Finite Group Databasebeta

← #29833474

Subgroups — #29833474

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458955C211yesmaximal, characteristic
2#21454737C211yesmaximal, characteristic, complemented
3#21454704C211yesmaximal, characteristic, complemented
4#21447719C214yesmaximal, complemented
5#21447712C214yesmaximal, complemented
6C2×#16839585C211yesmaximal, characteristic, complemented
7#21447569C211yesmaximal, characteristic, complemented
8C2×#16839573C211yesmaximal, characteristic, complemented
9C61×#2110C211yesmaximal, characteristic, complemented
10C2×#14519799C2211yescharacteristic
11C2×#14519799C2211yescharacteristic
12#16840528—21no—
13#16840242—21no—
14#16840205C412yescomplemented
15#16840205C2212yes—
16#16840200C412yescomplemented
17#16840200C2212yes—
18#16840194C414yescomplemented
19#16839885C2214yes—
20#16839880—22no—
21#16839875C2212yescomplemented
22#16839870C2212yescomplemented
23C4×#11758190—22no—
24C2×#14519713C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26#16839780—21no—
27#16839764—21no—
28#16839585C2214yescomplemented
29#16839579C2212yescomplemented
30#16839576C2212yescomplemented
31#16839573C2214yes—
32C2×M32×C61C2211yescharacteristic
33C2×M32×C61C2211yescharacteristic
34C61×#405C2211yescharacteristic
35C61×#347C2214yescomplemented
36C2×#11758222—21no—
37C2×#11758222—21no—
38#14519799—21no—
39#14519799—21no—
40#14519799—21no—
41#14519799—21no—
42#14519799C2×C412yes—
43#14519799C2×C412yes—
44#14519799C2×C412yescomplemented
45#14519799C2312yes—
46C2×Dic244C2×C411yescharacteristic, complemented
47C2×Dic244C2311yescharacteristic
48C2×#11758190—21no—
49C2×#11758190—21no—
50C2×D488C2×C411yescharacteristic, complemented
51C2×D488C2311yescharacteristic
52#14519785—42no—
53#14519783—24no—
54C2×#11758194—21no—
55M16×D61—42no—
56C4×Dic122—22no—
57C4×D244—22no—
58#14519737—42no—
59#14519735—41no—
60C2×C4⋊Dic61—21no—
61C4⋊C4×D61—41no—
62Dic488C2×C414yescomplemented
63#14519722C2×C414yescomplemented
64#14519722C2×C414yescomplemented
65D976C2×C414yescomplemented
66#14519713—21no—
67#14519713C2312yes—
68#14519709—21no—
69#14519709C2312yes—
70#14519705—24no—
71#14519702—22no—
72#14519699—22no—
73C2×M16×C61D411yescharacteristic
74C22×C8×C61C2311yescharacteristic
75#117×C61D411yescharacteristic
76M32×C61—21no—
77M32×C61C2×C412yescomplemented
78M32×C61C2312yes—
79M32×C61C2312yes—
80C2×C16×C61—21no—
81C2×C16×C61C2×C412yescomplemented
82#97×C61C2312yes—
83C4×C8×C61D412yes—
84C22×C4×C61C2×D411yescharacteristic
85C2×C22⋊D61—41no—
86C2×C22⋊D61—41no—
87C22×Dic61—41no—
88#11758222—22no—
89#11758222—22no—
90#11758222—22no—
91#11758222—22no—
92#11758222—41no—
93#11758222—41no—
94#11758222—41no—
95#11758222—41no—
96C2×D244—21no—
97C2×D244—21no—
98C2×C4×D61—41no—
99C2×C4×D61—41no—
100C2×C4×D61—42no—
101C2×Dic122—21no—
102C2×Dic122—21no—
103M16×C61—21no—
104M16×C61—21no—
105C2×C8×C61C22×C411yescharacteristic
106C2×C8×C61C22×C411yescharacteristic
107C2×C8×C61C2×D412yes—
108C2×C8×C61C2×D411yescharacteristic
109C2×C8×C61#4111yescharacteristic
110C2×C8×C61#4111yescharacteristic
111C2×C8×C61C2411yescharacteristic
112C4⋊C4×C61—21no—
113C22⋊C4×C61—21no—
114C42×C61C2×D412yes—
115#11758198—42no—
116C4⋊Dic61—22no—
117C4⋊Dic61—41no—
118#11758196—42no—
119C4×Dic61—42no—
120#11758194—22no—
121#11758194—41no—
122C2×#92418—41no—
123Dic244—21no—
124Dic244—21no—
125Dic244C22×C412yes—
126Dic244C22×C412yes—
127D488—21no—
128D488—21no—
129D488C22×C412yes—
130D488C22×C412yes—
131#11758190—22no—
132#11758190—22no—
133#11758190—22no—
134#11758190—22no—
135#11758189—44no—
136C8×D61—44no—
137C16×C61—22no—
138C16×C61C22×C414yes—
139C23×C61AGL(1,ℤ/8)11yescharacteristic
140C22×D61—41no—
141C22×D61—41no—
142C2×C4×C61—22no—
143C2×C4×C61C4×D411yescharacteristic
144C2×C4×C61C4×D411yescharacteristic
145C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
147C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C61C22×D411yescharacteristic
149C22⋊D61—42no—
150C22⋊D61—42no—
151C22⋊D61—42no—
152C22⋊D61—42no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—42no—
156C2×Dic61—161no—
157D244—22no—
158D244—22no—
159D244—41no—
160D244—41no—
161C4×D61—42no—
162C4×D61—42no—
163C4×D61—42no—
164C4×D61—42no—
165C4×D61—162no—
166Dic122—22no—
167Dic122—22no—
168Dic122—41no—
169Dic122—41no—
170C8×C61—22no—
171C8×C61C4×D411yescharacteristic
172C8×C61C4×D411yescharacteristic
173C8×C61C23×C411yescharacteristic
174C8×C61C2×#4111yescharacteristic
175#92418—42no—
176#33469—611nomaximal, Sylow
177C22×C61—41no—
178C22×C61#44211yescharacteristic
179C22×C61#44211yescharacteristic
180C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
