Finite Group Databasebeta

← #29654066

Subgroups — #29654066

Order 15616.

908 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456960C212yesmaximal, complemented
2#21451585C211yesmaximal, characteristic
3#21444827C214yesmaximal, complemented
4#21444810C214yesmaximal
5#21444383C212yesmaximal, complemented
6#21444133C211yesmaximal, characteristic
7C61×#3458C211yesmaximal, characteristic
8C2×#14519777C2211yescharacteristic
9C2×#14519756—21no—
10#16840158—22no—
11#16840112—22no—
12#16840090C2218yes—
13#16840047—24no—
14#16839815C2212yes—
15#16839692—21no—
16#16839692C2212yes—
17#16839687—21no—
18#16839687C2212yes—
19#16839668—21no—
20#16839628C2211yescharacteristic
21#16839499C2218yes—
22#16839488C2212yes—
23#16839486C2212yes—
24C61×#544—22no—
25C61×#544—22no—
26C61×#544—21no—
27C61×#544C2212yes—
28C61×#543—21no—
29C61×#543C2212yes—
30C61×#533—24no—
31C61×#533—21no—
32C61×#533C2212yes—
33C2×#117×C61—21no—
34C2×#117×C61C2211yescharacteristic
35C22×Dic122—21no—
36#14519777—22no—
37#14519777—22no—
38#14519756—44no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61C2312yes—
42C2×#11758196—22no—
43C2×C22⋊Dic61—22no—
44#14519708—28no—
45C4⋊#92418—22no—
46#14519701—28no—
47C22⋊#92418C2314yes—
48C4×#92418—22no—
49#14519684—21no—
50#14519684—22no—
51#14519684—22no—
52#14519684C2312yes—
53C23×C4×C61D411yescharacteristic
54C4⋊Q8×C61—24no—
55C4⋊Q8×C61—21no—
56C4⋊Q8×C61—21no—
57C4⋊Q8×C61—24no—
58C4⋊Q8×C61D412yes—
59C4⋊Q8×C61D412yes—
60#125×C61—24no—
61#125×C61—22no—
62#125×C61—24no—
63#125×C61—22no—
64C22⋊Q8×C61—24no—
65C22⋊Q8×C61—24no—
66C22⋊Q8×C61—24no—
67C22⋊Q8×C61—22no—
68C22⋊Q8×C61—22no—
69C22⋊Q8×C61—22no—
70C22⋊Q8×C61—24no—
71C22⋊Q8×C61C2314yes—
72C4×Q8×C61—22no—
73C4×Q8×C61—22no—
74C4×Q8×C61—22no—
75C4×Q8×C61D414yes—
76#117×C61—22no—
77#117×C61—22no—
78#117×C61—22no—
79#117×C61—22no—
80#117×C61—22no—
81#117×C61—22no—
82C2×C4⋊C4×C61—21no—
83C2×C4⋊C4×C61—21no—
84C2×C4⋊C4×C61D411yescharacteristic
85C2×C4⋊C4×C61C2311yescharacteristic
86C2×C22⋊C4×C61—22no—
87C2×C22⋊C4×C61D411yescharacteristic
88C2×C22⋊C4×C61C2311yescharacteristic
89C2×C42×C61—22no—
90C2×C42×C61D411yescharacteristic
91C2×C42×C61C2311yescharacteristic
92C24×C61C2×D411yescharacteristic
93C2×Q8×C61—22no—
94C2×Q8×C61—22no—
95C2×Q8×C61—22no—
96C2×Q8×C61C2×D414yes—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61SD1614yes—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2411yescharacteristic
109C22×Dic61—22no—
110C22×Dic61—22no—
111C2×Dic122—41no—
112C2×Dic122—41no—
113C2×Dic122—42no—
114C2×Dic122—42no—
115C4⋊C4×C61—22no—
116C4⋊C4×C61—22no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—24no—
119C4⋊C4×C61—24no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—22no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—24no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—24no—
128C4⋊C4×C61C2×D414yes—
129C22⋊C4×C61—24no—
130C22⋊C4×C61—24no—
131C22⋊C4×C61—22no—
132C22⋊C4×C61—22no—
133C42×C61—24no—
