Finite Group Databasebeta

← #29591060

Subgroups — #29591060

Order 15616.

764 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 99 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#21449175C211yesmaximal, characteristic
2#21445811C211yesmaximal, characteristic, complemented
3#21443823C214yesmaximal, complemented
4C61×#1459C211yesmaximal, characteristic, complemented
5C23×D244C411yescharacteristic
6C2×#14519751—21no—
7C2×#14519745—21no—
8#16839786—21no—
9C2×#14519694—21no—
10#16839679C2212yes—
11#16839671C412yes—
12#16839664—21no—
13#16839622C411yescharacteristic
14#16839617—21no—
15#16839531C2212yescomplemented
16C2×C4×D4×C61C2211yescharacteristic
17C2×C22⋊C8×C61—21no—
18C61×#324C2212yescomplemented
19C61×#323—21no—
20C24×D61—21no—
21C22×D244—21no—
22C22×D244—44no—
23C22×D244—41no—
24C22×D244—41no—
25C22×D244C2×C412yes—
26C22×C22⋊D61—22no—
27C2×#11758194—22no—
28#14519751—44no—
29#14519745—42no—
30#14519745—42no—
31C2×C4⋊Dic61—21no—
32C2×C22⋊Dic61—21no—
33C2×#11758198—22no—
34C2×#11758198—22no—
35C2×#11758198—22no—
36C4⋊#92418—41no—
37C4⋊#92418—41no—
38#14519694—42no—
39#14519684C2×C412yes—
40C22×D4×C61D411yescharacteristic
41C23×C4×C61C2×C411yescharacteristic
42C23×C4×C61D411yescharacteristic
43C22×C8×C61—41no—
44C4×D4×C61—41no—
45C4×D4×C61—41no—
46C2×C4⋊C4×C61C2311yescharacteristic
47C2×C22⋊C4×C61C2×C411yescharacteristic
48C2×C22⋊C4×C61D411yescharacteristic
49C2×C42×C61D411yescharacteristic
50C22⋊C8×C61—22no—
51C22⋊C8×C61—41no—
52C24×C61C22⋊C411yescharacteristic
53C24×C61C22⋊C411yescharacteristic
54C23×D61—22no—
55C23×D61—44no—
56C23×D61—42no—
57C23×D61—42no—
58C23×D61—42no—
59C23×D61—82no—
60C2×D4×C61—41no—
61C2×D4×C61—41no—
62C2×D4×C61D812yes—
63C2×D4×C61SD1612yes—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—42no—
69C22×C4×C61—42no—
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61C22⋊C411yescharacteristic
72C22×C4×C61C22×C411yescharacteristic
73C22×C4×C61C2×D411yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C2×C22⋊D61—44no—
76C2×C22⋊D61—44no—
77C2×C22⋊D61—82no—
78C22×Dic61—22no—
79C2×D244—24no—
80C2×D244—42no—
81C2×D244—42no—
82C2×D244—48no—
83C2×D244—44no—
84C2×D244—41no—
85C2×D244—41no—
86C2×D244—44no—
87C2×D244—44no—
88C2×C8×C61—42no—
89C2×C8×C61—82no—
90C4⋊C4×C61—41no—
91C22⋊C4×C61—41no—
92C22⋊C4×C61—41no—
93C42×C61—21no—
94C42×C61—21no—
95C22⋊Dic61—42no—
96#11758198—44no—
97#11758198—42no—
98#11758198—42no—
99#11758198—82no—
100C4⋊Dic61—42no—
101#11758194—42no—
102#11758194—42no—
103C2×#92418—42no—
104C23×C61—21no—
105C23×C61—21no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—41no—
109C23×C61—41no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61#10212yes—
113C23×C61#10212yes—
114C23×C61C2×C22⋊C411yescharacteristic
115C22×D61—24no—
116C22×D61—42no—
117C22×D61—42no—
118C22×D61—42no—
119C22×D61—48no—
120C22×D61—42no—
121C22×D61—42no—
122C22×D61—42no—
123C22×D61—88no—
124C22×D61—88no—
125C22×D61—84no—
126C22×D61—84no—
127D4×C61—42no—
128D4×C61—42no—
129C2×C4×C61—21no—
130C2×C4×C61—21no—
131C2×C4×C61—22no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61—42no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—81no—
142C2×C4×C61#9911yescharacteristic
143C2×C4×C61#9911yescharacteristic
144C2×C4×C61C2×D811yescharacteristic
145C2×C4×C61C2×SD1611yescharacteristic
146C22⋊D61—88no—
147C2×Dic61—42no—
148C2×Dic61—42no—
149C2×Dic61—42no—
150D244—44no—
151D244—44no—
152D244—44no—
153D244—44no—
154C8×C61—82no—
155#92418—82no—
156#11880—611nomaximal, Sylow
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—22no—
160C22×C61—22no—
161C22×C61—42no—
162C22×C61—42no—
163C22×C61—42no—
164C22×C61—41no—
165C22×C61—41no—
166C22×C61—41no—
167C22×C61—41no—
168C22×C61—81no—
169C22×C61C2×#9911yescharacteristic
170C22×C61C2×#10211yescharacteristic
171C22×C61#42111yescharacteristic
172C2×D61—44no—
173C2×D61—44no—
174C2×D61—44no—
175C2×D61—82no—
176C2×D61—82no—
177C2×D61—816no—
178C2×D61—82no—
179C2×D61—82no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—41no—
184C4×C61—41no—
185C4×C61—41no—
186C4×C61AΣL(1,𝔽16)12yes—
187Dic61—82no—
188#2408—611no—
