Finite Group Databasebeta

← #28372575

Subgroups — #28372575

Order 13952.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812719C211yesmaximal, characteristic, complemented
2#20295593C211yesmaximal, characteristic, complemented
3#20295509C211yesmaximal, characteristic, complemented
4#20295246C212yesmaximal, complemented
5#20295239C212yesmaximal, complemented
6#20295224C212yesmaximal, complemented
7#20295219C212yesmaximal, complemented
8#20295054C211yesmaximal, characteristic, complemented
9#20295028C211yesmaximal, characteristic, complemented
10C2×#16812620C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13#16812719—24no—
14C2×#13182019C2211yescharacteristic, complemented
15C2×#13182017C2211yescharacteristic, complemented
16C2×#13182016C2211yescharacteristic, complemented
17C2×#13182014C2211yescharacteristic, complemented
18C22×#11724784C2211yescharacteristic
19#16812695C2212yes—
20#16812686C2212yes—
21#16812670—21no—
22#16812670—21no—
23#16812668—22no—
24Q8×Dic109C2212yescomplemented
25D4×Dic109—22no—
26D4×Dic109C2212yescomplemented
27#16812658—21no—
28#16812658C2211yescharacteristic
29C2×C4⋊Dic109—21no—
30C2×C4⋊Dic109C2211yescharacteristic
31#16812628C2212yescomplemented
32#16812628C2212yescomplemented
33#16812624C2212yescomplemented
34#16812621C2212yescomplemented
35#16812620C2212yes—
36#16812620C2212yes—
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×Dic218C2311yescharacteristic
46C2×D436C2311yescharacteristic
47C2×C22⋊D109—21no—
48C2×C22⋊D109—22no—
49C2×C22⋊D109—22no—
50C22×Dic109—21no—
51C22×Dic109—22no—
52C2×C4×D109—21no—
53#13182019—22no—
54#13182017—22no—
55#13182016—22no—
56#13182014—22no—
57C2×#11724784—22no—
58C2×#11724784C2311yescharacteristic
59C2×#11724784C2311yescharacteristic
60C4⋊Dic109—21no—
61C4⋊Dic109—22no—
62C4⋊Dic109D411yescharacteristic
63C4⋊Dic109D411yescharacteristic
64C4⋊Dic109C2312yes—
65C22⋊Dic109—22no—
66C22⋊Dic109—22no—
67C22⋊Dic109—22no—
68C22⋊Dic109D412yes—
69C4×Dic109—22no—
70C4×Dic109C2312yes—
71C24×C109—21no—
72#41×C109—22no—
73#41×C109D414yescomplemented
74C2×Q8×C109C2311yescharacteristic
75C2×D4×C109—21no—
76C2×D4×C109—22no—
77C2×D4×C109—21no—
78C2×D4×C109—22no—
79C2×D4×C109C2312yes—
80C2×D4×C109C2311yescharacteristic
81C22×C4×C109—21no—
82C22×C4×C109C2311yescharacteristic
83SD16×C109—24no—
84D8×C109—24no—
85M16×C109—22no—
86C2×C8×C109C2312yes—
87C23×C109—21no—
88C23×C109—22no—
89C23×C109—42no—
90C23×C109—81no—
91C23×C109—81no—
92C23×C109C2×D411yescharacteristic
93C22×D109—21no—
94Q8×C109—21no—
95Q8×C109C2×D412yes—
96D4×C109—24no—
97D4×C109—22no—
98D4×C109—21no—
99D4×C109—22no—
100D4×C109—42no—
101D4×C109C2×D412yes—
102C2×C4×C109—21no—
103C2×C4×C109—24no—
104C2×C4×C109C2×D411yescharacteristic
105C2×C4×C109C2×D412yes—
106C2×C4×C109#4112yes—
107C2×C4×C109C2411yescharacteristic
108C22⋊D109—42no—
109C22⋊D109—82no—
110C22⋊D109—82no—
111C2×Dic109—22no—
112C2×Dic109—21no—
113C2×Dic109—82no—
114C2×Dic109—81no—
115C2×Dic109C2×D412yes—
116C2×Dic109C2×D412yes—
117D436—21no—
118D436—21no—
119C4×D109—42no—
120Dic218—21no—
121Dic218—21no—
122C8×C109—22no—
123#11724784—21no—
124#11724784—21no—
125C22×C109—21no—
126C22×C109—21no—
127C22×C109—22no—
128C22×C109—41no—
129C22×C109—41no—
130C22×C109—42no—
131C22×C109—42no—
132C22×C109—42no—
133C22×C109—84no—
134C22×C109#12112yes—
135C22×C109C22×D411yescharacteristic
136C2×D109—41no—
137C2×D109—41no—
138C2×D109—41no—
139C4×C109—22no—
140C4×C109#12112yes—
141C4×C109C22×D411yescharacteristic
142C4×C109C2×#4111yescharacteristic
143Dic109—22no—
144Dic109—22no—
145Dic109—41no—
146Dic109—42no—
147C2×C109—22no—
148C2×C109—42no—
149C2×C109—82no—
150C2×C109C2×#12111yescharacteristic
151C2×C109#57611yescharacteristic
152C2×C109#57711yescharacteristic
153D109—81no—
154#3045—1091nomaximal, Sylow
155C109#304511yescharacteristic, complemented, Sylow
