Finite Group Databasebeta

← #30603822

Subgroups — #30603822

Order 15616.

734 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 104 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×#16840605C212yesmaximal, complemented
2#21460746C212yesmaximal, complemented
3#21460744C212yesmaximal, complemented
4#21448394C211yesmaximal, characteristic
5C2×C22⋊#92420—22no—
6#16840605—22no—
7#16840605C2214yes—
8C23×C22⋊D61C411yescharacteristic
9C22×C22⋊Dic61C2211yescharacteristic
10#16839641—22no—
11#16839639—22no—
12#16839623C411yescharacteristic
13#16839623C2211yescharacteristic
14#16839617C411yescharacteristic
15#16839617C2211yescharacteristic
16C22×C22⋊C4×C61C411yescharacteristic
17C22×#92420—42no—
18C22⋊#92420—24no—
19C22⋊#92420—42no—
20C24×D61C2×C411yescharacteristic
21C22×C22⋊D61—21no—
22C22×C22⋊D61—21no—
23C22×C22⋊D61—22no—
24C22×C22⋊D61—42no—
25C22×C22⋊D61—42no—
26C22×C22⋊D61C814yescomplemented
27C23×Dic61C2×C411yescharacteristic
28C23×Dic61D412yes—
29C2×C22⋊Dic61—42no—
30C2×C22⋊Dic61C2×C411yescharacteristic
31C2×C22⋊Dic61D412yes—
32C2×C22⋊Dic61C2311yescharacteristic
33C2×#11758198—22no—
34C2×#11758198—22no—
35#14519684—22no—
36#14519684—22no—
37C25×C61C2×C411yescharacteristic
38C23×C4×C61—21no—
39C2×C22⋊C4×C61—21no—
40C2×C22⋊C4×C61—42no—
41C2×C22⋊C4×C61C2×C411yescharacteristic
42C2×C22⋊C4×C61C2×C411yescharacteristic
43C24×C61—22no—
44C24×C61—22no—
45C24×C61—82no—
46C24×C61C22⋊C411yescharacteristic
47C24×C61C22⋊C411yescharacteristic
48C24×C61C2×C812yes—
49C24×C61M1612yes—
50C24×C61C22×C411yescharacteristic
51C23×D61—21no—
52C23×D61—21no—
53C23×D61—41no—
54C23×D61—41no—
55C23×D61—82no—
56C23×D61C2×C812yescomplemented
57C22×C4×C61—21no—
58C22×C4×C61—21no—
59C22×C4×C61—42no—
60C22×C4×C61—82no—
61C2×C22⋊D61—24no—
62C2×C22⋊D61—24no—
63C2×C22⋊D61—48no—
64C2×C22⋊D61—42no—
65C2×C22⋊D61—42no—
66C2×C22⋊D61—42no—
67C2×C22⋊D61—42no—
68C2×C22⋊D61—48no—
69C22×Dic61—22no—
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61—22no—
73C22×Dic61—41no—
74C22×Dic61—41no—
75C22×Dic61—82no—
76C22×Dic61C22⋊C412yes—
77C22×Dic61C2×C812yescomplemented
78C22×Dic61C2×D412yes—
79C2×#92420—44no—
80C2×#92420—84no—
81C22⋊C4×C61—42no—
82C22⋊C4×C61—41no—
83C22⋊C4×C61—41no—
84C22⋊Dic61—41no—
85C22⋊Dic61—42no—
86C22⋊Dic61—41no—
87#11758198—82no—
88#11758198—82no—
89C23×C61—21no—
90C23×C61—21no—
91C23×C61—21no—
92C23×C61—21no—
93C23×C61—24no—
94C23×C61—24no—
95C23×C61—42no—
96C23×C61—42no—
97C23×C61—42no—
98C23×C61—82no—
99C23×C61—82no—
100C23×C61—84no—
101C23×C61—82no—
102C23×C61—82no—
103C23×C61C22⋊C812yes—
104C23×C61C22⋊C812yes—
105C23×C61#9912yes—
106C23×C61#10012yes—
107C23×C61C2×C22⋊C411yescharacteristic
108C23×C61C22×C811yescharacteristic
109C23×C61C2×M1611yescharacteristic
110C22×D61—22no—
111C22×D61—42no—
112C22×D61—42no—
113C22×D61—42no—
114C22×D61—41no—
115C22×D61—41no—
116C22×D61—42no—
117C22×D61—82no—
118C22×D61—82no—
119C22×D61—84no—
120C22×D61—84no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—84no—
126C22⋊D61—48no—
127C22⋊D61—48no—
128C2×Dic61—22no—
129C2×Dic61—22no—
130C2×Dic61—22no—
131C2×Dic61—44no—
132C2×Dic61—42no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—42no—
136C2×Dic61—42no—
137C2×Dic61—42no—
138C2×Dic61—82no—
139#92420—84no—
140#11426—611nomaximal, Sylow
141C22×C61—22no—
142C22×C61—22no—
143C22×C61—22no—
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—22no—
147C22×C61—42no—
148C22×C61—42no—
149C22×C61—41no—
150C22×C61—41no—
151C22×C61—42no—
152C22×C61—42no—
153C22×C61—42no—
154C22×C61—42no—
155C22×C61—42no—
156C22×C61—84no—
157C22×C61—84no—
158C22×C61#32314yes—
159C22×C61C2×C22⋊C811yescharacteristic
160C22×C61C2×#9911yescharacteristic
161C22×C61C2×#10011yescharacteristic
162C2×D61—42no—
163C2×D61—42no—
164C2×D61—42no—
165C2×D61—82no—
166C2×D61—84no—
167C2×D61—82no—
168C2×D61—84no—
169C4×C61—81no—
170C4×C61—81no—
171Dic61—42no—
172Dic61—42no—
173Dic61—42no—
174#2308—611no—
175C2×#323—612no—
176#1317—612no—
177#1316—612no—