134C42×C61—24no—
135C42×C61—21no—
136C42×C61—21no—
137C42×C61C2×D412yes—
138C42×C61C2×D412yes—
139C22⋊Dic61—44no—
140C4⋊Dic61—24no—
141C4⋊Dic61—42no—
142C4⋊Dic61—42no—
143#11758196—44no—
144C2×#92418—24no—
145C23×C61—21no—
146C23×C61—21no—
147C23×C61—41no—
148C23×C61—41no—
149C23×C61C2×SD1612yes—
150C23×C61C22×D411yescharacteristic
151Q8×C61—44no—
152Q8×C61—82no—
153Q8×C61—82no—
154Q8×C61—82no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—24no—
158C2×C4×C61—24no—
159C2×C4×C61—24no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—24no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—82no—
168C2×C4×C61—82no—
169C2×C4×C61#12012yes—
170C2×C4×C61#12011yescharacteristic
171C2×C4×C61#12011yescharacteristic
172C2×C4×C61C2×SD1612yes—
173C2×C4×C61C2×SD1612yes—
174C2×C4×C61#13712yes—
175C2×C4×C61C22×D412yes—
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179C2×Dic61—42no—
180C2×Dic61—42no—
181C2×Dic61—42no—
182Dic122—82no—
183Dic122—82no—
184#92418—44no—
185#18310—611nomaximal, Sylow
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—22no—
189C22×C61—42no—
190C22×C61—41no—
191C22×C61—41no—
192C22×C61—81no—
193C22×C61C2×#12011yescharacteristic
194C22×C61C22×SD1611yescharacteristic
195C22×C61C2×#13711yescharacteristic
196C4×C61—22no—
197C4×C61—22no—
198C4×C61—22no—
199C4×C61—41no—
200C4×C61—44no—
201C4×C61—41no—
202C4×C61—44no—
203C4×C61#44812yes—
204C4×C61#44812yes—
205Dic61—82no—
206Dic61—82no—
207#3458D6111yescharacteristic
208#3229—612no—
209#2621—611no—
210#1611—614no—
211#1608—614no—
212#1518—612no—
213#1458—611no—
214C2×C61—21no—
215C2×C61—21no—
216C2×C61—42no—
217C2×C61C2×#44811yescharacteristic
218C2×C61#302211yescharacteristic
219C2×C61#302811yescharacteristic
220#544—22no—
221#544—22no—
222#544—21no—
223#544C2×D6112yes—
224#543—21no—
225#543C2×D6112yes—
226#533—24no—
227#533—21no—
228#533C2×D6112yes—
229C2×C4⋊Q8—611no—
230C2×C22⋊Q8—1221no—
231C2×#117—21no—
232C2×#117C2×D6111yescharacteristic
233#498—1222no—
234#487—1222no—
235#483—618no—
236#475—1224no—
237#431—612no—
238#398—612no—
239#398—1221no—
240#397—612no—
241#397—1221no—
242#391—1221no—
243#382—611no—
244#328—618no—
245#324—612no—
246#323—612no—
247C61#1831011yescharacteristic, complemented, Sylow
248C22×Q8—1221no—
249C23×C4C22⋊D6111yescharacteristic
250C4⋊Q8—24no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8—24no—
254C4⋊Q8—1222no—
255C4⋊Q8—1222no—
256C4⋊Q8C22⋊D6112yes—
257C4⋊Q8C22⋊D6112yes—
258#125—24no—
259#125—22no—
260#125—24no—
261#125—22no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264C22⋊Q8—24no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267C22⋊Q8—22no—
268C22⋊Q8—24no—
269C22⋊Q8—2444no—
270C22⋊Q8C22×D6114yes—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—22no—
274C4×Q8C22⋊D6114yes—
275#117—22no—
276#117—22no—
277#117—22no—
278#117—22no—
279#117—22no—
280#117—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—612no—
284C2×C4⋊C4—1221no—
