Finite Group Databasebeta

← #30600305

Subgroups — #30600305

Order 15616.

560 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 100 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
1#21462670C211yesmaximal, characteristic
2#21461055C211yesmaximal, characteristic
3C2×#16840607C212yesmaximal, complemented
4#21460662C212yesmaximal, complemented
5#21448491C211yesmaximal, characteristic
6#16840793—24no—
7#16840782—21no—
8#16840732—21no—
9C2×C22⋊#92420C2212yes—
10C2×C22⋊#92420C2212yes—
11#16840607—21no—
12#16840607—24no—
13#16840605—21no—
14C2×D4×Dic61C2211yescharacteristic
15C22×C4⋊Dic61C2211yescharacteristic
16#16839661—21no—
17#16839655—21no—
18#16839648—21no—
19#16839642C411yescharacteristic
20#16839637—21no—
21#16839637C411yescharacteristic
22#16839625C411yescharacteristic
23C22⋊C4×Dic61C2211yescharacteristic
24C2×#123×C61C411yescharacteristic
25C2×#11758217—22no—
26C22×#92420—22no—
27C22×#92420—22no—
28#14519844—22no—
29C22⋊#92420—22no—
30C22⋊#92420—22no—
31C22⋊#92420—22no—
32C22⋊#92420—22no—
33C22⋊#92420D414yes—
34#14519832—22no—
35C23×Dic61C2×C411yescharacteristic
36C23×Dic61D411yescharacteristic
37C23×Dic61C2311yescharacteristic
38D4×Dic61—22no—
39D4×Dic61—22no—
40C2×C4⋊Dic61—22no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—22no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61D411yescharacteristic
46C2×C22⋊Dic61—21no—
47C2×C22⋊Dic61C2×C411yescharacteristic
48C2×C22⋊Dic61C2×C411yescharacteristic
49C2×C22⋊Dic61D411yescharacteristic
50C2×C4×Dic61—21no—
51C2×C4×Dic61D411yescharacteristic
52#14519684—21no—
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57C22×D4×C61C2×C411yescharacteristic
58C23×C4×C61C2×C411yescharacteristic
59#123×C61—22no—
60#123×C61—22no—
61C2×C4⋊C4×C61—21no—
62C2×C22⋊C4×C61—21no—
63C2×C22⋊C4×C61C2×C411yescharacteristic
64C24×C61C22⋊C411yescharacteristic
65C24×C61C22×C411yescharacteristic
66C2×D4×C61—22no—
67C2×D4×C61—21no—
68C2×D4×C61—21no—
69C2×D4×C61—22no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—22no—
73C22×C4×C61—21no—
74C22×C4×C61—22no—
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61M1612yescomplemented
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—22no—
82C22×Dic61—81no—
83C22×Dic61—81no—
84C22×Dic61—81no—
85C22×Dic61C22⋊C411yescharacteristic
86C22×Dic61C22⋊C411yescharacteristic
87C22×Dic61C2×D412yes—
88C22×Dic61C2×D411yescharacteristic
89C22×Dic61C2×D411yescharacteristic
90C22×Dic61#4112yes—
91#11758217—82no—
92C2×#92420—24no—
93C2×#92420—42no—
94C2×#92420—82no—
95C2×#92420—82no—
96C4⋊C4×C61—24no—
97C22⋊C4×C61—24no—
98C22⋊C4×C61—21no—
99C22⋊C4×C61M1612yescomplemented
100C22⋊Dic61—22no—
101C22⋊Dic61—22no—
102C22⋊Dic61—22no—
103C22⋊Dic61—81no—
104C4⋊Dic61—22no—
105C4⋊Dic61—41no—
106C4⋊Dic61—42no—
107C4⋊Dic61—41no—
108C4⋊Dic61—81no—
109C4×Dic61—21no—
110C4×Dic61—21no—
111C4×Dic61—81no—
112C23×C61—21no—
113C23×C61—21no—
114C23×C61—21no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61—41no—
119C23×C61C2×C22⋊C411yescharacteristic
120C23×C61C4×D412yes—
121C23×C61C2×M1611yescharacteristic
122C23×C61C2×M1611yescharacteristic
123C23×C61#13112yes—
124D4×C61—42no—
125D4×C61—42no—
126C2×C4×C61—21no—
127C2×C4×C61—22no—
128C2×C4×C61—22no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—24no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—42no—
137C2×Dic61—21no—
138C2×Dic61—21no—
139C2×Dic61—41no—
140C2×Dic61—42no—
141C2×Dic61—42no—
142C2×Dic61—41no—
143C2×Dic61—42no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—81no—
153C2×Dic61—82no—
154C2×Dic61#12011yescharacteristic
155C2×Dic61#12112yes—
156C2×Dic61#12311yescharacteristic
157#92420—44no—
158#92420—82no—
159#9110—611nomaximal, Sylow
160C22×C61—21no—
161C22×C61—22no—
162C22×C61—21no—
163C22×C61—22no—
164C22×C61—41no—
165C22×C61—41no—
166C22×C61—42no—
167C22×C61—42no—
168C22×C61—41no—
169C22×C61—41no—
170C22×C61—41no—
171C22×C61—81no—
172C22×C61#38611yescharacteristic
173C22×C61#40711yescharacteristic
174C22×C61#40811yescharacteristic
175C22×C61#43512yes—
176C22×C61#43612yes—
177C4×C61—42no—
178C4×C61—42no—
179C4×C61—41no—
180Dic61—21no—
