Finite Group Databasebeta

← #28372428

Subgroups — #28372428

Order 13952.

800 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 145 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×#16812744C211yesmaximal, characteristic, complemented
2#20295585C212yesmaximal, complemented
3#20295215C214yesmaximal, complemented
4#20295212C214yesmaximal, complemented
5C2×#16812614C211yesmaximal, characteristic, complemented
6C2×#16812613C212yesmaximal, complemented
7#534×C109C211yesmaximal, characteristic, complemented
8#16812744—24no—
9#16812744—22no—
10#16812744C2214yes—
11C2×#13182029—21no—
12C2×#13182028—22no—
13C22×C22⋊D109—21no—
14C22×C22⋊D109C2212yes—
15C2×D4×D109—22no—
16#16812682C2214yescomplemented
17#16812680—22no—
18#16812671—24no—
19#16812668C2214yescomplemented
20#16812667—22no—
21C2×C22⋊Dic109C2212yescomplemented
22#16812614C2214yescomplemented
23#16812613C2218yescomplemented
24C22×D4×C109C2211yescharacteristic
25C22×D4×C109C2211yescharacteristic, complemented
26#123×C109—21no—
27#123×C109C2212yescomplemented
28#121×C109—22no—
29#120×C109—21no—
30#120×C109C2212yescomplemented
31C2×C22⋊C4×C109C2211yescharacteristic, complemented
32C23×D109—21no—
33C23×D109D412yescomplemented
34#13182029—24no—
35#13182029—24no—
36#13182028—28no—
37#13182028—28no—
38C2×Dic218—21no—
39C2×D436—21no—
40C2×C22⋊D109—22no—
41C2×C22⋊D109—24no—
42C2×C22⋊D109—22no—
43C2×C22⋊D109—22no—
44C2×C22⋊D109—24no—
45C2×C22⋊D109—22no—
46C2×C22⋊D109—22no—
47C2×C22⋊D109—24no—
48C2×C22⋊D109D414yescomplemented
49C2×C22⋊D109C2314yes—
50D4×D109—28no—
51D4×D109—28no—
52C22×Dic109—21no—
53C22×Dic109—22no—
54C22×Dic109D412yescomplemented
55C2×C4×D109—22no—
56#13182008—24no—
57C22⋊Dic109—22no—
58C22⋊Dic109—24no—
59C22⋊Dic109C2314yescomplemented
60#13182006—24no—
61C24×C109D411yescharacteristic, complemented
62C24×C109C2311yescharacteristic
63C24×C109C2311yescharacteristic, complemented
64C2×D4×C109—22no—
65C2×D4×C109—22no—
66C2×D4×C109—21no—
67C2×D4×C109—21no—
68C2×D4×C109—22no—
69C2×D4×C109—22no—
70C2×D4×C109—22no—
71C2×D4×C109D412yescomplemented
72C2×D4×C109C2312yes—
73C22×C4×C109—21no—
74C22×C4×C109D411yescharacteristic, complemented
75C22×C4×C109C2311yescharacteristic, complemented
76C4⋊C4×C109—22no—
77C22⋊C4×C109—21no—
78C22⋊C4×C109—22no—
79C22⋊C4×C109—22no—
80C22⋊C4×C109C2312yescomplemented
81C23×C109—21no—
82C23×C109—21no—
83C23×C109—21no—
84C23×C109—22no—
85C23×C109—81no—
86C23×C109—81no—
87C23×C109—81no—
88C23×C109C2×D412yes—
89C23×C109C2×D412yes—
90C23×C109C2×D412yes—
91C23×C109C2×D411yescharacteristic
92C23×C109C2×D411yescharacteristic
93C23×C109C2411yescharacteristic
94C22×D109—22no—
95C22×D109—24no—
96C22×D109—22no—
97C22×D109—82no—
98C22×D109—82no—
99C22×D109C2×D414yes—
100D4×C109—24no—
101D4×C109—24no—
102D4×C109—41no—
103D4×C109—42no—
104D4×C109—41no—
105D4×C109—82no—
