Finite Group Databasebeta

← #30595566

Subgroups — #30595566

Order 15616.

874 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 102 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#21461546C212yesmaximal, complemented
2#21460636C212yesmaximal, complemented
3#21460635C212yesmaximal, complemented
4#390×D61C211yesmaximal, characteristic
5C2×#14519834—22no—
6C2×#14519832—22no—
7#16840628C2214yescomplemented
8C22×D4×D61C411yescharacteristic
9C2×C4⋊C4×D61C411yescharacteristic
10C2×C22⋊C4×D61—22no—
11C2×C42×D61C2211yescharacteristic
12#16839665C2211yescharacteristic
13#16839664C411yescharacteristic
14#16839663C2211yescharacteristic
15#16839662—22no—
16C61×#390C411yescharacteristic
17C22×#92420—24no—
18#14519834—28no—
19#14519832—44no—
20C24×D61—21no—
21C22×D244C2×C411yescharacteristic
22C22×C22⋊D61—21no—
23C2×D4×D61—22no—
24C2×D4×D61—22no—
25C2×D4×D61—42no—
26C2×D4×D61—42no—
27C22×C4×D61—21no—
28C22×C4×D61C2×C411yescharacteristic
29C22×C4×D61D412yes—
30C2×C4⋊Dic61C2×C411yescharacteristic
31C2×#11758196—21no—
32C4⋊C4×D61—42no—
33C2×C22⋊Dic61—22no—
34C2×#11758198—22no—
35C2×#11758198—22no—
36C22⋊C4×D61—48no—
37C2×C4×Dic61D412yes—
38C2×C4×Dic61C2311yescharacteristic
39C42×D61—42no—
40C22×D4×C61C2×C411yescharacteristic
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C22⋊C4×C61—22no—
43C2×C42×C61C2×C411yescharacteristic
44C24×C61—21no—
45C23×D61—24no—
46C23×D61—21no—
47C23×D61—42no—
48C23×D61—44no—
49C23×D61—42no—
50C23×D61—44no—
51C23×D61C22⋊C411yescharacteristic
52C2×D4×C61—21no—
53C2×D4×C61—41no—
54C2×D4×C61—41no—
55C2×D4×C61M1612yes—
56C22×C4×C61—21no—
57C22×C4×C61C22⋊C412yes—
58C22×C4×C61C22×C411yescharacteristic
59C2×C22⋊D61—24no—
60C2×C22⋊D61—42no—
61C2×C22⋊D61—42no—
62C22×Dic61—21no—
63C22×Dic61C22⋊C411yescharacteristic
64C22×Dic61C2×D412yes—
65D4×D61—48no—
66D4×D61—48no—
67C2×D244—21no—
68C2×D244—41no—
69C2×D244—41no—
70C2×D244M1612yes—
71C2×C4×D61—22no—
72C2×C4×D61—21no—
73C2×C4×D61—21no—
74C2×C4×D61—22no—
75C2×C4×D61—24no—
76C2×C4×D61—24no—
77C2×C4×D61—42no—
78C2×C4×D61—44no—
79C2×C4×D61—42no—
80C2×#92420—28no—
81C2×#92420—44no—
82C2×#92420—44no—
83C4⋊C4×C61—41no—
84C22⋊C4×C61—44no—
85C42×C61—41no—
86C22⋊Dic61—44no—
87#11758198—44no—
88#11758198—44no—
89C4⋊Dic61—41no—
90#11758196—42no—
91C4×Dic61—24no—
92C4×Dic61—41no—
93C23×C61—22no—
94C23×C61—41no—
95C23×C61—41no—
96C23×C61—42no—
97C23×C61—42no—
98C23×C61C2×C22⋊C411yescharacteristic
99C22×D61—21no—
100C22×D61—21no—
101C22×D61—22no—
102C22×D61—22no—
103C22×D61—48no—
104C22×D61—416no—
105C22×D61—41no—
106C22×D61—41no—
107C22×D61—42no—
108C22×D61—42no—
109C22×D61—84no—
110C22×D61—88no—
111C22×D61—84no—
112C22×D61#10012yes—
113C22×D61#10414yes—
114D4×C61—42no—
115D4×C61—42no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—22no—
119C2×C4×C61—22no—
120C2×C4×C61—22no—
121C2×C4×C61—41no—
122C2×C4×C61—42no—
123C2×C4×C61C2×M1611yescharacteristic
124C2×C4×C61C2×M1611yescharacteristic
125C22⋊D61—48no—
126C2×Dic61—21no—
127C2×Dic61—22no—
128C2×Dic61—21no—
129C2×Dic61—22no—
130C2×Dic61—41no—
131C2×Dic61—42no—
132C2×Dic61#9912yes—
133C2×Dic61AGL(1,ℤ/8)14yes—
134D244—42no—
135D244—42no—
136C4×D61—24no—
137C4×D61—24no—
138C4×D61—48no—
139C4×D61—84no—
140#92420—48no—
141#7626—611nomaximal, Sylow
142C22×C61—42no—
143C22×C61—44no—
144C22×C61—81no—
145C22×C61—82no—
146C22×C61—81no—
147C22×C61#40711yescharacteristic
148C22×C61C2×#9911yescharacteristic
149C22×C61C2×#10011yescharacteristic
150C22×C61#41812yes—
151C22×C61C2×#10412yes—
152C2×D61—22no—
153C2×D61—22no—
154C2×D61—24no—
155C2×D61—24no—
156C2×D61—42no—
157C2×D61—44no—
158C2×D61—816no—
159C2×D61—81no—
160C2×D61—82no—
161C2×D61—81no—
162C2×D61#32914yes—
163C4×C61—21no—
164C4×C61—21no—
165C4×C61—42no—
166C4×C61—81no—
167Dic61—22no—
168Dic61—24no—
169Dic61—81no—
170C2×#390—611no—
