Finite Group Databasebeta

← #29663898

Subgroups — #29663898

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×#16840134C211yesmaximal, characteristic, complemented
2#21455635C211yesmaximal, characteristic, complemented
3#21455295C211yesmaximal, characteristic
4#21454775C211yesmaximal, characteristic, complemented
5#21445166C214yesmaximal, complemented
6#21445159C214yesmaximal, complemented
7#21444574C211yesmaximal, characteristic, complemented
8#21444503C211yesmaximal, characteristic, complemented
9C61×#1545C211yesmaximal, characteristic, complemented
10C2×#14519803—21no—
11C22×#11758200—22no—
12C2×#14519777C2211yescharacteristic
13C4×#11758222C2211yescharacteristic
14#16840134—21no—
15#16840134—21no—
16#16840134C2214yes—
17#16840058—21no—
18#16840053—21no—
19#16840048C2212yescomplemented
20#16840043C2212yescomplemented
21#16839959—21no—
22#16839959C2212yes—
23#16839954—21no—
24#16839954C2212yes—
25#16839886C2212yescomplemented
26#16839827C2212yescomplemented
27#16839825—22no—
28#16839813—21no—
29#16839804—21no—
30#16839786C2211yescharacteristic
31#16839767—21no—
32#16839764—21no—
33C2×C4×#92418C2211yescharacteristic
34C4×C8×D61—21no—
35#16839512C2214yescomplemented
36#16839509C2214yescomplemented
37C2×#127×C61C2211yescharacteristic
38C61×#431C2211yescharacteristic
39C61×#423C2211yescharacteristic
40C61×#331C2214yes—
41C2×#11758222D411yescharacteristic
42C22×Dic122—21no—
43#14519803—44no—
44C2×#11758200—22no—
45C2×#11758200—24no—
46C2×#11758200—24no—
47C2×#11758200—48no—
48C2×#11758199—22no—
49C2×#11758194—21no—
50C22×#92418D412yes—
51C2×C8×D61—22no—
52#14519777—21no—
53#14519777—21no—
54#14519777—21no—
55#14519777C2312yes—
56C4×Dic122D411yescharacteristic
57C4×Dic122C2311yescharacteristic
58C4×D244D411yescharacteristic
59C4×D244C2311yescharacteristic
60C4×C22⋊D61—21no—
61C4×C22⋊D61—21no—
62#14519736D411yescharacteristic
63#14519734—21no—
64C2×C4⋊Dic61—21no—
65C2×C4⋊Dic61—21no—
66C42×D61—21no—
67#14519711—22no—
68#14519709—22no—
69C4⋊#92418—21no—
70C4⋊#92418C2312yes—
71#14519706—22no—
72#14519702—22no—
73#14519699—22no—
74#14519698—22no—
75#14519698—22no—
76#14519689—22no—
77C4×#92418D411yescharacteristic
78C4×#92418D411yescharacteristic
79C4×#92418C2312yes—
80C8×Dic61—22no—
81C22×D4×C61—21no—
82C22×D4×C61—21no—
83C22×D4×C61—21no—
84C2×M16×C61—21no—
85#127×C61—21no—
86#127×C61—21no—
87#127×C61—21no—
88#127×C61C2312yes—
89C2×C42×C61C2311yescharacteristic
90C4⋊C8×C61—21no—
91C4⋊C8×C61C2312yes—
92C4⋊C8×C61C2312yes—
93C22⋊C8×C61—21no—
94C4×C8×C61—21no—
95C24×C61—41no—
96C2×D4×C61—22no—
97C2×D4×C61—22no—
98C2×D4×C61—22no—
99C2×D4×C61—44no—
100C2×D4×C61—41no—
101C2×D4×C61—41no—
102C2×D4×C61—41no—
103C2×D4×C61—42no—
104C2×D4×C61—42no—
105C2×D4×C61—82no—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61#4111yescharacteristic
109C2×C22⋊D61—21no—
110C22×Dic61—41no—
111#11758222—22no—
112#11758222SD1614yescomplemented
113C2×D244C2×D411yescharacteristic
114C2×C4×D61—21no—
115C2×C4×D61—21no—
116C2×C4×D61—21no—
117C2×Dic122—22no—
118C2×Dic122—44no—
119C2×Dic122—41no—
