Finite Group Databasebeta

← #30598809

Subgroups — #30598809

Order 15616.

1102 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 1102 conjugacy classes of subgroups — too many to draw. Only its 112 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#21461066C212yesmaximal
4C22×#14519753C211yesmaximal, characteristic
5#16840732C2214yes—
6#16840642—24no—
7#16840607—24no—
8#16840606—24no—
9C23×C4×D61C411yescharacteristic
10C2×#14519753—24no—
11C2×#14519753—24no—
12C2×#14519753—24no—
13C22×C4⋊Dic61C2211yescharacteristic
14C22×#11758196C411yescharacteristic
15C22×#11758196C2211yescharacteristic
16C22×#11758198C411yescharacteristic
17C22×#11758198C2211yescharacteristic
18C23×Q8×C61C411yescharacteristic
19C2×#11758217—24no—
20C22⋊#92420—24no—
21C22⋊#92420—24no—
22C24×D61C2×C411yescharacteristic
23C23×Dic61C2×C411yescharacteristic
24C23×Dic61C2×C411yescharacteristic
25C23×Dic61C2311yescharacteristic
26C22×C4×D61—22no—
27C22×C4×D61—22no—
28C22×C4×D61—22no—
29C22×C4×D61—44no—
30#14519753—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×#11758198—22no—
41C2×#11758198—22no—
42C2×#11758198—22no—
43C2×#11758198—22no—
44C2×#11758198—22no—
45C2×#11758198D414yes—
46C22×Q8×C61—22no—
47C22×Q8×C61—22no—
48C22×Q8×C61—22no—
49C22×Q8×C61—44no—
50C23×C4×C61C2×C411yescharacteristic
51C23×C4×C61C2×C411yescharacteristic
52C23×C4×C61C2×C411yescharacteristic
53C24×C61C22×C411yescharacteristic
54C23×D61—21no—
55C23×D61—21no—
56C23×D61—24no—
57C23×D61—42no—
58C23×D61—42no—
59C23×D61C22⋊C412yes—
60C2×Q8×C61—28no—
61C2×Q8×C61—48no—
62C2×Q8×C61—48no—
63C2×Q8×C61—44no—
64C2×Q8×C61—44no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—24no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—42no—
73C22×C4×C61—42no—
74C22×C4×C61C22⋊C412yes—
75C22×C4×C61C22⋊C412yes—
76C22×C4×C61C22⋊C412yes—
77C22×Dic61—21no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—21no—
81C22×Dic61—42no—
82C22×Dic61—42no—
83C22×Dic61—42no—
84C22×Dic61C22⋊C412yes—
85C22×Dic61C22⋊C412yes—
86C22×Dic61C2×D412yes—
87C22×Dic61C2×D412yes—
88C22×Dic61C2×D412yes—
89C2×C4×D61—28no—
90C2×C4×D61—48no—
91C2×C4×D61—44no—
92C2×C4×D61—48no—
93C2×C4×D61—44no—
94#11758217—48no—
95C2×#92420—44no—
96#11758198—28no—
97#11758198—28no—
98C4⋊Dic61—28no—
99#11758196—28no—
100#11758196—28no—
101C23×C61—22no—
102C23×C61—22no—
103C23×C61—22no—
104C23×C61C2×C22⋊C411yescharacteristic
105C23×C61C2×C22⋊C411yescharacteristic
106C23×C61C2×C22⋊C411yescharacteristic
107C22×D61—24no—
108C22×D61—24no—
109C22×D61—22no—
110C22×D61—22no—
111C22×D61—42no—
112C22×D61—416no—
113C22×D61—42no—
114C22×D61—42no—
115C22×D61—42no—
116C22×D61—42no—
117C22×D61—42no—
118C22×D61#10114yes—
119Q8×C61—416no—
120C2×C4×C61—24no—
121C2×C4×C61—24no—
122C2×C4×C61—22no—
123C2×C4×C61—22no—
124C2×C4×C61—22no—
125C2×C4×C61—22no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61—42no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61#10014yes—
133C2×Dic61—24no—
134C2×Dic61—22no—
135C2×Dic61—22no—
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—
145C4×D61—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#37911yescharacteristic
156C22×C61#41012yes—
157C22×C61C2×#10012yes—
158C22×C61C2×#10112yes—
159C2×D61—24no—
160C2×D61—48no—
161C2×D61—48no—
162C2×D61—44no—
163C2×D61—42no—
164C2×D61—42no—
165C2×D61—44no—
166C4×C61—24no—
167C4×C61—42no—
168C4×C61—42no—
169Dic61—24no—
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#178211yescharacteristic
180C2×C61#178311yescharacteristic
181C2×C61#178411yescharacteristic
182D61—44no—
183C23×Q8#692311yescharacteristic
184C24×C4—611no—
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
198C25—611no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—44no—
203C23×C4—611no—
204C23×C4—611no—
205C23×C4—611no—
206C23×C4—1222no—
207C23×C4—1222no—
208C23×C4—1222no—
209C23×C4—2444no—
210C23×C4C2×#692311yescharacteristic
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—612no—
233C24—1224no—
234C24—1221no—
235C24—1221no—
236C24—2442no—
237C24—2442no—
238C24C22×#692311yescharacteristic
239C2×Q8—28no—
240C2×Q8—48no—
241C2×Q8—48no—
242C2×Q8—44no—
243C2×Q8—44no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—612no—
254C22×C4—612no—
255C22×C4—612no—
256C22×C4—612no—
257C22×C4—612no—
258C22×C4—1228no—
259C22×C4—1221no—
260C22×C4—1221no—
261C22×C4—1221no—
262C22×C4—1221no—
263C22×C4—2448no—
264C22×C4—2448no—
265C22×C4—2444no—
266C22×C4—2442no—
267C22×C4—2444no—
268C22×C4—2442no—
269C22×C4—2442no—
270C22×C4C22⋊#692312yes—
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—614no—
284C23—1224no—
285C23—1224no—
286C23—1222no—
287C23—1222no—
288C23—2442no—
289C23—2442no—
290C23—2442no—
291C23—24416no—
292C23—2442no—
293C23—2442no—
294C23—2442no—
295C23C2×C22⋊#692311yescharacteristic
296C23C2×C22⋊#692311yescharacteristic
297C23C2×C22⋊#692311yescharacteristic
298Q8—416no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—614no—
312C2×C4—1224no—
313C2×C4—1222no—
314C2×C4—1222no—
315C2×C4—24416no—
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—1224no—
334C22—2444no—
335C22—2448no—
336C22—2448no—
337C22—2444no—
338C22—2442no—
339C22—2442no—
340C22#1684070311yescharacteristic
341C22#1684074312yes—
342C22C2×#1451983712yes—
343C22D61⋊#10112yes—
344C4—24no—
345C4—42no—
346C4—42no—
347C4—1224no—
348C4—2442no—
349C4—2442no—
350C2—22no—
351C2—22no—
352C2—22no—
353C2—2444no—
354C2#2146165011yescharacteristic
355C2#2146165111yescharacteristic
356C2#2146165411yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC23×Q8×C61C4
socleC22×C61#379
prime radicalC24×D61C2×C4
cosocleC23×Dic61C23
hypercenterC23×Q8#6923
hypocenterC61#8555
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8

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
#214610662
C22×#145197532
#855561