Finite Group Databasebeta

← #29663149

Subgroups — #29663149

Order 15616.

714 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456947C211yesmaximal, characteristic, complemented
2#21456737C211yesmaximal, characteristic
3#21454532C211yesmaximal, characteristic, complemented
4#21445349C214yesmaximal, complemented
5#21445225C214yesmaximal, complemented
6#21444567C211yesmaximal, characteristic, complemented
7#21444500C211yesmaximal, characteristic, complemented
8#21444487C211yesmaximal, characteristic, complemented
9C2×C61×#486C211yesmaximal, characteristic, complemented
10C2×#14519783—21no—
11#16840312C2211yescharacteristic
12#16840158—21no—
13#16840138—21no—
14#16840110—21no—
15#16840108—21no—
16#16840080—22no—
17#16840049C2214yescomplemented
18#16839954C2214yes—
19#16839896—22no—
20#16839841—21no—
21#16839841C2212yescomplemented
22C8×D244—21no—
23C8×D244C2212yescomplemented
24#16839815C2211yescharacteristic
25#16839815C2211yescharacteristic
26#16839786C2211yescharacteristic
27#16839728—21no—
28#16839515C2212yescomplemented
29#16839513C2212yescomplemented
30#16839511C2214yescomplemented
31#16839506C2214yes—
32C2×C4⋊Q8×C61C2211yescharacteristic
33C2×#127×C61C2211yescharacteristic
34C61×#486—21no—
35C61×#486—21no—
36C61×#486C2212yes—
37C61×#486C2212yes—
38C2×#102×C61—21no—
39C2×#102×C61—21no—
40C2×C4×C8×C61C2211yescharacteristic
41C2×#11758222—21no—
42C2×#11758199—24no—
43#14519783—44no—
44C2×#11758194—21no—
45C2×#11758189—22no—
46C2×C8×D61—22no—
47#14519777D411yescharacteristic
48#14519774D411yescharacteristic
49#14519765—21no—
50#14519751—21no—
51#14519751D412yes—
52C4×D244—21no—
53C4×D244C2312yes—
54#14519708—24no—
55C4⋊#92418—21no—
56C4⋊#92418C2312yes—
57C4⋊#92418C2312yes—
58C4⋊#92418C2312yes—
59#14519698—24no—
60#14519697—24no—
61C22⋊#92418—21no—
62C22⋊#92418—21no—
63#14519689—22no—
64#14519689—22no—
65C4×#92418—21no—
66C4×#92418—21no—
67C22×Q8×C61—21no—
68C22×D4×C61—21no—
69C22×D4×C61—21no—
70C22×D4×C61—21no—
71C22×C8×C61D411yescharacteristic
72C22×C8×C61D411yescharacteristic
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61C2312yes—
77#127×C61—21no—
78#127×C61—21no—
79#127×C61—21no—
80#127×C61C2312yes—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—21no—
83C2×C42×C61C2311yescharacteristic
84#102×C61—22no—
85#102×C61—22no—
86#102×C61—22no—
87#102×C61—22no—
88C4×C8×C61D411yescharacteristic
89C4×C8×C61D411yescharacteristic
90C4×C8×C61C2311yescharacteristic
91C4×C8×C61C2311yescharacteristic
92C24×C61—41no—
93C2×Q8×C61—22no—
94C2×Q8×C61—41no—
95C2×Q8×C61—82no—
96C2×D4×C61—22no—
97C2×D4×C61—22no—
98C2×D4×C61—22no—
99C2×D4×C61—41no—
100C2×D4×C61—41no—
101C2×D4×C61—41no—
102C2×D4×C61—42no—
103C2×D4×C61—42no—
104C2×D4×C61—82no—
105C2×D4×C61—82no—
106C22×C4×C61—41no—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61#4111yescharacteristic
110C2×C22⋊D61—22no—
111#11758222—42no—
112#11758222—42no—
113C2×D244—21no—
114C2×D244—21no—
115C2×D244C2×D412yes—
116C2×C4×D61—22no—
117C2×Dic122—21no—
118C2×C8×C61—22no—
119C2×C8×C61—22no—
120C2×C8×C61C2×D411yescharacteristic
121C2×C8×C61C2×D411yescharacteristic
122C2×C8×C61C2×D411yescharacteristic
123C2×C8×C61C2×D411yescharacteristic
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—41no—
127C4⋊C4×C61—41no—
128C42×C61C2×D411yescharacteristic
129C42×C61C2×D411yescharacteristic
