Finite Group Databasebeta

← #29834558

Subgroups — #29834558

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×#16840203C211yesmaximal, characteristic, complemented
2C2×#16840179C211yesmaximal, characteristic, complemented
3#21455489C211yesmaximal, characteristic
4#21447934C212yesmaximal, complemented
5#21447925C212yesmaximal, complemented
6#21447917C212yesmaximal, complemented
7#21447908C212yesmaximal, complemented
8#21447608C211yesmaximal, characteristic, complemented
9#21447524C211yesmaximal, characteristic, complemented
10C2×#16839566C211yesmaximal, characteristic, complemented
11C61×#3059C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13#16840340—21no—
14#16840203—24no—
15C2×#14519723C2212yes—
16C2×#14519720C2212yes—
17#16840179—24no—
18C2×#14519717C2212yes—
19C2×#14519717C2211yescharacteristic
20C22×#11758185C2211yescharacteristic
21C22×#11758185C2212yes—
22#16840008—22no—
23#16840003C2212yes—
24#16839998C2212yes—
25C2×#14519711C2211yescharacteristic
26#16839764—21no—
27#16839589C2212yes—
28#16839587C2212yes—
29#16839566C2212yes—
30#16839566C2212yes—
31#16839565C2211yescharacteristic
32#16839565C2211yescharacteristic
33#16839562C2211yescharacteristic
34#16839562C2211yescharacteristic
35C22×Q16×C61C2211yescharacteristic
36C2×#131×C61C2211yescharacteristic
37C61×#536—21no—
38C61×#470—22no—
39C61×#467C2212yes—
40C61×#467C2212yes—
41C61×#427C2211yescharacteristic
42C61×#419—21no—
43C2×#11758222—21no—
44#14519799—22no—
45#14519799—22no—
46C2×Dic244C2311yescharacteristic
47C2×#11758190—21no—
48C2×D488C2311yescharacteristic
49#14519756—22no—
50#14519753—41no—
51#14519751—22no—
52#14519749—42no—
53#14519748—41no—
54C2×C4⋊Dic61—21no—
55#14519723—24no—
56#14519720—24no—
57#14519717—24no—
58#14519717D414yescomplemented
59C2×#11758185—22no—
60C2×#11758185—24no—
61C2×#11758185C2311yescharacteristic
62C2×#11758185C2311yescharacteristic
63C2×#11758185C2312yes—
64#14519711—21no—
65#14519711C2312yes—
66#14519702—22no—
67#14519699—22no—
68C2×#41×C61D411yescharacteristic
69C22×Q8×C61—21no—
70C2×Q16×C61—22no—
71C2×Q16×C61—21no—
72C2×Q16×C61—22no—
73C2×Q16×C61C2312yes—
74#131×C61—22no—
75#131×C61—22no—
76C2×M16×C61D412yes—
77C2×M16×C61D411yescharacteristic
78C22×C8×C61C2312yes—
79C22×C8×C61C2311yescharacteristic
80C4⋊Q8×C61—41no—
81#124×C61—41no—
82#123×C61—42no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—22no—
85#117×C61—21no—
86#107×C61C2311yescharacteristic
87#107×C61C2311yescharacteristic
88#106×C61—21no—
89#103×C61—22no—
90#103×C61—22no—
91#41×C61—22no—
92#41×C61—22no—
93C2×Q8×C61—22no—
94C2×Q8×C61—41no—
95C2×Q8×C61—42no—
96C2×Q8×C61—42no—
97C2×Q8×C61D811yescharacteristic
98C2×D4×C61D812yes—
99C2×D4×C61D811yescharacteristic
100C22×C4×C61—41no—
101C22×C4×C61C2×D412yes—
102C22×C4×C61C2×D411yescharacteristic
103C2×C22⋊D61—41no—
104C2×C22⋊D61—42no—
105C22×Dic61—41no—
106#11758222—42no—
107#11758222—42no—
108C2×D244—21no—
109C2×C4×D61—41no—
110C2×Dic122—21no—
111Q16×C61—22no—
112Q16×C61—22no—
113Q16×C61—42no—
114M16×C61—22no—
115M16×C61—24no—
116C2×C8×C61—22no—
117C2×C8×C61—22no—
118C2×C8×C61C2×D412yes—
