Finite Group Databasebeta

← #29663078

Subgroups — #29663078

Order 15616.

778 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456940C211yesmaximal, characteristic, complemented
2#21456739C211yesmaximal, characteristic
3C2×#16840136C211yesmaximal, characteristic, complemented
4#21454530C211yesmaximal, characteristic, complemented
5#21445287C214yesmaximal, complemented
6#21445163C214yesmaximal, complemented
7#21444565C211yesmaximal, characteristic, complemented
8#21444488C211yesmaximal, characteristic, complemented
9C61×#1546C211yesmaximal, characteristic, complemented
10C22×#11758201—21no—
11C22×#11758201—21no—
12C2×#14519785—21no—
13#16840312C2211yescharacteristic
14C2×#14519774C2211yescharacteristic
15#16840165—21no—
16#16840139—21no—
17#16840136—21no—
18#16840136—21no—
19#16840136C2212yes—
20#16840136C2212yes—
21#16840110—21no—
22#16840109—21no—
23#16840075—22no—
24#16840039C2214yescomplemented
25#16839958C2214yes—
26#16839898—22no—
27#16839839—21no—
28#16839839C2212yescomplemented
29#16839827—21no—
30#16839827C2212yescomplemented
31#16839815C2211yescharacteristic
32#16839730—21no—
33C2×C4×#92418C2211yescharacteristic
34#16839514C2212yescomplemented
35#16839512C2212yescomplemented
36#16839505C2214yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#332C2214yescomplemented
41C2×#11758222—21no—
42C22×D244—21no—
43C22×D244—21no—
44C22×D244—21no—
45C2×#11758201—22no—
46C2×#11758201—22no—
47C2×#11758201—22no—
48C2×#11758201—22no—
49C2×#11758201—44no—
50C2×#11758201—44no—
51C2×D488—24no—
52#14519785—44no—
53C2×#11758189—22no—
54C22×#92418D411yescharacteristic
55C22×#92418D411yescharacteristic
56C2×C8×D61—22no—
57#14519777D411yescharacteristic
58#14519774—21no—
59#14519774—21no—
60#14519774—21no—
61#14519774D411yescharacteristic
62#14519774C2312yes—
63#14519765—21no—
64#14519751—21no—
65#14519751D412yes—
66C4×D244—21no—
67C4×D244C2312yes—
68#14519710—24no—
69C4⋊#92418C2312yes—
70#14519699—24no—
71#14519697—24no—
72C22⋊#92418—21no—
73#14519689—22no—
74#14519689—22no—
75C4×#92418—21no—
76C4×#92418D411yescharacteristic
77C4×#92418D411yescharacteristic
78C4×#92418C2311yescharacteristic
79C4×#92418C2311yescharacteristic
80C22×Q8×C61—21no—
81C2×M16×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61—21no—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61C2312yes—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C42×C61C2311yescharacteristic
89C4⋊C8×C61—21no—
90C4⋊C8×C61C2312yes—
91C4⋊C8×C61C2312yes—
92C22⋊C8×C61—21no—
93C4×C8×C61—21no—
94C23×D61—41no—
95C2×Q8×C61—22no—
96C2×Q8×C61—41no—
97C2×Q8×C61—42no—
98C2×Q8×C61—42no—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61#4111yescharacteristic
103C2×C22⋊D61—22no—
104#11758222—42no—
105#11758222—42no—
106C2×D244—21no—
107C2×D244—21no—
108C2×D244—22no—
109C2×D244—22no—
110C2×D244—22no—
111C2×D244—42no—
112C2×D244—42no—
113C2×D244—41no—
114C2×D244—41no—
115C2×D244—41no—
116C2×D244—82no—
117C2×D244—82no—
118C2×D244C2×D412yes—
119C2×C4×D61—22no—
120C2×Dic122—21no—
121M16×C61—42no—
122C2×C8×C61—22no—
123C2×C8×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—41no—
127C4⋊C4×C61—41no—
128C42×C61C2×D411yescharacteristic
129C42×C61C2×D411yescharacteristic
130C42×C61#4111yescharacteristic
131C42×C61C2411yescharacteristic
132#11758201—44no—
133#11758201—44no—
134#11758201—44no—
135#11758201—44no—
136#11758198—22no—
137#11758198—22no—
138C4⋊Dic61—21no—
