Finite Group Databasebeta

← #30598807

Subgroups — #30598807

Order 15616.

1102 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 116 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#21461718C212yesmaximal, complemented
2#21461067C212yesmaximal, complemented
3#21461064C212yesmaximal
4C22×#14519756C211yesmaximal, characteristic
5#16840732C2214yes—
6#16840642—24no—
7#16840607—24no—
8#16840605—24no—
9C23×Dic122C411yescharacteristic
10C2×#14519756—24no—
11C2×#14519756—24no—
12C2×#14519756—24no—
13C22×C4⋊Dic61C2211yescharacteristic
14C22×#11758196C411yescharacteristic
15C22×#11758196C2211yescharacteristic
16C22×C22⋊Dic61C411yescharacteristic
17C22×C22⋊Dic61C2211yescharacteristic
18C24×C4×C61C411yescharacteristic
19C2×#11758217—24no—
20C22⋊#92420—24no—
21C22⋊#92420—24no—
22C22×Dic122—22no—
23C22×Dic122—22no—
24C22×Dic122—22no—
25C22×Dic122—44no—
26C23×Dic61C2×C411yescharacteristic
27C23×Dic61C2×C411yescharacteristic
28C23×Dic61C2×C411yescharacteristic
29C23×Dic61C2311yescharacteristic
30#14519756—232no—
31C2×C4⋊Dic61—22no—
32C2×C4⋊Dic61—22no—
33C2×C4⋊Dic61D414yes—
34C2×#11758196—22no—
35C2×#11758196—22no—
36C2×#11758196—22no—
37C2×#11758196—22no—
38C2×#11758196—22no—
39C2×#11758196D414yes—
40C2×C22⋊Dic61—22no—
41C2×C22⋊Dic61—22no—
42C2×C22⋊Dic61—22no—
43C2×C22⋊Dic61—22no—
44C2×C22⋊Dic61—22no—
45C2×C22⋊Dic61D414yes—
46C25×C61C2×C411yescharacteristic
47C23×C4×C61—22no—
48C23×C4×C61—22no—
49C23×C4×C61—22no—
50C23×C4×C61—44no—
51C23×C4×C61C2×C411yescharacteristic
52C23×C4×C61C2×C411yescharacteristic
53C24×C61—24no—
54C24×C61—21no—
55C24×C61—21no—
56C24×C61—42no—
57C24×C61—42no—
58C24×C61C22⋊C412yes—
59C24×C61C22×C411yescharacteristic
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—24no—
63C22×C4×C61—28no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—48no—
67C22×C4×C61—48no—
68C22×C4×C61—44no—
69C22×C4×C61—44no—
70C22×C4×C61—42no—
71C22×C4×C61C22⋊C412yes—
72C22×C4×C61C22⋊C412yes—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—42no—
80C22×Dic61—42no—
81C22×Dic61—42no—
82C22×Dic61—42no—
83C22×Dic61C22⋊C412yes—
84C22×Dic61C22⋊C412yes—
85C22×Dic61C22⋊C412yes—
86C22×Dic61C2×D412yes—
87C22×Dic61C2×D412yes—
88C22×Dic61C2×D412yes—
89C2×Dic122—28no—
90C2×Dic122—48no—
91C2×Dic122—48no—
92C2×Dic122—44no—
93C2×Dic122—44no—
94#11758217—48no—
95C2×#92420—44no—
96C22⋊Dic61—28no—
97C22⋊Dic61—28no—
98C4⋊Dic61—28no—
99#11758196—28no—
100#11758196—28no—
101C23×C61—22no—
102C23×C61—22no—
103C23×C61—22no—
104C23×C61—24no—
105C23×C61—24no—
106C23×C61—22no—
107C23×C61—22no—
108C23×C61—42no—
109C23×C61—416no—
110C23×C61—42no—
111C23×C61—42no—
112C23×C61—42no—
113C23×C61—42no—
114C23×C61—42no—
115C23×C61#10114yes—
116C23×C61C2×C22⋊C411yescharacteristic
117C23×C61C2×C22⋊C411yescharacteristic
118C23×C61C2×C22⋊C411yescharacteristic
119C2×C4×C61—24no—
120C2×C4×C61—24no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—22no—
124C2×C4×C61—416no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61#10014yes—
129C2×Dic61—24no—
130C2×Dic61—22no—
131C2×Dic61—22no—
132C2×Dic61—22no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61—42no—
136C2×Dic61—42no—
137C2×Dic61—42no—
138C2×Dic61—42no—
139C2×Dic61—42no—
140C2×Dic61—42no—
141C2×Dic61—42no—
142C2×Dic61—42no—
143C2×Dic61—42no—
144C2×Dic61#12014yes—
145Dic122—416no—
146#92420—84no—
147#8555—611nomaximal, Sylow
148C22×C61—28no—
149C22×C61—21no—
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—21no—
153C22×C61—21no—
154C22×C61—21no—
155C22×C61—24no—
156C22×C61—44no—
157C22×C61—48no—
158C22×C61—48no—
159C22×C61—44no—
160C22×C61—42no—
161C22×C61—42no—
162C22×C61#37911yescharacteristic
163C22×C61#41012yes—
164C22×C61C2×#10012yes—
165C22×C61C2×#10112yes—
166C4×C61—24no—
167C4×C61—42no—
168Dic61—24no—
169Dic61—42no—
170Dic61—42no—
171Dic61—42no—
