Finite Group Databasebeta

← #29602725

Subgroups — #29602725

Order 15616.

2230 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839630C211yesmaximal, characteristic, complemented
2C2×#16839608C211yesmaximal, characteristic, complemented
3#21444087C212yesmaximal, complemented
4#21444060C212yesmaximal
5#21444056C214yesmaximal, complemented
6#21444042C214yesmaximal, complemented
7C22×C4×D4×C61C211yesmaximal, characteristic
8C22×C22⋊Dic61C2218yes—
9C22×C4×Dic61C2212yes—
10#16839630—24no—
11#16839608—24no—
12C2×#14519684C2212yes—
13C2×#14519684C2212yes—
14C2×#14519684C2212yes—
15C2×#14519684C2212yes—
16C2×#14519684C2212yes—
17C2×#14519684C2218yes—
18C23×D4×C61C2211yescharacteristic
19C24×C4×C61C2212yes—
20C2×C4×D4×C61—24no—
21C2×C4×D4×C61—24no—
22C2×C4×D4×C61C418yescomplemented
23C22×C4⋊C4×C61C2211yescharacteristic
24C22×C22⋊C4×C61C2212yes—
25C22×C42×C61C2211yescharacteristic
26C23×Dic61C2314yes—
27C23×Dic61C2314yes—
28C2×C22⋊Dic61—216no—
29C2×C22⋊Dic61—216no—
30C2×C22⋊Dic61—48no—
31C2×C4×Dic61—24no—
32C2×C4×Dic61—28no—
33#14519684—24no—
34#14519684—24no—
35#14519684—24no—
36#14519684—24no—
37#14519684—24no—
38#14519684—48no—
39C25×C61C2312yes—
40C22×D4×C61—28no—
41C22×D4×C61—22no—
42C22×D4×C61—22no—
43C22×D4×C61C2×C414yes—
44C23×C4×C61—24no—
45C23×C4×C61—24no—
46C23×C4×C61—48no—
47C23×C4×C61C2×C418yes—
48C23×C4×C61C2312yes—
49C23×C4×C61C2311yescharacteristic
50C23×C4×C61C2311yescharacteristic
51C23×C4×C61C2311yescharacteristic
52C4×D4×C61—232no—
53C2×C4⋊C4×C61—22no—
54C2×C4⋊C4×C61—22no—
55C2×C4⋊C4×C61C2×C414yes—
56C2×C22⋊C4×C61—24no—
57C2×C22⋊C4×C61—24no—
58C2×C22⋊C4×C61C2×C418yes—
59C2×C42×C61C2×C414yes—
60C2×C42×C61D414yes—
61C2×C42×C61D414yes—
62C24×C61—22no—
63C24×C61—22no—
64C24×C61—24no—
65C24×C61—28no—
66C24×C61—28no—
67C24×C61C22×C414yes—
68C24×C61#4118yes—
69C24×C61C2411yescharacteristic
70C2×D4×C61—216no—
71C2×D4×C61—216no—
72C2×D4×C61—216no—
73C2×D4×C61—28no—
74C22×C4×C61—24no—
75C22×C4×C61—24no—
76C22×C4×C61—24no—
77C22×C4×C61—22no—
78C22×C4×C61—22no—
79C22×C4×C61—28no—
80C22×C4×C61—216no—
81C22×C4×C61—416no—
82C22×C4×C61—416no—
83C22×C4×C61—416no—
84C22×C4×C61C22×C414yes—
85C22×C4×C61C22×C412yes—
86C22×C4×C61C22×C412yes—
87C22×C4×C61C22×C412yes—
88C22×C4×C61C2×D412yes—
89C22×C4×C61C2×D412yes—
90C22×C4×C61C2×D412yes—
91C22×C4×C61C2×D412yes—
92C22×C4×C61C2×D412yes—
93C22×C4×C61C2×D412yes—
94C22×Dic61—24no—
95C22×Dic61—24no—
96C22×Dic61—28no—
97C22×Dic61—28no—
98C22×Dic61—24no—
99C22×Dic61—28no—
100C22×Dic61—28no—
101C22×Dic61—24no—
102C22×Dic61—24no—
103C22×Dic61#4118yes—
104C4⋊C4×C61—28no—
105C22⋊C4×C61—216no—
106C42×C61C22⋊C4116yescomplemented
107C22⋊Dic61—432no—
108C4×Dic61—216no—
109C23×C61—24no—
110C23×C61—24no—
111C23×C61—28no—
112C23×C61—28no—
113C23×C61—28no—
114C23×C61—28no—
115C23×C61—216no—
116C23×C61—216no—
117C23×C61—44no—
118C23×C61—44no—
119C23×C61—44no—
120C23×C61—44no—
121C23×C61—48no—
122C23×C61—48no—
