Finite Group Databasebeta

← #30605407

Subgroups — #30605407

Order 15616.

764 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 80 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#21461683C211yesmaximal, characteristic, complemented
2#21461054C211yesmaximal, characteristic, complemented
3#21460801C214yesmaximal, complemented
4#21449169C211yesmaximal, characteristic
5#16840765—21no—
6C2×#14519837—21no—
7D61⋊(C22⋊C8)—21no—
8#16840642C2212yescomplemented
9#16840608C2212yescomplemented
10#16840605—21no—
11C2×#14519748—21no—
12C2×#14519746—21no—
13C2×C4×C22⋊D61C2211yescharacteristic
14#16839676C2212yes—
15#16839670C412yes—
16#16839665—21no—
17#16839622C411yescharacteristic
18#16839618—21no—
19C23×D4×C61C411yescharacteristic
20C2×#11758217—22no—
21C2×D61⋊C8—41no—
22#14519844—41no—
23C4⋊#92420—41no—
24#14519837—42no—
25#14519834—41no—
26C22⋊#92420—22no—
27C22×C22⋊D61—22no—
28C22×C22⋊D61D411yescharacteristic
29C22×C4×D61D411yescharacteristic
30#14519748—44no—
31#14519746—42no—
32#14519746—42no—
33C4×C22⋊D61—41no—
34C4×C22⋊D61—41no—
35C2×C4⋊Dic61—21no—
36C2×#11758196C2311yescharacteristic
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—22no—
39C2×C22⋊Dic61—22no—
40C2×C22⋊Dic61—22no—
41C2×C22⋊Dic61D411yescharacteristic
42C2×#11758198C2×C411yescharacteristic
43C2×C4×Dic61D411yescharacteristic
44#14519684C2×C412yes—
45C25×C61—21no—
46C22×D4×C61—21no—
47C22×D4×C61—41no—
48C22×D4×C61—41no—
49C22×D4×C61—44no—
50C22×D4×C61C2×C412yes—
51C23×C4×C61C2×C411yescharacteristic
52C24×C61—22no—
53C24×C61—42no—
54C24×C61—44no—
55C24×C61—42no—
56C24×C61—42no—
57C24×C61—82no—
58C24×C61C22⋊C411yescharacteristic
59C23×D61C22⋊C411yescharacteristic
60C2×D4×C61—24no—
61C2×D4×C61—42no—
62C2×D4×C61—42no—
63C2×D4×C61—48no—
64C2×D4×C61—44no—
65C2×D4×C61—44no—
66C2×D4×C61—44no—
67C2×D4×C61—41no—
68C2×D4×C61—41no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—42no—
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22×C411yescharacteristic
74C2×C22⋊D61—41no—
75C2×C22⋊D61—41no—
76C2×C22⋊D61—44no—
77C2×C22⋊D61—44no—
78C2×C22⋊D61—82no—
79C2×C22⋊D61D812yes—
80C2×C22⋊D61SD1612yes—
81C22×Dic61—22no—
82C22×Dic61C22⋊C411yescharacteristic
83C22×Dic61C2×D411yescharacteristic
84C22×Dic61C2×D411yescharacteristic
85C2×C4×D61—21no—
86C2×C4×D61—21no—
87C2×C4×D61—42no—
88#11758217—42no—
89#11758217—42no—
90C2×#92420—42no—
91C2×#92420—42no—
92D61⋊C8—82no—
93C22⋊Dic61—41no—
94C22⋊Dic61—44no—
95C22⋊Dic61—42no—
96C22⋊Dic61—42no—
97C22⋊Dic61—42no—
98C22⋊Dic61—82no—
99#11758198—41no—
100C4⋊Dic61—42no—
101#11758196—41no—
102C4×Dic61—21no—
103C4×Dic61—21no—
104C23×C61—21no—
105C23×C61—21no—
106C23×C61—24no—
107C23×C61—41no—
108C23×C61—41no—
109C23×C61—42no—
110C23×C61—42no—
111C23×C61—42no—
112C23×C61—48no—
113C23×C61—42no—
114C23×C61—42no—
