Finite Group Databasebeta

← #29655703

Subgroups — #29655703

Order 15616.

996 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456141C211yesmaximal, characteristic, complemented
2#21455890C212yesmaximal, complemented
3#21445033C212yesmaximal, complemented
4#21445026C212yesmaximal, complemented
5#21445005C212yesmaximal, complemented
6#21444998C212yesmaximal, complemented
7#21444481C211yesmaximal, characteristic, complemented
8#21444201C212yesmaximal, complemented
9C61×#3481C211yesmaximal, characteristic
10#16840319C2211yescharacteristic, complemented
11#16840116—22no—
12#16840112C2212yescomplemented
13#16840108C2212yes—
14#16840103—22no—
15#16840047C2212yescomplemented
16#16840042C2212yes—
17#16839972C2212yescomplemented
18#16839967C2212yes—
19#16839924—22no—
20C4×#11758202—22no—
21C4×#11758200—22no—
22#16839815C2212yescomplemented
23#16839758—22no—
24C2×C4×#92418C2211yescharacteristic, complemented
25#16839511C2211yescharacteristic, complemented
26#16839511C2211yescharacteristic, complemented
27#16839506C2212yes—
28Q8⋊#92418C2212yescomplemented
29Q8⋊#92418C2212yescomplemented
30D4⋊#92418C2212yes—
31D4⋊#92418C2212yes—
32C61×#559C2211yescharacteristic
33C61×#554C2212yes—
34C61×#553—22no—
35C61×#551—22no—
36C61×#546—22no—
37D42×C61C2211yescharacteristic
38C61×#532C2211yescharacteristic
39C61×#529—22no—
40C61×#525—22no—
41C4×#41×C61C2212yes—
42C2×C4×D4×C61—22no—
43C2×#11758202—22no—
44C2×#11758200—22no—
45C22×#92418—21no—
46#14519777C2312yes—
47#14519769—21no—
48#14519756—22no—
49C4×Dic122—22no—
50#14519736—21no—
51#14519710—22no—
52#14519708—22no—
53C4⋊#92418C2312yes—
54C4⋊#92418C2312yes—
55#14519701—24no—
56#14519701—22no—
57#14519700—22no—
58#14519700—22no—
59#14519698—24no—
60#14519698—22no—
61C22⋊#92418—22no—
62C4×#92418—21no—
63C4×#92418—22no—
64C4×#92418C2312yes—
65C2×#41×C61D412yes—
66C2×#41×C61D412yes—
67C22×D4×C61—22no—
68C23×C4×C61—22no—
69C4⋊Q8×C61C2311yescharacteristic
70#127×C61C2311yescharacteristic
71#126×C61—24no—
72#125×C61—22no—
73#124×C61—22no—
74#124×C61D412yes—
75#123×C61—24no—
76#123×C61—22no—
77#123×C61—22no—
78C22⋊Q8×C61—22no—
79C22⋊Q8×C61—24no—
80#121×C61—24no—
81#121×C61—22no—
82#121×C61—22no—
83#120×C61—22no—
84#120×C61—22no—
85C4×Q8×C61D412yes—
86C4×Q8×C61C2312yes—
87C4×D4×C61—24no—
88C4×D4×C61—24no—
89C4×D4×C61—22no—
90C4×D4×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94C4×D4×C61D412yes—
95C4×D4×C61C2312yes—
96#117×C61—22no—
97#117×C61—22no—
98#117×C61D412yes—
99C2×C4⋊C4×C61—22no—
100C2×C22⋊C4×C61—24no—
101C2×C42×C61—22no—
102C2×C42×C61C2311yescharacteristic
103C24×C61—22no—
104#41×C61—24no—
105#41×C61—42no—
106#41×C61—44no—
107C2×Q8×C61C2×D412yes—
108C2×Q8×C61C2×D412yes—
109C2×D4×C61—24no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—44no—
114C2×D4×C61—44no—
115C2×D4×C61C2×D412yes—
116C2×D4×C61C2×D412yes—
117C22×C4×C61—24no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—22no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—84no—
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D411yescharacteristic
126C2×Dic122—22no—
127C4⋊C4×C61—24no—
128C4⋊C4×C61—24no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61C2×D412yes—
