Finite Group Databasebeta

← #27180849

Subgroups — #27180849

Order 12928.

452 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265975C211yesmaximal, characteristic, complemented
2C2×#15692275C211yesmaximal, characteristic, complemented
3C2×#15692262C211yesmaximal, characteristic, complemented
4#20265450C212yesmaximal, complemented
5#20265445C212yesmaximal, complemented
6#20265442C212yesmaximal, complemented
7#20265438C212yesmaximal, complemented
8C2×#15692190C211yesmaximal, characteristic, complemented
9#20265161C211yesmaximal, characteristic, complemented
10#20265159C211yesmaximal, characteristic, complemented
11#575×C101C211yesmaximal, characteristic, complemented
12C2×#13108074C2211yescharacteristic
13#15692286—21no—
14#15692285C2212yes—
15#15692281—21no—
16#15692280C2212yes—
17#15692275—21no—
18#15692275—21no—
19#15692275—21no—
20#15692275—21no—
21#15692275—22no—
22C2×#13108064C2211yescharacteristic, complemented
23C2×#13108062C2211yescharacteristic, complemented
24C2×#13108061C2211yescharacteristic, complemented
25C2×#13108059C2211yescharacteristic, complemented
26#15692262—21no—
27#15692262—21no—
28#15692262C2212yes—
29#15692262C2212yes—
30C2×#13108058—21no—
31C2×#13108058C2211yescharacteristic
32C2×#13108058C2211yescharacteristic
33C22×#11717579—21no—
34C22×#11717579C2211yescharacteristic
35#15692190C2212yescomplemented
36#15692190C2212yescomplemented
37#15692174C2212yescomplemented
38#15692173C2212yescomplemented
39#15692172C2212yescomplemented
40#15692171C2212yescomplemented
41C2×#41×C101C2211yescharacteristic
42C2×#41×C101C2211yescharacteristic
43#137×C101—21no—
44#137×C101C2212yes—
45AGL(1,ℤ/8)×C101—21no—
46AGL(1,ℤ/8)×C101C2212yes—
47#135×C101—22no—
48C2×M16×C101C2211yescharacteristic
49#13108074—21no—
50#13108074—21no—
51#13108074—21no—
52#13108074—21no—
53C2×Dic202C2311yescharacteristic
54C2×D404C2311yescharacteristic
55C2×C22⋊D101—21no—
56C2×C4×D101—21no—
57#13108064—22no—
58#13108064—21no—
59#13108064—21no—
60#13108062—22no—
61#13108062—21no—
62#13108062—21no—
63#13108061—22no—
64#13108061—21no—
65#13108061—21no—
66#13108059—22no—
67#13108059—21no—
68#13108059—21no—
69#13108058—21no—
70#13108058—21no—
71#13108058—22no—
72#13108058—22no—
73#13108058C2312yes—
74#13108058C2312yes—
75C2×#11717579—21no—
76C2×#11717579—21no—
77C2×#11717579—22no—
78C2×#11717579—22no—
79C2×#11717579D411yescharacteristic
80C2×#11717579D411yescharacteristic
81C2×#11717579D411yescharacteristic
82C2×#11717579D411yescharacteristic
83C2×#11717579C2311yescharacteristic
84C2×#11717579C2311yescharacteristic
85#41×C101—21no—
86#41×C101—21no—
87#41×C101—21no—
88#41×C101—21no—
89#41×C101—21no—
90#41×C101—21no—
91#41×C101D412yes—
92#41×C101C2312yes—
93C2×Q8×C101D411yescharacteristic
94C2×Q8×C101C2311yescharacteristic
95C2×D4×C101—21no—
96C2×D4×C101—21no—
97C2×D4×C101D411yescharacteristic
98C2×D4×C101C2311yescharacteristic
99C22×C4×C101—21no—
100C22×C4×C101—21no—
101C22×C4×C101C2311yescharacteristic
102Q16×C101—22no—
103SD16×C101—22no—
104SD16×C101—22no—