156C2×AGL(1,ℤ/8)D10911yescharacteristic, complemented
157C2×#135—1091no—
158#535—1091no—
159#520—1091no—
160#464—1092no—
161#463—1092no—
162#460—1092no—
163#459—1092no—
164#424—1091no—
165#417—1091no—
166C2×#102—1091no—
167C2×#41—1091no—
168C2×#41C2×D10911yescharacteristic
169C22×D4C2×D10911yescharacteristic
170AGL(1,ℤ/8)—24no—
171#135—2184no—
172C2×Q16—1091no—
173C2×SD16—1091no—
174C2×SD16—1091no—
175C2×SD16C2×D10912yes—
176C2×D8—1091no—
177C2×D8C2×D10912yes—
178C2×M16C2×D10911yescharacteristic
179C22×C8—1091no—
180#127—1092no—
181#124—1092no—
182#121—2181no—
183#121—2181no—
184#120—2182no—
185C4×Q8—1092no—
186C4×D4—1092no—
187C4×D4—2182no—
188#117—1091no—
189#117—2181no—
190C2×C4⋊C4—1091no—
191C2×C4⋊C4—2181no—
192C4⋊C8—1092no—
193C4⋊C8—1092no—
194#103—1092no—
195#102—1092no—
196#102—1092no—
197#102—1092no—
198C24—21no—
199#41—22no—
200#41—2182no—
201#41—2182no—
202#41C22⋊D10914yescomplemented
203C2×Q8—1091no—
204C2×Q8C22×D10911yescharacteristic
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—1091no—
210C2×D4—2182no—
211C2×D4—2182no—
212C2×D4—2181no—
213C2×D4C22×D10912yes—
214C2×D4C22×D10911yescharacteristic
215C22×C4—21no—
216C22×C4—2182no—
217C22×C4—2181no—
218C22×C4—2181no—
219C22×C4C22×D10911yescharacteristic
220Q16—2182no—
221SD16—24no—
222SD16—2182no—
223SD16—2182no—
224D8—24no—
225D8—2182no—
226M16—22no—
227C2×C8—1091no—
228C2×C8—1091no—
229C2×C8—2182no—
230C2×C8C22×D10912yes—
231C4⋊C4—1091no—
232C4⋊C4—1091no—
233C4⋊C4—1092no—
234C4⋊C4—2182no—
235C4⋊C4—2181no—
236C22⋊C4—1092no—
237C22⋊C4—2182no—
238C22⋊C4—2182no—
239C22⋊C4—2182no—
240C42—1092no—
241C42—2182no—
242C23—21no—
243C23—22no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—2181no—
248C23D4×D10911yescharacteristic
249Q8—21no—
250Q8—2181no—
251Q8—2181no—
252Q8C2×C22⋊D10912yes—
253D4—24no—
254D4—22no—
255D4—21no—
256D4—22no—
257D4—42no—
258D4—8722no—
259D4—8722no—
260D4—4362no—
261D4—2181no—
262D4—2181no—
263D4C2×C22⋊D10912yes—
264C2×C4—21no—
265C2×C4—24no—
266C2×C4—8721no—
267C2×C4—8722no—
268C2×C4—1092no—
269C2×C4—1092no—
270C2×C4—4362no—
271C2×C4—2182no—
272C2×C4—2181no—
273C2×C4D4×D10911yescharacteristic
274C2×C4C2×C22⋊D10912yes—
275C2×C4#1318202812yes—
276C2×C4C23×D10911yescharacteristic
277C8—22no—
278C8—2181no—
279C8—2181no—
280C22—21no—
281C22—21no—
282C22—22no—
283C22—41no—
284C22—41no—
285C22—42no—
286C22—42no—
287C22—42no—
288C22—84no—
289C22—4361no—
290C22—4361no—
291C22—4361no—
292C22#1681267112yes—
293C22C2×D4×D10911yescharacteristic
294C4—22no—
295C4—4361no—
296C4—4362no—
297C4—2182no—
298C4—2182no—
299C4#1681267112yes—
300C4C22×C22⋊D10911yescharacteristic
301C4C2×#1318202811yescharacteristic
302C2—22no—
303C2—42no—
304C2—82no—
305C2—8721no—
306C2C2×#1681267111yescharacteristic
307C2#2029583011yescharacteristic
308C2#2029583611yescharacteristic
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#3045
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×#16812719C2semidirect
2. #20295593C2semidirect
3. #20295509C2semidirect
4. #20295246C2semidirect
5. #20295239C2semidirect
6. #20295224C2semidirect
7. #20295219C2semidirect
8. #20295054C2semidirect
9. #20295028C2semidirect
10. C2×#16812620C2semidirect
11. C2×AGL(1,ℤ/8)×C109C2semidirect
14. C2×#13182019C22semidirect
15. C2×#13182017C22semidirect
16. C2×#13182016C22semidirect
17. C2×#13182014C22semidirect
24. Q8×Dic109C22semidirect
26. D4×Dic109C22semidirect
31. #16812628C22semidirect
32. #16812628C22semidirect
33. #16812624C22semidirect
34. #16812621C22semidirect
73. #41×C109D4semidirect
155. C109#3045semidirect
156. C2×AGL(1,ℤ/8)D109semidirect
202. #41C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127192
#202955932
#202955092
#202952462
#202952392
#202952242
#202952192
#202950542
#202950282
C2×#168126202
C2×AGL(1,ℤ/8)×C1092
#3045109