Finite Group Databasebeta

← #29652785

Subgroups — #29652785

Order 15616.

1386 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1386 conjugacy classes of subgroups — too many to draw. Only its 183 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#21455906C212yesmaximal, complemented
2#21449098C211yesmaximal, characteristic, complemented
3#21444839C214yesmaximal, complemented
4#21444737C214yesmaximal, complemented
5#21444327C212yesmaximal, complemented
6#21444160C211yesmaximal, characteristic, complemented
7C61×#3496C211yesmaximal, characteristic, complemented
8C2×C4×D244C2211yescharacteristic, complemented
9C2×C4×C22⋊D61—22no—
10#16840078C2214yescomplemented
11#16840049—22no—
12#16839954—22no—
13#16839895C2214yescomplemented
14#16839872—24no—
15#16839819C2212yescomplemented
16#16839698C2212yescomplemented
17#16839697—21no—
18#16839671C2212yescomplemented
19#16839670—21no—
20#16839608C2211yescharacteristic, complemented
21#16839496C2218yescomplemented
22#16839486C2214yescomplemented
23C2×H(5,𝔽2)×C61—21no—
24C61×#547—22no—
25C61×#546—24no—
26D42×C61—22no—
27C61×#537—21no—
28C61×#534C2214yes—
29C61×#532C2212yes—
30C2×#121×C61—22no—
31C2×C4×D4×C61C2211yescharacteristic
32C22×D244C2311yescharacteristic
33C22×C22⋊D61—22no—
34C22×C4×D61—21no—
35C2×#11758199—24no—
36C4×D244—22no—
37C4×D244—22no—
38C4×C22⋊D61—48no—
39C2×C4⋊Dic61C2311yescharacteristic
40C2×#11758196—22no—
41C2×C22⋊Dic61—22no—
42C2×#11758198—22no—
43C2×#11758198—21no—
44C2×C4×Dic61—21no—
45#14519708—24no—
46C4⋊#92418—22no—
47#14519698—24no—
48#14519697—24no—
49C22⋊#92418C2314yes—
50#14519688—22no—
51#14519684—21no—
52#14519684C2312yes—
53H(5,𝔽2)×C61—48no—
54C2×#41×C61—24no—
55C2×#41×C61—22no—
56C22×D4×C61—24no—
57C22×D4×C61—24no—
58C22×D4×C61—22no—
59C22×D4×C61D411yescharacteristic
60C23×C4×C61D412yes—
61C4⋊Q8×C61D412yescomplemented
62#127×C61D412yescomplemented
63#124×C61—22no—
64#123×C61—28no—
65C22⋊Q8×C61—24no—
66#121×C61—24no—
67#121×C61—24no—
68#121×C61—24no—
69#121×C61—22no—
70#121×C61—24no—
71#121×C61C2314yes—
72#120×C61—28no—
73C4×D4×C61—22no—
74C4×D4×C61—22no—
75C4×D4×C61—22no—
76C4×D4×C61D414yescomplemented
77C2×C4⋊C4×C61—22no—
78C2×C4⋊C4×C61D411yescharacteristic
79C2×C22⋊C4×C61—24no—
80C2×C22⋊C4×C61C2312yes—
81C2×C42×C61C2311yescharacteristic
82C24×C61—24no—
83C24×C61C2×D412yes—
84C23×D61—21no—
85#41×C61—48no—
86#41×C61—416no—
87C2×Q8×C61—22no—
88C2×D4×C61—24no—
89C2×D4×C61—28no—
90C2×D4×C61—24no—
91C2×D4×C61—24no—
92C2×D4×C61—22no—
93C2×D4×C61—22no—
94C2×D4×C61—22no—
95C2×D4×C61—416no—
96C2×D4×C61—48no—
97C2×D4×C61—48no—
98C2×D4×C61—84no—
99C2×D4×C61C2×D414yes—
100C22×C4×C61—24no—
101C22×C4×C61—22no—
102C22×C4×C61—24no—
103C22×C4×C61—22no—
104C22×C4×C61—44no—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D412yes—
108C22×C4×C61C2411yescharacteristic
109C2×C22⋊D61—44no—
110C2×C22⋊D61—44no—
111C2×C22⋊D61—44no—
112C22×Dic61—22no—
113C22×Dic61—21no—
114C2×D244—22no—
115C2×D244—42no—
116C2×C4×D61—44no—
117C2×C4×D61—41no—
118C2×C4×D61—41no—
119C4⋊C4×C61—24no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—22no—
122C4⋊C4×C61C2×D414yes—
123C22⋊C4×C61—28no—
124C22⋊C4×C61—24no—
125C22⋊C4×C61—24no—
126C22⋊C4×C61—24no—
127C42×C61C2×D412yes—
128C42×C61C2×D412yes—
