Finite Group Databasebeta

← #29835255

Subgroups — #29835255

Order 15616.

520 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456565C211yesmaximal, characteristic, complemented
2#21454724C211yesmaximal, characteristic
3#21448040C212yesmaximal, complemented
4#21447987C211yesmaximal, characteristic, complemented
5#21447901C211yesmaximal, characteristic, complemented
6#21447900C211yesmaximal, characteristic, complemented
7#21447771C212yesmaximal
8#21447709C212yesmaximal, complemented
9#21447707C212yesmaximal, complemented
10#21447630C211yesmaximal, characteristic, complemented
11C61×#2230C211yesmaximal, characteristic, complemented
12#16840559C2211yescharacteristic
13#16840527—21no—
14#16840317—21no—
15#16840205C2212yes—
16#16840202C2212yes—
17#16840142C2211yescharacteristic, complemented
18#16840114—21no—
19#16840021—21no—
20#16840021C2212yes—
21#16839884C2211yescharacteristic
22#16839883C2211yescharacteristic
23#16839882—21no—
24#16839880C2211yescharacteristic
25#16839876—21no—
26#16839868—21no—
27#16839868C2212yescomplemented
28#16839809C2211yescharacteristic
29#104×D61—21no—
30C4×#11758189C2211yescharacteristic
31#16839584C2212yescomplemented
32#16839583C2211yescharacteristic, complemented
33#16839579C2212yescomplemented
34#16839577C2212yescomplemented
35#16839571C2212yes—
36#16839543C2212yes—
37#16839539C2212yescomplemented
38C61×#498C2211yescharacteristic, complemented
39C61×#443C2211yescharacteristic
40C61×#365C2212yescomplemented
41C61×#364C2211yescharacteristic, complemented
42C61×#360C2212yes—
43#41×D61—21no—
44#14519809—21no—
45#14519806—21no—
46#14519804—21no—
47#14519801—21no—
48#14519799C2311yescharacteristic
49#14519797—21no—
50C2×Dic244—21no—
51C2×Dic244C2312yes—
52#14519785—21no—
53#14519785—41no—
54#14519784—21no—
55#14519783—21no—
56#14519783—41no—
57C2×#11758189—41no—
58C2×#11758189D411yescharacteristic
59C2×#11758189C2311yescharacteristic
60M16×D61—21no—
61M16×D61—41no—
62#14519777—21no—
63#14519768—41no—
64#14519757—22no—
65#14519753—41no—
66Q8×Dic61—41no—
67#14519742—22no—
68#14519735—41no—
69C42×D61D411yescharacteristic
70Dic488—22no—
71#14519723—22no—
72#14519722—22no—
73#14519721—22no—
74#14519717C2312yes—
75#14519713C2311yescharacteristic
76#14519712—21no—
77#14519705—21no—
78#14519704—21no—
79#14519703—21no—
80#14519700—22no—
81#14519700—22no—
82#14519687D411yescharacteristic
83#14519687C2311yescharacteristic
84C4×#92418D411yescharacteristic
85#135×C61C2311yescharacteristic
86#131×C61—21no—
87C4⋊Q8×C61—21no—
88M32×C61C2312yes—
89#108×C61C2311yescharacteristic
90#106×C61—21no—
91#106×C61C2312yes—
92#104×C61—21no—
93C4×C8×C61C2311yescharacteristic
94#41×C61—21no—
95C2×Q8×C61—41no—
96#11758226—21no—
97Q8×D61—21no—
98#11758224—21no—
99#11758224—41no—
100D4×D61—21no—
101D4×D61—41no—
102#11758222—21no—
103#11758222—41no—
104C2×C4×D61—21no—
105C2×C4×D61—41no—
106C2×C4×D61C2×D411yescharacteristic
107C2×Dic122—22no—
108Q16×C61#4111yescharacteristic
109SD16×C61—21no—
110D8×C61#4111yescharacteristic
111M16×C61—21no—
112M16×C61—41no—
113C2×C8×C61—41no—
114C2×C8×C61C2×D411yescharacteristic
115C2×C8×C61C2411yescharacteristic
116C4⋊C4×C61—22no—
117C4⋊C4×C61—41no—
118C42×C61C2×D411yescharacteristic
119#11758202—21no—
120#11758201—21no—
121#11758200—21no—
122#11758199—21no—
123#11758198—42no—
124C4⋊Dic61—41no—
125C4⋊Dic61—41no—
126C4⋊Dic61—42no—
127#11758196—42no—
128C4×Dic61—21no—
