Finite Group Databasebeta

← #30605397

Subgroups — #30605397

Order 15616.

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

normal subgroups only

This group has 644 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#21461680C211yesmaximal, characteristic, complemented
2#21461056C211yesmaximal, characteristic, complemented
3#21460797C214yesmaximal, complemented
4#21450766C211yesmaximal, characteristic
5C2×#14519836—21no—
6D61⋊(C22⋊C8)—21no—
7#16840704—21no—
8#16840640C2212yes—
9#16840607—21no—
10#16840606C2212yes—
11C2×#14519773—21no—
12C2×#14519748—21no—
13C2×#14519748—21no—
14C2×C4×D244C2211yescharacteristic
15#16839670C412yescomplemented
16#16839670C2212yes—
17#16839665—21no—
18C4×C4⋊Dic61C411yescharacteristic, complemented
19C2×#127×C61C411yescharacteristic, complemented
20C2×D61⋊C8—41no—
21C2×C4⋊#6923—41no—
22C2×C22⋊#6923—22no—
23#14519836—42no—
24#14519834—41no—
25C22⋊#92420—22no—
26(C2×#6923)⋊C4—41no—
27C22×D244D411yescharacteristic
28C22×C22⋊D61—22no—
29C22×C22⋊D61—22no—
30C22×C4×D61D411yescharacteristic
31C22×C4×D61D411yescharacteristic
32#14519773—42no—
33#14519773—42no—
34#14519748—44no—
35#14519748—44no—
36C4×D244—41no—
37C4×D244—41no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61D411yescharacteristic
40C2×C22⋊Dic61—22no—
41C2×C22⋊Dic61—22no—
42C2×#11758198C2×C411yescharacteristic
43C2×#11758198C2311yescharacteristic
44C2×C4×Dic61—21no—
45#14519684C2×C412yes—
46C22×D4×C61—21no—
47C22×D4×C61—21no—
48C22×D4×C61C2×C412yes—
49#127×C61—41no—
50#127×C61—41no—
51C2×C42×C61C2×C411yescharacteristic
52C24×C61—22no—
53C23×D61C22⋊C411yescharacteristic
54C23×D61C2×D411yescharacteristic
55C22×#6923—42no—
56C2×D4×C61—42no—
57C2×D4×C61—42no—
58C2×D4×C61—42no—
59C2×D4×C61—42no—
60C2×D4×C61—42no—
61C2×D4×C61—41no—
62C2×D4×C61—41no—
63C2×D4×C61—42no—
64C2×D4×C61—82no—
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22×C411yescharacteristic
68C2×C22⋊D61—44no—
69C2×C22⋊D61—44no—
70C2×C22⋊D61—44no—
71C2×C22⋊D61—42no—
72C2×C22⋊D61—42no—
73C2×C22⋊D61—82no—
74C2×C22⋊D61—82no—
75C22×Dic61—22no—
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C2×D411yescharacteristic
78C2×D244—41no—
79C2×D244—41no—
80C2×D244D812yes—
81C2×D244SD1612yes—
82C2×C4×D61—21no—
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—21no—
86C2×C4×D61—42no—
87C2×C4×D61—42no—
88C22⋊#6923—42no—
89C22⋊#6923—42no—
90C2×#92420—42no—
91C4⋊#6923—82no—
92D61⋊C8—82no—
93C42×C61—21no—
94C42×C61—21no—
95C22⋊Dic61—44no—
96C22⋊Dic61—44no—
97#11758198—41no—
98#11758198—41no—
99C4⋊Dic61—24no—
100C4⋊Dic61—41no—
101C4×Dic61—42no—
102C23×C61—42no—
103C23×C61—42no—
104C23×C61—42no—
105C23×C61—42no—
106C23×C61—42no—
107C23×C61—42no—
108C23×C61—42no—
109C23×C61C2×C22⋊C411yescharacteristic
110C22×D61—21no—
111C22×D61—21no—
