Finite Group Databasebeta

← #29577782

Subgroups — #29577782

Order 15616.

578 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 135 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#21450136C211yesmaximal, characteristic
2#21445910C212yesmaximal, complemented
3#21445807C211yesmaximal, characteristic
4#21443687C212yesmaximal, complemented
5C61×#2972C211yesmaximal, characteristic
6C22×C4⋊Dic61C2211yescharacteristic
7C2×#14519692—21no—
8#16839701C412yescomplemented
9#16839698C412yescomplemented
10#16839660—21no—
11#16839642—21no—
12#16839634—21no—
13#16839622—21no—
14#16839622C2212yes—
15#16839622C2211yescharacteristic
16#16839529—21no—
17#16839525—24no—
18C2×C4×D4×C61C2211yescharacteristic
19C8×D4×C61—24no—
20C61×#431—21no—
21C2×C22⋊C8×C61—21no—
22C2×C22⋊C8×C61C2212yes—
23C23×Dic61—21no—
24C2×C4⋊Dic61—22no—
25C2×C4⋊Dic61—22no—
26C2×C4⋊Dic61—21no—
27C2×C4⋊Dic61—21no—
28C2×C4⋊Dic61—22no—
29C2×C4⋊Dic61C2×C412yes—
30C2×#11758196—22no—
31C2×C22⋊Dic61—21no—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61—22no—
34#14519692—42no—
35#14519684—22no—
36#14519684—22no—
37#14519684—22no—
38#14519684C2×C412yes—
39C22×D4×C61D411yescharacteristic
40C23×C4×C61D411yescharacteristic
41C23×C4×C61C2311yescharacteristic
42C22×C8×C61—22no—
43C22×C8×C61D414yes—
44C4×D4×C61—22no—
45C4×D4×C61—22no—
46C2×C4⋊C4×C61C2×C411yescharacteristic
47C2×C22⋊C4×C61C2×C411yescharacteristic
48C2×C22⋊C4×C61D411yescharacteristic
49C2×C42×C61D411yescharacteristic
50C4⋊C8×C61—22no—
51C22⋊C8×C61—22no—
52C22⋊C8×C61—22no—
53C22⋊C8×C61—22no—
54C4×C8×C61—22no—
55C24×C61C22⋊C411yescharacteristic
56C24×C61C2×D411yescharacteristic
57C2×D4×C61—22no—
58C2×D4×C61—22no—
59C2×D4×C61SD1612yes—
60C2×D4×C61Q1612yes—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—42no—
64C22×C4×C61—42no—
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22×C411yescharacteristic
69C22×C4×C61C2×D412yes—
70C22×C4×C61C2×D411yescharacteristic
71C22×C4×C61#4112yes—
72C22×Dic61—22no—
73C22×Dic61—22no—
74C22×Dic61—42no—
75C22×Dic61—42no—
76C22×Dic61—82no—
77C2×C8×C61—24no—
78C2×C8×C61—44no—
79C2×C8×C61—82no—
80C4⋊C4×C61—22no—
81C22⋊C4×C61—22no—
82C22⋊C4×C61—22no—
83C42×C61—21no—
84C42×C61—21no—
85C22⋊Dic61—42no—
86C22⋊Dic61—42no—
87C22⋊Dic61—42no—
88C22⋊Dic61—82no—
89C4⋊Dic61—24no—
90C4⋊Dic61—42no—
91C4⋊Dic61—41no—
92C4⋊Dic61—41no—
93C4⋊Dic61—44no—
94#11758196—82no—
95C23×C61—21no—
96C23×C61—21no—
97C23×C61—41no—
98C23×C61—41no—
99C23×C61—41no—
100C23×C61—41no—
101C23×C61#10312yes—
102C23×C61C2×C22⋊C411yescharacteristic
103C23×C61#12112yes—
104C23×C61C2×SD1611yescharacteristic
105C23×C61C2×Q1611yescharacteristic
106C23×C61#13512yes—
107D4×C61—24no—
108D4×C61—24no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—24no—
112C2×C4×C61—22no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—41no—
116C2×C4×C61—42no—
117C2×C4×C61—42no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—81no—
121C2×C4×C61#10312yes—
122C2×C4×C61C4×D412yes—
123C2×C4×C61#12011yescharacteristic
124C2×C4×C61#12311yescharacteristic
125C2×Dic61—42no—
126C2×Dic61—42no—
127C2×Dic61—42no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130C2×Dic61—42no—
131C2×Dic61—84no—
132C2×Dic61—82no—
133C2×Dic61—82no—
134C2×Dic61—82no—
135C8×C61—44no—
136#9469—611nomaximal, Sylow
137C22×C61—21no—
138C22×C61—22no—
139C22×C61—21no—
140C22×C61—22no—
141C22×C61—22no—
142C22×C61—41no—
143C22×C61—41no—
144C22×C61—42no—
145C22×C61—42no—
146C22×C61—41no—
147C22×C61—41no—
148C22×C61—81no—
149C22×C61#38611yescharacteristic
150C22×C61C2×#10311yescharacteristic
151C22×C61#41611yescharacteristic
152C22×C61#46512yes—
153C22×C61#46712yes—
154C4×C61—21no—
155C4×C61—22no—
156C4×C61—22no—
157C4×C61—41no—
158C4×C61—41no—
159C4×C61#45812yes—
160Dic61—82no—
161Dic61—82no—
162Dic61—162no—
163#2972D6111yescharacteristic
