Finite Group Databasebeta

← #27182651

Subgroups — #27182651

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×#15692274C211yesmaximal, characteristic, complemented
2#20265826C211yesmaximal, characteristic, complemented
3#20265821C211yesmaximal, characteristic, complemented
4#20265568C211yesmaximal, characteristic, complemented
5#20265435C211yesmaximal, characteristic, complemented
6#20265403C211yesmaximal, characteristic, complemented
7#20265390C211yesmaximal, characteristic, complemented
8#20265380C211yesmaximal, characteristic, complemented
9#20265339C211yesmaximal, characteristic, complemented
10#20265336C211yesmaximal, characteristic, complemented
11#20265328C211yesmaximal, characteristic, complemented
12#20265319C211yesmaximal, characteristic, complemented
13#20265289C211yesmaximal, characteristic, complemented
14#20265246C211yesmaximal, characteristic, complemented
15C2×AGL(1,ℤ/8)×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C2×#13108073C2211yescharacteristic, complemented
18#15692274—24no—
19C2×Dic404C2211yescharacteristic, complemented
20C2×#13108064C2211yescharacteristic, complemented
21C2×#13108062C2211yescharacteristic
22C2×#13108062C2211yescharacteristic
23C2×#13108062C2211yescharacteristic, complemented
24C2×C8×D101C2211yescharacteristic, complemented
25#15692253C2211yescharacteristic, complemented
26#15692242C2211yescharacteristic
27#15692238—21no—
28#15692236—21no—
29#15692236—21no—
30#15692234C2211yescharacteristic
31#15692234C2211yescharacteristic
32#15692231C2211yescharacteristic, complemented
33#15692227—21no—
34#15692227—21no—
35#15692226C2211yescharacteristic
36#15692224—21no—
37D4×Dic101C2211yescharacteristic
38D4×Dic101C2211yescharacteristic, complemented
39C4×C22⋊D101C2211yescharacteristic
40C2×#13108053—21no—
41C2×C22⋊Dic101—21no—
42#15692187C2211yescharacteristic, complemented
43#15692184C2211yescharacteristic, complemented
44#15692180C2211yescharacteristic, complemented
45#15692180C2211yescharacteristic, complemented
46#15692179C2211yescharacteristic, complemented
47#15692176C2211yescharacteristic
48#15692176C2211yescharacteristic
49#15692176C2211yescharacteristic, complemented
50C22⋊#11717579C2211yescharacteristic
51#15692167C2211yescharacteristic, complemented
52#15692165C2211yescharacteristic, complemented
53C2×#41×C101C2211yescharacteristic
54C22×D4×C101C2211yescharacteristic
55AGL(1,ℤ/8)×C101—24no—
56C2×SD16×C101C2211yescharacteristic
57C2×SD16×C101C2211yescharacteristic
58C2×D8×C101C2211yescharacteristic
59C2×D8×C101C2211yescharacteristic
60C2×M16×C101C2211yescharacteristic
61#13108073—22no—
62#13108073—22no—
63#13108073—41no—
64#13108073—41no—
65C2×Dic202C2311yescharacteristic
66C2×Dic202C2311yescharacteristic
67C2×C22⋊D101—21no—
68C2×C22⋊D101—21no—
69C2×C22⋊D101—21no—
70C2×C22⋊D101D411yescharacteristic
71C22×Dic101—21no—
72C22×Dic101—21no—
73C22×Dic101—21no—
74C22×Dic101—21no—
75C2×C4×D101C2311yescharacteristic
76Dic404—22no—
77#13108064—41no—
78#13108062—22no—
79#13108062—22no—
80#13108062—41no—
81C2×#11717579C2311yescharacteristic
82C2×#11717579C2311yescharacteristic
83C8×D101—22no—
84C4⋊Dic101C2311yescharacteristic
85C4⋊Dic101C2311yescharacteristic
86#13108053—21no—
87#13108053—21no—
88#13108053—21no—
89#13108053—21no—
90#13108053—21no—
91#13108053D411yescharacteristic
92C22⋊Dic101—21no—
93C22⋊Dic101—21no—
94C22⋊Dic101—21no—
95C22⋊Dic101—21no—