139C4⋊Dic61C2×D412yes—
140C2×#92418—22no—
141C2×#92418—22no—
142C2×#92418D814yescomplemented
143C2×#92418C2×D411yescharacteristic
144C2×#92418C2×D411yescharacteristic
145C2×#92418C2×D411yescharacteristic
146C2×#92418C2×D411yescharacteristic
147D488—84no—
148#11758189—44no—
149C8×D61—44no—
150C23×C61#12111yescharacteristic
151C22×D61—22no—
152C22×D61—42no—
153C22×D61—82no—
154C22×D61—82no—
155C22×D61—82no—
156Q8×C61—42no—
157Q8×C61—42no—
158Q8×C61—82no—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
161C2×C4×C61—42no—
162C2×C4×C61—82no—
163C2×C4×C61#12111yescharacteristic
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61C2×D812yes—
167C2×C4×C61AGL(1,ℤ/8)12yes—
168C2×C4×C61C22×D411yescharacteristic
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C2×#4111yescharacteristic
171C22⋊D61—44no—
172C2×Dic61—22no—
173D244—24no—
174D244—42no—
175D244—42no—
176D244—44no—
177D244—42no—
178D244—42no—
179D244—82no—
180D244—82no—
181D244—82no—
182D244—82no—
183C4×D61—42no—
184C4×D61—42no—
185Dic122—42no—
186C8×C61—42no—
187C8×C61—42no—
188#92418—21no—
189#92418—21no—
190#92418—42no—
191#92418C2×D812yes—
192#92418C2×D812yes—
193#19186—611nomaximal, Sylow
194C22×C61—21no—
195C22×C61—21no—
196C22×C61#45912yes—
197C22×C61C2×#12111yescharacteristic
198C2×D61—42no—
199C2×D61—42no—
200C2×D61—42no—
201C2×D61—82no—
202C2×D61—82no—
203C2×D61—82no—
204C2×D61—164no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—82no—
208C4×C61#45912yes—
209C4×C61#47112yes—
210C4×C61C22×D811yescharacteristic
211C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
212C4×C61#57712yes—
213Dic61—42no—
214#3226—611no—
215#3205—611no—
216C2×#492—611no—
217#2931—611no—
218#1685—614no—
219#1665—614no—
220#1564—611no—
221#1546D6111yescharacteristic, complemented
222#1545—611no—
223C2×C61—22no—
224C2×C61C2×#45911yescharacteristic
225C2×C61#305711yescharacteristic
226C2×C61C2×#47111yescharacteristic
227D61—82no—
228D61—162no—
229C22×SD16—1221no—
230C22×SD16—1221no—
231C2×#131—1221no—
232#532—611no—
233C2×C4⋊Q8C2×D6111yescharacteristic
234C2×#127—611no—
235#500—1221no—
236#493—1221no—
237#492—612no—
238#492—612no—
239#492—1221no—
240#492—1221no—
241#486—1221no—
242#486—1221no—
243#480—1222no—
244#474—614no—
245#459—614no—
246#447—1222no—
247#436—612no—
248#436—1221no—
249C8×D4—612no—
250C8×D4—1221no—
251#431—611no—
252#431C2×D6111yescharacteristic
253#423C2×D6111yescharacteristic
254#408—1221no—
255C2×C4×C8—611no—
256#335—612no—
257#334—612no—
258#332C2×D6114yescomplemented
259#331—614no—
260C61#1918611yescharacteristic, complemented, Sylow
261C2×#41—1221no—
262C22×Q8—21no—
263C22×D4—1221no—
264C22×D4—1221no—
265C22×D4—1221no—
266C2×SD16—1222no—
267C2×SD16—1222no—
268C2×SD16—1222no—
269C2×SD16—1222no—
270C2×SD16—2444no—
271C2×SD16—2444no—
272C2×D8—1224no—
273#131—2444no—
274C2×M16—21no—
275C2×M16—1222no—
276C22×C8—611no—
277C22×C8—611no—
278C22×C8—1222no—
279C4⋊Q8—21no—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8—611no—
283C4⋊Q8C22×D6112yes—
284#127—611no—
285#127—612no—
286#127—1221no—
287#127—1221no—
288#127—1221no—
289#124—1221no—
290#121—612no—
291#121—1221no—
292C4×D4—612no—
293C4×D4—1221no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C42C22×D6111yescharacteristic
297#107—1224no—
298C4⋊C8—21no—
299C4⋊C8—612no—
300C4⋊C8C22×D6112yes—
301C4⋊C8C22×D6112yes—
302#102—1224no—
303#102—1224no—
304C22⋊C8—21no—
