Finite Group Databasebeta

← #29663981

Subgroups — #29663981

Order 15616.

754 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455302C211yesmaximal, characteristic, complemented
2#21455295C211yesmaximal, characteristic
3C8×#11758222C211yesmaximal, characteristic, complemented
4#21445260C214yesmaximal, complemented
5#21445253C214yesmaximal, complemented
6#21444578C211yesmaximal, characteristic, complemented
7#21444491C211yesmaximal, characteristic, complemented
8#21444487C211yesmaximal, characteristic, complemented
9C2×C61×#486C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11C2×#14519803—21no—
12C4×#11758222C2211yescharacteristic
13#16839969—21no—
14#16839969C2212yescomplemented
15#16839964—21no—
16#16839964C2212yescomplemented
17#16839959—21no—
18#16839959C2212yes—
19#16839954—21no—
20#16839954C2212yes—
21C8×Dic122C2212yescomplemented
22C8×D244C2212yescomplemented
23C8×C22⋊D61—22no—
24#16839815C2211yescharacteristic
25#16839813—21no—
26#16839786C2211yescharacteristic
27#16839786C2211yescharacteristic
28#16839771—21no—
29#16839767—21no—
30C4×C8×D61—21no—
31#16839515C2214yescomplemented
32#16839508C2214yescomplemented
33#16839506C2214yes—
34C2×C4⋊Q8×C61C2211yescharacteristic
35C2×#127×C61C2211yescharacteristic
36C61×#486—21no—
37C61×#486—21no—
38C61×#486C2214yes—
39C2×#102×C61—22no—
40C2×C4×C8×C61C2211yescharacteristic
41C2×#11758222D411yescharacteristic
42#14519808—44no—
43#14519803—44no—
44C2×#11758202—22no—
45C2×#11758200—22no—
46C2×#11758201—22no—
47C2×#11758199—22no—
48C2×#11758194—21no—
49C2×#11758194—21no—
50C2×C8×D61—22no—
51C4×Dic122D411yescharacteristic
52C4×Dic122C2311yescharacteristic
53C4×D244D411yescharacteristic
54C4×D244C2311yescharacteristic
55C4×C22⋊D61—21no—
56C4×C22⋊D61—21no—
57#14519736D411yescharacteristic
58#14519734—21no—
59C42×D61—21no—
60C4⋊#92418—21no—
61C4⋊#92418—21no—
62C4⋊#92418C2312yes—
63C4⋊#92418C2312yes—
64C4⋊#92418C2312yes—
65#14519706—22no—
66#14519701—22no—
67#14519701—22no—
68#14519698—22no—
69#14519698—22no—
70C22⋊#92418—21no—
71#14519689—22no—
72C4×#92418—21no—
73C8×Dic61—22no—
74C22×Q8×C61—21no—
75C22×D4×C61—21no—
76C22×D4×C61—21no—
77C22×D4×C61—21no—
78C22×C8×C61D412yes—
79C4⋊Q8×C61—21no—
80C4⋊Q8×C61—21no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61C2312yes—
83#127×C61—21no—
84#127×C61—21no—
85#127×C61—21no—
86#127×C61C2312yes—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—21no—
89C2×C42×C61C2311yescharacteristic
90#102×C61—24no—
91#102×C61—24no—
92C4×C8×C61D411yescharacteristic
93C4×C8×C61D411yescharacteristic
94C4×C8×C61C2312yes—
95C24×C61—41no—
96C2×Q8×C61—22no—
97C2×Q8×C61—41no—
98C2×Q8×C61—42no—
99C2×Q8×C61—42no—
100C2×D4×C61—22no—
101C2×D4×C61—22no—
102C2×D4×C61—22no—
103C2×D4×C61—44no—
104C2×D4×C61—41no—
105C2×D4×C61—41no—
106C2×D4×C61—41no—
107C2×D4×C61—42no—
108C2×D4×C61—42no—
109C2×D4×C61—82no—
110C22×C4×C61—41no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C2×C22⋊D61—21no—
115#11758222—22no—
116#11758222D814yes—
117C2×D244C2×D411yescharacteristic
118C2×C4×D61—21no—
119C2×C4×D61—21no—
120C2×C4×D61—21no—
121C2×Dic122C2×D411yescharacteristic
122C2×C8×C61—42no—
123C2×C8×C61C2×D412yes—
124C2×C8×C61C2×D412yes—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—41no—
128C4⋊C4×C61—41no—
129C42×C61C2×D411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C42×C61#4111yescharacteristic
132C42×C61C2411yescharacteristic
133C22⋊Dic61—21no—
134#11758202—44no—
