Finite Group Databasebeta

← #29665270

Subgroups — #29665270

Order 15616.

650 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456951C211yesmaximal, characteristic, complemented
2#21455827C211yesmaximal, characteristic, complemented
3#21454843C211yesmaximal, characteristic, complemented
4#21445254C212yesmaximal, complemented
5#21445226C212yesmaximal, complemented
6#21445216C212yesmaximal, complemented
7#21445196C212yesmaximal, complemented
8#21444600C211yesmaximal, characteristic, complemented
9#21444530C211yesmaximal, characteristic, complemented
10#21444481C211yesmaximal, characteristic, complemented
11C61×#3109C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840165C2211yescharacteristic, complemented
14#16840158C2211yescharacteristic, complemented
15#16840108C2211yescharacteristic, complemented
16#16840108C2211yescharacteristic, complemented
17#16840092—21no—
18#16840083—22no—
19#16840062—22no—
20#16840052C2212yes—
21#16839967C2212yes—
22#16839913—21no—
23C4×#11758200—22no—
24D4×#92418—22no—
25D4×#92418C2212yes—
26D4×#92418C2212yes—
27#16839815C2211yescharacteristic
28#16839815C2211yescharacteristic
29#16839815C2211yescharacteristic
30C2×C22⋊#92418—22no—
31#16839514C2212yescomplemented
32#16839512C2212yescomplemented
33#16839511C2212yescomplemented
34#16839510C2212yescomplemented
35#16839509C2212yescomplemented
36#16839506C2212yescomplemented
37C61×#532C2211yescharacteristic
38C2×C4×D4×C61C2211yescharacteristic
39C61×#481—21no—
40C61×#476C2212yes—
41C61×#459C2212yes—
42C61×#447—21no—
43C61×#442—22no—
44C61×#432C2211yescharacteristic
45C2×#11758200—22no—
46C22×#92418—22no—
47C22×#92418—22no—
48#14519777C2311yescharacteristic
49#14519777C2311yescharacteristic
50#14519769—21no—
51#14519756—21no—
52#14519756—21no—
53C4×Dic122—21no—
54C4×Dic122—21no—
55#14519736—21no—
56#14519710—22no—
57#14519708—22no—
58#14519708—22no—
59C4⋊#92418D412yes—
60C4⋊#92418C2312yes—
61C4⋊#92418C2312yes—
62#14519700—22no—
63#14519698—22no—
64#14519698—22no—
65#14519698—22no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C22⋊#92418—22no—
69C22⋊#92418—22no—
70C22⋊#92418D412yes—
71C4×#92418—21no—
72C4×#92418—21no—
73C4×#92418C2311yescharacteristic
74C4×#92418C2311yescharacteristic
75C2×#41×C61—21no—
76C22×D4×C61D411yescharacteristic
77C23×C4×C61—21no—
78C2×D8×C61—22no—
79C4⋊Q8×C61C2311yescharacteristic
80#127×C61C2311yescharacteristic
81#124×C61—21no—
82#121×C61—21no—
83#121×C61—21no—
84C4×D4×C61—21no—
85C4×D4×C61—21no—
86C4×D4×C61—21no—
87C4×D4×C61—21no—
88C4×D4×C61D412yes—
89C4×D4×C61C2312yes—
90C2×C4⋊C4×C61D411yescharacteristic
91C2×C22⋊C4×C61—21no—
92C2×C42×C61C2311yescharacteristic
93#106×C61—22no—
94C4⋊C8×C61C2312yes—
95#102×C61—22no—
96#102×C61—22no—
97C22⋊C8×C61—21no—
98#97×C61—21no—
99C24×C61—21no—
100#41×C61—82no—
101C2×Q8×C61—21no—
102C2×D4×C61—24no—
103C2×D4×C61—21no—
104C2×D4×C61—21no—
105C2×D4×C61—21no—
106C2×D4×C61—21no—
107C2×D4×C61—41no—
108C2×D4×C61C2×D412yes—
109C22×C4×C61—21no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61—41no—
113C22×C4×C61—82no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61#4111yescharacteristic
117C2×Dic122—21no—
118C2×Dic122—21no—
119D8×C61—44no—
120C2×C8×C61—22no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—21no—
123C4⋊C4×C61—21no—
124C4⋊C4×C61C2×D412yes—
125C22⋊C4×C61—22no—
126C22⋊C4×C61—22no—
