Finite Group Databasebeta

← #29663902

Subgroups — #29663902

Order 15616.

786 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840136C211yesmaximal, characteristic, complemented
2#21455628C211yesmaximal, characteristic, complemented
3#21455300C211yesmaximal, characteristic
4#21454775C211yesmaximal, characteristic, complemented
5#21445170C214yesmaximal, complemented
6#21445163C214yesmaximal, complemented
7#21444574C211yesmaximal, characteristic, complemented
8#21444486C211yesmaximal, characteristic, complemented
9C61×#1549C211yesmaximal, characteristic, complemented
10C2×#14519805—21no—
11C22×#11758201—22no—
12C2×#14519774C2211yescharacteristic
13C4×#11758222C2211yescharacteristic
14#16840136—21no—
15#16840136—21no—
16#16840136C2214yes—
17#16840054—21no—
18#16840049—21no—
19#16840044C2212yescomplemented
20#16840039C2212yescomplemented
21#16839963—21no—
22#16839963C2212yes—
23#16839958—21no—
24#16839958C2212yes—
25#16839886C2212yescomplemented
26#16839827C2212yescomplemented
27#16839825—22no—
28#16839813—21no—
29#16839802—21no—
30#16839786C2211yescharacteristic
31#16839768—21no—
32#16839761—21no—
33C2×C4×#92418C2211yescharacteristic
34C4×C8×D61—21no—
35#16839512C2214yescomplemented
36#16839505C2214yes—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#332C2214yescomplemented
41C2×#11758222D411yescharacteristic
42C22×D244—21no—
43C22×D244—21no—
44C22×D244—21no—
45#14519805—44no—
46C2×#11758190—22no—
47C2×#11758201—24no—
48C2×#11758201—24no—
49C2×#11758201—48no—
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—
67#14519710—22no—
68#14519708—22no—
69C4⋊#92418—21no—
70C4⋊#92418C2312yes—
71#14519706—22no—
72#14519700—22no—
73#14519699—22no—
74#14519699—22no—
75#14519697—22no—
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—44no—
98C2×Q8×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61#4111yescharacteristic
103C2×C22⋊D61—21no—
104#11758222—22no—
105#11758222SD1614yescomplemented
106C2×D244—22no—
107C2×D244—22no—
108C2×D244—22no—
109C2×D244—44no—
110C2×D244—42no—
111C2×D244—42no—
112C2×D244—41no—
113C2×D244—41no—
114C2×D244—41no—
115C2×D244—82no—
116C2×D244C2×D411yescharacteristic
117C2×C4×D61—21no—
118C2×C4×D61—21no—
119C2×C4×D61—21no—
120C2×Dic122C2×D411yescharacteristic
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
132C22⋊Dic61—21no—
133#11758201—48no—
134#11758201—48no—
135#11758198—21no—
136#11758198—21no—
137C4⋊Dic61C2×D411yescharacteristic
138C4⋊Dic61C2×D411yescharacteristic
139#11758196—21no—
140#11758196—21no—
141C4×Dic61—21no—
142C4×Dic61—21no—
143C4×Dic61—21no—
144#11758194—42no—
145C2×#92418—24no—
146C2×#92418—22no—
147C2×#92418C2×D412yes—
148C2×#92418C2×D412yes—
149D488—44no—
150#11758190—44no—
151C8×D61—44no—
152C23×C61#12111yescharacteristic
153C22×D61—21no—
154C22×D61—42no—
155C22×D61—82no—
156C22×D61—82no—
157C22×D61—82no—
158Q8×C61—44no—
159Q8×C61—82no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—42no—
163C2×C4×C61—82no—
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61C2×SD1612yes—
168C2×C4×C61#13512yes—
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C2×#4111yescharacteristic