130C42×C61#4111yescharacteristic
131C42×C61C2411yescharacteristic
132#11758199—84no—
133#11758198—22no—
134#11758198—22no—
135C4⋊Dic61—21no—
136C4⋊Dic61C2×D412yes—
137#11758194—42no—
138C2×#92418—22no—
139C2×#92418—22no—
140C2×#92418—22no—
141#11758189—44no—
142C8×D61—44no—
143C23×C61—42no—
144C23×C61—82no—
145C23×C61—82no—
146C23×C61—82no—
147C23×C61#12111yescharacteristic
148C22×D61—22no—
149Q8×C61—82no—
150Q8×C61—82no—
151D4×C61—42no—
152D4×C61—42no—
153D4×C61—44no—
154D4×C61—82no—
155D4×C61—82no—
156D4×C61—82no—
157D4×C61—82no—
158C2×C4×C61—22no—
159C2×C4×C61—22no—
160C2×C4×C61—42no—
161C2×C4×C61—82no—
162C2×C4×C61#12111yescharacteristic
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#13512yes—
166C2×C4×C61AGL(1,ℤ/8)12yes—
167C2×C4×C61C22×D411yescharacteristic
168C2×C4×C61C22×D411yescharacteristic
169C2×C4×C61C2×#4111yescharacteristic
170C22⋊D61—44no—
171C2×Dic61—22no—
172D244—24no—
173D244—42no—
174D244—42no—
175C4×D61—42no—
176C4×D61—42no—
177Dic122—42no—
178C8×C61—21no—
179C8×C61—21no—
180C8×C61—21no—
181C8×C61—21no—
182#92418—42no—
183#92418—42no—
184#92418—42no—
185#19196—611nomaximal, Sylow
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—82no—
189C22×C61—82no—
190C22×C61—82no—
191C22×C61—164no—
192C22×C61#46312yes—
193C22×C61C2×#12111yescharacteristic
194C2×D61—42no—
195C2×D61—42no—
196C2×D61—42no—
197C4×C61—21no—
198C4×C61—21no—
199C4×C61—82no—
200C4×C61#46312yes—
201C4×C61#47212yes—
202C4×C61C2×#13511yescharacteristic
203C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
204C4×C61#57712yes—
205Dic61—42no—
206#3227—611no—
207#3205—611no—
208C2×#486D6111yescharacteristic, complemented
209#2931—611no—
210#1697—614no—
211#1677—614no—
212#1564—611no—
213#1548—611no—
214#1545—611no—
215C2×C61—22no—
216C2×C61—162no—
217C2×C61C2×#46311yescharacteristic
218C2×C61#305711yescharacteristic
219C2×C61C2×#47211yescharacteristic
220D61—82no—
221C2×#131—1221no—
222#532—611no—
223C2×C4⋊Q8C2×D6111yescharacteristic
224C2×#127C2×D6111yescharacteristic
225#498—1221no—
226#493—1221no—
227#486—21no—
228#486—21no—
229#486—1221no—
230#486—1221no—
231#486C2×D6112yes—
232#486C2×D6112yes—
233#481—1222no—
234#476—614no—
235#459—614no—
236#447—1222no—
237#436—612no—
238#436—1221no—
239C8×D4—612no—
240C8×D4—1221no—
241#431—611no—
242#431—611no—
243#423—611no—
244C2×#102—21no—
245C2×#102—21no—
246#408—1221no—
247C2×C4×C8C2×D6111yescharacteristic
248#335—612no—
249#334—612no—
250#333—614no—
251#331—614no—
252C61#1919611yescharacteristic, complemented, Sylow
253C2×#41—1221no—
254C22×Q8—21no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—21no—
258C2×D8—1224no—
259#131—2444no—
260C2×M16—1221no—
261C2×M16—1222no—
262C22×C8—1222no—
263C22×C8C22⋊D6111yescharacteristic
264C22×C8C22⋊D6111yescharacteristic
265C4⋊Q8—21no—
266C4⋊Q8—21no—
267C4⋊Q8—21no—
268C4⋊Q8—611no—
269C4⋊Q8C22×D6112yes—
270#127—21no—
271#127—21no—
272#127—21no—
273#127—611no—
274#127C22×D6112yes—
275#124—1221no—
276#121—612no—
277#121—1221no—
278C4×D4—612no—
279C4×D4—1221no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C42C22×D6111yescharacteristic
283#106—1224no—
284C4⋊C8—612no—
285C4⋊C8—612no—
286C4⋊C8—612no—
287C4⋊C8—1221no—
288#102—22no—
289#102—22no—
290#102—22no—
291#102—22no—
292#102—1224no—
293#102—1224no—
294C22⋊C8—1221no—
295C22⋊C8—1221no—
296C22⋊C8—1222no—
297C22⋊C8—1222no—