106C2×C4×C109—21no—
107C2×C4×C109—21no—
108C2×C4×C109—22no—
109C2×C4×C109—21no—
110C2×C4×C109—22no—
111C2×C4×C109—81no—
112C2×C4×C109C2×D412yes—
113C22⋊D109—28no—
114C22⋊D109—24no—
115C22⋊D109—24no—
116C22⋊D109—24no—
117C22⋊D109—24no—
118C22⋊D109—84no—
119C2×Dic109—24no—
120C2×Dic109—22no—
121C2×Dic109—22no—
122C2×Dic109—24no—
123C2×Dic109—82no—
124C2×Dic109C2×D414yes—
125D436—24no—
126C4×D109—28no—
127Dic218—24no—
128C22×C109—22no—
129C22×C109—42no—
130C22×C109—41no—
131C22×C109—42no—
132C22×C109—41no—
133C22×C109—41no—
134C22×C109—42no—
135C22×C109—42no—
136C22×C109—44no—
137C22×C109—42no—
138C22×C109—41no—
139C22×C109—81no—
140C22×C109#12011yescharacteristic
141C22×C109#12012yes—
142C22×C109#12011yescharacteristic
143C22×C109C22×D412yes—
144C22×C109C22×D411yescharacteristic
145C2×D109—24no—
146C2×D109—44no—
147C2×D109—44no—
148C2×D109—82no—
149C2×D109—84no—
150C4×C109—22no—
151C4×C109—42no—
152C4×C109—41no—
153Dic109—24no—
154Dic109—44no—
155C2×C109—22no—
156C2×C109—21no—
157C2×C109—21no—
158C2×C109—22no—
159C2×C109—21no—
160C2×C109—42no—
161C2×C109—81no—
162C2×C109C2×#12011yescharacteristic
163D109—44no—
164#3026—1091nomaximal, Sylow
165C109#302611yescharacteristic, complemented, Sylow
166C2×H(5,𝔽2)—1091no—
167#534—1092no—
168#534D10911yescharacteristic, complemented
169#458—1094no—
170B(4,𝔽2)—1094no—
171C2×#99—1091no—
172C2×#99—1092no—
173H(5,𝔽2)—1094no—
174H(5,𝔽2)—2184no—
175H(5,𝔽2)—2182no—
176C2×#41—2182no—
177C2×#41—2181no—
178C22×D4—1092no—
179C22×D4—2182no—
180C22×D4—2181no—
181C22×D4C2×D10911yescharacteristic
182C22×D4C2×D10911yescharacteristic, complemented
183#123—21no—
184#123—1094no—
185#123—2182no—
186#123C2×D10912yescomplemented
187#121—22no—
188#121—2184no—
189#120—21no—
190#120—1094no—
191#120—2182no—
192#120C2×D10912yescomplemented
193C2×C22⋊C4—1092no—
194C2×C22⋊C4C2×D10911yescharacteristic, complemented
195#99—1094no—
196#99—1098no—
197C24—1092no—
198C24—2181no—
199C24C22⋊D10911yescharacteristic, complemented
200C24C22×D10911yescharacteristic
201C24C22×D10911yescharacteristic, complemented
202#41—2184no—
203#41—2188no—
204#41—2188no—
205#41—2184no—
206C2×Q8—2181no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—22no—
214C2×D4—1094no—
215C2×D4—1094no—
216C2×D4—2181no—
217C2×D4—2188no—
218C2×D4—2188no—
219C2×D4—2184no—
220C2×D4—2182no—
221C2×D4—2184no—
222C2×D4—2182no—
223C2×D4—2182no—
224C2×D4—2182no—
225C2×D4—2182no—
226C2×D4—2184no—
227C2×D4C22⋊D10912yescomplemented
228C2×D4C22×D10912yes—
229C22×C4—21no—
230C22×C4—1092no—
231C22×C4—2182no—
232C22×C4—2182no—
233C22×C4—2181no—
234C22×C4C22⋊D10911yescharacteristic, complemented
235C22×C4C22×D10911yescharacteristic, complemented
236C4⋊C4—22no—
237C4⋊C4—2184no—
238C22⋊C4—21no—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—1094no—
242C22⋊C4—2184no—
243C22⋊C4—2184no—
