Finite Group Databasebeta

← #29568800

Subgroups — #29568800

Order 15616.

740 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 95 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#21446955C212yesmaximal, complemented
2#21446901C211yesmaximal, characteristic, complemented
3#21443640C211yesmaximal, characteristic, complemented
4#21443640C212yesmaximal, complemented
5C61×#2679C211yesmaximal, characteristic
6C22×#11758200—21no—
7C2×#14519777C2211yescharacteristic, complemented
8C2×#14519700—21no—
9C2×#14519698—22no—
10#16839524—22no—
11#16839517C2212yescomplemented
12#16839517C2211yescharacteristic, complemented
13C2×#120×C61—21no—
14C2×#120×C61—22no—
15C61×#394—21no—
16C61×#394—22no—
17C61×#390C2212yes—
18C61×#390C2211yescharacteristic
19C22×Dic122—21no—
20C2×#11758200—42no—
21C2×#11758200—42no—
22C2×#11758200—42no—
23C22×#92418—21no—
24C22×#92418—22no—
25#14519777—42no—
26C2×C4⋊Dic61—22no—
27#14519700—42no—
28#14519698—44no—
29C25×C61—21no—
30C22×D4×C61—22no—
31C22×D4×C61—21no—
32C22×D4×C61D411yescharacteristic
33C22×D4×C61D412yes—
34#120×C61—42no—
35#120×C61—44no—
36#120×C61—42no—
37#120×C61—44no—
38C2×C4⋊C4×C61D411yescharacteristic
39C2×C4⋊C4×C61D412yes—
40C2×C22⋊C4×C61—21no—
41C2×C22⋊C4×C61—22no—
42C2×C22⋊C4×C61—22no—
43C2×C22⋊C4×C61—22no—
44C2×C22⋊C4×C61—22no—
45C2×C42×C61C2311yescharacteristic
46#95×C61—21no—
47C24×C61—21no—
48C24×C61—22no—
49C24×C61—42no—
50C24×C61—44no—
51C24×C61—81no—
52C24×C61—82no—
53C24×C61—81no—
54C2×D4×C61—21no—
55C2×D4×C61—22no—
56C2×D4×C61—42no—
57C2×D4×C61—42no—
58C2×D4×C61—41no—
59C2×D4×C61—41no—
60C2×D4×C61—44no—
61C2×D4×C61—44no—
62C2×D4×C61—44no—
63C2×D4×C61—81no—
64C2×D4×C61—82no—
65C2×D4×C61D812yes—
66C2×D4×C61SD1614yescomplemented
67C22×C4×C61—22no—
68C22×C4×C61—21no—
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D412yes—
71C22×Dic61—41no—
72C2×Dic122—22no—
73C2×Dic122—41no—
74C2×Dic122—41no—
75C2×Dic122—41no—
76C2×Dic122—41no—
77C2×Dic122—41no—
78C4⋊C4×C61—42no—
79C4⋊C4×C61—41no—
80C22⋊C4×C61—42no—
81C22⋊C4×C61—44no—
82C22⋊C4×C61—42no—
83C22⋊C4×C61—42no—
84C22⋊C4×C61—42no—
85C22⋊C4×C61—42no—
86C22⋊C4×C61—42no—
87C22⋊C4×C61—42no—
88C42×C61—41no—
89#11758200—48no—
90C4⋊Dic61—24no—
91C4⋊Dic61—42no—
92C2×#92418—22no—
93C2×#92418—41no—
94C2×#92418—41no—
95C2×#92418—42no—
96C2×#92418—42no—
97C2×#92418—42no—
98C23×C61—22no—
99C23×C61—24no—
100C23×C61—42no—
101C23×C61—42no—
102C23×C61—42no—
103C23×C61—41no—
104C23×C61—42no—
105C23×C61—42no—
106C23×C61—42no—
107C23×C61—42no—
108C23×C61—41no—
109C23×C61—42no—
110C23×C61—84no—
111C23×C61—88no—
112C23×C61—84no—
113C23×C61—88no—
114C23×C61—84no—
115C23×C61#12011yescharacteristic
116D4×C61—42no—
117D4×C61—44no—
118D4×C61—81no—
119D4×C61—82no—
120D4×C61—84no—
121D4×C61—88no—
122C2×C4×C61—21no—
123C2×C4×C61—22no—
124C2×C4×C61—22no—
125C2×C4×C61—21no—
126C2×C4×C61—42no—
127C2×C4×C61—42no—
128C2×C4×C61—42no—
129C2×C4×C61—41no—
130C2×C4×C61—42no—
131C2×C4×C61C2×D811yescharacteristic
132C2×C4×C61C2×SD1612yes—
133C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C61AGL(1,ℤ/8)12yes—
135C2×Dic61—42no—
136C2×Dic61—81no—
137C2×Dic61—81no—
138Dic122—42no—
139Dic122—42no—
140Dic122—81no—
141Dic122—81no—
142#92418—42no—
143#92418—82no—
144#7750—611nomaximal, Sylow
145C22×C61—42no—
146C22×C61—44no—
147C22×C61—44no—
148C22×C61—44no—
149C22×C61—44no—
150C22×C61—816no—
151C22×C61—82no—
152C22×C61—82no—
153C22×C61—82no—
154C22×C61—81no—
155C22×C61—82no—
156C22×C61—81no—
157C22×C61—82no—
158C22×C61#44711yescharacteristic
159C22×C61#45012yes—
160C22×C61#45311yescharacteristic
161C22×C61#45512yes—
162C22×C61B(4,𝔽2)11yescharacteristic
163C4×C61—21no—
164C4×C61—21no—
165C4×C61—22no—
166C4×C61—22no—
167C4×C61—81no—
168C4×C61—82no—
169Dic61—82no—
170#2679D6111yescharacteristic
171#2023—612no—
172#2013—611no—
173#1270—612no—
