Finite Group Databasebeta

← #30612949

Subgroups — #30612949

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462633C211yesmaximal, characteristic, complemented
2#21462544C211yesmaximal, characteristic, complemented
3#21461467C212yesmaximal, complemented
4#21461463C212yesmaximal, complemented
5#21461403C212yesmaximal, complemented
6#21461399C212yesmaximal, complemented
7#21461385C211yesmaximal, characteristic, complemented
8#21461180C211yesmaximal, characteristic, complemented
9#21461175C211yesmaximal, characteristic, complemented
10#21461017C211yesmaximal, characteristic, complemented
11#21456417C211yesmaximal, characteristic
12C2×#14519865—21no—
13#16840813C2211yescharacteristic
14Q8×#92420C2211yescharacteristic
15#16840797C2211yescharacteristic
16#16840796—21no—
17#16840796C2212yes—
18#16840794—21no—
19#16840790—21no—
20#16840790C2212yes—
21#16840789—21no—
22D4×#92420C2211yescharacteristic
23#16840786—21no—
24#16840784C2211yescharacteristic
25#16840780C2211yescharacteristic
26#16840766C2211yescharacteristic
27#16840737—21no—
28#16840726—21no—
29C4×D61⋊C8C2211yescharacteristic
30#16840626C2212yes—
31#16840624C2212yes—
32Q8⋊#92420C2212yes—
33#16840610C2212yes—
34#16840610C2212yes—
35D4⋊#92420C2212yes—
36#16840603C2212yes—
37C8×#92420C2212yes—
38#16840313C2211yescharacteristic
39#16840291C2211yescharacteristic
40#16840168C411yescharacteristic
41#16840152C411yescharacteristic
42#16840127C411yescharacteristic
43#16840014C412yes—
44#16840011C412yes—
45#16839847C2212yes—
46#16839842C2212yes—
47#16839714C2211yescharacteristic
48C61×#489C411yescharacteristic
49#14519865—44no—
50C2×#11758213—21no—
51C2×#11758213—22no—
52C2×#11758217—21no—
53C2×D61⋊C8—22no—
54C2×D61⋊C8D411yescharacteristic
55C2×D61⋊C8D411yescharacteristic
56C22×#92420—21no—
57#14519844—21no—
58#14519844—21no—
59C4⋊#92420—21no—
60C4⋊#92420—21no—
61C4⋊#92420C2312yes—
62C4⋊#92420C2311yescharacteristic
63C4⋊#92420C2311yescharacteristic
64#14519834—21no—
65#14519834—21no—
66#14519834—22no—
67#14519834—22no—
68C22⋊#92420—21no—
69C22⋊#92420—21no—
70#14519832—21no—
71#14519832—21no—
72C4×#92420—21no—
73C4×#92420D411yescharacteristic
74C4×#92420D411yescharacteristic
75C4×#92420C2312yes—
76C4×#92420C2311yescharacteristic
77C4×#92420C2311yescharacteristic
78C2×#11758226—21no—
79C2×#11758201C2×C412yes—
80C2×#11758199C2×C412yes—
81C2×#11758189C2×C412yes—
82#14519774C2×C411yescharacteristic
83#14519770—21no—
84#14519769C2×C411yescharacteristic
85#14519766—21no—
86#14519759—21no—
87#14519753C2×C411yescharacteristic
88#14519752—21no—
89#14519752C2×C412yes—
90#14519748C2×C411yescharacteristic
91Q8×Dic61C2311yescharacteristic
92D4×Dic61C2311yescharacteristic
93#14519739—21no—
94#14519739C2312yes—
95#14519737—21no—
96#14519733—21no—
97C42×D61C2311yescharacteristic
98#14519710C2×C412yes—
99#14519708C2×C412yes—
100#14519699C2×C412yes—
101#14519699C2×C412yes—
102#14519688C2×C411yescharacteristic
103C8×Dic61C2312yes—
104C4⋊Q8×C61C2×C411yescharacteristic
105#124×C61C2×C411yescharacteristic
106#103×C61C2×C412yes—
107#102×C61C2×C412yes—
108#97×C61C2×C411yescharacteristic
109C2×Q8×C61—21no—
110C2×Q8×C61C22×C411yescharacteristic
111C2×D4×C61C22×C411yescharacteristic
112C2×C22⋊D61—21no—
113C22×Dic61—21no—
114#11758226—42no—
115#11758226—42no—
116C2×D244—22no—
117C2×D244—21no—
118C2×D244—21no—
119C2×D244C22×C412yes—
120C2×C4×D61—22no—
121C2×C4×D61C22×C411yescharacteristic
122C2×C4×D61C2×D411yescharacteristic
123C2×C4×D61#4111yescharacteristic
124#11758217—42no—
125C2×#92420—21no—
126C2×#92420—21no—