181C2×D61—42no—
182C2×D61—42no—
183C2×D61—81no—
184C2×D61—81no—
185C2×D61—161no—
186C4×C61—42no—
187C4×C61#44211yescharacteristic
188C4×C61#44211yescharacteristic
189C4×C61C2×C4×D411yescharacteristic
190C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
191Dic61—42no—
192Dic61—42no—
193Dic61—82no—
194#3414—611no—
195#2957—611no—
196#2952—611no—
197AGL(1,ℤ/16)—614no—
198#2180—614no—
199C2×#365—611no—
200#2148—611no—
201C2×#360—611no—
202#2110D6111yescharacteristic, complemented
203C2×C61—21no—
204C2×C61—21no—
205C2×C61—21no—
206C2×C61C2×#44211yescharacteristic
207D61—82no—
208D61—82no—
209C2×#135—611no—
210C2×#135—611no—
211#568—1221no—
212#520—1221no—
213#510—612no—
214#510—612no—
215#509—612no—
216#509—612no—
217#508—614no—
218C2×M32C2×D6111yescharacteristic
219C2×M32C2×D6111yescharacteristic
220#444—614no—
221#443—1222no—
222#442—612no—
223#441—612no—
224C4×SD16—1222no—
225C2×#108—611no—
226C2×#106—611no—
227#421—1221no—
228#417—1221no—
229#405C2×D6111yescharacteristic
230#365—614no—
231#362—612no—
232#361—612no—
233#360—614no—
234#347C2×D6114yescomplemented
235C61#3346911yescharacteristic, complemented, Sylow
236C2×#41—1221no—
237C2×#41—1221no—
238#135—612no—
239#135—612no—
240#135—612no—
241#135—612no—
242#135—1221no—
243#135—1221no—
244#135—1221no—
245#135—1221no—
246C2×Q16—611no—
247C2×Q16—611no—
248C2×SD16—1221no—
249C2×SD16—1221no—
250C2×D8—611no—
251C2×D8—611no—
252#131—1224no—
253#131—2442no—
254C2×M16—1221no—
255C2×M16—2442no—
256C2×M16D24411yescharacteristic
257C22×C8C22×D6111yescharacteristic
258C4×Q8—1222no—
259C4×D4—1222no—
260C4×D4—2442no—
261#117—2441no—
262#117D24411yescharacteristic
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—2441no—
265Q32—614no—
266SD32—614no—
267SD32—614no—
268D16—614no—
269M32—21no—
270M32C4×D6112yescomplemented
271M32C22×D6112yes—
272M32C22×D6112yes—
273C2×C16—21no—
274C2×C16C4×D6112yescomplemented
275#108—612no—
276#108—1221no—
277#106—612no—
278#106—1221no—
279#104—1224no—
280#103—1222no—
281#102—1222no—
282#97C22×D6112yes—
283C4×C8D24412yes—
284#41—1222no—
285#41—1222no—
286#41—1222no—
287#41—1222no—
288#41—2441no—
289#41—2441no—
290#41—2441no—
291#41—2441no—
292C2×Q8—1221no—
293C2×Q8—1221no—
294C2×D4—1221no—
295C2×D4—1221no—
296C2×D4—2441no—
297C2×D4—2441no—
298C22×C4—2442no—
299C22×C4—2441no—
300C22×C4—2441no—
301C22×C4—2441no—
302C22×C4C2×D24411yescharacteristic
303Q16—612no—
304Q16—612no—
305Q16—1221no—
306Q16—1221no—
307SD16—1222no—
308SD16—1222no—
309SD16—1222no—
310SD16—1222no—
311D8—612no—
312D8—612no—
313D8—1221no—
314D8—1221no—
315M16—21no—
316M16—21no—
317M16—1222no—
318M16—2444no—
319M16—2441no—
320C2×C8—2444no—
321C2×C8—2441no—
322C2×C8C2×C4×D6111yescharacteristic
323C2×C8C2×C4×D6111yescharacteristic
324C2×C8C2×D24412yes—
325C2×C8C2×D24411yescharacteristic
326C2×C8#1175822211yescharacteristic
327C2×C8#1175822211yescharacteristic
328C2×C8C23×D6111yescharacteristic
329C4⋊C4—21no—
330C4⋊C4—1222no—
331C4⋊C4—2442no—
332C4⋊C4—2441no—
333C22⋊C4—21no—
334C22⋊C4—2442no—
335C42—2442no—
336C42C2×D24412yes—
337C16—22no—
338C16C2×C4×D6114yes—
339C23—2441no—
340C23—2441no—
341C23#1451980511yescharacteristic
342Q8—1222no—
343Q8—1222no—
344Q8—2441no—
345Q8—2441no—
346D4—1222no—
347D4—1222no—
348D4—2442no—
349D4—2442no—
350D4—2442no—
351D4—2442no—
352D4—2441no—
353D4—2441no—
354C2×C4—22no—
355C2×C4—9762no—
356C2×C4—9761no—
357C2×C4—2442no—
358C2×C4—2441no—
359C2×C4—2441no—
360C2×C4—2442no—
361C2×C4—2442no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4C4×D24411yescharacteristic
365C2×C4C4×D24411yescharacteristic
366C2×C4#1451980511yescharacteristic
367C2×C4#1451980511yescharacteristic
368C2×C4#1451980511yescharacteristic
369C2×C4C22×D24411yescharacteristic
370C8—22no—
371C8—2442no—
372C8C4×D24411yescharacteristic
373C8C4×D24411yescharacteristic
374C8C22×C4×D6111yescharacteristic
375C8C2×#1175822211yescharacteristic
376C22—41no—
377C22—9761no—
378C22—4881no—
379C22—4881no—
380C22—2442no—
381C22—2442no—
382C22#1683987511yescharacteristic
383C22#1683987511yescharacteristic
384C22C2×#1451980511yescharacteristic
385C4—42no—
386C4—4882no—
387C4—2442no—
388C4—2442no—
389C4#1683987511yescharacteristic
390C4#1683987511yescharacteristic
391C4C2×C4×D24411yescharacteristic
392C4C2×#1451980511yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—4882no—
397C2—4882no—
398C2C2×#1683987511yescharacteristic
subgroupG/H
centerC2C2×#16839875
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2110C2
socleC2×C61C2×#442
prime radical#21458955C2
cosocleC2×C8×C61C24
hypercenter#2110D61
hypocenterC61#33469
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2110

