Finite Group Databasebeta

← #29661229

Subgroups — #29661229

Order 15616.

1152 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1152 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456806C212yesmaximal, complemented
2#21456283C211yesmaximal, characteristic, complemented
3#21445125C214yesmaximal, complemented
4#21445108C214yesmaximal, complemented
5#21444443C211yesmaximal, characteristic, complemented
6#21444417C212yesmaximal, complemented
7C2×C61×H(3,𝔽4)C211yesmaximal, characteristic, complemented
8C2×#14519808—21no—
9C2×#14519803—21no—
10#16840312C2211yescharacteristic, complemented
11#16840157C2212yescomplemented
12#16840152C2212yescomplemented
13#16840152C2214yescomplemented
14#16840144—22no—
15#16840121C2214yescomplemented
16#16839974—24no—
17#16839901—22no—
18#16839896—22no—
19#16839819C2212yescomplemented
20C4×#11758194C2211yescharacteristic, complemented
21#16839502C2218yescomplemented
22#16839502C2214yescomplemented
23C61×H(3,𝔽4)—24no—
24C61×H(3,𝔽4)—22no—
25C61×H(3,𝔽4)C2214yes—
26C2×#124×C61—21no—
27C2×#124×C61—22no—
28C2×#124×C61C2212yes—
29C2×#124×C61C2211yescharacteristic
30C2×#120×C61—21no—
31C2×#120×C61—22no—
32C2×#11758222—21no—
33#14519808—44no—
34#14519803—44no—
35C2×#11758202—22no—
36C2×#11758200—22no—
37C2×#11758201—22no—
38C2×#11758201—24no—
39C2×#11758199—22no—
40C2×#11758199—24no—
41C2×#11758194—21no—
42#14519777C2311yescharacteristic, complemented
43#14519774C2311yescharacteristic, complemented
44#14519765—21no—
45#14519751—22no—
46C4×D244—22no—
47C4⋊#92418—22no—
48#14519701—24no—
49#14519698—24no—
50C22⋊#92418—22no—
51#14519688C2312yescomplemented
52#14519688C2314yescomplemented
53C4×#92418—21no—
54C25×C61—21no—
55C22×Q8×C61D411yescharacteristic
56C22×Q8×C61D412yes—
57C22×D4×C61—22no—
58C22×D4×C61—21no—
59C22×D4×C61D411yescharacteristic
60C22×D4×C61D412yes—
61#124×C61—28no—
62#124×C61—24no—
63#124×C61—28no—
64#124×C61—24no—
65#124×C61—22no—
66#124×C61—22no—
67#124×C61—22no—
68#124×C61D414yes—
69#124×C61D412yes—
70#124×C61C2314yes—
71#124×C61C2312yes—
72#120×C61—28no—
73#120×C61—24no—
74#120×C61—28no—
75#120×C61—24no—
76C2×C22⋊C4×C61—21no—
77C2×C22⋊C4×C61—22no—
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61—22no—
81C2×C42×C61—21no—
82C2×C42×C61C2311yescharacteristic
83C24×C61—24no—
84C24×C61—21no—
85C24×C61—22no—
86C24×C61—84no—
87C24×C61—82no—
88C2×Q8×C61—22no—
89C2×Q8×C61—21no—
90C2×Q8×C61—42no—
91C2×Q8×C61—82no—
92C2×Q8×C61C2×D414yes—
93C2×Q8×C61C2×D412yes—
94C2×D4×C61—24no—
95C2×D4×C61—28no—
96C2×D4×C61—22no—
97C2×D4×C61—21no—
98C2×D4×C61—42no—
99C2×D4×C61—82no—
100C2×D4×C61—84no—
101C2×D4×C61—82no—
102C2×D4×C61C2×D414yes—
103C2×D4×C61C2×D412yes—
104C22×C4×C61—22no—
105C22×C4×C61—21no—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D412yes—
108C2×C22⋊D61—41no—
109#11758222—42no—
110#11758222—42no—
111C2×D244—21no—
112C2×D244—22no—
113C2×C4×D61—41no—
114C2×Dic122—21no—
115C22⋊C4×C61—28no—
116C22⋊C4×C61—24no—
117C22⋊C4×C61—24no—
118C22⋊C4×C61—24no—
119C22⋊C4×C61—24no—
120C22⋊C4×C61—24no—
121C22⋊C4×C61—24no—
122C22⋊C4×C61—24no—
