#311997