133C4⋊C4×C61C2×D412yes—
134C22⋊C4×C61—24no—
135C22⋊C4×C61—24no—
136C22⋊C4×C61—22no—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—22no—
140C22⋊C4×C61—24no—
141C22⋊C4×C61—22no—
142C42×C61—22no—
143C42×C61—22no—
144C42×C61—22no—
145C42×C61C2×D412yes—
146C42×C61C2×D411yescharacteristic
147C42×C61C2411yescharacteristic
148C22⋊Dic61—41no—
149#11758202—44no—
150#11758200—44no—
151C4⋊Dic61—22no—
152C4⋊Dic61—22no—
153#11758196—42no—
154C4×Dic61—41no—
155C2×#92418—22no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—42no—
159C23×C61—22no—
160C23×C61—22no—
161C23×C61—48no—
162C23×C61—82no—
163C23×C61#12011yescharacteristic
164Q8×C61—22no—
165Q8×C61—22no—
166Q8×C61—42no—
167D4×C61—22no—
168D4×C61—42no—
169D4×C61—44no—
170D4×C61—48no—
171D4×C61—42no—
172D4×C61—42no—
173D4×C61—82no—
174D4×C61#13514yes—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—44no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—44no—
186C2×C4×C61—44no—
187C2×C4×C61—82no—
188C2×C4×C61—84no—
189C2×C4×C61#12012yes—
190C2×C4×C61#12011yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D412yes—
193C2×Dic61—42no—
194Dic122—42no—
195Dic122—42no—
196#92418—42no—
197#92418—42no—
198#92418—42no—
199#18485—611nomaximal, Sylow
200C22×C61—22no—
201C22×C61—24no—
202C22×C61—21no—
203C22×C61—42no—
204C22×C61—42no—
205C22×C61—44no—
206C22×C61—42no—
207C22×C61—88no—
208C22×C61C2×#12011yescharacteristic
209C4×C61—22no—
210C4×C61—21no—
211C4×C61—42no—
212C4×C61—42no—
213C4×C61—42no—
214C4×C61—42no—
215C4×C61#45512yes—
216C4×C61C2×#13512yes—
217C4×C61#57512yes—
218Dic61—82no—
219#3481D6111yescharacteristic
220#3131—611no—
221#3107—612no—
222#1644—612no—
223#1643—612no—
224#1640—612no—
225#1639—612no—
226#1543—611no—
227#1476—612no—
228C2×C61—41no—
229C2×C61—44no—
230C2×C61—82no—
231C2×C61#300212yes—
232C2×C61C2×#45511yescharacteristic
233#559C2×D6111yescharacteristic
234#554C2×D6112yes—
235#553—22no—
236#551—22no—
237#546—22no—
238D42C2×D6111yescharacteristic
239#533—611no—
240#532C2×D6111yescharacteristic
241#529—22no—
242#525—22no—
243C4×#41C2×D6112yes—
244C2×C4×D4—22no—
245#488—1222no—
246#487—612no—
247#486—612no—
248#485—1222no—
249#475—612no—
250#474—612no—
251#462—612no—
252#461—612no—
253#452—1222no—
254C4×Q16—1222no—
255C4×SD16—1222no—
256#431—612no—
257#416—1222no—
258C2×C4×C8—611no—
259#333—611no—
260#333—611no—
261#331—612no—
262Q8⋊C8—612no—
263Q8⋊C8—612no—
264D4⋊C8—612no—
265D4⋊C8—612no—
266C61#1848511yescharacteristic, complemented, Sylow
267C2×#41C22⋊D6112yes—
268C2×#41C22⋊D6112yes—
269C22×D4—22no—
270C23×C4—22no—
271C2×Q16—1222no—
272C2×SD16—1222no—
273C22×C8—1221no—
274C4⋊Q8—612no—
275C4⋊Q8C22×D6111yescharacteristic
276#127C22×D6111yescharacteristic
277#126—24no—
278#125—22no—
279#125—1221no—
280#124—22no—
281#124C22⋊D6112yes—
282#123—22no—
283#123—24no—
284#123—22no—
285C22⋊Q8—24no—
286C22⋊Q8—22no—
287C22⋊Q8—1222no—
288#121—24no—
289#121—22no—
290#121—22no—
291#120—22no—
292#120—22no—
293C4×Q8—1222no—
294C4×Q8C22⋊D6112yes—
295C4×Q8C22×D6112yes—
296C4×D4—24no—
297C4×D4—24no—
298C4×D4—22no—
299C4×D4—22no—
300C4×D4—22no—
301C4×D4—22no—
302C4×D4—22no—