135#11758201—44no—
136#11758200—44no—
137#11758199—44no—
138#11758198—21no—
139#11758198—21no—
140C4⋊Dic61C2×D411yescharacteristic
141C4⋊Dic61C2×D411yescharacteristic
142#11758196—21no—
143#11758196—21no—
144C4×Dic61—21no—
145C4×Dic61—21no—
146C4×Dic61—21no—
147#11758194—42no—
148#11758194—42no—
149C2×#92418—22no—
150C2×#92418—22no—
151C2×#92418—22no—
152C8×D61—44no—
153C23×C61—42no—
154C23×C61—82no—
155C23×C61—82no—
156C23×C61—82no—
157C23×C61#12111yescharacteristic
158C22×D61—21no—
159Q8×C61—42no—
160Q8×C61—42no—
161Q8×C61—82no—
162D4×C61—42no—
163D4×C61—42no—
164D4×C61—44no—
165D4×C61—82no—
166D4×C61—82no—
167D4×C61—82no—
168D4×C61—82no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—42no—
172C2×C4×C61—82no—
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61#12111yescharacteristic
175C2×C4×C61#12111yescharacteristic
176C2×C4×C61C2×D812yes—
177C2×C4×C61#13512yes—
178C2×C4×C61C22×D411yescharacteristic
179C2×C4×C61C22×D411yescharacteristic
180C2×C4×C61C2×#4111yescharacteristic
181C22⋊D61—24no—
182C2×Dic61—21no—
183C2×Dic61—21no—
184C2×Dic61—21no—
185D244—21no—
186D244C2×D812yes—
187C4×D61—24no—
188C4×D61—42no—
189C4×D61—42no—
190Dic122—21no—
191Dic122C2×D812yes—
192C8×C61—22no—
193C8×C61—22no—
194#92418—42no—
195#92418—42no—
196#92418—42no—
197#19276—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—82no—
201C22×C61—82no—
202C22×C61—82no—
203C22×C61—164no—
204C22×C61#46612yes—
205C22×C61C2×#12111yescharacteristic
206C2×D61—22no—
207C2×D61—41no—
208C2×D61—41no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—82no—
212C4×C61#46612yes—
213C4×C61#47112yes—
214C4×C61C22×D811yescharacteristic
215C4×C61C2×#13511yescharacteristic
216C4×C61#57512yes—
217Dic61—22no—
218Dic61—41no—
219Dic61—41no—
220C2×#486D6111yescharacteristic, complemented
221#3039—611no—
222#3038—611no—
223C8×#41—611no—
224#1682—614no—
225#1681—614no—
226#1567—611no—
227#1546—611no—
228#1545—611no—
229C2×C61—22no—
230C2×C61—162no—
231C2×C61#304011yescharacteristic
232C2×C61C2×#46611yescharacteristic
233C2×C61C2×#47111yescharacteristic
234D61—42no—
235C2×#137—1221no—
236C2×AGL(1,ℤ/8)—1221no—
237C2×C4⋊Q8C2×D6111yescharacteristic
238C2×#127C2×D6111yescharacteristic
239C4×#41—611no—
240#486—21no—
241#486—21no—
242#486C2×D6114yes—
243#462—612no—
244#462—1221no—
245#461—612no—
246#461—1221no—
247#460—612no—
248#460—1221no—
249#459—612no—
250#459—1221no—
251C8×Q8—612no—
252C8×D4—612no—
253C8×D4—1222no—
254#431—611no—
255#431—1221no—
256#423—611no—
257#423—611no—
258#419—1221no—
259#418—1221no—
260C2×#102—22no—
261C2×C4×C8—1221no—
262C2×C4×C8C2×D6111yescharacteristic
263#335—614no—
264#332—614no—
265#331—614no—
266C61#1927611yescharacteristic, complemented, Sylow
267C2×#41—611no—
268C22×Q8—21no—
269C22×D4—21no—
270C22×D4—21no—
271C22×D4—21no—
272#137—2444no—
273AGL(1,ℤ/8)—2444no—
274C2×Q16—1222no—
275C2×SD16—1222no—
276C2×SD16—1222no—
277C2×D8—1222no—
278C2×M16—1221no—
279C2×M16—1221no—
280C22×C8—1222no—
281C22×C8C22⋊D6112yes—
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8C22×D6112yes—
286#127—21no—
287#127—21no—
288#127—21no—
289#127C22×D6112yes—
290C4×Q8—611no—
291C4×Q8—611no—
292C4×D4—611no—
293C4×D4—611no—
294C4×D4—1221no—
295C4×D4—1221no—
296#117—611no—
297#117—1221no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C42—1221no—
301C2×C42C22×D6111yescharacteristic
302C4⋊C8—612no—
303C4⋊C8—612no—
304C4⋊C8—612no—
305C4⋊C8—1222no—
306C4⋊C8—1221no—
307C4⋊C8—1221no—