172C22⋊D61—24no—
173C2×Dic61—21no—
174C2×Dic61—21no—
175C2×Dic61—21no—
176D244—21no—
177D244—42no—
178D244—42no—
179D244—44no—
180D244—82no—
181D244—82no—
182D244—82no—
183D244—82no—
184D244C2×SD1612yescomplemented
185C4×D61—24no—
186C4×D61—42no—
187C4×D61—42no—
188Dic122—21no—
189Dic122C2×SD1612yescomplemented
190C8×C61—42no—
191C8×C61—42no—
192#92418—22no—
193#92418—42no—
194#92418AGL(1,ℤ/8)14yescomplemented
195#19266—611nomaximal, Sylow
196C22×C61—21no—
197C22×C61—21no—
198C22×C61#46512yes—
199C22×C61C2×#12111yescharacteristic
200C2×D61—22no—
201C2×D61—41no—
202C2×D61—41no—
203C2×D61—82no—
204C2×D61—82no—
205C2×D61—82no—
206C2×D61—164no—
207C4×C61—21no—
208C4×C61—21no—
209C4×C61—82no—
210C4×C61#46512yes—
211C4×C61#47112yes—
212C4×C61C22×SD1611yescharacteristic
213C4×C61C2×#13511yescharacteristic
214C4×C61C2×AGL(1,ℤ/8)12yes—
215Dic61—22no—
216Dic61—41no—
217Dic61—41no—
218C2×#492—611no—
219#3079—611no—
220#3038—611no—
221C8×#41—611no—
222#1666—614no—
223#1665—614no—
224#1566—611no—
225#1549D6111yescharacteristic, complemented
226#1545—611no—
227C2×C61—22no—
228C2×C61#303811yescharacteristic
229C2×C61C2×#46511yescharacteristic
230C2×C61C2×#47111yescharacteristic
231D61—42no—
232D61—162no—
233C2×AGL(1,ℤ/8)—1221no—
234C22×SD16—1222no—
235C2×C4⋊Q8C2×D6111yescharacteristic
236C2×#127—611no—
237C4×#41—611no—
238#492—614no—
239#492—1221no—
240#492—1221no—
241#477—1221no—
242#476—1221no—
243#475—612no—
244#474—612no—
245#460—612no—
246#460—1221no—
247#459—612no—
248#459—1221no—
249C8×Q8—612no—
250C8×D4—612no—
251C8×D4—1222no—
252#431—1221no—
253#431C2×D6111yescharacteristic
254#428—1221no—
255#423—611no—
256#423C2×D6111yescharacteristic
257#418—1221no—
258#417—1221no—
259C2×C4×C8—611no—
260C2×C4×C8—1221no—
261#334—614no—
262#332C2×D6114yescomplemented
263#331—614no—
264C61#1926611yescharacteristic, complemented, Sylow
265C2×#41—611no—
266C22×Q8—21no—
267C22×D4—1221no—
268C22×D4—1221no—
269C22×D4—1221no—
270AGL(1,ℤ/8)—2444no—
271C2×SD16—1222no—
272C2×SD16—1224no—
273C2×SD16—1224no—
274C2×SD16—2448no—
275C2×D8—1222no—
276C2×M16—21no—
277C2×M16—1221no—
278C22×C8—612no—
279C22×C8—1222no—
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8C22×D6112yes—
284#127—612no—
285#127—1221no—
286#127—1221no—
287#127—1221no—
288C4×Q8—611no—
289C4×Q8—611no—
290C4×D4—611no—
291C4×D4—611no—
292C4×D4—1221no—
293C4×D4—1221no—
294#117—611no—
295#117—1221no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C42—1221no—
299C2×C42C22×D6111yescharacteristic
300#107—1222no—
301#106—1222no—
302C4⋊C8—21no—
303C4⋊C8—612no—
304C4⋊C8—1222no—
305C4⋊C8—1221no—
306C4⋊C8C22×D6112yes—
307C4⋊C8C22×D6112yes—
308#103—1222no—
309#102—1222no—
310#102—1222no—
311#102—1222no—
312C22⋊C8—21no—
313C22⋊C8—1222no—
314C4×C8—21no—
315C4×C8—611no—
316C4×C8—611no—
317C4×C8—612no—
318C4×C8—1222no—
319C24—2441no—
320#41—614no—
321#41—1222no—
322C2×Q8—22no—
323C2×Q8—44no—
324C2×Q8—41no—
325C2×Q8—611no—
326C2×D4—611no—
327C2×D4—1221no—
328C2×D4—1222no—
329C2×D4—1222no—
330C2×D4—1222no—
331C2×D4—4882no—
332C2×D4—2444no—
333C2×D4—2441no—
334C2×D4—2441no—
335C2×D4—2441no—