129C4×Dic61—42no—
130C4×Dic61C2×D411yescharacteristic
131#11758194—21no—
132#11758194—41no—
133C2×#92418—41no—
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136Dic244—22no—
137Dic244—22no—
138Dic244—41no—
139Dic244—41no—
140Dic244#4111yescharacteristic
141D488#4111yescharacteristic
142#11758190—21no—
143#11758189—21no—
144#11758189—21no—
145#11758189—21no—
146#11758189—21no—
147#11758189—41no—
148#11758189—41no—
149#11758189—41no—
150C8×D61—41no—
151C8×D61—41no—
152C8×D61—41no—
153C16×C61—22no—
154#11758185—22no—
155C22×D61—41no—
156C22×D61AGL(1,ℤ/8)11yescharacteristic
157Q8×C61—21no—
158Q8×C61—82no—
159D4×C61—21no—
160D4×C61—41no—
161C2×C4×C61—21no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61C22×D411yescharacteristic
165C22⋊D61—41no—
166C22⋊D61—41no—
167C22⋊D61—41no—
168C2×Dic61—21no—
169C2×Dic61—41no—
170C2×Dic61—42no—
171C2×Dic61AGL(1,ℤ/8)11yescharacteristic
172D244—21no—
173D244—41no—
174C4×D61—21no—
175C4×D61—21no—
176C4×D61—41no—
177C4×D61—42no—
178C4×D61—41no—
179C4×D61—41no—
180Dic122—21no—
181Dic122—41no—
182Dic122—42no—
183Dic122—42no—
184C8×C61—41no—
185C8×C61AGL(1,ℤ/8)11yescharacteristic
186C8×C61AGL(1,ℤ/8)11yescharacteristic
187C8×C61C2×#4111yescharacteristic
188C8×C61C2×#4111yescharacteristic
189#92418—21no—
190#92418—21no—
191#92418—41no—
192#33689—611nomaximal, Sylow
193C22×C61—41no—
194C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
195C2×D61—21no—
196C2×D61—41no—
197C2×D61—41no—
198C2×D61—161no—
199C4×C61—22no—
200C4×C61—41no—
201C4×C61—82no—
202C4×C61#52511yescharacteristic
203C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
204Dic61—21no—
205Dic61—41no—
206Dic61—41no—
207Dic61—82no—
208#3185—611no—
209#2956—611no—
210#2239—612no—
211#2230—611no—
212#2230D6111yescharacteristic, complemented
213#2214—611no—
214#2214—611no—
215#2191—612no—
216#2180—612no—
217#2180—612no—
218#2165—611no—
219C2×C61—21no—
220C2×C61—81no—
221C2×C61#310911yescharacteristic
222D61—41no—
223D61—81no—
224#575—611no—
225#568—1221no—
226#533—1221no—
227#510—612no—
228#510—612no—
229#498C2×D6111yescharacteristic, complemented
230#494—611no—
231#487—1221no—
232#470—612no—
233#470—1221no—
234#444—611no—
235#444—611no—
236#444—1221no—
237#443—611no—
238#443—1221no—
239#443C2×D6111yescharacteristic
240#441—612no—
241#441—1221no—
242#430—611no—
243C2×#104—1221no—
244C4×M16—611no—
245#365—612no—
246#365C2×D6112yescomplemented
247#364—611no—
248#364C2×D6111yescharacteristic, complemented
249#362—612no—
250#362—612no—
251#360—612no—
252#360C2×D6112yes—
253#349—612no—
254#347—612no—
255C61#3368911yescharacteristic, complemented, Sylow
256C2×#41—1221no—
257#137—1221no—
258#137—1221no—
259AGL(1,ℤ/8)—1221no—
260AGL(1,ℤ/8)—1221no—
261#135—611no—
262#135—1221no—
263#135C22×D6111yescharacteristic
264C2×Q16—612no—
265C2×Q16—1221no—
266#131—21no—
267#131—1221no—
268#131—1221no—
269#131—1221no—
270#131—2441no—
271#131—2441no—
272C2×M16—611no—
273C2×M16—611no—
274C2×M16—1221no—
275C2×M16—2441no—
276C2×M16—2441no—
277C4⋊Q8—21no—
278C4⋊Q8—1221no—
279#125—2441no—
280C22⋊Q8—1222no—
281C22⋊Q8—2441no—
282C4×Q8—1222no—
283C4×Q8—2441no—
284#117—2441no—
285C2×C42—611no—
286Q32—1222no—
287Q32—1222no—
288SD32—1222no—
289SD32—1222no—
290M32—612no—
291M32C22×D6112yes—
292#108—611no—
293#108—1221no—
294#108C22×D6111yescharacteristic
295#106—21no—
296#106C22×D6112yes—
297#104—21no—
298#104—1221no—