115C23×C61—42no—
116C23×C61—84no—
117C23×C61—88no—
118C23×C61—88no—
119C23×C61—84no—
120C23×C61#10212yes—
121C23×C61C2×C22⋊C411yescharacteristic
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—41no—
125C22×D61—41no—
126C22×D61#10212yes—
127D4×C61—44no—
128D4×C61—44no—
129D4×C61—44no—
130D4×C61—44no—
131C2×C4×C61—22no—
132C2×C4×C61—21no—
133C2×C4×C61—42no—
134C2×C4×C61—42no—
135C2×C4×C61—41no—
136C2×C4×C61#9911yescharacteristic
137C2×C4×C61#9911yescharacteristic
138C22⋊D61—42no—
139C22⋊D61—42no—
140C22⋊D61—88no—
141C2×Dic61—21no—
142C2×Dic61—21no—
143C2×Dic61—21no—
144C2×Dic61—21no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—42no—
148C2×Dic61—42no—
149C2×Dic61—42no—
150C2×Dic61C2×D811yescharacteristic
151C2×Dic61C2×SD1611yescharacteristic
152C4×D61—42no—
153C4×D61—81no—
154#92420—82no—
155#92420—82no—
156#11880—611nomaximal, Sylow
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—22no—
160C22×C61—42no—
161C22×C61—42no—
162C22×C61—44no—
163C22×C61—44no—
164C22×C61—41no—
165C22×C61—41no—
166C22×C61—44no—
167C22×C61—82no—
168C22×C61—82no—
169C22×C61—816no—
170C22×C61—82no—
171C22×C61—82no—
172C22×C61C2×#9911yescharacteristic
173C22×C61C2×#10211yescharacteristic
174C22×C61#42111yescharacteristic
175C2×D61—22no—
176C2×D61—41no—
177C2×D61—41no—
178C2×D61—42no—
179C2×D61—81no—
180C4×C61—21no—
181C4×C61—41no—
182C4×C61AΣL(1,𝔽16)12yes—
183Dic61—21no—
184Dic61—21no—
185Dic61—41no—
186Dic61—41no—
187Dic61—82no—
188#2408—611no—
189#1800—611no—
190#1459—611no—
191#1345—614no—
192C2×C61—21no—
193C2×C61—21no—
194C2×C61—42no—
195C2×C61—84no—
196C2×C61#151911yescharacteristic
197C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
198C2×C61#213411yescharacteristic
199D61—42no—
200C23×D4#692311yescharacteristic
201C2×#121—1221no—
202C2×#120—1221no—
203C2×C4×D4—611no—
204#423—1221no—
205C2×#100—1221no—
206C2×C22⋊C8—1221no—
207#394—612no—
208#392—612no—
209#390—1221no—
210#380—611no—
211#379—1221no—
212#343—612no—
213#324—612no—
214#323—1221no—
215C61#1188011yescharacteristic, complemented, Sylow
216C25—21no—
217C22×D4—21no—
218C22×D4—41no—
219C22×D4—41no—
220C22×D4—44no—
221C22×D4—611no—
222C22×D4—1222no—
223C22×D4C2×#692312yes—
224C23×C4—611no—
225C23×C4C2×#692311yescharacteristic
226C2×M16—1222no—
227C22×C8—2441no—
228#121—2444no—
229#120—2442no—
230#120—2442no—
231C4×D4—2441no—
232C4×D4—2441no—
233C2×C4⋊C4—611no—
234C2×C4⋊C4—1221no—
235C2×C22⋊C4—611no—
236C2×C22⋊C4—611no—
237C2×C22⋊C4—1222no—
238C2×C22⋊C4—1222no—
239C2×C22⋊C4—1222no—
240C2×C22⋊C4—1221no—
241C2×C42—611no—
242C4⋊C8—2441no—
243C4⋊C8—2441no—
244#100—2442no—
245C22⋊C8—1222no—
246C22⋊C8—2441no—
247#95—612no—
248C24—22no—
249C24—42no—
250C24—44no—
251C24—42no—
252C24—42no—
253C24—82no—
254C24—611no—
255C24C22⋊#692311yescharacteristic