123C23×C61#11718yes—
124C23×C61C23×C411yescharacteristic
125C23×C61C22×D411yescharacteristic
126C23×C61C22×D411yescharacteristic
127C23×C61C2×#4114yes—
128C23×C61C2×#4114yes—
129C23×C61H(5,𝔽2)14yes—
130D4×C61—232no—
131C2×C4×C61—24no—
132C2×C4×C61—24no—
133C2×C4×C61—24no—
134C2×C4×C61—24no—
135C2×C4×C61—24no—
136C2×C4×C61—24no—
137C2×C4×C61—24no—
138C2×C4×C61—24no—
139C2×C4×C61—24no—
140C2×C4×C61—24no—
141C2×C4×C61—28no—
142C2×C4×C61—432no—
143C2×C4×C61—44no—
144C2×C4×C61—44no—
145C2×C4×C61C2×C22⋊C414yes—
146C2×C4×C61C2×C22⋊C414yes—
147C2×C4×C61C2×C22⋊C414yes—
148C2×Dic61—28no—
149C2×Dic61—216no—
150C2×Dic61—44no—
151C2×Dic61—44no—
152C2×Dic61—48no—
153C2×Dic61—44no—
154C2×Dic61—44no—
155C2×Dic61—44no—
156C2×Dic61—48no—
157C2×Dic61—48no—
158#13959—611nomaximal, Sylow
159C22×C61—216no—
160C22×C61—28no—
161C22×C61—44no—
162C22×C61—416no—
163C22×C61—416no—
164C22×C61—48no—
165C22×C61—44no—
166C22×C61C22×C22⋊C411yescharacteristic
167C22×C61C2×#11712yes—
168C22×C61C4×#4112yes—
169C22×C61#51812yes—
170C22×C61#52512yes—
171C22×C61#52512yes—
172C22×C61#52512yes—
173C22×C61#53212yes—
174C22×C61#53512yes—
175C22×C61#53512yes—
176C22×C61#55118yes—
177C22×C61#55318yes—
178C4×C61—24no—
179C4×C61—24no—
180C4×C61—24no—
181C4×C61—44no—
182Dic61—48no—
183Dic61—84no—
184C22×C4×D4D6111yescharacteristic
185C2×#383—611no—
186C2×#376—611no—
187#1445—612no—
188#1438—612no—
189#1437—614no—
190C22⋊#95—614no—
191C2×C61—48no—
192C2×C61#228011yescharacteristic
193C2×C61#229711yescharacteristic
194C2×C61#230911yescharacteristic
195C2×C61#235314yes—
196C2×C61#259814yes—
197C2×C61#260314yes—
198C23×D4C2×D6111yescharacteristic
199C24×C4C2×D6112yes—
200C2×C4×D4—24no—
201C2×C4×D4—24no—
202C2×C4×D4Dic6118yescomplemented
203C22×C4⋊C4C2×D6111yescharacteristic
204C22×C22⋊C4—618no—
205C22×C22⋊C4C2×D6112yes—
206C22×C42—612no—
207C22×C42C2×D6111yescharacteristic
208#383—1224no—
209#376—1224no—
210C2×#95—612no—
211C2×#95—612no—
212C2×#95—612no—
213C2×#95—612no—
214C2×#95—612no—
215C2×#95—618no—
216C61#1395911yescharacteristic, complemented, Sylow
217C25C22×D6112yes—
218C22×D4—28no—
219C22×D4—22no—
220C22×D4—22no—
221C22×D4C2×Dic6114yes—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—48no—
225C23×C4—614no—
226C23×C4—614no—
227C23×C4C2×Dic6118yes—
228C23×C4C22×D6112yes—
229C23×C4C22×D6111yescharacteristic
230C23×C4C22×D6111yescharacteristic
231C23×C4C22×D6111yescharacteristic
232C4×D4—232no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4C2×Dic6114yes—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—12216no—
239C2×C22⋊C4—12216no—
240C2×C22⋊C4—2448no—
241C2×C22⋊C4C2×Dic6118yes—
242C2×C42—1228no—
243C2×C42—1224no—
244C2×C42C2×Dic6114yes—
245C2×C42C22⋊D6114yes—
246C2×C42C22⋊D6114yes—
247#95—1224no—
248#95—1224no—
249#95—1224no—
250#95—1224no—
251#95—1224no—
252#95—2448no—
253C24—22no—
254C24—22no—
255C24—28no—
256C24—24no—
257C24—28no—
258C24#1175822218yes—
259C24C22×Dic6114yes—
