Finite Group Databasebeta

← #27182650

Subgroups — #27182650

Order 12928.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692279C211yesmaximal, characteristic, complemented
2C2×#15692275C211yesmaximal, characteristic, complemented
3#20265826C211yesmaximal, characteristic, complemented
4#20265824C211yesmaximal, characteristic, complemented
5#20265568C211yesmaximal, characteristic, complemented
6#20265434C211yesmaximal, characteristic, complemented
7#20265404C211yesmaximal, characteristic, complemented
8#20265390C211yesmaximal, characteristic, complemented
9#20265381C211yesmaximal, characteristic, complemented
10#20265341C211yesmaximal, characteristic, complemented
11#20265340C211yesmaximal, characteristic, complemented
12#20265326C211yesmaximal, characteristic, complemented
13#20265286C211yesmaximal, characteristic, complemented
14#20265245C211yesmaximal, characteristic, complemented
15#449×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C2×#13108073C2211yescharacteristic
18C2×D4×D101C2211yescharacteristic
19#15692279—24no—
20#15692275—24no—
21C2×Dic404C2211yescharacteristic, complemented
22C2×#13108064C2211yescharacteristic, complemented
23C2×#13108063C2211yescharacteristic
24C2×#13108062C2211yescharacteristic
25C2×#13108062C2211yescharacteristic, complemented
26C2×#13108061C2211yescharacteristic
27C2×#13108059C2211yescharacteristic
28C2×#13108057C2211yescharacteristic
29C22×#11717579C2211yescharacteristic, complemented
30#15692253C2211yescharacteristic, complemented
31#15692242C2211yescharacteristic
32#15692238—21no—
33#15692237—21no—
34#15692236—21no—
35#15692235C2211yescharacteristic
36#15692234C2211yescharacteristic, complemented
37#15692231C2211yescharacteristic
38#15692227—21no—
39#15692226—21no—
40#15692223—21no—
41D4×Dic101C2211yescharacteristic, complemented
42#15692217C2211yescharacteristic
43C4×C22⋊D101C2211yescharacteristic
44C2×#13108053—21no—
45C22⋊C4×D101—21no—
46#15692186C2211yescharacteristic, complemented
47#15692184C2211yescharacteristic, complemented
48#15692180C2211yescharacteristic, complemented
49#15692178C2211yescharacteristic, complemented
50#15692177C2211yescharacteristic
51#15692176C2211yescharacteristic
52#15692167C2211yescharacteristic
53#15692165C2211yescharacteristic, complemented
54C2×#41×C101C2211yescharacteristic, complemented
55C2×SD16×C101C2211yescharacteristic
56C2×D8×C101C2211yescharacteristic
57#121×C101C2211yescharacteristic
58#103×C101C2211yescharacteristic, complemented
59#102×C101C2211yescharacteristic, complemented
60C22⋊C8×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62#13108074—22no—
63#13108074—41no—
64#13108073—22no—
65#13108073—41no—
66C2×Dic202C2311yescharacteristic
67C2×Dic202C2311yescharacteristic
68C2×D404C2311yescharacteristic
69C2×C22⋊D101—21no—
70C2×C22⋊D101—21no—
71C2×C22⋊D101—21no—
72C2×C22⋊D101D411yescharacteristic
73D4×D101—22no—
74D4×D101—41no—
75C22×Dic101—21no—
76C22×Dic101—21no—
77C2×C4×D101—21no—
78C2×C4×D101—21no—
79C2×C4×D101C2311yescharacteristic
80Dic404—41no—
81#13108064—22no—
82#13108063—22no—
83#13108062—22no—
84#13108062—22no—
85#13108061—22no—
86#13108059D414yescomplemented
87#13108057—22no—
88C2×#11717579—22no—
89C2×#11717579C2311yescharacteristic
90C2×#11717579C2311yescharacteristic
91C4⋊Dic101—21no—
92C4⋊Dic101C2311yescharacteristic
93#13108053—21no—
94#13108053—21no—
95#13108053—21no—
96#13108053—21no—
97#13108053D411yescharacteristic
98C22⋊Dic101—21no—
99C22⋊Dic101—21no—
100C22⋊Dic101—21no—
101C22⋊Dic101—21no—
102C22⋊Dic101D411yescharacteristic
103#13108051—21no—
104#13108051—21no—
105#13108051—21no—
106#13108051D411yescharacteristic
107C4×Dic101C2311yescharacteristic