129C22⋊Dic61—44no—
130#11758199—48no—
131#11758198—44no—
132#11758198—42no—
133C4⋊Dic61—22no—
134#11758196—44no—
135C4×Dic61—24no—
136C2×#92418—24no—
137C23×C61—22no—
138C23×C61—22no—
139C23×C61—24no—
140C23×C61—42no—
141C23×C61—42no—
142C23×C61—48no—
143C23×C61—84no—
144C23×C61—84no—
145C23×C61AGL(1,ℤ/8)14yes—
146C23×C61C22×D411yescharacteristic
147C22×D61—22no—
148C22×D61—42no—
149C22×D61—41no—
150C22×D61—41no—
151C22×D61—42no—
152Q8×C61—44no—
153D4×C61—28no—
154D4×C61—44no—
155D4×C61—44no—
156D4×C61—416no—
157D4×C61—48no—
158D4×C61—44no—
159D4×C61—84no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—24no—
164C2×C4×C61—41no—
165C2×C4×C61—48no—
166C2×C4×C61—48no—
167C2×C4×C61—44no—
168C2×C4×C61—42no—
169C2×C4×C61—82no—
170C2×C4×C61#12012yes—
171C2×C4×C61#12011yescharacteristic
172C2×C4×C61#12011yescharacteristic
173C2×C4×C61C22×D412yes—
174C22⋊D61—416no—
175C2×Dic61—24no—
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61—42no—
179C2×Dic61—41no—
180D244—24no—
181D244—42no—
182C4×D61—82no—
183C4×D61—82no—
184#92418—44no—
185#18178—611nomaximal, Sylow
186C22×C61—21no—
187C22×C61—21no—
188C22×C61—24no—
189C22×C61—44no—
190C22×C61—42no—
191C22×C61—42no—
192C22×C61—44no—
193C22×C61—44no—
194C22×C61—44no—
195C22×C61—82no—
196C22×C61—168no—
197C22×C61C2×#12011yescharacteristic
198C22×C61C2×AGL(1,ℤ/8)12yes—
199C2×D61—44no—
200C2×D61—42no—
201C2×D61—82no—
202C2×D61—81no—
203C2×D61—81no—
204C4×C61—22no—
205C4×C61—21no—
206C4×C61—22no—
207C4×C61—44no—
208C4×C61—42no—
209C4×C61#45312yes—
210Dic61—44no—
211Dic61—81no—
212#3496D6111yescharacteristic, complemented
213#3109—612no—
214#2397—611no—
215#1613—614no—
216#1595—614no—
217#1505—612no—
218#1466—611no—
219C2×C61—22no—
220C2×C61—44no—
221C2×C61—84no—
222C2×C61#300311yescharacteristic
223C2×C61C2×#45311yescharacteristic
224C2×C61#302511yescharacteristic
225D61—82no—
226C2×H(5,𝔽2)—21no—
227#547—22no—
228#546—24no—
229D42—22no—
230#537—21no—
231#534C2×D6114yes—
232#532C2×D6112yes—
233C2×#121—22no—
234C2×C4×D4—611no—
235C2×C4×D4—1222no—
236C2×C4×D4C2×D6111yescharacteristic
237#481—614no—
238#476—1222no—
239#459—1222no—
240#447—614no—
241#442—1224no—
242#432—612no—
243#399—612no—
244#399—1221no—
245#392—612no—
246#392—1221no—
247#376—611no—
248#327—618no—
249#323—614no—
250C61#1817811yescharacteristic, complemented, Sylow
251H(5,𝔽2)—48no—
252C2×#41—24no—
253C2×#41—22no—
254C22×D4—24no—
255C22×D4—24no—
256C22×D4—22no—
257C22×D4—611no—
258C22×D4—1222no—
259C22×D4C22⋊D6111yescharacteristic
260C23×C4—1221no—
261C23×C4C22⋊D6112yes—
262C2×D8—1224no—
263C4⋊Q8C22⋊D6112yescomplemented
264#127C22⋊D6112yescomplemented
265#124—22no—
266#123—28no—
267C22⋊Q8—24no—
268#121—24no—
269#121—24no—
270#121—24no—
271#121—22no—
272#121—24no—
273#121C22×D6114yes—
274#120—28no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—1222no—
279C4×D4—1222no—
280C4×D4—2448no—
281C4×D4C22⋊D6114yescomplemented
282C2×C4⋊C4—22no—
283C2×C4⋊C4—611no—
284C2×C4⋊C4—1222no—
285C2×C4⋊C4C22⋊D6111yescharacteristic
286C2×C22⋊C4—24no—
287C2×C22⋊C4—1222no—
288C2×C22⋊C4—1221no—
289C2×C22⋊C4—1222no—
290C2×C22⋊C4C22×D6112yes—
291C2×C42—1221no—
292C2×C42C22×D6111yescharacteristic
293#106—1224no—
294C4⋊C8—1222no—
295#102—1224no—
296#102—1224no—
