Finite Group Databasebeta

← #29835268

Subgroups — #29835268

Order 15616.

524 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454723C211yesmaximal, characteristic, complemented
2#21448036C211yesmaximal, characteristic, complemented
3#21448031C211yesmaximal, characteristic, complemented
4#21447987C212yesmaximal, complemented
5#21447885C212yesmaximal, complemented
6#21447769C211yesmaximal, characteristic, complemented
7#21447764C211yesmaximal, characteristic
8#21447715C212yesmaximal, complemented
9#21447708C212yesmaximal, complemented
10#21447635C211yesmaximal, characteristic, complemented
11C61×#3173C211yesmaximal, characteristic, complemented
12#16840562C2211yescharacteristic
13C2×#14519783—21no—
14#16840203C2212yes—
15#16840198C2212yes—
16#16839885—21no—
17#16839885C2212yes—
18#16839880—21no—
19#16839880C2212yes—
20#16839811C2211yescharacteristic
21#16839780—21no—
22#16839584C2212yescomplemented
23#16839584C2212yescomplemented
24#16839583C2212yescomplemented
25#16839578C2212yescomplemented
26#16839575C2212yescomplemented
27#16839571C2212yes—
28#16839571C2212yes—
29#16839544C2211yescharacteristic
30#16839544C2211yescharacteristic
31#16839540C2212yescomplemented
32C61×#542—21no—
33C61×#498C2211yescharacteristic, complemented
34C61×#495C2211yescharacteristic, complemented
35C61×#489—21no—
36C61×#470C2211yescharacteristic
37C61×#469C2211yescharacteristic
38C61×#465—21no—
39C61×#442C2211yescharacteristic, complemented
40C61×#441C2211yescharacteristic, complemented
41C4×SD16×C61—21no—
42C61×#405C2211yescharacteristic
43C2×#11758222—21no—
44#14519810—22no—
45#14519805—22no—
46#14519799—21no—
47#14519799C2312yes—
48#14519783—22no—
49#14519783—22no—
50#14519783—41no—
51#14519783—41no—
52C2×#11758194—21no—
53C2×#11758189—41no—
54C2×C8×D61—41no—
55#14519723—22no—
56#14519721—22no—
57#14519720—22no—
58#14519718—22no—
59#14519717C2312yes—
60#14519717C2312yes—
61#14519713—21no—
62#14519713C2312yes—
63#14519705—22no—
64#14519705—22no—
65C2×Q16×C61C2311yescharacteristic
66C2×SD16×C61—21no—
67C2×D8×C61C2311yescharacteristic
68C2×M16×C61C2311yescharacteristic
69C22×C8×C61D411yescharacteristic
70C4⋊Q8×C61—21no—
71#126×C61—41no—
72#124×C61—21no—
73#124×C61—41no—
74#123×C61—41no—
75C22⋊Q8×C61—21no—
76C22⋊Q8×C61—41no—
77#121×C61—21no—
78C4×Q8×C61—21no—
79C4×D4×C61—21no—
80#117×C61D411yescharacteristic
81#106×C61—21no—
82#106×C61C2311yescharacteristic
83#106×C61C2311yescharacteristic
84#103×C61—21no—
85#103×C61—21no—
86#102×C61—21no—
87#102×C61—21no—
88#97×C61D411yescharacteristic
89#97×C61C2311yescharacteristic
90C4×C8×C61D411yescharacteristic
91C4×C8×C61C2311yescharacteristic
92C2×Q8×C61—21no—
93C2×Q8×C61—41no—
94C2×D4×C61—21no—
95C2×D4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61C2×D411yescharacteristic
98C2×C22⋊D61—41no—
99#11758222—22no—
100#11758222—22no—
101#11758222—41no—
102#11758222—41no—
103C2×D244—21no—
104C2×C4×D61—41no—
105C2×Dic122—21no—
106Q16×C61—22no—
107SD16×C61—41no—
108SD16×C61—41no—
109D8×C61—22no—
110M16×C61—22no—
111C2×C8×C61—21no—
112C2×C8×C61—21no—
113C2×C8×C61C2×D411yescharacteristic
114C2×C8×C61C2×D411yescharacteristic
115C2×C8×C61C2×D411yescharacteristic
116C2×C8×C61C2411yescharacteristic
117C4⋊C4×C61—21no—
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—41no—
121C4⋊C4×C61—41no—
