Finite Group Databasebeta

← #28373517

Subgroups — #28373517

Order 13952.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812727C211yesmaximal, characteristic, complemented
2C2×#16812719C211yesmaximal, characteristic, complemented
3C22×#13182014C211yesmaximal, characteristic, complemented
4#20295593C211yesmaximal, characteristic, complemented
5#20295590C211yesmaximal, characteristic, complemented
6#20295414C212yesmaximal, complemented
7#20295412C212yesmaximal, complemented
8#20295407C212yesmaximal, complemented
9#20295401C212yesmaximal, complemented
10#20294984C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13C22×D436C2211yescharacteristic
14#16812727—24no—
15#16812719—24no—
16C2×#13182019C2211yescharacteristic, complemented
17C2×#13182018C2212yescomplemented
18C2×#13182017C2211yescharacteristic, complemented
19C2×#13182016C2211yescharacteristic, complemented
20C2×D872C2212yescomplemented
21C2×#13182014—24no—
22C2×#13182014C2212yes—
23C2×#13182014C2212yes—
24C2×#13182014C2211yescharacteristic, complemented
25C22×#11724784C2211yescharacteristic
26#16812695C2212yescomplemented
27#16812695C2212yescomplemented
28#16812688C2212yescomplemented
29#16812686C2212yescomplemented
30#16812673—22no—
31#16812670—22no—
32#16812668—22no—
33#16812667—22no—
34#16812658C2211yescharacteristic
35#16812609C2212yescomplemented
36C8×Dic109C2212yescomplemented
37C2×#41×C109C2211yescharacteristic
38C22×D4×C109C2211yescharacteristic
39AGL(1,ℤ/8)×C109—24no—
40C2×SD16×C109C2212yescomplemented
41C2×D8×C109C2212yescomplemented
42C2×M16×C109C2211yescharacteristic
43C23×D109—21no—
44#13182029—22no—
45#13182029—22no—
46C2×Dic218C2311yescharacteristic
47C2×D436—22no—
48C2×D436—22no—
49C2×D436—21no—
50C2×D436C2312yes—
51C2×D436C2311yescharacteristic
52C2×C22⋊D109—21no—
53C2×C22⋊D109—24no—
54C2×C22⋊D109—24no—
55C2×C4×D109—21no—
56#13182019—22no—
57#13182018—24no—
58#13182017—22no—
59#13182016—22no—
60D872—24no—
61#13182014—24no—
62#13182014—24no—
63#13182014—22no—
64C2×#11724784D414yescomplemented
65C2×#11724784C2311yescharacteristic
66C2×#11724784C2311yescharacteristic
67C4⋊Dic109D412yes—
68C22⋊Dic109—24no—
69C22⋊Dic109D412yes—
70#13182006—24no—
71C4×Dic109C2312yes—
72C24×C109—21no—
73#41×C109—22no—
74#41×C109—22no—
75C2×Q8×C109C2311yescharacteristic
76C2×D4×C109—21no—
77C2×D4×C109—22no—
78C2×D4×C109—22no—
79C2×D4×C109—21no—
80C2×D4×C109C2311yescharacteristic
81C2×D4×C109C2312yes—
82C22×C4×C109—21no—
83C22×C4×C109C2311yescharacteristic
84SD16×C109—24no—
85D8×C109—24no—
86M16×C109D414yescomplemented
87C2×C8×C109C2312yes—
88C23×C109—21no—
89C23×C109—22no—
90C23×C109—42no—
91C23×C109—82no—
92C23×C109C2×D411yescharacteristic
93C22×D109—22no—
94C22×D109—21no—
95C22×D109—42no—
96C22×D109—82no—
97Q8×C109—21no—
98Q8×C109—21no—
99D4×C109—21no—
100D4×C109—22no—
101D4×C109—22no—
102D4×C109—21no—
103D4×C109—42no—
104D4×C109—42no—
105C2×C4×C109—21no—
106C2×C4×C109—42no—
107C2×C4×C109C2×D411yescharacteristic
108C2×C4×C109C2×D412yes—
109C2×C4×C109C2×D412yes—
110C2×C4×C109C2411yescharacteristic
111C22⋊D109—42no—
112C22⋊D109—84no—
113C22⋊D109—84no—
114C2×Dic109—21no—
115C2×Dic109C2×D414yes—
116D436—22no—
117D436—21no—
118D436—21no—
119D436—22no—
120D436—42no—
121C4×D109—42no—
122Dic218—21no—
123Dic218—21no—
124C8×C109C2×D414yes—
125#11724784C2×D412yes—
126#11724784C2×D412yes—
127C22×C109—22no—
128C22×C109—41no—
129C22×C109—42no—
130C22×C109—42no—
131C22×C109—44no—
132C22×C109—84no—
133C22×C109#12712yes—
134C22×C109C22×D411yescharacteristic
135C2×D109—41no—
136C2×D109—42no—
137C2×D109—42no—
138C2×D109—44no—
139C2×D109—84no—
140C4×C109—41no—
141C4×C109#12712yes—
142C4×C109C22×D411yescharacteristic
143C4×C109C22×D411yescharacteristic
144Dic109—24no—
145Dic109—41no—
146C2×C109—22no—
147C2×C109—81no—
148C2×C109—82no—
149C2×C109C2×#12711yescharacteristic
150C2×C109#57612yes—
151D109—81no—
152D109—82no—
153#3155—1091nomaximal, Sylow
154C109#315511yescharacteristic, complemented, Sylow
155C2×AGL(1,ℤ/8)—1091no—
