Finite Group Databasebeta

← #29665567

Subgroups — #29665567

Order 15616.

768 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455279C211yesmaximal, characteristic, complemented
2C2×#16839954C211yesmaximal, characteristic, complemented
3#21454859C211yesmaximal, characteristic, complemented
4#21445377C214yesmaximal, complemented
5#21445295C214yesmaximal, complemented
6#21444590C211yesmaximal, characteristic, complemented
7#21444487C211yesmaximal, characteristic, complemented
8#21444468C211yesmaximal, characteristic, complemented
9C61×#3205C211yesmaximal, characteristic, complemented
10C2×#14519802—21no—
11C22×#11758199—21no—
12C2×C4×D244C2211yescharacteristic
13#16839974—22no—
14#16839964C2212yescomplemented
15#16839954—22no—
16#16839954C2214yes—
17#16839954C2212yescomplemented
18#16839841C2214yescomplemented
19#16839839—22no—
20#16839815C2211yescharacteristic
21#16839786C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23#16839763—21no—
24C2×#14519698—21no—
25#16839724—22no—
26#16839513C2214yescomplemented
27#16839508C2212yescomplemented
28#16839506C2214yescomplemented
29#16839506C2212yescomplemented
30C61×#532C2211yescharacteristic
31C2×#127×C61C2211yescharacteristic
32C61×#493C2212yescomplemented
33C61×#486C2212yescomplemented
34C61×#459—22no—
35C61×#447—22no—
36C61×#431C2211yescharacteristic
37C22×D244D411yescharacteristic
38C22×C4×D61—21no—
39#14519802—44no—
40C2×#11758201—22no—
41C2×#11758199—24no—
42C2×#11758199—22no—
43C2×#11758199—44no—
44C2×#11758194—21no—
45C2×#11758189—24no—
46C22×#92418—21no—
47C4×D244—21no—
48C4×D244—21no—
49C4×D244D412yes—
50C4×D244C2312yes—
51C2×C4⋊Dic61D411yescharacteristic
52C2×#11758198—21no—
53C4⋊#92418—21no—
54C4⋊#92418—21no—
55C4⋊#92418C2312yes—
56C4⋊#92418C2312yes—
57C4⋊#92418C2312yes—
58#14519701—22no—
59#14519698—24no—
60#14519698—22no—
61C22⋊#92418—21no—
62#14519689—24no—
63C4×#92418—21no—
64C2×#41×C61—21no—
65C22×D4×C61—21no—
66C22×D4×C61—21no—
67C22×D4×C61—21no—
68C2×D8×C61—24no—
69C4⋊Q8×C61C2311yescharacteristic
70#127×C61—21no—
71#127×C61—21no—
72#127×C61—21no—
73#127×C61C2312yes—
74#127×C61C2311yescharacteristic
75#124×C61—21no—
76#121×C61—21no—
77#121×C61—21no—
78C4×D4×C61—21no—
79C4×D4×C61—21no—
80C2×C42×C61C2311yescharacteristic
81C4⋊C8×C61D412yes—
82#102×C61—24no—
83C22⋊C8×C61D412yes—
84C4×C8×C61C2312yes—
85C24×C61—41no—
86C23×D61—21no—
87#41×C61—42no—
88#41×C61—42no—
89C2×Q8×C61—21no—
90C2×D4×C61—22no—
91C2×D4×C61—21no—
92C2×D4×C61—21no—
93C2×D4×C61—22no—
94C2×D4×C61—21no—
95C2×D4×C61—22no—
96C2×D4×C61—41no—
97C2×D4×C61—41no—
98C2×D4×C61—41no—
99C2×D4×C61—41no—
100C2×D4×C61—42no—
101C2×D4×C61—42no—
102C2×D4×C61—82no—
103C2×D4×C61—82no—
104C22×C4×C61—41no—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61#4111yescharacteristic
108C22×Dic61—21no—
109C2×D244—22no—
110C2×D244—22no—
111C2×D244—21no—
112C2×D244C2×D412yes—
113C2×C4×D61—22no—
114C2×C4×D61—22no—
115C2×C4×D61—82no—
116D8×C61—84no—
117C2×C8×C61C2×D414yes—
118C4⋊C4×C61—21no—
119C4⋊C4×C61—21no—
120C22⋊C4×C61—41no—
121C22⋊C4×C61—41no—
122C42×C61C2×D411yescharacteristic
123C42×C61C2×D411yescharacteristic
124C42×C61#4111yescharacteristic
125C42×C61C2411yescharacteristic
126#11758201—44no—
127#11758199—48no—
128#11758199—44no—
129#11758198—22no—
130#11758198—22no—
131C4⋊Dic61—21no—
132C4⋊Dic61C2×D412yes—
133#11758194—42no—
