Finite Group Databasebeta

← #29708152

Subgroups — #29708152

Order 15616.

1334 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840388C211yesmaximal, characteristic
2#21448433C212yesmaximal
3#21448431C211yesmaximal, characteristic
4#21447173C218yesmaximal, complemented
5C2×#16839528C212yesmaximal
6C2×C61×#342C211yesmaximal, characteristic
7#16840388—24no—
8#16840388—22no—
9#16840388C2214yes—
10C2×#14519756—22no—
11C2×#14519754—24no—
12C22×C22⋊Dic61C2211yescharacteristic
13C22×C22⋊Dic61C2212yes—
14C22×C22⋊Dic61C2211yescharacteristic
15#16839637—24no—
16#16839637C2218yes—
17#16839635—22no—
18#16839634C2214yes—
19#16839622—24no—
20#16839620—22no—
21#16839528C2218yes—
22C22×C22⋊C4×C61C2211yescharacteristic
23C22×C22⋊C4×C61C2212yes—
24C61×#342C2214yes—
25C22×Dic122—22no—
26C23×Dic61—21no—
27C23×Dic61D414yes—
28C23×Dic61C2312yes—
29#14519756—28no—
30#14519756—28no—
31#14519754—216no—
32#14519754—216no—
33C2×C4⋊Dic61—22no—
34C2×C4⋊Dic61—28no—
35C2×#11758196—24no—
36C2×#11758196—24no—
37C2×C22⋊Dic61—22no—
38C2×C22⋊Dic61—22no—
39C2×C22⋊Dic61—22no—
40C2×C22⋊Dic61—22no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—22no—
43C2×C22⋊Dic61—21no—
44C2×C22⋊Dic61—22no—
45C2×C22⋊Dic61—21no—
46C2×C22⋊Dic61—22no—
47C2×C22⋊Dic61D414yes—
48C2×C22⋊Dic61Q814yescomplemented
49C2×C22⋊Dic61C2314yes—
50C2×C22⋊Dic61C2312yes—
51#14519684—24no—
52#14519684—28no—
53C25×C61C2311yescharacteristic
54C23×C4×C61—22no—
55C23×C4×C61D412yes—
56C2×C22⋊C4×C61—22no—
57C2×C22⋊C4×C61—22no—
58C2×C22⋊C4×C61—22no—
59C2×C22⋊C4×C61—24no—
60C2×C22⋊C4×C61D412yes—
61C2×C22⋊C4×C61Q818yescomplemented
62C2×C22⋊C4×C61C2312yes—
63C2×C22⋊C4×C61C2314yes—
64C24×C61—82no—
65C24×C61C2×D411yescharacteristic
66C24×C61C2×D411yescharacteristic
67C24×C61C2×D412yes—
68C24×C61C2×D412yes—
69C24×C61C2×Q814yes—
70C24×C61C2×Q812yes—
71C24×C61#4112yes—
72C24×C61C2411yescharacteristic
73C22×C4×C61—22no—
74C22×C4×C61—24no—
75C22×C4×C61—22no—
76C22×C4×C61—28no—
77C22×C4×C61—82no—
78C22×C4×C61—84no—
79C22×C4×C61C2×D414yes—
80C22×Dic61—24no—
81C22×Dic61—22no—
82C22×Dic61—24no—
83C22×Dic61—24no—
84C22×Dic61—22no—
85C22×Dic61—24no—
86C22×Dic61—82no—
87C22×Dic61—84no—
88C22×Dic61—82no—
89C22×Dic61C2×D418yes—
90C2×Dic122—28no—
91C2×Dic122—84no—
92C22⋊C4×C61—24no—
93C22⋊C4×C61—28no—
94C22⋊C4×C61—24no—
95C22⋊C4×C61—24no—
96C22⋊C4×C61C2×Q818yescomplemented
97C22⋊Dic61—24no—
98C22⋊Dic61—24no—
99C22⋊Dic61—24no—
100C22⋊Dic61—24no—
101C22⋊Dic61—24no—
102C22⋊Dic61—22no—
103C22⋊Dic61—41no—
104C22⋊Dic61—42no—
105C22⋊Dic61—41no—
106C22⋊Dic61C2×Q814yescomplemented
107C4⋊Dic61—28no—
108C4⋊Dic61—88no—
109#11758196—216no—
110#11758196—84no—
111C23×C61—44no—
112C23×C61—42no—
113C23×C61—42no—
114C23×C61—42no—
115C23×C61—44no—
116C23×C61—82no—
117C23×C61—84no—
118C23×C61—82no—
119C23×C61#12011yescharacteristic
120C23×C61#12012yes—
121C23×C61#12011yescharacteristic