96C22⋊Dic101—21no—
97C22⋊Dic101—21no—
98C22⋊Dic101—21no—
99C22⋊Dic101D411yescharacteristic
100#13108051—21no—
101#13108051D411yescharacteristic
102C4×Dic101C2311yescharacteristic
103C24×C101—21no—
104#41×C101—22no—
105#41×C101—41no—
106C2×Q8×C101C2311yescharacteristic
107C2×D4×C101—22no—
108C2×D4×C101—21no—
109C2×D4×C101—21no—
110C2×D4×C101—41no—
111C2×D4×C101C2311yescharacteristic
112C2×D4×C101C2311yescharacteristic
113C2×D4×C101C2311yescharacteristic
114C22×C4×C101—21no—
115C22×C4×C101C2311yescharacteristic
116SD16×C101—22no—
117SD16×C101—22no—
118D8×C101—22no—
119D8×C101D414yescomplemented
120M16×C101—22no—
121C2×C8×C101C2311yescharacteristic
122C2×C8×C101C2311yescharacteristic
123C23×C101—21no—
124C23×C101—21no—
125C23×C101—21no—
126C23×C101—42no—
127C23×C101—42no—
128C23×C101—81no—
129C23×C101C2×D411yescharacteristic
130C22×D101C2×D411yescharacteristic
131Q8×C101—21no—
132Q8×C101—21no—
133D4×C101—21no—
134D4×C101—21no—
135D4×C101—21no—
136D4×C101—21no—
137D4×C101—42no—
138D4×C101—42no—
139D4×C101C2×D412yes—
140D4×C101C2×D412yes—
141C2×C4×C101—21no—
142C2×C4×C101—21no—
143C2×C4×C101—21no—
144C2×C4×C101—42no—
145C2×C4×C101C2×D411yescharacteristic
146C2×C4×C101C2411yescharacteristic
147C22⋊D101—41no—
148C22⋊D101—42no—
149C22⋊D101—42no—
150C22⋊D101—81no—
151C2×Dic101—21no—
152C2×Dic101—21no—
153C2×Dic101—21no—
154C2×Dic101—21no—
155C2×Dic101—42no—
156C2×Dic101—42no—
157C2×Dic101—42no—
158C2×Dic101—81no—
159C2×Dic101C2×D411yescharacteristic
160C2×Dic101C2×D411yescharacteristic
161C2×Dic101C2×D411yescharacteristic
162C4×D101—21no—
163C4×D101—21no—
164Dic202—21no—
165Dic202—21no—
166Dic202—21no—
167Dic202—21no—
168C8×C101—21no—
169C8×C101C2×D412yes—
170#11717579—21no—
171#11717579—21no—
172C22×C101—22no—
173C22×C101—22no—
174C22×C101—21no—
175C22×C101—21no—
176C22×C101—21no—
177C22×C101—41no—
178C22×C101—41no—
179C22×C101—41no—
180C22×C101—41no—
181C22×C101—41no—
182C22×C101—41no—
183C22×C101—41no—
184C22×C101—84no—
185C22×C101C22×D411yescharacteristic
186C2×D101—21no—
187C2×D101—21no—
188C2×D101—21no—
189C4×C101—21no—
190C4×C101—41no—
191C4×C101C22×D411yescharacteristic
192C4×C101H(5,𝔽2)11yescharacteristic
193Dic101—21no—
194Dic101—21no—
195Dic101—21no—
196Dic101—41no—
197Dic101—41no—
198Dic101—41no—
199Dic101—41no—
200C2×C101—41no—
201C2×C101—42no—
202C2×C101—42no—
203C2×C101—81no—
204C2×C101D4211yescharacteristic
205C2×C101#57511yescharacteristic
206C2×C101#57711yescharacteristic
207D101—41no—
208#3267—1011nomaximal, Sylow
209C101#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)D10111yescharacteristic, complemented
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#449—1011no—
223#442—1011no—
224#435—1011no—
225C2×#41—1011no—
226C2×#41—1011no—
227C2×#41C2×D10111yescharacteristic
228C22×D4C2×D10111yescharacteristic
229AGL(1,ℤ/8)—24no—
230#135—2024no—
231C2×Q16—1011no—
232C2×Q16—1011no—
233C2×SD16—1011no—
234C2×SD16—1011no—
235C2×SD16—1011no—
236C2×SD16C2×D10111yescharacteristic
237C2×SD16C2×D10111yescharacteristic
238C2×D8C2×D10111yescharacteristic
239C2×D8C2×D10111yescharacteristic
240C2×M16C2×D10111yescharacteristic
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—1011no—
251#121—2021no—
252#121—2021no—
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#103—1011no—
264#102—1011no—
265#102—1011no—
266#102—1011no—
267C22⋊C8—1011no—
268C22⋊C8—1011no—
269#97—1011no—
270C24—21no—
