Finite Group Databasebeta

← #29665236

Subgroups — #29665236

Order 15616.

696 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840049C211yesmaximal, characteristic, complemented
2#21455279C211yesmaximal, characteristic
3#21454842C211yesmaximal, characteristic, complemented
4#21445225C214yesmaximal, complemented
5#21445191C214yesmaximal, complemented
6#21444590C211yesmaximal, characteristic, complemented
7#21444500C211yesmaximal, characteristic, complemented
8#21444468C211yesmaximal, characteristic, complemented
9C61×#3227C211yesmaximal, characteristic, complemented
10C2×#14519802—21no—
11C2×C4×D244C2211yescharacteristic
12#16840049—22no—
13#16840049C2214yes—
14#16839974—22no—
15#16839964C2212yes—
16#16839954C2212yes—
17C8×D244C2214yes—
18#16839827—22no—
19#16839815C2211yescharacteristic
20C2×#14519708—21no—
21#16839786C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23#16839763—21no—
24C2×#14519697—21no—
25C22⋊C8×D61—22no—
26#16839513C2214yescomplemented
27#16839511C2214yescomplemented
28#16839508C2212yescomplemented
29#16839506C2212yescomplemented
30C61×#532C2211yescharacteristic
31C2×C4⋊Q8×C61C2211yescharacteristic
32C61×#498C2212yescomplemented
33C61×#486C2212yescomplemented
34C61×#481—22no—
35C61×#476—22no—
36C61×#431C2211yescharacteristic
37C22×D244D411yescharacteristic
38C22×C4×D61—21no—
39#14519802—44no—
40C2×#11758201—22no—
41C2×#11758199—22no—
42C2×#11758194—21no—
43C22×#92418—21no—
44C2×C8×D61—24no—
45C4×D244—21no—
46C4×D244—21no—
47C4×D244D412yes—
48C4×D244C2312yes—
49C2×C4⋊Dic61D411yescharacteristic
50C2×#11758198—21no—
51#14519708—24no—
52C4⋊#92418—21no—
53C4⋊#92418—21no—
54C4⋊#92418C2312yes—
55C4⋊#92418C2312yes—
56C4⋊#92418C2312yes—
57#14519701—22no—
58#14519698—22no—
59#14519697—24no—
60C22⋊#92418—21no—
61#14519689—24no—
62C4×#92418—21no—
63C2×#41×C61—21no—
64C22×Q8×C61—21no—
65C4⋊Q8×C61—21no—
66C4⋊Q8×C61—21no—
67C4⋊Q8×C61—21no—
68C4⋊Q8×C61C2312yes—
69C4⋊Q8×C61C2311yescharacteristic
70#127×C61C2311yescharacteristic
71#124×C61—21no—
72#121×C61—21no—
73#121×C61—21no—
74C4×D4×C61—21no—
75C4×D4×C61—21no—
76C2×C4⋊C4×C61—21no—
77C2×C4⋊C4×C61—21no—
78C2×C42×C61C2311yescharacteristic
79#106×C61—24no—
80C4⋊C8×C61D412yes—
81#102×C61—24no—
82C22⋊C8×C61D412yes—
83C4×C8×C61C2312yes—
84C23×D61—21no—
85#41×C61—42no—
86#41×C61—42no—
87C2×Q8×C61—21no—
88C2×Q8×C61—22no—
89C2×Q8×C61—41no—
90C2×Q8×C61—82no—
91C2×D4×C61—21no—
92C2×D4×C61—21no—
93C2×D4×C61—21no—
94C2×D4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61#4111yescharacteristic
100C22×Dic61—21no—
101C2×D244—21no—
102C2×D244—22no—
103C2×D244SD1614yes—
104C2×D244C2×D412yes—
105C2×C4×D61—22no—
106C2×C4×D61—22no—
107C2×C4×D61—82no—
108C2×C8×C61C2×D414yes—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—21no—
111C4⋊C4×C61—22no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C22⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C42×C61C2×D411yescharacteristic
118C42×C61C2×D411yescharacteristic
119C42×C61#4111yescharacteristic
120C42×C61C2411yescharacteristic
121#11758201—44no—
122#11758199—44no—
123#11758198—22no—
124#11758198—22no—
125C4⋊Dic61—21no—