119C2×C8×C61C2×D411yescharacteristic
120C2×C8×C61C2×D412yes—
121C2×C8×C61#4112yes—
122C2×C8×C61C2411yescharacteristic
123C4⋊C4×C61—21no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—42no—
126C4⋊C4×C61—42no—
127C22⋊C4×C61—41no—
128C22⋊C4×C61—42no—
129C22⋊C4×C61—42no—
130C42×C61—41no—
131C22⋊Dic61—42no—
132#11758198—42no—
133C4⋊Dic61—22no—
134C4⋊Dic61—41no—
135#11758196—42no—
136Dic244—21no—
137Dic244—21no—
138D488—21no—
139D488—21no—
140#11758190—42no—
141#11758185—21no—
142#11758185—21no—
143#11758185C2×D414yes—
144C23×C61C2×D812yes—
145C23×C61C2×D811yescharacteristic
146C22×D61—41no—
147Q8×C61—22no—
148Q8×C61—42no—
149Q8×C61—42no—
150Q8×C61—82no—
151D4×C61—24no—
152D4×C61—22no—
153C2×C4×C61—22no—
154C2×C4×C61—22no—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—42no—
158C2×C4×C61—82no—
159C2×C4×C61#12112yes—
160C2×C4×C61C2×D812yes—
161C2×C4×C61C2×D811yescharacteristic
162C2×C4×C61#13512yes—
163C2×C4×C61C22×D411yescharacteristic
164C22⋊D61—82no—
165C22⋊D61—162no—
166C2×Dic61—41no—
167C2×Dic61—42no—
168C2×Dic61—161no—
169D244—41no—
170D244—41no—
171C4×D61—82no—
172Dic122—41no—
173Dic122—41no—
174C8×C61—22no—
175C8×C61#12112yes—
176C8×C61C22×D411yescharacteristic
177C8×C61C2×#4111yescharacteristic
178#33613—611nomaximal, Sylow
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—22no—
182C22×C61—22no—
183C22×C61—22no—
184C22×C61#46612yes—
185C22×C61C22×D811yescharacteristic
186C2×D61—81no—
187C2×D61—81no—
188C2×D61—81no—
189C4×C61—22no—
190C4×C61—81no—
191C4×C61—81no—
192C4×C61—82no—
193C4×C61#46612yes—
194C4×C61C2×#12111yescharacteristic
195C4×C61C2×#13511yescharacteristic
196Dic61—82no—
197Dic61—81no—
198C2×#510—611no—
199C2×#504—611no—
200#3061—611no—
201#3059D6111yescharacteristic, complemented
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—42no—
211C2×C61C2×#46611yescharacteristic
212C2×C61#341611yescharacteristic
213C2×C61#341711yescharacteristic
214D61—161no—
215C2×#135—611no—
216C22×Q16C2×D6111yescharacteristic
217C2×#131C2×D6111yescharacteristic
218#537—1221no—
219#536—21no—
220#510—1224no—
221C2×Q32—612no—
222C2×SD32—612no—
223#504—1224no—
224C2×M32—611no—
225C2×M32—612no—
226C22×C16—611no—
227C22×C16—612no—
228#470—22no—
229#468—1222no—
230#467—612no—
231#467C2×D6112yes—
232#467C2×D6112yes—
233#466—612no—
234#427C2×D6111yescharacteristic
235C2×#107—611no—
236#419—21no—
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—1221no—
248C2×#41C22⋊D6111yescharacteristic
249C22×Q8—21no—
250#135—1222no—
251#135—1222no—
252C2×Q16—22no—
253C2×Q16—22no—
254C2×Q16—21no—
255C2×Q16—611no—
256C2×Q16C22×D6112yes—
257C2×SD16—1221no—
258C2×D8—611no—
259#131—22no—
260#131—22no—
261C2×M16C22⋊D6111yescharacteristic
262C2×M16C22⋊D6112yes—
263C22×C8C22×D6112yes—
264C22×C8C22×D6111yescharacteristic
265C4⋊Q8—41no—
266#124—41no—
267#123—42no—
268C22⋊Q8—22no—
269C22⋊Q8—22no—
270C22⋊Q8—1222no—
271C22⋊Q8—2441no—
272#121—1222no—
273#121—2441no—
274#121—2442no—
275#117—21no—
276C2×C4⋊C4—1221no—
277Q32—1224no—
278SD32—1224no—
279M32—614no—
280M32—1224no—
281C2×C16—612no—
282C2×C16—611no—
283C2×C16—611no—
284C2×C16—1224no—
285C2×C16—1222no—
