Finite Group Databasebeta

← #29656235

Subgroups — #29656235

Order 15616.

918 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840482C211yesmaximal, characteristic
2#21456366C211yesmaximal, characteristic, complemented
3#21455945C211yesmaximal, characteristic, complemented
4#21445088C214yesmaximal, complemented
5#21445073C214yesmaximal, complemented
6#21444537C211yesmaximal, characteristic, complemented
7#21444265C211yesmaximal, characteristic, complemented
8#21444262C211yesmaximal, characteristic
9C61×#3110C211yesmaximal, characteristic
10#16840482—22no—
11#16840482—22no—
12#16840482—22no—
13#16840482C2214yes—
14C2×#14519777C2211yescharacteristic
15C2×#14519770—22no—
16C2×#14519764—21no—
17C2×#14519756—21no—
18C2×C4×Dic122C2211yescharacteristic
19C2×#14519742—21no—
20C2×#14519736—21no—
21#16840172—22no—
22#16840165C2212yescomplemented
23#16840158C2212yescomplemented
24#16840090—21no—
25#16840044C2212yescomplemented
26#16839969C2212yescomplemented
27#16839917—21no—
28#16839867—22no—
29#16839819C2211yescharacteristic
30#16839805—22no—
31C4×#11758194C2211yescharacteristic
32#16839509C2214yescomplemented
33#16839492C2214yescomplemented
34D4⋊#92418C2214yes—
35C61×#533C2211yescharacteristic
36C2×C4×D4×C61C2211yescharacteristic
37C61×#484—21no—
38C61×#474C2212yes—
39C61×#461C2212yes—
40C61×#450—21no—
41C61×#440—22no—
42C61×#432C2211yescharacteristic
43C22×Dic122—21no—
44C22×Dic122D411yescharacteristic
45C2×#11758202—22no—
46C2×#11758194—21no—
47#14519777—21no—
48#14519777—21no—
49#14519777D412yes—
50#14519777C2312yes—
51#14519770—28no—
52#14519770—28no—
53#14519764—24no—
54#14519764—24no—
55#14519756—24no—
56#14519756—24no—
57C4×Dic122—21no—
58C4×Dic122—21no—
59C4×Dic122D412yes—
60C4×Dic122C2312yes—
61#14519742—24no—
62#14519742—24no—
63#14519736—24no—
64#14519736—24no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61D411yescharacteristic
67C2×C4⋊Dic61D412yes—
68C2×#11758196—22no—
69C2×#11758196—21no—
70C2×C22⋊Dic61—22no—
71C2×C22⋊Dic61—22no—
72C2×C4×Dic61—22no—
73C2×C4×Dic61—21no—
74#14519710—24no—
75#14519708—22no—
76#14519708—24no—
77C4⋊#92418C2312yes—
78#14519701—22no—
79#14519700—22no—
80C22⋊#92418—21no—
81#14519688—21no—
82#14519688—21no—
83C4×#92418C2312yes—
84C22×D4×C61D411yescharacteristic
85C23×C4×C61—21no—
86C2×SD16×C61—22no—
87C4⋊Q8×C61C2311yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89#125×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×D4×C61—21no—
95C4×D4×C61—21no—
96C4×D4×C61D412yes—
97C4×D4×C61C2312yes—
98#117×C61—21no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C22⋊C4×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#107×C61—22no—
103C4⋊C8×C61C2312yes—
104#103×C61—22no—
105#102×C61—22no—
106C22⋊C8×C61—21no—
107#97×C61—21no—
108C24×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—42no—
113C2×D4×C61C2×D412yes—
114C22×C4×C61—21no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—44no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×Dic61—22no—
122C22×Dic61—21no—
123C22×Dic61—21no—
124C2×Dic122—22no—
125C2×Dic122—21no—
126C2×Dic122—22no—
127C2×Dic122—42no—
128C2×Dic122—44no—
129C2×Dic122C2×D412yes—
130SD16×C61—44no—
131C2×C8×C61—22no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—41no—
138C4⋊C4×C61—41no—
139C4⋊C4×C61C2×D412yes—
140C22⋊C4×C61—22no—
141C22⋊C4×C61—22no—
