Finite Group Databasebeta

← #29663988

Subgroups — #29663988

Order 15616.

754 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840110C211yesmaximal, characteristic, complemented
2#21455302C211yesmaximal, characteristic, complemented
3#21455300C211yesmaximal, characteristic
4#21454775C211yesmaximal, characteristic, complemented
5#21445266C214yesmaximal, complemented
6#21445259C214yesmaximal, complemented
7#21444577C211yesmaximal, characteristic, complemented
8#21444486C211yesmaximal, characteristic, complemented
9C61×#1546C211yesmaximal, characteristic, complemented
10C2×#14519808—21no—
11C2×#14519805—21no—
12C2×#14519774C2211yescharacteristic
13C4×#11758222C2211yescharacteristic
14#16840110—21no—
15#16840110—21no—
16#16840110C2214yes—
17#16839969—21no—
18#16839969C2212yescomplemented
19#16839964—21no—
20#16839964C2212yescomplemented
21#16839963—21no—
22#16839963C2212yes—
23#16839958—21no—
24#16839958C2212yes—
25#16839886C2212yescomplemented
26#16839827C2212yescomplemented
27#16839825—22no—
28#16839813—21no—
29#16839786C2211yescharacteristic
30#16839771—21no—
31#16839768—21no—
32C2×#14519697—22no—
33C2×C4×#92418C2211yescharacteristic
34C4×C8×D61—21no—
35#16839514C2214yescomplemented
36#16839505C2214yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#332C2214yescomplemented
41C2×#11758222D411yescharacteristic
42C22×D244—21no—
43C22×D244—21no—
44C22×D244—21no—
45#14519808—44no—
46#14519805—44no—
47C2×#11758202—22no—
48C2×#11758190—22no—
49C2×#11758201—22no—
50C2×D488—22no—
51C2×#11758194—21no—
52C22×#92418D412yes—
53C2×C8×D61—22no—
54#14519774—21no—
55#14519774—21no—
56#14519774—21no—
57#14519774C2312yes—
58C4×Dic122D411yescharacteristic
59C4×Dic122C2311yescharacteristic
60C4×D244D411yescharacteristic
61C4×D244C2311yescharacteristic
62C4×C22⋊D61—21no—
63C4×C22⋊D61—21no—
64#14519736D411yescharacteristic
65#14519734—21no—
66C42×D61—21no—
67C4⋊#92418—21no—
68C4⋊#92418C2312yes—
69#14519706—22no—
70#14519701—22no—
71#14519701—22no—
72#14519699—22no—
73#14519699—22no—
74#14519697—24no—
75#14519697—24no—
76#14519689—22no—
77C4×#92418D411yescharacteristic
78C4×#92418D411yescharacteristic
79C4×#92418C2312yes—
80C8×Dic61—22no—
81C22×Q8×C61—21no—
82C2×M16×C61—21no—
83C4⋊Q8×C61—21no—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61C2312yes—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—21no—
89C2×C42×C61C2311yescharacteristic
90C4⋊C8×C61—21no—
91C4⋊C8×C61C2312yes—
92C4⋊C8×C61C2312yes—
93C22⋊C8×C61—21no—
94C4×C8×C61—21no—
95C23×D61—41no—
96C2×Q8×C61—22no—
97C2×Q8×C61—41no—
98C2×Q8×C61—42no—
99C2×Q8×C61—42no—
100C22×C4×C61—41no—
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61#4111yescharacteristic
104C2×C22⋊D61—21no—
105#11758222—22no—
106#11758222D814yes—
107C2×D244—22no—
108C2×D244—22no—
109C2×D244—22no—
110C2×D244—44no—
111C2×D244—42no—
112C2×D244—42no—
113C2×D244—41no—
114C2×D244—41no—
115C2×D244—41no—
116C2×D244—82no—
117C2×D244C2×D411yescharacteristic
118C2×C4×D61—21no—
119C2×C4×D61—21no—
120C2×C4×D61—21no—
121C2×Dic122C2×D411yescharacteristic
122M16×C61—42no—
123C2×C8×C61—22no—
124C2×C8×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—41no—
128C4⋊C4×C61—41no—
129C42×C61C2×D411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C42×C61#4111yescharacteristic
132C42×C61C2411yescharacteristic
133C22⋊Dic61—21no—
134#11758202—44no—
135#11758201—44no—
136#11758198—21no—
137#11758198—21no—
