Finite Group Databasebeta

← #29662753

Subgroups — #29662753

Order 15616.

642 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#21455873C212yesmaximal, complemented
2#21454785C211yesmaximal, characteristic, complemented
3#21445356C212yesmaximal, complemented
4#21445349C212yesmaximal, complemented
5#21445344C211yesmaximal, characteristic, complemented
6#21445339C211yesmaximal, characteristic, complemented
7#21445300C211yesmaximal, characteristic, complemented
8#21445295C211yesmaximal, characteristic, complemented
9#21444567C211yesmaximal, characteristic, complemented
10#21444481C212yesmaximal, complemented
11C61×#3169C211yesmaximal, characteristic, complemented
12C4×#11758222C2211yescharacteristic
13#16840083—22no—
14#16840054C2212yes—
15#16840049C2212yes—
16#16839959C2212yes—
17#16839954C2212yes—
18#16839905—22no—
19#16839892C2211yescharacteristic
20#16839892C2211yescharacteristic
21C4×#11758200—22no—
22C4×#11758199—22no—
23#16839841C2211yescharacteristic
24#16839841C2211yescharacteristic
25#16839833—21no—
26#16839833—21no—
27#16839815C2212yes—
28#16839815C2211yescharacteristic
29#16839813—21no—
30C4×#11758189—21no—
31#16839515C2212yes—
32#16839513C2211yescharacteristic
33#16839513C2211yescharacteristic
34#16839511C2212yes—
35#16839511C2212yes—
36#16839506C2212yes—
37#16839506C2212yes—
38C61×#532C2212yes—
39C61×#500C2211yescharacteristic
40C61×#493C2211yescharacteristic
41C61×#486C2212yes—
42C61×#466—21no—
43C61×#466—21no—
44C4×D8×C61—21no—
45C4×D8×C61—21no—
46C2×C4×C8×C61C2211yescharacteristic
47C2×#11758222D411yescharacteristic
48C2×#11758200—22no—
49C2×#11758199—22no—
50C2×#11758189—21no—
51C2×#11758189—21no—
52C4×Dic122D411yescharacteristic
53C4×Dic122C2311yescharacteristic
54C4×D244D411yescharacteristic
55C4×D244C2311yescharacteristic
56C4×C22⋊D61—21no—
57C4×C22⋊D61—21no—
58#14519736D411yescharacteristic
59#14519734—21no—
60C42×D61—21no—
61#14519710—22no—
62#14519708—22no—
63C4⋊#92418C2312yes—
64C4⋊#92418C2312yes—
65C4⋊#92418C2311yescharacteristic
66C4⋊#92418C2311yescharacteristic
67#14519706—21no—
68#14519706—21no—
69#14519700—22no—
70#14519698—22no—
71#14519698—22no—
72#14519697—22no—
73C22⋊#92418—22no—
74C22⋊#92418—21no—
75#14519689—21no—
76#14519689—21no—
77#14519687—21no—
78#14519687—21no—
79C4×#92418—21no—
80C4×#92418—22no—
81C2×#41×C61—22no—
82C2×D8×C61—21no—
83C2×D8×C61—21no—
84C22×C8×C61D411yescharacteristic
85C22×C8×C61D411yescharacteristic
86C4⋊Q8×C61C2312yes—
87#127×C61C2312yes—
88#124×C61—22no—
89#121×C61—22no—
90#121×C61—22no—
91C4×D4×C61—22no—
92C4×D4×C61—22no—
93C2×C42×C61C2311yescharacteristic
94#107×C61—21no—
95#107×C61—21no—
96#102×C61—22no—
97#102×C61—22no—
98C4×C8×C61D411yescharacteristic
99C4×C8×C61D411yescharacteristic
100C4×C8×C61C2311yescharacteristic
101C4×C8×C61C2311yescharacteristic
102#41×C61—84no—
103C2×Q8×C61—22no—
104C2×D4×C61—22no—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—42no—
108C22×C4×C61—42no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C2×C22⋊D61—21no—
113#11758222—42no—
114C2×D244C2×D411yescharacteristic
115C2×C4×D61—21no—
116C2×C4×D61—21no—
117C2×C4×D61—21no—
118C2×Dic122C2×D411yescharacteristic
119D8×C61—42no—
120D8×C61—42no—
121C2×C8×C61—22no—
122C2×C8×C61—41no—
123C2×C8×C61C2×D411yescharacteristic
124C2×C8×C61C2×D411yescharacteristic
125C2×C8×C61C2×D411yescharacteristic