171#1749—612no—
172#1270—612no—
173#1270—612no—
174C2×C61—82no—
175C2×C61#146211yescharacteristic
176C2×C61#149012yes—
177C2×C61#150712yes—
178C2×C61C2×#32911yescharacteristic
179C2×C61#155511yescharacteristic
180D61—24no—
181D61—82no—
182C23×D4—611no—
183C22×C4⋊C4—611no—
184C22×C22⋊C4—1222no—
185C22×C42—611no—
186C2×C22⋊C8—1222no—
187C2×#97—1222no—
188#390—611no—
189#390—611no—
190#390—611no—
191#390—1222no—
192#390#692311yescharacteristic
193#336—614no—
194C61#762611yescharacteristic, complemented, Sylow
195C25—1221no—
196C22×D4—611no—
197C22×D4—1221no—
198C22×D4—1222no—
199C22×D4—1222no—
200C22×D4—2442no—
201C22×D4—2442no—
202C22×D4C2×#692311yescharacteristic
203C23×C4—611no—
204C23×C4—612no—
205C23×C4—1221no—
206C22×C8—1224no—
207C2×C4⋊C4—611no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—2442no—
210C2×C4⋊C4C2×#692311yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—1222no—
213C2×C22⋊C4—1222no—
214C2×C22⋊C4—1222no—
215C2×C22⋊C4—2448no—
216C2×C42—612no—
217C2×C42—611no—
218C2×C42—2442no—
219C2×C42C2×#692311yescharacteristic
220C22⋊C8—1228no—
221#97—2444no—
222C24—21no—
223C24—611no—
224C24—1224no—
225C24—1221no—
226C24—2442no—
227C24—2444no—
228C24—2444no—
229C24—2442no—
230C2×D4—21no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—612no—
234C2×D4—1224no—
235C2×D4—1221no—
236C2×D4—2441no—
237C2×D4—2441no—
238C2×D4—2442no—
239C2×D4—2448no—
240C2×D4—2448no—
241C2×D4—2442no—
242C2×D4#1175821312yes—
243C22×C4—21no—
244C22×C4—612no—
245C22×C4—611no—
246C22×C4—1222no—
247C22×C4—1224no—
248C22×C4—1222no—
249C22×C4—1224no—
250C22×C4—1221no—
251C22×C4—1221no—
252C22×C4—1221no—
253C22×C4—2442no—
254C22×C4—2442no—
255C22×C4—2444no—
256C22×C4C22⋊#692312yes—
257C22×C4C22×#692311yescharacteristic
258C2×C8—1228no—
259C2×C8—2444no—
260C2×C8—2444no—
261C4⋊C4—41no—
262C4⋊C4—2441no—
263C4⋊C4—2442no—
264C22⋊C4—44no—
265C22⋊C4—2444no—
266C22⋊C4—2444no—
267C22⋊C4—2444no—
268C42—41no—
269C42—1224no—
270C42—2441no—
271C23—22no—
272C23—41no—
273C23—41no—
274C23—42no—
275C23—42no—
276C23—612no—
277C23—614no—
278C23—1221no—
279C23—1221no—
280C23—1222no—
281C23—1222no—
282C23—4884no—
283C23—4884no—
284C23—4888no—
285C23—2448no—
286C23—24416no—
287C23—2441no—
288C23—2441no—
289C23—2442no—
290C23—2442no—
291C23C2×C22⋊#692311yescharacteristic
292D4—42no—
293D4—42no—
294D4—2442no—
295D4—2442no—
296D4—2448no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—612no—
305C2×C4—614no—
306C2×C4—1224no—
307C2×C4—1224no—
308C2×C4—1221no—
309C2×C4—1222no—
310C2×C4—1221no—
311C2×C4—1222no—
312C2×C4—4884no—
313C2×C4—2448no—
314C2×C4—2441no—
315C2×C4—2442no—
316C2×C4D61⋊M1611yescharacteristic
317C2×C4C2×#1175821311yescharacteristic
318C8—2448no—
319C22—42no—
320C22—44no—
321C22—81no—
322C22—81no—
323C22—82no—
324C22—614no—
325C22—1222no—
326C22—1222no—
327C22—1224no—
328C22—1224no—
329C22—48816no—
330C22—4881no—
331C22—4881no—
332C22—4882no—
333C22—2442no—
334C22—2444no—
335C22#1684073311yescharacteristic
336C22D61⋊#9911yescharacteristic
337C22D61⋊#10011yescharacteristic
338C22#1684075412yes—
339C22D61⋊#10412yes—
340C4—21no—
341C4—21no—
342C4—42no—
343C4—81no—
344C4—1222no—
345C4—1224no—
346C4—4881no—
347C2—82no—
348C2—1224no—
349C2—4882no—
350C2#2146106511yescharacteristic
351C2#2146115912yes—
352C2#2146121412yes—
353C2D61⋊#32911yescharacteristic
354C2#2146134711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radicalC22×D4×D61C4
cosocleC2×C4×Dic61C23
hypercenter#390#6923
hypocenterC61#7626
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #21461546C2semidirect
2. #21460636C2semidirect
3. #21460635C2semidirect
7. #16840628C22semidirect
194. C61#7626semidirect

Maximal subgroups

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

subgroupindex
#214615462
#214606362
#214606352
#390×D612
#762661