Finite Group Databasebeta

← #29708135

Subgroups — #29708135

Order 15616.

1054 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839805C212yesmaximal
2#21448439C211yesmaximal, characteristic
3#21447166C214yesmaximal, complemented
4#21447161C214yesmaximal, complemented
5#21445558C211yesmaximal, characteristic
6#21445528C212yesmaximal
7C61×#3458C211yesmaximal, characteristic
8C22×#11758194C2212yes—
9C2×#14519736C2211yescharacteristic
10C22×C4⋊Dic61C2211yescharacteristic
11#16839805—24no—
12#16839805C2218yes—
13C2×#14519710—22no—
14C2×#14519708—22no—
15#16839660—22no—
16#16839634—22no—
17#16839631C2212yes—
18#16839627C2212yes—
19#16839531C2212yes—
20#16839531C2212yes—
21#16839521C2218yes—
22C61×#544—22no—
23C61×#544—22no—
24C61×#544C2212yes—
25C61×#544C2212yes—
26C61×#543—22no—
27C61×#533—24no—
28C61×#533—22no—
29C2×#117×C61C2212yes—
30C2×#117×C61C2211yescharacteristic
31C23×Dic61—21no—
32C2×#11758194—22no—
33C2×#11758194—24no—
34C2×#11758194Q818yescomplemented
35C2×#11758194C2314yes—
36C22×#92418—22no—
37#14519736—22no—
38#14519736Q814yescomplemented
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61—22no—
41C2×C4⋊Dic61—22no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61C2312yes—
44C2×#11758196—24no—
45C2×C22⋊Dic61—21no—
46C2×C4×Dic61C2312yes—
47#14519710—28no—
48#14519708—28no—
49#14519684—24no—
50C23×C4×C61C2311yescharacteristic
51C4⋊Q8×C61—22no—
52C4⋊Q8×C61—22no—
53C4⋊Q8×C61—24no—
54C4⋊Q8×C61—24no—
55#125×C61—24no—
56#125×C61—24no—
57#125×C61—24no—
58C22⋊Q8×C61—28no—
59C22⋊Q8×C61—28no—
60C22⋊Q8×C61—24no—
61C22⋊Q8×C61—24no—
62C4×Q8×C61—24no—
63C4×Q8×C61—24no—
64#117×C61—22no—
65#117×C61—22no—
66#117×C61—22no—
67#117×C61—22no—
68#117×C61D414yes—
69#117×C61C2314yes—
70C2×C4⋊C4×C61D412yes—
71C2×C4⋊C4×C61D412yes—
72C2×C4⋊C4×C61C2311yescharacteristic
73C2×C4⋊C4×C61C2311yescharacteristic
74C2×C22⋊C4×C61—22no—
75C2×C22⋊C4×C61D412yes—
76C2×C42×C61—22no—
77C2×C42×C61D412yes—
78C24×C61C2×D411yescharacteristic
79C2×Q8×C61—24no—
80C2×Q8×C61—24no—
81C22×C4×C61—22no—
82C22×C4×C61—22no—
83C22×C4×C61C2×D412yes—
84C22×C4×C61C2×D412yes—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D412yes—
87C22×C4×C61C2×D412yes—
88C22×C4×C61C2×Q814yes—
89C22×C4×C61C2×Q812yes—
90C22×C4×C61#4112yes—
91C22×C4×C61C2411yescharacteristic
92C22×Dic61—22no—
93C22×Dic61—22no—
94C22×Dic61—42no—
95C22×Dic61—82no—
96C4⋊C4×C61—28no—
97C4⋊C4×C61—28no—
98C4⋊C4×C61—24no—
99C4⋊C4×C61—24no—
100C4⋊C4×C61—22no—
101C4⋊C4×C61—22no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61C2×D414yes—
105C4⋊C4×C61C2×D414yes—
106C22⋊C4×C61—24no—
107C22⋊C4×C61—24no—
108C22⋊C4×C61—24no—
109C42×C61—24no—
110C42×C61—22no—
111C42×C61—22no—
112C42×C61—24no—
113C22⋊Dic61—24no—
114C4⋊Dic61—24no—
115C4⋊Dic61—24no—
116C4⋊Dic61—41no—
117C4⋊Dic61—41no—
118#11758196—84no—
119C4×Dic61—22no—
120C4×Dic61C2×Q814yes—
