Finite Group Databasebeta

← #29663963

Subgroups — #29663963

Order 15616.

722 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455628C211yesmaximal, characteristic, complemented
2#21455295C211yesmaximal, characteristic
3C8×#11758222C211yesmaximal, characteristic, complemented
4#21445232C214yesmaximal, complemented
5#21445225C214yesmaximal, complemented
6#21444575C211yesmaximal, characteristic, complemented
7#21444500C211yesmaximal, characteristic, complemented
8#21444487C211yesmaximal, characteristic, complemented
9C2×C61×#486C211yesmaximal, characteristic, complemented
10C2×#14519803—21no—
11C4×#11758222C2211yescharacteristic
12#16840054C2212yescomplemented
13#16840049C2212yescomplemented
14#16840044—21no—
15#16840039—21no—
16#16839959—21no—
17#16839959C2212yes—
18#16839954—21no—
19#16839954C2212yes—
20C8×Dic122C2212yescomplemented
21C8×D244C2212yescomplemented
22C8×C22⋊D61—22no—
23#16839815C2211yescharacteristic
24#16839813—21no—
25#16839802—21no—
26#16839786C2211yescharacteristic
27#16839786C2211yescharacteristic
28#16839767—21no—
29#16839761—21no—
30C4×C8×D61—21no—
31#16839513C2214yescomplemented
32#16839511C2214yescomplemented
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#14519803—44no—
43C2×#11758200—22no—
44C2×#11758199—22no—
45C2×#11758194—21no—
46C2×#11758194—21no—
47C2×C8×D61—22no—
48C4×Dic122D411yescharacteristic
49C4×Dic122C2311yescharacteristic
50C4×D244D411yescharacteristic
51C4×D244C2311yescharacteristic
52C4×C22⋊D61—21no—
53C4×C22⋊D61—21no—
54#14519736D411yescharacteristic
55#14519734—21no—
56C42×D61—21no—
57#14519710—22no—
58#14519708—22no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61C4⋊#92418C2312yes—
62C4⋊#92418C2312yes—
63C4⋊#92418C2312yes—
64#14519706—22no—
65#14519700—22no—
66#14519698—22no—
67#14519698—22no—
68#14519697—22no—
69C22⋊#92418—21no—
70#14519689—22no—
71C4×#92418—21no—
72C8×Dic61—22no—
73C22×Q8×C61—21no—
74C22×D4×C61—21no—
75C22×D4×C61—21no—
76C22×D4×C61—21no—
77C22×C8×C61D412yes—
78C4⋊Q8×C61—21no—
79C4⋊Q8×C61—21no—
80C4⋊Q8×C61—21no—
81C4⋊Q8×C61C2312yes—
82#127×C61—21no—
83#127×C61—21no—
84#127×C61—21no—
85#127×C61C2312yes—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C42×C61C2311yescharacteristic
89#102×C61—24no—
90#102×C61—24no—
91C4×C8×C61D411yescharacteristic
92C4×C8×C61D411yescharacteristic
93C4×C8×C61C2312yes—
94C24×C61—41no—
95C2×Q8×C61—22no—
96C2×Q8×C61—41no—
97C2×Q8×C61—82no—
98C2×D4×C61—22no—
99C2×D4×C61—22no—
100C2×D4×C61—22no—
101C2×D4×C61—44no—
102C2×D4×C61—41no—
103C2×D4×C61—41no—
104C2×D4×C61—41no—
105C2×D4×C61—42no—
106C2×D4×C61—42no—
107C2×D4×C61—82no—
108C22×C4×C61—41no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C2×C22⋊D61—21no—
113#11758222—22no—
114#11758222SD1614yes—
115C2×D244C2×D411yescharacteristic
116C2×C4×D61—21no—
117C2×C4×D61—21no—
118C2×C4×D61—21no—
119C2×Dic122C2×D411yescharacteristic
120C2×C8×C61—42no—
121C2×C8×C61C2×D412yes—
122C2×C8×C61C2×D412yes—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—41no—
126C4⋊C4×C61—41no—
127C42×C61C2×D411yescharacteristic
128C42×C61C2×D411yescharacteristic
129C42×C61#4111yescharacteristic
130C42×C61C2411yescharacteristic
131C22⋊Dic61—21no—
132#11758200—44no—
133#11758199—44no—
134#11758198—21no—
135#11758198—21no—
136C4⋊Dic61C2×D411yescharacteristic