122C23×C61C22⋊Q814yes—
123C23×C61C22⋊Q814yes—
124C23×C61C22⋊Q812yes—
125C23×C61C22⋊Q812yes—
126C23×C61C4⋊Q814yes—
127C23×C61C4⋊Q814yes—
128C23×C61C4⋊Q812yes—
129C23×C61C4⋊Q812yes—
130C23×C61C22×D412yes—
131C23×C61C22×D411yescharacteristic
132C23×C61C22×Q811yescharacteristic
133C23×C61C22×Q812yes—
134C23×C61C2×#4111yescharacteristic
135C2×C4×C61—24no—
136C2×C4×C61—28no—
137C2×C4×C61—48no—
138C2×C4×C61—44no—
139C2×C4×C61—44no—
140C2×C4×C61—82no—
141C2×C4×C61—88no—
142C2×Dic61—28no—
143C2×Dic61—24no—
144C2×Dic61—44no—
145C2×Dic61—48no—
146C2×Dic61—48no—
147C2×Dic61—44no—
148C2×Dic61—48no—
149C2×Dic61—42no—
150C2×Dic61—84no—
151C2×Dic61—84no—
152Dic122—88no—
153#23565—611nomaximal, Sylow
154C22×C61—22no—
155C22×C61—24no—
156C22×C61—22no—
157C22×C61—24no—
158C22×C61—22no—
159C22×C61—42no—
160C22×C61—44no—
161C22×C61—44no—
162C22×C61—42no—
163C22×C61—44no—
164C22×C61—44no—
165C22×C61—42no—
166C22×C61—42no—
167C22×C61—41no—
168C22×C61—42no—
169C22×C61—41no—
170C22×C61#39512yes—
171C22×C61#39514yes—
172C22×C61#39512yes—
173C22×C61C2×#12011yescharacteristic
174C22×C61C2×C22⋊Q812yes—
175C22×C61C2×C22⋊Q811yescharacteristic
176C22×C61C2×C4⋊Q812yes—
177C22×C61C2×C4⋊Q811yescharacteristic
178C4×C61—48no—
179C4×C61—44no—
180Dic61—44no—
181Dic61—48no—
182Dic61—82no—
183C2×#543—611no—
184#2315—611no—
185#2315—612no—
186#2062—618no—
187C2×#342—612no—
188C2×#342D6111yescharacteristic
189C2×C61—21no—
190C2×C61—22no—
191C2×C61—22no—
192C2×C61—21no—
193C2×C61—22no—
194C2×C61—24no—
195C2×C61—22no—
196C2×C61C2×#39511yescharacteristic
197C2×C61#302612yes—
198#543—614no—
199#543—1224no—
200#543—1222no—
201C2×C22⋊Q8—1224no—
202C2×C22⋊Q8—1222no—
203C22×C22⋊C4—611no—
204C22×C22⋊C4—612no—
205C22×C22⋊C4—611no—
206C22×C22⋊C4C2×D6111yescharacteristic
207C22×C22⋊C4C2×D6112yes—
208#385—614no—
209#385—618no—
210#385—1224no—
211#385—1222no—
212#380—1224no—
213#380—1222no—
214#342—618no—
215#342C2×D6114yes—
216C61#2356511yescharacteristic, complemented, Sylow
217C25C22×D6111yescharacteristic
218C22×Q8—1222no—
219C23×C4—22no—
220C23×C4—614no—
221C23×C4—612no—
222C23×C4—1221no—
223C23×C4D24412yes—
224C22⋊Q8—12216no—
225C22⋊Q8—12216no—
226C22⋊Q8—1228no—
227C22⋊Q8—1228no—
228C2×C4⋊C4—1224no—
229C2×C4⋊C4—1222no—
230C2×C4⋊C4—1228no—
231C2×C4⋊C4—1224no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—614no—
237C2×C22⋊C4—612no—
238C2×C22⋊C4—614no—
239C2×C22⋊C4—614no—
240C2×C22⋊C4—1222no—
241C2×C22⋊C4—1222no—
242C2×C22⋊C4—1221no—
243C2×C22⋊C4—1222no—
244C2×C22⋊C4—1222no—
245C2×C22⋊C4—1222no—
246C2×C22⋊C4—1221no—
247C2×C22⋊C4—1221no—
248C2×C22⋊C4—1222no—
249C2×C22⋊C4—1222no—
250C2×C22⋊C4Dic12218yescomplemented
251C2×C22⋊C4D24412yes—
252C2×C22⋊C4C22×D6112yes—
253C2×C22⋊C4C22×D6114yes—
254#95—1228no—
255#95—1224no—
256C24—82no—
257C24C2×Dic12214yes—
258C24C2×D24411yescharacteristic
259C24C2×D24411yescharacteristic
260C24D4×D6112yes—
261C24D4×D6112yes—
