Finite Group Databasebeta

← #29577303

Subgroups — #29577303

Order 15616.

492 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 89 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#21456818C211yesmaximal, characteristic
2#21448632C211yesmaximal, characteristic
3#21444416C211yesmaximal, characteristic
4#21444370C211yesmaximal, characteristic, complemented
5#21443684C211yesmaximal, characteristic, complemented
6#21443675C211yesmaximal, characteristic, complemented
7C61×#1986C211yesmaximal, characteristic, complemented
8C2×#14519765C2211yescharacteristic
9#16840154C412yescomplemented
10#16840124C412yescomplemented
11#16839984—22no—
12#16839917—21no—
13#16839912—21no—
14#16839819C2211yescharacteristic
15#16839819C2211yescharacteristic
16#16839666—21no—
17#16839653—21no—
18#16839652—21no—
19#16839640—21no—
20#16839633—21no—
21#16839630C2211yescharacteristic
22C4×#11758198—21no—
23#16839501—22no—
24#16839499—21no—
25#16839499—21no—
26C61×#533C2211yescharacteristic
27C4×M16×C61C2211yescharacteristic
28C4×C4⋊C4×C61C2211yescharacteristic
29C61×#339—21no—
30C61×#339—21no—
31C22×Dic122D411yescharacteristic
32C22×D244D411yescharacteristic
33C22×C4×D61—21no—
34C2×#11758202—22no—
35C2×#11758201—22no—
36#14519765—21no—
37#14519765—21no—
38#14519765C2×C412yes—
39#14519765D412yes—
40C2×#11758196—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×C4×Dic61—21no—
47C4⋊#92418—21no—
48C4⋊#92418—21no—
49#14519700—22no—
50#14519697—22no—
51C22⋊#92418—21no—
52C22⋊#92418—21no—
53#14519688—21no—
54#14519688C2×C412yescomplemented
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684D411yescharacteristic
60#14519684D411yescharacteristic
61C2×M16×C61—21no—
62C4⋊Q8×C61C2×C411yescharacteristic, complemented
63C4⋊Q8×C61C2×C411yescharacteristic, complemented
64#125×C61—21no—
65C22⋊Q8×C61—21no—
66C22⋊Q8×C61—21no—
67C4×Q8×C61—21no—
68C4×Q8×C61—21no—
69#117×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C42×C61D411yescharacteristic
72C2×C42×C61D411yescharacteristic
73C2×C42×C61C2311yescharacteristic
74#97×C61—21no—
75C4×C8×C61—21no—
76#95×C61—21no—
77C23×D61—21no—
78C2×Q8×C61—21no—
79C2×Q8×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C2×D244—21no—
89C2×D244—81no—
90C2×D244C22⋊C412yes—
91C2×C4×D61—42no—
92C2×C4×D61—81no—
93C2×C4×D61—81no—
94C2×Dic122—21no—
95C2×Dic122—81no—
96C2×Dic122C22⋊C412yes—
97M16×C61—22no—
98M16×C61—41no—
99C2×C8×C61—41no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—42no—
105C4⋊C4×C61—41no—
106C4⋊C4×C61—41no—
107C22⋊C4×C61—41no—
108C42×C61—41no—
109C42×C61—41no—
110C42×C61—41no—
111C42×C61C22×C411yescharacteristic
112C42×C61C2×D411yescharacteristic
113C42×C61#4111yescharacteristic
114C42×C61#4111yescharacteristic
115#11758202—82no—
116#11758201—82no—
117#11758198—22no—
118#11758198—22no—
119#11758198—22no—
120#11758198—22no—
121#11758198—42no—
122#11758198—81no—
123#11758198—81no—
124#11758196—81no—
125C4×Dic61—81no—
126C2×#92418—22no—
127C2×#92418—41no—
128C23×C61#12011yescharacteristic
129C22×D61—22no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135C22×D61—41no—
136Q8×C61—42no—
137Q8×C61—81no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—22no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—41no—
150C2×C4×C61C2×C22⋊C411yescharacteristic
151C2×C4×C61C4×D411yescharacteristic
152C2×C4×C61C4×D411yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12111yescharacteristic
155C2×C4×C61#12311yescharacteristic
156C2×Dic61—22no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164C2×Dic61—41no—
165D244—42no—
