Finite Group Databasebeta

← #29665508

Subgroups — #29665508

Order 15616.

608 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455628C211yesmaximal, characteristic
2#21455601C211yesmaximal, characteristic
3#21454927C211yesmaximal, characteristic
4#21445274C212yesmaximal, complemented
5#21445267C212yesmaximal, complemented
6#21445220C212yesmaximal, complemented
7#21445215C212yesmaximal, complemented
8#21444594C211yesmaximal, characteristic, complemented
9#21444509C211yesmaximal, characteristic, complemented
10C2×#16839511C211yesmaximal, characteristic, complemented
11C61×#3230C211yesmaximal, characteristic, complemented
12C4×#11758222C2211yescharacteristic
13#16840066—21no—
14#16840059—21no—
15#16840054C2212yes—
16#16840054C2211yescharacteristic
17#16840049C2212yes—
18#16840049C2211yescharacteristic
19#16840044—21no—
20#16840044C2211yescharacteristic
21#16840039—21no—
22#16840039C2211yescharacteristic
23C8×Dic122C2212yes—
24C8×D244C2212yes—
25#16839825—22no—
26#16839814—21no—
27#16839802—21no—
28#16839798—21no—
29#16839786C2211yescharacteristic
30C2×C4⋊#92418C2211yescharacteristic
31C2×C4⋊#92418C2211yescharacteristic
32C4⋊C8×D61—21no—
33#16839761—21no—
34#16839756—21no—
35#16839515C2212yes—
36#16839515C2212yes—
37#16839511C2212yes—
38#16839511C2212yes—
39#16839511C2212yes—
40#16839508C2212yes—
41C61×#533C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#500C2212yes—
44C61×#487C2212yes—
45C61×#484—22no—
46C61×#477—21no—
47C61×#477—21no—
48C61×#431C2211yescharacteristic
49C2×#11758222D411yescharacteristic
50C2×#11758194—21no—
51C22×#92418—21no—
52C22×#92418—21no—
53C2×C8×D61—22no—
54C4×Dic122D411yescharacteristic
55C4×Dic122C2311yescharacteristic
56C4×D244D411yescharacteristic
57C4×D244C2311yescharacteristic
58C4×C22⋊D61—21no—
59C4×C22⋊D61—21no—
60#14519736D411yescharacteristic
61#14519734—21no—
62C42×D61—21no—
63#14519710—22no—
64#14519710—21no—
65#14519710—21no—
66#14519708—22no—
67#14519708—21no—
68#14519708—21no—
69C4⋊#92418—21no—
70C4⋊#92418—21no—
71C4⋊#92418—21no—
72C4⋊#92418C2312yes—
73C4⋊#92418C2312yes—
74C4⋊#92418C2311yescharacteristic
75C4⋊#92418C2311yescharacteristic
76#14519706—22no—
77#14519700—21no—
78#14519700—21no—
79#14519700—22no—
80#14519697—21no—
81#14519697—21no—
82#14519697—22no—
83#14519689—22no—
84C8×Dic61—22no—
85C22×Q8×C61—21no—
86C4⋊Q8×C61—21no—
87C4⋊Q8×C61—21no—
88C4⋊Q8×C61—21no—
89C4⋊Q8×C61C2312yes—
90C4⋊Q8×C61C2311yescharacteristic
91C4⋊Q8×C61C2311yescharacteristic
92#125×C61—21no—
93C22⋊Q8×C61—21no—
94C22⋊Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×Q8×C61—21no—
97#117×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C42×C61C2311yescharacteristic
101#107×C61—22no—
102#107×C61—22no—
103C4⋊C8×C61D411yescharacteristic
104C4⋊C8×C61D411yescharacteristic
105#103×C61—22no—
106#103×C61—22no—
107C22⋊C8×C61D412yes—
108C4×C8×C61C2312yes—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—41no—
113C2×Q8×C61—82no—
114C22×C4×C61—41no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61#4111yescharacteristic
118C2×C22⋊D61—21no—
119#11758222—22no—
120#11758222Q1614yes—
121C2×D244C2×D411yescharacteristic
122C2×C4×D61—21no—
123C2×C4×D61—21no—
124C2×C4×D61—21no—
125C2×Dic122C2×D411yescharacteristic
126C2×C8×C61C2×D412yes—
127C2×C8×C61C2×D412yes—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—41no—
135C4⋊C4×C61—41no—
