Finite Group Databasebeta

← #29834402

Subgroups — #29834402

Order 15616.

696 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458953C212yesmaximal, complemented
2C2×#16839880C211yesmaximal, characteristic, complemented
3#21447897C212yesmaximal, complemented
4#21447892C212yesmaximal, complemented
5#21447885C214yesmaximal, complemented
6C2×#16839583C211yesmaximal, characteristic, complemented
7#21447568C212yesmaximal, complemented
8C61×#3147C211yesmaximal, characteristic, complemented
9C2×#14519799C2211yescharacteristic
10C2×#14519783—21no—
11#16840203C2214yes—
12#16840202—22no—
13#16840198C2214yes—
14#16840197—22no—
15#16840192—24no—
16C2×#14519717C2212yes—
17#16839880—21no—
18#16839880—21no—
19#16839880C2212yes—
20#16839880C2212yes—
21C2×#14519713C2211yescharacteristic
22C2×#14519705—21no—
23#16839583C2212yescomplemented
24#16839583C2212yescomplemented
25#16839578C2214yescomplemented
26#16839575C2214yescomplemented
27C2×#135×C61—21no—
28C2×#135×C61C2212yes—
29C61×#532—21no—
30C61×#495—21no—
31C61×#495C2212yescomplemented
32C61×#494C2211yescharacteristic, complemented
33C61×#493C2211yescharacteristic, complemented
34C61×#492—21no—
35C2×C4×C8×C61C2211yescharacteristic
36C2×#11758222—21no—
37#14519799—21no—
38#14519799—21no—
39#14519799D412yescomplemented
40#14519799C2312yes—
41C2×Dic244D411yescharacteristic, complemented
42C2×#11758190—21no—
43C2×D488D411yescharacteristic, complemented
44#14519783—22no—
45#14519783—22no—
46#14519783—41no—
47#14519783—41no—
48C2×#11758194—21no—
49C2×#11758189—41no—
50C2×C8×D61—41no—
51#14519723—24no—
52#14519721—24no—
53#14519720—24no—
54#14519718—24no—
55#14519717—22no—
56#14519717C2314yes—
57C2×#11758185—22no—
58#14519713—21no—
59#14519713C2312yes—
60#14519705—22no—
61#14519705—22no—
62C2×#41×C61—22no—
63#135×C61—22no—
64#135×C61—22no—
65#135×C61—22no—
66#135×C61—22no—
67#135×C61—41no—
68#135×C61—41no—
69#135×C61—41no—
70#135×C61—41no—
71C2×Q16×C61—21no—
72C2×Q16×C61C2312yes—
73C2×SD16×C61—22no—
74C2×SD16×C61—22no—
75C2×D8×C61—21no—
76C2×D8×C61C2312yes—
77C22×C8×C61D411yescharacteristic
78C22×C8×C61C2311yescharacteristic
79C4⋊Q8×C61—21no—
80#127×C61—21no—
81#124×C61—22no—
82#121×C61—42no—
83C4×D4×C61—42no—
84C2×C42×C61D411yescharacteristic
85C4×C8×C61D411yescharacteristic
86C4×C8×C61D411yescharacteristic
87C4×C8×C61C2311yescharacteristic
88C4×C8×C61C2311yescharacteristic
89#41×C61—42no—
90#41×C61—42no—
91#41×C61—42no—
92#41×C61—42no—
93C2×Q8×C61—22no—
94C2×D4×C61—22no—
95C2×D4×C61—42no—
96C2×D4×C61—42no—
97C22×C4×C61—21no—
98C22×C4×C61—42no—
99C22×C4×C61C2×D411yescharacteristic
100C2×C22⋊D61—41no—
101#11758222—22no—
102#11758222—22no—
103#11758222—41no—
104#11758222—41no—
105C2×D244—21no—
106C2×C4×D61—41no—
107C2×Dic122—21no—
108Q16×C61—22no—
109Q16×C61—22no—
110Q16×C61—41no—
111Q16×C61—41no—
112SD16×C61—42no—
113SD16×C61—42no—
114SD16×C61—42no—
115SD16×C61—42no—
116D8×C61—22no—
117D8×C61—22no—
118D8×C61—41no—
119D8×C61—41no—
120C2×C8×C61D811yescharacteristic
121C2×C8×C61D811yescharacteristic
122C2×C8×C61D811yescharacteristic
123C2×C8×C61D811yescharacteristic
124C2×C8×C61C2×D411yescharacteristic
125C2×C8×C61C2×D411yescharacteristic
126C2×C8×C61C2×D411yescharacteristic
127C2×C8×C61C2×D411yescharacteristic
128C2×C8×C61C2×D411yescharacteristic
129C2×C8×C61#4111yescharacteristic
130C2×C8×C61#4111yescharacteristic
131C2×C8×C61C2411yescharacteristic
132C4⋊C4×C61—42no—
133C22⋊C4×C61—44no—
134C42×C61—21no—
135C42×C61C2×D411yescharacteristic
136C42×C61C2×D411yescharacteristic
137#11758194—22no—
138#11758194—41no—
139C2×#92418—41no—
