Finite Group Databasebeta

← #29665280

Subgroups — #29665280

Order 15616.

628 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456980C211yesmaximal, characteristic, complemented
2#21455919C211yesmaximal, characteristic
3#21455828C211yesmaximal, characteristic
4#21454842C211yesmaximal, characteristic
5#21445271C212yesmaximal, complemented
6#21445243C212yesmaximal, complemented
7#21445209C212yesmaximal, complemented
8#21445195C212yesmaximal, complemented
9#21444530C211yesmaximal, characteristic, complemented
10#21444483C211yesmaximal, characteristic, complemented
11C61×#1572C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840319C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16840167C2211yescharacteristic, complemented
16#16840160C2211yescharacteristic, complemented
17#16840113C2211yescharacteristic, complemented
18#16840113C2211yescharacteristic, complemented
19#16840104—22no—
20#16840098—21no—
21#16840093—21no—
22#16840070—22no—
23#16840053C2212yes—
24#16840043C2212yes—
25#16839968C2212yes—
26#16839968C2212yes—
27#16839915—21no—
28#16839915—21no—
29#16839864—22no—
30C4×#11758190—22no—
31C8×D244C2212yes—
32C8×D244C2212yes—
33#16839827—22no—
34#16839815C2211yescharacteristic
35C22⋊C8×D61—22no—
36#16839510C2212yescomplemented
37#16839509C2212yescomplemented
38#16839509C2212yescomplemented
39#16839507C2212yescomplemented
40C61×#432C2211yescharacteristic
41C61×#431C2211yescharacteristic
42C61×#431C2211yescharacteristic
43C61×#335C2212yescomplemented
44C61×#334C2212yescomplemented
45C22×D244D411yescharacteristic
46C22×C4×D61—21no—
47C2×#11758190—22no—
48C2×#11758190—22no—
49C2×C8×D61—22no—
50C2×C8×D61—22no—
51#14519777C2311yescharacteristic
52#14519777C2311yescharacteristic
53#14519777C2311yescharacteristic
54#14519777C2311yescharacteristic
55#14519769—21no—
56#14519769—21no—
57#14519756—21no—
58#14519756—21no—
59#14519756—21no—
60#14519756—21no—
61C4×Dic122—21no—
62C4×Dic122—21no—
63C4×Dic122—21no—
64C4×Dic122—21no—
65C4×D244—21no—
66C4×D244—21no—
67C4×D244D412yes—
68C4×D244C2312yes—
69#14519736—21no—
70#14519736—21no—
71C2×C4⋊Dic61D411yescharacteristic
72C2×#11758198—21no—
73#14519711—22no—
74#14519711—22no—
75#14519709—22no—
76#14519709—22no—
77C4⋊#92418C2312yes—
78#14519702—22no—
79#14519702—22no—
80#14519702—22no—
81#14519702—22no—
82#14519699—22no—
83#14519699—22no—
84C22⋊#92418—21no—
85#14519689—22no—
86#14519689—22no—
87C4×#92418—21no—
88C2×C42×C61C2311yescharacteristic
89C4⋊C8×C61D412yes—
90C4⋊C8×C61C2312yes—
91C4⋊C8×C61C2312yes—
92C22⋊C8×C61—21no—
93C22⋊C8×C61—21no—
94C22⋊C8×C61D412yes—
95#97×C61—21no—
96C4×C8×C61—21no—
97C4×C8×C61C2311yescharacteristic
98C4×C8×C61C2311yescharacteristic
99C23×D61—21no—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61#4111yescharacteristic
103C22×Dic61—21no—
104C2×D244—21no—
105C2×D244—24no—
106C2×D244C2×D412yes—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—82no—
110C2×Dic122—21no—
111C2×Dic122—21no—
112C2×Dic122—21no—
113C2×Dic122—21no—
114C2×C8×C61—22no—
115C2×C8×C61—22no—
116C2×C8×C61C2×D412yes—
117C2×C8×C61C2×D412yes—
118C42×C61C2×D411yescharacteristic
119C42×C61C2×D411yescharacteristic
120C42×C61#4111yescharacteristic
121C42×C61C2411yescharacteristic
122C22⋊Dic61—41no—
123C22⋊Dic61—41no—
124#11758198—22no—
125#11758198—22no—
