Finite Group Databasebeta

← #29834556

Subgroups — #29834556

Order 15616.

662 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
1C2×#16840180C211yesmaximal, characteristic, complemented
2#21455485C211yesmaximal, characteristic
3#21455470C211yesmaximal, characteristic, complemented
4#21447929C212yesmaximal, complemented
5#21447924C212yesmaximal, complemented
6#21447912C212yesmaximal, complemented
7#21447907C212yesmaximal, complemented
8#21447607C211yesmaximal, characteristic, complemented
9#21447526C211yesmaximal, characteristic, complemented
10C2×#16839566C211yesmaximal, characteristic, complemented
11C2×C61×#510C211yesmaximal, characteristic, complemented
12C2×#14519783C2211yescharacteristic
13#16840340—21no—
14#16840336—21no—
15#16840180—24no—
16C2×#14519716C2212yes—
17C22×#11758185C2211yescharacteristic
18C2×C16×D61C2212yes—
19#16840017—22no—
20#16840005—22no—
21#16840000C2212yes—
22#16840000C2212yes—
23#16840000C2212yes—
24#16839995C2212yes—
25#16839800C2211yescharacteristic
26C2×#14519711C2211yescharacteristic
27#16839771—21no—
28#16839763—21no—
29#16839590C2212yes—
30#16839588C2212yes—
31#16839566C2212yes—
32#16839566C2212yes—
33#16839566C2211yescharacteristic
34#16839566C2211yescharacteristic
35#16839563C2211yescharacteristic
36#16839563C2211yescharacteristic
37C2×#135×C61C2211yescharacteristic
38C22×Q16×C61C2211yescharacteristic
39C61×#510—24no—
40C2×Q32×C61C2212yes—
41C2×SD32×C61C2212yes—
42C2×M32×C61C2211yescharacteristic
43C2×#11758222D411yescharacteristic
44#14519783—22no—
45#14519783—22no—
46C2×#11758194D411yescharacteristic
47C2×#11758189D412yes—
48C2×C8×D61C2312yes—
49#14519775—41no—
50#14519766—41no—
51#14519760—42no—
52#14519756—41no—
53#14519753—22no—
54#14519753—22no—
55#14519753—22no—
56#14519751—41no—
57#14519749—42no—
58#14519748—22no—
59#14519736—21no—
60C2×C4⋊Dic61—21no—
61#14519716—24no—
62C2×#11758185—22no—
63C2×#11758185C2311yescharacteristic
64C2×#11758185C2311yescharacteristic
65C16×D61—24no—
66#14519711—21no—
67#14519711C2312yes—
68#14519711C2311yescharacteristic
69#14519711C2311yescharacteristic
70#14519709—21no—
71#14519701—22no—
72#14519701—22no—
73#14519701—22no—
74#14519698—22no—
75C2×#41×C61—21no—
76C22×Q8×C61—21no—
77#135×C61—22no—
78#135×C61—22no—
79C2×Q16×C61—22no—
80C2×Q16×C61—21no—
81C2×Q16×C61—22no—
82C2×Q16×C61C2312yes—
83C2×Q16×C61C2311yescharacteristic
84C2×SD16×C61—21no—
85C2×D8×C61C2311yescharacteristic
86C22×C8×C61C2311yescharacteristic
87Q32×C61—24no—
88SD32×C61—24no—
89M32×C61D414yescomplemented
90C2×C16×C61C2312yes—
91#41×C61—42no—
92#41×C61—42no—
93C2×Q8×C61—21no—
94C2×Q8×C61—22no—
95C2×Q8×C61—41no—
96C2×Q8×C61—42no—
97C2×Q8×C61—42no—
98C2×D4×C61—21no—
99C2×D4×C61—41no—
100C22×C4×C61—41no—
101C22×C4×C61—41no—
102C22×C4×C61C2×D411yescharacteristic
103C2×C22⋊D61—42no—
104C2×C22⋊D61D812yes—
105C22×Dic61—41no—
106#11758222—22no—
107#11758222—22no—
108C2×D244D811yescharacteristic
109C2×C4×D61C2×D412yes—
110C2×Dic122D811yescharacteristic
111Q16×C61—22no—
112Q16×C61—21no—
113Q16×C61—22no—
114Q16×C61—21no—
115Q16×C61—42no—
116SD16×C61—42no—
117D8×C61—21no—
118D8×C61—21no—
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D412yes—