138C4⋊Dic61C2×D411yescharacteristic
139C4⋊Dic61C2×D411yescharacteristic
140#11758196—21no—
141#11758196—21no—
142C4×Dic61—21no—
143C4×Dic61—21no—
144C4×Dic61—21no—
145#11758194—42no—
146C2×#92418—22no—
147C2×#92418—42no—
148C2×#92418C2×D412yes—
149C2×#92418C2×D412yes—
150D488—44no—
151#11758190—44no—
152C8×D61—44no—
153C23×C61#12111yescharacteristic
154C22×D61—21no—
155C22×D61—42no—
156C22×D61—82no—
157C22×D61—82no—
158C22×D61—82no—
159Q8×C61—42no—
160Q8×C61—42no—
161Q8×C61—82no—
162C2×C4×C61—22no—
163C2×C4×C61—22no—
164C2×C4×C61—42no—
165C2×C4×C61—82no—
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61C2×D812yes—
170C2×C4×C61#13512yes—
171C2×C4×C61C22×D411yescharacteristic
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C2×#4111yescharacteristic
174C22⋊D61—24no—
175C2×Dic61—21no—
176C2×Dic61—21no—
177C2×Dic61—21no—
178D244—21no—
179D244—42no—
180D244—42no—
181D244—44no—
182D244—82no—
183D244—82no—
184D244—82no—
185D244—82no—
186D244C2×D812yes—
187C4×D61—24no—
188C4×D61—42no—
189C4×D61—42no—
190Dic122—21no—
191Dic122C2×D812yes—
192C8×C61—42no—
193C8×C61—42no—
194#92418—22no—
195#92418—22no—
196#92418—42no—
197#19276—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61#46612yes—
201C22×C61C2×#12111yescharacteristic
202C2×D61—22no—
203C2×D61—41no—
204C2×D61—41no—
205C2×D61—82no—
206C2×D61—82no—
207C2×D61—82no—
208C2×D61—164no—
209C4×C61—21no—
210C4×C61—21no—
211C4×C61—82no—
212C4×C61#46612yes—
213C4×C61#47112yes—
214C4×C61C22×D811yescharacteristic
215C4×C61C2×#13511yescharacteristic
216C4×C61#57512yes—
217Dic61—22no—
218Dic61—41no—
219Dic61—41no—
220C2×#486—611no—
221#3039—611no—
222#3038—611no—
223C8×#41—611no—
224#1682—614no—
225#1681—614no—
226#1567—611no—
227#1546D6111yescharacteristic, complemented
228#1545—611no—
229C2×C61—22no—
230C2×C61#304011yescharacteristic
231C2×C61C2×#46611yescharacteristic
232C2×C61C2×#47111yescharacteristic
233D61—42no—
234D61—162no—
235C2×#137—1221no—
236C2×AGL(1,ℤ/8)—1221no—
237C2×C4⋊Q8C2×D6111yescharacteristic
238C2×#127—611no—
239C4×#41—611no—
240#486—614no—
241#486—1221no—
242#486—1221no—
243#462—612no—
244#462—1221no—
245#461—612no—
246#461—1221no—
247#460—612no—
248#460—1221no—
249#459—612no—
250#459—1221no—
251C8×Q8—612no—
252C8×D4—612no—
253C8×D4—1222no—
254#431—1221no—
255#431C2×D6111yescharacteristic
256#423—611no—
257#423C2×D6111yescharacteristic
258#419—1221no—
259#418—1221no—
260C2×#102—1222no—
261C2×C4×C8—611no—
262C2×C4×C8—1221no—
263#335—614no—
264#332C2×D6114yescomplemented
265#331—614no—
266C61#1927611yescharacteristic, complemented, Sylow
267C2×#41—611no—
268C22×Q8—21no—
269C22×D4—1221no—
270C22×D4—1221no—
271C22×D4—1221no—
272#137—2444no—
273AGL(1,ℤ/8)—2444no—
274C2×Q16—1222no—
275C2×SD16—1222no—
276C2×SD16—1222no—
277C2×D8—1222no—
278C2×M16—21no—
279C2×M16—1221no—
280C22×C8—612no—
281C22×C8—1222no—
282C4⋊Q8—21no—
283C4⋊Q8—21no—
284C4⋊Q8—21no—
285C4⋊Q8C22×D6112yes—
286#127—612no—
287#127—1221no—
288#127—1221no—
289#127—1221no—
290C4×Q8—611no—
291C4×Q8—611no—
292C4×D4—611no—
293C4×D4—611no—
294C4×D4—1221no—
295C4×D4—1221no—
296#117—611no—
297#117—1221no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C42—1221no—
301C2×C42C22×D6111yescharacteristic
302C4⋊C8—21no—
303C4⋊C8—612no—
304C4⋊C8—1222no—
305C4⋊C8—1221no—
306C4⋊C8C22×D6112yes—
307C4⋊C8C22×D6112yes—
308#103—1222no—
309#103—1222no—
310#102—1224no—
311#102—1224no—
312#102—1222no—
