Finite Group Databasebeta

← #27182657

Subgroups — #27182657

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
2#20265826C211yesmaximal, characteristic, complemented
3#20265815C211yesmaximal, characteristic, complemented
4#20265570C211yesmaximal, characteristic, complemented
5#20265432C211yesmaximal, characteristic, complemented
6#20265406C211yesmaximal, characteristic, complemented
7#20265394C211yesmaximal, characteristic, complemented
8#20265380C211yesmaximal, characteristic, complemented
9#20265339C211yesmaximal, characteristic, complemented
10#20265336C211yesmaximal, characteristic, complemented
11#20265326C211yesmaximal, characteristic, complemented
12#20265322C211yesmaximal, characteristic, complemented
13#20265289C211yesmaximal, characteristic, complemented
14#20265248C211yesmaximal, characteristic, complemented
15C2×#135×C101C211yesmaximal, characteristic, complemented
16C2×#13108073C2211yescharacteristic
17C2×D4×D101C2211yescharacteristic
18#15692279—24no—
19C2×#13108064C2211yescharacteristic, complemented
20C2×#13108063C2211yescharacteristic
21C2×#13108062C2211yescharacteristic
22C2×#13108061C2211yescharacteristic
23C2×#13108059C2211yescharacteristic
24C2×#13108057C2211yescharacteristic
25#15692253C2211yescharacteristic, complemented
26#15692244C2211yescharacteristic
27#15692238—21no—
28#15692238—21no—
29#15692236—21no—
30#15692234C2211yescharacteristic, complemented
31#15692231C2211yescharacteristic
32#15692231C2211yescharacteristic
33#15692229C2211yescharacteristic
34#15692227—21no—
35#15692227—21no—
36#15692223—21no—
37D4×Dic101C2211yescharacteristic
38D4×Dic101C2211yescharacteristic, complemented
39C4×C22⋊D101C2211yescharacteristic
40C2×#13108053—21no—
41C2×#13108051—21no—
42#15692187C2211yescharacteristic, complemented
43#15692184C2211yescharacteristic, complemented
44#15692180C2211yescharacteristic, complemented
45#15692179C2211yescharacteristic, complemented
46#15692179C2211yescharacteristic, complemented
47#15692177C2211yescharacteristic
48#15692176C2211yescharacteristic
49#15692176C2211yescharacteristic, complemented
50C22⋊#11717579C2211yescharacteristic, complemented
51#15692167C2211yescharacteristic
52#15692165C2211yescharacteristic, complemented
53C2×#41×C101C2211yescharacteristic
54C2×#41×C101C2211yescharacteristic, complemented
55#135×C101—24no—
56C2×Q16×C101C2211yescharacteristic, complemented
57C2×SD16×C101C2211yescharacteristic
58C2×SD16×C101C2211yescharacteristic, complemented
59C2×D8×C101C2211yescharacteristic
60C22×C8×C101C2211yescharacteristic, complemented
61C23×D101—21no—
62#13108073—22no—
63#13108073—41no—
64C2×Dic202C2311yescharacteristic
65C2×D404C2311yescharacteristic
66C2×C22⋊D101—21no—
67C2×C22⋊D101—21no—
68C2×C22⋊D101—21no—
69C2×C22⋊D101D411yescharacteristic
70D4×D101—22no—
71D4×D101—41no—
72C22×Dic101—21no—
73C22×Dic101—21no—
74C22×Dic101—21no—
75C2×C4×D101C2311yescharacteristic
76#13108064—41no—
77#13108063—22no—
78#13108062—22no—
79#13108061—41no—
80#13108059—22no—
81#13108057—22no—
82C2×#11717579C2311yescharacteristic
83C2×#11717579C2311yescharacteristic
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⋊Dic101D411yescharacteristic
96#13108051—21no—
97#13108051—21no—
98#13108051—21no—
99#13108051—21no—
100#13108051—21no—
101#13108051D411yescharacteristic
102C4×Dic101C2311yescharacteristic
103#41×C101—22no—
104#41×C101—22no—
105#41×C101—41no—
106#41×C101—41no—
107C2×Q8×C101C2311yescharacteristic
108C2×Q8×C101C2311yescharacteristic
109C2×D4×C101—21no—
110C2×D4×C101—21no—
111C2×D4×C101C2311yescharacteristic
112C2×D4×C101C2311yescharacteristic
113C22×C4×C101—21no—
114C22×C4×C101—21no—
115C22×C4×C101C2311yescharacteristic
116Q16×C101—22no—
117SD16×C101—22no—
118SD16×C101—22no—
119D8×C101D414yescomplemented
120C2×C8×C101—22no—
121C2×C8×C101C2311yescharacteristic
122C2×C8×C101C2311yescharacteristic
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—
133Q8×C101—21no—
134Q8×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—21no—
145C2×C4×C101—42no—
146C2×C4×C101—42no—
147C2×C4×C101C2×D411yescharacteristic
148C2×C4×C101C2411yescharacteristic
149C22⋊D101—41no—
150C22⋊D101—42no—
151C22⋊D101—42no—
152C22⋊D101—81no—
153C2×Dic101—21no—
154C2×Dic101—21no—
155C2×Dic101—21no—
156C2×Dic101—42no—
157C2×Dic101—42no—
158C2×Dic101—81no—
159C2×Dic101C2×D411yescharacteristic
160C2×Dic101C2×D411yescharacteristic