140Dic244—21no—
141Dic244C2×D412yes—
142D488—21no—
143D488C2×D412yes—
144#11758190—22no—
145#11758190—22no—
146#11758189—42no—
147#11758189—42no—
148C8×D61—42no—
149C8×D61—42no—
150#11758185—24no—
151C23×C61—42no—
152C23×C61AGL(1,ℤ/8)11yescharacteristic
153C22×D61—41no—
154Q8×C61—42no—
155Q8×C61—42no—
156D4×C61—42no—
157D4×C61—42no—
158D4×C61—84no—
159D4×C61—82no—
160D4×C61—82no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61—82no—
166C2×C4×C61—82no—
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61C2×D811yescharacteristic
170C2×C4×C61C2×D811yescharacteristic
171C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C61C22×D411yescharacteristic
173C22⋊D61—42no—
174C22⋊D61—42no—
175C2×Dic61—41no—
176D244—22no—
177D244—41no—
178C4×D61—42no—
179C4×D61—42no—
180Dic122—22no—
181Dic122—41no—
182C8×C61#12111yescharacteristic
183C8×C61#12111yescharacteristic
184C8×C61C2×D811yescharacteristic
185C8×C61C2×D811yescharacteristic
186C8×C61C2×D811yescharacteristic
187C8×C61C2×D811yescharacteristic
188C8×C61C22×D411yescharacteristic
189C8×C61C2×#4111yescharacteristic
190#92418—42no—
191#33594—611nomaximal, Sylow
192C22×C61—41no—
193C22×C61—84no—
194C22×C61—82no—
195C22×C61#45911yescharacteristic
196C22×C61#45911yescharacteristic
197C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
198C2×D61—42no—
199C2×D61—82no—
200C4×C61—22no—
201C4×C61—41no—
202C4×C61—82no—
203C4×C61#45911yescharacteristic
204C4×C61#45911yescharacteristic
205C4×C61C2×#12111yescharacteristic
206C4×C61C22×D811yescharacteristic
207Dic61—42no—
208#3414—612no—
209#3147D6111yescharacteristic, complemented
210C2×#443—611no—
211#2214—612no—
212#2213—612no—
213#2212—614no—
214C2×#364—611no—
215#2148—612no—
216C2×C61—21no—
217C2×C61—21no—
218C2×C61—21no—
219C2×C61—162no—
220C2×C61C2×#45911yescharacteristic
221D61—82no—
222C2×#135—21no—
223C2×#135—611no—
224C2×#135C2×D6112yes—
225C2×#131—1221no—
226#532—21no—
227#510—614no—
228#510—1222no—
229#509—614no—
230#509—1222no—
231#508—1224no—
232C2×M32—612no—
233#495—21no—
234#495C2×D6112yescomplemented
235#494C2×D6111yescharacteristic, complemented
236#493C2×D6111yescharacteristic, complemented
237#492—21no—
238#443—612no—
239#443—612no—
240#443—1221no—
241#443—1221no—
242C2×#108—611no—
243C2×#104—1221no—
244C2×C4×C8C2×D6111yescharacteristic
245#364—612no—
246#364—612no—
247#362—614no—
248#361—614no—
249C61#3359411yescharacteristic, complemented, Sylow
250C2×#41—22no—
251C2×#41—1221no—
252#135—22no—
253#135—22no—
254#135—22no—
255#135—22no—
256#135—41no—
257#135—41no—
258#135—41no—
259#135—41no—
260#135—612no—
261#135—612no—
262#135—1221no—
263#135—1221no—
264C2×Q16—21no—
265C2×Q16—611no—
266C2×Q16C22×D6112yes—
267C2×SD16—22no—
268C2×SD16—22no—
269C2×SD16—1221no—
270C2×D8—21no—
271C2×D8—611no—
272C2×D8C22×D6112yes—
273#131—1222no—
274#131—1222no—
275#131—2441no—
276#131—2441no—
277C2×M16—1221no—
278C2×M16—2441no—
279C22×C8—2441no—
280C22×C8C22⋊D6111yescharacteristic
281C22×C8C22×D6111yescharacteristic
282C4⋊Q8—21no—
283#127—21no—
284#124—22no—
285#121—42no—
286C4×D4—42no—
287C2×C42C22⋊D6111yescharacteristic
288Q32—1224no—
289SD32—1224no—
290SD32—1224no—
291D16—1224no—
292M32—614no—
293M32—1222no—
294C2×C16—1222no—
295#108—612no—
296#108—1221no—
297#104—1222no—
298#104—1222no—
299C4×C8C22⋊D6111yescharacteristic
300C4×C8C22⋊D6111yescharacteristic
301C4×C8C22×D6111yescharacteristic
302C4×C8C22×D6111yescharacteristic
303#41—42no—
304#41—42no—
305#41—42no—
306#41—42no—
307#41—1222no—
308#41—1222no—
309#41—2441no—
310#41—2441no—
311C2×Q8—22no—