305C22⋊C8—1222no—
306C22⋊C8—1222no—
307C22⋊C8—1221no—
308C4×C8—21no—
309C4×C8—611no—
310C4×C8—611no—
311C4×C8—611no—
312C4×C8—611no—
313C4×C8—1221no—
314C24—2441no—
315#41—2442no—
316#41—2442no—
317C2×Q8—22no—
318C2×Q8—41no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—1221no—
322C2×D4—612no—
323C2×D4—1222no—
324C2×D4—1222no—
325C2×D4—1222no—
326C2×D4—1222no—
327C2×D4—1221no—
328C2×D4—1221no—
329C2×D4—4882no—
330C2×D4—4882no—
331C2×D4—2441no—
332C2×D4—2441no—
333C2×D4—2441no—
334C2×D4—2442no—
335C2×D4—2442no—
336C22×C4—41no—
337C22×C4—1222no—
338C22×C4#1175822211yescharacteristic
339C22×C4D4×D6111yescharacteristic
340C22×C4D4×D6111yescharacteristic
341SD16—2444no—
342SD16—2444no—
343SD16—2444no—
344SD16—2444no—
345D8—4884no—
346M16—42no—
347M16—2444no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—611no—
351C2×C8—611no—
352C2×C8—614no—
353C2×C8—611no—
354C2×C8—611no—
355C2×C8—1222no—
356C2×C8—1222no—
357C2×C8—2444no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—612no—
363C4⋊C4—1221no—
364C22⋊C4—1222no—
365C22⋊C4—1222no—
366C42#1175822211yescharacteristic
367C42D4×D6111yescharacteristic
368C42D4×D6111yescharacteristic
369C42C23×D6111yescharacteristic
370C23—1222no—
371C23—4882no—
372C23—4882no—
373C23—4882no—
374C23—2442no—
375C23#1451975011yescharacteristic
376Q8—42no—
377Q8—42no—
378Q8—82no—
379Q8—2442no—
380D4—1224no—
381D4—4882no—
382D4—4882no—
383D4—4882no—
384D4—4882no—
385D4—2444no—
386D4—2442no—
387D4—2442no—
388D4—2442no—
389D4—2442no—
390D4—2444no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—42no—
394C2×C4—82no—
395C2×C4—1222no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4#1451975011yescharacteristic
399C2×C4#1451975011yescharacteristic
400C2×C4#1451975011yescharacteristic
401C2×C4D8×D6112yes—
402C2×C4#1451980412yes—
403C2×C4C2×D4×D6111yescharacteristic
404C2×C4C2×D4×D6111yescharacteristic
405C2×C4C2×#1175822211yescharacteristic
406C8—42no—
407C8—42no—
408C8—612no—
409C8—612no—
410C8—1221no—
411C8—1221no—
412C8—2442no—
413C22—21no—
414C22—21no—
415C22—9764no—
416C22—4882no—
417C22—4882no—
418C22—4882no—
419C22—2442no—
420C22—2442no—
421C22—2442no—
422C22#1683995612yes—
423C22C2×#1451975011yescharacteristic
424C4—21no—
425C4—21no—
426C4—82no—
427C4—2442no—
428C4#1683995612yes—
429C4#1684002812yes—
430C4C2×D8×D6111yescharacteristic
431C4C2×#1451980411yescharacteristic
432C4#1684057112yes—
433C2—22no—
434C2—9762no—
435C2—4882no—
436C2C2×#1683995611yescharacteristic
437C2#2145546111yescharacteristic
438C2C2×#1684002811yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1546C2
socleC22×C61C2×#121
prime radical#21456739C2
hypercenter#1546D61
hypocenterC61#19186
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
1. #21456940C2semidirect
3. C2×#16840136C2semidirect
4. #21454530C2semidirect
5. #21445287C2semidirect
6. #21445163C2semidirect
7. #21444565C2semidirect
8. #21444488C2semidirect
9. C61×#1546C2semidirect
24. #16840039C22semidirect
28. #16839839C22semidirect
30. #16839827C22semidirect
34. #16839514C22semidirect
35. #16839512C22semidirect
40. C61×#332C22semidirect
142. C2×#92418D8semidirect
221. #1546D61semidirect
258. #332C2×D61semidirect
260. C61#19186semidirect

Maximal subgroups

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

subgroupindex
#214569402
#214567392
C2×#168401362
#214545302
#214452872
#214451632
#214445652
#214444882
C61×#15462
#1918661