134C2×#92418—22no—
135C2×#92418—22no—
136C2×#92418—22no—
137C2×#92418—42no—
138#11758189—84no—
139C23×C61—41no—
140C23×C61—42no—
141C23×C61—82no—
142C23×C61—82no—
143C23×C61—82no—
144C23×C61#12111yescharacteristic
145C22×D61—22no—
146C22×D61—42no—
147C22×D61—81no—
148C22×D61—81no—
149Q8×C61—41no—
150Q8×C61—41no—
151D4×C61—42no—
152D4×C61—41no—
153D4×C61—42no—
154D4×C61—41no—
155D4×C61—42no—
156D4×C61—44no—
157D4×C61—81no—
158D4×C61—82no—
159D4×C61—82no—
160D4×C61—82no—
161D4×C61—82no—
162D4×C61—82no—
163C2×C4×C61—22no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—82no—
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61AGL(1,ℤ/8)12yes—
171C2×C4×C61AGL(1,ℤ/8)12yes—
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C2×#4111yescharacteristic
175C2×Dic61—22no—
176C2×Dic61—81no—
177D244—22no—
178D244—22no—
179D244—42no—
180C4×D61—44no—
181C4×D61—82no—
182C8×C61—24no—
183#92418—42no—
184#92418—42no—
185#92418—42no—
186#19422—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—82no—
190C22×C61—81no—
191C22×C61—81no—
192C22×C61—82no—
193C22×C61—82no—
194C22×C61—81no—
195C22×C61—164no—
196C22×C61#46912yes—
197C22×C61C2×#12111yescharacteristic
198C2×D61—42no—
199C2×D61—42no—
200C2×D61—42no—
201C2×D61—84no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—81no—
206C4×C61#46912yes—
207C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
208C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
209C4×C61#57612yes—
210Dic61—42no—
211#3205D6111yescharacteristic, complemented
212#3036—611no—
213C2×#459—611no—
214#2975—611no—
215#1701—614no—
216#1687—614no—
217#1570—611no—
218#1545—611no—
219#1540—611no—
220C2×C61—22no—
221C2×C61—161no—
222C2×C61—162no—
223C2×C61#304111yescharacteristic
224C2×C61C2×#46911yescharacteristic
225C2×C61#306811yescharacteristic
226D61—82no—
227C2×AGL(1,ℤ/8)—1221no—
228C22×D8—1221no—
229#532C2×D6111yescharacteristic
230C2×#127C2×D6111yescharacteristic
231C2×C4×D4—611no—
232#493C2×D6112yescomplemented
233#486C2×D6112yescomplemented
234#463—1222no—
235#461—612no—
236#459—22no—
237#459—614no—
238#459—612no—
239#459—1222no—
240#447—22no—
241#436—614no—
242#436—1222no—
243#431—611no—
244#431C2×D6111yescharacteristic
245#423—611no—
246C2×C4⋊C8—611no—
247#417—1221no—
248C2×#102—1221no—
249#407—1222no—
250#334—614no—
251#332—612no—
252#331—612no—
253#331—614no—
254C61#1942211yescharacteristic, complemented, Sylow
255C2×#41—21no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—21no—
259C22×D4—611no—
260C23×C4—1221no—
261AGL(1,ℤ/8)—2444no—
262C2×SD16—1222no—
263C2×D8—24no—
264C2×D8—1224no—
265C2×D8—1222no—
266C2×D8—2444no—
267C2×M16—1221no—
268C2×M16—1224no—
269C22×C8—1221no—
270C4⋊Q8C22×D6111yescharacteristic
271#127—21no—
272#127—21no—
273#127—21no—
274#127C22×D6112yes—
275#127C22×D6111yescharacteristic
276#124—21no—
277#121—21no—
278#121—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—612no—
282C4×D4—612no—
283C4×D4—1221no—
284C4×D4—1221no—
285C2×C4⋊C4—611no—
286C2×C22⋊C4—1221no—
287C2×C42C22×D6111yescharacteristic
288C4⋊C8—612no—
289C4⋊C8—612no—
290C4⋊C8—612no—
291C4⋊C8—1221no—
292C4⋊C8—1221no—
293C4⋊C8C22⋊D6112yes—
294#103—1222no—
295#102—24no—
296#102—1224no—
297#102—1222no—
298C22⋊C8—1221no—
299C22⋊C8—1224no—
300C22⋊C8C22⋊D6112yes—
301C4×C8—1221no—
302C4×C8C22×D6112yes—
303C24—41no—
304C24—1221no—