303C4×D4C22⋊D6112yes—
304C4×D4C22×D6112yes—
305#117—22no—
306#117—22no—
307#117—1221no—
308#117C22⋊D6112yes—
309C2×C4⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C42—22no—
312C2×C42C22×D6111yescharacteristic
313#107—1222no—
314#106—1222no—
315C4⋊C8—612no—
316C4⋊C8—612no—
317#103—1224no—
318#103—1222no—
319#103—1222no—
320#103—1222no—
321#102—1224no—
322#102—1222no—
323C22⋊C8—1222no—
324C4×C8—612no—
325C4×C8—1221no—
326C4×C8—1222no—
327C24—22no—
328#41—24no—
329#41—42no—
330#41—44no—
331C2×Q8—1222no—
332C2×Q8C2×C22⋊D6112yes—
333C2×Q8C2×C22⋊D6112yes—
334C2×D4—24no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4C2×C22⋊D6112yes—
341C2×D4C2×C22⋊D6112yes—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—84no—
349C22×C4C2×C22⋊D6112yes—
350C22×C4C2×C22⋊D6111yescharacteristic
351Q16—2444no—
352SD16—2444no—
353C2×C8—1222no—
354C2×C8—1222no—
355C2×C8—1222no—
356C2×C8—2442no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—1222no—
363C4⋊C4—1222no—
364C4⋊C4—2442no—
365C4⋊C4C2×C22⋊D6112yes—
366C4⋊C4C2×C22⋊D6112yes—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—24no—
375C22⋊C4—2441no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—2441no—
380C42C2×C22⋊D6111yescharacteristic
381C42C2×C22⋊D6112yes—
382C42C23×D6111yescharacteristic
383C23—22no—
384C23—22no—
385C23—48no—
386C23—82no—
387C23#1451974611yescharacteristic
388Q8—22no—
389Q8—22no—
390Q8—42no—
391Q8—2442no—
392Q8—2442no—
393D4—22no—
394D4—42no—
395D4—44no—
396D4—48no—
397D4—42no—
398D4—42no—
399D4—82no—
400D4#1451979714yes—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—82no—
414C2×C4—84no—
415C2×C4—2442no—
416C2×C4#1451974612yes—
417C2×C4#1451974611yescharacteristic
418C2×C4C22×C22⋊D6111yescharacteristic
419C2×C4C22×C22⋊D6112yes—
420C8—2442no—
421C8—2442no—
422C8—2442no—
423C22—21no—
424C22—22no—
425C22—24no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—44no—
430C22—88no—
431C22C2×#1451974611yescharacteristic
432C4—21no—
433C4—22no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—42no—
438C4—4882no—
439C4#1683994012yes—
440C4C2×#1451979712yes—
441C4#1684056012yes—
442C2—41no—
443C2—44no—
444C2—82no—
445C2#2145503712yes—
446C2C2×#1683994011yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3481C2
socleC22×C61C2×#120
hypercenter#3481D61
hypocenterC61#18485
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3481

Complements

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

NH ≅ G/Ntype
1. #21456141C2semidirect
2. #21455890C2semidirect
3. #21445033C2semidirect
4. #21445026C2semidirect
5. #21445005C2semidirect
6. #21444998C2semidirect
7. #21444481C2semidirect
8. #21444201C2semidirect
10. #16840319C22semidirect
12. #16840112C22semidirect
15. #16840047C22semidirect
17. #16839972C22semidirect
22. #16839815C22semidirect
24. C2×C4×#92418C22semidirect
25. #16839511C22semidirect
26. #16839511C22semidirect
28. Q8⋊#92418C22semidirect
29. Q8⋊#92418C22semidirect
266. C61#18485semidirect

Maximal subgroups

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

subgroupindex
#214561412
#214558902
#214450332
#214450262
#214450052
#214449982
#214444812
#214442012
C61×#34812
#1848561