127C22⋊C4×C61—41no—
128C22⋊C4×C61—41no—
129C42×C61C2×D411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C42×C61#4111yescharacteristic
132C42×C61C2411yescharacteristic
133C22⋊Dic61—41no—
134#11758200—44no—
135C4⋊Dic61—21no—
136C4⋊Dic61—21no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139#11758196—41no—
140#11758196—41no—
141C4×Dic61—41no—
142C2×#92418—22no—
143C2×#92418—22no—
144C2×#92418—82no—
145C2×#92418—82no—
146C2×#92418C2×D412yes—
147C2×#92418C2×D412yes—
148C23×C61—24no—
149C23×C61—22no—
150C23×C61—41no—
151C23×C61—81no—
152C23×C61—81no—
153C23×C61#12111yescharacteristic
154Q8×C61—81no—
155D4×C61—24no—
156D4×C61—22no—
157D4×C61—42no—
158D4×C61—42no—
159D4×C61—81no—
160D4×C61—82no—
161D4×C61#13514yes—
162C2×C4×C61—22no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61—41no—
166C2×C4×C61—42no—
167C2×C4×C61—42no—
168C2×C4×C61—41no—
169C2×C4×C61—82no—
170C2×C4×C61—82no—
171C2×C4×C61—81no—
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61#12111yescharacteristic
175C2×C4×C61C22×D411yescharacteristic
176C2×C4×C61C22×D411yescharacteristic
177C2×C4×C61C2×#4111yescharacteristic
178C2×Dic61—41no—
179C2×Dic61—41no—
180Dic122—42no—
181Dic122—81no—
182C8×C61—42no—
183#92418—22no—
184#92418—22no—
185#92418—42no—
186#92418—42no—
187#19394—611nomaximal, Sylow
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—24no—
191C22×C61—21no—
192C22×C61—42no—
193C22×C61—42no—
194C22×C61—84no—
195C22×C61—81no—
196C22×C61—81no—
197C22×C61—81no—
198C22×C61C2×#12111yescharacteristic
199C4×C61—21no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—42no—
204C4×C61—81no—
205C4×C61C2×#13512yes—
206C4×C61#57712yes—
207Dic61—81no—
208Dic61—81no—
209#3228—611no—
210#3109D6111yescharacteristic
211#3101—611no—
212#2972—611no—
213#1681—612no—
214#1677—612no—
215#1675—612no—
216#1671—612no—
217#1572—611no—
218#1554—611no—
219#1543—611no—
220C2×C61—41no—
221C2×C61—44no—
222C2×C61—161no—
223C2×C61#304211yescharacteristic
224C2×C61#305011yescharacteristic
225C2×C61#306811yescharacteristic
226#533—611no—
227#532C2×D6111yescharacteristic
228C2×C4×D4C2×D6111yescharacteristic
229#500—611no—
230#498—611no—
231#486—611no—
232#486—611no—
233#483—1221no—
234#482—1222no—
235#481—21no—
236#478—1222no—
237#476—612no—
238#476C2×D6112yes—
239#461—612no—
240#459C2×D6112yes—
241#450—1221no—
242#447—21no—
243#442—22no—
244C4×SD16—1222no—
245C8×D4—612no—
246C8×D4—612no—
247C8×D4—1222no—
248#432C2×D6111yescharacteristic
249#431—611no—
250#431—611no—
251#431—611no—
252C2×C22⋊C8—1222no—
253#335—612no—
254#334—612no—
255#333—612no—
256#333—612no—
257#332—612no—
258#331—612no—
259C61#1939411yescharacteristic, complemented, Sylow
260C2×#41—21no—
261C22×D4C22⋊D6111yescharacteristic
262C23×C4—21no—
263C2×SD16—1222no—
264C2×D8—22no—
265C22×C8—1222no—
266C22×C8—1222no—
267C4⋊Q8—611no—
268C4⋊Q8—611no—
269C4⋊Q8C22×D6111yescharacteristic
270#127C22×D6111yescharacteristic
271#125—1221no—
272#124—21no—
273C22⋊Q8—1221no—
274C22⋊Q8—1221no—
275#121—21no—
276#121—21no—
277C4×Q8—1221no—
278C4×Q8—1221no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4—21no—
283C4×D4C22⋊D6112yes—
284C4×D4C22×D6112yes—
285#117—1221no—
286C2×C4⋊C4C22⋊D6111yescharacteristic
287C2×C22⋊C4—21no—
288C2×C42C22×D6111yescharacteristic
289#107—1222no—
290#106—22no—
291#106—1222no—
292#106—1222no—
293C4⋊C8—612no—