137C4⋊Dic61C2×D411yescharacteristic
138#11758196—21no—
139#11758196—21no—
140C4×Dic61—21no—
141C4×Dic61—21no—
142C4×Dic61—21no—
143#11758194—42no—
144#11758194—42no—
145C2×#92418—22no—
146C2×#92418—22no—
147C2×#92418—22no—
148C8×D61—44no—
149C23×C61—42no—
150C23×C61—82no—
151C23×C61—82no—
152C23×C61—82no—
153C23×C61#12111yescharacteristic
154C22×D61—21no—
155Q8×C61—82no—
156Q8×C61—82no—
157D4×C61—42no—
158D4×C61—42no—
159D4×C61—44no—
160D4×C61—82no—
161D4×C61—82no—
162D4×C61—82no—
163D4×C61—82no—
164C2×C4×C61—22no—
165C2×C4×C61—22no—
166C2×C4×C61—42no—
167C2×C4×C61—82no—
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61C2×SD1612yes—
172C2×C4×C61#13512yes—
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C2×C4×C61C2×#4111yescharacteristic
176C22⋊D61—24no—
177C2×Dic61—21no—
178C2×Dic61—21no—
179C2×Dic61—21no—
180D244—21no—
181D244C2×SD1612yes—
182C4×D61—24no—
183C4×D61—42no—
184C4×D61—42no—
185Dic122—21no—
186Dic122C2×SD1612yes—
187C8×C61—22no—
188C8×C61—22no—
189#92418—42no—
190#92418—42no—
191#92418—42no—
192#19274—611nomaximal, Sylow
193C22×C61—21no—
194C22×C61—21no—
195C22×C61—82no—
196C22×C61—82no—
197C22×C61—82no—
198C22×C61—164no—
199C22×C61#46512yes—
200C22×C61C2×#12111yescharacteristic
201C2×D61—22no—
202C2×D61—41no—
203C2×D61—41no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—82no—
207C4×C61#46512yes—
208C4×C61#47212yes—
209C4×C61C22×SD1611yescharacteristic
210C4×C61C2×#13511yescharacteristic
211C4×C61#57512yes—
212Dic61—22no—
213Dic61—41no—
214Dic61—41no—
215C2×#486D6111yescharacteristic, complemented
216#3079—611no—
217#3038—611no—
218C8×#41—611no—
219#1678—614no—
220#1677—614no—
221#1566—611no—
222#1548—611no—
223#1545—611no—
224C2×C61—22no—
225C2×C61—162no—
226C2×C61#304011yescharacteristic
227C2×C61C2×#46511yescharacteristic
228C2×C61C2×#47211yescharacteristic
229D61—42no—
230C2×AGL(1,ℤ/8)—1221no—
231C2×C4⋊Q8C2×D6111yescharacteristic
232C2×#127C2×D6111yescharacteristic
233C4×#41—611no—
234#486—21no—
235#486—21no—
236#486C2×D6114yes—
237#477—612no—
238#476—612no—
239#475—1221no—
240#474—1221no—
241#460—612no—
242#460—1221no—
243#459—612no—
244#459—1221no—
245C8×Q8—612no—
246C8×D4—612no—
247C8×D4—1222no—
248#431—611no—
249#431—1221no—
250#428—1221no—
251#423—611no—
252#423—611no—
253#418—1221no—
254#417—1221no—
255C2×#102—22no—
256C2×C4×C8—1221no—
257C2×C4×C8C2×D6111yescharacteristic
258#334—614no—
259#333—614no—
260#331—614no—
261C61#1927411yescharacteristic, complemented, Sylow
262C2×#41—611no—
263C22×Q8—21no—
264C22×D4—21no—
265C22×D4—21no—
266C22×D4—21no—
267AGL(1,ℤ/8)—2444no—
268C2×SD16—1222no—
269C2×D8—1222no—
270C2×M16—1221no—
271C2×M16—1221no—
272C22×C8—1222no—
273C22×C8C22⋊D6112yes—
274C4⋊Q8—21no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8C22×D6112yes—
278#127—21no—
279#127—21no—
280#127—21no—
281#127C22×D6112yes—
282C4×Q8—611no—
283C4×Q8—611no—
284C4×D4—611no—
285C4×D4—611no—
286C4×D4—1221no—
287C4×D4—1221no—
288#117—611no—
289#117—1221no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C42—1221no—
293C2×C42C22×D6111yescharacteristic
294#107—1222no—
295#106—1222no—
296C4⋊C8—612no—
297C4⋊C8—612no—
298C4⋊C8—612no—
299C4⋊C8—1222no—
300C4⋊C8—1221no—
301C4⋊C8—1221no—
302#103—1222no—
303#102—24no—
304#102—24no—
305#102—1222no—
306#102—1222no—
307#102—1222no—
308C22⋊C8—1221no—
