Finite Group Databasebeta

← #29685123

Subgroups — #29685123

Order 15616.

1430 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840312C211yesmaximal, characteristic
2C4×#14519736C211yesmaximal, characteristic, complemented
3#21446770C212yesmaximal, complemented
4#21446745C212yesmaximal, complemented
5#21446139C214yesmaximal, complemented
6#21445645C214yesmaximal, complemented
7C2×C61×#432C211yesmaximal, characteristic, complemented
8C22×#11758222—21no—
9#16840312—24no—
10#16840312C418yescomplemented
11C2×#14519777C2211yescharacteristic
12C2×#14519774C2211yescharacteristic
13C2×#14519765—21no—
14C2×#14519751—21no—
15C2×#14519751C2212yes—
16C2×C4×D244—21no—
17C2×C4×D244C2212yes—
18C2×#14519736C2212yes—
19C2×#14519699C2218yes—
20C4×C4⋊Dic61C2212yes—
21C4×C4⋊Dic61C2212yes—
22C4×C22⋊Dic61—21no—
23C4×C22⋊Dic61—21no—
24#16839608—21no—
25C42×Dic61—21no—
26#16839525C2218yes—
27C22×C42×C61C2211yescharacteristic
28C61×#432—24no—
29C2×C4⋊C8×C61C2212yes—
30C2×C22⋊C8×C61C2212yes—
31C2×#97×C61C2212yes—
32C2×#11758222—28no—
33C2×#11758222—48no—
34C22×Dic122—21no—
35C22×D244—21no—
36C22×D244—21no—
37C22×D244C2312yes—
38C22×C22⋊D61—22no—
39C22×C4×D61—22no—
40#14519777—22no—
41#14519777C2×C414yes—
42#14519774—22no—
43#14519774C2×C414yes—
44#14519765—28no—
45#14519751—28no—
46#14519751—24no—
47#14519751C2×C418yes—
48C4×D244—28no—
49C4×D244—24no—
50C4×D244C2×C418yes—
51#14519736—28no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C4⋊Dic61C2314yes—
55C2×C22⋊Dic61—22no—
56C2×#11758198—22no—
57C2×#11758198—22no—
58C2×C4×Dic61—22no—
59C2×C4×Dic61—22no—
60C2×C4×Dic61—22no—
61#14519699—216no—
62#14519684—22no—
63#14519684—22no—
64C23×C4×C61D411yescharacteristic
65C23×C4×C61D411yescharacteristic
66C23×C4×C61C2311yescharacteristic
67C22×C8×C61C2314yes—
68C2×C42×C61C2×C414yes—
69C2×C42×C61D411yescharacteristic
70C2×C42×C61D411yescharacteristic
71C2×C42×C61D414yes—
72C2×C42×C61C2311yescharacteristic
73C2×C42×C61C2311yescharacteristic
74C4⋊C8×C61—24no—
75C22⋊C8×C61—24no—
76#97×C61—24no—
77C24×C61C2×D411yescharacteristic
78C23×D61—22no—
79C22×C4×C61—24no—
80C22×C4×C61—24no—
81C22×C4×C61—42no—
82C22×C4×C61C22⋊C414yes—
83C22×C4×C61C22⋊C414yes—
84C22×C4×C61C22×C412yes—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C22×C4×C61C2×D412yes—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2411yescharacteristic
92C2×C22⋊D61—28no—
93C2×C22⋊D61—48no—
94C22×Dic61—22no—
95C22×Dic61—24no—
96#11758222—432no—
97C2×D244—28no—
98C2×D244—24no—
99C2×D244—24no—
100C2×D244—22no—
101C2×D244—44no—
102C2×D244—44no—
103C2×D244C22×C414yes—
104C2×C4×D61—28no—
105C2×C4×D61—48no—
106C2×Dic122—24no—
107C2×Dic122—44no—
108C2×C8×C61—24no—
109C2×C8×C61—28no—
110C42×C61C22⋊C414yes—
111C42×C61C22⋊C414yes—
112C42×C61C22×C412yes—
113C42×C61C22×C412yes—
114C42×C61C2×D412yes—
115C42×C61C2×D412yes—
116C22⋊Dic61—28no—
117#11758198—28no—
118#11758198—28no—
119C4⋊Dic61—24no—
120C4⋊Dic61—22no—
121C4⋊Dic61—24no—
122C4⋊Dic61C22×C414yes—
123C4⋊Dic61#4118yescomplemented
124C4×Dic61—28no—