244C22⋊C4—2182no—
245C22⋊C4C22×D10912yescomplemented
246C23—21no—
247C23—21no—
248C23—22no—
249C23—21no—
250C23—81no—
251C23—81no—
252C23—81no—
253C23—8722no—
254C23—8722no—
255C23—1094no—
256C23—2184no—
257C23—2182no—
258C23—2182no—
259C23D4×D10912yes—
260C23D4×D10912yes—
261C23C2×C22⋊D10912yes—
262C23C2×C22⋊D10911yescharacteristic
263C23C2×C22⋊D10911yescharacteristic
264C23C23×D10911yescharacteristic
265Q8—2184no—
266D4—24no—
267D4—24no—
268D4—41no—
269D4—41no—
270D4—42no—
271D4—82no—
272D4—8724no—
273D4—2184no—
274D4—2188no—
275D4—2184no—
276D4—2184no—
277D4—2184no—
278D4—2184no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—81no—
285C2×C4—8722no—
286C2×C4—1094no—
287C2×C4—2188no—
288C2×C4—2184no—
289C2×C4—2184no—
290C2×C4—2182no—
291C2×C4—2182no—
292C2×C4C2×C22⋊D10912yes—
293C22—22no—
294C22—41no—
295C22—42no—
296C22—41no—
297C22—41no—
298C22—42no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—44no—
303C22—41no—
304C22—81no—
305C22—8722no—
306C22—8724no—
307C22—4364no—
308C22—4364no—
309C22—2184no—
310C22#1681266711yescharacteristic
311C22#1681266711yescharacteristic
312C22#1681266712yes—
313C22C2×D4×D10912yes—
314C22C22×C22⋊D10911yescharacteristic
315C4—22no—
316C4—42no—
317C4—41no—
318C4—4364no—
319C4—2184no—
320C2—21no—
321C2—22no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—42no—
326C2—81no—
327C2—4364no—
328C2C2×#1681266711yescharacteristic
subgroupG/H
centerC2C2×#16812667
commutator, cosocleC23×C109C24
FrattiniC23C23×D109
Fitting#534×C109C2
socleC2×C109C2×#120
hypercenter#534D109
hypocenterC109#3026
LCSG ▷ C23×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. C2×#16812744C2semidirect
2. #20295585C2semidirect
3. #20295215C2semidirect
4. #20295212C2semidirect
5. C2×#16812614C2semidirect
6. C2×#16812613C2semidirect
7. #534×C109C2semidirect
16. #16812682C22semidirect
19. #16812668C22semidirect
21. C2×C22⋊Dic109C22semidirect
22. #16812614C22semidirect
23. #16812613C22semidirect
25. C22×D4×C109C22semidirect
27. #123×C109C22semidirect
30. #120×C109C22semidirect
31. C2×C22⋊C4×C109C22semidirect
33. C23×D109D4semidirect
48. C2×C22⋊D109D4semidirect
54. C22×Dic109D4semidirect
59. C22⋊Dic109C23semidirect
61. C24×C109D4semidirect
63. C24×C109C23semidirect
71. C2×D4×C109D4semidirect
74. C22×C4×C109D4semidirect
75. C22×C4×C109C23semidirect
80. C22⋊C4×C109C23semidirect
165. C109#3026semidirect
168. #534D109semidirect
182. C22×D4C2×D109semidirect
186. #123C2×D109semidirect
192. #120C2×D109semidirect
194. C2×C22⋊C4C2×D109semidirect
199. C24C22⋊D109semidirect
201. C24C22×D109semidirect
227. C2×D4C22⋊D109semidirect
234. C22×C4C22⋊D109semidirect
235. C22×C4C22×D109semidirect
245. C22⋊C4C22×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127442
#202955852
#202952152
#202952122
C2×#168126142
C2×#168126132
#534×C1092
#3026109