105D8×C101—22no—
106M16×C101—21no—
107M16×C101C2312yes—
108C2×C8×C101—21no—
109C23×C101—21no—
110C23×C101—21no—
111C23×C101#4111yescharacteristic
112C22×D101—21no—
113Q8×C101—21no—
114Q8×C101—21no—
115Q8×C101—21no—
116Q8×C101C2×D412yes—
117D4×C101—22no—
118D4×C101—22no—
119D4×C101—21no—
120D4×C101—21no—
121D4×C101—21no—
122D4×C101—41no—
123D4×C101—41no—
124D4×C101C2×D412yes—
125C2×C4×C101—21no—
126C2×C4×C101—22no—
127C2×C4×C101—22no—
128C2×C4×C101—21no—
129C2×C4×C101—41no—
130C2×C4×C101—41no—
131C2×C4×C101C2×D411yescharacteristic
132C2×C4×C101C2×D411yescharacteristic
133C2×C4×C101C2×D411yescharacteristic
134C2×C4×C101C2×D411yescharacteristic
135C2×C4×C101#4111yescharacteristic
136C2×C4×C101C2411yescharacteristic
137C22⋊D101—41no—
138C22⋊D101—41no—
139C2×Dic101—21no—
140D404—21no—
141D404—21no—
142C4×D101—41no—
143C4×D101—41no—
144Dic202—21no—
145Dic202—21no—
146C8×C101—22no—
147#11717579—22no—
148#11717579—22no—
149#11717579C2×D411yescharacteristic
150#11717579C2×D411yescharacteristic
151#11717579C2×D411yescharacteristic
152#11717579C2×D411yescharacteristic
153C22×C101—22no—
154C22×C101—41no—
155C22×C101—42no—
156C22×C101—41no—
157C22×C101—81no—
158C22×C101#12111yescharacteristic
159C22×C101#12111yescharacteristic
160C22×C101C2×#4111yescharacteristic
161C2×D101—41no—
162C2×D101—81no—
163C4×C101—22no—
164C4×C101—41no—
165C4×C101#12111yescharacteristic
166C4×C101#12111yescharacteristic
167C4×C101C22×D411yescharacteristic
168C4×C101C22×D411yescharacteristic
169Dic101—41no—
170C2×C101—21no—
171C2×C101—21no—
172C2×C101—21no—
173C2×C101—42no—
174C2×C101—81no—
175C2×C101C2×#12111yescharacteristic
176D101—81no—
177#3067—1011nomaximal, Sylow
178C101#306711yescharacteristic, complemented, Sylow
179#575—1011no—
180#575D10111yescharacteristic, complemented
181C2×#135—1011no—
182C2×#131—1011no—
183#473—1012no—
184#472—1012no—
185#471—1012no—
186#471—1012no—
187C2×#108—1011no—
188#413—1011no—
189#413—1011no—
190C2×#41—1011no—
191C2×#41C2×D10111yescharacteristic
192C2×#41C2×D10111yescharacteristic
193#137—21no—
194#137—1012no—
195#137—2021no—
196#137C2×D10112yes—
197AGL(1,ℤ/8)—21no—
198AGL(1,ℤ/8)—1012no—
199AGL(1,ℤ/8)—2021no—
200AGL(1,ℤ/8)C2×D10112yes—
201#135—22no—
202#135—2021no—
203#135—2021no—
204#135—2021no—
205#135—2021no—
206#135—2022no—
207C2×Q16—1011no—
208C2×SD16—1011no—
209C2×SD16—1011no—
210C2×D8—1011no—
211#131—1012no—
212#131—1012no—
213#131—2021no—
214#131—2021no—
215C2×M16—1011no—
216C2×M16—1011no—
217C2×M16—2021no—
218C2×M16C2×D10111yescharacteristic
219C22×C8—1011no—
220C22×C8—2021no—
221#108—1012no—
222#108—1012no—
223#101—1012no—
224#101—1012no—
225#100—1012no—
226#100—1012no—
227#41—21no—
228#41—21no—
229#41—21no—
230#41—21no—
231#41—21no—
232#41—21no—
233#41—2021no—
234#41—2021no—
235#41—2021no—
236#41—2021no—
237#41C22⋊D10112yes—
238#41C22×D10112yes—
239C2×Q8—1011no—
240C2×Q8C22⋊D10111yescharacteristic
241C2×Q8C22×D10111yescharacteristic
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—1011no—
245C2×D4—2021no—