126C2×C8×C61C2×D411yescharacteristic
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C22⋊C4×C61—42no—
130C22⋊C4×C61—42no—
131C42×C61C2×D411yescharacteristic
132C42×C61C2×D411yescharacteristic
133C42×C61#4111yescharacteristic
134C42×C61C2411yescharacteristic
135C22⋊Dic61—21no—
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—
147C2×#92418—21no—
148C2×#92418—21no—
149C2×#92418—22no—
150C2×#92418—22no—
151#11758189—42no—
152#11758189—42no—
153C23×C61—42no—
154C23×C61#12111yescharacteristic
155C22×D61—21no—
156Q8×C61—82no—
157D4×C61—42no—
158D4×C61—42no—
159D4×C61—42no—
160D4×C61—42no—
161D4×C61—82no—
162D4×C61—84no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61—42no—
168C2×C4×C61—84no—
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C2×#4111yescharacteristic
175C22⋊D61—42no—
176C2×Dic61—21no—
177C2×Dic61—21no—
178C2×Dic61—21no—
179D244—22no—
180C4×D61—42no—
181C4×D61—41no—
182C4×D61—41no—
183C4×D61—42no—
184Dic122—22no—
185C8×C61—21no—
186C8×C61—21no—
187C8×C61—21no—
188C8×C61#13512yes—
189#92418—41no—
190#92418—41no—
191#92418—42no—
192#92418—42no—
193#19160—611nomaximal, Sylow
194C22×C61—21no—
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—82no—
198C22×C61—82no—
199C22×C61—82no—
200C22×C61C2×#12111yescharacteristic
201C2×D61—41no—
202C2×D61—41no—
203C2×D61—41no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—82no—
208C4×C61#47212yes—
209C4×C61C2×#13511yescharacteristic
210C4×C61#57512yes—
211C4×C61#57511yescharacteristic
212Dic61—41no—
213Dic61—41no—
214Dic61—41no—
215#3169D6111yescharacteristic, complemented
216#3106—612no—
217#2965—611no—
218#1698—612no—
219#1697—612no—
220#1696—611no—
221#1695—611no—
222#1688—611no—
223#1687—611no—
224#1564—611no—
225#1543—612no—
226C2×C61—41no—
227C2×C61—162no—
228C2×C61#303511yescharacteristic
229C2×C61#305411yescharacteristic
230C2×C61C2×#47211yescharacteristic
231D61—81no—
232#532C2×D6112yes—
233C4×#41—611no—
234#500C2×D6111yescharacteristic
235#493C2×D6111yescharacteristic
236#486C2×D6112yes—
237#482—1222no—
238#477—612no—
239#476—612no—
240#466—21no—
241#466—21no—
242#460—612no—
243#459—612no—
244#449—1222no—
245#446—611no—
246#446—611no—
247C4×SD16—1222no—
248C4×D8—21no—
249C4×D8—21no—
250C4×D8—1222no—
251#436—611no—
252#436—611no—
253#435—1221no—
254#435—1221no—
255#431—611no—
256#431—612no—
257#431—1221no—
258C4×M16—1221no—
259C2×C4×C8C2×D6111yescharacteristic
260#335—612no—
261#334—611no—
262#334—611no—
263#333—612no—
264#333—612no—
265#331—612no—
266#331—612no—
267C61#1916011yescharacteristic, complemented, Sylow
268C2×#41—22no—
269C2×#41—611no—
270C2×SD16—1222no—
271C2×D8—21no—
272C2×D8—21no—
273C2×D8—1222no—
274C2×M16—1221no—
275C2×M16—1221no—
276C22×C8C22⋊D6111yescharacteristic
277C22×C8C22⋊D6111yescharacteristic
278C4⋊Q8C22×D6112yes—
279#127C22×D6112yes—
280#124—22no—
281#121—22no—
282#121—22no—
283C4×Q8—611no—
284C4×Q8—611no—
285C4×D4—22no—
286C4×D4—22no—
287C4×D4—611no—
288C4×D4—611no—
289C4×D4—1221no—
290C4×D4—1221no—
291#117—611no—
292#117—1221no—
293C2×C42—1221no—
294C2×C42C22×D6111yescharacteristic
295#107—21no—
296#107—21no—
297#107—1222no—
298#106—1222no—
299C4⋊C8—611no—
300C4⋊C8—611no—
301C4⋊C8—612no—
302C4⋊C8—612no—
303C4⋊C8—1221no—
304C4⋊C8—1221no—
305#103—1222no—
306#102—22no—
307#102—22no—
308#102—1222no—
309#102—1222no—
