Finite Group Databasebeta

← #29589139

Subgroups — #29589139

Order 15616.

504 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 89 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#21454261C211yesmaximal, characteristic
2#349×D61C212yesmaximal, complemented
3#21443757C211yesmaximal, characteristic, complemented
4#21443756C212yesmaximal, complemented
5C61×#1321C211yesmaximal, characteristic, complemented
6C2×#41×D61C411yescharacteristic
7C22×#11758189C411yescharacteristic
8C2×M16×D61C2211yescharacteristic
9M32×D61—22no—
10#16839730C2211yescharacteristic
11#16839729C411yescharacteristic
12#16839728C411yescharacteristic
13#16839727—22no—
14#16839724—22no—
15#16839544C2212yescomplemented
16#16839543—22no—
17C61×#408C2211yescharacteristic
18C61×#349C2212yescomplemented
19C2×#11758226C811yescharacteristic
20C2×#11758222—21no—
21C2×#11758222C2×C411yescharacteristic
22C2×#11758224—21no—
23C2×#11758224C811yescharacteristic
24#41×D61—42no—
25#41×D61—42no—
26C2×Q8×D61C811yescharacteristic
27C2×D4×D61—21no—
28C2×D4×D61C811yescharacteristic
29C22×C4×D61—21no—
30C22×C4×D61C2×C411yescharacteristic
31C2×#11758194C2×C411yescharacteristic
32C2×#11758189—21no—
33C2×#11758189—21no—
34C2×#11758189—42no—
35C2×#11758189D412yes—
36M16×D61—42no—
37C22×#92418C2×C411yescharacteristic
38C2×C8×D61D412yes—
39#14519717—22no—
40#14519716—42no—
41C16×D61—42no—
42C22⋊#92418—22no—
43#14519689—22no—
44#14519689—22no—
45C2×#41×C61C2×C411yescharacteristic
46C2×M16×C61C2311yescharacteristic
47C22×C8×C61C2×C411yescharacteristic
48M32×C61—22no—
49C22⋊C8×C61—22no—
50C23×D61—21no—
51C23×D61M1611yescharacteristic
52#41×C61—41no—
53#41×C61—41no—
54C2×Q8×C61C2×C811yescharacteristic
55C2×D4×C61—21no—
56C2×D4×C61C2×C811yescharacteristic
57C22×C4×C61—21no—
58C22×C4×C61C22×C411yescharacteristic
59C2×C22⋊D61—21no—
60C2×C22⋊D61—21no—
61C2×C22⋊D61—21no—
62C22×Dic61—21no—
63C22×Dic61M1611yescharacteristic
64#11758226—42no—
65Q8×D61—42no—
66#11758224—42no—
67#11758224—44no—
68D4×D61—42no—
69D4×D61—44no—
70#11758222—42no—
71#11758222—42no—
72#11758222—41no—
73#11758222—41no—
74C2×D244—21no—
75C2×D244C2×C811yescharacteristic
76C2×C4×D61—22no—
77C2×C4×D61—22no—
78C2×C4×D61—21no—
79C2×C4×D61—21no—
80C2×C4×D61—21no—
81C2×C4×D61—42no—
82C2×C4×D61—42no—
83C2×C4×D61C22⋊C411yescharacteristic
84C2×C4×D61C22⋊C411yescharacteristic
85C2×C4×D61C2×C811yescharacteristic
86C2×C4×D61C2×C811yescharacteristic
87C2×Dic122—21no—
88C2×Dic122C2×C811yescharacteristic
89M16×C61—41no—
90C2×C8×C61—21no—
91C2×C8×C61—41no—
92C2×C8×C61C2×D412yes—
93#11758194—41no—
94C2×#92418—21no—
95C2×#92418—41no—
96C2×#92418C22⋊C412yes—
97#11758189—42no—
98#11758189—42no—
