Finite Group Databasebeta

← #29590474

Subgroups — #29590474

Order 15616.

500 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 91 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#21445552C211yesmaximal, characteristic, complemented
2#21444153C211yesmaximal, characteristic, complemented
3#21443843C212yesmaximal, complemented
4#21443816C212yesmaximal
5C61×#2598C211yesmaximal, characteristic
6C2×#14519736—21no—
7#16839819—21no—
8#16839529—21no—
9#16839529C2212yes—
10#16839488—21no—
11#16839486C2212yes—
12C2×C4×D4×C61C411yescharacteristic, complemented
13C2×#117×C61C2211yescharacteristic
14C61×#400C411yescharacteristic, complemented
15C61×#399C2211yescharacteristic
16C61×#394C411yescharacteristic, complemented
17C61×#392C2211yescharacteristic
18C61×#388—21no—
19C61×#386—21no—
20C61×#386—21no—
21C61×#385—21no—
22C61×#383C411yescharacteristic, complemented
23#14519736—24no—
24#14519736—42no—
25C2×C4⋊Dic61—41no—
26C2×C22⋊Dic61—22no—
27C2×C4×Dic61—41no—
28C4⋊#92418—41no—
29C22⋊#92418—22no—
30#14519688—41no—
31C22×D4×C61—21no—
32C22×D4×C61C2×C411yescharacteristic
33C23×C4×C61—21no—
34C23×C4×C61D411yescharacteristic
35C4×D4×C61—41no—
36C4×D4×C61—41no—
37#117×C61—41no—
38#117×C61—41no—
39C2×C4⋊C4×C61—21no—
40C2×C4⋊C4×C61C2×C411yescharacteristic
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C4⋊C4×C61D411yescharacteristic
43C2×C22⋊C4×C61—21no—
44C2×C22⋊C4×C61—21no—
45C2×C22⋊C4×C61—21no—
46C2×C22⋊C4×C61—21no—
47C2×C22⋊C4×C61—21no—
48C2×C22⋊C4×C61D411yescharacteristic
49C2×C22⋊C4×C61C2311yescharacteristic
50C2×C42×C61C2×C411yescharacteristic
51C2×C42×C61D411yescharacteristic
52#95×C61—21no—
53#95×C61—21no—
54#95×C61—21no—
55#95×C61—21no—
56#95×C61C2×C411yescharacteristic
57#95×C61C2×C411yescharacteristic
58C24×C61—21no—
59C24×C61C2×D411yescharacteristic
60C2×D4×C61—41no—
61C2×D4×C61—42no—
62C2×D4×C61—42no—
63C2×D4×C61—41no—
64C2×D4×C61—41no—
65C2×D4×C61—81no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—22no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—42no—
75C22×C4×C61—81no—
76C22×C4×C61—81no—
77C22×C4×C61C22⋊C411yescharacteristic
78C22×C4×C61C22⋊C411yescharacteristic
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22×C411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C22×Dic61—42no—
84C4⋊C4×C61—41no—
85C4⋊C4×C61—41no—
86C4⋊C4×C61—41no—
87C4⋊C4×C61—81no—
88C22⋊C4×C61—41no—
89C22⋊C4×C61—41no—
90C22⋊C4×C61—42no—
91C22⋊C4×C61—41no—
92C22⋊C4×C61—41no—
93C22⋊C4×C61—42no—
94C22⋊C4×C61—81no—
95C22⋊C4×C61—81no—
96C42×C61—21no—
97C42×C61—21no—
98C42×C61—21no—
99C42×C61—21no—
100C22⋊Dic61—24no—
101C22⋊Dic61—42no—
102C4⋊Dic61—44no—
103C4×Dic61—44no—
104C2×#92418—42no—
105C23×C61—21no—
106C23×C61—21no—
107C23×C61—22no—
108C23×C61—41no—
109C23×C61—41no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61C2×C22⋊C411yescharacteristic
117C23×C61AGL(1,ℤ/8)11yescharacteristic
118C23×C61#13711yescharacteristic
119D4×C61—42no—
120D4×C61—42no—
121D4×C61—82no—
122D4×C61—82no—
123C2×C4×C61—22no—
124C2×C4×C61—22no—
125C2×C4×C61—22no—
126C2×C4×C61—22no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—84no—
149C2×C4×C61#9911yescharacteristic
150C2×C4×C61#9911yescharacteristic
151C2×C4×C61#10412yes—
152C2×Dic61—44no—
153C2×Dic61—82no—
154C2×Dic61—82no—
155#92418—82no—
156#11754—611nomaximal, Sylow
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—41no—
162C22×C61—41no—
163C22×C61—42no—
164C22×C61—42no—
165C22×C61—41no—
166C22×C61—41no—
167C22×C61—42no—
168C22×C61—81no—
169C22×C61—81no—
170C22×C61—81no—
171C22×C61—81no—
172C22×C61—81no—
173C22×C61C2×#9911yescharacteristic
174C22×C61#41711yescharacteristic
175C22×C61C2×#10411yescharacteristic
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—22no—
179C4×C61—41no—
180C4×C61—41no—
181C4×C61—41no—
182C4×C61—41no—
183C4×C61—81no—
184C4×C61—81no—
185C4×C61—81no—
186Dic61—84no—
187#2598D6111yescharacteristic
188#1740—611no—
189#1464—611no—
190#1352—612no—
