Finite Group Databasebeta

← #30599921

Subgroups — #30599921

Order 15616.

626 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 126 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#21462666C211yesmaximal, characteristic
2#21461709C212yesmaximal, complemented
3#21461055C211yesmaximal, characteristic
4#21460662C212yesmaximal, complemented
5#123×Dic61C211yesmaximal, characteristic
6D4×#92420—24no—
7#16840778—21no—
8C2×C22⋊#92420—21no—
9C2×C22⋊#92420C2212yes—
10C2×C22⋊#92420C2212yes—
11#16840628—24no—
12#16840607—21no—
13#16840605—21no—
14C2×D4×Dic61C2211yescharacteristic
15C22×C4×Dic61C2211yescharacteristic
16#16839655—21no—
17#16839648—21no—
18#16839642C411yescharacteristic
19#16839637C411yescharacteristic
20#16839625C411yescharacteristic
21C4⋊C4×Dic61—21no—
22C22⋊C4×Dic61—21no—
23C22⋊C4×Dic61C2211yescharacteristic
24C2×#123×C61C411yescharacteristic
25C22×#92420—22no—
26C22×#92420—22no—
27C22×#92420D414yes—
28C4⋊#92420—22no—
29C22⋊#92420—22no—
30C22⋊#92420—22no—
31C22⋊#92420—22no—
32C22⋊#92420—22no—
33C22⋊#92420—22no—
34C4×#92420—22no—
35C23×Dic61C2×C411yescharacteristic
36C23×Dic61D411yescharacteristic
37C23×Dic61C2311yescharacteristic
38D4×Dic61—22no—
39D4×Dic61—22no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61D411yescharacteristic
42C2×C22⋊Dic61—21no—
43C2×C22⋊Dic61C2×C411yescharacteristic
44C2×C22⋊Dic61C2×C411yescharacteristic
45C2×C22⋊Dic61D411yescharacteristic
46C2×C4×Dic61—22no—
47C2×C4×Dic61—21no—
48C2×C4×Dic61—21no—
49C2×C4×Dic61—21no—
50C2×C4×Dic61—22no—
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×C61C814yescomplemented
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—22no—
68C2×D4×C61C2×C812yes—
69C2×D4×C61M1612yes—
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×C61C2×C812yescomplemented
77C22×Dic61—21no—
78C22×Dic61—22no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—21no—
82C22×Dic61—42no—
83C22×Dic61—42no—
84C22×Dic61—42no—
85C22×Dic61C22⋊C411yescharacteristic
86C22×Dic61C22⋊C411yescharacteristic
87C22×Dic61C2×D412yes—
88C22×Dic61C2×D411yescharacteristic
89C22×Dic61C2×D411yescharacteristic
90C22×Dic61#4112yes—
91C2×#92420—24no—
92C2×#92420—44no—
93C2×#92420—44no—
94C2×#92420—42no—
95C2×#92420—82no—
96C4⋊C4×C61—24no—
97C22⋊C4×C61—24no—
98C22⋊C4×C61—21no—
99C22⋊C4×C61C2×C812yescomplemented
100C22⋊Dic61—22no—
101C22⋊Dic61—22no—
102C22⋊Dic61—22no—
103C22⋊Dic61—42no—
104C4⋊Dic61—22no—
105C4⋊Dic61—42no—
106C4×Dic61—21no—
107C4×Dic61—21no—
108C4×Dic61—42no—
109C4×Dic61—41no—
110C4×Dic61—42no—
111C4×Dic61—41no—
112C23×C61—21no—
113C23×C61—21no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61C22⋊C812yes—
119C23×C61C2×C22⋊C411yescharacteristic
120C23×C61C4×D412yes—
121C23×C61C22×C811yescharacteristic
122C23×C61C2×M1611yescharacteristic
123C23×C61#13112yes—
124D4×C61—24no—
125D4×C61—24no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—24no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—42no—
136C2×C4×C61C22⋊C812yes—
137C2×Dic61—22no—
138C2×Dic61—24no—
139C2×Dic61—22no—
140C2×Dic61—24no—
141C2×Dic61—21no—
142C2×Dic61—21no—
143C2×Dic61—41no—
144C2×Dic61—42no—
145C2×Dic61—41no—
146C2×Dic61—42no—
147C2×Dic61—41no—
148C2×Dic61—42no—
149C2×Dic61—41no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—81no—
155C2×Dic61#12011yescharacteristic
156C2×Dic61#12112yes—
157C2×Dic61#12311yescharacteristic
158#92420—44no—
159#92420—82no—
160#8966—611nomaximal, Sylow
161C22×C61—21no—
162C22×C61—22no—
163C22×C61—21no—
164C22×C61—22no—
165C22×C61—22no—
166C22×C61—41no—
167C22×C61—41no—
168C22×C61—42no—
169C22×C61—42no—
170C22×C61—41no—
171C22×C61—41no—
172C22×C61—81no—
173C22×C61#38611yescharacteristic
174C22×C61C2×C22⋊C811yescharacteristic
175C22×C61#40811yescharacteristic
176C22×C61C8×D412yes—
177C22×C61#43512yes—
178C4×C61—22no—
179C4×C61—42no—
180C4×C61—42no—
181Dic61—22no—
182Dic61—21no—
183Dic61—41no—
184Dic61—42no—
185Dic61—41no—
186Dic61—81no—
187Dic61#45812yes—
188#2972—611no—
189C4×#123—611no—
190#1817—612no—
191#1459—611no—
192#1282—612no—
193C2×C61—21no—
194C2×C61—21no—
195C2×C61—22no—
196C2×C61—22no—
197C2×C61—42no—
198C2×C61#186911yescharacteristic
199C2×C61#189011yescharacteristic