122C4⋊C4×C61—41no—
123C22⋊C4×C61—21no—
124C22⋊C4×C61—41no—
125C22⋊C4×C61—41no—
126C22⋊C4×C61—41no—
127C22⋊C4×C61—41no—
128C22⋊C4×C61—41no—
129C42×C61—41no—
130C42×C61C2×D411yescharacteristic
131C42×C61C2×D411yescharacteristic
132#11758194—22no—
133#11758194—41no—
134C2×#92418—41no—
135Dic244—21no—
136Dic244#4112yes—
137D488—21no—
138D488#4112yes—
139#11758190—22no—
140#11758190—22no—
141#11758189—42no—
142#11758189—42no—
143C8×D61—42no—
144C8×D61—42no—
145#11758185—22no—
146#11758185—22no—
147C23×C61—41no—
148C23×C61AGL(1,ℤ/8)11yescharacteristic
149C22×D61—41no—
150Q8×C61—41no—
151Q8×C61—41no—
152Q8×C61—161no—
153D4×C61—41no—
154D4×C61—41no—
155D4×C61—161no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—161no—
164C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C61C22×D411yescharacteristic
166C22⋊D61—42no—
167C22⋊D61—42no—
168C2×Dic61—41no—
169D244—22no—
170D244—41no—
171C4×D61—42no—
172C4×D61—42no—
173Dic122—22no—
174Dic122—41no—
175C8×C61—21no—
176C8×C61—21no—
177C8×C61AGL(1,ℤ/8)11yescharacteristic
178C8×C61AGL(1,ℤ/8)11yescharacteristic
179C8×C61C2×#4112yes—
180#92418—42no—
181#33691—611nomaximal, Sylow
182C22×C61—21no—
183C22×C61—41no—
184C22×C61—81no—
185C22×C61—81no—
186C22×C61—81no—
187C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
188C2×D61—42no—
189C2×D61—82no—
190C4×C61—21no—
191C4×C61—41no—
192C4×C61—41no—
193C4×C61—81no—
194C4×C61—81no—
195C4×C61—81no—
196C4×C61#52511yescharacteristic
197C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
198Dic61—42no—
199#3173D6111yescharacteristic, complemented
200#2955—611no—
201#2239—611no—
202#2238—611no—
203#2230—612no—
204#2212—612no—
205#2191—611no—
206#2190—611no—
207AGL(1,ℤ/16)—612no—
208#2180—612no—
209#2166—611no—
210C2×C61—21no—
211C2×C61—41no—
212C2×C61—161no—
213C2×C61#310911yescharacteristic
214D61—82no—
215#575—611no—
216C2×#131—1221no—
217#542—21no—
218#510—612no—
219#509—612no—
220#498C2×D6111yescharacteristic, complemented
221#495C2×D6111yescharacteristic, complemented
222#489—21no—
223#470C2×D6111yescharacteristic
224#469C2×D6111yescharacteristic
225#465—21no—
226#444—612no—
227#444—1221no—
228#443—612no—
229#443—1221no—
230#442C2×D6111yescharacteristic, complemented
231#441C2×D6111yescharacteristic, complemented
232C4×SD16—21no—
233#430—611no—
234#421—1221no—
235#405C2×D6111yescharacteristic
236#365—612no—
237#365—612no—
238#364—612no—
239#362—612no—
240#361—612no—
241#360—612no—
242#360—612no—
243#349—611no—
244#349—611no—
245#347—612no—
246C61#3369111yescharacteristic, complemented, Sylow
247C2×#41—1221no—
248#137—1222no—
249AGL(1,ℤ/8)—1222no—
250#135—612no—
251#135—1221no—
252C2×Q16C22×D6111yescharacteristic
253C2×SD16—21no—
254C2×D8C22×D6111yescharacteristic
255#131—1222no—
256#131—1222no—
257#131—2441no—
258#131—2441no—
259C2×M16—1221no—
260C2×M16—2441no—
261C2×M16C22×D6111yescharacteristic
262C22×C8—2441no—
263C22×C8C22⋊D6111yescharacteristic
264C4⋊Q8—21no—
265#126—41no—
266#124—21no—
267#124—41no—
268#123—41no—
269C22⋊Q8—21no—
270C22⋊Q8—41no—
271#121—21no—
272C4×Q8—21no—
273C4×D4—21no—
274#117C22⋊D6111yescharacteristic
275Q32—1222no—
276SD32—1222no—
277SD32—1222no—
278D16—1222no—
279M32—612no—
280M32—612no—
281#108—612no—
282#108—1221no—
283#106—21no—
284#106C22×D6111yescharacteristic
285#106C22×D6111yescharacteristic