127C2×#92420—22no—
128C2×#92420—22no—
129C2×#92420—42no—
130C2×#92420C2×D411yescharacteristic
131C2×#92420C2×D411yescharacteristic
132C2×#92420C2×D411yescharacteristic
133C2×#92420C2×D411yescharacteristic
134#11758213—44no—
135#11758213—42no—
136D61⋊C8—22no—
137D61⋊C8—41no—
138D61⋊C8—44no—
139C2×C8×C61C22×C412yes—
140C4⋊C4×C61—21no—
141C4⋊C4×C61C22×C412yes—
142C22⋊C4×C61—21no—
143C22⋊C4×C61—21no—
144C42×C61C22×C411yescharacteristic
145C22⋊Dic61—21no—
146C22⋊Dic61—21no—
147#11758201—24no—
148#11758199—24no—
149#11758198—21no—
150#11758198—22no—
151#11758198—22no—
152C4⋊Dic61—21no—
153C4⋊Dic61C22×C411yescharacteristic
154C4⋊Dic61C22×C411yescharacteristic
155#11758196—21no—
156#11758196—21no—
157#11758196—21no—
158C4×Dic61—21no—
159C4×Dic61C2×D411yescharacteristic
160C4×Dic61#4111yescharacteristic
161C4×Dic61C2411yescharacteristic
162C2×#92418C22×C412yes—
163#11758189—24no—
164C23×C61—21no—
165C22×D61—22no—
166C22×D61C4×D411yescharacteristic
167Q8×C61—22no—
168Q8×C61—42no—
169D4×C61—22no—
170C2×C4×C61—21no—
171C2×C4×C61—22no—
172C2×C4×C61C4×D411yescharacteristic
173C2×C4×C61C4×D411yescharacteristic
174C2×C4×C61C23×C411yescharacteristic
175C22⋊D61—42no—
176C2×Dic61—21no—
177C2×Dic61—21no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61C4×D411yescharacteristic
181C2×Dic61C22×D411yescharacteristic
182C2×Dic61C2×#4111yescharacteristic
183D244—22no—
184D244—22no—
185D244—42no—
186D244—42no—
187D244—44no—
188C4×D61—22no—
189C4×D61—21no—
190C4×D61—21no—
191C4×D61—41no—
192C4×D61—42no—
193C4×D61—42no—
194#92420—21no—
195#92420—21no—
196#92420—21no—
197#92420—42no—
198#92420—42no—
199#92420—41no—
200#92420—41no—
201#92420AGL(1,ℤ/8)12yes—
202C8×C61—22no—
203#92418#13114yescomplemented
204#17964—611nomaximal, Sylow
205C22×C61—41no—
206C22×C61—41no—
207C22×C61—41no—
208C22×C61C2×C4×D411yescharacteristic
209C2×D61—21no—
210C2×D61—21no—
211C2×D61—21no—
212C2×D61—42no—
213C2×D61—42no—
214C2×D61—42no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—41no—
218C4×C61—42no—
219C4×C61C2×#13112yes—
220Dic61—21no—
221Dic61—21no—
222Dic61—41no—
223Dic61—41no—
224Dic61C2×AGL(1,ℤ/8)11yescharacteristic
225Dic61#57511yescharacteristic
226#3171—611no—
227#2966—611no—
228#2931—611no—
229#1594—612no—
230#1592—612no—
231C8×SD16—612no—
232C8×D8—612no—
233#1569—611no—
234#1496—611no—
235#1495—611no—
236C8×M16—611no—
237C2×C61—81no—
238C2×C61#293111yescharacteristic
239C2×C61C4×AGL(1,ℤ/8)11yescharacteristic
240C2×C61#295511yescharacteristic
241D61—41no—
242D61—82no—
243C2×#131—1221no—
244#532—611no—
245#529—611no—
246#501—611no—
247#496—611no—
248#490—611no—
249#489#692311yescharacteristic
250#469—612no—
251#468—612no—
252#446—611no—
253C8×Q8—611no—
254C4×SD16—612no—
255C4×D8—612no—
256#436—612no—
257#436—611no—
258#436—1221no—
259#435—1221no—
260C8×D4—611no—
261C8×D4—612no—
262C8×D4—1221no—
263C8×D4—1221no—
264#433—1221no—
265#432—611no—
266#431—611no—
267#423—611no—
268#408—1221no—
269#405—1221no—
270C4×M16—611no—
271C2×C4×C8—611no—
272#335—612no—
273#334—612no—
274Q8⋊C8—612no—
275D4⋊C8—612no—
276D4⋊C8—612no—
277D4⋊C8—612no—
278#322—612no—
279C82—612no—
280C61#1796411yescharacteristic, complemented, Sylow
281C2×#41—1221no—
282C2×SD16—612no—
283C2×D8—612no—
284#131—2444no—
285C2×M16—612no—
286C2×M16—1221no—
287C2×M16—1222no—
288C2×M16—1221no—
289C22×C8—611no—
290C22×C8—611no—
291C22×C8—1221no—
292C22×C8—1222no—
293C4⋊Q8C2×#692311yescharacteristic
294#127—611no—
295#126—1221no—
296#125—611no—
297#124—1221no—
298#124C2×#692311yescharacteristic
299#123—1221no—
300C22⋊Q8—611no—