121C2×C8×C61#4112yes—
122C2×C8×C61C2411yescharacteristic
123C22⋊Dic61—41no—
124C22⋊Dic61—42no—
125#11758198—42no—
126#11758198—42no—
127#11758198—42no—
128C4⋊Dic61—22no—
129C4⋊Dic61—21no—
130C4⋊Dic61—21no—
131C4⋊Dic61—41no—
132#11758196—42no—
133#11758196—42no—
134#11758196—42no—
135C4×Dic61—41no—
136#11758194—22no—
137C2×#92418C2×D412yes—
138#11758189—24no—
139C8×D61—22no—
140C8×D61—22no—
141C16×C61C2×D414yes—
142#11758185—21no—
143#11758185—21no—
144C23×C61—41no—
145C23×C61C2×D811yescharacteristic
146C22×D61C2×D812yes—
147Q8×C61—41no—
148Q8×C61—42no—
149Q8×C61—42no—
150Q8×C61—41no—
151Q8×C61—82no—
152D4×C61—41no—
153D4×C61—41no—
154D4×C61—82no—
155C2×C4×C61—41no—
156C2×C4×C61—42no—
157C2×C4×C61—82no—
158C2×C4×C61—82no—
159C2×C4×C61#12112yes—
160C2×C4×C61C2×D811yescharacteristic
161C2×C4×C61#13512yes—
162C2×C4×C61C22×D411yescharacteristic
163C22⋊D61—24no—
164C22⋊D61—162no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—42no—
168C2×Dic61—161no—
169C2×Dic61C2×D812yes—
170D244—22no—
171C4×D61—22no—
172C4×D61—22no—
173Dic122—22no—
174C8×C61#12112yes—
175C8×C61C22×D411yescharacteristic
176C8×C61C2×#4111yescharacteristic
177#92418—22no—
178#33613—611nomaximal, Sylow
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—81no—
182C22×C61—81no—
183C22×C61—81no—
184C22×C61#46612yes—
185C22×C61C22×D811yescharacteristic
186C2×D61—22no—
187C2×D61—22no—
188C2×D61—22no—
189C4×C61—81no—
190C4×C61—82no—
191C4×C61#46612yes—
192C4×C61C2×#12111yescharacteristic
193C4×C61C2×#13511yescharacteristic
194Dic61—22no—
195Dic61—81no—
196Dic61—82no—
197Dic61—81no—
198C2×#510D6111yescharacteristic, complemented
199C2×#504—611no—
200#3061—611no—
201#3059—611no—
202#2220—612no—
203#2219—612no—
204#2217—612no—
205#2216—612no—
206#2159—611no—
207#2138—611no—
208C2×#358—611no—
209C2×C61—22no—
210C2×C61—161no—
211C2×C61C2×#46611yescharacteristic
212C2×C61#341611yescharacteristic
213C2×C61#341711yescharacteristic
214D61—42no—
215C2×#135C2×D6111yescharacteristic
216C22×Q16C2×D6111yescharacteristic
217C2×#131—611no—
218#537—1221no—
219#536—1221no—
220#510—24no—
221C2×Q32C2×D6112yes—
222C2×SD32C2×D6112yes—
223#504—1224no—
224C2×M32—612no—
225C2×M32C2×D6111yescharacteristic
226C22×C16—611no—
227C22×C16—612no—
228#470—1222no—
229#468—1222no—
230#467—612no—
231#467—612no—
232#467—612no—
233#466—612no—
234#427—611no—
235C2×#107—611no—
236#419—1221no—
237#417—1221no—
238#367—612no—
239#366—612no—
240#358—612no—
241#358—612no—
242#358—611no—
243#358—611no—
244#357—611no—
245#357—611no—
246C61#3361311yescharacteristic, complemented, Sylow
247C2×#41—21no—
248C2×#41—611no—
249C22×Q8—21no—
250#135—22no—
251#135—22no—
252C2×Q16—21no—
253C2×Q16—22no—
254C2×Q16—22no—
255C2×Q16C22×D6112yes—
256C2×Q16C22×D6111yescharacteristic
257C2×SD16—21no—
258C2×D8C22×D6111yescharacteristic
259#131—1222no—
260#131—1222no—
261C2×M16—611no—
262C2×M16—612no—
263C22×C8—612no—
264C22×C8C22×D6111yescharacteristic
265C4⋊Q8—2441no—
266#124—2441no—
267#123—2442no—
268C22⋊Q8—1222no—
269C22⋊Q8—1222no—
270C22⋊Q8—1222no—
271C22⋊Q8—2441no—
272#121—1222no—
273#121—2441no—
274#121—2442no—
275#117—1221no—
276C2×C4⋊C4—1221no—
277Q32—24no—
278SD32—24no—
279M32—1224no—
