Finite Group Databasebeta

← #29653922

Subgroups — #29653922

Order 15616.

1220 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456576C212yesmaximal, complemented
2#21451632C211yesmaximal, characteristic, complemented
3#21444856C214yesmaximal, complemented
4#21444788C214yesmaximal, complemented
5#21444387C212yesmaximal, complemented
6C2×#16839488C211yesmaximal, characteristic, complemented
7C2×C61×#537C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic
9#16840092—22no—
10#16840078C2214yescomplemented
11#16839989—24no—
12#16839913—22no—
13#16839900C2214yescomplemented
14C2×C22⋊#92418C2212yes—
15#16839701C2212yescomplemented
16#16839695—22no—
17#16839682C2212yescomplemented
18#16839635—21no—
19#16839625—21no—
20#16839622C2211yescharacteristic
21#16839496C2218yescomplemented
22#16839488C2214yescomplemented
23C22×#41×C61—21no—
24C61×#537—24no—
25C61×#537—22no—
26C61×#537C2214yes—
27C2×C22⋊Q8×C61—22no—
28C2×#121×C61—22no—
29C2×#121×C61—21no—
30C2×#121×C61C2212yes—
31C22×C4⋊C4×C61C2211yescharacteristic
32C22×Dic122C2311yescharacteristic
33C2×#11758200—24no—
34C22×#92418—22no—
35#14519756—22no—
36#14519756—22no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61C2311yescharacteristic
39C2×#11758196—21no—
40C2×#11758196—22no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43#14519708—24no—
44#14519700—24no—
45#14519698—24no—
46C22⋊#92418—22no—
47C22⋊#92418C2314yes—
48#14519684—22no—
49#14519684—22no—
50#14519684—22no—
51#14519684C2312yes—
52C2×#41×C61—24no—
53C2×#41×C61—24no—
54C2×#41×C61—44no—
55C2×#41×C61—44no—
56C22×Q8×C61—21no—
57C22×D4×C61—21no—
58C22×D4×C61—22no—
59C22×D4×C61—21no—
60C22×D4×C61D412yes—
61C23×C4×C61—22no—
62C23×C4×C61D411yescharacteristic
63C23×C4×C61D411yescharacteristic
64C23×C4×C61C2311yescharacteristic
65C22⋊Q8×C61—28no—
66C22⋊Q8×C61—28no—
67#121×C61—24no—
68#121×C61—28no—
69#121×C61—28no—
70#121×C61—22no—
71#121×C61—22no—
72#121×C61—24no—
73#121×C61D414yes—
74#121×C61C2314yes—
75C2×C4⋊C4×C61—21no—
76C2×C4⋊C4×C61—22no—
77C2×C4⋊C4×C61—22no—
78C2×C4⋊C4×C61D412yes—
79C2×C4⋊C4×C61D412yes—
80C2×C4⋊C4×C61C2312yes—
81C2×C22⋊C4×C61—22no—
82C2×C22⋊C4×C61—22no—
83C24×C61—22no—
84C24×C61C2×D411yescharacteristic
85#41×C61—48no—
86#41×C61—416no—
87#41×C61—48no—
88C2×Q8×C61—24no—
89C2×Q8×C61—44no—
90C2×D4×C61—24no—
91C2×D4×C61—24no—
92C2×D4×C61—24no—
93C2×D4×C61—24no—
94C2×D4×C61—22no—
95C2×D4×C61—44no—
96C2×D4×C61—44no—
97C2×D4×C61—44no—
98C2×D4×C61—82no—
99C2×D4×C61—82no—
100C2×D4×C61C2×D414yes—
101C22×C4×C61—21no—
102C22×C4×C61—21no—
103C22×C4×C61—22no—
104C22×C4×C61—28no—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61—21no—
108C22×C4×C61—48no—
109C22×C4×C61—81no—
110C22×C4×C61—81no—
111C22×C4×C61SD1614yescomplemented
112C22×C4×C61C2×D412yes—
113C22×C4×C61C2×D412yes—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D412yes—
116C22×C4×C61C2×D412yes—
117C22×C4×C61C2411yescharacteristic
118C22×Dic61—21no—
119C22×Dic61—21no—
120C22×Dic61—22no—
121C2×Dic122—22no—
122C2×Dic122—81no—
123C4⋊C4×C61—24no—
124C4⋊C4×C61—24no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—24no—
127C4⋊C4×C61C2×D414yes—
128C22⋊C4×C61—28no—
129C22⋊C4×C61—24no—
130C22⋊C4×C61—24no—
131C22⋊Dic61—41no—
132C22⋊Dic61—41no—
