Finite Group Databasebeta

← #29654005

Subgroups — #29654005

Order 15616.

912 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456764C212yesmaximal, complemented
2#21451640C211yesmaximal, characteristic
3#21444793C214yesmaximal, complemented
4#21444776C214yesmaximal
5#21444383C212yesmaximal, complemented
6#21444133C211yesmaximal, characteristic
7C61×#3458C211yesmaximal, characteristic
8C2×#14519777C2211yescharacteristic
9#16840141—22no—
10#16840114—22no—
11#16839972—24no—
12#16839917C2218yes—
13#16839815C2212yes—
14#16839701—21no—
15#16839698—21no—
16#16839682C2212yes—
17#16839674C2212yes—
18#16839660—21no—
19#16839628C2211yescharacteristic
20#16839622—21no—
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×Dic122C2312yes—
36C2×#11758202—28no—
37#14519777—22no—
38#14519777—22no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×#11758196—22no—
43C2×C22⋊Dic61—22no—
44C4⋊#92418—22no—
45#14519700—28no—
46C22⋊#92418C2314yes—
47C4×#92418—22no—
48#14519684—22no—
49#14519684—21no—
50#14519684—22no—
51#14519684C2312yes—
52C23×C4×C61D411yescharacteristic
53C4⋊Q8×C61—24no—
54C4⋊Q8×C61—21no—
55C4⋊Q8×C61—21no—
56C4⋊Q8×C61—24no—
57C4⋊Q8×C61D412yes—
58C4⋊Q8×C61D412yes—
59#125×C61—24no—
60#125×C61—22no—
61#125×C61—24no—
62#125×C61—22no—
63C22⋊Q8×C61—24no—
64C22⋊Q8×C61—24no—
65C22⋊Q8×C61—24no—
66C22⋊Q8×C61—22no—
67C22⋊Q8×C61—22no—
68C22⋊Q8×C61—22no—
69C22⋊Q8×C61—24no—
70C22⋊Q8×C61C2314yes—
71C4×Q8×C61—22no—
72C4×Q8×C61—22no—
73C4×Q8×C61—22no—
74C4×Q8×C61D414yes—
75#117×C61—22no—
76#117×C61—22no—
77#117×C61—22no—
78#117×C61—22no—
79#117×C61—22no—
80#117×C61—22no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—21no—
83C2×C4⋊C4×C61D411yescharacteristic
84C2×C4⋊C4×C61C2311yescharacteristic
85C2×C22⋊C4×C61—22no—
86C2×C22⋊C4×C61D411yescharacteristic
87C2×C22⋊C4×C61C2311yescharacteristic
88C2×C42×C61—22no—
89C2×C42×C61D411yescharacteristic
90C2×C42×C61C2311yescharacteristic
91C24×C61C2×D411yescharacteristic
92C2×Q8×C61—22no—
93C2×Q8×C61—22no—
94C2×Q8×C61—22no—
95C2×Q8×C61C2×D414yes—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—22no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61Q1614yescomplemented
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2411yescharacteristic
108C22×Dic61—22no—
109C22×Dic61—22no—
110C2×Dic122—24no—
111C2×Dic122—82no—
112C4⋊C4×C61—22no—
113C4⋊C4×C61—22no—
114C4⋊C4×C61—22no—
115C4⋊C4×C61—24no—
116C4⋊C4×C61—24no—
117C4⋊C4×C61—22no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61—24no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—24no—
125C4⋊C4×C61C2×D414yes—
126C22⋊C4×C61—24no—
127C22⋊C4×C61—24no—
128C22⋊C4×C61—22no—
129C22⋊C4×C61—22no—
130C42×C61—24no—
131C42×C61—24no—
132C42×C61—21no—
133C42×C61—21no—
134C42×C61C2×D412yes—
135C42×C61C2×D412yes—
136C22⋊Dic61—82no—
137#11758202—88no—
138C4⋊Dic61—42no—
139C4⋊Dic61—41no—
140C4⋊Dic61—41no—
141C4⋊Dic61—42no—
142#11758196—82no—
143C2×#92418—24no—
144C23×C61—21no—
145C23×C61—21no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61C2×Q1612yes—
149C23×C61C22×D411yescharacteristic
150Q8×C61—44no—
151Q8×C61—82no—
152Q8×C61—82no—
153Q8×C61—82no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—24no—
157C2×C4×C61—24no—
158C2×C4×C61—24no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—24no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—82no—
167C2×C4×C61—82no—
168C2×C4×C61#12012yes—
169C2×C4×C61#12011yescharacteristic
170C2×C4×C61#12011yescharacteristic
171C2×C4×C61C2×Q1612yes—
172C2×C4×C61C2×Q1612yes—
173C2×C4×C61#13712yes—
174C2×C4×C61C22×D412yes—
175C2×Dic61—42no—
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179C2×Dic61—42no—
180C2×Dic61—42no—
181Dic122—48no—
182#92418—44no—
183#18303—611nomaximal, Sylow
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—22no—
187C22×C61—42no—
188C22×C61—41no—
189C22×C61—41no—
190C22×C61—81no—
191C22×C61C2×#12011yescharacteristic
192C22×C61C22×Q1611yescharacteristic
193C22×C61C2×#13711yescharacteristic
194C4×C61—22no—
195C4×C61—22no—
196C4×C61—22no—
197C4×C61—41no—
198C4×C61—44no—
199C4×C61—41no—
200C4×C61—44no—
201C4×C61#45112yes—
202C4×C61#45112yes—
203Dic61—82no—
204Dic61—82no—
205#3458D6111yescharacteristic
206#3209—612no—
207#2628—611no—
208#1605—614no—