280M32C22⋊D6114yescomplemented
281C2×C16—611no—
282C2×C16—611no—
283C2×C16—1222no—
284C2×C16—1224no—
285C2×C16C22×D6112yes—
286#107—611no—
287#107—611no—
288#107—612no—
289#107—1221no—
290#106—1221no—
291#103—1222no—
292#103—1222no—
293#103—1222no—
294#102—1222no—
295#41—42no—
296#41—42no—
297#41—1222no—
298#41—1222no—
299C2×Q8—21no—
300C2×Q8—22no—
301C2×Q8—41no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×Q8—611no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—612no—
308C2×D4—611no—
309C2×D4—2442no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—612no—
313C22×C4—2441no—
314C22×C4D4×D6111yescharacteristic
315Q16—22no—
316Q16—21no—
317Q16—22no—
318Q16—21no—
319Q16—42no—
320SD16—42no—
321D8—21no—
322D8—21no—
323M16—1222no—
324M16—1224no—
325C2×C8—612no—
326C2×C8—1222no—
327C2×C8—1222no—
328C2×C8D4×D6111yescharacteristic
329C2×C8#1175822412yes—
330C2×C8C2×C22⋊D6112yes—
331C2×C8C23×D6111yescharacteristic
332C4⋊C4—1222no—
333C4⋊C4—1221no—
334C4⋊C4—1221no—
335C4⋊C4—2442no—
336C4⋊C4—2442no—
337C4⋊C4—2442no—
338C4⋊C4—2441no—
339C22⋊C4—2442no—
340C22⋊C4—2442no—
341C22⋊C4—2442no—
342C22⋊C4—2442no—
343C22⋊C4—2441no—
344C42—2441no—
345C16—1221no—
346C16—1221no—
347C16C2×C22⋊D6114yes—
348C23—41no—
349C23—612no—
350C23D8×D6111yescharacteristic
351Q8—41no—
352Q8—42no—
353Q8—42no—
354Q8—41no—
355Q8—82no—
356Q8—1222no—
357D4—41no—
358D4—41no—
359D4—82no—
360D4—1224no—
361D4—1222no—
362D4—9762no—
363C2×C4—41no—
364C2×C4—42no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—612no—
368C2×C4—1222no—
369C2×C4—1222no—
370C2×C4—9761no—
371C2×C4—2441no—
372C2×C4—2441no—
373C2×C4—2442no—
374C2×C4#1451974812yes—
375C2×C4D8×D6111yescharacteristic
376C2×C4#1451979612yes—
377C2×C4C2×D4×D6111yescharacteristic
378C8—1222no—
379C8#1451974812yes—
380C8C22×C22⋊D6111yescharacteristic
381C8C2×#1175822411yescharacteristic
382C22—21no—
383C22—21no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—1222no—
388C22—1222no—
389C22—1222no—
390C22#1683999512yes—
391C22C2×D8×D6111yescharacteristic
392C4—81no—
393C4—82no—
394C4—1222no—
395C4—4882no—
396C4—4881no—
397C4—4881no—
398C4#1683999512yes—
399C4C2×#1451974811yescharacteristic
400C4C2×#1451979611yescharacteristic
401C2—22no—
402C2—161no—
403C2—2442no—
404C2C2×#1683999511yescharacteristic
405C2#2145896411yescharacteristic
406C2#2145897111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC2×C61×#510C2
socleC22×C61C22×D8
prime radical#21455485C2
hypercenterC2×#510D61
hypocenterC61#33613
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#510

Complements

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

NH ≅ G/Ntype
1. C2×#16840180C2semidirect
3. #21455470C2semidirect
4. #21447929C2semidirect
5. #21447924C2semidirect
6. #21447912C2semidirect
7. #21447907C2semidirect
8. #21447607C2semidirect
9. #21447526C2semidirect
10. C2×#16839566C2semidirect
11. C2×C61×#510C2semidirect
89. M32×C61D4semidirect
198. C2×#510D61semidirect
246. C61#33613semidirect
280. M32C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401802
#214554852
#214554702
#214479292
#214479242
#214479122
#214479072
#214476072
#214475262
C2×#168395662
C2×C61×#5102
#3361361