133C22⋊Dic61—42no—
134#11758200—84no—
135C4⋊Dic61—22no—
136C4⋊Dic61—81no—
137#11758196—42no—
138#11758196—82no—
139C2×#92418—24no—
140C2×#92418—82no—
141C23×C61—24no—
142C23×C61—41no—
143C23×C61—41no—
144C23×C61—44no—
145C23×C61—44no—
146C23×C61—44no—
147C23×C61#12011yescharacteristic
148C23×C61#12011yescharacteristic
149C23×C61C2×SD1612yes—
150C23×C61AGL(1,ℤ/8)12yes—
151C23×C61C22×D411yescharacteristic
152Q8×C61—48no—
153D4×C61—44no—
154D4×C61—48no—
155D4×C61—44no—
156D4×C61—44no—
157D4×C61—48no—
158D4×C61—82no—
159D4×C61—84no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—24no—
163C2×C4×C61—24no—
164C2×C4×C61—22no—
165C2×C4×C61—24no—
166C2×C4×C61—42no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—42no—
170C2×C4×C61—416no—
171C2×C4×C61—42no—
172C2×C4×C61—44no—
173C2×C4×C61—42no—
174C2×C4×C61—81no—
175C2×C4×C61#12012yes—
176C2×C4×C61C2×SD1614yes—
177C2×C4×C61C22×D412yes—
178C2×Dic61—42no—
179C2×Dic61—42no—
180C2×Dic61—42no—
181C2×Dic61—42no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—42no—
185Dic122—44no—
186#92418—44no—
187#18293—611nomaximal, Sylow
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—42no—
195C22×C61—42no—
196C22×C61—44no—
197C22×C61—44no—
198C22×C61—44no—
199C22×C61—88no—
200C22×C61#45012yes—
201C22×C61#45012yes—
202C22×C61C2×#12011yescharacteristic
203C22×C61C22×SD1611yescharacteristic
204C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
205C4×C61—24no—
206C4×C61—22no—
207C4×C61—22no—
208C4×C61—44no—
209C4×C61—42no—
210Dic61—82no—
211Dic61—81no—
212Dic61—81no—
213C2×#537D6111yescharacteristic
214#3187—612no—
215#2627—611no—
216#1616—614no—
217#1604—614no—
218#1519—612no—
219C2×#324—611no—
220C2×C61—21no—
221C2×C61—22no—
222C2×C61—21no—
223C2×C61—22no—
224C2×C61—84no—
225C2×C61C2×#45011yescharacteristic
226C2×C61#302011yescharacteristic
227C2×C61#302611yescharacteristic
228C22×#41—21no—
229#537—24no—
230#537—22no—
231#537C2×D6114yes—
232C2×C22⋊Q8—22no—
233C2×C22⋊Q8—611no—
234C2×#121—22no—
235C2×#121—21no—
236C2×#121C2×D6112yes—
237C22×C4⋊C4C2×D6111yescharacteristic
238#483—1222no—
239#481—614no—
240#465—1224no—
241#450—1222no—
242#448—614no—
243C2×C22⋊C8—612no—
244#400—612no—
245#398—1222no—
246#395—612no—
247#385—1221no—
248#381—1221no—
249#380—611no—
250#327—618no—
251#324—614no—
252C61#1829311yescharacteristic, complemented, Sylow
253C2×#41—24no—
254C2×#41—24no—
255C2×#41—44no—
256C2×#41—44no—
257C22×Q8—21no—
258C22×Q8—611no—
259C22×D4—21no—
260C22×D4—22no—
261C22×D4—21no—
262C22×D4C22⋊D6112yes—
263C23×C4—22no—
264C23×C4C22⋊D6111yescharacteristic
265C23×C4C22⋊D6111yescharacteristic
266C23×C4C22×D6111yescharacteristic
267C2×SD16—1224no—
268C22×C8—1222no—
269C22⋊Q8—28no—
270C22⋊Q8—28no—
271C22⋊Q8—1222no—
272C22⋊Q8—1222no—
273#121—24no—
274#121—28no—
275#121—28no—
276#121—24no—
277#121—22no—
278#121—22no—
279#121C22⋊D6114yes—
280#121C22×D6114yes—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—611no—
285C2×C4⋊C4—1221no—
286C2×C4⋊C4—1222no—
287C2×C4⋊C4—1221no—
288C2×C4⋊C4C22⋊D6112yes—
289C2×C4⋊C4C22⋊D6112yes—
290C2×C4⋊C4C22×D6112yes—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—1221no—
294C2×C22⋊C4—1221no—
295#106—1224no—
296#103—1224no—
297#102—1224no—
298C22⋊C8—614no—