123C42×C61—24no—
124C42×C61C2×D411yescharacteristic
125C42×C61C2×D412yes—
126C42×C61C2411yescharacteristic
127#11758202—44no—
128#11758201—44no—
129#11758201—84no—
130#11758200—44no—
131#11758199—44no—
132#11758199—84no—
133#11758198—42no—
134C4⋊Dic61—22no—
135#11758194—24no—
136C2×#92418—22no—
137C2×#92418—24no—
138C23×C61—22no—
139C23×C61—24no—
140C23×C61—44no—
141C23×C61—44no—
142C23×C61—44no—
143C23×C61—42no—
144C23×C61—44no—
145C23×C61—84no—
146C23×C61—88no—
147C23×C61—816no—
148C23×C61#12011yescharacteristic
149C22×D61—41no—
150Q8×C61—42no—
151Q8×C61—44no—
152Q8×C61—81no—
153Q8×C61—82no—
154D4×C61—42no—
155D4×C61—44no—
156D4×C61—81no—
157D4×C61—82no—
158D4×C61—84no—
159D4×C61—88no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—24no—
163C2×C4×C61—42no—
164C2×C4×C61—44no—
165C2×C4×C61—42no—
166C2×C4×C61#12012yes—
167C2×C4×C61#12011yescharacteristic
168C2×C4×C61C22×D412yes—
169C2×C4×C61C22×D411yescharacteristic
170C22⋊D61—82no—
171C2×Dic61—41no—
172D244—41no—
173D244—44no—
174D244—41no—
175C4×D61—82no—
176Dic122—41no—
177Dic122—41no—
178#92418—24no—
179#92418—44no—
180#18992—611nomaximal, Sylow
181C22×C61—22no—
182C22×C61—42no—
183C22×C61—44no—
184C22×C61—44no—
185C22×C61—44no—
186C22×C61—44no—
187C22×C61—84no—
188C22×C61—88no—
189C22×C61—816no—
190C22×C61#45312yes—
191C22×C61C2×#12011yescharacteristic
192C2×D61—81no—
193C2×D61—82no—
194C4×C61—21no—
195C4×C61—21no—
196C4×C61—22no—
197C4×C61—22no—
198C4×C61—42no—
199C4×C61—44no—
200Dic61—81no—
201C2×H(3,𝔽4)D6111yescharacteristic, complemented
202#3213—612no—
203#3153—611no—
204#1659—614no—
205#1655—614no—
206#1533—611no—
207#1527—612no—
208C2×C61—22no—
209C2×C61—82no—
210C2×C61—84no—
211C2×C61C2×#45311yescharacteristic
212C2×C61#301912yes—
213D61—161no—
214C2×#137—1221no—
215C2×AGL(1,ℤ/8)—1221no—
216H(3,𝔽4)—24no—
217H(3,𝔽4)—22no—
218H(3,𝔽4)C2×D6114yes—
219#532—611no—
220C2×#124—21no—
221C2×#124—22no—
222C2×#124C2×D6111yescharacteristic
223C2×#124C2×D6112yes—
224C2×#120—21no—
225C2×#120—22no—
226#497—612no—
227#496—612no—
228#496—614no—
229#495—1222no—
230#489—614no—
231#463—1224no—
232#448—1222no—
233#447—1222no—
234#432—612no—
235C4×M16—611no—
236#329—618no—
237#329—614no—
238C61#1899211yescharacteristic, complemented, Sylow
239C25—21no—
240C2×#41—1221no—
241C22×Q8C22⋊D6111yescharacteristic
242C22×Q8C22⋊D6112yes—
243C22×D4—22no—
244C22×D4—21no—
245C22×D4C22⋊D6111yescharacteristic
246C22×D4C22⋊D6112yes—
247#137—2444no—
248AGL(1,ℤ/8)—2444no—
249C2×Q16—1222no—
250C2×SD16—1224no—
251C2×SD16—1222no—
252C2×SD16—1222no—
253C2×D8—1224no—
254C2×D8—1222no—
255C2×M16—1221no—
256C4⋊Q8—611no—
257#127—611no—
258#124—28no—
259#124—24no—
260#124—28no—
261#124—24no—
262#124—22no—
263#124—22no—
264#124—22no—
265#124—1221no—
266#124C22⋊D6114yes—
267#124C22⋊D6112yes—
268#124C22×D6114yes—
269#124C22×D6112yes—
270#121—1222no—
271#120—28no—
272#120—24no—
273#120—28no—
274#120—24no—
275C4×D4—1222no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C42—21no—
282C2×C42C22×D6111yescharacteristic
283C4⋊C8—1222no—
284#103—1224no—