142C22⋊C4×C61—41no—
143C42×C61—41no—
144C42×C61C2×D411yescharacteristic
145C42×C61C2×D411yescharacteristic
146C42×C61C2×D411yescharacteristic
147C42×C61C2411yescharacteristic
148C22⋊Dic61—28no—
149C22⋊Dic61—28no—
150#11758202—44no—
151C4⋊Dic61—22no—
152C4⋊Dic61—21no—
153C4⋊Dic61—22no—
154C4⋊Dic61—22no—
155C4⋊Dic61C2×D412yes—
156C4⋊Dic61C2×D414yes—
157#11758196—28no—
158#11758196—22no—
159#11758196—22no—
160C4×Dic61—28no—
161C4×Dic61—22no—
162C4×Dic61—22no—
163#11758194—24no—
164C2×#92418—22no—
165C2×#92418—22no—
166C23×C61—22no—
167C23×C61—42no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61#12011yescharacteristic
171Q8×C61—42no—
172Q8×C61—42no—
173D4×C61—22no—
174D4×C61—22no—
175D4×C61—42no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—44no—
181C2×C4×C61—42no—
182C2×C4×C61—44no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61#12011yescharacteristic
186C2×C4×C61#12011yescharacteristic
187C2×C4×C61#12011yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C2×Dic61—22no—
192C2×Dic61—22no—
193C2×Dic61—24no—
194C2×Dic61—42no—
195C2×Dic61—44no—
196C2×Dic61—42no—
197Dic122—22no—
198Dic122—22no—
199Dic122—42no—
200Dic122—44no—
201Dic122—44no—
202C8×C61—42no—
203#92418—24no—
204#92418—42no—
205#18542—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—42no—
209C22×C61—42no—
210C22×C61—42no—
211C22×C61—84no—
212C22×C61#45612yes—
213C22×C61C2×#12011yescharacteristic
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—42no—
219C4×C61—81no—
220C4×C61—81no—
221C4×C61#57512yes—
222C4×C61#57512yes—
223Dic61—42no—
224Dic61—44no—
225Dic61—42no—
226C2×#556—611no—
227#3165—611no—
228#3113—611no—
229#3110D6111yescharacteristic
230#1652—614no—
231#1650—614no—
232#1556—611no—
233#1491—611no—
234#1490—611no—
235C2×C61—22no—
236C2×C61—82no—
237C2×C61#300511yescharacteristic
238C2×C61#300511yescharacteristic
239C2×C61C2×#45611yescharacteristic
240#556—614no—
241#556—1222no—
242#556—1222no—
243#556—1222no—
244#533C2×D6111yescharacteristic
245C2×C4⋊Q8—611no—
246C2×#126—1222no—
247C2×#124—1221no—
248C2×C22⋊Q8—1221no—
249C2×C4×Q8—611no—
250C2×C4×Q8—1221no—
251C2×C4×D4C2×D6111yescharacteristic
252C2×#117—1221no—
253#501—1222no—
254#500—612no—
255#498—612no—
256#484—21no—
257#483—1221no—
258#475—612no—
259#474C2×D6112yes—
260#462—612no—
261#461C2×D6112yes—
262#451—1221no—
263#450—21no—
264#441—1222no—
265#440—22no—
266#432—611no—
267#432C2×D6111yescharacteristic
268#428—1222no—
269C4×M16—611no—
270#332—614no—
271Q8⋊C8—614no—
272D4⋊C8—614no—
273C61#1854211yescharacteristic, complemented, Sylow
274C22×Q8—611no—
275C22×Q8—1221no—
276C22×D4C22⋊D6111yescharacteristic
277C23×C4—21no—
278C2×Q16—1222no—
279C2×SD16—22no—
280C2×M16—1221no—
281C4⋊Q8—612no—
282C4⋊Q8—612no—
283C4⋊Q8—1221no—
284C4⋊Q8—1221no—
285C4⋊Q8C22×D6111yescharacteristic
286C4⋊Q8C22×D6111yescharacteristic
287#126—1228no—
288#126—1228no—
289#125—21no—
290#124—1224no—
291#124—1224no—
292C22⋊Q8—21no—
293C22⋊Q8—21no—
294C22⋊Q8—1224no—
295C22⋊Q8—1224no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8—612no—
299C4×Q8—612no—
300C4×Q8—1224no—
301C4×Q8—1224no—
302C4×Q8—1221no—
303C4×Q8—1221no—
304C4×D4—21no—
305C4×D4—21no—
306C4×D4C22⋊D6112yes—
307C4×D4C22×D6112yes—
308#117—21no—
309#117—1224no—
310#117—1224no—
311C2×C4⋊C4—611no—
312C2×C4⋊C4—612no—
313C2×C4⋊C4—1222no—