308#103—1222no—
309#103—1222no—
310#102—24no—
311#102—24no—
312#102—1222no—
313#102—1222no—
314C22⋊C8—1221no—
315C22⋊C8—1222no—
316C4×C8—1222no—
317C4×C8—1221no—
318C4×C8C22⋊D6111yescharacteristic
319C4×C8C22⋊D6111yescharacteristic
320C4×C8C22×D6112yes—
321C24—41no—
322#41—614no—
323#41—1222no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—611no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—44no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—82no—
339C2×D4—611no—
340C2×D4—1221no—
341C22×C4—41no—
342C22×C4—1221no—
343C22×C4—1221no—
344C22×C4—1221no—
345C22×C4D4×D6111yescharacteristic
346C22×C4#1175822411yescharacteristic
347C22×C4C2×C22⋊D6111yescharacteristic
348Q16—2444no—
349SD16—2444no—
350SD16—2444no—
351D8—2444no—
352M16—2442no—
353M16—2442no—
354C2×C8—42no—
355C2×C8—1222no—
356C2×C8—1222no—
357C2×C8—1222no—
358C2×C8—2444no—
359C2×C8C2×C22⋊D6112yes—
360C2×C8C2×C22⋊D6112yes—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—611no—
366C4⋊C4—611no—
367C4⋊C4—1221no—
368C4⋊C4—1221no—
369C22⋊C4—1221no—
370C22⋊C4—1221no—
371C22⋊C4—1221no—
372C42—1221no—
373C42—1221no—
374C42—1221no—
375C42D4×D6111yescharacteristic
376C42#1175822411yescharacteristic
377C42C2×C22⋊D6111yescharacteristic
378C42C23×D6111yescharacteristic
379C23—42no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—1221no—
384C23#1451974811yescharacteristic
385Q8—42no—
386Q8—42no—
387Q8—82no—
388Q8—612no—
389Q8—1221no—
390D4—42no—
391D4—42no—
392D4—44no—
393D4—82no—
394D4—82no—
395D4—82no—
396D4—82no—
397D4—612no—
398D4—1224no—
399D4—1221no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—42no—
403C2×C4—82no—
404C2×C4—1221no—
405C2×C4—1224no—
406C2×C4—1221no—
407C2×C4—1221no—
408C2×C4—2442no—
409C2×C4—2442no—
410C2×C4#1451974811yescharacteristic
411C2×C4#1451974811yescharacteristic
412C2×C4#1451974811yescharacteristic
413C2×C4D8×D6112yes—
414C2×C4#1451979612yes—
415C2×C4C2×D4×D6111yescharacteristic
416C2×C4C22×C22⋊D6111yescharacteristic
417C2×C4C2×#1175822411yescharacteristic
418C8—22no—
419C8—22no—
420C8—2442no—
421C8—2442no—
422C8—2442no—
423C22—21no—
424C22—21no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—164no—
429C22—1222no—
430C22—2441no—
431C22—2441no—
432C22#1683999512yes—
433C22C2×#1451974811yescharacteristic
434C4—21no—
435C4—21no—
436C4—82no—
437C4—1222no—
438C4—2441no—
439C4—2441no—
440C4#1683999512yes—
441C4#1684002212yes—
442C4C2×D8×D6111yescharacteristic
443C4C2×#1451979611yescharacteristic
444C4#1684056012yes—
445C2—22no—
446C2—162no—
447C2—2442no—
448C2#2145530911yescharacteristic
449C2C2×#1683999511yescharacteristic
450C2C2×#1684002211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#486C2
socleC22×C61C2×#121
prime radical#21455295C2
hypercenterC2×#486D61
hypocenterC61#19276
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. #21455302C2semidirect
3. C8×#11758222C2semidirect
4. #21445260C2semidirect
5. #21445253C2semidirect
6. #21444578C2semidirect
7. #21444491C2semidirect
8. #21444487C2semidirect
9. C2×C61×#486C2semidirect
14. #16839969C22semidirect
16. #16839964C22semidirect
21. C8×Dic122C22semidirect
22. C8×D244C22semidirect
31. #16839515C22semidirect
32. #16839508C22semidirect
220. C2×#486D61semidirect
266. C61#19276semidirect

Maximal subgroups

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

subgroupindex
#214553022
#214552952
C8×#117582222
#214452602
#214452532
#214445782
#214444912
#214444872
C2×C61×#4862
#1927661