Finite Group Databasebeta

← #29663170

Subgroups — #29663170

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456746C211yesmaximal, characteristic, complemented
2#21456737C211yesmaximal, characteristic
3C2×#16840108C211yesmaximal, characteristic, complemented
4#21454530C211yesmaximal, characteristic, complemented
5#21445379C214yesmaximal, complemented
6#21445255C214yesmaximal, complemented
7#21444565C211yesmaximal, characteristic, complemented
8#21444505C211yesmaximal, characteristic, complemented
9C61×#1545C211yesmaximal, characteristic, complemented
10C2×#14519785—21no—
11#16840312C2211yescharacteristic
12C2×#14519777C2211yescharacteristic
13#16840138—21no—
14#16840135—21no—
15#16840113—21no—
16#16840108—21no—
17#16840108—21no—
18#16840108—21no—
19#16840108C2212yes—
20#16840108C2212yes—
21#16839968C2214yescomplemented
22#16839954C2214yes—
23#16839903—22no—
24#16839896—22no—
25#16839839—21no—
26#16839839C2212yescomplemented
27#16839827—21no—
28#16839827C2212yescomplemented
29#16839815C2211yescharacteristic
30C2×#14519698—21no—
31C2×#14519698—21no—
32#16839730—21no—
33C2×C4×#92418C2211yescharacteristic
34#16839514C2212yescomplemented
35#16839512C2212yescomplemented
36#16839507C2214yescomplemented
37C2×#127×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#331C2214yes—
41C2×#11758222—21no—
42C22×Dic122—21no—
43C2×#11758190—24no—
44C2×#11758199—24no—
45#14519785—44no—
46C2×#11758189—22no—
47C22×#92418D411yescharacteristic
48C22×#92418D411yescharacteristic
49C2×C8×D61—22no—
50#14519777—21no—
51#14519777—21no—
52#14519777—21no—
53#14519777D411yescharacteristic
54#14519777C2312yes—
55#14519774D411yescharacteristic
56#14519765—21no—
57#14519751—21no—
58#14519751D412yes—
59C4×D244—21no—
60C4×D244C2312yes—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61—21no—
63C4⋊#92418C2312yes—
64#14519702—24no—
65#14519698—24no—
66#14519698—22no—
67#14519698—22no—
68#14519698—22no—
69#14519698—22no—
70C22⋊#92418—21no—
71#14519689—22no—
72#14519689—22no—
73C4×#92418—21no—
74C4×#92418D411yescharacteristic
75C4×#92418D411yescharacteristic
76C4×#92418C2311yescharacteristic
77C4×#92418C2311yescharacteristic
78C22×D4×C61—21no—
79C22×D4×C61—21no—
80C22×D4×C61—21no—
81C2×M16×C61—21no—
82#127×C61—21no—
83#127×C61—21no—
84#127×C61—21no—
85#127×C61C2312yes—
86C2×C42×C61C2311yescharacteristic
87C4⋊C8×C61—21no—
88C4⋊C8×C61C2312yes—
89C4⋊C8×C61C2312yes—
90C22⋊C8×C61—21no—
91C4×C8×C61—21no—
92C24×C61—41no—
93C2×D4×C61—22no—
94C2×D4×C61—22no—
95C2×D4×C61—22no—
96C2×D4×C61—41no—
97C2×D4×C61—41no—
98C2×D4×C61—41no—
99C2×D4×C61—42no—
100C2×D4×C61—42no—
101C2×D4×C61—82no—
102C2×D4×C61—82no—
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61#4111yescharacteristic
106C2×C22⋊D61—22no—
107C22×Dic61—41no—
108#11758222—42no—
109#11758222—42no—
110C2×D244—21no—
111C2×D244—21no—
112C2×D244C2×D412yes—
113C2×C4×D61—22no—
114C2×Dic122—21no—
115C2×Dic122—22no—
116C2×Dic122—41no—
117C2×Dic122—82no—
118M16×C61—42no—
119C2×C8×C61—22no—
120C2×C8×C61—22no—
121C42×C61C2×D411yescharacteristic
122C42×C61C2×D411yescharacteristic
123C42×C61#4111yescharacteristic
124C42×C61C2411yescharacteristic
125#11758199—84no—
126#11758198—22no—
127#11758198—22no—
128C4⋊Dic61—21no—
129C4⋊Dic61—22no—
130C4⋊Dic61—22no—
131C4⋊Dic61—41no—
