Finite Group Databasebeta

← #29568563

Subgroups — #29568563

Order 15616.

470 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 107 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#21445799C211yesmaximal, characteristic
3#21443644C212yesmaximal, complemented
4#21443641C212yesmaximal, complemented
5C61×#1749C211yesmaximal, characteristic
6C2×#14519756C411yescharacteristic
7C2×C4⋊#92418—21no—
8C2×C22⋊#92418—21no—
9#16839668C411yescharacteristic
10#16839668C2211yescharacteristic
11#16839660C411yescharacteristic
12#16839660C2211yescharacteristic
13#16839646—21no—
14#16839646—21no—
15#16839635—21no—
16#16839625—21no—
17#16839622C411yescharacteristic
18#16839517C2212yes—
19C22×C42×C61C2211yescharacteristic
20C2×C22⋊C8×C61—21no—
21C2×#97×C61—21no—
22C61×#336C2212yes—
23C22×Dic122C2×C411yescharacteristic
24C22×#92418—22no—
25#14519756—42no—
26C2×C4⋊Dic61C2×C411yescharacteristic
27C2×C4⋊Dic61C2×C411yescharacteristic
28C2×C4⋊Dic61C2×C411yescharacteristic
29C2×#11758196—21no—
30C2×#11758196—21no—
31C2×#11758196—21no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—21no—
34C4⋊#92418—42no—
35C22⋊#92418—42no—
36#14519684—21no—
37#14519684—21no—
38#14519684—21no—
39#14519684—21no—
40#14519684—21no—
41#14519684—21no—
42#14519684—21no—
43C23×C4×C61C2×C411yescharacteristic
44C23×C4×C61D411yescharacteristic
45C23×C4×C61D411yescharacteristic
46C22×C8×C61—22no—
47C2×C42×C61—42no—
48C2×C42×C61C2×C411yescharacteristic
49C2×C42×C61D411yescharacteristic
50C2×C42×C61D411yescharacteristic
51C2×C42×C61C2311yescharacteristic
52C22⋊C8×C61—24no—
53#97×C61—42no—
54C24×C61C22⋊C411yescharacteristic
55C22×C4×C61—22no—
56C22×C4×C61—22no—
57C22×C4×C61—22no—
58C22×C4×C61—21no—
59C22×C4×C61—21no—
60C22×C4×C61—22no—
61C22×C4×C61—41no—
62C22×C4×C61—41no—
63C22×C4×C61—41no—
64C22×C4×C61—41no—
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22×C411yescharacteristic
69C22×C4×C61C2×D411yescharacteristic
70C22×C4×C61C2×D411yescharacteristic
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C2×Dic122—22no—
76C2×Dic122—41no—
77C2×Dic122—41no—
78C2×C8×C61—24no—
79C2×C8×C61—42no—
80C2×C8×C61—42no—
81C42×C61—22no—
82C42×C61—22no—
83C42×C61—41no—
84C42×C61—41no—
85C22⋊Dic61—42no—
86C22⋊Dic61—42no—
87C4⋊Dic61—22no—
88C4⋊Dic61—41no—
89C4⋊Dic61—41no—
90#11758196—42no—
91#11758196—42no—
92#11758196—41no—
93#11758196—41no—
94C2×#92418—42no—
95C2×#92418—42no—
96C2×#92418—42no—
97C23×C61—21no—
98C23×C61—21no—
99C23×C61—21no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61#10412yes—
104C23×C61C2×C22⋊C411yescharacteristic
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—44no—
120C2×C4×C61—44no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61#9912yes—
124C2×C4×C61#13112yes—
125C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C61#13711yescharacteristic
127C2×C4×C61#13712yes—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140Dic122—82no—
141C8×C61—44no—
142#92418—82no—
143#7701—611nomaximal, Sylow
144C22×C61—22no—
145C22×C61—22no—
146C22×C61—22no—
147C22×C61—41no—
148C22×C61—41no—
149C22×C61—41no—
150C22×C61—41no—
151C22×C61#40811yescharacteristic
152C22×C61C2×#9911yescharacteristic
153C22×C61#41311yescharacteristic
154C22×C61#41711yescharacteristic
155C22×C61#41911yescharacteristic
156C22×C61C2×#10411yescharacteristic
157C22×C61#42111yescharacteristic
158C4×C61—22no—
159C4×C61—22no—
160C4×C61—22no—
161C4×C61—21no—
162C4×C61—21no—
163C4×C61—41no—
164C4×C61—41no—
165Dic61—81no—
166Dic61—81no—
167Dic61—81no—
168Dic61—81no—
169#2855—611no—
170#1797—611no—
171#1749D6111yescharacteristic
172#1272—612no—
