Finite Group Databasebeta

← #29835341

Subgroups — #29835341

Order 15616.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456359C211yesmaximal, characteristic, complemented
2#21456300C211yesmaximal, characteristic
3#21448059C212yesmaximal, complemented
4#21448054C212yesmaximal, complemented
5#21448049C212yesmaximal, complemented
6#21448041C212yesmaximal, complemented
7#21447532C211yesmaximal, characteristic, complemented
8#21447524C211yesmaximal, characteristic, complemented
9C2×#16839563C211yesmaximal, characteristic, complemented
10#21447418C211yesmaximal, characteristic, complemented
11C61×#2140C211yesmaximal, characteristic, complemented
12C2×#14519803—21no—
13C2×#14519799C2211yescharacteristic
14#16840390—21no—
15#16840329—21no—
16C22×#11758185C2211yescharacteristic
17#16840169—22no—
18#16840166C2212yescomplemented
19#16840163C2212yescomplemented
20#16840150—22no—
21#16840145C2212yes—
22#16840137C2212yes—
23#16839805—21no—
24C2×#14519711C2211yescharacteristic
25#16839717C2211yescharacteristic
26#16839567C2212yescomplemented
27#16839565C2211yescharacteristic, complemented
28#16839565C2211yescharacteristic, complemented
29#16839564C2212yescomplemented
30#16839563C2212yes—
31#16839563C2212yes—
32#16839562C2211yescharacteristic, complemented
33#16839562C2211yescharacteristic, complemented
34#16839537C2212yescomplemented
35C16×Dic61C2212yescomplemented
36C22×D8×C61C2211yescharacteristic
37C2×M32×C61C2211yescharacteristic
38C61×#427C2211yescharacteristic
39C61×#357C2212yes—
40C61×#357C2212yes—
41C2×#11758222—21no—
42#14519803—44no—
43#14519799—22no—
44#14519799—22no—
45C2×Dic244C2311yescharacteristic
46C2×#11758190—21no—
47C2×#11758200—22no—
48C2×D488C2311yescharacteristic
49C2×#11758199—22no—
50C2×#11758194D411yescharacteristic
51#14519777—41no—
52#14519776—22no—
53#14519773—22no—
54#14519769—41no—
55#14519768—22no—
56#14519764—22no—
57#14519754—42no—
58#14519748—41no—
59#14519748—41no—
60#14519746—42no—
61#14519736D411yescharacteristic
62C2×C4⋊Dic61—21no—
63C2×#11758185—22no—
64C2×#11758185C2311yescharacteristic
65C2×#11758185C2311yescharacteristic
66#14519711—21no—
67#14519711C2312yes—
68#14519710—22no—
69#14519708—22no—
70#14519687D412yes—
71C8×Dic61C2312yes—
72C22×D4×C61—21no—
73C2×D8×C61—21no—
74C2×D8×C61—22no—
75C2×D8×C61—22no—
76C2×D8×C61C2312yes—
77C22×C8×C61C2311yescharacteristic
78#117×C61—21no—
79M32×C61—22no—
80C2×C16×C61C2312yes—
81#107×C61C2311yescharacteristic
82#107×C61C2311yescharacteristic
83#106×C61—21no—
84C24×C61—41no—
85C2×D4×C61—22no—
86C2×D4×C61—42no—
87C2×D4×C61—41no—
88C2×D4×C61—42no—
89C22×C4×C61C2×D411yescharacteristic
90C2×C22⋊D61—41no—
91C2×C22⋊D61—42no—
92C2×C22⋊D61—42no—
93C22×Dic61—41no—
94#11758222—42no—
95#11758222—42no—
96C2×D244—21no—
97C2×C4×D61—41no—
98C2×Dic122—21no—
99C2×Dic122—42no—
100D8×C61—22no—
101D8×C61—22no—
102D8×C61—42no—
103C2×C8×C61C2×D411yescharacteristic
104C2×C8×C61#4112yes—
105C2×C8×C61#4112yes—
106C2×C8×C61C2411yescharacteristic
107C4⋊C4×C61—21no—
108C4⋊C4×C61—21no—
109C22⋊C4×C61—41no—
110C42×C61—41no—
111C22⋊Dic61—42no—
112C22⋊Dic61—42no—
113C22⋊Dic61D812yes—
114#11758200—44no—
115#11758199—44no—