285#102—1224no—
286C22⋊C8—1222no—
287#97—612no—
288#97—614no—
289C4×C8—1221no—
290C24—24no—
291C24—21no—
292C24—22no—
293C24—84no—
294C24—82no—
295#41—2442no—
296#41—2442no—
297C2×Q8—22no—
298C2×Q8—21no—
299C2×Q8—42no—
300C2×Q8—82no—
301C2×Q8—1221no—
302C2×Q8C2×C22⋊D6114yes—
303C2×Q8C2×C22⋊D6112yes—
304C2×D4—24no—
305C2×D4—28no—
306C2×D4—22no—
307C2×D4—21no—
308C2×D4—42no—
309C2×D4—82no—
310C2×D4—84no—
311C2×D4—82no—
312C2×D4—1222no—
313C2×D4—1221no—
314C2×D4—2441no—
315C2×D4C2×C22⋊D6114yes—
316C2×D4C2×C22⋊D6112yes—
317C22×C4—22no—
318C22×C4—21no—
319C22×C4—2441no—
320C22×C4C2×C22⋊D6111yescharacteristic
321C22×C4C2×C22⋊D6112yes—
322Q16—2444no—
323SD16—4884no—
324SD16—2444no—
325SD16—2444no—
326D8—4884no—
327D8—2444no—
328M16—1224no—
329C2×C8—1222no—
330C2×C8—1224no—
331C4⋊C4—1222no—
332C22⋊C4—28no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—2442no—
341C42—24no—
342C42C2×C22⋊D6111yescharacteristic
343C42C2×C22⋊D6112yes—
344C42C23×D6111yescharacteristic
345C23—22no—
346C23—24no—
347C23—44no—
348C23—44no—
349C23—44no—
350C23—42no—
351C23—44no—
352C23—84no—
353C23—88no—
354C23—816no—
355C23—2441no—
356C23#1451974611yescharacteristic
357Q8—44no—
358Q8—42no—
359Q8—81no—
360Q8—82no—
361Q8—2441no—
362Q8—2441no—
363D4—44no—
364D4—42no—
365D4—81no—
366D4—82no—
367D4—84no—
368D4—88no—
369D4—4882no—
370D4—2444no—
371D4—2441no—
372D4—2441no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—4882no—
380C2×C4—2441no—
381C2×C4#1451974612yes—
382C2×C4#1451974611yescharacteristic
383C2×C4C22×C22⋊D6112yes—
384C2×C4C22×C22⋊D6111yescharacteristic
385C8—1224no—
386C8—2444no—
387C22—22no—
388C22—42no—
389C22—44no—
390C22—44no—
391C22—44no—
392C22—44no—
393C22—84no—
394C22—88no—
395C22—816no—
396C22—4881no—
397C22—4882no—
398C22#1683993012yes—
399C22C2×#1451974611yescharacteristic
400C4—21no—
401C4—21no—
402C4—22no—
403C4—22no—
404C4—42no—
405C4—44no—
406C4—4881no—
407C2—22no—
408C2—82no—
409C2—84no—
410C2—9761no—
411C2C2×#1683993011yescharacteristic
412C2#2145517612yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×H(3,𝔽4)C2
socleC22×C61C2×#120
hypercenterC2×H(3,𝔽4)D61
hypocenterC61#18992
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×H(3,𝔽4)

Complements

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

NH ≅ G/Ntype
1. #21456806C2semidirect
2. #21456283C2semidirect
3. #21445125C2semidirect
4. #21445108C2semidirect
5. #21444443C2semidirect
6. #21444417C2semidirect
7. C2×C61×H(3,𝔽4)C2semidirect
10. #16840312C22semidirect
11. #16840157C22semidirect
12. #16840152C22semidirect
13. #16840152C22semidirect
15. #16840121C22semidirect
19. #16839819C22semidirect
20. C4×#11758194C22semidirect
21. #16839502C22semidirect
22. #16839502C22semidirect
42. #14519777C23semidirect
43. #14519774C23semidirect
51. #14519688C23semidirect
52. #14519688C23semidirect
201. C2×H(3,𝔽4)D61semidirect
238. C61#18992semidirect

Maximal subgroups

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

subgroupindex
#214568062
#214562832
#214451252
#214451082
#214444432
#214444172
C2×C61×H(3,𝔽4)2
#1899261