181Dic61—22no—
182Dic61—21no—
183Dic61—41no—
184Dic61—41no—
185Dic61—81no—
186Dic61—81no—
187#2973—611no—
188#2322—611no—
189#1459—611no—
190C2×#324—612no—
191#1282—612no—
192C2×C61—21no—
193C2×C61—21no—
194C2×C61—22no—
195C2×C61—22no—
196C2×C61—81no—
197C2×C61#187011yescharacteristic
198C2×C61#191311yescharacteristic
199C2×C61#191511yescharacteristic
200C2×#123#692311yescharacteristic
201C2×C4×D4—611no—
202C22×C4⋊C4—611no—
203#435—1224no—
204#432—1221no—
205#407—1221no—
206C2×C22⋊C8—612no—
207C2×C22⋊C8—612no—
208#389—1221no—
209#388—1221no—
210#387—1221no—
211#386—611no—
212#385—611no—
213#385—1221no—
214#381—611no—
215C4×C22⋊C4—611no—
216#324—1224no—
217#324—1221no—
218#323—1221no—
219C61#911011yescharacteristic, complemented, Sylow
220C22×D4C2×#692311yescharacteristic
221C23×C4—611no—
222C23×C4—611no—
223C23×C4—611no—
224C23×C4C2×#692311yescharacteristic
225C2×M16—1222no—
226C22×C8—1222no—
227C22×C8—1222no—
228#123—22no—
229#123—22no—
230C4×D4—1222no—
231C4×D4—1222no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—611no—
234C2×C4⋊C4—1221no—
235C2×C4⋊C4—1222no—
236C2×C4⋊C4—1221no—
237C2×C4⋊C4—1221no—
238C2×C4⋊C4—1222no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—611no—
241C2×C22⋊C4—611no—
242C2×C22⋊C4—611no—
243C2×C22⋊C4—1221no—
244C2×C22⋊C4C2×#692311yescharacteristic
245C2×C42—611no—
246C2×C42—1221no—
247C4⋊C8—1222no—
248C22⋊C8—614no—
249C22⋊C8—1222no—
250C22⋊C8—1222no—
251C22⋊C8—1222no—
252C22⋊C8—1222no—
253#97—1222no—
254#95—1221no—
255#95—1221no—
256#95—1221no—
257#95—1221no—
258#95—1221no—
259C24C22⋊#692311yescharacteristic
260C24C22×#692311yescharacteristic
261C2×D4—22no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—611no—
271C22×C4—611no—
272C22×C4—612no—
273C22×C4—612no—
274C22×C4—611no—
275C22×C4—611no—
276C22×C4—1221no—
277C22×C4—1221no—
278C22×C4—1222no—
279C22×C4—1221no—
280C22×C4—1221no—
281C22×C4—4881no—
282C22×C4—4881no—
283C22×C4—4881no—
284C22×C4#1175821712yescomplemented
285C22×C4C22⋊#692311yescharacteristic
286M16—4882no—
287C2×C8—1224no—
288C2×C8—4882no—
289C2×C8—4882no—
290C2×C8—2442no—
291C4⋊C4—24no—
292C4⋊C4—1222no—
293C4⋊C4—4881no—
294C4⋊C4—2441no—
295C4⋊C4—2442no—
296C4⋊C4—2441no—
297C22⋊C4—24no—
298C22⋊C4—21no—
299C22⋊C4—1222no—
300C22⋊C4—1222no—
301C22⋊C4—1222no—
302C22⋊C4—4881no—
303C22⋊C4#1175821712yescomplemented
304C42—1221no—
305C42—1221no—
306C42—4881no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23C2×C22⋊#692311yescharacteristic
315C23D4×#692312yes—
316C23C2×#1175821711yescharacteristic
317C23C2×#1175821711yescharacteristic
318C23#1451986412yes—
319D4—42no—
320D4—42no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—24no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—611no—
333C2×C4—612no—
334C2×C4—611no—
335C2×C4—1221no—
336C2×C4—1221no—
337C2×C4—4881no—
338C2×C4—4882no—
339C2×C4—2441no—
340C2×C4—2442no—
341C2×C4—2442no—
342C2×C4—2441no—
343C2×C4—2442no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4—2441no—
351C2×C4—2441no—
352C8—4882no—
353C8—2444no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22#1684071011yescharacteristic
367C22#1684073211yescharacteristic
368C22#1684073511yescharacteristic
369C22#1684079312yes—
370C22#1684079712yes—
371C4—42no—
372C4—42no—
373C4—41no—
374C4—1222no—
375C4—1221no—
376C4—1221no—
377C4—4881no—
378C4—4881no—
379C4—2441no—
380C4—2441no—
381C2—21no—
382C2—21no—
383C2—22no—
384C2—22no—
385C2—81no—
386C2#2146179611yescharacteristic
387C2#2146186911yescharacteristic
388C2#2146187111yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC2×#123×C61C4
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC23×Dic61C23
hypercenterC2×#123#6923
hypocenterC61#9110
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
3. C2×#16840607C2semidirect
4. #21460662C2semidirect
76. C22×C4×C61M16semidirect
99. C22⋊C4×C61M16semidirect
219. C61#9110semidirect
284. C22×C4#11758217semidirect
303. C22⋊C4#11758217semidirect

Maximal subgroups

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

subgroupindex
#214626702
#214610552
C2×#168406072
#214606622
#214484912
#911061