286#107—612no—
287#107—1221no—
288#107C22×D6111yescharacteristic
289#107C22×D6111yescharacteristic
290#106—21no—
291#103—22no—
292#103—22no—
293#103—1222no—
294#102—1222no—
295#41—22no—
296#41—22no—
297#41—2442no—
298#41—2442no—
299C2×Q8—22no—
300C2×Q8—41no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—1221no—
304C2×Q8#1175819911yescharacteristic
305C2×D4—1221no—
306C2×D4—2442no—
307C2×D4—2441no—
308C2×D4#1175819912yes—
309C2×D4#1175819911yescharacteristic
310C22×C4—41no—
311C22×C4—2441no—
312C22×C4—2441no—
313C22×C4C2×C22⋊D6112yes—
314C22×C4C2×C22⋊D6111yescharacteristic
315Q16—22no—
316Q16—22no—
317Q16—42no—
318Q16—1221no—
319Q16—1221no—
320SD16—2442no—
321D8—1221no—
322D8—1221no—
323M16—22no—
324M16—24no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8D4×D6112yes—
328C2×C8#1175822412yes—
329C2×C8C2×C22⋊D6112yes—
330C2×C8C2×C22⋊D6111yescharacteristic
331C2×C8C23×D6111yescharacteristic
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—1222no—
337C4⋊C4—2441no—
338C4⋊C4—2442no—
339C22⋊C4—41no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—2442no—
343C22⋊C4—2442no—
344C42—41no—
345C16—614no—
346C16—1221no—
347C16—1221no—
348C23—2441no—
349C23C2×#1175819912yes—
350C23C2×#1175819911yescharacteristic
351Q8—22no—
352Q8—42no—
353Q8—42no—
354Q8—82no—
355Q8—2441no—
356Q8—2441no—
357D4—24no—
358D4—22no—
359D4—9762no—
360D4—4882no—
361D4—2441no—
362D4—2441no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—82no—
369C2×C4—9761no—
370C2×C4—4882no—
371C2×C4—2442no—
372C2×C4—2441no—
373C2×C4#1451974912yes—
374C2×C4C2×#1175819912yes—
375C2×C4C2×#1175819911yescharacteristic
376C2×C4#1451979712yes—
377C2×C4C22×C22⋊D6111yescharacteristic
378C8—22no—
379C8#1451974912yes—
380C8C2×D4×D6111yescharacteristic
381C8C2×#1175822411yescharacteristic
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—4881no—
388C22—4881no—
389C22—4881no—
390C22#1683999612yes—
391C22C22×#1175819911yescharacteristic
392C4—22no—
393C4—81no—
394C4—82no—
395C4—81no—
396C4—4882no—
397C4—4881no—
398C4#1683999612yes—
399C4C2×#1451974911yescharacteristic
400C4C2×#1451979711yescharacteristic
401C2—22no—
402C2—42no—
403C2—9761no—
404C2C2×#1683999611yescharacteristic
405C2#2145896311yescharacteristic
406C2#2145897011yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3059C2
socleC22×C61C22×D8
prime radical#21455489C2
hypercenter#3059D61
hypocenterC61#33613
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ #3059

Complements

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

NH ≅ G/Ntype
1. C2×#16840203C2semidirect
2. C2×#16840179C2semidirect
4. #21447934C2semidirect
5. #21447925C2semidirect
6. #21447917C2semidirect
7. #21447908C2semidirect
8. #21447608C2semidirect
9. #21447524C2semidirect
10. C2×#16839566C2semidirect
11. C61×#3059C2semidirect
58. #14519717D4semidirect
201. #3059D61semidirect
246. C61#33613semidirect

Maximal subgroups

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

subgroupindex
C2×#168402032
C2×#168401792
#214554892
#214479342
#214479252
#214479172
#214479082
#214476082
#214475242
C2×#168395662
C61×#30592
#3361361