120C2×Dic122C2×D411yescharacteristic
121M16×C61—42no—
122C2×C8×C61—22no—
123C2×C8×C61—22no—
124C42×C61C2×D411yescharacteristic
125C42×C61C2×D411yescharacteristic
126C42×C61#4111yescharacteristic
127C42×C61C2411yescharacteristic
128C22⋊Dic61—21no—
129#11758200—48no—
130#11758200—44no—
131#11758200—48no—
132#11758199—44no—
133#11758198—21no—
134#11758198—21no—
135C4⋊Dic61—22no—
136C4⋊Dic61—22no—
137C4⋊Dic61—41no—
138C4⋊Dic61—41no—
139C4⋊Dic61C2×D411yescharacteristic
140C4⋊Dic61C2×D411yescharacteristic
141#11758196—21no—
142#11758196—21no—
143C4×Dic61—21no—
144C4×Dic61—21no—
145C4×Dic61—21no—
146#11758194—42no—
147C2×#92418—24no—
148C2×#92418—22no—
149C2×#92418C2×D412yes—
150C2×#92418C2×D412yes—
151C8×D61—44no—
152C23×C61—42no—
153C23×C61—82no—
154C23×C61—82no—
155C23×C61—82no—
156C23×C61#12111yescharacteristic
157C22×D61—21no—
158D4×C61—42no—
159D4×C61—42no—
160D4×C61—44no—
161D4×C61—82no—
162D4×C61—82no—
163D4×C61—82no—
164D4×C61—82no—
165C2×C4×C61—22no—
166C2×C4×C61—22no—
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61C2×SD1612yes—
171C2×C4×C61#13512yes—
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C2×#4111yescharacteristic
175C22⋊D61—24no—
176C2×Dic61—21no—
177C2×Dic61—21no—
178C2×Dic61—21no—
179C2×Dic61—42no—
180C2×Dic61—82no—
181D244—21no—
182D244C2×SD1612yescomplemented
183C4×D61—24no—
184C4×D61—42no—
185C4×D61—42no—
186Dic122—21no—
187Dic122—44no—
188Dic122—82no—
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—82no—
199C22×C61—82no—
200C22×C61—82no—
201C22×C61—164no—
202C22×C61#46512yes—
203C22×C61C2×#12111yescharacteristic
204C2×D61—22no—
205C2×D61—41no—
206C2×D61—41no—
207C4×C61—21no—
208C4×C61—21no—
209C4×C61#46512yes—
210C4×C61#47112yes—
211C4×C61C22×SD1611yescharacteristic
212C4×C61C2×#13511yescharacteristic
213C4×C61C2×AGL(1,ℤ/8)12yes—
214Dic61—22no—
215Dic61—41no—
216Dic61—41no—
217Dic61—82no—
218C2×#492—611no—
219#3079—611no—
220#3038—611no—
221C8×#41—611no—
222#1666—614no—
223#1665—614no—
224#1566—611no—
225#1549—611no—
226#1545D6111yescharacteristic, complemented
227C2×C61—22no—
228C2×C61—162no—
229C2×C61#303811yescharacteristic
230C2×C61C2×#46511yescharacteristic
231C2×C61C2×#47111yescharacteristic
232D61—42no—
233C2×AGL(1,ℤ/8)—1221no—
234C22×SD16—1222no—
235C2×C4⋊Q8—611no—
236C2×#127C2×D6111yescharacteristic
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#332—614no—
263#331C2×D6114yes—
264C61#1926611yescharacteristic, complemented, Sylow
265C2×#41—611no—
266C22×Q8—1221no—
267C22×D4—21no—
268C22×D4—21no—
269C22×D4—21no—
270AGL(1,ℤ/8)—2444no—
271C2×SD16—1224no—
272C2×SD16—1224no—
273C2×SD16—1222no—
274C2×SD16—2448no—
275C2×D8—1222no—
276C2×M16—21no—
277C2×M16—1221no—
278C22×C8—612no—
279C22×C8—1222no—
280C4⋊Q8—612no—
281C4⋊Q8—1221no—
282C4⋊Q8—1221no—
283C4⋊Q8—1221no—
284#127—21no—
285#127—21no—
286#127—21no—
287#127C22×D6112yes—
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—1221no—
297C2×C4⋊C4—1221no—
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—41no—
320#41—614no—
321#41—1222no—
322C2×Q8—611no—
323C2×Q8—1222no—