116C4⋊Dic61—22no—
117C4⋊Dic61—41no—
118C4⋊Dic61—42no—
119C4⋊Dic61D811yescharacteristic
120C4⋊Dic61D811yescharacteristic
121#11758196—42no—
122#11758196—42no—
123C4×Dic61C2×D412yes—
124#11758194SD1614yes—
125C2×#92418C2×D412yes—
126Dic244—21no—
127Dic244—21no—
128D488—21no—
129D488—21no—
130#11758190—42no—
131C16×C61—22no—
132#11758185—21no—
133#11758185—21no—
134C23×C61—42no—
135C23×C61—82no—
136C23×C61—161no—
137C23×C61—161no—
138C23×C61C2×D811yescharacteristic
139C22×D61—41no—
140D4×C61—42no—
141D4×C61—42no—
142D4×C61—82no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61#12412yes—
146C2×C4×C61C2×D811yescharacteristic
147C2×C4×C61C2×SD1612yes—
148C2×C4×C61C22×D411yescharacteristic
149C22⋊D61—82no—
150C22⋊D61—162no—
151C22⋊D61—162no—
152C2×Dic61—42no—
153C2×Dic61—41no—
154C2×Dic61—161no—
155C2×Dic61C2×D812yes—
156C2×Dic61C2×D812yes—
157D244—41no—
158D244—41no—
159C4×D61—82no—
160Dic122—41no—
161Dic122—41no—
162Dic122—162no—
163C8×C61#12412yes—
164C8×C61C2×#4111yescharacteristic
165C8×C61C2×#4111yescharacteristic
166#92418C2×SD1614yes—
167#33702—611nomaximal, Sylow
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—82no—
171C22×C61—82no—
172C22×C61—82no—
173C22×C61—164no—
174C22×C61#48612yes—
175C22×C61C22×D811yescharacteristic
176C2×D61—81no—
177C2×D61—81no—
178C2×D61—81no—
179C4×C61—81no—
180C4×C61—81no—
181C4×C61#48612yes—
182C4×C61C2×#12411yescharacteristic
183C4×C61C22×SD1611yescharacteristic
184Dic61—22no—
185Dic61—22no—
186Dic61—81no—
187Dic61—82no—
188#3164—611no—
189#3155—611no—
190#2244—612no—
191#2243—612no—
192#2242—612no—
193#2240—612no—
194#2140D6111yescharacteristic, complemented
195#2139—611no—
196#2138—611no—
197C2×#357—611no—
198#2108—611no—
199C2×C61—22no—
200C2×C61—162no—
201C2×C61C2×#48611yescharacteristic
202C2×C61#341511yescharacteristic
203C2×C61#341611yescharacteristic
204D61—161no—
205C2×AGL(1,ℤ/8)—1221no—
206C2×#135—611no—
207C22×D8C2×D6111yescharacteristic
208#544—1221no—
209#535—1221no—
210C2×M32C2×D6111yescharacteristic
211C22×C16—611no—
212#501—1222no—
213#500—612no—
214#499—612no—
215#496—1222no—
216#495—612no—
217#493—612no—
218#428—1221no—
219#427C2×D6111yescharacteristic
220C2×#107—611no—
221#405—611no—
222#358—612no—
223#358—611no—
224#358—611no—
225#357—612no—
226#357—612no—
227#357—612no—
228#357—611no—
229#357—611no—
230#357C2×D6112yes—
231#357C2×D6112yes—
232#346—612no—
233C4×C16—612no—
234C61#3370211yescharacteristic, complemented, Sylow
235C2×#41—1221no—
236C22×D4—21no—
237AGL(1,ℤ/8)—2444no—
238#135—1222no—
239#135—1222no—
240C2×Q16—611no—
241C2×SD16—1222no—
242C2×SD16—1221no—
243C2×D8—22no—
244C2×D8—21no—
245C2×D8—22no—
246C2×D8—611no—
247C2×D8—1222no—
248C2×D8C22×D6112yes—
249C2×M16—611no—
250C22×C8C22×D6111yescharacteristic
251C4⋊Q8—1222no—
252C4⋊Q8—2441no—
253#127—1222no—
254#125—1222no—
255#125—2441no—
256#124—1222no—
257C22⋊Q8—2442no—
258#121—2441no—
259#121—2441no—
260#120—2442no—
261#117—21no—
262#117—611no—
263C2×C4⋊C4—1221no—
264M32—22no—
265C2×C16—611no—
266C2×C16—611no—
267C2×C16—1222no—
268C2×C16C22×D6112yes—
269#107—612no—
270#107—1222no—
271#107—1221no—
272#107C22×D6111yescharacteristic
