Finite Group Databasebeta

← #28373506

Subgroups — #28373506

Order 13952.

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

normal subgroups only

This group has 564 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×#16812732C211yesmaximal, characteristic, complemented
2C2×#16812719C211yesmaximal, characteristic, complemented
3C22×#13182017C211yesmaximal, characteristic, complemented
4#20295597C211yesmaximal, characteristic, complemented
5#20295593C211yesmaximal, characteristic, complemented
6#20295417C212yesmaximal, complemented
7#20295414C212yesmaximal, complemented
8#20295409C212yesmaximal, complemented
9#20295397C212yesmaximal, complemented
10#20294984C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13C22×Dic218C2211yescharacteristic
14#16812732—24no—
15#16812719—24no—
16C2×Dic436C2212yescomplemented
17C2×#13182019C2211yescharacteristic, complemented
18C2×#13182018C2212yescomplemented
19C2×#13182017—24no—
20C2×#13182017C2212yes—
21C2×#13182017C2212yes—
22C2×#13182017C2211yescharacteristic, complemented
23C2×#13182016C2211yescharacteristic, complemented
24C2×#13182014C2211yescharacteristic, complemented
25C22×#11724784C2211yescharacteristic
26#16812697C2212yescomplemented
27#16812695C2212yes—
28#16812686C2212yes—
29#16812686C2212yescomplemented
30#16812680—22no—
31#16812678—22no—
32#16812670—22no—
33#16812668—22no—
34#16812658C2211yescharacteristic
35#16812609C2212yescomplemented
36C8×Dic109C2212yescomplemented
37C2×#41×C109C2211yescharacteristic
38C22×D4×C109C2211yescharacteristic
39AGL(1,ℤ/8)×C109—24no—
40C2×SD16×C109C2212yes—
41C2×D8×C109C2212yes—
42C2×M16×C109C2211yescharacteristic
43#13182029—22no—
44#13182029—22no—
45C2×Dic218—22no—
46C2×Dic218—22no—
47C2×Dic218—21no—
48C2×Dic218C2311yescharacteristic
49C2×Dic218C2312yes—
50C2×D436C2311yescharacteristic
51C2×C22⋊D109—21no—
52C2×C22⋊D109—24no—
53C22×Dic109—21no—
54C2×C4×D109—21no—
55Dic436—24no—
56#13182019—22no—
57#13182018—24no—
58#13182017—24no—
59#13182017—24no—
60#13182017—22no—
61#13182016—22no—
62#13182014—22no—
63C2×#11724784D414yescomplemented
64C2×#11724784C2311yescharacteristic
65C2×#11724784C2311yescharacteristic
66C4⋊Dic109D412yes—
67#13182008—24no—
68C22⋊Dic109—24no—
69C22⋊Dic109—24no—
70C22⋊Dic109D412yes—
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—21no—
94Q8×C109—21no—
95Q8×C109—21no—
96D4×C109—21no—
97D4×C109—22no—
98D4×C109—22no—
99D4×C109—21no—
100D4×C109—42no—
101D4×C109—42no—
102C2×C4×C109—21no—
103C2×C4×C109—42no—
104C2×C4×C109C2×D411yescharacteristic
105C2×C4×C109C2×D412yes—
106C2×C4×C109C2×D412yes—
107C2×C4×C109C2411yescharacteristic
108C22⋊D109—42no—
109C22⋊D109—84no—
110C2×Dic109—21no—
111C2×Dic109—22no—
112C2×Dic109—42no—
113C2×Dic109C2×D414yes—
114D436—21no—
115D436—21no—
116C4×D109—42no—
117Dic218—22no—
118Dic218—21no—
119Dic218—22no—
120Dic218—21no—
121Dic218—42no—
122C8×C109C2×D414yes—
123#11724784C2×D412yes—
124#11724784C2×D412yes—
125C22×C109—22no—
126C22×C109—41no—
127C22×C109—42no—
128C22×C109—44no—
129C22×C109—42no—
130C22×C109—84no—
131C22×C109#12712yes—
132C22×C109C22×D411yescharacteristic
133C2×D109—41no—
134C2×D109—42no—
135C4×C109—41no—
136C4×C109#12712yes—
137C4×C109C22×D411yescharacteristic
138C4×C109C22×D411yescharacteristic
139Dic109—24no—
140Dic109—42no—
141Dic109—41no—
142C2×C109—22no—
143C2×C109—81no—
144C2×C109—82no—
145C2×C109C2×#12711yescharacteristic
146C2×C109#57712yes—
147D109—81no—
148#3154—1091nomaximal, Sylow
149C109#315411yescharacteristic, complemented, Sylow
150C2×#137—1091no—
151C2×AGL(1,ℤ/8)D10911yescharacteristic, complemented
152C2×#135—1091no—
153C22×SD16—1091no—