126C4⋊Dic61C2×D412yes—
127#11758194—42no—
128C2×#92418—22no—
129C2×#92418—22no—
130C2×#92418—22no—
131C2×#92418—42no—
132C8×D61—84no—
133C23×C61—41no—
134C23×C61#12111yescharacteristic
135C22×D61—22no—
136C22×D61—24no—
137C22×D61—81no—
138C22×D61—81no—
139Q8×C61—41no—
140Q8×C61—41no—
141Q8×C61—82no—
142Q8×C61—82no—
143D4×C61—41no—
144D4×C61—41no—
145D4×C61—42no—
146D4×C61—81no—
147D4×C61—82no—
148C2×C4×C61—22no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—42no—
152C2×C4×C61—82no—
153C2×C4×C61—82no—
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61#12111yescharacteristic
156C2×C4×C61#12111yescharacteristic
157C2×C4×C61C2×SD1612yes—
158C2×C4×C61#13512yes—
159C2×C4×C61C22×D411yescharacteristic
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61C2×#4111yescharacteristic
162C2×Dic61—22no—
163C2×Dic61—81no—
164D244—22no—
165D244—24no—
166D244C2×SD1614yes—
167C4×D61—44no—
168C4×D61—82no—
169C8×C61—24no—
170#92418—42no—
171#92418—42no—
172#92418—42no—
173#19391—611nomaximal, Sylow
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—81no—
177C22×C61—81no—
178C22×C61—81no—
179C22×C61#46512yes—
180C22×C61C2×#12111yescharacteristic
181C2×D61—24no—
182C2×D61—42no—
183C2×D61—42no—
184C2×D61—84no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—81no—
189C4×C61—82no—
190C4×C61#46512yes—
191C4×C61C22×SD1611yescharacteristic
192C4×C61C2×#13511yescharacteristic
193C4×C61#57712yes—
194Dic61—42no—
195#3227D6111yescharacteristic, complemented
196C2×#476—611no—
197#3036—611no—
198#2972—611no—
199#1677—614no—
200#1671—614no—
201#1570—611no—
202#1548—611no—
203#1540—611no—
204C2×C61—22no—
205C2×C61—161no—
206C2×C61#304211yescharacteristic
207C2×C61C2×#46511yescharacteristic
208C2×C61#306811yescharacteristic
209D61—44no—
210C2×AGL(1,ℤ/8)—1221no—
211#532C2×D6111yescharacteristic
212C2×C4⋊Q8C2×D6111yescharacteristic
213C2×C4×D4—611no—
214#498C2×D6112yescomplemented
215#486C2×D6112yescomplemented
216#481—22no—
217#476—22no—
218#476—614no—
219#476—1222no—
220#463—1222no—
221#461—612no—
222#459—612no—
223C8×D4—614no—
224C8×D4—1222no—
225#431—611no—
226#431C2×D6111yescharacteristic
227C2×#106—1221no—
228#423—611no—
229C2×C4⋊C8—611no—
230#417—1221no—
231C2×#102—1221no—
232C2×C22⋊C8—1222no—
233#334—614no—
234#333—614no—
235#332—612no—
236#331—612no—
237C61#1939111yescharacteristic, complemented, Sylow
238C2×#41—21no—
239C22×Q8—21no—
240C22×D4—611no—
241C23×C4—1221no—
242AGL(1,ℤ/8)—2444no—
243C2×SD16—1222no—
244C2×D8—1222no—
245C2×M16—1221no—
246C22×C8—1221no—
247C22×C8—1224no—
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8C22×D6112yes—
252C4⋊Q8C22×D6111yescharacteristic
253#127C22×D6111yescharacteristic
254#124—21no—
255#121—21no—
256#121—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4—612no—
260C4×D4—612no—
261C4×D4—1221no—
262C4×D4—1221no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—611no—
266C2×C22⋊C4—1221no—
267C2×C42C22×D6111yescharacteristic
268#106—24no—
269#106—1224no—
270C4⋊C8—612no—
271C4⋊C8—612no—
272C4⋊C8—612no—
273C4⋊C8—1221no—
274C4⋊C8—1221no—
275C4⋊C8C22⋊D6112yes—
276#103—1222no—
277#102—24no—