121#11758194—28no—
122#11758194—24no—
123#11758194C2×Q818yes—
124C2×#92418—28no—
125C2×#92418—24no—
126C23×C61—41no—
127C23×C61—41no—
128C23×C61#12012yes—
129C23×C61C22⋊Q812yes—
130C23×C61C4⋊Q812yes—
131C23×C61C22×D411yescharacteristic
132Q8×C61—84no—
133Q8×C61—84no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—24no—
137C2×C4×C61—24no—
138C2×C4×C61—42no—
139C2×C4×C61—42no—
140C2×C4×C61—82no—
141C2×C4×C61—82no—
142C2×C4×C61#12011yescharacteristic
143C2×C4×C61#12011yescharacteristic
144C2×C4×C61C22⋊Q814yes—
145C2×C4×C61C22⋊Q814yes—
146C2×C4×C61C22⋊Q812yes—
147C2×C4×C61C4⋊Q814yes—
148C2×C4×C61C4⋊Q814yes—
149C2×C4×C61C4⋊Q812yes—
150C2×C4×C61C22×D412yes—
151C2×C4×C61C22×Q812yes—
152C2×C4×C61C22×Q811yescharacteristic
153C2×C4×C61C2×#4111yescharacteristic
154C2×Dic61—24no—
155C2×Dic61—42no—
156C2×Dic61—42no—
157C2×Dic61—44no—
158C2×Dic61—42no—
159C2×Dic61—84no—
160#92418—28no—
161#23562—611nomaximal, Sylow
162C22×C61—21no—
163C22×C61—22no—
164C22×C61—21no—
165C22×C61—22no—
166C22×C61—42no—
167C22×C61—42no—
168C22×C61#39514yes—
169C22×C61C2×#12011yescharacteristic
170C22×C61C2×C22⋊Q811yescharacteristic
171C22×C61C2×C4⋊Q811yescharacteristic
172C4×C61—42no—
173C4×C61—44no—
174C4×C61—42no—
175C4×C61—44no—
176C4×C61#39512yes—
177C4×C61#39512yes—
178C4×C61C2×C22⋊Q812yes—
179C4×C61C2×C4⋊Q812yes—
180Dic61—44no—
181Dic61—82no—
182#3458D6111yescharacteristic
183C2×#428—612no—
184#2317—611no—
185#2061—614no—
186#2060—614no—
187#1742—611no—
188#1733—612no—
189C2×C61—21no—
190C2×C61—21no—
191C2×C61—24no—
192C2×C61C2×#39511yescharacteristic
193C2×C61#302111yescharacteristic
194C2×C61#302211yescharacteristic
195C22×M16—612no—
196#544—22no—
197#544—22no—
198#544C2×D6112yes—
199#544C2×D6112yes—
200#543—22no—
201#533—24no—
202#533—22no—
203C2×#117—611no—
204C2×#117C2×D6112yes—
205C2×#117C2×D6111yescharacteristic
206C22×C4⋊C4—611no—
207#428—618no—
208#428—1224no—
209C2×#107—1222no—
210C2×#106—1222no—
211#389—1222no—
212#385—1222no—
213#384—612no—
214#382—612no—
215#343—612no—
216#343—612no—
217#339—618no—
218C61#2356211yescharacteristic, complemented, Sylow
219C23×C4—1221no—
220C23×C4C22×D6111yescharacteristic
221C2×M16—618no—
222C2×M16—614no—
223C2×M16—1224no—
224C2×M16—1222no—
225C22×C8—1222no—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228C4⋊Q8—24no—
229C4⋊Q8—24no—
230#125—24no—
231#125—24no—
232#125—24no—
233C22⋊Q8—28no—
234C22⋊Q8—28no—
235C22⋊Q8—24no—
236C22⋊Q8—24no—
237C4×Q8—24no—
238C4×Q8—24no—
239#117—22no—
240#117—22no—
241#117—22no—
242#117—22no—
243#117—614no—
244#117—1222no—
245#117C22⋊D6114yes—
246#117C22×D6114yes—
247C2×C4⋊C4—612no—
248C2×C4⋊C4—1221no—
249C2×C4⋊C4—1222no—
250C2×C4⋊C4—1222no—
251C2×C4⋊C4—1224no—
252C2×C4⋊C4—1221no—
253C2×C4⋊C4C22⋊D6112yes—
254C2×C4⋊C4C22⋊D6112yes—
255C2×C4⋊C4C22×D6111yescharacteristic
256C2×C4⋊C4C22×D6111yescharacteristic