273#107C22×D6111yescharacteristic
274#106—21no—
275#106—1222no—
276#97—612no—
277C4×C8—612no—
278C24—41no—
279#41—2442no—
280#41—2442no—
281C2×Q8—1221no—
282C2×Q8—2442no—
283C2×D4—22no—
284C2×D4—42no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—1221no—
288C2×D4—2442no—
289C2×D4—2442no—
290C2×D4—2441no—
291C22×C4—2441no—
292C22×C4—2441no—
293C22×C4D4×D6111yescharacteristic
294Q16—1221no—
295Q16—1221no—
296SD16—2444no—
297SD16—2442no—
298D8—22no—
299D8—22no—
300D8—42no—
301D8—1221no—
302D8—1221no—
303D8—2444no—
304M16—614no—
305C2×C8—612no—
306C2×C8#1175822212yes—
307C2×C8D4×D6111yescharacteristic
308C2×C8#1175822412yes—
309C2×C8C23×D6111yescharacteristic
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—611no—
313C4⋊C4—611no—
314C4⋊C4—1222no—
315C4⋊C4—2442no—
316C4⋊C4—2442no—
317C4⋊C4—2442no—
318C4⋊C4—2441no—
319C22⋊C4—41no—
320C22⋊C4—612no—
321C22⋊C4—2442no—
322C22⋊C4—2442no—
323C42—41no—
324C42—612no—
325C16—22no—
326C16—1221no—
327C16—1221no—
328C23—42no—
329C23—82no—
330C23—161no—
331C23—161no—
332C23—2441no—
333C23D8×D6111yescharacteristic
334Q8—9762no—
335Q8—2441no—
336Q8—2441no—
337D4—42no—
338D4—42no—
339D4—82no—
340D4—9762no—
341D4—9762no—
342D4—4882no—
343D4—2441no—
344D4—2441no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—612no—
348C2×C4—612no—
349C2×C4—9761no—
350C2×C4—4882no—
351C2×C4—2441no—
352C2×C4—2442no—
353C2×C4#1451976312yes—
354C2×C4D8×D6111yescharacteristic
355C2×C4SD16×D6112yes—
356C2×C4C2×D4×D6111yescharacteristic
357C8—614no—
358C8#1451976312yes—
359C8C2×#1175822411yescharacteristic
360C8C2×#1175822211yescharacteristic
361C22—21no—
362C22—21no—
363C22—82no—
364C22—82no—
365C22—82no—
366C22—164no—
367C22—4881no—
368C22—4881no—
369C22—4881no—
370C22#1684010712yes—
371C22C2×D8×D6111yescharacteristic
372C4—81no—
373C4—81no—
374C4—1222no—
375C4—1222no—
376C4—4882no—
377C4—4881no—
378C4#1684010712yes—
379C4C2×#1451976311yescharacteristic
380C4C2×SD16×D6111yescharacteristic
381C2—22no—
382C2—162no—
383C2—9761no—
384C2C2×#1684010711yescharacteristic
385C2#2145895911yescharacteristic
386C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2140C2
socleC22×C61C22×D8
prime radical#21456300C2
hypercenter#2140D61
hypocenterC61#33702
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2140

Complements

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

NH ≅ G/Ntype
1. #21456359C2semidirect
3. #21448059C2semidirect
4. #21448054C2semidirect
5. #21448049C2semidirect
6. #21448041C2semidirect
7. #21447532C2semidirect
8. #21447524C2semidirect
9. C2×#16839563C2semidirect
10. #21447418C2semidirect
11. C61×#2140C2semidirect
18. #16840166C22semidirect
19. #16840163C22semidirect
26. #16839567C22semidirect
27. #16839565C22semidirect
28. #16839565C22semidirect
29. #16839564C22semidirect
32. #16839562C22semidirect
33. #16839562C22semidirect
34. #16839537C22semidirect
35. C16×Dic61C22semidirect
194. #2140D61semidirect
234. C61#33702semidirect

Maximal subgroups

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

subgroupindex
#214563592
#214563002
#214480592
#214480542
#214480492
#214480412
#214475322
#214475242
C2×#168395632
#214474182
C61×#21402
#3370261