314C2×C4⋊C4—1221no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4C22⋊D6111yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—1222no—
319C2×C22⋊C4—1222no—
320C2×C42—1222no—
321C2×C42—1221no—
322C2×C42C22×D6111yescharacteristic
323#107—22no—
324#107—1224no—
325#106—1224no—
326#106—1222no—
327C4⋊C8—612no—
328C4⋊C8C22×D6112yes—
329#103—22no—
330#103—1222no—
331#103—1222no—
332#102—22no—
333C22⋊C8—21no—
334C22⋊C8—1221no—
335#97—21no—
336#97—1221no—
337#97—1221no—
338C4×C8—612no—
339C24—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—612no—
343C2×Q8—1222no—
344C2×Q8—1221no—
345C2×Q8—1222no—
346C2×Q8—2442no—
347C2×Q8—2444no—
348C2×D4—21no—
349C2×D4—42no—
350C2×D4C2×C22⋊D6112yes—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—44no—
355C22×C4—1221no—
356C22×C4—1222no—
357C22×C4—1221no—
358C22×C4D4×D6111yescharacteristic
359C22×C4D4×D6111yescharacteristic
360C22×C4C2×C22⋊D6111yescharacteristic
361Q16—2444no—
362SD16—44no—
363M16—1224no—
364C2×C8—22no—
365C2×C8—1222no—
366C2×C8—1222no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—612no—
375C4⋊C4—614no—
376C4⋊C4—1228no—
377C4⋊C4—1222no—
378C4⋊C4—1222no—
379C4⋊C4—1221no—
380C4⋊C4—1222no—
381C4⋊C4—1222no—
382C4⋊C4—1222no—
383C4⋊C4C2×C22⋊D6112yes—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—41no—
387C22⋊C4—1228no—
388C22⋊C4—1228no—
389C42—41no—
390C42—1228no—
391C42—1222no—
392C42—1222no—
393C42D4×D6111yescharacteristic
394C42D4×D6111yescharacteristic
395C42C2×C22⋊D6111yescharacteristic
396C42C23×D6111yescharacteristic
397C23—22no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23#1451974511yescharacteristic
402Q8—42no—
403Q8—42no—
404Q8—1222no—
405Q8—1222no—
406Q8—2442no—
407Q8—2444no—
408Q8—2444no—
409D4—22no—
410D4—22no—
411D4—42no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—1222no—
422C2×C4—1224no—
423C2×C4—1222no—
424C2×C4—2442no—
425C2×C4—2444no—
426C2×C4—2442no—
427C2×C4#1451974511yescharacteristic
428C2×C4#1451974511yescharacteristic
429C2×C4#1451974511yescharacteristic
430C2×C4C2×D4×D6111yescharacteristic
431C2×C4C2×D4×D6111yescharacteristic
432C2×C4C22×C22⋊D6111yescharacteristic
433C8—42no—
434C8—1224no—
435C8—2442no—
436C22—21no—
437C22—21no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—84no—
442C22#1683994312yes—
443C22C2×#1451974511yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—42no—
449C4—81no—
450C4—81no—
451C4—2442no—
452C4—2442no—
453C4—2444no—
454C4#1684055812yes—
455C4#1684056012yes—
456C2—22no—
457C2—82no—
458C2#2145505811yescharacteristic
459C2#2145505911yescharacteristic
460C2C2×#1683994311yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3110C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenter#3110D61
hypocenterC61#18542
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
2. #21456366C2semidirect
3. #21455945C2semidirect
4. #21445088C2semidirect
5. #21445073C2semidirect
6. #21444537C2semidirect
7. #21444265C2semidirect
22. #16840165C22semidirect
23. #16840158C22semidirect
25. #16840044C22semidirect
26. #16839969C22semidirect
32. #16839509C22semidirect
33. #16839492C22semidirect
273. C61#18542semidirect

Maximal subgroups

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

subgroupindex
C2×#168404822
#214563662
#214559452
#214450882
#214450732
#214445372
#214442652
#214442622
C61×#31102
#1854261