126C4⋊Dic61—21no—
127C4⋊Dic61—21no—
128C4⋊Dic61—21no—
129C4⋊Dic61—21no—
130C4⋊Dic61—21no—
131C4⋊Dic61—21no—
132C4⋊Dic61—21no—
133C4⋊Dic61—21no—
134C4⋊Dic61—21no—
135C4⋊Dic61C2×D412yes—
136#11758196—41no—
137#11758196—41no—
138#11758196—41no—
139#11758196—41no—
140C4×Dic61—41no—
141C4×Dic61—41no—
142C2×#92418—22no—
143#11758190—44no—
144#11758190—44no—
145C8×D61—82no—
146C8×D61—82no—
147C23×C61#12111yescharacteristic
148C22×D61—22no—
149C22×D61—24no—
150C22×D61—81no—
151C22×D61—81no—
152C2×C4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—41no—
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61#12111yescharacteristic
158C2×C4×C61C22×D411yescharacteristic
159C2×C4×C61C22×D411yescharacteristic
160C2×C4×C61C2×#4111yescharacteristic
161C2×Dic61—22no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—81no—
167D244—24no—
168D244—22no—
169D244#13514yes—
170C4×D61—42no—
171C4×D61—42no—
172C4×D61—82no—
173Dic122—42no—
174Dic122—42no—
175Dic122—42no—
176Dic122—81no—
177C8×C61—22no—
178C8×C61—22no—
179C8×C61—42no—
180C8×C61—42no—
181#92418—42no—
182#19395—611nomaximal, Sylow
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61C2×#12111yescharacteristic
187C2×D61—24no—
188C2×D61—42no—
189C2×D61—42no—
190C2×D61—84no—
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61C2×#13512yes—
196C4×C61#57712yes—
197Dic61—42no—
198Dic61—81no—
199Dic61—81no—
200Dic61—81no—
201Dic61—81no—
202#3231—611no—
203#3110—611no—
204#3101—611no—
205#2972—611no—
206#1683—612no—
207#1679—612no—
208#1673—612no—
209#1671—612no—
210#1572D6111yescharacteristic, complemented
211#1554—611no—
212#1544—611no—
213C2×C61—41no—
214C2×C61#304211yescharacteristic
215C2×C61#305011yescharacteristic
216C2×C61#306911yescharacteristic
217D61—44no—
218#533—611no—
219#533—611no—
220C2×C4×D4—611no—
221#500—611no—
222#498—611no—
223#487—611no—
224#487—611no—
225#485—1222no—
226#484—1221no—
227#483—1221no—
228#479—1222no—
229#476—612no—
230#474—612no—
231#461—612no—
232#461—612no—
233#450—1221no—
234#450—1221no—
235#440—1222no—
236C4×SD16—1222no—
237C8×D4—612no—
238C8×D4—612no—
239C8×D4—1222no—
240#432C2×D6111yescharacteristic
241#431—611no—
242#431C2×D6111yescharacteristic
243#431C2×D6111yescharacteristic
244C2×C22⋊C8—1222no—
245#335C2×D6112yescomplemented
246#334C2×D6112yescomplemented
247#333—612no—
248#332—612no—
249#332—612no—
250#332—612no—
251C61#1939511yescharacteristic, complemented, Sylow
252C22×D4—611no—
253C23×C4—1221no—
254C2×SD16—1222no—
255C2×SD16—1222no—
256C22×C8—1222no—
257C22×C8—1222no—
258C4⋊Q8—611no—
259C4⋊Q8—611no—
260C4⋊Q8—611no—
261C4⋊Q8—611no—
262#125—1221no—
263#125—1221no—
264C22⋊Q8—1221no—
265C22⋊Q8—1221no—
266C22⋊Q8—1221no—
267C22⋊Q8—1221no—
268C4×Q8—1221no—
269C4×Q8—1221no—
270C4×Q8—1221no—
271C4×Q8—1221no—
272C4×D4—612no—
273C4×D4—612no—
274C4×D4—1221no—
275C4×D4—1221no—
276#117—1221no—
277#117—1221no—
278C2×C4⋊C4—611no—
279C2×C22⋊C4—1221no—
280C2×C42C22×D6111yescharacteristic
281#107—1222no—
282#107—1222no—
283#106—1222no—
284#106—1222no—
285C4⋊C8—612no—
286C4⋊C8D24412yes—
287C4⋊C8C22×D6112yes—
288C4⋊C8C22×D6112yes—
289#103—1222no—
290#103—1222no—
291#103—1222no—
292#103—1222no—
293#102—1222no—
294#102—1222no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—1221no—
298C22⋊C8—1222no—
299C22⋊C8—1222no—
300C22⋊C8D24412yes—