313#102—1222no—
314C22⋊C8—21no—
315C22⋊C8—1222no—
316C4×C8—21no—
317C4×C8—611no—
318C4×C8—611no—
319C4×C8—612no—
320C4×C8—1222no—
321C24—2441no—
322#41—614no—
323#41—1222no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—611no—
329C2×D4—611no—
330C2×D4—1221no—
331C2×D4—1222no—
332C2×D4—1222no—
333C2×D4—1222no—
334C2×D4—4882no—
335C2×D4—2444no—
336C2×D4—2441no—
337C2×D4—2441no—
338C2×D4—2441no—
339C2×D4—2442no—
340C2×D4—2442no—
341C22×C4—41no—
342C22×C4—1221no—
343C22×C4—1221no—
344C22×C4—1221no—
345C22×C4#1175822211yescharacteristic
346C22×C4D4×D6111yescharacteristic
347C22×C4D4×D6111yescharacteristic
348Q16—2444no—
349SD16—2444no—
350SD16—2444no—
351D8—2444no—
352M16—42no—
353M16—2442no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—612no—
357C2×C8—612no—
358C2×C8—1222no—
359C2×C8—2444no—
360C2×C8—2442no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—611no—
366C4⋊C4—611no—
367C4⋊C4—1221no—
368C4⋊C4—1221no—
369C22⋊C4—1221no—
370C22⋊C4—1221no—
371C22⋊C4—1221no—
372C42—1221no—
373C42—1221no—
374C42—1221no—
375C42#1175822211yescharacteristic
376C42D4×D6111yescharacteristic
377C42D4×D6111yescharacteristic
378C42C23×D6111yescharacteristic
379C23—1221no—
380C23—4882no—
381C23—4882no—
382C23—4882no—
383C23—2442no—
384C23#1451975011yescharacteristic
385Q8—42no—
386Q8—42no—
387Q8—82no—
388Q8—612no—
389Q8—1221no—
390D4—612no—
391D4—1224no—
392D4—1221no—
393D4—4882no—
394D4—4882no—
395D4—4882no—
396D4—4882no—
397D4—2442no—
398D4—2442no—
399D4—2444no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—42no—
403C2×C4—82no—
404C2×C4—1221no—
405C2×C4—1224no—
406C2×C4—1221no—
407C2×C4—1221no—
408C2×C4—2442no—
409C2×C4—2442no—
410C2×C4#1451975011yescharacteristic
411C2×C4#1451975011yescharacteristic
412C2×C4#1451975011yescharacteristic
413C2×C4D8×D6112yes—
414C2×C4#1451979812yes—
415C2×C4C2×D4×D6111yescharacteristic
416C2×C4C2×D4×D6111yescharacteristic
417C2×C4C2×#1175822211yescharacteristic
418C8—42no—
419C8—42no—
420C8—1222no—
421C8—1222no—
422C8—2442no—
423C22—21no—
424C22—21no—
425C22—1222no—
426C22—9764no—
427C22—4882no—
428C22—4882no—
429C22—4882no—
430C22—2441no—
431C22—2441no—
432C22#1683999712yes—
433C22C2×#1451975011yescharacteristic
434C4—21no—
435C4—21no—
436C4—82no—
437C4—1222no—
438C4—2441no—
439C4—2441no—
440C4#1683999712yes—
441C4#1684002812yes—
442C4C2×D8×D6111yescharacteristic
443C4C2×#1451979811yescharacteristic
444C4#1684056112yes—
445C2—22no—
446C2—9762no—
447C2—2442no—
448C2#2145531611yescharacteristic
449C2C2×#1683999711yescharacteristic
450C2C2×#1684002811yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1546C2
socleC22×C61C2×#121
prime radical#21455300C2
hypercenter#1546D61
hypocenterC61#19276
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. C2×#16840110C2semidirect
2. #21455302C2semidirect
4. #21454775C2semidirect
5. #21445266C2semidirect
6. #21445259C2semidirect
7. #21444577C2semidirect
8. #21444486C2semidirect
9. C61×#1546C2semidirect
18. #16839969C22semidirect
20. #16839964C22semidirect
25. #16839886C22semidirect
26. #16839827C22semidirect
35. #16839514C22semidirect
40. C61×#332C22semidirect
227. #1546D61semidirect
264. #332C2×D61semidirect
266. C61#19276semidirect

Maximal subgroups

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

subgroupindex
C2×#168401102
#214553022
#214553002
#214547752
#214452662
#214452592
#214445772
#214444862
C61×#15462
#1927661