Finite Group Databasebeta

← #29663152

Subgroups — #29663152

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#21456949C211yesmaximal, characteristic, complemented
2#21456737C211yesmaximal, characteristic
3C2×#16840108C211yesmaximal, characteristic, complemented
4#21454530C211yesmaximal, characteristic, complemented
5#21445351C214yesmaximal, complemented
6#21445227C214yesmaximal, complemented
7#21444565C211yesmaximal, characteristic, complemented
8#21444501C211yesmaximal, characteristic, complemented
9C61×#1545C211yesmaximal, characteristic, complemented
10C2×#14519785—21no—
11#16840312C2211yescharacteristic
12C2×#14519777C2211yescharacteristic
13#16840160—21no—
14#16840138—21no—
15#16840109—21no—
16#16840108—21no—
17#16840108—21no—
18#16840108—21no—
19#16840108C2212yes—
20#16840108C2212yes—
21#16840081—22no—
22#16840053C2214yescomplemented
23#16839954C2214yes—
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#16839510C2214yescomplemented
37C2×#127×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#331C2214yes—
41C2×#11758222—21no—
42C22×Dic122—21no—
43C2×#11758199—24no—
44#14519785—44no—
45C2×#11758189—22no—
46C22×#92418D411yescharacteristic
47C22×#92418D411yescharacteristic
48C2×C8×D61—22no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777D411yescharacteristic
53#14519777C2312yes—
54#14519774D411yescharacteristic
55#14519765—21no—
56#14519751—21no—
57#14519751D412yes—
58C4×D244—21no—
59C4×D244C2312yes—
60C2×C4⋊Dic61—21no—
61C2×C4⋊Dic61—21no—
62#14519709—24no—
63C4⋊#92418C2312yes—
64#14519699—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#11758189—44no—
142C8×D61—44no—
143C23×C61—42no—
144C23×C61—82no—
145C23×C61—82no—
146C23×C61—82no—
147C23×C61#12111yescharacteristic
148C22×D61—22no—
149D4×C61—42no—
150D4×C61—42no—
151D4×C61—44no—
152D4×C61—82no—
153D4×C61—82no—
154D4×C61—82no—
155D4×C61—82no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61#12111yescharacteristic
159C2×C4×C61#12111yescharacteristic
160C2×C4×C61#12111yescharacteristic
161C2×C4×C61#13512yes—
162C2×C4×C61AGL(1,ℤ/8)12yes—
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C2×#4111yescharacteristic
166C22⋊D61—44no—
167C2×Dic61—22no—
168C2×Dic61—42no—
169C2×Dic61—82no—
170D244—24no—
171D244—42no—
172D244—42no—
173C4×D61—42no—
174C4×D61—42no—
175Dic122—42no—
176Dic122—82no—
177Dic122—82no—
178C8×C61—42no—
179C8×C61—42no—
180#92418—21no—
181#92418—21no—
182#92418—21no—
183#92418—21no—
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#46312yes—
200C4×C61#47212yes—
201C4×C61C2×#13511yescharacteristic
202C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
203C4×C61#57712yes—
204Dic61—42no—
205Dic61—82no—
206#3227—611no—
207#3205—611no—
208C2×#486—611no—
209#2931—611no—
210#1697—614no—
211#1677—614no—
212#1564—611no—
213#1548—611no—
214#1545D6111yescharacteristic, complemented
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⋊Q8—611no—
224C2×#127C2×D6111yescharacteristic
225#498—1221no—
226#493—1221no—
227#486—612no—
228#486—612no—
229#486—1221no—
230#486—1221no—
231#486—1221no—
232#486—1221no—
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#431C2×D6111yescharacteristic
243#423C2×D6111yescharacteristic
244C2×#102—1221no—
245C2×#102—1221no—
246#408—1221no—
247C2×C4×C8—611no—
248#335—612no—
249#334—612no—
250#333—614no—
251#331C2×D6114yes—
252C61#1919611yescharacteristic, complemented, Sylow
253C2×#41—1221no—
254C22×Q8—1221no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4—21no—
