Finite Group Databasebeta

← #29575773

Subgroups — #29575773

Order 15616.

630 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 147 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#21454158C211yesmaximal, characteristic
2#21445887C211yesmaximal, characteristic
3C2×#16839498C212yesmaximal
4#21443698C212yesmaximal
5C2×C61×#382C211yesmaximal, characteristic
6C2×#14519756C2211yescharacteristic
7C2×C22⋊#92418C2212yes—
8#16839668C412yescomplemented
9#16839660C412yescomplemented
10#16839646—22no—
11#16839635—21no—
12#16839625—21no—
13#16839622C2211yescharacteristic
14#16839517—22no—
15#16839498—24no—
16C22×C4⋊C4×C61—21no—
17C22×C42×C61C2211yescharacteristic
18C61×#382—24no—
19C2×#95×C61—21no—
20C2×#95×C61C2212yes—
21C22×Dic122C2×C411yescharacteristic
22C22×#92418—22no—
23#14519756—22no—
24#14519756—22no—
25C2×C4⋊Dic61C2×C411yescharacteristic
26C2×C4⋊Dic61C2×C412yes—
27C2×#11758196—22no—
28C2×#11758196—21no—
29C2×C22⋊Dic61—21no—
30C2×C22⋊Dic61—21no—
31C22⋊#92418—22no—
32C22⋊#92418—22no—
33#14519684—22no—
34#14519684—22no—
35#14519684—22no—
36#14519684—21no—
37C23×C4×C61—22no—
38C23×C4×C61D411yescharacteristic
39C23×C4×C61D411yescharacteristic
40C23×C4×C61C2311yescharacteristic
41C2×C4⋊C4×C61—24no—
42C2×C4⋊C4×C61—42no—
43C2×C4⋊C4×C61—42no—
44C2×C42×C61—22no—
45C2×C42×C61—22no—
46C2×C42×C61C2×C412yes—
47C2×C42×C61D412yes—
48#95×C61—24no—
49#95×C61—22no—
50#95×C61D414yes—
51C24×C61C2×D411yescharacteristic
52C22×C4×C61—21no—
53C22×C4×C61—21no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—21no—
57C22×C4×C61—21no—
58C22×C4×C61—24no—
59C22×C4×C61—42no—
60C22×C4×C61—42no—
61C22×C4×C61—44no—
62C22×C4×C61—44no—
63C22×C4×C61—42no—
64C22×C4×C61—42no—
65C22×C4×C61C22⋊C412yes—
66C22×C4×C61C22⋊C412yes—
67C22×C4×C61SD1612yes—
68C22×C4×C61Q1612yes—
69C22×C4×C61C22×C411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61C2×D412yes—
72C22×C4×C61#4112yes—
73C22×Dic61—21no—
74C22×Dic61—22no—
75C22×Dic61—21no—
76C2×Dic122—22no—
77C2×Dic122—42no—
78C4⋊C4×C61—48no—
79C42×C61—24no—
80C42×C61—24no—
81C22⋊Dic61—41no—
82C22⋊Dic61—41no—
83C22⋊Dic61—41no—
84C22⋊Dic61—41no—
85C4⋊Dic61—22no—
86C4⋊Dic61—42no—
87#11758196—44no—
88#11758196—42no—
89C2×#92418—44no—
90C2×#92418—42no—
91C23×C61—22no—
92C23×C61—22no—
93C23×C61C2×C22⋊C411yescharacteristic
94C23×C61#12011yescharacteristic
95C23×C61#12311yescharacteristic
96C23×C61C2×SD1611yescharacteristic
97C23×C61C2×Q1611yescharacteristic
98C23×C61AGL(1,ℤ/8)11yescharacteristic
99C23×C61#13711yescharacteristic
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61—22no—
103C2×C4×C61—22no—
104C2×C4×C61—22no—
105C2×C4×C61—22no—
106C2×C4×C61—22no—
107C2×C4×C61—22no—
108C2×C4×C61—22no—
109C2×C4×C61—22no—
110C2×C4×C61—22no—
111C2×C4×C61—42no—
112C2×C4×C61—42no—
113C2×C4×C61—42no—
114C2×C4×C61—42no—
115C2×C4×C61—44no—
116C2×C4×C61—44no—
117C2×C4×C61—44no—
118C2×C4×C61—84no—
119C2×C4×C61—84no—
120C2×C4×C61#10314yes—
121C2×C4×C61C4×D412yes—
122C2×C4×C61#12112yes—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—42no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130Dic122—82no—
131#92418—82no—
132#9152—611nomaximal, Sylow
133C22×C61—21no—
134C22×C61—21no—
135C22×C61—21no—
136C22×C61—21no—
137C22×C61—22no—
138C22×C61—22no—
139C22×C61—22no—
140C22×C61—22no—
141C22×C61#38611yescharacteristic
142C22×C61#41012yes—
143C22×C61C2×#10311yescharacteristic
144C22×C61#41711yescharacteristic
145C22×C61#42112yes—
146C22×C61#45011yescharacteristic
147C22×C61#45111yescharacteristic
148C22×C61#48311yescharacteristic
149C22×C61#48411yescharacteristic
150C4×C61—24no—
151C4×C61—22no—
152C4×C61—22no—
153C4×C61—42no—
154C4×C61—84no—
155Dic61—81no—