112C22×D61—21no—
113C22×D61—21no—
114C22×D61—41no—
115C22×D61—41no—
116C22×D61—41no—
117C22×D61—41no—
118C22×D61#10212yes—
119C22×D61C2×D811yescharacteristic
120C22×D61C2×SD1611yescharacteristic
121C2×#6923—82no—
122C2×#6923—82no—
123C2×#6923—82no—
124D4×C61—82no—
125D4×C61—84no—
126D4×C61—84no—
127D4×C61—82no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—22no—
131C2×C4×C61—21no—
132C2×C4×C61—41no—
133C2×C4×C61#9911yescharacteristic
134C2×C4×C61#9911yescharacteristic
135C2×C4×C61#10212yes—
136C22⋊D61—88no—
137C22⋊D61—84no—
138C22⋊D61—84no—
139C2×Dic61—24no—
140C2×Dic61—21no—
141C2×Dic61—21no—
142C2×Dic61—42no—
143C2×Dic61—42no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146D244—42no—
147D244—42no—
148C4×D61—42no—
149C4×D61—42no—
150C4×D61—81no—
151C4×D61—81no—
152#92420—82no—
153#11877—611nomaximal, Sylow
154C22×C61—21no—
155C22×C61—21no—
156C22×C61—82no—
157C22×C61—82no—
158C22×C61—82no—
159C22×C61—82no—
160C22×C61—82no—
161C22×C61—82no—
162C22×C61—82no—
163C22×C61C2×#9911yescharacteristic
164C22×C61C2×#10211yescharacteristic
165C22×C61#42111yescharacteristic
166C2×D61—21no—
167C2×D61—22no—
168C2×D61—21no—
169C2×D61—41no—
170C2×D61—41no—
171C2×D61—41no—
172C2×D61—41no—
173C2×D61—42no—
174C2×D61—41no—
175C2×D61—41no—
176C2×D61—81no—
177C2×D61—81no—
178#6923—162no—
179C4×C61—21no—
180C4×C61—22no—
181C4×C61—41no—
182C4×C61#35312yes—
183Dic61—44no—
184Dic61—41no—
185Dic61—41no—
186#2541—611no—
187#1799—611no—
188#1459—611no—
189#1343—614no—
190C2×C61—21no—
191C2×C61—21no—
192C2×C61—162no—
193C2×C61#151911yescharacteristic
194C2×C61#212211yescharacteristic
195C2×C61C2×#35311yescharacteristic
196D61—41no—
197D61—42no—
198D61—41no—
199C2×#127—1221no—
200C2×#127#692311yescharacteristic, complemented
201C2×#121—1221no—
202C2×#121—1221no—
203C2×C4×D4—611no—
204C2×#99—1221no—
205C2×C22⋊C8—1221no—
206#392—612no—
207#392—612no—
208#390—1221no—
209#380—1221no—
210C4×C4⋊C4—611no—
211#342—612no—
212#324—1221no—
213#323—612no—
214C61#1187711yescharacteristic, complemented, Sylow
215C22×D4—21no—
216C22×D4—21no—
217C22×D4—611no—
218C22×D4—1222no—
219C22×D4—1222no—
220C22×D4C2×#692312yes—
221C23×C4—611no—
222C23×C4—611no—
223C22×C8—2441no—
224#127—41no—
225#127—41no—
226#127—2442no—
227#127—2442no—
228#121—2444no—
229#121—2444no—
230C4×D4—2441no—
231C4×D4—2441no—
232C2×C4⋊C4—611no—
233C2×C4⋊C4—1221no—
234C2×C4⋊C4—2441no—
235C2×C22⋊C4—611no—
236C2×C22⋊C4—611no—
237C2×C22⋊C4—1222no—
238C2×C22⋊C4—1222no—
239C2×C22⋊C4—1222no—
240C2×C42—1221no—
241C2×C42C2×#692311yescharacteristic
242#99—2442no—
243C22⋊C8—1222no—
244C22⋊C8—2441no—
245#95—612no—
246#95—2441no—
247C24—22no—
248C24—611no—
249C24—611no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—82no—
259C2×D4—612no—
260C2×D4—612no—