301#121—611no—
302#121—612no—
303#121—1221no—
304C4×Q8—611no—
305C4×D4—611no—
306C4×D4—612no—
307C4×D4—1221no—
308C4×D4—1221no—
309#117—1221no—
310C2×C42—611no—
311#107—612no—
312#106—612no—
313C4⋊C8—611no—
314C4⋊C8—611no—
315C4⋊C8—612no—
316C4⋊C8—1221no—
317C4⋊C8—1221no—
318C4⋊C8—1221no—
319C4⋊C8—1221no—
320#103C2×#692312yes—
321#102—612no—
322#102—612no—
323#102C2×#692312yes—
324C22⋊C8—1221no—
325C22⋊C8—1221no—
326C22⋊C8—1222no—
327C22⋊C8—1222no—
328C22⋊C8—1221no—
329C22⋊C8—1221no—
330#97—611no—
331#97—1221no—
332#97—1221no—
333#97C2×#692311yescharacteristic
334C4×C8—611no—
335C4×C8—611no—
336C4×C8—611no—
337C4×C8—611no—
338C4×C8—612no—
339C4×C8—612no—
340C4×C8—1221no—
341#41—2442no—
342#41—2442no—
343C2×Q8—21no—
344C2×Q8C22×#692311yescharacteristic
345C2×D4—612no—
346C2×D4—1222no—
347C2×D4—1221no—
348C2×D4—1221no—
349C2×D4—1221no—
350C2×D4C22×#692311yescharacteristic
351C22×C4—611no—
352C22×C4—611no—
353C22×C4—611no—
354C22×C4—1222no—
355C22×C4—1221no—
356SD16—1224no—
357D8—1224no—
358M16—1224no—
359M16—2444no—
360M16—2442no—
361M16—2442no—
362C2×C8—611no—
363C2×C8—612no—
364C2×C8—611no—
365C2×C8—611no—
366C2×C8—611no—
367C2×C8—1222no—
368C2×C8—1222no—
369C2×C8—1221no—
370C2×C8—1221no—
371C2×C8—1222no—
372C2×C8—2444no—
373C2×C8—2442no—
374C2×C8—2441no—
375C2×C8C22×#692312yes—
376C4⋊C4—21no—
377C4⋊C4—611no—
378C4⋊C4—611no—
379C4⋊C4—1221no—
380C4⋊C4—1221no—
381C4⋊C4—1221no—
382C4⋊C4—1221no—
383C4⋊C4C22×#692312yes—
384C22⋊C4—21no—
385C22⋊C4—21no—
386C22⋊C4—1222no—
387C22⋊C4—1222no—
388C22⋊C4—1221no—
389C22⋊C4—1221no—
390C22⋊C4—1221no—
391C42—611no—
392C42—611no—
393C42—611no—
394C42—1221no—
395C42C22×#692311yescharacteristic
396C23—21no—
397C23—611no—
398C23—1222no—
399Q8—22no—
400Q8—42no—
401D4—22no—
402D4—1222no—
403D4—1222no—
404D4—2442no—
405D4—2442no—
406D4—2444no—
407D4—2442no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—611no—
411C2×C4—611no—
412C2×C4—611no—
413C2×C4—1221no—
414C2×C4—1221no—
415C2×C4—1222no—
416C2×C4—1221no—
417C2×C4—1221no—
418C2×C4—2441no—
419C2×C4—2442no—
420C2×C4—2442no—
421C2×C4—2441no—
422C2×C4—2441no—
423C2×C4D4×#692311yescharacteristic
424C2×C4D4×#692311yescharacteristic
425C2×C4C23×#692311yescharacteristic
426C8—22no—
427C8—612no—
428C8—614no—
429C8—1221no—
430C8—1221no—
431C8—1221no—
432C8—2441no—
433C8—2442no—
434C8—2441no—
435C8—2442no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—1221no—
440C22—1221no—
441C22—1221no—
442C22—2442no—
443C22—2442no—
444C22—2442no—
445C22C2×D4×#692311yescharacteristic
446C4—21no—
447C4—21no—
448C4—41no—
449C4—42no—
450C4—611no—
451C4—611no—
452C4—1221no—
453C4—1221no—
454C4—2441no—
455C4—2441no—
456C4D61⋊#13112yes—
457C2—81no—
458C2—4882no—
459C2—2441no—
460C2#2146254311yescharacteristic
461C2AGL(1,ℤ/8)×#692311yescharacteristic
462C2#2146259211yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#489C4
socleC22×C61C2×C4×D4
prime radical#21456417C2
cosocleC4×Dic61C24
hypercenter#489#6923
hypocenterC61#17964
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #21462633C2semidirect
2. #21462544C2semidirect
3. #21461467C2semidirect
4. #21461463C2semidirect
5. #21461403C2semidirect
6. #21461399C2semidirect
7. #21461385C2semidirect
8. #21461180C2semidirect
9. #21461175C2semidirect
10. #21461017C2semidirect
203. #92418#131semidirect
280. C61#17964semidirect

Maximal subgroups

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

subgroupindex
#214626332
#214625442
#214614672
#214614632
#214614032
#214613992
#214613852
#214611802
#214611752
#214610172
#214564172
#1796461