297C22⋊C8—614no—
298#97—1222no—
299#95—612no—
300#95—1221no—
301C24—24no—
302C24—1221no—
303C24C2×C22⋊D6112yes—
304#41—48no—
305#41—416no—
306C2×Q8—22no—
307C2×D4—24no—
308C2×D4—28no—
309C2×D4—24no—
310C2×D4—24no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—416no—
315C2×D4—48no—
316C2×D4—48no—
317C2×D4—84no—
318C2×D4—1222no—
319C2×D4—2442no—
320C2×D4—2444no—
321C2×D4—2444no—
322C2×D4—2444no—
323C2×D4C2×C22⋊D6114yes—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—44no—
329C22×C4—1222no—
330C22×C4—1221no—
331C22×C4—2444no—
332C22×C4—2441no—
333C22×C4—2441no—
334C22×C4C2×C22⋊D6111yescharacteristic
335C22×C4C2×C22⋊D6111yescharacteristic
336C22×C4C2×C22⋊D6112yes—
337C22×C4C23×D6111yescharacteristic
338D8—2448no—
339C2×C8—1224no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—1222no—
344C4⋊C4—2444no—
345C4⋊C4C2×C22⋊D6114yes—
346C22⋊C4—28no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—2444no—
351C22⋊C4—2444no—
352C22⋊C4—2442no—
353C42—1224no—
354C42C2×C22⋊D6112yes—
355C42C2×C22⋊D6112yes—
356C23—22no—
357C23—22no—
358C23—24no—
359C23—42no—
360C23—42no—
361C23—48no—
362C23—84no—
363C23—84no—
364C23—1222no—
365C23—2442no—
366C23—2441no—
367C23—2441no—
368C23—2442no—
369C23#1451980314yes—
370C23C22×C22⋊D6111yescharacteristic
371Q8—44no—
372D4—28no—
373D4—44no—
374D4—44no—
375D4—416no—
376D4—48no—
377D4—44no—
378D4—84no—
379D4—1224no—
380D4—2442no—
381D4—24416no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—41no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—82no—
392C2×C4—1224no—
393C2×C4—4882no—
394C2×C4—4882no—
395C2×C4—2442no—
396C2×C4—2441no—
397C2×C4—2442no—
398C2×C4—2442no—
399C2×C4#1451974611yescharacteristic
400C2×C4#1451974612yes—
401C2×C4#1451974611yescharacteristic
402C2×C4C22×C22⋊D6112yes—
403C8—2444no—
404C22—21no—
405C22—21no—
406C22—24no—
407C22—44no—
408C22—42no—
409C22—42no—
410C22—44no—
411C22—44no—
412C22—44no—
413C22—82no—
414C22—168no—
415C22—4882no—
416C22—4881no—
417C22—4881no—
418C22—2444no—
419C22—2442no—
420C22C2×#1451974611yescharacteristic
421C22C2×#1451980312yes—
422C4—21no—
423C4—22no—
424C4—22no—
425C4—42no—
426C4—44no—
427C4—4881no—
428C4—2444no—
429C4#1683993012yes—
430C2—22no—
431C2—44no—
432C2—84no—
433C2—4882no—
434C2#2145504611yescharacteristic
435C2C2×#1683993011yescharacteristic
436C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3496C2
socleC22×C61C2×#120
hypercenter#3496D61
hypocenterC61#18178
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. #21455906C2semidirect
2. #21449098C2semidirect
3. #21444839C2semidirect
4. #21444737C2semidirect
5. #21444327C2semidirect
6. #21444160C2semidirect
7. C61×#3496C2semidirect
8. C2×C4×D244C22semidirect
10. #16840078C22semidirect
13. #16839895C22semidirect
15. #16839819C22semidirect
16. #16839698C22semidirect
18. #16839671C22semidirect
20. #16839608C22semidirect
21. #16839496C22semidirect
22. #16839486C22semidirect
61. C4⋊Q8×C61D4semidirect
62. #127×C61D4semidirect
76. C4×D4×C61D4semidirect
212. #3496D61semidirect
250. C61#18178semidirect
263. C4⋊Q8C22⋊D61semidirect
264. #127C22⋊D61semidirect
281. C4×D4C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214559062
#214490982
#214448392
#214447372
#214443272
#214441602
C61×#34962
#1817861