108#41×C101—22no—
109#41×C101—41no—
110C2×Q8×C101C2311yescharacteristic
111C2×D4×C101—21no—
112C2×D4×C101—21no—
113C2×D4×C101C2311yescharacteristic
114C2×D4×C101C2311yescharacteristic
115C22×C4×C101—21no—
116C22×C4×C101C2311yescharacteristic
117SD16×C101—41no—
118D8×C101—22no—
119C2×C8×C101C2311yescharacteristic
120C2×C8×C101C2311yescharacteristic
121C4⋊C4×C101C2311yescharacteristic
122C22⋊C4×C101—21no—
123C23×C101—21no—
124C23×C101—21no—
125C23×C101C2×D411yescharacteristic
126C22×D101—21no—
127C22×D101—42no—
128C22×D101—42no—
129C22×D101—81no—
130C22×D101C2×D411yescharacteristic
131Q8×C101—21no—
132Q8×C101—21no—
133D4×C101—21no—
134D4×C101—21no—
135D4×C101—21no—
136D4×C101—42no—
137D4×C101—81no—
138D4×C101C2×D412yes—
139C2×C4×C101—21no—
140C2×C4×C101—21no—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—42no—
144C2×C4×C101C2×D411yescharacteristic
145C2×C4×C101C2411yescharacteristic
146C22⋊D101—41no—
147C22⋊D101—42no—
148C22⋊D101—42no—
149C22⋊D101—42no—
150C2×Dic101—21no—
151C2×Dic101—21no—
152C2×Dic101—21no—
153C2×Dic101—42no—
154C2×Dic101—42no—
155C2×Dic101C2×D411yescharacteristic
156C2×Dic101C2×D411yescharacteristic
157C2×Dic101C2×D411yescharacteristic
158D404—21no—
159D404C2×D412yes—
160C4×D101—21no—
161C4×D101—21no—
162C4×D101—42no—
163C4×D101—81no—
164Dic202—21no—
165Dic202—21no—
166Dic202—21no—
167Dic202—21no—
168C8×C101—21no—
169C8×C101—21no—
170#11717579—21no—
171#11717579C2×D412yes—
172C22×C101—22no—
173C22×C101—21no—
174C22×C101—21no—
175C22×C101—21no—
176C22×C101—41no—
177C22×C101—41no—
178C22×C101—41no—
179C22×C101—41no—
180C22×C101—41no—
181C22×C101C22×D411yescharacteristic
182C2×D101—21no—
183C2×D101—22no—
184C2×D101—21no—
185C2×D101—21no—
186C2×D101—41no—
187C2×D101—41no—
188C2×D101—84no—
189C4×C101—21no—
190C4×C101—41no—
191C4×C101—41no—
192C4×C101C22×D411yescharacteristic
193C4×C101H(5,𝔽2)11yescharacteristic
194Dic101—21no—
195Dic101—21no—
196Dic101—21no—
197Dic101—41no—
198Dic101—41no—
199Dic101—41no—
200C2×C101—41no—
201C2×C101—42no—
202C2×C101—81no—
203C2×C101D4211yescharacteristic
204C2×C101#57511yescharacteristic
205C2×C101#57711yescharacteristic
206D101—41no—
207D101—42no—
208#3267—1011nomaximal, Sylow
209C101#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—1011no—
211C2×#135—1011no—
212#547—1011no—
213#546—1011no—
214#497—1011no—
215#470—1011no—
216#465—1011no—
217#463—1011no—
218#461—1011no—
219#452—1011no—
220#452—1011no—
221#450—1011no—
222#449D10111yescharacteristic, complemented
223#442—1011no—
224#435—1011no—
225C2×#41—1011no—
226C2×#41—1011no—
227C2×#41C2×D10111yescharacteristic, complemented
228C22×D4—1011no—
229AGL(1,ℤ/8)—2024no—
230#135—2024no—
231C2×Q16—1011no—
232C2×Q16—1011no—
233C2×SD16—1011no—
234C2×SD16—1011no—
235C2×SD16—1011no—
236C2×SD16—1011no—
237C2×SD16C2×D10111yescharacteristic
238C2×D8—1011no—
239C2×D8C2×D10111yescharacteristic
240C2×M16—1011no—
241C22×C8—1011no—
242C4⋊Q8—1011no—
243#124—1011no—
244#123—2021no—
245#123—2021no—
246#123—2021no—
247C22⋊Q8—1011no—
248C22⋊Q8—1011no—
249C22⋊Q8—1011no—
250#121—2021no—
251#121—2021no—
252#121C2×D10111yescharacteristic
253#120—2021no—
254C4×D4—1011no—
255C4×D4—1011no—
256C4×D4—1011no—
257C2×C4⋊C4—2021no—
258C2×C22⋊C4—2021no—
259#106—1011no—
260C4⋊C8—1011no—
261#103—1011no—
262#103—1011no—
263#103C2×D10111yescharacteristic, complemented
264#102—1011no—
265#102—1011no—
266#102C2×D10111yescharacteristic, complemented
267C22⋊C8—1011no—
268C22⋊C8C2×D10111yescharacteristic, complemented
269#97—1011no—
270C24—2021no—
271#41—22no—
272#41—41no—
273#41—4041no—
274#41—4041no—
275#41—2022no—
276#41—2022no—
277C2×Q8—1011no—
278C2×Q8—1011no—
279C2×Q8C22×D10111yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—1011no—