324C2×Q8—2444no—
325C2×Q8—2441no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—44no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—82no—
336C2×D4—611no—
337C2×D4—1221no—
338C22×C4—1221no—
339C22×C4—1221no—
340C22×C4—1221no—
341C22×C4—2441no—
342C22×C4#1175822211yescharacteristic
343C22×C4D4×D6111yescharacteristic
344C22×C4D4×D6111yescharacteristic
345SD16—2448no—
346SD16—2448no—
347SD16—2444no—
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—611no—
359C4⋊C4—611no—
360C4⋊C4—1221no—
361C4⋊C4—1221no—
362C4⋊C4—1222no—
363C4⋊C4—1222no—
364C4⋊C4—2441no—
365C4⋊C4—2441no—
366C22⋊C4—1221no—
367C22⋊C4—1221no—
368C22⋊C4—1221no—
369C42—1221no—
370C42—1221no—
371C42—1221no—
372C42#1175822211yescharacteristic
373C42D4×D6111yescharacteristic
374C42D4×D6111yescharacteristic
375C42C23×D6111yescharacteristic
376C23—42no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—1221no—
381C23#1451975011yescharacteristic
382Q8—612no—
383Q8—1221no—
384Q8—4882no—
385Q8—2444no—
386D4—42no—
387D4—42no—
388D4—44no—
389D4—82no—
390D4—82no—
391D4—82no—
392D4—82no—
393D4—612no—
394D4—1224no—
395D4—1221no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—1221no—
399C2×C4—1224no—
400C2×C4—1221no—
401C2×C4—1221no—
402C2×C4—4882no—
403C2×C4—2442no—
404C2×C4—2442no—
405C2×C4—2442no—
406C2×C4#1451975011yescharacteristic
407C2×C4#1451975011yescharacteristic
408C2×C4#1451975011yescharacteristic
409C2×C4SD16×D6112yes—
410C2×C4#1451979612yes—
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—82no—
422C22—82no—
423C22—82no—
424C22—164no—
425C22—1222no—
426C22—2441no—
427C22—2441no—
428C22#1683999012yes—
429C22C2×#1451975011yescharacteristic
430C4—21no—
431C4—21no—
432C4—1222no—
433C4—4882no—
434C4—2441no—
435C4—2441no—
436C4#1683999012yes—
437C4#1684002412yes—
438C4C2×SD16×D6111yescharacteristic
439C4C2×#1451979611yescharacteristic
440C4AGL(1,ℤ/8)×D6112yes—
441C2—22no—
442C2—162no—
443C2—2442no—
444C2#2145529811yescharacteristic
445C2C2×#1683999011yescharacteristic
446C2C2×#1684002411yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1545C2
socleC22×C61C2×#121
prime radical#21455295C2
hypercenter#1545D61
hypocenterC61#19266
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1545

Complements

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

NH ≅ G/Ntype
1. C2×#16840134C2semidirect
2. #21455635C2semidirect
4. #21454775C2semidirect
5. #21445166C2semidirect
6. #21445159C2semidirect
7. #21444574C2semidirect
8. #21444503C2semidirect
9. C61×#1545C2semidirect
19. #16840048C22semidirect
20. #16840043C22semidirect
25. #16839886C22semidirect
26. #16839827C22semidirect
35. #16839512C22semidirect
36. #16839509C22semidirect
112. #11758222SD16semidirect
182. D244C2×SD16semidirect
189. Dic122C2×SD16semidirect
194. #92418AGL(1,ℤ/8)semidirect
226. #1545D61semidirect
264. C61#19266semidirect

Maximal subgroups

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

subgroupindex
C2×#168401342
#214556352
#214552952
#214547752
#214451662
#214451592
#214445742
#214445032
C61×#15452
#1926661