258C2×D8—1224no—
259#131—2444no—
260C2×M16—21no—
261C2×M16—1222no—
262C22×C8—611no—
263C22×C8—611no—
264C22×C8—1222no—
265C4⋊Q8—611no—
266C4⋊Q8—612no—
267C4⋊Q8—1221no—
268C4⋊Q8—1221no—
269C4⋊Q8—1221no—
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—1221no—
281C2×C4⋊C4—1221no—
282C2×C42C22×D6111yescharacteristic
283#106—1224no—
284C4⋊C8—21no—
285C4⋊C8—612no—
286C4⋊C8C22×D6112yes—
287C4⋊C8C22×D6112yes—
288#102—1224no—
289#102—1224no—
290#102—1222no—
291#102—1222no—
292#102—1222no—
293#102—1222no—
294C22⋊C8—21no—
295C22⋊C8—1222no—
296C22⋊C8—1222no—
297C22⋊C8—1221no—
298C4×C8—21no—
299C4×C8—611no—
300C4×C8—611no—
301C4×C8—611no—
302C4×C8—611no—
303C4×C8—1221no—
304C24—41no—
305#41—2442no—
306#41—2442no—
307C2×Q8—1221no—
308C2×Q8—1222no—
309C2×Q8—4882no—
310C2×Q8—2441no—
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—1222no—
326C22×C4—2441no—
327C22×C4#1175822211yescharacteristic
328C22×C4D4×D6111yescharacteristic
329C22×C4D4×D6111yescharacteristic
330D8—4884no—
331M16—42no—
332M16—2444no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—611no—
336C2×C8—611no—
337C2×C8—611no—
338C2×C8—611no—
339C2×C8—1222no—
340C2×C8—1222no—
341C2×C8—1222no—
342C2×C8—2444no—
343C4⋊C4—612no—
344C4⋊C4—1221no—
345C4⋊C4—1222no—
346C4⋊C4—1222no—
347C4⋊C4—2441no—
348C4⋊C4—2441no—
349C22⋊C4—1222no—
350C22⋊C4—1222no—
351C42#1175822211yescharacteristic
352C42D4×D6111yescharacteristic
353C42D4×D6111yescharacteristic
354C42C23×D6111yescharacteristic
355C23—42no—
356C23—82no—
357C23—82no—
358C23—82no—
359C23—1222no—
360C23#1451975011yescharacteristic
361Q8—4882no—
362Q8—4882no—
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—1222no—
378C2×C4—4882no—
379C2×C4—2442no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4#1451975011yescharacteristic
383C2×C4#1451975011yescharacteristic
384C2×C4#1451975011yescharacteristic
385C2×C4#1451979812yes—
386C2×C4#1451980112yes—
387C2×C4C2×D4×D6111yescharacteristic
388C2×C4C2×D4×D6111yescharacteristic
389C2×C4C2×#1175822211yescharacteristic
390C8—42no—
391C8—42no—
392C8—1221no—
393C8—1221no—
394C8—1221no—
395C8—1221no—
396C8—2442no—
397C22—21no—
398C22—21no—
399C22—82no—
400C22—82no—
401C22—82no—
402C22—164no—
403C22—2442no—
404C22—2442no—
405C22—2442no—
406C22#1683997612yes—
407C22C2×#1451975011yescharacteristic
408C4—21no—
409C4—21no—
410C4—4882no—
411C4—2442no—
412C4#1683997612yes—
413C4#1684003112yes—
414C4C2×#1451979811yescharacteristic
415C4C2×#1451980111yescharacteristic
416C4#1684057212yes—
417C2—22no—
418C2—162no—
419C2—4882no—
420C2C2×#1683997611yescharacteristic
421C2#2145545711yescharacteristic
422C2C2×#1684003111yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1545C2
socleC22×C61C2×#121
prime radical#21456737C2
hypercenter#1545D61
hypocenterC61#19196
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. #21456949C2semidirect
3. C2×#16840108C2semidirect
4. #21454530C2semidirect
5. #21445351C2semidirect
6. #21445227C2semidirect
7. #21444565C2semidirect
8. #21444501C2semidirect
9. C61×#1545C2semidirect
22. #16840053C22semidirect
26. #16839839C22semidirect
28. #16839827C22semidirect
34. #16839514C22semidirect
35. #16839512C22semidirect
36. #16839510C22semidirect
214. #1545D61semidirect
252. C61#19196semidirect

Maximal subgroups

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

subgroupindex
#214569492
#214567372
C2×#168401082
#214545302
#214453512
#214452272
#214445652
#214445012
C61×#15452
#1919661