132C4⋊Dic61—41no—
133C4⋊Dic61C2×D412yes—
134C2×#92418—22no—
135C2×#92418—22no—
136C2×#92418—22no—
137C2×#92418C2×D411yescharacteristic
138C2×#92418C2×D411yescharacteristic
139C2×#92418C2×D411yescharacteristic
140C2×#92418C2×D411yescharacteristic
141#11758190—84no—
142#11758189—44no—
143C8×D61—44no—
144C23×C61—42no—
145C23×C61—82no—
146C23×C61—82no—
147C23×C61—82no—
148C23×C61#12111yescharacteristic
149C22×D61—22no—
150D4×C61—42no—
151D4×C61—42no—
152D4×C61—44no—
153D4×C61—82no—
154D4×C61—82no—
155D4×C61—82no—
156D4×C61—82no—
157C2×C4×C61—22no—
158C2×C4×C61—22no—
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#12111yescharacteristic
162C2×C4×C61#13512yes—
163C2×C4×C61AGL(1,ℤ/8)12yes—
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C22×D411yescharacteristic
166C2×C4×C61C2×#4111yescharacteristic
167C22⋊D61—44no—
168C2×Dic61—22no—
169C2×Dic61—42no—
170C2×Dic61—82no—
171D244—24no—
172D244—42no—
173D244—42no—
174C4×D61—42no—
175C4×D61—42no—
176Dic122—42no—
177Dic122—44no—
178Dic122—82no—
179C8×C61—42no—
180C8×C61—42no—
181#92418—21no—
182#92418—21no—
183#92418—21no—
184#92418—21no—
185#92418—42no—
186#19198—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—82no—
190C22×C61—82no—
191C22×C61—82no—
192C22×C61—164no—
193C22×C61#46312yes—
194C22×C61C2×#12111yescharacteristic
195C2×D61—42no—
196C2×D61—42no—
197C2×D61—42no—
198C4×C61—21no—
199C4×C61—21no—
200C4×C61#46312yes—
201C4×C61#47112yes—
202C4×C61C2×#13511yescharacteristic
203C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
204C4×C61#57612yes—
205Dic61—42no—
206Dic61—82no—
207#3206—611no—
208#3205—611no—
209C2×#486—611no—
210#2931—611no—
211#1701—614no—
212#1681—614no—
213#1564—611no—
214#1549—611no—
215#1545D6111yescharacteristic, complemented
216C2×C61—22no—
217C2×C61—162no—
218C2×C61C2×#46311yescharacteristic
219C2×C61#305811yescharacteristic
220C2×C61C2×#47111yescharacteristic
221D61—82no—
222C2×#131—1221no—
223#532—611no—
224C2×C4⋊Q8—611no—
225C2×#127C2×D6111yescharacteristic
226#493—1221no—
227#492—1221no—
228#487—1221no—
229#486—612no—
230#486—612no—
231#486—1221no—
232#486—1221no—
233#486—1221no—
234#461—614no—
235#459—614no—
236#448—1222no—
237#447—1222no—
238#436—612no—
239#436—1221no—
240C8×D4—612no—
241C8×D4—1221no—
242#431—611no—
243#431C2×D6111yescharacteristic
244#423C2×D6111yescharacteristic
245C2×#102—1221no—
246C2×#102—1221no—
247#408—1221no—
248C2×C4×C8—611no—
249#335—612no—
250#334—612no—
251#332—614no—
252#331C2×D6114yes—
253C61#1919811yescharacteristic, complemented, Sylow
254C2×#41—1221no—
255C22×Q8—1221no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—21no—
259C2×SD16—1224no—
260C2×D8—1224no—
261#131—2444no—
262C2×M16—21no—
263C2×M16—1222no—
264C22×C8—611no—
265C22×C8—611no—
266C22×C8—1222no—
267C4⋊Q8—611no—
268C4⋊Q8—612no—
269C4⋊Q8—1221no—
270C4⋊Q8—1221no—
271C4⋊Q8—1221no—
272#127—21no—
273#127—21no—
274#127—21no—
275#127—611no—
276#127C22×D6112yes—
277#124—1221no—
278#121—612no—
279#121—1221no—
280C4×D4—612no—
281C4×D4—1221no—
282C2×C4⋊C4—1221no—
283C2×C4⋊C4—1221no—
284C2×C42C22×D6111yescharacteristic
285C4⋊C8—21no—
286C4⋊C8—612no—
287C4⋊C8C22×D6112yes—
288C4⋊C8C22×D6112yes—
289#103—1224no—
290#102—1224no—
291#102—1222no—
292#102—1222no—
293#102—1222no—
294#102—1222no—
295C22⋊C8—21no—