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—22no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4—1011no—
285C2×D4—2021no—
286C2×D4—2021no—
287C2×D4—2021no—
288C2×D4C22×D10111yescharacteristic
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—22no—
301SD16—22no—
302SD16—4041no—
303SD16—2022no—
304SD16—2022no—
305D8—22no—
306D8C22⋊D10114yescomplemented
307M16—22no—
308C2×C8—1011no—
309C2×C8—1011no—
310C2×C8—2022no—
311C2×C8C22×D10111yescharacteristic
312C2×C8C22×D10111yescharacteristic
313C4⋊C4—1011no—
314C4⋊C4—1011no—
315C4⋊C4—1011no—
316C4⋊C4—2021no—
317C4⋊C4—2021no—
318C4⋊C4—2021no—
319C4⋊C4—2021no—
320C4⋊C4—2021no—
321C22⋊C4—1011no—
322C22⋊C4—1011no—
323C22⋊C4—2021no—
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—21no—
335C23—42no—
336C23—42no—
337C23—81no—
338C23—1011no—
339C23D4×D10111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—2021no—
343Q8—2021no—
344Q8—2021no—
345Q8—2021no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—42no—
351D4—42no—
352D4—8081no—
353D4—4042no—
354D4—4042no—
355D4—4041no—
356D4C2×C22⋊D10112yes—
357D4C2×C22⋊D10112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—8081no—
363C2×C4—1011no—
364C2×C4—1011no—
365C2×C4—1011no—
366C2×C4—4042no—
367C2×C4—4042no—
368C2×C4—4042no—
369C2×C4—2021no—
370C2×C4—2021no—
371C2×C4—2021no—
372C2×C4—2021no—
373C2×C4—2021no—
374C2×C4—2021no—
375C2×C4D4×D10111yescharacteristic
376C2×C4C23×D10111yescharacteristic
377C8—21no—
378C8—2021no—
379C8—2021no—
380C8C2×C22⋊D10112yes—
381C22—21no—
382C22—21no—
383C22—22no—
384C22—22no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—84no—
394C22—2021no—
395C22—2021no—
396C22—2021no—
397C22C2×D4×D10111yescharacteristic
398C4—21no—
399C4—41no—
400C4—4041no—
401C4—4041no—
402C4—4041no—
403C4—4041no—
404C4—2021no—
405C4—2021no—
406C4—2021no—
407C4C22×C22⋊D10111yescharacteristic
408C4#1569230011yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—81no—
413C2—4041no—
414C2D4×C22⋊D10111yescharacteristic
415C2#2026597311yescharacteristic
416C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×AGL(1,ℤ/8)×C101C2
socleC22×C101C22×D4
hypercenterC2×AGL(1,ℤ/8)D101
hypocenterC101#3267
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#15692274C2semidirect
2. #20265826C2semidirect
3. #20265821C2semidirect
4. #20265568C2semidirect
5. #20265435C2semidirect
6. #20265403C2semidirect
7. #20265390C2semidirect
8. #20265380C2semidirect
9. #20265339C2semidirect
10. #20265336C2semidirect
11. #20265328C2semidirect
12. #20265319C2semidirect
13. #20265289C2semidirect
14. #20265246C2semidirect
15. C2×AGL(1,ℤ/8)×C101C2semidirect
17. C2×#13108073C22semidirect
19. C2×Dic404C22semidirect
20. C2×#13108064C22semidirect
23. C2×#13108062C22semidirect
24. C2×C8×D101C22semidirect
25. #15692253C22semidirect
32. #15692231C22semidirect
38. D4×Dic101C22semidirect
42. #15692187C22semidirect
43. #15692184C22semidirect
44. #15692180C22semidirect
45. #15692180C22semidirect
46. #15692179C22semidirect
49. #15692176C22semidirect
51. #15692167C22semidirect
52. #15692165C22semidirect
119. D8×C101D4semidirect
209. C101#3267semidirect
210. C2×AGL(1,ℤ/8)D101semidirect
306. D8C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922742
#202658262
#202658212
#202655682
#202654352
#202654032
#202653902
#202653802
#202653392
#202653362
#202653282
#202653192
#202652892
#202652462
C2×AGL(1,ℤ/8)×C1012
#3267101