310#102—1222no—
311C22⋊C8—1222no—
312C22⋊C8—1221no—
313C22⋊C8—1221no—
314C22⋊C8—1221no—
315#97—1221no—
316#97—1221no—
317C4×C8—1222no—
318C4×C8—1221no—
319C4×C8C22⋊D6111yescharacteristic
320C4×C8C22⋊D6111yescharacteristic
321C4×C8C22×D6111yescharacteristic
322C4×C8C22×D6111yescharacteristic
323#41—84no—
324#41—2442no—
325C2×Q8—22no—
326C2×Q8—611no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—42no—
331C2×D4—611no—
332C2×D4—1221no—
333C22×C4—42no—
334C22×C4—1221no—
335C22×C4—1221no—
336C22×C4—1221no—
337C22×C4D4×D6111yescharacteristic
338C22×C4#1175822411yescharacteristic
339C22×C4C2×C22⋊D6111yescharacteristic
340SD16—2444no—
341D8—42no—
342D8—42no—
343D8—2444no—
344M16—2442no—
345M16—2442no—
346C2×C8—22no—
347C2×C8—41no—
348C2×C8—1222no—
349C2×C8—1222no—
350C2×C8—1221no—
351C2×C8—1221no—
352C2×C8C2×C22⋊D6111yescharacteristic
353C2×C8C2×C22⋊D6111yescharacteristic
354C2×C8C2×C22⋊D6111yescharacteristic
355C2×C8C2×C22⋊D6111yescharacteristic
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—611no—
359C4⋊C4—611no—
360C4⋊C4—1221no—
361C4⋊C4—1221no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—1221no—
365C22⋊C4—1221no—
366C22⋊C4—1221no—
367C42—1221no—
368C42—1221no—
369C42—1221no—
370C42D4×D6111yescharacteristic
371C42#1175822411yescharacteristic
372C42C2×C22⋊D6111yescharacteristic
373C42C23×D6111yescharacteristic
374C23—42no—
375C23—1221no—
376C23#1451974811yescharacteristic
377Q8—82no—
378Q8—1222no—
379D4—42no—
380D4—42no—
381D4—42no—
382D4—42no—
383D4—82no—
384D4—84no—
385D4—1222no—
386D4—2442no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—42no—
392C2×C4—84no—
393C2×C4—1221no—
394C2×C4—1221no—
395C2×C4—1221no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4—2441no—
399C2×C4—2441no—
400C2×C4#1451974811yescharacteristic
401C2×C4#1451974811yescharacteristic
402C2×C4#1451974811yescharacteristic
403C2×C4C2×D4×D6111yescharacteristic
404C2×C4C22×C22⋊D6111yescharacteristic
405C2×C4C2×#1175822411yescharacteristic
406C8—21no—
407C8—21no—
408C8—21no—
409C8—2441no—
410C8—2442no—
411C8—2442no—
412C8—2441no—
413C8#1451979712yes—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—2441no—
421C22—2441no—
422C22—2441no—
423C22C2×#1451974811yescharacteristic
424C4—21no—
425C4—21no—
426C4—21no—
427C4—82no—
428C4—2441no—
429C4—2441no—
430C4—2441no—
431C4#1684002912yes—
432C4C2×#1451979711yescharacteristic
433C4#1684055912yes—
434C4#1684056011yescharacteristic
435C2—41no—
436C2—162no—
437C2—4881no—
438C2#2145526811yescharacteristic
439C2#2145542411yescharacteristic
440C2C2×#1684002911yescharacteristic
subgroupG/H
centerC2×C4#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3169C2
socleC22×C61C2×#121
hypercenter#3169D61
hypocenterC61#19160
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. #21455873C2semidirect
2. #21454785C2semidirect
3. #21445356C2semidirect
4. #21445349C2semidirect
5. #21445344C2semidirect
6. #21445339C2semidirect
7. #21445300C2semidirect
8. #21445295C2semidirect
9. #21444567C2semidirect
10. #21444481C2semidirect
11. C61×#3169C2semidirect
215. #3169D61semidirect
267. C61#19160semidirect

Maximal subgroups

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

subgroupindex
#214558732
#214547852
#214453562
#214453492
#214453442
#214453392
#214453002
#214452952
#214445672
#214444812
C61×#31692
#1916061