Finite Group Databasebeta

← #29834565

Subgroups — #29834565

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×#16840202C211yesmaximal, characteristic, complemented
2C2×#16840178C211yesmaximal, characteristic, complemented
3#21455485C211yesmaximal, characteristic
4#21455475C211yesmaximal, characteristic, complemented
5#21447931C212yesmaximal, complemented
6#21447926C212yesmaximal, complemented
7#21447914C212yesmaximal, complemented
8#21447909C212yesmaximal, complemented
9#21447608C211yesmaximal, characteristic, complemented
10C2×#16839567C211yesmaximal, characteristic, complemented
11C61×#2138C211yesmaximal, characteristic, complemented
12C22×Dic244C2211yescharacteristic
13C2×#14519783C2211yescharacteristic
14#16840340—21no—
15#16840338—21no—
16#16840202—24no—
17C2×#14519723C2212yes—
18C2×#14519721C2212yes—
19#16840178—24no—
20C2×#14519717C2211yescharacteristic
21C2×#14519716C2212yes—
22C2×C16×D61C2212yes—
23#16840021—22no—
24#16840005—22no—
25#16840004C2212yes—
26#16840004C2212yes—
27#16840000C2212yes—
28#16839995C2212yes—
29C2×#14519711C2211yescharacteristic
30#16839770—21no—
31#16839763—21no—
32#16839589C2212yes—
33#16839587C2212yes—
34#16839567C2212yes—
35#16839567C2212yes—
36C2×#135×C61C2211yescharacteristic
37C22×C16×C61C2211yescharacteristic
38C61×#427C2211yescharacteristic
39C61×#358C2211yescharacteristic
40C61×#358C2211yescharacteristic
41C61×#357C2211yescharacteristic
42C61×#357C2211yescharacteristic
43C2×#11758222D411yescharacteristic
44C22×Dic122—21no—
45C2×Dic244—22no—
46C2×Dic244—22no—
47C2×Dic244—21no—
48C2×Dic244C2312yes—
49#14519783—22no—
50#14519783—22no—
51C2×#11758194D411yescharacteristic
52C2×#11758189D412yes—
53C2×C8×D61C2312yes—
54#14519777—41no—
55#14519765—41no—
56#14519761—42no—
57#14519757—22no—
58#14519757—22no—
59#14519756—41no—
60#14519753—22no—
61#14519751—41no—
62#14519749—42no—
63#14519748—22no—
64C2×C4⋊Dic61—21no—
65#14519723—24no—
66#14519721—24no—
67#14519717D414yescomplemented
68#14519716—24no—
69C2×#11758185C2312yes—
70C16×D61—24no—
71#14519711—21no—
72#14519711C2312yes—
73#14519701—22no—
74#14519700—22no—
75#14519700—22no—
76#14519698—22no—
77C2×#41×C61—21no—
78#135×C61—22no—
79#135×C61—22no—
80C2×Q16×C61C2311yescharacteristic
81C2×SD16×C61—21no—
82C2×D8×C61C2311yescharacteristic
83C22×C8×C61C2311yescharacteristic
84#117×C61—21no—
85C2×C16×C61—22no—
86C2×C16×C61C2311yescharacteristic
87C2×C16×C61C2311yescharacteristic
88#107×C61C2311yescharacteristic
89#107×C61C2311yescharacteristic
90#106×C61—21no—
91#41×C61—42no—
92#41×C61—42no—
93C2×Q8×C61—21no—
94C2×D4×C61—21no—
95C2×D4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61C2×D411yescharacteristic
98C2×C22⋊D61—42no—
99C2×C22⋊D61D812yes—
100C22×Dic61—41no—
101C22×Dic61—41no—
102#11758222—22no—
103#11758222—22no—
104C2×D244D811yescharacteristic
105C2×C4×D61C2×D412yes—
106C2×Dic122—22no—
107C2×Dic122—42no—
108C2×Dic122—42no—
109C2×Dic122—41no—