257C2×C22⋊C4—22no—
258C2×C22⋊C4—1221no—
259C2×C22⋊C4C22⋊D6112yes—
260C2×C42—22no—
261C2×C42—612no—
262C2×C42C22⋊D6112yes—
263#107—1228no—
264#106—1228no—
265#95—1224no—
266C24C2×C22⋊D6111yescharacteristic
267C2×Q8—24no—
268C2×Q8—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—1222no—
272C22×C4—1222no—
273C22×C4—4882no—
274C22×C4—2442no—
275C22×C4Q8×D6114yes—
276C22×C4Q8×D6112yes—
277C22×C4#1175822612yes—
278C22×C4C2×C22⋊D6112yes—
279C22×C4C2×C22⋊D6112yes—
280C22×C4C2×C22⋊D6112yes—
281C22×C4C2×C22⋊D6112yes—
282C22×C4C2×C22⋊D6111yescharacteristic
283C22×C4C23×D6111yescharacteristic
284M16—618no—
285M16—1224no—
286M16—1228no—
287C2×C8—1224no—
288C2×C8—1228no—
289C4⋊C4—28no—
290C4⋊C4—28no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—1224no—
298C4⋊C4—1224no—
299C4⋊C4—4884no—
300C4⋊C4—2441no—
301C4⋊C4—2441no—
302C4⋊C4C2×C22⋊D6114yes—
303C4⋊C4C2×C22⋊D6114yes—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—24no—
307C22⋊C4—1224no—
308C42—24no—
309C42—24no—
310C42—22no—
311C42—22no—
312C42—614no—
313C42—1222no—
314C23—41no—
315C23—41no—
316C23#1451974612yes—
317C23#1451975312yes—
318C23#1451977512yes—
319C23C22×C22⋊D6111yescharacteristic
320Q8—84no—
321Q8—84no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—1224no—
331C2×C4—4884no—
332C2×C4—2444no—
333C2×C4—2442no—
334C2×C4—2442no—
335C2×C4—2442no—
336C2×C4#1451974611yescharacteristic
337C2×C4#1451974611yescharacteristic
338C2×C4#1451975312yes—
339C2×C4#1451975314yes—
340C2×C4#1451975314yes—
341C2×C4#1451977512yes—
342C2×C4#1451977514yes—
343C2×C4#1451977514yes—
344C2×C4C22×C22⋊D6112yes—
345C2×C4C2×Q8×D6112yes—
346C2×C4C2×Q8×D6111yescharacteristic
347C2×C4C2×#1175822611yescharacteristic
348C8—1228no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—22no—
353C22—42no—
354C22—42no—
355C22#1683968114yes—
356C22C2×#1451974611yescharacteristic
357C22C2×#1451975311yescharacteristic
358C22C2×#1451977511yescharacteristic
359C4—42no—
360C4—44no—
361C4—42no—
362C4—44no—
363C4—4882no—
364C4—2444no—
365C4#1683968112yes—
366C4#1683968112yes—
367C4C2×#1451975312yes—
368C4C2×#1451977512yes—
369C2—21no—
370C2—24no—
371C2—21no—
372C2C2×#1683968111yescharacteristic
373C2#2145519111yescharacteristic
374C2#2145519811yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3458C2
socleC22×C61C2×#120
prime radicalC24×C61C2×D4
hypercenter#3458D61
hypocenterC61#23562
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3458

Complements

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

NH ≅ G/Ntype
3. #21447166C2semidirect
4. #21447161C2semidirect
34. C2×#11758194Q8semidirect
38. #14519736Q8semidirect
218. C61#23562semidirect

Maximal subgroups

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

subgroupindex
C2×#168398052
#214484392
#214471662
#214471612
#214455582
#214455282
C61×#34582
#2356261