Finite Group Databasebeta

← #29708133

Subgroups — #29708133

Order 15616.

1074 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840393C211yesmaximal, characteristic
2#21454409C212yesmaximal
3#21448443C211yesmaximal, characteristic
4#21447170C214yesmaximal, complemented
5#21447165C214yesmaximal, complemented
6#21445530C212yesmaximal
7C2×C61×#343C211yesmaximal, characteristic
8#16840393—24no—
9#16840393—22no—
10#16840393C2214yes—
11C2×#14519776—22no—
12C2×#14519768—22no—
13C2×#14519756—22no—
14C2×#14519736C2212yes—
15C2×#14519736C2211yescharacteristic
16C2×#14519736C2211yescharacteristic
17#16839804—22no—
18#16839804—22no—
19#16839804C2214yes—
20#16839804C2214yes—
21#16839800—24no—
22#16839658—22no—
23#16839635—22no—
24#16839631C2212yes—
25#16839627C2212yes—
26#16839523C2218yes—
27C22×M16×C61C2212yes—
28C22×C4⋊C4×C61C2211yescharacteristic
29C61×#343C2214yes—
30C22×Dic122—22no—
31#14519776—28no—
32#14519776—28no—
33#14519768—28no—
34#14519768—28no—
35#14519756—28no—
36#14519756—28no—
37#14519736—22no—
38#14519736—22no—
39#14519736—22no—
40#14519736—22no—
41#14519736—22no—
42#14519736D414yes—
43#14519736Q814yescomplemented
44#14519736C2314yes—
45C2×C4⋊Dic61—22no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61D412yes—
48C2×C4⋊Dic61D412yes—
49C2×C4⋊Dic61C2311yescharacteristic
50C2×C4⋊Dic61C2311yescharacteristic
51C2×#11758196—24no—
52C2×#11758196—24no—
53C2×C22⋊Dic61—22no—
54C2×C22⋊Dic61—21no—
55C2×C22⋊Dic61—21no—
56C2×C4×Dic61—22no—
57C2×C4×Dic61—21no—
58C2×C4×Dic61C2312yes—
59#14519711—24no—
60#14519711—24no—
61#14519709—24no—
62#14519709—24no—
63#14519684—24no—
64C23×C4×C61D412yes—
65C23×C4×C61C2311yescharacteristic
66C2×M16×C61—22no—
67C2×M16×C61—24no—
68C2×M16×C61Q818yescomplemented
69C2×M16×C61C2314yes—
70C22×C8×C61—22no—
71C2×C4⋊C4×C61—22no—
72C2×C4⋊C4×C61—22no—
73C2×C4⋊C4×C61D412yes—
74C2×C4⋊C4×C61C2312yes—
75C24×C61C2×D411yescharacteristic
76C22×C4×C61—22no—
77C22×C4×C61—22no—
78C22×C4×C61—82no—
79C22×C4×C61C2×D414yes—
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D412yes—
82C22×C4×C61C2×D412yes—
83C22×C4×C61C2×Q814yes—
84C22×C4×C61C2×Q812yes—
85C22×C4×C61#4112yes—
86C22×C4×C61C2411yescharacteristic
87C22×Dic61—22no—
88C22×Dic61—22no—
89C22×Dic61—22no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C2×Dic122—28no—
93C2×Dic122—84no—
94M16×C61—28no—
95M16×C61—24no—
96M16×C61C2×Q818yescomplemented
97C2×C8×C61—28no—
98C2×C8×C61—24no—
99C4⋊C4×C61—24no—
100C4⋊C4×C61—24no—
101C22⋊Dic61—24no—
102C22⋊Dic61—24no—
103C22⋊Dic61—24no—
104C22⋊Dic61—41no—
105C22⋊Dic61—41no—
106C4⋊Dic61—28no—
107C4⋊Dic61—22no—
108C4⋊Dic61—22no—
109C4⋊Dic61—24no—
110C4⋊Dic61—22no—
111C4⋊Dic61—22no—
112C4⋊Dic61C2×D414yes—
113C4⋊Dic61C2×D414yes—
114#11758196—216no—
115#11758196—84no—
116C4×Dic61—24no—
117C4×Dic61—24no—
118C4×Dic61—22no—
119C4×Dic61—42no—
120C4×Dic61C2×Q814yescomplemented