189#1800—611no—
190#1459D6111yescharacteristic, complemented
191#1345—614no—
192C2×C61—21no—
193C2×C61—21no—
194C2×C61—42no—
195C2×C61—42no—
196C2×C61#151911yescharacteristic
197C2×C61C2×AΣL(1,𝔽16)11yescharacteristic
198C2×C61#213411yescharacteristic
199D61—84no—
200C23×D4—611no—
201C2×#121—1221no—
202C2×#120—1221no—
203C2×C4×D4C2×D6111yescharacteristic
204#423—1221no—
205C2×#100—1221no—
206C2×C22⋊C8—21no—
207#394—612no—
208#392—612no—
209#390—1221no—
210#380—611no—
211#379—1221no—
212#343—612no—
213#324C2×D6112yescomplemented
214#323—21no—
215C61#1188011yescharacteristic, complemented, Sylow
216C25—1221no—
217C22×D4—612no—
218C22×D4—1222no—
219C22×D4—1221no—
220C22×D4—2441no—
221C22×D4—2441no—
222C22×D4—2444no—
223C22×D4D24411yescharacteristic
224C23×C4C4×D6111yescharacteristic
225C23×C4C22⋊D6111yescharacteristic
226C2×M16—1222no—
227C22×C8—41no—
228#121—2444no—
229#120—2442no—
230#120—2442no—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4—1221no—
234C2×C4⋊C4C22×D6111yescharacteristic
235C2×C22⋊C4—1222no—
236C2×C22⋊C4—1221no—
237C2×C22⋊C4—1222no—
238C2×C22⋊C4—1222no—
239C2×C22⋊C4C4×D6111yescharacteristic
240C2×C22⋊C4C22⋊D6111yescharacteristic
241C2×C42D24411yescharacteristic
242C4⋊C8—2441no—
243C4⋊C8—2441no—
244#100—2442no—
245C22⋊C8—22no—
246C22⋊C8—41no—
247#95—612no—
248C24—1222no—
249C24—4882no—
250C24—2442no—
251C24—2442no—
252C24—2444no—
253C24—2442no—
254C24#1175819811yescharacteristic
255C24#1175819811yescharacteristic
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—1224no—
259C2×D4—4882no—
260C2×D4—2442no—
261C2×D4—2442no—
262C2×D4—2448no—
263C2×D4—2444no—
264C2×D4—2444no—
265C2×D4—2444no—
266C2×D4—2444no—
267C2×D4—2444no—
268C2×D4—2441no—
269C2×D4—2441no—
270C2×D4#1175819012yes—
271C2×D4D48812yes—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—1222no—
279C22×C4#1175819811yescharacteristic
280C22×C4#1175819811yescharacteristic
281C22×C4C2×C4×D6111yescharacteristic
282C22×C4C2×D24411yescharacteristic
283C22×C4C2×C22⋊D6111yescharacteristic
284M16—2442no—
285M16—2442no—
286C2×C8—42no—
287C2×C8—82no—
288C2×C8—2442no—
289C4⋊C4—41no—
290C4⋊C4—2442no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—4882no—
294C22⋊C4—2444no—
295C22⋊C4—2442no—
296C22⋊C4—2442no—
297C22⋊C4—2442no—
298C42—21no—
299C42—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—1224no—
309C23—4884no—
310C23—4884no—
311C23—4888no—
312C23—4888no—
313C23—2442no—
314C23—2442no—
315C23—2442no—
316C23—2448no—
317C23—2442no—
318C23—2442no—
319C23—2442no—
320C23#1451969912yes—
321C23#1451969912yes—
322C23C2×#1175819811yescharacteristic
323D4—42no—
324D4—42no—
325D4—4888no—
326D4—2444no—
327D4—2444no—
328D4—2444no—
329D4—2444no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—81no—
343C2×C4—2442no—
344C2×C4—2442no—
345C2×C4—2442no—
346C2×C4#1451969111yescharacteristic
347C2×C4#1451969111yescharacteristic
348C2×C4C2×D48811yescharacteristic
349C2×C4C2×#1175819011yescharacteristic
350C8—82no—
351C8—4882no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—81no—
364C22—4882no—
365C22—4882no—
366C22—48816no—
367C22—4882no—
368C22—4882no—
369C22—2444no—
370C22—2444no—
371C22—2444no—
372C22C2×#1451969111yescharacteristic
373C22C2×#1451969911yescharacteristic
374C22#1683977811yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—4882no—
382C4#1683954812yes—
383C2—21no—
384C2—21no—
385C2—42no—
386C2—42no—
387C2—4884no—
388C2#2144438811yescharacteristic
389C2C2×#1683954811yescharacteristic
390C2#2144751011yescharacteristic
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC2×C4⋊C4C22×D61
FittingC61×#1459C2
socleC22×C61C2×#99
prime radical#21449175C2
cosocleC2×C4⋊C4×C61C23
hypercenter#1459D61
hypocenterC61#11880
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
2. #21445811C2semidirect
3. #21443823C2semidirect
4. C61×#1459C2semidirect
15. #16839531C22semidirect
18. C61×#324C22semidirect
190. #1459D61semidirect
213. #324C2×D61semidirect
215. C61#11880semidirect

Maximal subgroups

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

subgroupindex
#214491752
#214458112
#214438232
C61×#14592
#1188061