99#11758189—42no—
100C8×D61—22no—
101C8×D61—22no—
102C16×C61—42no—
103#11758185—42no—
104C23×C61—21no—
105C23×C61C2×M1611yescharacteristic
106C22×D61—22no—
107C22×D61—21no—
108C22×D61—42no—
109C22×D61—82no—
110C22×D61—81no—
111C22×D61C22⋊C811yescharacteristic
112C22×D61C22⋊C811yescharacteristic
113Q8×C61—41no—
114D4×C61—41no—
115D4×C61—42no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61C2×C22⋊C411yescharacteristic
122C2×C4×C61C22×C811yescharacteristic
123C22⋊D61—42no—
124C22⋊D61—42no—
125C22⋊D61—42no—
126C2×Dic61—22no—
127C2×Dic61—21no—
128C2×Dic61—42no—
129C2×Dic61C22⋊C811yescharacteristic
130C2×Dic61C22⋊C811yescharacteristic
131D244—41no—
132D244—42no—
133C4×D61—22no—
134C4×D61—21no—
135C4×D61—21no—
136C4×D61—22no—
137C4×D61—42no—
138C4×D61—42no—
139C4×D61—41no—
140C4×D61—41no—
141C4×D61#9912yes—
142C4×D61#10012yes—
143Dic122—42no—
144Dic122—41no—
145C8×C61—22no—
146C8×C61—41no—
147#92418—22no—
148#92418—41no—
149#11438—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—41no—
152C22×C61—42no—
153C22×C61—41no—
154C22×C61C2×C22⋊C811yescharacteristic
155C2×D61—21no—
156C2×D61—41no—
157C2×D61—41no—
158C2×D61—42no—
159C2×D61—41no—
160C2×D61—82no—
161C2×D61—84no—
162C2×D61#32312yes—
163C4×C61—21no—
164C4×C61—41no—
165C4×C61C2×#9911yescharacteristic
166C4×C61C2×#10011yescharacteristic
167Dic61—21no—
168Dic61—41no—
169Dic61#32312yes—
170#2879—611no—
171C2×#349—612no—
172#1321—611no—
173#1321—612no—
174#1321D6111yescharacteristic, complemented
175C2×C61—21no—
176C2×C61—41no—
177C2×C61—81no—
178C2×C61C2×#32311yescharacteristic
179D61—22no—
180D61—41no—
181D61—81no—
182C22×#41—611no—
183C22×M16—611no—
184C22×M16—611no—
185C2×M32—1222no—
186#408—611no—
187#408—611no—
188#408—611no—
189#408—1222no—
190#408C2×D6111yescharacteristic
191#407—1222no—
192#349—612no—
193#349—1222no—
194#349C2×D6112yescomplemented
195C61#1143811yescharacteristic, complemented, Sylow
196C2×#41—611no—
197C2×#41—611no—
198C2×#41—611no—
199C2×#41—1221no—
200C2×#41—1221no—
201C2×#41—2442no—
202C2×#41—2442no—
203C2×#41C4×D6111yescharacteristic
204C22×Q8—611no—
205C22×D4—611no—
206C22×D4—1221no—
207C23×C4—611no—
208C23×C4—1221no—
209C2×M16—611no—
210C2×M16—612no—
211C2×M16—1221no—
212C2×M16—1221no—
213C2×M16—2442no—
214C2×M16—2442no—
215C2×M16C22×D6111yescharacteristic
216C22×C8—612no—
217C22×C8—611no—
218C22×C8C4×D6111yescharacteristic
219M32—22no—
220M32—1222no—
221M32—2442no—
222C2×C16—2442no—
223C22⋊C8—22no—
224C22⋊C8—1222no—
225C22⋊C8—1222no—
226C22⋊C8—1222no—
227C24—611no—
228C24—1221no—
229#41—41no—
230#41—41no—
231#41—2441no—
232#41—2441no—
233#41—2442no—
234#41—2442no—
235#41—2444no—
236#41—2442no—
237#41—2442no—
