Finite Group Databasebeta

← #28372284

Subgroups — #28372284

Order 13952.

806 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 169 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×#16812724C212yesmaximal, complemented
2C2×#16812673C211yesmaximal, characteristic, complemented
3#20295171C214yesmaximal, complemented
4#20295165C212yesmaximal, complemented
5#20295160C212yesmaximal, complemented
6#20295027C212yesmaximal, complemented
7#20294990C211yesmaximal, characteristic, complemented
8C22×#41×C109C211yesmaximal, characteristic, complemented
9C22×D436C2211yescharacteristic, complemented
10C22×C22⋊D109—21no—
11#16812724—28no—
12C2×#13182016C2214yescomplemented
13C2×#13182014C2214yescomplemented
14C2×#13182013C2212yescomplemented
15#16812673—21no—
16#16812673—21no—
17#16812673C2212yes—
18#16812673C2212yescomplemented
19C2×C4⋊Dic109C2211yescharacteristic, complemented
20C2×#13182006—21no—
21#16812623C2214yescomplemented
22#16812620C2214yescomplemented
23C22⋊#11724784C2212yescomplemented
24C22⋊#11724784C2212yescomplemented
25C2×#41×C109—28no—
26C2×#41×C109—21no—
27C2×#41×C109—21no—
28C2×#41×C109C2212yes—
29C2×#41×C109C2212yescomplemented
30C22×Q8×C109C2211yescharacteristic, complemented
31C22×D4×C109—21no—
32C22×D4×C109C2211yescharacteristic, complemented
33C23×C4×C109—21no—
34C23×C4×C109C2211yescharacteristic
35C23×D109—21no—
36C2×D436—24no—
37C2×D436—21no—
38C2×D436C2312yes—
39C2×C22⋊D109—22no—
40C2×C22⋊D109—22no—
41C2×C22⋊D109—44no—
42C22×Dic109—21no—
43#13182016—28no—
44#13182014—28no—
45#13182013—24no—
46C2×#11724784C2314yes—
47C4⋊Dic109—21no—
48C4⋊Dic109C2312yes—
49#13182006—22no—
50#13182006—22no—
51C24×C109—21no—
52C24×C109D411yescharacteristic
53#41×C109—28no—
54#41×C109—28no—
55#41×C109—24no—
56#41×C109—24no—
57#41×C109—24no—
58#41×C109D418yescomplemented
59C2×Q8×C109—24no—
60C2×Q8×C109—21no—
61C2×Q8×C109C2312yes—
62C2×D4×C109—28no—
63C2×D4×C109—22no—
64C2×D4×C109—22no—
65C2×D4×C109—24no—
66C2×D4×C109—21no—
67C2×D4×C109C2312yes—
68C22×C4×C109—28no—
69C22×C4×C109—21no—
70C22×C4×C109—22no—
71C22×C4×C109—22no—
72C22×C4×C109—24no—
73C22×C4×C109D411yescharacteristic
74C22×C4×C109D411yescharacteristic
75C22×C4×C109D411yescharacteristic
76C22×C4×C109C2311yescharacteristic
77C22×C4×C109C2311yescharacteristic
78C22×C4×C109C2311yescharacteristic
79C23×C109—24no—
80C23×C109—22no—
81C23×C109—22no—
82C23×C109—22no—
83C23×C109—41no—
84C23×C109—42no—
85C23×C109—42no—
86C23×C109C2×D411yescharacteristic
87C23×C109C2×D411yescharacteristic
88C23×C109C2×D411yescharacteristic
89C22×D109—22no—
90C22×D109—42no—
91C22×D109—42no—
92C22×D109—42no—
93Q8×C109—24no—
94Q8×C109—22no—
95Q8×C109C2×D414yescomplemented
96D4×C109—28no—
97D4×C109—24no—
98D4×C109—28no—
99D4×C109—22no—
100D4×C109C2×D414yescomplemented
101C2×C4×C109—24no—