110C2×Dic122D811yescharacteristic
111Q16×C61—21no—
112Q16×C61—21no—
113SD16×C61—42no—
114D8×C61—21no—
115D8×C61—21no—
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61C2×D412yes—
118C2×C8×C61#4112yes—
119C2×C8×C61C2411yescharacteristic
120C4⋊C4×C61—21no—
121C4⋊C4×C61—21no—
122C22⋊C4×C61—41no—
123C42×C61—41no—
124C22⋊Dic61—42no—
125#11758198—42no—
126#11758198—42no—
127#11758198—42no—
128C4⋊Dic61—22no—
129C4⋊Dic61—41no—
130C4⋊Dic61—42no—
131C4⋊Dic61—42no—
132#11758196—42no—
133#11758194—22no—
134C2×#92418C2×D412yes—
135Dic244—22no—
136Dic244—22no—
137Dic244—42no—
138#11758189—24no—
139C8×D61—22no—
140C8×D61—22no—
141C16×C61—21no—
142C16×C61—21no—
143#11758185C2×D414yes—
144C23×C61—41no—
145C23×C61C2×D811yescharacteristic
146C22×D61C2×D812yes—
147Q8×C61—41no—
148Q8×C61—41no—
149D4×C61—41no—
150D4×C61—41no—
151D4×C61—82no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—82no—
156C2×C4×C61#12112yes—
157C2×C4×C61C2×D811yescharacteristic
158C2×C4×C61#13512yes—
159C2×C4×C61C22×D411yescharacteristic
160C22⋊D61—24no—
161C22⋊D61—162no—
162C2×Dic61—42no—
163C2×Dic61—42no—
164C2×Dic61—82no—
165C2×Dic61—161no—
166C2×Dic61C2×D812yes—
167D244—22no—
168C4×D61—22no—
169C4×D61—22no—
170Dic122—22no—
171Dic122—42no—
172Dic122—42no—
173Dic122—82no—
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—81no—
191C4×C61—81no—
192C4×C61#46612yes—
193C4×C61C2×#12111yescharacteristic
194C4×C61C2×#13511yescharacteristic
195Dic61—22no—
196Dic61—82no—
197Dic61—82no—
198C2×#510—611no—
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#2138D6111yescharacteristic, complemented
208C2×#358—611no—
209C2×C61—22no—
210C2×C61—161no—
211C2×C61C2×#46611yescharacteristic
212C2×C61#341611yescharacteristic
213C2×C61#341711yescharacteristic
214D61—42no—
215C2×#135C2×D6111yescharacteristic
216C22×Q16—611no—
217C2×#131—611no—
218#537—1221no—
219#536—1221no—
220#510—1224no—
221C2×Q32—612no—
222C2×SD32—612no—
223#504—1224no—
224C2×M32—611no—
225C2×M32—612no—
226C22×C16—612no—
227C22×C16C2×D6111yescharacteristic
228#470—1222no—
229#468—1222no—
230#467—612no—
231#467—612no—
232#467—612no—
233#466—612no—
234#427C2×D6111yescharacteristic
235C2×#107—611no—
236#419—1221no—
237#417—1221no—
238#367—612no—
239#366—612no—
240#358—612no—
241#358—612no—
242#358C2×D6111yescharacteristic
243#358C2×D6111yescharacteristic
244#357C2×D6111yescharacteristic
245#357C2×D6111yescharacteristic
246C61#3361311yescharacteristic, complemented, Sylow
247C2×#41—21no—
248C2×#41—611no—
249C22×Q8—1221no—
250#135—22no—
251#135—22no—
252C2×Q16—612no—
253C2×Q16—1222no—
254C2×Q16—1222no—
255C2×Q16—1221no—
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—21no—
276C2×C4⋊C4—1221no—
277Q32—1224no—
278SD32—1224no—
279M32—614no—
280M32—1224no—
281C2×C16—22no—
282C2×C16—612no—
283C2×C16—1224no—
284C2×C16C22×D6111yescharacteristic
285C2×C16C22×D6111yescharacteristic
286#107—612no—
287#107—1221no—