260C24C23×D6111yescharacteristic
261C2×D4—216no—
262C2×D4—216no—
263C2×D4—216no—
264C2×D4—28no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—28no—
271C22×C4—216no—
272C22×C4—416no—
273C22×C4—416no—
274C22×C4—416no—
275C22×C4—618no—
276C22×C4—1224no—
277C22×C4—1224no—
278C22×C4—1228no—
279C22×C4—1228no—
280C22×C4—1224no—
281C22×C4—1228no—
282C22×C4—1228no—
283C22×C4—1224no—
284C22×C4—1224no—
285C22×C4C22×Dic6114yes—
286C22×C4C22×Dic6112yes—
287C22×C4C22×Dic6112yes—
288C22×C4C22×Dic6112yes—
289C22×C4C2×C22⋊D6112yes—
290C22×C4C2×C22⋊D6112yes—
291C22×C4C2×C22⋊D6112yes—
292C22×C4C2×C22⋊D6112yes—
293C22×C4C2×C22⋊D6112yes—
294C22×C4C2×C22⋊D6112yes—
295C4⋊C4—28no—
296C22⋊C4—216no—
297C22⋊C4—24432no—
298C42—12216no—
299C42C22⋊Dic61116yescomplemented
300C23—216no—
301C23—24no—
302C23—24no—
303C23—28no—
304C23—28no—
305C23—28no—
306C23—28no—
307C23—216no—
308C23—48no—
309C23—44no—
310C23—44no—
311C23—44no—
312C23—44no—
313C23—48no—
314C23#1451973618yes—
315C23C23×Dic6111yescharacteristic
316C23C22×C22⋊D6111yescharacteristic
317C23C22×C22⋊D6111yescharacteristic
318C23#41×D6114yes—
319C23C2×#1175822214yes—
320C23#1451982214yes—
321D4—232no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—28no—
333C2×C4—432no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—1228no—
337C2×C4—12216no—
338C2×C4—2448no—
339C2×C4—2444no—
340C2×C4—2444no—
341C2×C4—2444no—
342C2×C4—2448no—
343C2×C4—2448no—
344C2×C4—2444no—
345C2×C4—2444no—
346C2×C4C2×C22⋊Dic6114yes—
347C2×C4C2×C22⋊Dic6114yes—
348C2×C4C2×C22⋊Dic6114yes—
349C22—28no—
350C22—216no—
351C22—48no—
352C22—416no—
353C22—44no—
354C22—416no—
355C22—44no—
356C22C22×C22⋊Dic6111yescharacteristic
357C22C2×#1451973612yes—
358C22#41×Dic6112yes—
359C22#1684023712yes—
360C22#1684026912yes—
361C22#1684027312yes—
362C22#1684027312yes—
363C22#1684030812yes—
364C22#1684032912yes—
365C22#1684032912yes—
366C22#1684044118yes—
367C22#1684046118yes—
368C4—24no—
369C4—24no—
370C4—24no—
371C4—44no—
372C4—4884no—
373C4—2448no—
374C2—48no—
375C2#2144820211yescharacteristic
376C2#2144831011yescharacteristic
377C2#2144840111yescharacteristic
378C2#2144875914yes—
379C2#2145137914yes—
380C2#2145144014yes—
subgroupG/H
center, FrattiniC24C23×D61
commutatorC23×C61C23×C4
FittingC22×C4×D4×C61C2
socle, cosocleC24×C61C24
prime radicalC23×D4×C61C22
hypercenterC22×C4×D4D61
hypocenterC61#13959
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C24 ◁ C22×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16839630C2semidirect
2. C2×#16839608C2semidirect
3. #21444087C2semidirect
5. #21444056C2semidirect
6. #21444042C2semidirect
22. C2×C4×D4×C61C4semidirect
106. C42×C61C22⋊C4semidirect
202. C2×C4×D4Dic61semidirect
216. C61#13959semidirect
299. C42C22⋊Dic61semidirect

Maximal subgroups

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

subgroupindex
C2×#168396302
C2×#168396082
#214440872
#214440602
#214440562
#214440422
C22×C4×D4×C612
#1395961