286#104—1222no—
287#104—1222no—
288#103—21no—
289#103—21no—
290#102—21no—
291#102—21no—
292#97C22⋊D6111yescharacteristic
293#97C22×D6111yescharacteristic
294C4×C8C22⋊D6111yescharacteristic
295C4×C8C22×D6111yescharacteristic
296#41—1222no—
297#41—1222no—
298#41—2441no—
299#41—2441no—
300C2×Q8—21no—
301C2×Q8—41no—
302C2×Q8—1221no—
303C2×D4—21no—
304C2×D4—41no—
305C2×D4—1221no—
306C2×D4—2441no—
307C22×C4—41no—
308C22×C4—2441no—
309C22×C4C2×C22⋊D6111yescharacteristic
310Q16—22no—
311Q16—612no—
312Q16—1221no—
313SD16—41no—
314SD16—41no—
315SD16—1222no—
316SD16—1222no—
317D8—22no—
318D8—612no—
319D8—1221no—
320M16—22no—
321M16—1222no—
322M16—2442no—
323M16—2441no—
324M16—2442no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—2442no—
328C2×C8—2441no—
329C2×C8—2442no—
330C2×C8C2×C22⋊D6111yescharacteristic
331C2×C8C2×C22⋊D6111yescharacteristic
332C2×C8C2×C22⋊D6111yescharacteristic
333C2×C8C23×D6111yescharacteristic
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C22⋊C4—21no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C42—41no—
347C42C2×C22⋊D6111yescharacteristic
348C42C2×C22⋊D6111yescharacteristic
349C16—1222no—
350C16—1222no—
351C23—41no—
352C23—2441no—
353C23#1451980311yescharacteristic
354Q8—41no—
355Q8—41no—
356Q8—161no—
357Q8—1222no—
358Q8—2441no—
359D4—41no—
360D4—41no—
361D4—161no—
362D4—1222no—
363D4—2441no—
364D4—2442no—
365D4—2442no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—161no—
374C2×C4—2442no—
375C2×C4—2441no—
376C2×C4—2442no—
377C2×C4#1451980311yescharacteristic
378C2×C4C22×C22⋊D6111yescharacteristic
379C8—21no—
380C8—21no—
381C8—2442no—
382C8#1451980211yescharacteristic
383C8#1451980211yescharacteristic
384C8#41×D6112yes—
385C22—21no—
386C22—41no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—4882no—
391C22—2442no—
392C22C2×#1451980311yescharacteristic
393C4—21no—
394C4—41no—
395C4—41no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—2442no—
400C4#1684026911yescharacteristic
401C4C2×#1451980211yescharacteristic
402C2—21no—
403C2—41no—
404C2—161no—
405C2—4882no—
406C2#2145590611yescharacteristic
subgroupG/H
centerC2#21455906
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3173C2
socleC2×C61#3109
prime radical#21447764C2
hypercenter#3173D61
hypocenterC61#33691
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #3173

Complements

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

NH ≅ G/Ntype
1. #21454723C2semidirect
2. #21448036C2semidirect
3. #21448031C2semidirect
4. #21447987C2semidirect
5. #21447885C2semidirect
6. #21447769C2semidirect
8. #21447715C2semidirect
9. #21447708C2semidirect
10. #21447635C2semidirect
11. C61×#3173C2semidirect
22. #16839584C22semidirect
23. #16839584C22semidirect
24. #16839583C22semidirect
25. #16839578C22semidirect
26. #16839575C22semidirect
31. #16839540C22semidirect
33. C61×#498C22semidirect
34. C61×#495C22semidirect
39. C61×#442C22semidirect
40. C61×#441C22semidirect
199. #3173D61semidirect
220. #498C2×D61semidirect
221. #495C2×D61semidirect
230. #442C2×D61semidirect
231. #441C2×D61semidirect
246. C61#33691semidirect

Maximal subgroups

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

subgroupindex
#214547232
#214480362
#214480312
#214479872
#214478852
#214477692
#214477642
#214477152
#214477082
#214476352
C61×#31732
#3369161