121C23×C61—41no—
122C23×C61—41no—
123C23×C61#12011yescharacteristic
124C23×C61#12011yescharacteristic
125C23×C61C22⋊Q812yes—
126C23×C61C4⋊Q812yes—
127C23×C61C22×D411yescharacteristic
128C2×C4×C61—24no—
129C2×C4×C61—44no—
130C2×C4×C61—44no—
131C2×C4×C61—82no—
132C2×C4×C61#12012yes—
133C2×C4×C61C22⋊Q814yes—
134C2×C4×C61C22⋊Q814yes—
135C2×C4×C61C22⋊Q812yes—
136C2×C4×C61C4⋊Q814yes—
137C2×C4×C61C4⋊Q814yes—
138C2×C4×C61C4⋊Q812yes—
139C2×C4×C61C22×D412yes—
140C2×C4×C61C22×Q812yes—
141C2×C4×C61C22×Q811yescharacteristic
142C2×C4×C61C2×#4111yescharacteristic
143C2×Dic61—24no—
144C2×Dic61—24no—
145C2×Dic61—24no—
146C2×Dic61—42no—
147C2×Dic61—44no—
148C2×Dic61—42no—
149C2×Dic61—44no—
150C2×Dic61—42no—
151C2×Dic61—42no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154Dic122—88no—
155C8×C61—28no—
156#23561—611nomaximal, Sylow
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—21no—
162C22×C61—21no—
163C22×C61—42no—
164C22×C61—42no—
165C22×C61#39512yes—
166C22×C61#39512yes—
167C22×C61C2×#12011yescharacteristic
168C22×C61C2×C22⋊Q811yescharacteristic
169C22×C61C2×C4⋊Q811yescharacteristic
170C4×C61—44no—
171C4×C61#39514yes—
172C4×C61C2×C22⋊Q812yes—
173C4×C61C2×C4⋊Q812yes—
174Dic61—44no—
175Dic61—44no—
176Dic61—44no—
177Dic61—81no—
178Dic61—81no—
179C2×#544—611no—
180#2911—612no—
181#2317—611no—
182#2061—614no—
183#2060—614no—
184C2×#343D6111yescharacteristic
185#1733—612no—
186C2×C61—21no—
187C2×C61—21no—
188C2×C61—22no—
189C2×C61—22no—
190C2×C61C2×#39511yescharacteristic
191C2×C61#301911yescharacteristic
192C2×C61#302011yescharacteristic
193C22×M16C2×D6112yes—
194#544—614no—
195#544—1224no—
196#544—1222no—
197C2×C4⋊Q8—1222no—
198C2×#125—1222no—
199C2×C22⋊Q8—1222no—
200C2×#117—611no—
201C2×#117—612no—
202C2×#117—611no—
203C22×C4⋊C4C2×D6111yescharacteristic
204#428—614no—
205#428—614no—
206#428—1222no—
207#428—1222no—
208#427—1224no—
209#389—1222no—
210#385—1222no—
211#384—612no—
212#382—612no—
213#343C2×D6114yes—
214#339—618no—
215C61#2356111yescharacteristic, complemented, Sylow
216C22×Q8—1222no—
217C23×C4D24412yes—
218C23×C4C22×D6111yescharacteristic
219C2×M16—22no—
220C2×M16—24no—
221C2×M16Dic12218yescomplemented
222C2×M16C22×D6114yes—
223C22×C8—22no—
224C4⋊Q8—1228no—
225C4⋊Q8—1228no—
226#125—1228no—
227#125—1228no—
228C22⋊Q8—1228no—
229C22⋊Q8—1228no—
230#117—614no—
231#117—614no—
232#117—614no—
233#117—1222no—
234#117—1222no—
235#117—1222no—
236#117—1222no—
237#117—1222no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—612no—
241C2×C4⋊C4—612no—
242C2×C4⋊C4—611no—
243C2×C4⋊C4—611no—
244C2×C4⋊C4—1222no—
245C2×C4⋊C4—1224no—
246C2×C4⋊C4—1221no—
247C2×C4⋊C4—1224no—
248C2×C4⋊C4D24412yes—
249C2×C4⋊C4C22×D6112yes—
250C2×C22⋊C4—1222no—
251C2×C22⋊C4—1221no—
252C2×C22⋊C4—1221no—
253C2×C42—612no—
254C2×C42—1222no—
255C2×C42—1221no—
256#107—1224no—
257#107—1224no—
258#106—1224no—