209#1602—614no—
210#1518—612no—
211#1458—611no—
212C2×C61—21no—
213C2×C61—21no—
214C2×C61—42no—
215C2×C61C2×#45111yescharacteristic
216C2×C61#302211yescharacteristic
217C2×C61#302711yescharacteristic
218#544—22no—
219#544—22no—
220#544—21no—
221#544C2×D6112yes—
222#543—21no—
223#543C2×D6112yes—
224#533—24no—
225#533—21no—
226#533C2×D6112yes—
227C2×C4⋊Q8—611no—
228C2×#117—21no—
229C2×#117C2×D6111yescharacteristic
230#494—1222no—
231#487—1222no—
232#462—1224no—
233#451—618no—
234#431—612no—
235#400—1221no—
236#399—1221no—
237#395—612no—
238#393—612no—
239#389—1221no—
240#382—611no—
241#380—1221no—
242#328—618no—
243#324—612no—
244#323—612no—
245C61#1830311yescharacteristic, complemented, Sylow
246C22×Q8—612no—
247C23×C4C22⋊D6111yescharacteristic
248C2×Q16—1228no—
249C4⋊Q8—24no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—24no—
253C4⋊Q8—1222no—
254C4⋊Q8—1222no—
255C4⋊Q8C22⋊D6112yes—
256C4⋊Q8C22⋊D6112yes—
257#125—24no—
258#125—22no—
259#125—24no—
260#125—22no—
261C22⋊Q8—24no—
262C22⋊Q8—24no—
263C22⋊Q8—24no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267C22⋊Q8—24no—
268C22⋊Q8C22×D6114yes—
269C4×Q8—22no—
270C4×Q8—22no—
271C4×Q8—22no—
272C4×Q8C22⋊D6114yes—
273#117—22no—
274#117—22no—
275#117—22no—
276#117—22no—
277#117—22no—
278#117—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—1222no—
282C2×C4⋊C4—1221no—
283C2×C4⋊C4—1221no—
284C2×C4⋊C4—1221no—
285C2×C4⋊C4C22⋊D6111yescharacteristic
286C2×C4⋊C4C22×D6111yescharacteristic
287C2×C22⋊C4—22no—
288C2×C22⋊C4—1222no—
289C2×C22⋊C4C22⋊D6111yescharacteristic
290C2×C22⋊C4C22×D6111yescharacteristic
291C2×C42—22no—
292C2×C42C22⋊D6111yescharacteristic
293C2×C42C22×D6111yescharacteristic
294C4⋊C8—1222no—
295#103—1228no—
296C22⋊C8—614no—
297C4×C8—1222no—
298#95—612no—
299#95—1222no—
300#95—1221no—
301#95—1222no—
302C24C2×C22⋊D6111yescharacteristic
303C2×Q8—22no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—1224no—
307C2×Q8—4882no—
308C2×Q8C2×C22⋊D6114yes—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—1222no—
315C22×C4—1222no—
316C22×C4#1175820214yescomplemented
317C22×C4C2×C22⋊D6111yescharacteristic
318C22×C4C2×C22⋊D6111yescharacteristic
319C22×C4C2×C22⋊D6111yescharacteristic
320C22×C4C2×C22⋊D6111yescharacteristic
321C22×C4C2×C22⋊D6111yescharacteristic
322C22×C4C23×D6111yescharacteristic
323Q16—4888no—
324C2×C8—1224no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—24no—
338C4⋊C4—4882no—
339C4⋊C4—2442no—
340C4⋊C4—2442no—
341C4⋊C4—2441no—
342C4⋊C4—2441no—
343C4⋊C4C2×C22⋊D6114yes—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—4882no—
349C42—24no—
350C42—24no—
351C42—21no—
352C42—21no—
353C42C2×C22⋊D6112yes—
354C42C2×C22⋊D6112yes—
355C23—21no—
356C23—21no—
357C23—41no—
358C23—41no—
359C23C2×#1175820212yes—
360C23C22×C22⋊D6111yescharacteristic
361Q8—44no—
362Q8—82no—
363Q8—82no—
364Q8—82no—
365Q8—2448no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—24no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4—2442no—
386C2×C4#1451974612yes—
387C2×C4#1451974611yescharacteristic
388C2×C4#1451974611yescharacteristic
389C2×C4C2×#1175820212yes—
390C2×C4C2×#1175820212yes—
391C2×C4#1451980812yes—
392C2×C4C22×C22⋊D6112yes—
393C8—2444no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—42no—
398C22—41no—
399C22—41no—
400C22—81no—
401C22C2×#1451974611yescharacteristic
402C22C22×#1175820211yescharacteristic
403C22C2×#1451980811yescharacteristic
404C4—22no—
405C4—22no—
406C4—22no—
407C4—41no—
408C4—44no—
409C4—41no—
410C4—44no—
411C4—4882no—
412C4—4882no—
413C4#1683991812yes—
414C4#1683991812yes—
415C2—21no—
416C2—21no—
417C2—42no—
418C2C2×#1683991811yescharacteristic
419C2#2145519811yescharacteristic
420C2#2145522111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3458C2
socleC22×C61C2×#120
prime radicalC24×C61C2×D4
hypercenter#3458D61
hypocenterC61#18303
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. #21456764C2semidirect
3. #21444793C2semidirect
5. #21444383C2semidirect
101. C22×C4×C61Q16semidirect
245. C61#18303semidirect
316. C22×C4#11758202semidirect

Maximal subgroups

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

subgroupindex
#214567642
#214516402
#214447932
#214447762
#214443832
#214441332
C61×#34582
#1830361