305#41—42no—
306#41—42no—
307C2×Q8—21no—
308C2×D4—21no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—22no—
314C2×D4—41no—
315C2×D4—41no—
316C2×D4—41no—
317C2×D4—41no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4—82no—
321C2×D4—82no—
322C2×D4—612no—
323C2×D4—1221no—
324C2×D4—1222no—
325C2×D4—1222no—
326C22×C4—41no—
327C22×C4—1221no—
328C22×C4—1222no—
329C22×C4—1222no—
330C22×C4—4882no—
331C22×C4D4×D6111yescharacteristic
332C22×C4#1175822411yescharacteristic
333C22×C4C2×C22⋊D6111yescharacteristic
334SD16—2444no—
335D8—84no—
336D8—2448no—
337D8—2444no—
338M16—4884no—
339M16—2442no—
340C2×C8—1222no—
341C2×C8—1222no—
342C2×C8—1222no—
343C2×C8—2442no—
344C2×C8C2×C22⋊D6114yes—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—612no—
348C4⋊C4—1221no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—1222no—
352C22⋊C4—1222no—
353C42D4×D6111yescharacteristic
354C42#1175822411yescharacteristic
355C42C2×C22⋊D6111yescharacteristic
356C42C23×D6111yescharacteristic
357C23—41no—
358C23—42no—
359C23—82no—
360C23—82no—
361C23—82no—
362C23—1222no—
363C23—4881no—
364C23—4881no—
365C23—2442no—
366C23#1451974811yescharacteristic
367Q8—41no—
368Q8—41no—
369D4—42no—
370D4—41no—
371D4—42no—
372D4—41no—
373D4—42no—
374D4—44no—
375D4—81no—
376D4—82no—
377D4—82no—
378D4—82no—
379D4—82no—
380D4—82no—
381D4—1222no—
382D4—1222no—
383D4—2442no—
384C2×C4—22no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—82no—
388C2×C4—1222no—
389C2×C4—4882no—
390C2×C4—4881no—
391C2×C4—2444no—
392C2×C4#1451974811yescharacteristic
393C2×C4#1451974811yescharacteristic
394C2×C4#1451974811yescharacteristic
395C2×C4#1451980112yes—
396C2×C4#1451980112yes—
397C2×C4C2×D4×D6111yescharacteristic
398C2×C4C22×C22⋊D6111yescharacteristic
399C2×C4C2×#1175822411yescharacteristic
400C8—24no—
401C8—2442no—
402C8—2442no—
403C8—2442no—
404C22—21no—
405C22—21no—
406C22—81no—
407C22—82no—
408C22—82no—
409C22—81no—
410C22—82no—
411C22—81no—
412C22—164no—
413C22—4884no—
414C22—2442no—
415C22—2442no—
416C22—2442no—
417C22#1684001212yes—
418C22C2×#1451974811yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—81no—
423C4—2442no—
424C4#1684001212yes—
425C4C2×#1451980111yescharacteristic
426C4C2×#1451980111yescharacteristic
427C4#1684056512yes—
428C2—22no—
429C2—162no—
430C2—161no—
431C2—4882no—
432C2#2145531811yescharacteristic
433C2C2×#1684001211yescharacteristic
434C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3205C2
socleC22×C61C2×#121
hypercenter#3205D61
hypocenterC61#19422
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #21455279C2semidirect
2. C2×#16839954C2semidirect
3. #21454859C2semidirect
4. #21445377C2semidirect
5. #21445295C2semidirect
6. #21444590C2semidirect
7. #21444487C2semidirect
8. #21444468C2semidirect
9. C61×#3205C2semidirect
14. #16839964C22semidirect
17. #16839954C22semidirect
18. #16839841C22semidirect
26. #16839513C22semidirect
27. #16839508C22semidirect
28. #16839506C22semidirect
29. #16839506C22semidirect
32. C61×#493C22semidirect
33. C61×#486C22semidirect
211. #3205D61semidirect
232. #493C2×D61semidirect
233. #486C2×D61semidirect
254. C61#19422semidirect

Maximal subgroups

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

subgroupindex
#214552792
C2×#168399542
#214548592
#214453772
#214452952
#214445902
#214444872
#214444682
C61×#32052
#1942261