154#536—1091no—
155#535—1091no—
156#497—1092no—
157#496—1092no—
158#495—1092no—
159#492—1092no—
160#405—1091no—
161C2×#41—1091no—
162C2×#41C2×D10911yescharacteristic
163C22×Q8—1091no—
164C22×D4C2×D10911yescharacteristic
165#137—2184no—
166AGL(1,ℤ/8)—24no—
167#135—2184no—
168C2×Q16—1091no—
169C2×Q16—1092no—
170C2×SD16—1091no—
171C2×SD16—1092no—
172C2×SD16—1092no—
173C2×SD16—1091no—
174C2×SD16—1092no—
175C2×SD16—2184no—
176C2×SD16C2×D10912yes—
177C2×D8—1091no—
178C2×D8C2×D10912yes—
179C2×M16C2×D10911yescharacteristic
180C22×C8—1091no—
181C4⋊Q8—1092no—
182#127—1092no—
183#124—1092no—
184#124—1092no—
185#123—2182no—
186C22⋊Q8—2182no—
187#121—2182no—
188#120—2182no—
189#117—1091no—
190#97—1092no—
191C4×C8—1092no—
192C24—21no—
193#41—22no—
194#41—22no—
195#41—2182no—
196#41—2182no—
197C2×Q8—1091no—
198C2×Q8—1092no—
199C2×Q8—2182no—
200C2×Q8—2182no—
201C2×Q8—2181no—
202C2×Q8C22×D10911yescharacteristic
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—1091no—
208C2×D4—2184no—
209C2×D4—2181no—
210C2×D4C22×D10911yescharacteristic
211C2×D4C22×D10912yes—
212C22×C4—21no—
213C22×C4—2181no—
214C22×C4—2181no—
215C22×C4C22×D10911yescharacteristic
216Q16—2182no—
217Q16—2184no—
218SD16—24no—
219SD16—2184no—
220SD16—2184no—
221SD16—2182no—
222SD16—2182no—
223SD16—2184no—
224D8—24no—
225D8—2182no—
226M16C22⋊D10914yescomplemented
227C2×C8—1094no—
228C2×C8—1091no—
229C2×C8—1091no—
230C2×C8C22×D10912yes—
231C4⋊C4—1092no—
232C4⋊C4—2184no—
233C22⋊C4—1092no—
234C22⋊C4—2184no—
235C22⋊C4—2184no—
236C42—1092no—
237C23—21no—
238C23—22no—
239C23—42no—
240C23—82no—
241C23—2181no—
242C23D4×D10911yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—4362no—
246Q8—2182no—
247Q8—2181no—
248Q8—2182no—
249Q8—2181no—
250D4—21no—
251D4—22no—
252D4—21no—
253D4—22no—
254D4—42no—
255D4—42no—
256D4—8724no—
257D4—4362no—
258D4—2181no—
259D4—2181no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—1094no—
263C2×C4—4362no—
264C2×C4—4362no—
265C2×C4—2181no—
266C2×C4—2182no—
267C2×C4D4×D10911yescharacteristic
268C2×C4D4×D10912yes—
269C2×C4C2×C22⋊D10912yes—
270C2×C4C23×D10911yescharacteristic
271C8—1092no—
272C8—1092no—
273C8C2×C22⋊D10914yes—
274C22—22no—
275C22—41no—
276C22—42no—
277C22—44no—
278C22—42no—
279C22—84no—
280C22—4361no—
281C22—4362no—
282C22#1681269512yes—
283C22C2×D4×D10911yescharacteristic
284C4—41no—
285C4—4361no—
286C4—4362no—
287C4—2184no—
288C4#1681269512yes—
289C4C2×D4×D10911yescharacteristic
290C4C22×C22⋊D10911yescharacteristic
291C2—22no—
292C2—82no—
293C2—81no—
294C2—8721no—
295C2C2×#1681269511yescharacteristic
296C2#2029583612yes—
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#3154
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×#16812732C2semidirect
2. C2×#16812719C2semidirect
3. C22×#13182017C2semidirect
4. #20295597C2semidirect
5. #20295593C2semidirect
6. #20295417C2semidirect
7. #20295414C2semidirect
8. #20295409C2semidirect
9. #20295397C2semidirect
10. #20294984C2semidirect
11. C2×AGL(1,ℤ/8)×C109C2semidirect
16. C2×Dic436C22semidirect
17. C2×#13182019C22semidirect
18. C2×#13182018C22semidirect
22. C2×#13182017C22semidirect
23. C2×#13182016C22semidirect
24. C2×#13182014C22semidirect
26. #16812697C22semidirect
29. #16812686C22semidirect
35. #16812609C22semidirect
36. C8×Dic109C22semidirect
63. C2×#11724784D4semidirect
86. M16×C109D4semidirect
149. C109#3154semidirect
151. C2×AGL(1,ℤ/8)D109semidirect
226. M16C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127322
C2×#168127192
C22×#131820172
#202955972
#202955932
#202954172
#202954142
#202954092
#202953972
#202949842
C2×AGL(1,ℤ/8)×C1092
#3154109