161C2×Dic101C2×D411yescharacteristic
162D404—21no—
163D404—21no—
164C4×D101—21no—
165C4×D101—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×C101C22×D411yescharacteristic
182C2×D101—21no—
183C2×D101—21no—
184C2×D101—21no—
185C2×D101—41no—
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—42no—
201C2×C101—41no—
202C2×C101—42no—
203C2×C101D4211yescharacteristic
204C2×C101#57511yescharacteristic
205C2×C101#57711yescharacteristic
206D101—41no—
207D101—81no—
208#3267—1011nomaximal, Sylow
209C101#326711yescharacteristic, complemented, Sylow
210C2×AGL(1,ℤ/8)—1011no—
211C2×#135D10111yescharacteristic, complemented
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×#41C2×D10111yescharacteristic
227C2×#41C2×D10111yescharacteristic, complemented
228C22×D4—1011no—
229AGL(1,ℤ/8)—2024no—
230#135—24no—
231C2×Q16—1011no—
232C2×Q16C2×D10111yescharacteristic, complemented
233C2×SD16—1011no—
234C2×SD16—1011no—
235C2×SD16—1011no—
236C2×SD16C2×D10111yescharacteristic
237C2×SD16C2×D10111yescharacteristic, complemented
238C2×D8—1011no—
239C2×D8C2×D10111yescharacteristic
240C2×M16—1011no—
241C22×C8C2×D10111yescharacteristic, complemented
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—2021no—
271#41—22no—
272#41—22no—
273#41—41no—
274#41—41no—
275#41—4041no—
276#41—2022no—
277C2×Q8—1011no—
278C2×Q8C22×D10111yescharacteristic
279C2×Q8C22×D10111yescharacteristic
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—1011no—
283C2×D4—1011no—
284C2×D4—4041no—
285C2×D4—2021no—
286C2×D4—2021no—
287C2×D4—2021no—
288C2×D4—2022no—
289C2×D4C22×D10111yescharacteristic
290C2×D4C22×D10111yescharacteristic
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—1011no—
294C22×C4—2021no—
295C22×C4—2021no—
296C22×C4—2021no—
297C22×C4C22×D10111yescharacteristic
298Q16—22no—
299Q16—4041no—
300SD16—22no—
301SD16—22no—
302SD16—4041no—
303SD16—2022no—
304SD16—2022no—
305D8—2022no—
306D8C22⋊D10114yescomplemented
307M16—2022no—
308C2×C8—22no—
309C2×C8—1011no—
310C2×C8—1011no—
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—8081no—
335C23—1011no—
336C23—4042no—
337C23—4042no—
338C23—2021no—
339C23D4×D10111yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—2021no—
345Q8—2021no—
346D4—21no—
347D4—21no—
348D4—42no—
349D4—42no—
350D4—8081no—
351D4—4042no—
352D4—4042no—
353D4—4041no—
354D4—2021no—
355D4—2021no—
356D4C2×C22⋊D10112yes—
357D4C2×C22⋊D10112yes—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—8081no—
365C2×C4—1011no—
366C2×C4—1011no—
367C2×C4—1011no—
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—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—8084no—
391C22—4041no—
392C22—4041no—
393C22—4041no—
394C22—2021no—
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—
407C4C22×C22⋊D10111yescharacteristic
408C4#1569230011yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—8081no—
413C2—4041no—
414C2D4×C22⋊D10111yescharacteristic
415C2#2026597411yescharacteristic
416C2#2026598411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#135×C101C2
socleC22×C101C22×D4
hypercenterC2×#135D101
hypocenterC101#3267
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#15692279C2semidirect
2. #20265826C2semidirect
3. #20265815C2semidirect
4. #20265570C2semidirect
5. #20265432C2semidirect
6. #20265406C2semidirect
7. #20265394C2semidirect
8. #20265380C2semidirect
9. #20265339C2semidirect
10. #20265336C2semidirect
11. #20265326C2semidirect
12. #20265322C2semidirect
13. #20265289C2semidirect
14. #20265248C2semidirect
15. C2×#135×C101C2semidirect
19. C2×#13108064C22semidirect
25. #15692253C22semidirect
30. #15692234C22semidirect
38. D4×Dic101C22semidirect
42. #15692187C22semidirect
43. #15692184C22semidirect
44. #15692180C22semidirect
45. #15692179C22semidirect
46. #15692179C22semidirect
49. #15692176C22semidirect
50. C22⋊#11717579C22semidirect
52. #15692165C22semidirect
54. C2×#41×C101C22semidirect
56. C2×Q16×C101C22semidirect
58. C2×SD16×C101C22semidirect
60. C22×C8×C101C22semidirect
119. D8×C101D4semidirect
209. C101#3267semidirect
211. C2×#135D101semidirect
227. C2×#41C2×D101semidirect
232. C2×Q16C2×D101semidirect
237. C2×SD16C2×D101semidirect
241. C22×C8C2×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×#156922792
#202658262
#202658152
#202655702
#202654322
#202654062
#202653942
#202653802
#202653392
#202653362
#202653262
#202653222
#202652892
#202652482
C2×#135×C1012
#3267101