283C2×D4—1011no—
284C2×D4—4041no—
285C2×D4—2021no—
286C2×D4—2022no—
287C2×D4—2021no—
288C2×D4—2021no—
289C2×D4C22×D10111yescharacteristic
290C2×D4C22×D10111yescharacteristic
291C22×C4—21no—
292C22×C4—1011no—
293C22×C4—2021no—
294C22×C4—2021no—
295C22×C4—2021no—
296C22×C4—2021no—
297C22×C4C22×D10111yescharacteristic
298Q16—4041no—
299Q16—2022no—
300SD16—41no—
301SD16—2022no—
302SD16—2022no—
303SD16—2022no—
304SD16—2022no—
305D8—22no—
306D8—1014no—
307M16—2022no—
308C2×C8—1011no—
309C2×C8—1011no—
310C2×C8—2022no—
311C2×C8C22×D10111yescharacteristic
312C2×C8C22×D10111yescharacteristic
313C4⋊C4—1011no—
314C4⋊C4—1011no—
315C4⋊C4—2021no—
316C4⋊C4—2021no—
317C4⋊C4—2021no—
318C4⋊C4—2021no—
319C4⋊C4—2021no—
320C4⋊C4C22×D10111yescharacteristic
321C22⋊C4—21no—
322C22⋊C4—1011no—
323C22⋊C4—1011no—
324C22⋊C4—2021no—
325C22⋊C4—2021no—
326C22⋊C4—2021no—
327C22⋊C4—2021no—
328C22⋊C4—2021no—
329C22⋊C4—2021no—
330C22⋊C4—2021no—
331C42—1011no—
332C23—21no—
333C23—21no—
334C23—8081no—
335C23—1011no—
336C23—4042no—
337C23—4042no—
338C23—2021no—
339C23D4×D10111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—2021no—
343Q8—2021no—
344Q8—2021no—
345Q8—2021no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—81no—
351D4—1012no—
352D4—4042no—
353D4—4042no—
354D4—4041no—
355D4—4042no—
356D4—2021no—
357D4D4×D10112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—8081no—
364C2×C4—1011no—
365C2×C4—1011no—
366C2×C4—1011no—
367C2×C4—4042no—
368C2×C4—4042no—
369C2×C4—4042no—
370C2×C4—2021no—
371C2×C4—2021no—
372C2×C4—2021no—
373C2×C4—2021no—
374C2×C4—2021no—
375C2×C4D4×D10111yescharacteristic
376C2×C4C23×D10111yescharacteristic
377C8—21no—
378C8—21no—
379C8—1012no—
380C8—2021no—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—8084no—
391C22—4041no—
392C22—4041no—
393C22—2021no—
394C22—2022no—
395C22—2021no—
396C22—2021no—
397C22C2×D4×D10111yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—4041no—
402C4—4041no—
403C4—4041no—
404C4—2021no—
405C4—2021no—
406C4—2021no—
407C4C2×D4×D10111yescharacteristic
408C4#1569230011yescharacteristic
409C2—41no—
410C2—42no—
411C2—81no—
412C2—4041no—
413C2—4042no—
414C2#2026580911yescharacteristic
415C2#2026597411yescharacteristic
416C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#449×C101C2
socleC22×C101C22×D4
hypercenter#449D101
hypocenterC101#3267
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. C2×#15692279C2semidirect
2. C2×#15692275C2semidirect
3. #20265826C2semidirect
4. #20265824C2semidirect
5. #20265568C2semidirect
6. #20265434C2semidirect
7. #20265404C2semidirect
8. #20265390C2semidirect
9. #20265381C2semidirect
10. #20265341C2semidirect
11. #20265340C2semidirect
12. #20265326C2semidirect
13. #20265286C2semidirect
14. #20265245C2semidirect
15. #449×C101C2semidirect
21. C2×Dic404C22semidirect
22. C2×#13108064C22semidirect
25. C2×#13108062C22semidirect
29. C22×#11717579C22semidirect
30. #15692253C22semidirect
36. #15692234C22semidirect
41. D4×Dic101C22semidirect
46. #15692186C22semidirect
47. #15692184C22semidirect
48. #15692180C22semidirect
49. #15692178C22semidirect
53. #15692165C22semidirect
54. C2×#41×C101C22semidirect
58. #103×C101C22semidirect
59. #102×C101C22semidirect
60. C22⋊C8×C101C22semidirect
86. #13108059D4semidirect
209. C101#3267semidirect
222. #449D101semidirect
227. C2×#41C2×D101semidirect
263. #103C2×D101semidirect
266. #102C2×D101semidirect
268. C22⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922792
C2×#156922752
#202658262
#202658242
#202655682
#202654342
#202654042
#202653902
#202653812
#202653412
#202653402
#202653262
#202652862
#202652452
#449×C1012
#3267101