312C2×Q8—1221no—
313C2×D4—22no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—1221no—
317C2×D4—2441no—
318C22×C4—21no—
319C22×C4—42no—
320C22×C4—2441no—
321C22×C4C2×C22⋊D6111yescharacteristic
322Q16—22no—
323Q16—22no—
324Q16—41no—
325Q16—41no—
326Q16—612no—
327Q16—1221no—
328SD16—42no—
329SD16—42no—
330SD16—42no—
331SD16—42no—
332SD16—1222no—
333SD16—1222no—
334D8—22no—
335D8—22no—
336D8—41no—
337D8—41no—
338D8—612no—
339D8—1221no—
340M16—1222no—
341M16—2441no—
342M16—2442no—
343M16—2442no—
344C2×C8—2441no—
345C2×C8—2442no—
346C2×C8—2442no—
347C2×C8#1175819911yescharacteristic
348C2×C8#1175819911yescharacteristic
349C2×C8#1175819911yescharacteristic
350C2×C8#1175819911yescharacteristic
351C2×C8D4×D6111yescharacteristic
352C2×C8D4×D6111yescharacteristic
353C2×C8#1175822411yescharacteristic
354C2×C8#1175822411yescharacteristic
355C2×C8C2×C22⋊D6111yescharacteristic
356C2×C8C2×C22⋊D6111yescharacteristic
357C2×C8C2×C22⋊D6111yescharacteristic
358C2×C8C23×D6111yescharacteristic
359C4⋊C4—42no—
360C22⋊C4—44no—
361C42—21no—
362C42C2×C22⋊D6111yescharacteristic
363C42C2×C22⋊D6111yescharacteristic
364C16—1224no—
365C23—42no—
366C23—2441no—
367C23#1451980311yescharacteristic
368Q8—42no—
369Q8—42no—
370Q8—1222no—
371Q8—2441no—
372D4—42no—
373D4—42no—
374D4—84no—
375D4—82no—
376D4—82no—
377D4—1222no—
378D4—2442no—
379D4—2442no—
380D4—2441no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—2441no—
388C2×C4—2442no—
389C2×C4—2442no—
390C2×C4#1451974811yescharacteristic
391C2×C4#1451974811yescharacteristic
392C2×C4C2×#1175819911yescharacteristic
393C2×C4C2×#1175819911yescharacteristic
394C2×C4#1451980311yescharacteristic
395C2×C4C22×C22⋊D6111yescharacteristic
396C8—2442no—
397C8#1451974811yescharacteristic
398C8#1451974811yescharacteristic
399C8C2×#1175819911yescharacteristic
400C8C2×#1175819911yescharacteristic
401C8C2×#1175819911yescharacteristic
402C8C2×#1175819911yescharacteristic
403C8C2×D4×D6111yescharacteristic
404C8C2×#1175822411yescharacteristic
405C22—41no—
406C22—84no—
407C22—82no—
408C22—4882no—
409C22—2442no—
410C22#1683995411yescharacteristic
411C22#1683995411yescharacteristic
412C22C2×#1451980311yescharacteristic
413C4—22no—
414C4—41no—
415C4—82no—
416C4—2442no—
417C4#1683995411yescharacteristic
418C4#1683995411yescharacteristic
419C4C2×#1451974811yescharacteristic
420C4C22×#1175819911yescharacteristic
421C2—21no—
422C2—21no—
423C2—21no—
424C2—162no—
425C2—4882no—
426C2C2×#1683995411yescharacteristic
subgroupG/H
centerC4#16839954
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3147C2
socleC2×C61C2×#459
hypercenter#3147D61
hypocenterC61#33594
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ #3147

Complements

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

NH ≅ G/Ntype
1. #21458953C2semidirect
2. C2×#16839880C2semidirect
3. #21447897C2semidirect
4. #21447892C2semidirect
5. #21447885C2semidirect
6. C2×#16839583C2semidirect
7. #21447568C2semidirect
8. C61×#3147C2semidirect
23. #16839583C22semidirect
24. #16839583C22semidirect
25. #16839578C22semidirect
26. #16839575C22semidirect
31. C61×#495C22semidirect
32. C61×#494C22semidirect
33. C61×#493C22semidirect
39. #14519799D4semidirect
41. C2×Dic244D4semidirect
43. C2×D488D4semidirect
209. #3147D61semidirect
234. #495C2×D61semidirect
235. #494C2×D61semidirect
236. #493C2×D61semidirect
249. C61#33594semidirect

Maximal subgroups

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

subgroupindex
#214589532
C2×#168398802
#214478972
#214478922
#214478852
C2×#168395832
#214475682
C61×#31472
#3359461