Complements

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

NH ≅ G/Ntype
2. #21454737C2semidirect
3. #21454704C2semidirect
4. #21447719C2semidirect
5. #21447712C2semidirect
6. C2×#16839585C2semidirect
7. #21447569C2semidirect
8. C2×#16839573C2semidirect
9. C61×#2110C2semidirect
14. #16840205C4semidirect
16. #16840200C4semidirect
18. #16840194C4semidirect
21. #16839875C22semidirect
22. #16839870C22semidirect
28. #16839585C22semidirect
29. #16839579C22semidirect
30. #16839576C22semidirect
35. C61×#347C22semidirect
44. #14519799C2×C4semidirect
46. C2×Dic244C2×C4semidirect
50. C2×D488C2×C4semidirect
62. Dic488C2×C4semidirect
63. #14519722C2×C4semidirect
64. #14519722C2×C4semidirect
65. D976C2×C4semidirect
77. M32×C61C2×C4semidirect
81. C2×C16×C61C2×C4semidirect
202. #2110D61semidirect
234. #347C2×D61semidirect
235. C61#33469semidirect
270. M32C4×D61semidirect
274. C2×C16C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214589552
#214547372
#214547042
#214477192
#214477122
C2×#168395852
#214475692
C2×#168395732
C61×#21102
#3346961