173#1271—612no—
174C2×C61—42no—
175C2×C61#146811yescharacteristic
176C2×C61#148311yescharacteristic
177C2×C61#149711yescharacteristic
178C2×C61#150811yescharacteristic
179C2×C61#151311yescharacteristic
180C2×C61#153211yescharacteristic
181C2×C61#155911yescharacteristic
182C2×C22⋊Q8—611no—
183C22×C42C2×D6111yescharacteristic
184C2×C4⋊C8—1221no—
185C2×C22⋊C8—21no—
186C2×C22⋊C8—1221no—
187C2×#97—21no—
188#391—611no—
189#391—611no—
190#389—611no—
191#389—611no—
192#387—1221no—
193#387—1221no—
194#385—1221no—
195#381—1221no—
196#380—611no—
197#336—612no—
198#336C2×D6112yes—
199C61#770111yescharacteristic, complemented, Sylow
200C22×Q8—611no—
201C23×C4C4×D6111yescharacteristic
202C23×C4D24411yescharacteristic
203C23×C4C22⋊D6111yescharacteristic
204C22×C8—22no—
205C22×C8—1222no—
206C22⋊Q8—2442no—
207C2×C4⋊C4—611no—
208C2×C4⋊C4—611no—
209C2×C4⋊C4—611no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C4⋊C4—1221no—
213C2×C22⋊C4—1221no—
214C2×C22⋊C4—1221no—
215C2×C42—42no—
216C2×C42C4×D6111yescharacteristic
217C2×C42D24411yescharacteristic
218C2×C42C22⋊D6111yescharacteristic
219C2×C42C22×D6111yescharacteristic
220C4⋊C8—2442no—
221C22⋊C8—24no—
222C22⋊C8—2442no—
223#97—42no—
224#95—1221no—
225#95—1221no—
226#95—1221no—
227#95—1221no—
228#95—1221no—
229#95—1221no—
230#95—1221no—
231C24#1175819811yescharacteristic
232C2×Q8—1222no—
233C2×Q8—2441no—
234C2×Q8—2441no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—1221no—
246C22×C4—1221no—
247C22×C4—1221no—
248C22×C4—1221no—
249C22×C4#1175819811yescharacteristic
250C22×C4#1175819811yescharacteristic
251C22×C4#1175819811yescharacteristic
252C22×C4C2×C4×D6111yescharacteristic
253C22×C4C2×D24411yescharacteristic
254C22×C4C2×C22⋊D6111yescharacteristic
255C2×C8—24no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—2442no—
259C2×C8—2442no—
260C2×C8—2442no—
261C4⋊C4—1222no—
262C4⋊C4—2441no—
263C4⋊C4—2441no—
264C4⋊C4—2442no—
265C4⋊C4—2441no—
266C4⋊C4—2442no—
267C4⋊C4—2441no—
268C22⋊C4—2442no—
269C22⋊C4—2442no—
270C42—22no—
271C42—22no—
272C42—41no—
273C42—41no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23#1451970512yes—
281C23C2×#1175819811yescharacteristic
282Q8—4882no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4#1451969112yes—
314C2×C4#1451978512yes—
315C2×C4#1451980511yescharacteristic
316C2×C4#1451980712yes—
317C2×C4#1451981011yescharacteristic
318C8—44no—
319C8—4882no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22#1683973011yescharacteristic
328C22C2×#1451969111yescharacteristic
329C22#1683974411yescharacteristic
330C22#1683976411yescharacteristic
331C22#1683977011yescharacteristic
332C22C2×#1451970511yescharacteristic
333C22#1683977811yescharacteristic
334C4—21no—
335C4—21no—
336C4—22no—
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—4881no—
342C4—4881no—
343C4—4881no—
344C4—4881no—
345C2—42no—
346C2#2144416811yescharacteristic
347C2#2144423311yescharacteristic
348C2#2144429111yescharacteristic
349C2#2144433811yescharacteristic
350C2#2144436311yescharacteristic
351C2#2144443711yescharacteristic
352C2#2144455011yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1749C2
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1749D61
hypocenterC61#7701
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1749

Complements

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

NH ≅ G/Ntype
3. #21443644C2semidirect
4. #21443641C2semidirect
199. C61#7701semidirect

Maximal subgroups

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

subgroupindex
#214541582
#214457992
#214436442
#214436412
C61×#17492
#770161