288#107C22×D6111yescharacteristic
289#107C22×D6111yescharacteristic
290#106—21no—
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—611no—
301C2×Q8—1222no—
302C2×Q8—2442no—
303C2×Q8—2442no—
304C2×Q8—2441no—
305C2×D4—21no—
306C2×D4—41no—
307C2×D4—612no—
308C2×D4—611no—
309C2×D4—2442no—
310C22×C4—41no—
311C22×C4—612no—
312C22×C4—2441no—
313C22×C4—2441no—
314C22×C4D4×D6111yescharacteristic
315Q16—21no—
316Q16—21no—
317Q16—1222no—
318Q16—1222no—
319Q16—2442no—
320SD16—42no—
321D8—21no—
322D8—21no—
323M16—1222no—
324M16—1224no—
325C2×C8—612no—
326C2×C8—1222no—
327C2×C8—1222no—
328C2×C8#1175822212yes—
329C2×C8D4×D6112yes—
330C2×C8D4×D6111yescharacteristic
331C2×C8C23×D6111yescharacteristic
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—1222no—
335C4⋊C4—2442no—
336C4⋊C4—2442no—
337C4⋊C4—2442no—
338C4⋊C4—2441no—
339C22⋊C4—41no—
340C22⋊C4—2442no—
341C22⋊C4—2442no—
342C22⋊C4—2442no—
343C22⋊C4—2442no—
344C42—41no—
345C16—21no—
346C16—21no—
347C16—614no—
348C23—41no—
349C23—612no—
350C23D8×D6111yescharacteristic
351Q8—41no—
352Q8—41no—
353Q8—1222no—
354Q8—4882no—
355Q8—2442no—
356Q8—2442no—
357D4—41no—
358D4—41no—
359D4—82no—
360D4—1224no—
361D4—1222no—
362D4—9762no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—82no—
367C2×C4—612no—
368C2×C4—1222no—
369C2×C4—1222no—
370C2×C4—9761no—
371C2×C4—4882no—
372C2×C4—2442no—
373C2×C4—2442no—
374C2×C4#1451975012yes—
375C2×C4D8×D6111yescharacteristic
376C2×C4#1451979812yes—
377C2×C4C2×D4×D6111yescharacteristic
378C8—1222no—
379C8#1451975012yes—
380C8C2×D4×D6111yescharacteristic
381C8C2×#1175822211yescharacteristic
382C22—21no—
383C22—21no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—1222no—
388C22—1222no—
389C22—1222no—
390C22#1683999712yes—
391C22C2×D8×D6111yescharacteristic
392C4—81no—
393C4—81no—
394C4—81no—
395C4—1222no—
396C4—4882no—
397C4—4882no—
398C4#1683999712yes—
399C4C2×#1451975011yescharacteristic
400C4C2×#1451979811yescharacteristic
401C2—22no—
402C2—161no—
403C2—2442no—
404C2C2×#1683999711yescharacteristic
405C2#2145896211yescharacteristic
406C2#2145896911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2138C2
socleC22×C61C22×D8
prime radical#21455485C2
hypercenter#2138D61
hypocenterC61#33613
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2138

Complements

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

NH ≅ G/Ntype
1. C2×#16840202C2semidirect
2. C2×#16840178C2semidirect
4. #21455475C2semidirect
5. #21447931C2semidirect
6. #21447926C2semidirect
7. #21447914C2semidirect
8. #21447909C2semidirect
9. #21447608C2semidirect
10. C2×#16839567C2semidirect
11. C61×#2138C2semidirect
67. #14519717D4semidirect
207. #2138D61semidirect
246. C61#33613semidirect

Maximal subgroups

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

subgroupindex
C2×#168402022
C2×#168401782
#214554852
#214554752
#214479312
#214479262
#214479142
#214479092
#214476082
C2×#168395672
C61×#21382
#3361361