336C2×D4—2442no—
337C2×D4—2442no—
338C22×C4—41no—
339C22×C4—1221no—
340C22×C4—1221no—
341C22×C4—1221no—
342C22×C4#1175822211yescharacteristic
343C22×C4D4×D6111yescharacteristic
344C22×C4D4×D6111yescharacteristic
345SD16—2444no—
346SD16—2448no—
347SD16—2448no—
348D8—2444no—
349M16—42no—
350M16—2442no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—612no—
354C2×C8—612no—
355C2×C8—1224no—
356C2×C8—1222no—
357C2×C8—2444no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—611no—
363C4⋊C4—611no—
364C4⋊C4—1221no—
365C4⋊C4—1221no—
366C22⋊C4—1221no—
367C22⋊C4—1221no—
368C22⋊C4—1221no—
369C42—1221no—
370C42—1221no—
371C42—1221no—
372C42#1175822211yescharacteristic
373C42D4×D6111yescharacteristic
374C42D4×D6111yescharacteristic
375C42C23×D6111yescharacteristic
376C23—1221no—
377C23—4882no—
378C23—4882no—
379C23—4882no—
380C23—2442no—
381C23#1451975011yescharacteristic
382Q8—44no—
383Q8—82no—
384Q8—612no—
385Q8—1221no—
386D4—612no—
387D4—1224no—
388D4—1221no—
389D4—4882no—
390D4—4882no—
391D4—4882no—
392D4—4882no—
393D4—2442no—
394D4—2442no—
395D4—2444no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—42no—
399C2×C4—82no—
400C2×C4—1221no—
401C2×C4—1224no—
402C2×C4—1221no—
403C2×C4—1221no—
404C2×C4—2442no—
405C2×C4—2442no—
406C2×C4#1451975011yescharacteristic
407C2×C4#1451975011yescharacteristic
408C2×C4#1451975011yescharacteristic
409C2×C4SD16×D6112yes—
410C2×C4#1451980012yes—
411C2×C4C2×D4×D6111yescharacteristic
412C2×C4C2×D4×D6111yescharacteristic
413C2×C4C2×#1175822211yescharacteristic
414C8—42no—
415C8—42no—
416C8—614no—
417C8—1222no—
418C8—2442no—
419C22—21no—
420C22—21no—
421C22—1222no—
422C22—9764no—
423C22—4882no—
424C22—4882no—
425C22—4882no—
426C22—2441no—
427C22—2441no—
428C22#1683999412yes—
429C22C2×#1451975011yescharacteristic
430C4—21no—
431C4—21no—
432C4—82no—
433C4—1222no—
434C4—2441no—
435C4—2441no—
436C4#1683999412yes—
437C4#1684002812yes—
438C4C2×SD16×D6111yescharacteristic
439C4C2×#1451980011yescharacteristic
440C4AGL(1,ℤ/8)×D6112yes—
441C2—22no—
442C2—9762no—
443C2—2442no—
444C2#2145529811yescharacteristic
445C2C2×#1683999411yescharacteristic
446C2C2×#1684002811yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1549C2
socleC22×C61C2×#121
prime radical#21455300C2
hypercenter#1549D61
hypocenterC61#19266
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
1. C2×#16840136C2semidirect
2. #21455628C2semidirect
4. #21454775C2semidirect
5. #21445170C2semidirect
6. #21445163C2semidirect
7. #21444574C2semidirect
8. #21444486C2semidirect
9. C61×#1549C2semidirect
19. #16840044C22semidirect
20. #16840039C22semidirect
25. #16839886C22semidirect
26. #16839827C22semidirect
35. #16839512C22semidirect
40. C61×#332C22semidirect
105. #11758222SD16semidirect
184. D244C2×SD16semidirect
189. Dic122C2×SD16semidirect
194. #92418AGL(1,ℤ/8)semidirect
225. #1549D61semidirect
262. #332C2×D61semidirect
264. C61#19266semidirect

Maximal subgroups

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

subgroupindex
C2×#168401362
#214556282
#214553002
#214547752
#214451702
#214451632
#214445742
#214444862
C61×#15492
#1926661