294C4⋊C8—612no—
295C4⋊C8—612no—
296C4⋊C8C22×D6112yes—
297#103—1222no—
298#102—22no—
299#102—22no—
300#102—1222no—
301#102—1222no—
302#102—1222no—
303C22⋊C8—21no—
304C22⋊C8—612no—
305C22⋊C8—1222no—
306C22⋊C8—1222no—
307C22⋊C8—1221no—
308C22⋊C8—1221no—
309#97—21no—
310C4×C8—611no—
311C4×C8—611no—
312C4×C8—1221no—
313C4×C8—1221no—
314C24—21no—
315#41—82no—
316C2×Q8—21no—
317C2×Q8—1221no—
318C2×Q8—1221no—
319C2×D4—24no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—41no—
325C2×D4C2×C22⋊D6112yes—
326C22×C4—21no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—41no—
330C22×C4—82no—
331C22×C4D4×D6111yescharacteristic
332C22×C4#1175822411yescharacteristic
333C22×C4C2×C22⋊D6111yescharacteristic
334SD16—2444no—
335D8—44no—
336C2×C8—22no—
337C2×C8—612no—
338C2×C8—612no—
339C2×C8—1222no—
340C2×C8—1222no—
341C2×C8—4882no—
342C2×C8—4882no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—1221no—
347C4⋊C4—1221no—
348C4⋊C4—1221no—
349C4⋊C4—1221no—
350C4⋊C4—2441no—
351C4⋊C4—2441no—
352C4⋊C4C2×C22⋊D6112yes—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—2441no—
358C42—2441no—
359C42D4×D6111yescharacteristic
360C42#1175822411yescharacteristic
361C42C2×C22⋊D6111yescharacteristic
362C42C23×D6111yescharacteristic
363C23—24no—
364C23—22no—
365C23—41no—
366C23—81no—
367C23—81no—
368C23#1451974911yescharacteristic
369Q8—81no—
370Q8—4881no—
371Q8—2442no—
372D4—24no—
373D4—22no—
374D4—42no—
375D4—42no—
376D4—81no—
377D4—82no—
378D4#1451979714yes—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—81no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4#1451974911yescharacteristic
392C2×C4#1451974911yescharacteristic
393C2×C4#1451974911yescharacteristic
394C2×C4C2×D4×D6111yescharacteristic
395C2×C4C22×C22⋊D6111yescharacteristic
396C2×C4C2×#1175822411yescharacteristic
397C8—42no—
398C8—1222no—
399C8—1222no—
400C8—2442no—
401C8—2442no—
402C22—21no—
403C22—24no—
404C22—21no—
405C22—21no—
406C22—42no—
407C22—42no—
408C22—84no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22C2×#1451974911yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—42no—
418C4—81no—
419C4—4881no—
420C4—4881no—
421C4C2×#1451979712yes—
422C4#1684057212yes—
423C2—41no—
424C2—44no—
425C2—161no—
426C2#2145533311yescharacteristic
427C2#2145539711yescharacteristic
428C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3109C2
socleC22×C61C2×#121
hypercenter#3109D61
hypocenterC61#19394
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3109

Complements

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

NH ≅ G/Ntype
1. #21456951C2semidirect
2. #21455827C2semidirect
3. #21454843C2semidirect
4. #21445254C2semidirect
5. #21445226C2semidirect
6. #21445216C2semidirect
7. #21445196C2semidirect
8. #21444600C2semidirect
9. #21444530C2semidirect
10. #21444481C2semidirect
13. #16840165C22semidirect
14. #16840158C22semidirect
15. #16840108C22semidirect
16. #16840108C22semidirect
31. #16839514C22semidirect
32. #16839512C22semidirect
33. #16839511C22semidirect
34. #16839510C22semidirect
35. #16839509C22semidirect
36. #16839506C22semidirect
259. C61#19394semidirect

Maximal subgroups

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

subgroupindex
#214569512
#214558272
#214548432
#214452542
#214452262
#214452162
#214451962
#214446002
#214445302
#214444812
C61×#31092
#1939461