Finite Group Databasebeta

← #29833472

Subgroups — #29833472

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#21458953C211yesmaximal, characteristic
2#21454704C211yesmaximal, characteristic, complemented
3#21447717C214yesmaximal, complemented
4#21447710C214yesmaximal, complemented
5C2×#16839586C211yesmaximal, characteristic, complemented
6#21447566C211yesmaximal, characteristic, complemented
7C2×#16839572C211yesmaximal, characteristic, complemented
8#21447424C211yesmaximal, characteristic, complemented
9C61×#2957C211yesmaximal, characteristic, complemented
10C2×#14519799C2211yescharacteristic
11#16840242—21no—
12#16840203C412yescomplemented
13#16840202C2212yes—
14#16840198C412yescomplemented
15#16840197C2212yes—
16#16840192C414yescomplemented
17C2×#14519717C2211yescharacteristic
18C2×#14519717C2211yescharacteristic
19#16839875C2212yescomplemented
20#16839870C2212yescomplemented
21C4×#11758190—22no—
22C2×#14519709C2211yescharacteristic
23#16839764—21no—
24#16839586C2214yescomplemented
25#16839577C2212yescomplemented
26#16839574C2212yescomplemented
27#16839572C2214yes—
28#16839540C2214yescomplemented
29C2×#135×C61C2211yescharacteristic
30C61×#568—21no—
31C61×#444C2214yes—
32C61×#443—22no—
33C2×#108×C61C2211yescharacteristic
34C61×#421—21no—
35C61×#405C2211yescharacteristic
36C2×#11758222—21no—
37#14519799—21no—
38#14519799—21no—
39#14519799C2×C412yes—
40#14519799C2×C412yes—
41C2×Dic244C2311yescharacteristic
42C2×#11758190—21no—
43C2×D488C2311yescharacteristic
44C4×Dic122—22no—
45C4×D244—22no—
46#14519737—42no—
47#14519735—41no—
48C2×C4⋊Dic61—21no—
49C4⋊C4×D61—41no—
50#14519723C2×C414yescomplemented
51#14519721C2×C414yescomplemented
52#14519720C2×C414yescomplemented
53#14519718C2×C414yescomplemented
54#14519717—21no—
55#14519717C2×C412yescomplemented
56#14519717C2312yes—
57#14519717C2312yes—
58C2×#11758185—21no—
59C2×#11758185C2×C412yescomplemented
60#14519709—21no—
61#14519709C2312yes—
62#14519702—22no—
63#14519699—22no—
64C2×#41×C61—21no—
65#135×C61—21no—
66#135×C61—21no—
67#135×C61C2×C412yescomplemented
68#135×C61C2312yes—
69C2×Q16×C61C2×C411yescharacteristic, complemented
70C2×SD16×C61—21no—
71C2×D8×C61C2×C411yescharacteristic, complemented
72#131×C61—24no—
73#131×C61—42no—
74C2×M16×C61—21no—
75C2×M16×C61—42no—
76C2×M16×C61D411yescharacteristic
77C22×C8×C61C2311yescharacteristic
78#117×C61D411yescharacteristic
79#108×C61—21no—
80#108×C61C2312yes—
81#104×C61—24no—
82#97×C61C2312yes—
83C4×C8×C61D412yes—
84#41×C61—22no—
85#41×C61—22no—
86#41×C61—41no—
87#41×C61—41no—
88C2×Q8×C61—21no—
89C2×D4×C61—21no—
90C2×D4×C61—41no—
91C22×C4×C61—41no—
92C22×C4×C61C2×D411yescharacteristic
93C2×C22⋊D61—41no—
94C22×Dic61—41no—
95#11758222—22no—
96#11758222—22no—
97#11758222—41no—
98#11758222—41no—
99C2×D244—21no—
100C2×C4×D61—41no—
101C2×C4×D61—42no—
102C2×Dic122—21no—
103Q16×C61—21no—
104Q16×C61C22×C412yes—
105SD16×C61—22no—
106SD16×C61—22no—
107D8×C61—21no—
108D8×C61C22×C412yes—
109M16×C61—21no—
110M16×C61—21no—
111M16×C61—22no—
112M16×C61—44no—
113M16×C61—41no—
114C2×C8×C61—44no—
115C2×C8×C61—41no—
116C2×C8×C61C22×C411yescharacteristic
117C2×C8×C61C22×C411yescharacteristic
118C2×C8×C61C2×D412yes—
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61#4111yescharacteristic
121C2×C8×C61#4111yescharacteristic
122C2×C8×C61C2411yescharacteristic
123C4⋊C4×C61—21no—
124C22⋊C4×C61—21no—
125C42×C61C2×D412yes—
126#11758198—42no—
127C4⋊Dic61—22no—
128C4⋊Dic61—41no—
129#11758196—42no—
130C4×Dic61—42no—
131Dic244—21no—
132Dic244C22×C412yes—
133D488—21no—
134D488C22×C412yes—
135#11758190—22no—
136#11758190—22no—
137#11758185—22no—
138#11758185C22×C414yes—
139C23×C61—41no—
140C23×C61AGL(1,ℤ/8)11yescharacteristic
141C22×D61—41no—
142Q8×C61—22no—
143Q8×C61—41no—
144D4×C61—22no—
145D4×C61—42no—
146D4×C61—42no—
147D4×C61—41no—
148C2×C4×C61—22no—
149C2×C4×C61—41no—
150C2×C4×C61—42no—
151C2×C4×C61—42no—
152C2×C4×C61C4×D411yescharacteristic
153C2×C4×C61C4×D411yescharacteristic
154C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
157C2×C4×C61C22×D411yescharacteristic
158C22⋊D61—42no—
159C22⋊D61—42no—
160C2×Dic61—41no—
161C2×Dic61—42no—
162C2×Dic61—161no—
163D244—22no—
164D244—41no—
165C4×D61—42no—
166C4×D61—42no—
167C4×D61—162no—
168Dic122—22no—
169Dic122—41no—
170C8×C61—22no—
171C8×C61—42no—
172C8×C61C4×D411yescharacteristic
173C8×C61C4×D411yescharacteristic
174C8×C61C23×C411yescharacteristic
175C8×C61C2×#4111yescharacteristic
176#33469—611nomaximal, Sylow
177C22×C61—41no—
178C22×C61—42no—
179C22×C61—161no—
180C22×C61#44211yescharacteristic
181C22×C61#44211yescharacteristic
182C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