278#102—1224no—
279#102—1222no—
280C22⋊C8—1221no—
281C22⋊C8—1224no—
282C22⋊C8C22⋊D6112yes—
283C4×C8—1221no—
284C4×C8C22×D6112yes—
285C24—1221no—
286#41—42no—
287#41—42no—
288C2×Q8—21no—
289C2×Q8—22no—
290C2×Q8—41no—
291C2×Q8—82no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—612no—
297C2×D4—614no—
298C2×D4—1221no—
299C2×D4—1222no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—1221no—
303C22×C4—1222no—
304C22×C4—1222no—
305C22×C4—4882no—
306C22×C4D4×D6111yescharacteristic
307C22×C4#1175822411yescharacteristic
308C22×C4C2×C22⋊D6111yescharacteristic
309SD16—2444no—
310D8—2444no—
311M16—2442no—
312C2×C8—1222no—
313C2×C8—1222no—
314C2×C8—1222no—
315C2×C8—4884no—
316C2×C8—2442no—
317C2×C8C2×C22⋊D6114yes—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—612no—
325C4⋊C4—1221no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—1222no—
329C22⋊C4—1222no—
330C42D4×D6111yescharacteristic
331C42#1175822411yescharacteristic
332C42C2×C22⋊D6111yescharacteristic
333C42C23×D6111yescharacteristic
334C23—41no—
335C23—1224no—
336C23—1222no—
337C23—4881no—
338C23—4881no—
339C23#1451974811yescharacteristic
340Q8—41no—
341Q8—41no—
342Q8—82no—
343Q8—82no—
344D4—41no—
345D4—41no—
346D4—42no—
347D4—81no—
348D4—82no—
349D4—614no—
350D4—1224no—
351D4—1222no—
352C2×C4—22no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—1222no—
359C2×C4—4882no—
360C2×C4—4881no—
361C2×C4—2444no—
362C2×C4#1451974811yescharacteristic
363C2×C4#1451974811yescharacteristic
364C2×C4#1451974811yescharacteristic
365C2×C4SD16×D6112yes—
366C2×C4#1451979812yes—
367C2×C4C2×D4×D6111yescharacteristic
368C2×C4C22×C22⋊D6111yescharacteristic
369C2×C4C2×#1175822411yescharacteristic
370C8—24no—
371C8—2442no—
372C8—2442no—
373C8—2442no—
374C22—21no—
375C22—21no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—1224no—
380C22—4884no—
381C22—2442no—
382C22—2442no—
383C22#1683998812yes—
384C22C2×#1451974811yescharacteristic
385C4—21no—
386C4—21no—
387C4—21no—
388C4—81no—
389C4—82no—
390C4—2442no—
391C4#1683998812yes—
392C4C2×SD16×D6111yescharacteristic
393C4C2×#1451979811yescharacteristic
394C4#1684057012yes—
395C2—22no—
396C2—161no—
397C2—2444no—
398C2#2145532711yescharacteristic
399C2C2×#1683998811yescharacteristic
400C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3227C2
socleC22×C61C2×#121
prime radical#21455279C2
hypercenter#3227D61
hypocenterC61#19391
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
1. C2×#16840049C2semidirect
3. #21454842C2semidirect
4. #21445225C2semidirect
5. #21445191C2semidirect
6. #21444590C2semidirect
7. #21444500C2semidirect
8. #21444468C2semidirect
9. C61×#3227C2semidirect
26. #16839513C22semidirect
27. #16839511C22semidirect
28. #16839508C22semidirect
29. #16839506C22semidirect
32. C61×#498C22semidirect
33. C61×#486C22semidirect
195. #3227D61semidirect
214. #498C2×D61semidirect
215. #486C2×D61semidirect
237. C61#19391semidirect

Maximal subgroups

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

subgroupindex
C2×#168400492
#214552792
#214548422
#214452252
#214451912
#214445902
#214445002
#214444682
C61×#32272
#1939161