172C22×C22⋊Q8—611no—
173#1821—612no—
174#1463—612no—
175#1462—612no—
176C2×C61—22no—
177C2×C61—22no—
178C2×C61—22no—
179C2×C61—44no—
180C2×C61#178211yescharacteristic
181C2×C61#178311yescharacteristic
182C2×C61#178411yescharacteristic
183C23×Q8—611no—
184C24×C4#692311yescharacteristic
185C2×C22⋊Q8—1224no—
186C2×C22⋊Q8—1224no—
187C2×C22⋊Q8—1224no—
188C22×C4⋊C4—611no—
189C22×C4⋊C4—611no—
190C22×C4⋊C4—611no—
191C22×C22⋊C4—611no—
192C22×C22⋊C4—611no—
193#407—614no—
194#343—1224no—
195#324—1224no—
196#323—1224no—
197C61#855511yescharacteristic, complemented, Sylow
198C25C2×#692311yescharacteristic
199C22×Q8—1222no—
200C22×Q8—1222no—
201C22×Q8—1222no—
202C22×Q8—2444no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—44no—
207C23×C4—611no—
208C23×C4—611no—
209C23×C4—611no—
210C23×C4—611no—
211C23×C4C2×#692311yescharacteristic
212C23×C4C2×#692311yescharacteristic
213C2×M16—1224no—
214C22⋊Q8—12232no—
215C2×C4⋊C4—614no—
216C2×C4⋊C4—614no—
217C2×C4⋊C4—1222no—
218C2×C4⋊C4—1222no—
219C2×C4⋊C4—1222no—
220C2×C4⋊C4—1222no—
221C2×C4⋊C4—1222no—
222C2×C4⋊C4—1222no—
223C2×C4⋊C4—1222no—
224C2×C22⋊C4—614no—
225C2×C22⋊C4—1222no—
226C2×C22⋊C4—1222no—
227C2×C22⋊C4—1222no—
228C2×C22⋊C4—1222no—
229C2×C22⋊C4—1222no—
230C22⋊C8—1224no—
231C22⋊C8—1224no—
232C24—24no—
233C24—21no—
234C24—21no—
235C24—42no—
236C24—42no—
237C24C22⋊#692312yes—
238C24C22×#692311yescharacteristic
239C2×Q8—1228no—
240C2×Q8—2448no—
241C2×Q8—2448no—
242C2×Q8—2444no—
243C2×Q8—2444no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—28no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—48no—
251C22×C4—48no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—612no—
256C22×C4—612no—
257C22×C4—612no—
258C22×C4—612no—
259C22×C4—612no—
260C22×C4—612no—
261C22×C4—1221no—
262C22×C4—1221no—
263C22×C4—1221no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4—2442no—
268C22×C4—2442no—
269C22×C4—2442no—
270C22×C4—2442no—
271C22×C4C22⋊#692312yes—
272C22×C4C22⋊#692312yes—
273M16—2448no—
274C2×C8—2444no—
275C4⋊C4—1228no—
276C4⋊C4—1228no—
277C4⋊C4—1228no—
278C22⋊C4—1228no—
279C22⋊C4—1228no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—24no—
284C23—24no—
285C23—22no—
286C23—22no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—416no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23#1451983814yes—
295C23C2×C22⋊#692311yescharacteristic
296C23C2×C22⋊#692311yescharacteristic
297C23C2×C22⋊#692311yescharacteristic
298Q8—24416no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—416no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—614no—
309C2×C4—1224no—
310C2×C4—1222no—
311C2×C4—1222no—
312C2×C4—1222no—
313C2×C4—2442no—
314C2×C4—2442no—
315C2×C4—2442no—
316C2×C4—2442no—
317C2×C4—2442no—
318C2×C4—2442no—
319C2×C4—2442no—
320C2×C4—2442no—
321C2×C4—2442no—
322C2×C4—2442no—
323C2×C4—2442no—
324C2×C4#1451983714yes—
325C8—4884no—
326C22—28no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—24no—
334C22—44no—
335C22—48no—
336C22—48no—
337C22—44no—
338C22—42no—
339C22—42no—
340C22#1684070311yescharacteristic
341C22#1684074112yes—
342C22C2×#1451983712yes—
343C22C2×#1451983812yes—
344C4—24no—
345C4—42no—
346C4—1224no—
347C4—2442no—
348C4—2442no—
349C4—2442no—
350C2—22no—
351C2—22no—
352C2—22no—
353C2—44no—
354C2#2146164811yescharacteristic
355C2#2146165111yescharacteristic
356C2#2146165311yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC24×C4×C61C4
socleC22×C61#379
prime radicalC25×C61C2×C4
cosocleC23×Dic61C23
hypercenterC24×C4#6923
hypocenterC61#8555
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
1. #21461718C2semidirect
2. #21461067C2semidirect
197. C61#8555semidirect

Maximal subgroups

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

subgroupindex
#214617182
#214610672
#214610642
C22×#145197562
#855561