156Dic61—81no—
157Dic61—82no—
158#2855—611no—
159C2×#382D6111yescharacteristic
160#1817—611no—
161C2×#328—612no—
162#1294—612no—
163C2×C61—21no—
164C2×C61—21no—
165C2×C61—21no—
166C2×C61—21no—
167C2×C61#151212yes—
168C2×C61#151812yes—
169C2×C61#187211yescharacteristic
170C2×C61#189711yescharacteristic
171C2×C61#189911yescharacteristic
172C2×C22⋊Q8—611no—
173C22×C4⋊C4—21no—
174C22×C42C2×D6111yescharacteristic
175C2×C22⋊C8—612no—
176#391—612no—
177#389—612no—
178#387—1222no—
179#385—1221no—
180#382—24no—
181#381—1221no—
182#380—611no—
183C2×#95—21no—
184C2×#95C2×D6112yes—
185#336—1222no—
186#328—1224no—
187C61#915211yescharacteristic, complemented, Sylow
188C22×Q8—611no—
189C23×C4—22no—
190C23×C4C22⋊D6111yescharacteristic
191C23×C4C22⋊D6111yescharacteristic
192C23×C4C22×D6111yescharacteristic
193C22×C8—1222no—
194C22⋊Q8—1222no—
195C22⋊Q8—1222no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—611no—
200C2×C4⋊C4—612no—
201C2×C4⋊C4—1222no—
202C2×C4⋊C4—1221no—
203C2×C22⋊C4—1221no—
204C2×C22⋊C4—1221no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C4×D6112yes—
208C2×C42D24412yes—
209C22⋊C8—1222no—
210C22⋊C8—1222no—
211#95—24no—
212#95—22no—
213#95—1222no—
214#95—1222no—
215#95—1222no—
216#95—1221no—
217#95C22⋊D6114yes—
218C24C2×C22⋊D6111yescharacteristic
219C2×Q8—1222no—
220C2×Q8—2442no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—1222no—
235C22×C4—1221no—
236C22×C4—1221no—
237C22×C4#1175819812yes—
238C22×C4#1175819812yes—
239C22×C4#1175820012yes—
240C22×C4#1175820212yes—
241C22×C4C2×C4×D6111yescharacteristic
242C22×C4C2×D24411yescharacteristic
243C22×C4#1175822212yes—
244C22×C4C2×C22⋊D6112yes—
245C2×C8—2444no—
246C2×C8—2442no—
247C4⋊C4—48no—
248C4⋊C4—1222no—
249C4⋊C4—2442no—
250C4⋊C4—2444no—
251C4⋊C4—2442no—
252C22⋊C4—2441no—
253C22⋊C4—2441no—
254C22⋊C4—2441no—
255C22⋊C4—2441no—
256C42—24no—
257C42—24no—
258C23—22no—
259C23—22no—
260C23C2×#1175819811yescharacteristic
261C23#1451974611yescharacteristic
262C23#1451976011yescharacteristic
263C23C2×#1175820011yescharacteristic
264C23C2×#1175820211yescharacteristic
265C23#1451980311yescharacteristic
266C23#1451980811yescharacteristic
267Q8—4882no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—44no—
286C2×C4—84no—
287C2×C4—84no—
288C2×C4—2442no—
289C2×C4—2442no—
290C2×C4—2442no—
291C2×C4—2442no—
292C2×C4—2442no—
293C2×C4—2441no—
294C2×C4—2441no—
295C2×C4#1451970014yes—
296C2×C4C4×C22⋊D6112yes—
297C2×C4#1451975112yes—
298C8—4882no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—22no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22#1683964111yescharacteristic
308C22#1683973512yes—
309C22C2×#1451970011yescharacteristic
310C22#1683976111yescharacteristic
311C22#1683978012yes—
312C22#1683991311yescharacteristic
313C22#1683991811yescharacteristic
314C22#1684009211yescharacteristic
315C22#1684009711yescharacteristic
316C4—22no—
317C4—22no—
318C4—24no—
319C4—42no—
320C4—84no—
321C4—4882no—
322C4—4881no—
323C4—4881no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2#2144435512yes—
329C2#2144438212yes—
330C2#2144612611yescharacteristic
331C2#2144626711yescharacteristic
332C2#2144627711yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC2×C61×#382C2
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenterC2×#382D61
hypocenterC61#9152
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#382

Complements

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

NH ≅ G/Ntype
8. #16839668C4semidirect
9. #16839660C4semidirect
187. C61#9152semidirect

Maximal subgroups

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

subgroupindex
#214541582
#214458872
C2×#168394982
#214436982
C2×C61×#3822
#915261