183C2×D61—42no—
184C2×D61—81no—
185C2×D61—81no—
186C4×C61—42no—
187C4×C61—42no—
188C4×C61#44211yescharacteristic
189C4×C61#44211yescharacteristic
190C4×C61C2×C4×D411yescharacteristic
191C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
192Dic61—42no—
193Dic61—82no—
194#3414—611no—
195#2957D6111yescharacteristic, complemented
196#2952—611no—
197AGL(1,ℤ/16)—614no—
198#2180—614no—
199C2×#365—611no—
200#2148—611no—
201C2×#360—611no—
202#2110—611no—
203C2×C61—21no—
204C2×C61—21no—
205C2×C61—21no—
206C2×C61—82no—
207C2×C61C2×#44211yescharacteristic
208D61—82no—
209C2×#135—611no—
210C2×#135C2×D6111yescharacteristic
211#568—21no—
212#520—1221no—
213#510—612no—
214#510—612no—
215#509—612no—
216#509—612no—
217#508—614no—
218C2×M32—611no—
219C2×M32—611no—
220#444C2×D6114yes—
221#443—22no—
222#442—612no—
223#441—612no—
224C4×SD16—1222no—
225C2×#108C2×D6111yescharacteristic
226C2×#106—611no—
227#421—21no—
228#417—1221no—
229#405C2×D6111yescharacteristic
230#365—614no—
231#362—612no—
232#361—612no—
233#360—614no—
234#347—614no—
235C61#3346911yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C2×#41—1221no—
238#135—21no—
239#135—21no—
240#135—612no—
241#135—612no—
242#135—1221no—
243#135—1221no—
244#135C4×D6112yescomplemented
245#135C22×D6112yes—
246C2×Q16—611no—
247C2×Q16C4×D6111yescharacteristic, complemented
248C2×SD16—21no—
249C2×SD16—1221no—
250C2×D8—611no—
251C2×D8C4×D6111yescharacteristic, complemented
252#131—24no—
253#131—42no—
254C2×M16—21no—
255C2×M16—42no—
256C2×M16C22⋊D6111yescharacteristic
257C22×C8C22×D6111yescharacteristic
258C4×Q8—1222no—
259C4×D4—1222no—
260C4×D4—2442no—
261#117—2441no—
262#117C22⋊D6111yescharacteristic
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—2441no—
265Q32—614no—
266SD32—614no—
267SD32—614no—
268D16—614no—
269M32—612no—
270M32—612no—
271M32—612no—
272M32—1221no—
273C2×C16—612no—
274C2×C16—1221no—
275#108—21no—
276#108C22×D6112yes—
277#106—612no—
278#106—1221no—
279#104—24no—
280#103—1222no—
281#102—1222no—
282#97C22×D6112yes—
283C4×C8C22⋊D6112yes—
284#41—22no—
285#41—22no—
286#41—41no—
287#41—41no—
288#41—1222no—
289#41—1222no—
290#41—2441no—
291#41—2441no—
292C2×Q8—21no—
293C2×Q8—1221no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—1221no—
297C2×D4—2441no—
298C22×C4—41no—
299C22×C4—2442no—
300C22×C4—2441no—
301C22×C4—2441no—
302C22×C4C2×C22⋊D6111yescharacteristic
303Q16—21no—
304Q16—612no—
305Q16—1221no—
306Q16C2×C4×D6112yes—
307SD16—22no—
308SD16—22no—
309SD16—1222no—
310SD16—1222no—
311D8—21no—
312D8—612no—
313D8—1221no—
314D8C2×C4×D6112yes—
315M16—21no—
316M16—21no—
317M16—22no—
318M16—44no—
319M16—41no—
320C2×C8—44no—
321C2×C8—41no—
322C2×C8C2×C4×D6111yescharacteristic
323C2×C8C2×C4×D6111yescharacteristic
324C2×C8#1175822211yescharacteristic
325C2×C8#1175822211yescharacteristic
326C2×C8C2×C22⋊D6112yes—
327C2×C8C2×C22⋊D6111yescharacteristic
328C2×C8C23×D6111yescharacteristic
329C4⋊C4—21no—
330C4⋊C4—1222no—
331C4⋊C4—2442no—
332C4⋊C4—2441no—
333C22⋊C4—21no—
334C22⋊C4—2442no—
335C42—2442no—
336C42C2×C22⋊D6112yes—
337C16—614no—
338C16—1222no—
339C23—41no—
340C23—2441no—
341C23#1451980211yescharacteristic
342Q8—22no—
343Q8—41no—
344Q8—1222no—
345Q8—2441no—
346D4—22no—
347D4—42no—
348D4—42no—
349D4—41no—
350D4—1222no—
351D4—2442no—
352D4—2442no—
353D4—2441no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—9762no—
359C2×C4—9761no—
360C2×C4—2442no—
361C2×C4—2441no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4C4×C22⋊D6111yescharacteristic
365C2×C4C4×C22⋊D6111yescharacteristic
366C2×C4#1451980211yescharacteristic
367C2×C4#1451980311yescharacteristic
368C2×C4#1451980311yescharacteristic
369C2×C4C22×C22⋊D6111yescharacteristic
370C8—22no—
371C8—42no—
372C8C4×C22⋊D6111yescharacteristic
373C8C4×C22⋊D6111yescharacteristic
374C8C22×C4×D6111yescharacteristic
375C8C2×#1175822211yescharacteristic
376C22—41no—
377C22—42no—
378C22—161no—
379C22—4881no—
380C22—4881no—
381C22—2442no—
382C22#1683987211yescharacteristic
383C22#1683987211yescharacteristic
384C22C2×#1451980211yescharacteristic
385C4—42no—
386C4—42no—
387C4—4882no—
388C4—2442no—
389C4#1683987211yescharacteristic
390C4#1683987211yescharacteristic
391C4C2×C4×C22⋊D6111yescharacteristic
392C4C2×#1451980311yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—4882no—
398C2C2×#1683987211yescharacteristic
subgroupG/H
centerC2C2×#16839872
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2957C2
socleC2×C61C2×#442
prime radical#21458953C2
cosocleC2×C8×C61C24
hypercenter#2957D61
hypocenterC61#33469
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2957