136C4⋊C4×C61—41no—
137C4⋊C4×C61—41no—
138C22⋊C4×C61—41no—
139C42×C61—41no—
140C42×C61C2×D411yescharacteristic
141C42×C61C2×D411yescharacteristic
142C42×C61#4111yescharacteristic
143C42×C61C2411yescharacteristic
144C22⋊Dic61—21no—
145#11758198—21no—
146#11758198—21no—
147C4⋊Dic61C2×D411yescharacteristic
148C4⋊Dic61C2×D411yescharacteristic
149#11758196—21no—
150#11758196—21no—
151C4×Dic61—21no—
152C4×Dic61—21no—
153C4×Dic61—21no—
154#11758194—42no—
155C2×#92418—22no—
156C2×#92418—21no—
157C2×#92418—22no—
158C2×#92418—21no—
159C2×#92418—42no—
160C2×#92418—81no—
161C8×D61—82no—
162C23×C61#12111yescharacteristic
163C22×D61—21no—
164Q8×C61—41no—
165Q8×C61—41no—
166Q8×C61—82no—
167Q8×C61—81no—
168Q8×C61—82no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—42no—
175C2×C4×C61—82no—
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61C2×Q1612yes—
180C2×C4×C61#13512yes—
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C2×#4111yescharacteristic
184C22⋊D61—24no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—21no—
188D244—21no—
189D244C2×Q1612yes—
190C4×D61—24no—
191C4×D61—41no—
192C4×D61—41no—
193C4×D61—81no—
194Dic122—21no—
195Dic122C2×Q1612yes—
196C8×C61—22no—
197C8×C61—22no—
198#92418—41no—
199#92418—41no—
200#92418—42no—
201#92418—42no—
202#19417—611nomaximal, Sylow
203C22×C61—21no—
204C22×C61—21no—
205C22×C61#46712yes—
206C22×C61C2×#12111yescharacteristic
207C2×D61—22no—
208C2×D61—41no—
209C2×D61—41no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—81no—
214C4×C61—82no—
215C4×C61—81no—
216C4×C61#46712yes—
217C4×C61C22×Q1611yescharacteristic
218C4×C61C2×#13511yescharacteristic
219C4×C61#57711yescharacteristic
220C4×C61#57811yescharacteristic
221Dic61—22no—
222Dic61—41no—
223Dic61—41no—
224#3230D6111yescharacteristic, complemented
225#3079—611no—
226#3076—611no—
227#2988—611no—
228#1684—612no—
229#1683—612no—
230#1676—612no—
231#1675—612no—
232#1571—611no—
233#1550—611no—
234C2×#333—611no—
235C2×C61—22no—
236C2×C61#304611yescharacteristic
237C2×C61C2×#46711yescharacteristic
238C2×C61#306811yescharacteristic
239D61—42no—
240#533C2×D6111yescharacteristic
241C2×C4⋊Q8C2×D6111yescharacteristic
242C4×#41—611no—
243#500C2×D6112yes—
244#487C2×D6112yes—
245#484—22no—
246#479—1221no—
247#478—1221no—
248#477—21no—
249#477—21no—
250#477—611no—
251#477—612no—
252#476—611no—
253#476—612no—
254#475—611no—
255#475—1221no—
256#474—611no—
257#474—1221no—
258C8×Q8—612no—
259C8×D4—612no—
260C8×D4—1222no—
261#431—1221no—
262#431C2×D6111yescharacteristic
263#428—1221no—
264#427—1221no—
265#423—611no—
266C2×C4⋊C8—611no—
267C2×C4⋊C8—611no—
268C2×C4⋊C8—1221no—
269#417—1221no—
270#416—1221no—
271#335—612no—
272#335—612no—
273#333—612no—
274#333—612no—
275#333—612no—
276#332—612no—
277C61#1941711yescharacteristic, complemented, Sylow
278C2×#41—611no—
279C22×Q8—21no—
280C2×M16—1221no—
281C22×C8—1221no—
282C22×C8—1221no—
283C22×C8—1222no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—21no—
287C4⋊Q8C22×D6112yes—
288C4⋊Q8C22×D6111yescharacteristic
289C4⋊Q8C22×D6111yescharacteristic
290#125—21no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8—611no—
296C4×Q8—611no—
297C4×D4—611no—
298C4×D4—611no—
299C4×D4—1221no—
300C4×D4—1221no—
301#117—21no—
302#117—611no—
303#117—1221no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C42—1221no—
307C2×C42C22×D6111yescharacteristic
308#107—22no—