191#1343—612no—
192C2×C61—21no—
193C2×C61—21no—
194C2×C61—41no—
195C2×C61—41no—
196C2×C61—82no—
197C2×C61#150411yescharacteristic
198C2×C61#212011yescharacteristic
199C2×C61#212811yescharacteristic
200C2×C4×D4Dic6111yescharacteristic, complemented
201C2×#117—1221no—
202C2×#117C2×D6111yescharacteristic
203#432—1221no—
204#400Dic6111yescharacteristic, complemented
205#399C2×D6111yescharacteristic
206#394Dic6111yescharacteristic, complemented
207#392C2×D6111yescharacteristic
208#388—21no—
209#386—21no—
210#386—21no—
211#385—21no—
212#383Dic6111yescharacteristic, complemented
213#342—612no—
214#342—1221no—
215#324—1221no—
216#323—612no—
217C61#1175411yescharacteristic, complemented, Sylow
218C22×D4—21no—
219C22×D4C2×Dic6111yescharacteristic
220C23×C4—21no—
221C23×C4C22⋊D6111yescharacteristic
222C4×D4—41no—
223C4×D4—41no—
224#117—41no—
225#117—41no—
226#117—1224no—
227#117—2442no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—2441no—
230C2×C4⋊C4C2×Dic6111yescharacteristic
231C2×C4⋊C4C2×Dic6111yescharacteristic
232C2×C4⋊C4C22⋊D6111yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—1222no—
239C2×C22⋊C4C22⋊D6111yescharacteristic
240C2×C22⋊C4C22×D6111yescharacteristic
241C2×C42—2441no—
242C2×C42C2×Dic6111yescharacteristic
243C2×C42C22⋊D6111yescharacteristic
244C4⋊C8—2441no—
245C22⋊C8—1222no—
246#97—2441no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95C2×Dic6111yescharacteristic
252#95C2×Dic6111yescharacteristic
253C24—21no—
254C24C2×C22⋊D6111yescharacteristic
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—81no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—42no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4—2442no—
273C22×C4C22⋊Dic6111yescharacteristic
274C22×C4C22⋊Dic6111yescharacteristic
275C22×C4C22⋊Dic6111yescharacteristic
276C22×C4C22⋊Dic6111yescharacteristic
277C22×C4C22×Dic6111yescharacteristic
278C22×C4C2×C22⋊D6111yescharacteristic
279C2×C8—2442no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—81no—
284C4⋊C4—2444no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C22⋊C4—1224no—
294C22⋊C4—2442no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—2444no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23C2×C22⋊Dic6111yescharacteristic
312C23#1451980211yescharacteristic
313C23#1451980711yescharacteristic
314D4—42no—
315D4—42no—
316D4—82no—
317D4—82no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—84no—
344C2×C4—4882no—
345C2×C4—4882no—
346C2×C4—2444no—
347C2×C4#1451969211yescharacteristic
348C2×C4#1451969211yescharacteristic
349C2×C4#1451970412yes—
350C8—4882no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—41no—
357C22—42no—
358C22—42no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22C2×#1451969211yescharacteristic
368C22#1683976311yescharacteristic
369C22C2×#1451970411yescharacteristic
370C4—21no—
371C4—21no—
372C4—22no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—4884no—
381C2—21no—
382C2—21no—
383C2—41no—
384C2—41no—
385C2—82no—
386C2#2144432311yescharacteristic
387C2#2144746011yescharacteristic
388C2#2144749111yescharacteristic
subgroupG/H
centerC22C2×#14519692
commutatorC22×C4×C61C22×C4
FrattiniC2×C22⋊C4C22×D61
FittingC61×#2598C2
socleC22×C61C2×#99
prime radicalC61×#392C22
cosocleC2×C22⋊C4×C61C23
hypercenter#2598D61
hypocenterC61#11754
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2598

Complements

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

NH ≅ G/Ntype
1. #21445552C2semidirect
2. #21444153C2semidirect
3. #21443843C2semidirect
12. C2×C4×D4×C61C4semidirect
14. C61×#400C4semidirect
16. C61×#394C4semidirect
22. C61×#383C4semidirect
200. C2×C4×D4Dic61semidirect
204. #400Dic61semidirect
206. #394Dic61semidirect
212. #383Dic61semidirect
217. C61#11754semidirect

Maximal subgroups

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

subgroupindex
#214455522
#214441532
#214438432
#214438162
C61×#25982
#1175461