238C2×Q8—611no—
239C2×Q8—1221no—
240C2×Q8—2442no—
241C2×Q8C8×D6111yescharacteristic
242C2×D4—21no—
243C2×D4—611no—
244C2×D4—1221no—
245C2×D4—1221no—
246C2×D4—1221no—
247C2×D4—1221no—
248C2×D4—2442no—
249C2×D4—2444no—
250C2×D4C8×D6111yescharacteristic
251C22×C4—21no—
252C22×C4—611no—
253C22×C4—611no—
254C22×C4—611no—
255C22×C4—611no—
256C22×C4—611no—
257C22×C4—1222no—
258C22×C4—1222no—
259C22×C4—1221no—
260C22×C4—1221no—
261C22×C4—1221no—
262C22×C4—1221no—
263C22×C4—2442no—
264C22×C4—2442no—
265C22×C4C2×C4×D6111yescharacteristic
266M16—41no—
267M16—2442no—
268M16—2442no—
269M16—2442no—
270M16—2441no—
271C2×C8—21no—
272C2×C8—41no—
273C2×C8—612no—
274C2×C8—1222no—
275C2×C8—1222no—
276C2×C8—1221no—
277C2×C8—2441no—
278C2×C8D4×D6112yes—
279C16—42no—
280C16—2442no—
281C23—21no—
282C23—611no—
283C23—611no—
284C23—1222no—
285C23—1221no—
286C23—4882no—
287C23—4881no—
288C23—2442no—
289C23M16×D6111yescharacteristic
290Q8—41no—
291Q8—2441no—
292Q8—2442no—
293D4—41no—
294D4—42no—
295D4—2441no—
296D4—2442no—
297D4—2442no—
298D4—2442no—
299D4—2442no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—611no—
306C2×C4—612no—
307C2×C4—612no—
308C2×C4—611no—
309C2×C4—1222no—
310C2×C4—1222no—
311C2×C4—1222no—
312C2×C4—1221no—
313C2×C4—1221no—
314C2×C4—1221no—
315C2×C4—2442no—
316C2×C4—2442no—
317C2×C4—2442no—
318C2×C4—2441no—
319C2×C4—2441no—
320C2×C4C22⋊C4×D6111yescharacteristic
321C2×C4C2×C8×D6111yescharacteristic
322C8—22no—
323C8—41no—
324C8—1222no—
325C8—2441no—
326C22—21no—
327C22—41no—
328C22—42no—
329C22—41no—
330C22—612no—
331C22—1221no—
332C22—4882no—
333C22—4884no—
334C22—2441no—
335C22—2442no—
336C22—2441no—
337C22—2441no—
338C22C22⋊C8×D6111yescharacteristic
339C4—21no—
340C4—41no—
341C4—612no—
342C4—1221no—
343C4—2441no—
344C4#99×D6111yescharacteristic
345C4#100×D6111yescharacteristic
346C2—21no—
347C2—41no—
348C2—81no—
349C2—1222no—
350C2—4881no—
351C2—2441no—
352C2#323×D6111yescharacteristic
subgroupG/H
centerC4#99×D61
commutatorC2×C4×C61C22×C8
FrattiniC2×M16C22×D61
FittingC61×#1321C2
socleC2×C61C2×#323
prime radicalC2×#41×D61C4
cosocleC2×M16×C61C23
hypercenter#1321D61
hypocenterC61#11438
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1321

Complements

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

NH ≅ G/Ntype
2. #349×D61C2semidirect
3. #21443757C2semidirect
4. #21443756C2semidirect
5. C61×#1321C2semidirect
15. #16839544C22semidirect
18. C61×#349C22semidirect
174. #1321D61semidirect
194. #349C2×D61semidirect
195. C61#11438semidirect

Maximal subgroups

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

subgroupindex
#214542612
#349×D612
#214437572
#214437562
C61×#13212
#1143861