301#97—21no—
302C4×C8—21no—
303C4×C8—1221no—
304C4×C8C22×D6111yescharacteristic
305C4×C8C22×D6111yescharacteristic
306C24—1221no—
307C2×Q8—1221no—
308C2×Q8—1221no—
309C2×Q8—1221no—
310C2×Q8—1221no—
311C2×D4—612no—
312C2×D4—1224no—
313C2×D4—1221no—
314C22×C4—1221no—
315C22×C4—1222no—
316C22×C4—1222no—
317C22×C4—4882no—
318C22×C4C2×D24411yescharacteristic
319C22×C4D4×D6111yescharacteristic
320C22×C4#1175822611yescharacteristic
321SD16—2444no—
322SD16—2444no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—1222no—
326C2×C8—4882no—
327C2×C8—4882no—
328C2×C8C2×D24412yes—
329C2×C8C2×D24412yes—
330C4⋊C4—612no—
331C4⋊C4—1221no—
332C4⋊C4—1221no—
333C4⋊C4—1221no—
334C4⋊C4—1221no—
335C4⋊C4—1221no—
336C4⋊C4—1221no—
337C4⋊C4—1221no—
338C4⋊C4—1221no—
339C4⋊C4—1221no—
340C4⋊C4—2441no—
341C4⋊C4—2441no—
342C4⋊C4—2441no—
343C4⋊C4—2441no—
344C22⋊C4—1222no—
345C22⋊C4—1222no—
346C22⋊C4—2441no—
347C22⋊C4—2441no—
348C42—2441no—
349C42—2441no—
350C42C2×D24411yescharacteristic
351C42D4×D6111yescharacteristic
352C42#1175822611yescharacteristic
353C42C23×D6111yescharacteristic
354C23—1224no—
355C23—1222no—
356C23—4881no—
357C23—4881no—
358C23#1451975211yescharacteristic
359Q8—4881no—
360Q8—2442no—
361Q8—2442no—
362Q8—2442no—
363D4—614no—
364D4—1224no—
365D4—1222no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—1222no—
370C2×C4—4882no—
371C2×C4—4881no—
372C2×C4—2441no—
373C2×C4—2441no—
374C2×C4—2441no—
375C2×C4—2441no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4#1451975211yescharacteristic
379C2×C4#1451975211yescharacteristic
380C2×C4#1451975211yescharacteristic
381C2×C4C2×D4×D6111yescharacteristic
382C2×C4C22×D24411yescharacteristic
383C2×C4C2×#1175822611yescharacteristic
384C8—22no—
385C8—22no—
386C8—42no—
387C8—42no—
388C8—2442no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—1224no—
393C22—4884no—
394C22—2442no—
395C22—2442no—
396C22C2×#1451975211yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—4881no—
402C4—4881no—
403C4—4881no—
404C4—4881no—
405C4—2442no—
406C4#135×D6112yes—
407C4#1684057412yes—
408C2—41no—
409C2—2444no—
410C2#2145533211yescharacteristic
411C2#2145540111yescharacteristic
412C2#2145555711yescharacteristic
subgroupG/H
centerC22C2×#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1572C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#1572D61
hypocenterC61#19395
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1572

Complements

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

NH ≅ G/Ntype
1. #21456980C2semidirect
5. #21445271C2semidirect
6. #21445243C2semidirect
7. #21445209C2semidirect
8. #21445195C2semidirect
9. #21444530C2semidirect
10. #21444483C2semidirect
11. C61×#1572C2semidirect
15. #16840167C22semidirect
16. #16840160C22semidirect
17. #16840113C22semidirect
18. #16840113C22semidirect
36. #16839510C22semidirect
37. #16839509C22semidirect
38. #16839509C22semidirect
39. #16839507C22semidirect
43. C61×#335C22semidirect
44. C61×#334C22semidirect
210. #1572D61semidirect
245. #335C2×D61semidirect
246. #334C2×D61semidirect
251. C61#19395semidirect

Maximal subgroups

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

subgroupindex
#214569802
#214559192
#214558282
#214548422
#214452712
#214452432
#214452092
#214451952
#214445302
#214444832
C61×#15722
#1939561