259#106—1224no—
260#95—1224no—
261C24C2×D24411yescharacteristic
262C2×Q8—1228no—
263C2×Q8—4884no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—82no—
267C22×C4—1222no—
268C22×C4—1222no—
269C22×C4—1222no—
270C22×C4—1221no—
271C22×C4—1221no—
272C22×C4C2×Dic12214yes—
273C22×C4C2×D24414yes—
274C22×C4C2×D24411yescharacteristic
275C22×C4D4×D6112yes—
276C22×C4D4×D6112yes—
277C22×C4#1175822412yes—
278C22×C4Q8×D6112yes—
279C22×C4C23×D6111yescharacteristic
280M16—24no—
281M16—28no—
282M16C2×Dic12218yescomplemented
283C2×C8—24no—
284C2×C8—28no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—614no—
288C4⋊C4—614no—
289C4⋊C4—12216no—
290C4⋊C4—1228no—
291C4⋊C4—1222no—
292C4⋊C4—1222no—
293C4⋊C4—1224no—
294C4⋊C4—1222no—
295C4⋊C4—1222no—
296C4⋊C4—4884no—
297C22⋊C4—1224no—
298C22⋊C4—1224no—
299C22⋊C4—1224no—
300C22⋊C4—2441no—
301C22⋊C4—2441no—
302C42—614no—
303C42—1224no—
304C42—1222no—
305C42—1224no—
306C42—2442no—
307C23—41no—
308C23—41no—
309C23#1451974711yescharacteristic
310C23#1451974711yescharacteristic
311C23#1451975712yes—
312C23#1451977712yes—
313C23C22×D24411yescharacteristic
314Q8—4888no—
315C2×C4—24no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—82no—
319C2×C4—1224no—
320C2×C4—1224no—
321C2×C4—1224no—
322C2×C4—2442no—
323C2×C4—2442no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4—2442no—
327C2×C4—2442no—
328C2×C4—2444no—
329C2×C4—2444no—
330C2×C4#1451974712yes—
331C2×C4#1451975414yes—
332C2×C4#1451975414yes—
333C2×C4#1451975712yes—
334C2×C4#1451977614yes—
335C2×C4#1451977614yes—
336C2×C4#1451977712yes—
337C2×C4C2×D4×D6112yes—
338C2×C4C2×Q8×D6111yescharacteristic
339C2×C4C22×Dic12212yes—
340C2×C4C2×#1175822411yescharacteristic
341C8—28no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—42no—
349C22—42no—
350C22#1683968312yes—
351C22#1683968312yes—
352C22C2×#1451974711yescharacteristic
353C22C2×#1451975711yescharacteristic
354C22C2×#1451977711yescharacteristic
355C4—44no—
356C4—4881no—
357C4—4881no—
358C4—2444no—
359C4—2444no—
360C4—2444no—
361C4#1683968314yes—
362C4C2×#1451975412yes—
363C4C2×#1451977612yes—
364C2—21no—
365C2—22no—
366C2—22no—
367C2—21no—
368C2C2×#1683968311yescharacteristic
369C2#2145517911yescharacteristic
370C2#2145518611yescharacteristic
subgroupG/H
centerC22C2×#14519747
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#343C2
socleC22×C61C2×#120
prime radicalC24×C61C2×D4
hypercenterC2×#343D61
hypocenterC61#23561
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
4. #21447170C2semidirect
5. #21447165C2semidirect
43. #14519736Q8semidirect
68. C2×M16×C61Q8semidirect
96. M16×C61C2×Q8semidirect
120. C4×Dic61C2×Q8semidirect
215. C61#23561semidirect
221. C2×M16Dic122semidirect
282. M16C2×Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168403932
#214544092
#214484432
#214471702
#214471652
#214455302
C2×C61×#3432
#2356161