309#107—22no—
310#107—1221no—
311#107—1221no—
312#107—1222no—
313#106—1221no—
314#106—1221no—
315#106—1222no—
316C4⋊C8—612no—
317C4⋊C8—612no—
318C4⋊C8—611no—
319C4⋊C8—611no—
320C4⋊C8—1222no—
321C4⋊C8—1221no—
322C4⋊C8—1221no—
323C4⋊C8—1221no—
324C4⋊C8C22⋊D6111yescharacteristic
325C4⋊C8C22⋊D6111yescharacteristic
326#103—22no—
327#103—22no—
328#103—1221no—
329#103—1221no—
330#103—1222no—
331#102—1221no—
332#102—1221no—
333#102—1222no—
334C22⋊C8—1222no—
335C22⋊C8C22⋊D6112yes—
336C4×C8—1222no—
337C4×C8C22×D6112yes—
338#41—614no—
339#41—1222no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—22no—
343C2×Q8—41no—
344C2×Q8—82no—
345C2×Q8—611no—
346C2×D4—611no—
347C2×D4—1221no—
348C22×C4—41no—
349C22×C4—1221no—
350C22×C4—1221no—
351C22×C4—1221no—
352C22×C4D4×D6111yescharacteristic
353C22×C4#1175822411yescharacteristic
354C22×C4C2×C22⋊D6111yescharacteristic
355M16—2442no—
356C2×C8—1221no—
357C2×C8—1222no—
358C2×C8—1221no—
359C2×C8—1222no—
360C2×C8—4882no—
361C2×C8—4881no—
362C2×C8—2442no—
363C2×C8C2×C22⋊D6112yes—
364C2×C8C2×C22⋊D6112yes—
365C4⋊C4—22no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—611no—
376C4⋊C4—611no—
377C4⋊C4—1221no—
378C4⋊C4—1221no—
379C22⋊C4—41no—
380C22⋊C4—1221no—
381C22⋊C4—1221no—
382C22⋊C4—1221no—
383C42—41no—
384C42—1221no—
385C42—1221no—
386C42—1221no—
387C42D4×D6111yescharacteristic
388C42#1175822411yescharacteristic
389C42C2×C22⋊D6111yescharacteristic
390C42C23×D6111yescharacteristic
391C23—1221no—
392C23#1451974811yescharacteristic
393Q8—41no—
394Q8—41no—
395Q8—81no—
396Q8—82no—
397Q8—82no—
398Q8—612no—
399Q8—1221no—
400D4—612no—
401D4—1224no—
402D4—1221no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—82no—
410C2×C4—1221no—
411C2×C4—1224no—
412C2×C4—1221no—
413C2×C4—1221no—
414C2×C4—4881no—
415C2×C4—2441no—
416C2×C4—2441no—
417C2×C4#1451974811yescharacteristic
418C2×C4#1451974811yescharacteristic
419C2×C4#1451974811yescharacteristic
420C2×C4Q16×D6112yes—
421C2×C4#1451980012yes—
422C2×C4C2×D4×D6111yescharacteristic
423C2×C4C22×C22⋊D6111yescharacteristic
424C2×C4C2×#1175822411yescharacteristic
425C8—22no—
426C8—22no—
427C8—2441no—
428C8—2441no—
429C8—2442no—
430C8—2442no—
431C22—21no—
432C22—21no—
433C22—1222no—
434C22—2441no—
435C22—2441no—
436C22#1684000012yes—
437C22C2×#1451974811yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—81no—
442C4—82no—
443C4—81no—
444C4—1222no—
445C4—2441no—
446C4—2441no—
447C4#1684000012yes—
448C4C2×Q16×D6111yescharacteristic
449C4C2×#1451980011yescharacteristic
450C4#1684057011yescharacteristic
451C4#1684057711yescharacteristic
452C2—22no—
453C2—2442no—
454C2#2145536511yescharacteristic
455C2C2×#1684000011yescharacteristic
456C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3230C2
socleC22×C61C2×#121
prime radicalC2×#11758222D4
hypercenter#3230D61
hypocenterC61#19417
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
4. #21445274C2semidirect
5. #21445267C2semidirect
6. #21445220C2semidirect
7. #21445215C2semidirect
8. #21444594C2semidirect
9. #21444509C2semidirect
10. C2×#16839511C2semidirect
11. C61×#3230C2semidirect
224. #3230D61semidirect
277. C61#19417semidirect

Maximal subgroups

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

subgroupindex
#214556282
#214556012
#214549272
#214452742
#214452672
#214452202
#214452152
#214445942
#214445092
C2×#168395112
C61×#32302
#1941761