102C2×C4×C109—28no—
103C2×C4×C109—22no—
104C2×C4×C109—22no—
105C2×C4×C109—22no—
106C2×C4×C109—28no—
107C2×C4×C109—22no—
108C2×C4×C109—22no—
109C2×C4×C109—22no—
110C2×C4×C109C2×D411yescharacteristic
111C2×C4×C109C2×D411yescharacteristic
112C2×C4×C109C2×D411yescharacteristic
113C2×C4×C109C2×D414yes—
114C2×C4×C109C2411yescharacteristic
115C22⋊D109—44no—
116C22⋊D109—44no—
117C2×Dic109—22no—
118C2×Dic109—42no—
119D436—24no—
120D436—42no—
121#11724784—24no—
122C22×C109—21no—
123C22×C109—21no—
124C22×C109—21no—
125C22×C109—21no—
126C22×C109—21no—
127C22×C109—24no—
128C22×C109—21no—
129C22×C109—21no—
130C22×C109—44no—
131C22×C109—48no—
132C22×C109—42no—
133C22×C109—42no—
134C22×C109#12012yes—
135C22×C109AGL(1,ℤ/8)12yescomplemented
136C22×C109C22×D411yescharacteristic
137C2×D109—42no—
138C2×D109—42no—
139C2×D109—42no—
140C2×D109—84no—
141C4×C109—21no—
142C4×C109—24no—
143C4×C109—21no—
144C4×C109#12012yes—
145C4×C109C22×D412yes—
146Dic109—42no—
147C2×C109—22no—
148C2×C109—22no—
149C2×C109—41no—
150C2×C109—44no—
151C2×C109C2×#12011yescharacteristic
152C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
153C2×C109#57511yescharacteristic
154D109—82no—
155#3006—1091nomaximal, Sylow
156C109#300611yescharacteristic, complemented, Sylow
157C22×#41D10911yescharacteristic, complemented
158C2×AGL(1,ℤ/8)—1092no—
159C2×#121—1091no—
160#449—1094no—
161#448—1092no—
162#447—1092no—
163#417—1092no—
164#407—1091no—
165C2×#41—28no—
166C2×#41—21no—
167C2×#41—21no—
168C2×#41C2×D10912yes—
169C2×#41C2×D10912yescomplemented
170C22×Q8C2×D10911yescharacteristic, complemented
171C22×D4—21no—
172C22×D4—1091no—
173C22×D4—2181no—
174C22×D4C2×D10911yescharacteristic, complemented
175C23×C4—21no—
176C23×C4C2×D10911yescharacteristic
177AGL(1,ℤ/8)—2188no—
178C2×SD16—1094no—
179C2×D8—1094no—
180C2×M16—1092no—
181#121—1092no—
182#121—1092no—
183#121—2181no—
184#121—2181no—
185C2×C4⋊C4—1091no—
186C2×C22⋊C4—2181no—
187#103—1094no—
188#102—1094no—
189C22⋊C8—1092no—
190C22⋊C8—1092no—
191C24—21no—
192C24—2181no—
193C24C22⋊D10911yescharacteristic
194#41—28no—
195#41—28no—
196#41—24no—
197#41—24no—
198#41—24no—
199#41C22⋊D10918yescomplemented
200C2×Q8—24no—
201C2×Q8—21no—
202C2×Q8C22×D10912yes—
203C2×D4—28no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—24no—
207C2×D4—21no—
208C2×D4—1092no—
209C2×D4—4364no—
210C2×D4—2182no—
211C2×D4—2182no—
212C2×D4—2181no—
213C2×D4—2184no—
214C2×D4C22×D10912yes—
215C22×C4—28no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—2181no—
221C22×C4C22⋊D10911yescharacteristic
222C22×C4C22⋊D10911yescharacteristic
223C22×C4C22⋊D10911yescharacteristic
224C22×C4C22×D10911yescharacteristic
225C22×C4C22×D10911yescharacteristic
226C22×C4C22×D10911yescharacteristic
227SD16—2188no—
228D8—2188no—
229M16—2184no—