298C4×C8—1221no—
299C4×C8—1221no—
300C4×C8C22⋊D6111yescharacteristic
301C4×C8C22⋊D6111yescharacteristic
302C4×C8C22×D6111yescharacteristic
303C4×C8C22×D6111yescharacteristic
304C24—41no—
305#41—2442no—
306#41—2442no—
307C2×Q8—22no—
308C2×Q8—41no—
309C2×Q8—82no—
310C2×Q8—1221no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—82no—
320C2×D4—82no—
321C2×D4—612no—
322C2×D4—1222no—
323C2×D4—1221no—
324C2×D4—1221no—
325C22×C4—41no—
326C22×C4—1222no—
327C22×C4D4×D6111yescharacteristic
328C22×C4#1175822411yescharacteristic
329C22×C4C2×C22⋊D6111yescharacteristic
330D8—4884no—
331M16—2444no—
332M16—2442no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—1222no—
336C2×C8—1222no—
337C2×C8—1222no—
338C2×C8—2444no—
339C2×C8C2×C22⋊D6111yescharacteristic
340C2×C8C2×C22⋊D6111yescharacteristic
341C2×C8C2×C22⋊D6111yescharacteristic
342C2×C8C2×C22⋊D6111yescharacteristic
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—612no—
348C4⋊C4—1221no—
349C22⋊C4—1222no—
350C22⋊C4—1222no—
351C42D4×D6111yescharacteristic
352C42#1175822411yescharacteristic
353C42C2×C22⋊D6111yescharacteristic
354C42C23×D6111yescharacteristic
355C23—42no—
356C23—82no—
357C23—82no—
358C23—82no—
359C23—1222no—
360C23#1451974811yescharacteristic
361Q8—82no—
362Q8—82no—
363Q8—2442no—
364D4—42no—
365D4—42no—
366D4—44no—
367D4—82no—
368D4—82no—
369D4—82no—
370D4—82no—
371D4—1224no—
372D4—2444no—
373D4—2442no—
374D4—2442no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—42no—
378C2×C4—82no—
379C2×C4—1222no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4#1451974811yescharacteristic
383C2×C4#1451974811yescharacteristic
384C2×C4#1451974811yescharacteristic
385C2×C4#1451979712yes—
386C2×C4#1451980212yes—
387C2×C4C2×D4×D6111yescharacteristic
388C2×C4C22×C22⋊D6111yescharacteristic
389C2×C4C2×#1175822411yescharacteristic
390C8—21no—
391C8—21no—
392C8—21no—
393C8—21no—
394C8—2442no—
395C8—2442no—
396C8—2442no—
397C22—21no—
398C22—21no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—164no—
403C22—2442no—
404C22—2442no—
405C22—2442no—
406C22#1683997412yes—
407C22C2×#1451974811yescharacteristic
408C4—21no—
409C4—21no—
410C4—82no—
411C4—2442no—
412C4#1683997412yes—
413C4#1684002912yes—
414C4C2×#1451979711yescharacteristic
415C4C2×#1451980211yescharacteristic
416C4#1684056812yes—
417C2—22no—
418C2—162no—
419C2—4882no—
420C2C2×#1683997411yescharacteristic
421C2#2145545411yescharacteristic
422C2C2×#1684002911yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#486C2
socleC22×C61C2×#121
prime radical#21456737C2
hypercenterC2×#486D61
hypocenterC61#19196
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#486

Complements

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

NH ≅ G/Ntype
1. #21456947C2semidirect
3. #21454532C2semidirect
4. #21445349C2semidirect
5. #21445225C2semidirect
6. #21444567C2semidirect
7. #21444500C2semidirect
8. #21444487C2semidirect
9. C2×C61×#486C2semidirect
17. #16840049C22semidirect
21. #16839841C22semidirect
23. C8×D244C22semidirect
28. #16839515C22semidirect
29. #16839513C22semidirect
30. #16839511C22semidirect
208. C2×#486D61semidirect
252. C61#19196semidirect

Maximal subgroups

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

subgroupindex
#214569472
#214567372
#214545322
#214453492
#214452252
#214445672
#214445002
#214444872
C2×C61×#4862
#1919661