299#104—1221no—
300#104—1221no—
301#103—1222no—
302#103—1222no—
303#97—611no—
304#97—611no—
305C4×C8—611no—
306C4×C8C22×D6111yescharacteristic
307#41—21no—
308#41—1221no—
309#41—1221no—
310#41—1221no—
311#41—2441no—
312#41—2441no—
313C2×Q8—41no—
314C2×Q8—1221no—
315C2×Q8—1222no—
316C2×D4—1221no—
317C2×D4—2441no—
318C22×C4—611no—
319C22×C4—1221no—
320C22×C4—2441no—
321Q16—611no—
322Q16—1221no—
323Q16—1222no—
324Q16—1222no—
325Q16—2441no—
326Q16—2441no—
327Q16#1175822611yescharacteristic
328SD16—21no—
329SD16—1221no—
330SD16—1221no—
331SD16—1221no—
332D8—611no—
333D8—1221no—
334D8#1175822611yescharacteristic
335M16—21no—
336M16—41no—
337M16—1221no—
338M16—1221no—
339M16—1221no—
340M16—1221no—
341M16—1221no—
342M16—2441no—
343M16—2441no—
344M16—2441no—
345M16—2441no—
346C2×C8—41no—
347C2×C8—611no—
348C2×C8—611no—
349C2×C8—2441no—
350C2×C8—2441no—
351C2×C8—2441no—
352C2×C8—2441no—
353C2×C8D4×D6111yescharacteristic
354C2×C8C23×D6111yescharacteristic
355C4⋊C4—22no—
356C4⋊C4—41no—
357C4⋊C4—2441no—
358C4⋊C4—2441no—
359C4⋊C4—2442no—
360C4⋊C4—2442no—
361C22⋊C4—2442no—
362C42—611no—
363C42—1221no—
364C42—2442no—
365C42D4×D6111yescharacteristic
366C16—22no—
367C16—1222no—
368C23—611no—
369C23—2441no—
370Q8—21no—
371Q8—82no—
372Q8—1221no—
373Q8—2441no—
374Q8—2442no—
375Q8—2442no—
376D4—21no—
377D4—41no—
378D4—1221no—
379D4—2441no—
380D4—2441no—
381D4—2441no—
382D4—2441no—
383C2×C4—21no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—611no—
387C2×C4—1221no—
388C2×C4—1221no—
389C2×C4—1221no—
390C2×C4—2442no—
391C2×C4—2441no—
392C2×C4—2441no—
393C2×C4—2441no—
394C2×C4—2442no—
395C2×C4—2441no—
396C2×C4C2×D4×D6111yescharacteristic
397C8—41no—
398C8—1221no—
399C8—1221no—
400C8—2441no—
401C8#1451980411yescharacteristic
402C8#1451980411yescharacteristic
403C8#41×D6111yescharacteristic
404C8C2×#1175822611yescharacteristic
405C22—41no—
406C22—1221no—
407C22—9761no—
408C22—2441no—
409C22—2441no—
410C22AGL(1,ℤ/8)×D6111yescharacteristic
411C4—22no—
412C4—41no—
413C4—82no—
414C4—1221no—
415C4—4882no—
416C4—2441no—
417C4—2441no—
418C4#1684027511yescharacteristic
419C4C2×#1451980411yescharacteristic
420C2—21no—
421C2—81no—
422C2—4881no—
423C2—2441no—
424C2#2145591211yescharacteristic
subgroupG/H
centerC2#21455912
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2230C2
socleC2×C61#3109
prime radical#16840559C22
hypercenter#2230D61
hypocenterC61#33689
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ #2230

Complements

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

NH ≅ G/Ntype
1. #21456565C2semidirect
3. #21448040C2semidirect
4. #21447987C2semidirect
5. #21447901C2semidirect
6. #21447900C2semidirect
8. #21447709C2semidirect
9. #21447707C2semidirect
10. #21447630C2semidirect
11. C61×#2230C2semidirect
17. #16840142C22semidirect
27. #16839868C22semidirect
31. #16839584C22semidirect
32. #16839583C22semidirect
33. #16839579C22semidirect
34. #16839577C22semidirect
37. #16839539C22semidirect
38. C61×#498C22semidirect
40. C61×#365C22semidirect
41. C61×#364C22semidirect
212. #2230D61semidirect
229. #498C2×D61semidirect
246. #365C2×D61semidirect
248. #364C2×D61semidirect
255. C61#33689semidirect

Maximal subgroups

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

subgroupindex
#214565652
#214547242
#214480402
#214479872
#214479012
#214479002
#214477712
#214477092
#214477072
#214476302
C61×#22302
#3368961