156C2×AGL(1,ℤ/8)D10911yescharacteristic, complemented
157C2×#135—1091no—
158C22×D8—1091no—
159#535—1091no—
160#535—1091no—
161#496—1092no—
162#496—1092no—
163#495—1092no—
164#493—1092no—
165#405—1091no—
166C2×#41—1091no—
167C2×#41C2×D10911yescharacteristic
168C22×D4—1091no—
169C22×D4C2×D10911yescharacteristic
170AGL(1,ℤ/8)—24no—
171AGL(1,ℤ/8)—2184no—
172#135—2184no—
173C2×Q16—1091no—
174C2×SD16—1091no—
175C2×SD16—1091no—
176C2×SD16—1092no—
177C2×SD16C2×D10912yescomplemented
178C2×D8—1092no—
179C2×D8—1092no—
180C2×D8—1091no—
181C2×D8—1092no—
182C2×D8—2184no—
183C2×D8C2×D10912yescomplemented
184C2×M16C2×D10911yescharacteristic
185C22×C8—1091no—
186#127—1092no—
187#127—1092no—
188#124—1092no—
189#124—1092no—
190#121—2182no—
191#121—2182no—
192#120—2182no—
193#120—2182no—
194#117—1091no—
195#97—1092no—
196C4×C8—1092no—
197C24—21no—
198C24—2181no—
199#41—22no—
200#41—22no—
201#41—2182no—
202#41—2182no—
203C2×Q8—1091no—
204C2×Q8C22×D10911yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—1092no—
210C2×D4—1091no—
211C2×D4—2184no—
212C2×D4—2184no—
213C2×D4—2181no—
214C2×D4—2181no—
215C2×D4—2182no—
216C2×D4—2182no—
217C2×D4C22×D10912yes—
218C2×D4C22×D10911yescharacteristic
219C22×C4—21no—
220C22×C4—2181no—
221C22×C4C22×D10911yescharacteristic
222Q16—2182no—
223SD16—24no—
224SD16—2182no—
225SD16—2182no—
226SD16—2184no—
227D8—24no—
228D8—2184no—
229D8—2184no—
230D8—2182no—
231D8—2184no—
232M16C22⋊D10914yescomplemented
233C2×C8—1094no—
234C2×C8—1091no—
235C2×C8—1091no—
236C2×C8C22×D10912yes—
237C4⋊C4—1092no—
238C22⋊C4—1092no—
239C22⋊C4—2184no—
240C22⋊C4—2184no—
241C42—1092no—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—8722no—
247C23—4362no—
248C23—2182no—
249C23—2181no—
250C23D4×D10911yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—2181no—
254Q8—2181no—
255D4—22no—
256D4—21no—
257D4—21no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—8724no—
262D4—8724no—
263D4—4362no—
264D4—4362no—
265D4—2182no—
266D4—2181no—
267D4—2181no—
268D4—2182no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—1094no—
272C2×C4—4362no—
273C2×C4—2181no—
274C2×C4D4×D10911yescharacteristic
275C2×C4D4×D10912yes—
276C2×C4C2×C22⋊D10912yes—
277C2×C4C23×D10911yescharacteristic
278C8—1092no—
279C8—1092no—
280C8C2×C22⋊D10914yes—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—8724no—
288C22—4361no—
289C22—4362no—
290C22—4362no—
291C22—4364no—
292C22#1681269512yes—
293C22C2×D4×D10911yescharacteristic
294C4—41no—
295C4—4361no—
296C4—2184no—
297C4#1681269512yes—
298C4C2×D4×D10911yescharacteristic
299C4C22×C22⋊D10911yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—8722no—
304C2—8721no—
305C2C2×#1681269511yescharacteristic
306C2#2029583012yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC2×AGL(1,ℤ/8)×C109C2
socleC22×C109C22×D4
hypercenterC2×AGL(1,ℤ/8)D109
hypocenterC109#3155
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
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×#16812727C2semidirect
2. C2×#16812719C2semidirect
3. C22×#13182014C2semidirect
4. #20295593C2semidirect
5. #20295590C2semidirect
6. #20295414C2semidirect
7. #20295412C2semidirect
8. #20295407C2semidirect
9. #20295401C2semidirect
10. #20294984C2semidirect
11. C2×AGL(1,ℤ/8)×C109C2semidirect
16. C2×#13182019C22semidirect
17. C2×#13182018C22semidirect
18. C2×#13182017C22semidirect
19. C2×#13182016C22semidirect
20. C2×D872C22semidirect
24. C2×#13182014C22semidirect
26. #16812695C22semidirect
27. #16812695C22semidirect
28. #16812688C22semidirect
29. #16812686C22semidirect
35. #16812609C22semidirect
36. C8×Dic109C22semidirect
40. C2×SD16×C109C22semidirect
41. C2×D8×C109C22semidirect
64. C2×#11724784D4semidirect
86. M16×C109D4semidirect
154. C109#3155semidirect
156. C2×AGL(1,ℤ/8)D109semidirect
177. C2×SD16C2×D109semidirect
183. C2×D8C2×D109semidirect
232. M16C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127272
C2×#168127192
C22×#131820142
#202955932
#202955902
#202954142
#202954122
#202954072
#202954012
#202949842
C2×AGL(1,ℤ/8)×C1092
#3155109