256C2×D4—24no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—48no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—612no—
266C2×D4—612no—
267C2×D4—4882no—
268C2×D4—2441no—
269C2×D4—2441no—
270C2×D4—2444no—
271C2×D4—2444no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—42no—
275C22×C4—611no—
276C22×C4—611no—
277C22×C4—611no—
278C22×C4—1221no—
279C22×C4—1221no—
280C22×C4—1222no—
281C22×C4—2442no—
282C22×C4C22⋊#692311yescharacteristic
283C22×C4C22×#692311yescharacteristic
284M16—2442no—
285M16—2442no—
286C2×C8—4882no—
287C2×C8—2442no—
288C2×C8—2442no—
289C4⋊C4—2441no—
290C4⋊C4—2442no—
291C22⋊C4—4882no—
292C22⋊C4—2441no—
293C22⋊C4—2442no—
294C22⋊C4—2442no—
295C22⋊C4—2441no—
296C22⋊C4—2444no—
297C22⋊C4—2442no—
298C42—1221no—
299C42—1221no—
300C23—21no—
301C23—21no—
302C23—24no—
303C23—41no—
304C23—41no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—48no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—84no—
313C23—88no—
314C23—88no—
315C23—84no—
316C23—612no—
317C23—1221no—
318C23—1221no—
319C23—2441no—
320C23—2441no—
321C23#1451983912yes—
322C23C2×C22⋊#692311yescharacteristic
323D4—44no—
324D4—44no—
325D4—44no—
326D4—44no—
327D4—4888no—
328D4—2442no—
329D4—2442no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—611no—
336C2×C4—611no—
337C2×C4—1221no—
338C2×C4—1221no—
339C2×C4—1221no—
340C2×C4—1221no—
341C2×C4—4881no—
342C2×C4—2442no—
343C2×C4—2442no—
344C2×C4—2442no—
345C2×C4—2442no—
346C2×C4—2441no—
347C2×C4—2441no—
348C2×C4#1451983511yescharacteristic
349C2×C4#1451983511yescharacteristic
350C8—4882no—
351C8—4882no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—42no—
356C22—42no—
357C22—44no—
358C22—44no—
359C22—41no—
360C22—41no—
361C22—44no—
362C22—82no—
363C22—82no—
364C22—816no—
365C22—82no—
366C22—82no—
367C22—1222no—
368C22—4881no—
369C22—2442no—
370C22—2441no—
371C22—2441no—
372C22C2×#1451983511yescharacteristic
373C22C2×#1451983911yescharacteristic
374C22#1684076011yescharacteristic
375C4—21no—
376C4—41no—
377C4—1221no—
378C4—1221no—
379C4—4882no—
380C4—2441no—
381C4—2441no—
382C4#1684066212yes—
383C2—21no—
384C2—21no—
385C2—42no—
386C2—84no—
387C2—2442no—
388C2#2146125511yescharacteristic
389C2C2×#1684066211yescharacteristic
390C2#2146211811yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC23×D4×C61C4
socleC22×C61C2×#99
prime radical#21449169C2
cosocleC2×#11758196C23
hypercenterC23×D4#6923
hypocenterC61#11880
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. #21461683C2semidirect
2. #21461054C2semidirect
3. #21460801C2semidirect
8. #16840642C22semidirect
9. #16840608C22semidirect
215. C61#11880semidirect

Maximal subgroups

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

subgroupindex
#214616832
#214610542
#214608012
#214491692
#1188061