246C2×D4C22⋊D10111yescharacteristic
247C2×D4C22×D10111yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—2021no—
251C22×C4C22×D10111yescharacteristic
252Q16—22no—
253Q16—2021no—
254Q16—2021no—
255Q16—2022no—
256SD16—22no—
257SD16—22no—
258SD16—2021no—
259SD16—2021no—
260SD16—2021no—
261SD16—2021no—
262SD16—2022no—
263SD16—2022no—
264D8—22no—
265D8—2021no—
266D8—2021no—
267D8—2022no—
268M16—21no—
269M16—1012no—
270M16—1012no—
271M16—2022no—
272M16—2022no—
273M16—2021no—
274M16—2021no—
275M16C22×D10112yes—
276C2×C8—21no—
277C2×C8—1011no—
278C2×C8—1011no—
279C2×C8—1011no—
280C2×C8—1011no—
281C2×C8—1011no—
282C2×C8—1011no—
283C2×C8—2022no—
284C2×C8—2022no—
285C2×C8—2021no—
286C2×C8—2021no—
287C23—21no—
288C23—21no—
289C23—2021no—
290C23#1310807311yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—2021no—
295Q8—2021no—
296Q8C2×C22⋊D10112yes—
297D4—22no—
298D4—22no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—41no—
303D4—41no—
304D4—4041no—
305D4—4041no—
306D4—2021no—
307D4—2021no—
308D4C2×C22⋊D10112yes—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—4041no—
316C2×C4—4041no—
317C2×C4—2021no—
318C2×C4D4×D10111yescharacteristic
319C2×C4D4×D10111yescharacteristic
320C2×C4C2×C22⋊D10111yescharacteristic
321C2×C4C2×C22⋊D10111yescharacteristic
322C2×C4#1310807311yescharacteristic
323C2×C4C23×D10111yescharacteristic
324C8—22no—
325C8—1011no—
326C8—1011no—
327C8—1011no—
328C8—1011no—
329C8—2022no—
330C8—2022no—
331C22—22no—
332C22—41no—
333C22—42no—
334C22—41no—
335C22—81no—
336C22—8081no—
337C22—4041no—
338C22#1569222711yescharacteristic
339C22#1569222711yescharacteristic
340C22C2×#1310807311yescharacteristic
341C4—22no—
342C4—41no—
343C4—4041no—
344C4#1569222711yescharacteristic
345C4#1569222711yescharacteristic
346C4C2×D4×D10111yescharacteristic
347C4C22×C22⋊D10111yescharacteristic
348C2—21no—
349C2—21no—
350C2—21no—
351C2—42no—
352C2—81no—
353C2—8081no—
354C2C2×#1569222711yescharacteristic
subgroupG/H
centerC4#15692227
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#575×C101C2
socleC2×C101C2×#121
hypercenter#575D101
hypocenterC101#3067
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #20265975C2semidirect
2. C2×#15692275C2semidirect
3. C2×#15692262C2semidirect
4. #20265450C2semidirect
5. #20265445C2semidirect
6. #20265442C2semidirect
7. #20265438C2semidirect
8. C2×#15692190C2semidirect
9. #20265161C2semidirect
10. #20265159C2semidirect
11. #575×C101C2semidirect
22. C2×#13108064C22semidirect
23. C2×#13108062C22semidirect
24. C2×#13108061C22semidirect
25. C2×#13108059C22semidirect
35. #15692190C22semidirect
36. #15692190C22semidirect
37. #15692174C22semidirect
38. #15692173C22semidirect
39. #15692172C22semidirect
40. #15692171C22semidirect
178. C101#3067semidirect
180. #575D101semidirect

Maximal subgroups

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

subgroupindex
#202659752
C2×#156922752
C2×#156922622
#202654502
#202654452
#202654422
#202654382
C2×#156921902
#202651612
#202651592
#575×C1012
#3067101