125C4×Dic61—44no—
126C4×Dic61—44no—
127C23×C61—22no—
128C23×C61—22no—
129C23×C61—22no—
130C23×C61C2×C22⋊C412yes—
131C23×C61C22×D411yescharacteristic
132C22×D61—24no—
133C22×D61—44no—
134C22×D61—44no—
135C22×D61—44no—
136C2×C4×C61—22no—
137C2×C4×C61—22no—
138C2×C4×C61—22no—
139C2×C4×C61—22no—
140C2×C4×C61—24no—
141C2×C4×C61—24no—
142C2×C4×C61—28no—
143C2×C4×C61—44no—
144C2×C4×C61C2×C22⋊C412yes—
145C2×C4×C61C2×C22⋊C412yes—
146C2×C4×C61C2×C22⋊C412yes—
147C2×C4×C61C2×C22⋊C412yes—
148C2×C4×C61C2×C22⋊C412yes—
149C2×C4×C61AGL(1,ℤ/8)14yes—
150C2×C4×C61C23×C411yescharacteristic
151C2×C4×C61C22×D411yescharacteristic
152C2×C4×C61C2×#4114yes—
153C22⋊D61—416no—
154C2×Dic61—24no—
155C2×Dic61—28no—
156C2×Dic61—44no—
157C2×Dic61—44no—
158C2×Dic61—44no—
159D244—28no—
160D244—48no—
161D244—44no—
162D244—48no—
163C4×D61—416no—
164Dic122—48no—
165C8×C61—44no—
166#21393—611nomaximal, Sylow
167C22×C61—22no—
168C22×C61—22no—
169C22×C61—22no—
170C22×C61—44no—
171C22×C61C22×C22⋊C411yescharacteristic
172C22×C61#52512yes—
173C22×C61C2×AGL(1,ℤ/8)12yes—
174C22×C61#57512yes—
175C2×D61—44no—
176C2×D61—44no—
177C2×D61—44no—
178C2×D61—88no—
179C4×C61—22no—
180C4×C61—22no—
181C4×C61—22no—
182C4×C61—22no—
183C4×C61#41814yes—
184C4×C61C4×#4112yes—
185C4×C61#53212yes—
186Dic61—48no—
187Dic61—44no—
188C2×#532—611no—
189C2×#432D6111yescharacteristic, complemented
190C4×#117—611no—
191#1988—612no—
192#1982—612no—
193#1875—614no—
194#1762—614no—
195C2×C61—42no—
196C2×C61#229711yescharacteristic
197C2×C61C2×#41811yescharacteristic
198C2×C61#289611yescharacteristic
199C2×C61#311514yes—
200D61—84no—
201C22×#41—1221no—
202#532—618no—
203#532—1224no—
204C2×C4⋊Q8—611no—
205C2×#127—611no—
206C2×#124—1221no—
207C2×#121—612no—
208C2×#121—1221no—
209C2×C4×D4—612no—
210C2×C4×D4—1221no—
211C2×#117—612no—
212C22×C42C2×D6111yescharacteristic
213#432—24no—
214C2×C4⋊C8C2×D6112yes—
215C2×#102—618no—
216C2×C22⋊C8C2×D6112yes—
217C2×#97C2×D6112yes—
218C4×C4⋊C4—612no—
219C4×C4⋊C4—612no—
220C4×C22⋊C4—1221no—
221C4×C22⋊C4—1221no—
222#376—1221no—
223C43—1221no—
224#340—618no—
225C61#2139311yescharacteristic, complemented, Sylow
226C2×#41—1228no—
227C2×#41—2448no—
228C22×Q8—1221no—
229C22×D4—612no—
230C22×D4—1222no—
231C22×D4—1221no—
232C22×D4—1221no—
233C23×C4—1222no—
234C23×C4D24411yescharacteristic
235C23×C4D24411yescharacteristic
236C23×C4C22×D6111yescharacteristic
237C22×C8C22×D6114yes—
238C4⋊Q8—614no—
239C4⋊Q8—1222no—
240#127—614no—
241#127—1222no—
242#124—1228no—
243#121—618no—
244#121—1228no—
245#121—1224no—
246C4×D4—618no—
247C4×D4—1228no—
248C4×D4—1224no—
249#117—1228no—
250C2×C4⋊C4—614no—
251C2×C4⋊C4—612no—
252C2×C4⋊C4—1221no—
253C2×C22⋊C4—1222no—
254C2×C22⋊C4—1222no—
255C2×C22⋊C4—1222no—
256C2×C42—1222no—
257C2×C42—1222no—
258C2×C42—1222no—
259C2×C42C4×D6114yes—
260C2×C42D24411yescharacteristic
261C2×C42D24411yescharacteristic
262C2×C42C22⋊D6114yes—
263C2×C42C22×D6111yescharacteristic
264C2×C42C22×D6111yescharacteristic
265C4⋊C8—24no—
266#102—12216no—
267C22⋊C8—24no—