262C24#1175822412yes—
263C24Q8×D6112yes—
264C24C23×D6111yescharacteristic
265C2×Q8—1228no—
266C2×Q8—4884no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—28no—
271C22×C4—82no—
272C22×C4—84no—
273C22×C4—618no—
274C22×C4—1224no—
275C22×C4—1224no—
276C22×C4—1222no—
277C22×C4—1224no—
278C22×C4—1224no—
279C22×C4—1222no—
280C22×C4—4882no—
281C22×C4—4884no—
282C22×C4—4882no—
283C22×C4C2×D24414yes—
284C4⋊C4—12216no—
285C4⋊C4—1228no—
286C4⋊C4—4884no—
287C4⋊C4—4888no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—28no—
292C22⋊C4—614no—
293C22⋊C4—1224no—
294C22⋊C4—1224no—
295C22⋊C4—1222no—
296C22⋊C4—1224no—
297C22⋊C4—1224no—
298C22⋊C4—1224no—
299C22⋊C4—2442no—
300C22⋊C4—2441no—
301C22⋊C4—2441no—
302C22⋊C4C2×Dic12218yescomplemented
303C23—42no—
304C23—44no—
305C23—42no—
306C23—44no—
307C23—42no—
308C23—84no—
309C23—82no—
310C23—82no—
311C23#1451974712yes—
312C23#1451974711yescharacteristic
313C23#1451974711yescharacteristic
314C23#1451975414yes—
315C23#1451975414yes—
316C23#1451975712yes—
317C23#1451975712yes—
318C23#1451977614yes—
319C23#1451977614yes—
320C23#1451977712yes—
321C23#1451977712yes—
322C23C2×D4×D6112yes—
323C23C22×D24411yescharacteristic
324C23C2×Q8×D6111yescharacteristic
325C23C22×Dic12212yes—
326C23C2×#1175822411yescharacteristic
327Q8—4888no—
328C2×C4—24no—
329C2×C4—28no—
330C2×C4—48no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—82no—
334C2×C4—88no—
335C2×C4—1228no—
336C2×C4—1224no—
337C2×C4—4884no—
338C2×C4—4884no—
339C2×C4—2448no—
340C2×C4—2444no—
341C2×C4—2442no—
342C2×C4—2444no—
343C2×C4—2448no—
344C2×C4—2448no—
345C22—24no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—24no—
350C22—44no—
351C22—42no—
352C22—41no—
353C22—42no—
354C22—41no—
355C22—44no—
356C22—42no—
357C22—44no—
358C22—44no—
359C22—42no—
360C22—42no—
361C22#1683968314yes—
362C22#1683968312yes—
363C22#1683968312yes—
364C22C2×#1451974711yescharacteristic
365C22C2×#1451975412yes—
366C22C2×#1451975711yescharacteristic
367C22C2×#1451977612yes—
368C22C2×#1451977711yescharacteristic
369C4—48no—
370C4—44no—
371C4—4882no—
372C4—2444no—
373C4—2448no—
374C2—22no—
375C2—21no—
376C2—22no—
377C2—22no—
378C2—24no—
379C2—22no—
380C2—21no—
381C2C2×#1683968311yescharacteristic
382C2#2145521812yes—
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC2×C61×#342C2
socleC22×C61C2×#120
prime radicalC25×C61C23
hypercenterC2×#342D61
hypocenterC61#23565
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C25 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
4. #21447173C2semidirect
48. C2×C22⋊Dic61Q8semidirect
61. C2×C22⋊C4×C61Q8semidirect
96. C22⋊C4×C61C2×Q8semidirect
106. C22⋊Dic61C2×Q8semidirect
216. C61#23565semidirect
250. C2×C22⋊C4Dic122semidirect
302. C22⋊C4C2×Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168403882
#214484332
#214484312
#214471732
C2×#168395282
C2×C61×#3422
#2356561