174#1270—611no—
175C2×C61—82no—
176C2×C61—84no—
177C2×C61#159511yescharacteristic
178C2×C61#160112yes—
179C2×C61#162911yescharacteristic
180C2×C61#163912yes—
181C2×C61#165911yescharacteristic
182C22×SD16—1221no—
183C2×C4⋊Q8—611no—
184C2×#120—21no—
185C2×#120—22no—
186C2×#103—1221no—
187C2×#102—1222no—
188#394—22no—
189#394—21no—
190#390C2×D6111yescharacteristic
191#390C2×D6112yes—
192#340—1222no—
193#336—611no—
194#336—612no—
195C61#775011yescharacteristic, complemented, Sylow
196C25—21no—
197C22×Q8—1221no—
198C22×D4—22no—
199C22×D4—21no—
200C22×D4C22⋊D6111yescharacteristic
201C22×D4C22⋊D6112yes—
202C2×SD16—2442no—
203C2×SD16—2442no—
204C2×SD16—2442no—
205C22×C8—1221no—
206C22×C8—1222no—
207C4⋊Q8—2442no—
208#120—42no—
209#120—44no—
210#120—42no—
211#120—44no—
212C2×C4⋊C4—1222no—
213C2×C4⋊C4C22⋊D6112yes—
214C2×C4⋊C4C22⋊D6111yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C42C22×D6111yescharacteristic
221#103—2442no—
222#102—2444no—
223#95—21no—
224C24—21no—
225C24—22no—
226C24—42no—
227C24—44no—
228C24—82no—
229C24—81no—
230C24—81no—
231C2×Q8—1222no—
232C2×Q8—2441no—
233C2×Q8—2441no—
234C2×Q8—2441no—
235C2×Q8—2441no—
236C2×Q8—2441no—
237C2×D4—21no—
238C2×D4—22no—
239C2×D4—42no—
240C2×D4—41no—
241C2×D4—41no—
242C2×D4—42no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—44no—
246C2×D4—82no—
247C2×D4—81no—
248C2×D4#1175819912yes—
249C2×D4#1175820014yescomplemented
250C22×C4—22no—
251C22×C4—21no—
252C22×C4—2441no—
253C22×C4C2×C22⋊D6111yescharacteristic
254C22×C4C2×C22⋊D6112yes—
255SD16—2448no—
256C2×C8—1222no—
257C2×C8—2442no—
258C2×C8—2442no—
259C2×C8—2442no—
260C2×C8—2441no—
261C2×C8—2441no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—1224no—
265C4⋊C4—2442no—
266C22⋊C4—42no—
267C22⋊C4—44no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C42—41no—
275C23—22no—
276C23—24no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—41no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—41no—
286C23—42no—
287C23—84no—
288C23—88no—
289C23—84no—
290C23—88no—
291C23—84no—
292C23#1451974611yescharacteristic
293Q8—4881no—
294Q8—4881no—
295Q8—2442no—
296Q8—2442no—
297D4—42no—
298D4—44no—
299D4—81no—
300D4—82no—
301D4—84no—
302D4—88no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—41no—
312C2×C4—4881no—
313C2×C4—4881no—
314C2×C4—2442no—
315C2×C4C2×#1175819911yescharacteristic
316C2×C4C2×#1175820012yes—
317C2×C4#1451980311yescharacteristic
318C2×C4#1451980312yes—
319C8—4882no—
320C8—2442no—
321C22—44no—
322C22—42no—
323C22—44no—
324C22—44no—
325C22—44no—
326C22—816no—
327C22—82no—
328C22—81no—
329C22—81no—
330C22—82no—
331C22—82no—
332C22—82no—
333C22—82no—
334C22#1683989611yescharacteristic
335C22#1683991312yes—
336C22#1683993011yescharacteristic
337C22#1683994012yes—
338C22#1683994911yescharacteristic
339C4—21no—
340C4—21no—
341C4—22no—
342C4—22no—
343C4—81no—
344C4—82no—
345C4—4882no—
346C2—84no—
347C2—82no—
348C2#2144473711yescharacteristic
349C2#2144477112yes—
350C2#2144493511yescharacteristic
351C2#2144499812yes—
352C2#2144512511yescharacteristic
subgroupG/H
centerC23#14519746
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
Fitting, prime radicalC61×#2679C2
socleC23×C61#120
hypercenter#2679D61
hypocenterC61#7750
LCSG ▷ C2×C42×C61 ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2679

Complements

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

NH ≅ G/Ntype
1. #21446955C2semidirect
2. #21446901C2semidirect
3. #21443640C2semidirect
4. #21443640C2semidirect
7. C2×#14519777C22semidirect
11. #16839517C22semidirect
12. #16839517C22semidirect
66. C2×D4×C61SD16semidirect
195. C61#7750semidirect
249. C2×D4#11758200semidirect

Maximal subgroups

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

subgroupindex
#214469552
#214469012
#214436402
C61×#26792
#775061