299C22⋊C8—1222no—
300#95—612no—
301#95—1222no—
302#95—1222no—
303#95—1222no—
304C24—22no—
305C24C2×C22⋊D6111yescharacteristic
306#41—48no—
307#41—416no—
308#41—48no—
309C2×Q8—24no—
310C2×Q8—44no—
311C2×Q8—1222no—
312C2×Q8—4881no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—22no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—44no—
321C2×D4—82no—
322C2×D4—82no—
323C2×D4C2×C22⋊D6114yes—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—28no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—48no—
332C22×C4—81no—
333C22×C4—81no—
334C22×C4—1222no—
335C22×C4—1221no—
336C22×C4—1221no—
337C22×C4#1175820014yescomplemented
338C22×C4C2×C22⋊D6112yes—
339C22×C4C2×C22⋊D6112yes—
340C22×C4C2×C22⋊D6112yes—
341C22×C4C2×C22⋊D6112yes—
342C22×C4C2×C22⋊D6111yescharacteristic
343C22×C4C23×D6111yescharacteristic
344SD16—4884no—
345C2×C8—1224no—
346C2×C8—4882no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—1222no—
352C4⋊C4—4882no—
353C4⋊C4—4881no—
354C4⋊C4—2442no—
355C4⋊C4C2×C22⋊D6114yes—
356C22⋊C4—28no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—2441no—
360C22⋊C4—2441no—
361C22⋊C4—2442no—
362C23—24no—
363C23—41no—
364C23—41no—
365C23—44no—
366C23—44no—
367C23—44no—
368C23#1451974611yescharacteristic
369C23#1451974611yescharacteristic
370C23C2×#1175820012yes—
371C23#1451980312yes—
372C23C22×C22⋊D6111yescharacteristic
373Q8—48no—
374Q8—2444no—
375D4—44no—
376D4—48no—
377D4—44no—
378D4—44no—
379D4—48no—
380D4—82no—
381D4—84no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—416no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—81no—
397C2×C4—2442no—
398C2×C4—2442no—
399C2×C4—2442no—
400C2×C4—2442no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4—2442no—
404C2×C4#1451974612yes—
405C2×C4C2×#1175820014yes—
406C2×C4C22×C22⋊D6112yes—
407C8—2444no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—42no—
415C22—42no—
416C22—44no—
417C22—44no—
418C22—44no—
419C22—88no—
420C22#1683991312yes—
421C22#1683991312yes—
422C22C2×#1451974611yescharacteristic
423C22C22×#1175820011yescharacteristic
424C22C2×#1451980311yescharacteristic
425C4—22no—
426C4—22no—
427C4—24no—
428C4—42no—
429C4—44no—
430C4—4882no—
431C4—4881no—
432C4—4881no—
433C2—21no—
434C2—22no—
435C2—21no—
436C2—22no—
437C2—84no—
438C2C2×#1683991311yescharacteristic
439C2#2145518311yescharacteristic
440C2#2145521611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC2×C61×#537C2
socleC22×C61C2×#120
hypercenterC2×#537D61
hypocenterC61#18293
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#537

Complements

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

NH ≅ G/Ntype
1. #21456576C2semidirect
2. #21451632C2semidirect
3. #21444856C2semidirect
4. #21444788C2semidirect
5. #21444387C2semidirect
6. C2×#16839488C2semidirect
10. #16840078C22semidirect
13. #16839900C22semidirect
15. #16839701C22semidirect
17. #16839682C22semidirect
21. #16839496C22semidirect
22. #16839488C22semidirect
111. C22×C4×C61SD16semidirect
252. C61#18293semidirect
337. C22×C4#11758200semidirect

Maximal subgroups

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

subgroupindex
#214565762
#214516322
#214448562
#214447882
#214443872
C2×#168394882
C2×C61×#5372
#1829361