261C2×D4—4882no—
262C2×D4—4882no—
263C2×D4—2441no—
264C2×D4—2441no—
265C2×D4—2444no—
266C2×D4—2444no—
267C2×D4—2444no—
268C2×D4—2442no—
269C2×D4—2442no—
270C22×C4—611no—
271C22×C4—611no—
272C22×C4—1222no—
273C22×C4—1221no—
274C22×C4—1221no—
275C22×C4—1221no—
276C22×C4—1221no—
277C22×C4—2442no—
278C22×C4—2442no—
279C22×C4—2442no—
280C22×C4C22⋊#692311yescharacteristic
281C22×C4C22⋊#692311yescharacteristic
282C22×C4C22×#692311yescharacteristic
283C2×C8—4882no—
284C2×C8—2442no—
285C4⋊C4—1224no—
286C4⋊C4—4882no—
287C4⋊C4—2441no—
288C22⋊C4—2441no—
289C22⋊C4—2441no—
290C22⋊C4—2444no—
291C22⋊C4—2444no—
292C22⋊C4—2442no—
293C22⋊C4—2442no—
294C42—21no—
295C42—21no—
296C42—2442no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—612no—
305C23—611no—
306C23—611no—
307C23—1221no—
308C23—1221no—
309C23—1221no—
310C23—1221no—
311C23—2441no—
312C23—2441no—
313C23—2441no—
314C23—2441no—
315C23C2×C22⋊#692311yescharacteristic
316D4—84no—
317D4—82no—
318D4—84no—
319D4—82no—
320D4—4884no—
321D4—4888no—
322D4—4884no—
323D4—2442no—
324D4—2442no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—41no—
330C2×C4—1224no—
331C2×C4—1221no—
332C2×C4—1221no—
333C2×C4—4881no—
334C2×C4—4881no—
335C2×C4—4882no—
336C2×C4—4882no—
337C2×C4—4882no—
338C2×C4—2442no—
339C2×C4—2442no—
340C2×C4—2442no—
341C2×C4—2442no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4#1451983511yescharacteristic
345C2×C4#1451983511yescharacteristic
346C2×C4#1451983912yes—
347C8—4882no—
348C22—21no—
349C22—21no—
350C22—82no—
351C22—82no—
352C22—82no—
353C22—82no—
354C22—82no—
355C22—82no—
356C22—82no—
357C22—1221no—
358C22—1222no—
359C22—1221no—
360C22—4881no—
361C22—4881no—
362C22—2441no—
363C22—2442no—
364C22—2441no—
365C22—2441no—
366C22—2441no—
367C22—2441no—
368C22—2441no—
369C22C2×#1451983511yescharacteristic
370C22C2×#1451983911yescharacteristic
371C22#1684076011yescharacteristic
372C4—22no—
373C4—21no—
374C4—41no—
375C4—9762no—
376C4—2444no—
377C4—2441no—
378C4—2441no—
379C4#1684066612yes—
380C2—21no—
381C2—21no—
382C2—162no—
383C2—2441no—
384C2—2441no—
385C2—2442no—
386C2#2146125511yescharacteristic
387C2#2146207811yescharacteristic
388C2C2×#1684066611yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#127×C61C4
socleC22×C61C2×#99
prime radical#21450766C2
cosocleC2×#11758198C23
hypercenterC2×#127#6923
hypocenterC61#11877
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #21461680C2semidirect
2. #21461056C2semidirect
3. #21460797C2semidirect
15. #16839670C4semidirect
18. C4×C4⋊Dic61C4semidirect
19. C2×#127×C61C4semidirect
200. C2×#127#6923semidirect
214. C61#11877semidirect

Maximal subgroups

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

subgroupindex
#214616802
#214610562
#214607972
#214507662
#1187761