Complements

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

NH ≅ G/Ntype
2. #21454704C2semidirect
3. #21447717C2semidirect
4. #21447710C2semidirect
5. C2×#16839586C2semidirect
6. #21447566C2semidirect
7. C2×#16839572C2semidirect
8. #21447424C2semidirect
9. C61×#2957C2semidirect
12. #16840203C4semidirect
14. #16840198C4semidirect
16. #16840192C4semidirect
19. #16839875C22semidirect
20. #16839870C22semidirect
24. #16839586C22semidirect
25. #16839577C22semidirect
26. #16839574C22semidirect
28. #16839540C22semidirect
50. #14519723C2×C4semidirect
51. #14519721C2×C4semidirect
52. #14519720C2×C4semidirect
53. #14519718C2×C4semidirect
55. #14519717C2×C4semidirect
59. C2×#11758185C2×C4semidirect
67. #135×C61C2×C4semidirect
69. C2×Q16×C61C2×C4semidirect
71. C2×D8×C61C2×C4semidirect
195. #2957D61semidirect
235. C61#33469semidirect
244. #135C4×D61semidirect
247. C2×Q16C4×D61semidirect
251. C2×D8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214589532
#214547042
#214477172
#214477102
C2×#168395862
#214475662
C2×#168395722
#214474242
C61×#29572
#3346961