309C22⋊C8—1222no—
310C4×C8—1222no—
311C4×C8—1221no—
312C4×C8C22⋊D6111yescharacteristic
313C4×C8C22⋊D6111yescharacteristic
314C4×C8C22×D6112yes—
315C24—41no—
316#41—614no—
317#41—1222no—
318C2×Q8—22no—
319C2×Q8—41no—
320C2×Q8—82no—
321C2×Q8—611no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—44no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—82no—
332C2×D4—611no—
333C2×D4—1221no—
334C22×C4—41no—
335C22×C4—1221no—
336C22×C4—1221no—
337C22×C4—1221no—
338C22×C4D4×D6111yescharacteristic
339C22×C4#1175822411yescharacteristic
340C22×C4C2×C22⋊D6111yescharacteristic
341SD16—2444no—
342D8—2444no—
343M16—2442no—
344M16—2442no—
345C2×C8—42no—
346C2×C8—1222no—
347C2×C8—1222no—
348C2×C8—1222no—
349C2×C8—2444no—
350C2×C8C2×C22⋊D6112yes—
351C2×C8C2×C22⋊D6112yes—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—611no—
357C4⋊C4—611no—
358C4⋊C4—1221no—
359C4⋊C4—1221no—
360C22⋊C4—1221no—
361C22⋊C4—1221no—
362C22⋊C4—1221no—
363C42—1221no—
364C42—1221no—
365C42—1221no—
366C42D4×D6111yescharacteristic
367C42#1175822411yescharacteristic
368C42C2×C22⋊D6111yescharacteristic
369C42C23×D6111yescharacteristic
370C23—42no—
371C23—82no—
372C23—82no—
373C23—82no—
374C23—1221no—
375C23#1451974811yescharacteristic
376Q8—82no—
377Q8—82no—
378Q8—612no—
379Q8—1221no—
380D4—42no—
381D4—42no—
382D4—44no—
383D4—82no—
384D4—82no—
385D4—82no—
386D4—82no—
387D4—612no—
388D4—1224no—
389D4—1221no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—42no—
393C2×C4—82no—
394C2×C4—1221no—
395C2×C4—1224no—
396C2×C4—1221no—
397C2×C4—1221no—
398C2×C4—2442no—
399C2×C4—2442no—
400C2×C4#1451974811yescharacteristic
401C2×C4#1451974811yescharacteristic
402C2×C4#1451974811yescharacteristic
403C2×C4SD16×D6112yes—
404C2×C4#1451979812yes—
405C2×C4C2×D4×D6111yescharacteristic
406C2×C4C22×C22⋊D6111yescharacteristic
407C2×C4C2×#1175822411yescharacteristic
408C8—22no—
409C8—22no—
410C8—2442no—
411C8—2442no—
412C8—2442no—
413C22—21no—
414C22—21no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22—164no—
419C22—1222no—
420C22—2441no—
421C22—2441no—
422C22#1683998812yes—
423C22C2×#1451974811yescharacteristic
424C4—21no—
425C4—21no—
426C4—82no—
427C4—1222no—
428C4—2441no—
429C4—2441no—
430C4#1683998812yes—
431C4#1684002912yes—
432C4C2×SD16×D6111yescharacteristic
433C4C2×#1451979811yescharacteristic
434C4#1684056012yes—
435C2—22no—
436C2—162no—
437C2—2442no—
438C2#2145530911yescharacteristic
439C2C2×#1683998811yescharacteristic
440C2C2×#1684002911yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#486C2
socleC22×C61C2×#121
prime radical#21455295C2
hypercenterC2×#486D61
hypocenterC61#19274
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. #21455628C2semidirect
3. C8×#11758222C2semidirect
4. #21445232C2semidirect
5. #21445225C2semidirect
6. #21444575C2semidirect
7. #21444500C2semidirect
8. #21444487C2semidirect
9. C2×C61×#486C2semidirect
12. #16840054C22semidirect
13. #16840049C22semidirect
20. C8×Dic122C22semidirect
21. C8×D244C22semidirect
31. #16839513C22semidirect
32. #16839511C22semidirect
215. C2×#486D61semidirect
261. C61#19274semidirect

Maximal subgroups

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

subgroupindex
#214556282
#214552952
C8×#117582222
#214452322
#214452252
#214445752
#214445002
#214444872
C2×C61×#4862
#1927461