268#97—24no—
269#95—1222no—
270#95—1222no—
271C24—1222no—
272C24C2×D24411yescharacteristic
273#41—24432no—
274C2×Q8—1224no—
275C2×Q8—2444no—
276C2×D4—614no—
277C2×D4—1228no—
278C2×D4—1222no—
279C2×D4—1228no—
280C2×D4—1224no—
281C2×D4—1224no—
282C2×D4—2444no—
283C2×D4—2448no—
284C2×D4—2444no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—42no—
288C22×C4—1224no—
289C22×C4—1222no—
290C22×C4—1228no—
291C22×C4—2448no—
292C22×C4#1175819814yes—
293C22×C4#1175819814yes—
294C22×C4C2×C4×D6112yes—
295C22×C4C2×D24411yescharacteristic
296C22×C4C2×D24411yescharacteristic
297C22×C4C2×D24411yescharacteristic
298C22×C4C2×D24411yescharacteristic
299C22×C4C2×D24411yescharacteristic
300C22×C4C2×C22⋊D6112yes—
301C22×C4C23×D6111yescharacteristic
302C2×C8—24no—
303C2×C8—28no—
304C4⋊C4—614no—
305C4⋊C4—618no—
306C4⋊C4—1222no—
307C4⋊C4—1224no—
308C4⋊C4—1224no—
309C22⋊C4—1228no—
310C22⋊C4—1228no—
311C22⋊C4—1228no—
312C42—1228no—
313C42—2444no—
314C42—2444no—
315C42#1175819814yes—
316C42#1175819814yes—
317C42C2×C4×D6112yes—
318C42C2×C4×D6112yes—
319C42C2×C22⋊D6112yes—
320C42C2×C22⋊D6112yes—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—1224no—
325C23—2444no—
326C23—2444no—
327C23—2444no—
328C23C2×#1175819812yes—
329C23C22×D24411yescharacteristic
330Q8—2448no—
331D4—1228no—
332D4—2448no—
333D4—24416no—
334D4—2448no—
335D4—2444no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—28no—
342C2×C4—24no—
343C2×C4—44no—
344C2×C4—1224no—
345C2×C4—1228no—
346C2×C4—24416no—
347C2×C4—2444no—
348C2×C4—2444no—
349C2×C4—2444no—
350C2×C4C2×#1175819812yes—
351C2×C4C2×#1175819812yes—
352C2×C4C2×#1175819812yes—
353C2×C4C2×#1175819812yes—
354C2×C4C2×#1175819812yes—
355C2×C4#1451980514yes—
356C2×C4C22×C4×D6111yescharacteristic
357C2×C4C22×C22⋊D6111yescharacteristic
358C2×C4#41×D6114yes—
359C8—44no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—44no—
364C22—4888no—
365C22—2444no—
366C22—2444no—
367C22—2444no—
368C22C22×#1175819811yescharacteristic
369C22#1684027412yes—
370C22C2×#1451980512yes—
371C22#1684056212yes—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—2448no—
377C4—2444no—
378C4#1683976814yes—
379C4#1684023012yes—
380C4#1684030812yes—
381C2—42no—
382C2—4884no—
383C2#2144830711yescharacteristic
384C2C2×#1683976811yescharacteristic
385C2#2145433911yescharacteristic
386C2#2145597114yes—
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#432C2
socleC23×C61C22×D4
prime radicalC2×#16840312C2
cosocleC22×C4×C61C24
hypercenterC2×#432D61
hypocenterC61#21393
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
2. C4×#14519736C2semidirect
3. #21446770C2semidirect
4. #21446745C2semidirect
5. #21446139C2semidirect
6. #21445645C2semidirect
7. C2×C61×#432C2semidirect
10. #16840312C4semidirect
123. C4⋊Dic61#41semidirect
189. C2×#432D61semidirect
225. C61#21393semidirect

Maximal subgroups

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

subgroupindex
C2×#168403122
C4×#145197362
#214467702
#214467452
#214461392
#214456452
C2×C61×#4322
#2139361