164#2487—611no—
165#1824—612no—
166#1799—611no—
167#1289—612no—
168C2×C61—21no—
169C2×C61—22no—
170C2×C61—21no—
171C2×C61—22no—
172C2×C61—42no—
173C2×C61#189011yescharacteristic
174C2×C61#189211yescharacteristic
175C2×C61#191011yescharacteristic
176C2×C4×D4C2×D6111yescharacteristic
177C22×C4⋊C4—611no—
178C8×D4—24no—
179#431—21no—
180C2×#99—1221no—
181C2×C22⋊C8—21no—
182C2×C22⋊C8C2×D6112yes—
183#400—612no—
184#399—612no—
185#389—1221no—
186#386—1221no—
187#385—1221no—
188#380—611no—
189#380—612no—
190#380—1221no—
191#342—1221no—
192#340—1224no—
193C61#946911yescharacteristic, complemented, Sylow
194C22×D4D24411yescharacteristic
195C23×C4—1221no—
196C23×C4C22⋊D6111yescharacteristic
197C23×C4C22×D6111yescharacteristic
198C22×C8—22no—
199C22×C8C22⋊D6114yes—
200C4×D4—22no—
201C4×D4—22no—
202C2×C4⋊C4—612no—
203C2×C4⋊C4—1222no—
204C2×C4⋊C4—1222no—
205C2×C4⋊C4—1222no—
206C2×C4⋊C4—1221no—
207C2×C4⋊C4—1221no—
208C2×C4⋊C4—1222no—
209C2×C4⋊C4C4×D6111yescharacteristic
210C2×C22⋊C4—1222no—
211C2×C22⋊C4—1221no—
212C2×C22⋊C4—1222no—
213C2×C22⋊C4C4×D6111yescharacteristic
214C2×C22⋊C4D24411yescharacteristic
215C2×C42C22⋊D6111yescharacteristic
216C4⋊C8—22no—
217#99—2442no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C4×C8—22no—
222#95—612no—
223#95—1222no—
224#95—1222no—
225#95—1222no—
226C24#1175819811yescharacteristic
227C24C2×D24411yescharacteristic
228C2×D4—22no—
229C2×D4—22no—
230C2×D4#1175819012yes—
231C2×D4Dic24412yes—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—1222no—
237C22×C4—1222no—
238C22×C4—4882no—
239C22×C4—2442no—
240C22×C4—2442no—
241C22×C4#1175819811yescharacteristic
242C22×C4#1175819811yescharacteristic
243C22×C4#1175819811yescharacteristic
244C22×C4C2×C4×D6111yescharacteristic
245C22×C4#1175822212yes—
246C22×C4C2×C22⋊D6112yes—
247C22×C4C2×C22⋊D6111yescharacteristic
248C2×C8—24no—
249C2×C8—44no—
250C2×C8—82no—
251C4⋊C4—22no—
252C4⋊C4—1224no—
253C4⋊C4—4882no—
254C4⋊C4—2444no—
255C4⋊C4—2442no—
256C4⋊C4—2441no—
257C4⋊C4—2441no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—4882no—
261C22⋊C4—2442no—
262C22⋊C4—2442no—
263C22⋊C4—2442no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23#1451970212yes—
273C23C2×#1175819811yescharacteristic
274C23#1451975112yes—
275C23C2×#1175819011yescharacteristic
276C23C2×Dic24411yescharacteristic
277C23#1451979912yes—
278D4—24no—
279D4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—81no—
292C2×C4—4882no—
293C2×C4—4882no—
294C2×C4—4882no—
295C2×C4—4884no—
296C2×C4—2442no—
297C2×C4—2442no—
298C2×C4—2442no—
299C2×C4—2442no—
300C2×C4—2442no—
301C2×C4—2442no—
302C2×C4#1451970212yes—
303C2×C4C4×C22⋊D6112yes—
304C2×C4#1451974611yescharacteristic
305C2×C4#1451976011yescharacteristic
306C8—44no—
307C22—21no—
308C22—22no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—41no—
313C22—41no—
314C22—42no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—81no—
319C22#1683964111yescharacteristic
320C22C2×#1451970211yescharacteristic
321C22#1683975911yescharacteristic
322C22#1683999112yes—
323C22#1684000312yes—
324C4—21no—
325C4—22no—
326C4—22no—
327C4—41no—
328C4—41no—
329C4—9762no—
330C4—4882no—
331C4—4882no—
332C4#1683995212yes—
333C2—22no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—42no—
338C2#2144623211yescharacteristic
339C2#2144624211yescharacteristic
340C2#2144634611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2972C2
socleC22×C61#386
prime radicalC22×D4×C61D4
cosocleC23×C4×C61C23
hypercenter#2972D61
hypocenterC61#9469
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #21445910C2semidirect
4. #21443687C2semidirect
8. #16839701C4semidirect
9. #16839698C4semidirect
193. C61#9469semidirect

Maximal subgroups

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

subgroupindex
#214501362
#214459102
#214458072
#214436872
C61×#29722
#946961