178C2×C61—21no—
179C2×C61—21no—
180C2×C61—24no—
181C2×C61—41no—
182C2×C61—41no—
183C2×C61—42no—
184C2×C61C2×#32311yescharacteristic
185C2×C61#212112yes—
186D61—82no—
187C23×D4—611no—
188C22×C22⋊C4—611no—
189C22×C22⋊C4#692311yescharacteristic
190C2×C22⋊C8—1222no—
191#386—1222no—
192#386—1222no—
193#381—611no—
194#381—611no—
195#379—611no—
196#379—611no—
197#323—614no—
198#323—1222no—
199C61#1142611yescharacteristic, complemented, Sylow
200C25—611no—
201C25C2×#692311yescharacteristic
202C22×D4—614no—
203C22×D4—1222no—
204C22×D4—1221no—
205C22×D4—1221no—
206C22×D4—2442no—
207C22×D4—2442no—
208C23×C4—21no—
209C23×C4—612no—
210C23×C4—611no—
211C22×C8—2442no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—612no—
215C2×C22⋊C4—611no—
216C2×C22⋊C4—611no—
217C2×C22⋊C4—1222no—
218C2×C22⋊C4—1222no—
219C2×C22⋊C4—2442no—
220C2×C22⋊C4C2×#692311yescharacteristic
221C2×C22⋊C4C2×#692311yescharacteristic
222C22⋊C8—1224no—
223C22⋊C8—2442no—
224#95—1222no—
225#95—1222no—
226C24—22no—
227C24—22no—
228C24—82no—
229C24—612no—
230C24—1221no—
231C24—1221no—
232C24—4882no—
233C24—2441no—
234C24—2441no—
235C24D61⋊C812yes—
236C24#1175821312yes—
237C24C22⋊#692311yescharacteristic
238C24C22⋊#692311yescharacteristic
239C24C22×#692311yescharacteristic
240C2×D4—1224no—
241C2×D4—1224no—
242C2×D4—2448no—
243C2×D4—2442no—
244C2×D4—2442no—
245C2×D4—2442no—
246C2×D4—2442no—
247C2×D4—2448no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—42no—
251C22×C4—82no—
252C22×C4—612no—
253C22×C4—612no—
254C22×C4—612no—
255C22×C4—1222no—
256C22×C4—1222no—
257C22×C4—1221no—
258C22×C4—1221no—
259C22×C4—4882no—
260C22×C4—2441no—
261C22×C4—2441no—
262C2×C8—4884no—
263C2×C8—2444no—
264C22⋊C4—42no—
265C22⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—4882no—
268C22⋊C4—4882no—
269C22⋊C4—2441no—
270C22⋊C4—2442no—
271C22⋊C4—2441no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—24no—
277C23—24no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—82no—
282C23—82no—
283C23—82no—
284C23—82no—
285C23—84no—
286C23—1222no—
287C23—4884no—
288C23—4884no—
289C23—4882no—
290C23—4882no—
291C23—2442no—
292C23—2442no—
293C23—2442no—
294C23—2442no—
295C23—2441no—
296C23—2441no—
297C23#1451983412yes—
298C23#1451983412yes—
299C23#1451983612yes—
300C23#1451983712yes—
301C23C2×C22⋊#692311yescharacteristic
302C23C2×D61⋊C811yescharacteristic
303C23C2×#1175821311yescharacteristic
304D4—2448no—
305D4—2448no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—84no—
311C2×C4—1222no—
312C2×C4—1222no—
313C2×C4—1222no—
314C2×C4—4882no—
315C2×C4—2444no—
316C2×C4—2442no—
317C2×C4—2441no—
318C2×C4—2441no—
319C2×C4—2442no—
320C2×C4—2442no—
321C2×C4—2442no—
322C8—4884no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—21no—
327C22—21no—
328C22—22no—
329C22—42no—
330C22—42no—
331C22—41no—
332C22—41no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—84no—
339C22—84no—
340C22—4882no—
341C22—4884no—
342C22—4884no—
343C22—4882no—
344C22—2442no—
345C22—2442no—
346C22—2442no—
347C22#1684060614yes—
348C22C2×#1451983411yescharacteristic
349C22C2×#1451983611yescharacteristic
350C22C2×#1451983711yescharacteristic
351C4—81no—
352C4—81no—
353C4—2442no—
354C4—2442no—
355C4—2442no—
356C2—21no—
357C2—21no—
358C2—24no—
359C2—41no—
360C2—41no—
361C2—42no—
362C2—4882no—
363C2C2×#1684060611yescharacteristic
364C2#2146207612yes—
subgroupG/H
centerC22C2×#14519836
commutatorC23×C61C22×C8
FrattiniC24C22×#6923
FittingC22×C22⋊C4×C61C4
socleC22×C61C2×#99
prime radicalC23×C22⋊D61C4
cosocleC2×C22⋊Dic61C23
hypercenterC22×C22⋊C4#6923
hypocenterC61#11426
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16840605C2semidirect
2. #21460746C2semidirect
3. #21460744C2semidirect
26. C22×C22⋊D61C8semidirect
56. C23×D61C2×C8semidirect
77. C22×Dic61C2×C8semidirect
199. C61#11426semidirect

Maximal subgroups

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

subgroupindex
C2×#168406052
#214607462
#214607442
#214483942
#1142661