296C22⋊C8—1222no—
297C22⋊C8—1222no—
298C22⋊C8—1221no—
299C4×C8—21no—
300C4×C8—611no—
301C4×C8—611no—
302C4×C8—611no—
303C4×C8—611no—
304C4×C8—1221no—
305C24—41no—
306#41—2442no—
307#41—2442no—
308C2×Q8—1221no—
309C2×Q8—1222no—
310C2×Q8—4882no—
311C2×Q8—2441no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—82no—
321C2×D4—82no—
322C2×D4—612no—
323C2×D4—1222no—
324C2×D4—1221no—
325C2×D4—1221no—
326C22×C4—1222no—
327C22×C4—2441no—
328C22×C4#1175822211yescharacteristic
329C22×C4D4×D6111yescharacteristic
330C22×C4D4×D6111yescharacteristic
331SD16—4884no—
332D8—4884no—
333M16—42no—
334M16—2444no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—611no—
338C2×C8—611no—
339C2×C8—611no—
340C2×C8—611no—
341C2×C8—1222no—
342C2×C8—1222no—
343C2×C8—1222no—
344C2×C8—2444no—
345C4⋊C4—612no—
346C4⋊C4—1221no—
347C4⋊C4—1222no—
348C4⋊C4—1222no—
349C4⋊C4—2441no—
350C4⋊C4—2441no—
351C22⋊C4—1222no—
352C22⋊C4—1222no—
353C42#1175822211yescharacteristic
354C42D4×D6111yescharacteristic
355C42D4×D6111yescharacteristic
356C42C23×D6111yescharacteristic
357C23—42no—
358C23—82no—
359C23—82no—
360C23—82no—
361C23—1222no—
362C23#1451975011yescharacteristic
363Q8—4882no—
364Q8—2444no—
365Q8—2442no—
366D4—42no—
367D4—42no—
368D4—44no—
369D4—82no—
370D4—82no—
371D4—82no—
372D4—82no—
373D4—1224no—
374D4—2444no—
375D4—2442no—
376D4—2442no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—1222no—
380C2×C4—4882no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4—2442no—
384C2×C4#1451975011yescharacteristic
385C2×C4#1451975011yescharacteristic
386C2×C4#1451975011yescharacteristic
387C2×C4#1451979812yes—
388C2×C4#1451980112yes—
389C2×C4C2×D4×D6111yescharacteristic
390C2×C4C2×D4×D6111yescharacteristic
391C2×C4C2×#1175822211yescharacteristic
392C8—42no—
393C8—42no—
394C8—1221no—
395C8—1221no—
396C8—1221no—
397C8—1221no—
398C8—2442no—
399C22—21no—
400C22—21no—
401C22—82no—
402C22—82no—
403C22—82no—
404C22—164no—
405C22—2442no—
406C22—2442no—
407C22—2442no—
408C22#1683997612yes—
409C22C2×#1451975011yescharacteristic
410C4—21no—
411C4—21no—
412C4—4882no—
413C4—2442no—
414C4#1683997612yes—
415C4#1684002412yes—
416C4C2×#1451979811yescharacteristic
417C4C2×#1451980111yescharacteristic
418C4#1684056612yes—
419C2—22no—
420C2—162no—
421C2—4882no—
422C2C2×#1683997611yescharacteristic
423C2#2145546611yescharacteristic
424C2C2×#1684002411yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1545C2
socleC22×C61C2×#121
prime radical#21456737C2
hypercenter#1545D61
hypocenterC61#19198
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1545

Complements

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

NH ≅ G/Ntype
1. #21456746C2semidirect
3. C2×#16840108C2semidirect
4. #21454530C2semidirect
5. #21445379C2semidirect
6. #21445255C2semidirect
7. #21444565C2semidirect
8. #21444505C2semidirect
9. C61×#1545C2semidirect
21. #16839968C22semidirect
26. #16839839C22semidirect
28. #16839827C22semidirect
34. #16839514C22semidirect
35. #16839512C22semidirect
36. #16839507C22semidirect
215. #1545D61semidirect
253. C61#19198semidirect

Maximal subgroups

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

subgroupindex
#214567462
#214567372
C2×#168401082
#214545302
#214453792
#214452552
#214445652
#214445052
C61×#15452
#1919861