230C2×C8—1094no—
231C4⋊C4—1092no—
232C4⋊C4—2181no—
233C22⋊C4—2182no—
234C22⋊C4—2182no—
235C23—24no—
236C23—22no—
237C23—22no—
238C23—22no—
239C23—41no—
240C23—42no—
241C23—42no—
242C23—4362no—
243C23—4362no—
244C23—4362no—
245C23—2182no—
246C23C2×C22⋊D10911yescharacteristic
247C23C2×C22⋊D10911yescharacteristic
248C23C2×C22⋊D10911yescharacteristic
249Q8—24no—
250Q8—22no—
251Q8C2×C22⋊D10914yescomplemented
252D4—28no—
253D4—24no—
254D4—28no—
255D4—22no—
256D4—4364no—
257D4—4364no—
258D4—4362no—
259D4—2184no—
260D4C2×C22⋊D10914yescomplemented
261C2×C4—24no—
262C2×C4—28no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—28no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—4362no—
271C2×C4—2182no—
272C2×C4C2×C22⋊D10911yescharacteristic
273C2×C4C2×C22⋊D10911yescharacteristic
274C2×C4C2×C22⋊D10911yescharacteristic
275C2×C4C2×C22⋊D10914yes—
276C2×C4C23×D10911yescharacteristic
277C8—2184no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—24no—
284C22—21no—
285C22—21no—
286C22—44no—
287C22—48no—
288C22—42no—
289C22—42no—
290C22—8724no—
291C22—4362no—
292C22—4362no—
293C22—4362no—
294C22#1681266812yes—
295C22#1681272412yescomplemented
296C22C22×C22⋊D10911yescharacteristic
297C4—21no—
298C4—24no—
299C4—21no—
300C4—4362no—
301C4#1681266812yes—
302C4C22×C22⋊D10912yes—
303C2—22no—
304C2—22no—
305C2—41no—
306C2—44no—
307C2—8722no—
308C2C2×#1681266811yescharacteristic
309C2C2×#1681272411yescharacteristic
310C2#2029582411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC22×#41×C109C2
socleC22×C109C22×D4
hypercenterC22×#41D109
hypocenterC109#3006
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#16812724C2semidirect
2. C2×#16812673C2semidirect
3. #20295171C2semidirect
4. #20295165C2semidirect
5. #20295160C2semidirect
6. #20295027C2semidirect
7. #20294990C2semidirect
8. C22×#41×C109C2semidirect
9. C22×D436C22semidirect
12. C2×#13182016C22semidirect
13. C2×#13182014C22semidirect
14. C2×#13182013C22semidirect
18. #16812673C22semidirect
19. C2×C4⋊Dic109C22semidirect
21. #16812623C22semidirect
22. #16812620C22semidirect
23. C22⋊#11724784C22semidirect
24. C22⋊#11724784C22semidirect
29. C2×#41×C109C22semidirect
30. C22×Q8×C109C22semidirect
32. C22×D4×C109C22semidirect
58. #41×C109D4semidirect
95. Q8×C109C2×D4semidirect
100. D4×C109C2×D4semidirect
135. C22×C109AGL(1,ℤ/8)semidirect
156. C109#3006semidirect
157. C22×#41D109semidirect
169. C2×#41C2×D109semidirect
170. C22×Q8C2×D109semidirect
174. C22×D4C2×D109semidirect
199. #41C22⋊D109semidirect
251. Q8C2×C22⋊D109semidirect
260. D4C2×C22⋊D109semidirect
295. C22#16812724semidirect

Maximal subgroups

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

subgroupindex
C2×#168127242
C2×#168126732
#202951712
#202951652
#202951602
#202950272
#202949902
C22×#41×C1092
#3006109