200C2×C61#189111yescharacteristic
201C2×#123#692311yescharacteristic
202C2×C4×D4—611no—
203C22×C42—611no—
204C8×D4—1224no—
205#431—1221no—
206C2×C22⋊C8—612no—
207C2×C22⋊C8—612no—
208C2×C22⋊C8—1221no—
209#388—1221no—
210#387—1221no—
211#386—611no—
212#385—611no—
213#381—611no—
214C4×C4⋊C4—1221no—
215C4×C22⋊C4—611no—
216C4×C22⋊C4—1221no—
217#336—1224no—
218#324—1221no—
219#323—1221no—
220C61#896611yescharacteristic, complemented, Sylow
221C22×D4C2×#692311yescharacteristic
222C23×C4—611no—
223C23×C4—611no—
224C23×C4—611no—
225C23×C4C2×#692311yescharacteristic
226C22×C8—614no—
227C22×C8—1222no—
228C22×C8—1222no—
229#123—22no—
230#123#9242014yescomplemented
231C4×D4—1222no—
232C4×D4—1222no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—611no—
235C2×C4⋊C4—1221no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—611no—
238C2×C22⋊C4—611no—
239C2×C22⋊C4—611no—
240C2×C22⋊C4—1221no—
241C2×C22⋊C4C2×#692311yescharacteristic
242C2×C42—611no—
243C2×C42—1222no—
244C2×C42—1221no—
245C2×C42—1221no—
246C2×C42—1221no—
247C2×C42—1222no—
248C4⋊C8—1222no—
249C22⋊C8—1222no—
250C22⋊C8—1222no—
251C22⋊C8—1222no—
252C22⋊C8—1222no—
253C22⋊C8—1222no—
254C4×C8—1222no—
255#95—1221no—
256#95—1221no—
257#95—1221no—
258#95—1221no—
259#95—1221no—
260C24C22⋊#692311yescharacteristic
261C24C22×#692311yescharacteristic
262C2×D4—22no—
263C2×D4—22no—
264C2×D4C2×#9242012yes—
265C2×D4#1175821712yes—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—611no—
272C22×C4—612no—
273C22×C4—611no—
274C22×C4—612no—
275C22×C4—611no—
276C22×C4—611no—
277C22×C4—1221no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—1222no—
281C22×C4—1221no—
282C22×C4—2442no—
283C22×C4—2442no—
284C22×C4—2442no—
285C22×C4C2×#9242012yescomplemented
286C22×C4C22⋊#692311yescharacteristic
287C2×C8—1224no—
288C2×C8—4882no—
289C2×C8—2442no—
290C2×C8—2444no—
291C2×C8—2444no—
292C4⋊C4—24no—
293C4⋊C4—1222no—
294C4⋊C4—2442no—
295C22⋊C4—24no—
296C22⋊C4—21no—
297C22⋊C4—1222no—
298C22⋊C4—1222no—
299C22⋊C4—1222no—
300C22⋊C4—2442no—
301C22⋊C4C2×#9242012yescomplemented
302C42—1221no—
303C42—1221no—
304C42—2441no—
305C42—2442no—
306C42—2441no—
307C42—2442no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23C22⋊#9242012yes—
315C23C2×C22⋊#692311yescharacteristic
316C23D4×#692312yes—
317C23C22×#9242011yescharacteristic
318C23C2×#1175821711yescharacteristic
319C23#1451986412yes—
320D4—24no—
321D4—24no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
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—1222no—
336C2×C4—1224no—
337C2×C4—1222no—
338C2×C4—1224no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—4881no—
342C2×C4—2442no—
343C2×C4—2441no—
344C2×C4—2442no—
345C2×C4—2441no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4—2441no—
349C2×C4—2442no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4—2441no—
353C2×C4C22⋊#9242012yes—
354C8—4882no—
355C8—2444no—
356C22—21no—
357C22—22no—
358C22—21no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—42no—
364C22—42no—
365C22—41no—
366C22—41no—
367C22—81no—
368C22#1684071011yescharacteristic
369C22C2×C22⋊#9242011yescharacteristic
370C22#1684073511yescharacteristic
371C22D4×#9242012yes—
372C22#1684079312yes—
373C4—22no—
374C4—42no—
375C4—42no—
376C4—612no—
377C4—1222no—
378C4—1221no—
379C4—4881no—
380C4—2441no—
381C4—2442no—
382C4—2441no—
383C2—21no—
384C2—21no—
385C2—22no—
386C2—22no—
387C2—42no—
388C2#2146179211yescharacteristic
389C2#2146182911yescharacteristic
390C2#2146183111yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC23×C61C22×C8
FrattiniC24C22×#6923
FittingC2×#123×C61C4
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC23×Dic61C23
hypercenterC2×#123#6923
hypocenterC61#8966
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. #21461709C2semidirect
4. #21460662C2semidirect
60. #123×C61C8semidirect
76. C22×C4×C61C2×C8semidirect
99. C22⋊C4×C61C2×C8semidirect
220. C61#8966semidirect
230. #123#92420semidirect
285. C22×C4C2×#92420semidirect
301. C22⋊C4C2×#92420semidirect

Maximal subgroups

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

subgroupindex
#214626662
#214617092
#214610552
#214606622
#123×Dic612
#896661