285C2×C4⋊C4—1222no—
286C2×C4⋊C4—1221no—
287C2×C4⋊C4C22⋊D6111yescharacteristic
288C2×C4⋊C4C22×D6111yescharacteristic
289C2×C22⋊C4—22no—
290C2×C22⋊C4—1222no—
291C2×C22⋊C4C22⋊D6111yescharacteristic
292C2×C22⋊C4C22×D6111yescharacteristic
293C2×C42—22no—
294C2×C42C22⋊D6111yescharacteristic
295C2×C42C22×D6111yescharacteristic
296#106—1228no—
297C4⋊C8—1222no—
298#103—1228no—
299C22⋊C8—614no—
300C4×C8—1222no—
301#95—612no—
302#95—1221no—
303#95—1222no—
304#95—1222no—
305C24C2×C22⋊D6111yescharacteristic
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—2442no—
310C2×Q8—2441no—
311C2×Q8—2441no—
312C2×Q8—2442no—
313C2×Q8C2×C22⋊D6114yes—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—1222no—
320C22×C4—1222no—
321C22×C4#1175820114yes—
322C22×C4C2×C22⋊D6111yescharacteristic
323C22×C4C2×C22⋊D6111yescharacteristic
324C22×C4C2×C22⋊D6111yescharacteristic
325C22×C4C2×C22⋊D6111yescharacteristic
326C22×C4C2×C22⋊D6111yescharacteristic
327C22×C4C23×D6111yescharacteristic
328C2×C8—1224no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—1224no—
343C4⋊C4—2442no—
344C4⋊C4—2444no—
345C4⋊C4—2442no—
346C4⋊C4C2×C22⋊D6114yes—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—2444no—
352C42—24no—
353C42—24no—
354C42—21no—
355C42—21no—
356C42C2×C22⋊D6112yes—
357C42C2×C22⋊D6112yes—
358C23—21no—
359C23—21no—
360C23—41no—
361C23—41no—
362C23C2×#1175820112yes—
363C23C22×C22⋊D6111yescharacteristic
364Q8—44no—
365Q8—82no—
366Q8—82no—
367Q8—82no—
368Q8—4882no—
369Q8—4882no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—24no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—2442no—
385C2×C4—2442no—
386C2×C4—2442no—
387C2×C4—2442no—
388C2×C4—2442no—
389C2×C4—2442no—
390C2×C4#1451974612yes—
391C2×C4#1451974611yescharacteristic
392C2×C4#1451974611yescharacteristic
393C2×C4C2×#1175820112yes—
394C2×C4C2×#1175820112yes—
395C2×C4#1451980812yes—
396C2×C4C22×C22⋊D6112yes—
397C8—2444no—
398C22—21no—
399C22—21no—
400C22—22no—
401C22—42no—
402C22—41no—
403C22—41no—
404C22—81no—
405C22C2×#1451974611yescharacteristic
406C22C22×#1175820111yescharacteristic
407C22C2×#1451980811yescharacteristic
408C4—22no—
409C4—22no—
410C4—22no—
411C4—41no—
412C4—44no—
413C4—41no—
414C4—44no—
415C4—4882no—
416C4—4882no—
417C4#1683990112yes—
418C4#1683990112yes—
419C2—21no—
420C2—21no—
421C2—42no—
422C2C2×#1683990111yescharacteristic
423C2#2145519811yescharacteristic
424C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3458C2
socleC22×C61C2×#120
prime radicalC24×C61C2×D4
hypercenter#3458D61
hypocenterC61#18310
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3458

Complements

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

NH ≅ G/Ntype
1. #21456960C2semidirect
3. #21444827C2semidirect
5. #21444383C2semidirect
247. C61#18310semidirect

Maximal subgroups

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

subgroupindex
#214569602
#214515852
#214448272
#214448102
#214443832
#214441332
C61×#34582
#1831061