166D244—81no—
167C4×D61—84no—
168Dic122—42no—
169Dic122—81no—
170C8×C61—42no—
171#92418—42no—
172#92418—81no—
173#9379—611nomaximal, Sylow
174C22×C61—21no—
175C22×C61—21no—
176C22×C61#38611yescharacteristic
177C22×C61#45511yescharacteristic
178C22×C61#45611yescharacteristic
179C2×D61—42no—
180C2×D61—42no—
181C2×D61—42no—
182C2×D61—81no—
183C2×D61—81no—
184C2×D61—81no—
185C2×D61—81no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—21no—
192C4×C61—42no—
193C4×C61—81no—
194C4×C61—81no—
195C4×C61—81no—
196Dic61—42no—
197Dic61—81no—
198Dic61—81no—
199#3215—611no—
200#2337—611no—
201#1986D6111yescharacteristic, complemented
202#1527—611no—
203#1515—611no—
204#1288—611no—
205#1285—611no—
206C2×C61—21no—
207C2×C61—21no—
208C2×C61#188211yescharacteristic
209C2×C61#190611yescharacteristic
210C2×C61#191111yescharacteristic
211D61—82no—
212#533C2×D6111yescharacteristic
213C2×#124—611no—
214#497—612no—
215#489—612no—
216#464—1222no—
217#451—1221no—
218#450—1221no—
219#432—611no—
220#432—611no—
221C4×M16C2×D6111yescharacteristic
222#391—1221no—
223#388—1221no—
224#388—1221no—
225#386—1221no—
226#385—1221no—
227#383—611no—
228C4×C4⋊C4C2×D6111yescharacteristic
229C4×C22⋊C4—1221no—
230#339—21no—
231#339—21no—
232#329—1222no—
233#328—1221no—
234#328—1221no—
235C61#937911yescharacteristic, complemented, Sylow
236C22×Q8—611no—
237C22×D4—611no—
238C23×C4—1221no—
239C2×Q16—1222no—
240C2×SD16—1222no—
241C2×M16—21no—
242C4⋊Q8C4×D6111yescharacteristic, complemented
243C4⋊Q8C4×D6111yescharacteristic, complemented
244#125—21no—
245#124—612no—
246#124—612no—
247#124—1221no—
248#124—1221no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253#117—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—1221no—
256C2×C22⋊C4—1221no—
257C2×C22⋊C4—1221no—
258C2×C22⋊C4—1221no—
259C2×C22⋊C4—1221no—
260C2×C22⋊C4—1221no—
261C2×C42—1221no—
262C2×C42C22⋊D6111yescharacteristic
263C2×C42C22⋊D6111yescharacteristic
264C2×C42C22×D6111yescharacteristic
265C4⋊C8—1221no—
266C4⋊C8—1221no—
267#103—1222no—
268#102—1222no—
269C22⋊C8—1221no—
270C22⋊C8—1221no—
271#97—21no—
272#97—612no—
273#97—1221no—
274C4×C8—21no—
275#95—21no—
276#95—611no—
277#95—611no—
278#95—1221no—
279#95—1221no—
280#95—1221no—
281#95—1221no—
282C24—1221no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—612no—
286C2×Q8—1221no—
287C2×Q8—4881no—
288C2×D4—612no—
289C2×D4—1221no—
290C2×D4—4881no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—1221no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4—4881no—
297C22×C4—4881no—
298C22×C4—2442no—
299C22×C4D4×D6111yescharacteristic
300C22×C4D4×D6111yescharacteristic
301C22×C4C2×C22⋊D6111yescharacteristic
302Q16—4882no—
303SD16—4882no—
304M16—22no—
305M16—41no—
306C2×C8—41no—
307C2×C8—1222no—
308C2×C8—2441no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—4881no—
317C22⋊C4—41no—
318C22⋊C4—1222no—
319C22⋊C4—1222no—
320C22⋊C4—1222no—
321C22⋊C4—1222no—
322C22⋊C4—4881no—
323C22⋊C4—4881no—
324C22⋊C4—2442no—
325C42—41no—
326C42—41no—
327C42—41no—
328C42—4881no—
329C42C2×C4×D6111yescharacteristic
330C42#1175822211yescharacteristic
331C42D4×D6111yescharacteristic
332C42#1175822411yescharacteristic
333C23—1222no—
334C23—2441no—
335C23—2441no—
336C23—2441no—
337C23—2441no—
338C23—2441no—
339C23—2441no—
340C23#1451974511yescharacteristic
341Q8—42no—
342Q8—81no—
343Q8—4881no—
344Q8—2442no—
345D4—4881no—
346D4—2442no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—1222no—
360C2×C4—4884no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4—2441no—
364C2×C4—2441no—
365C2×C4—2441no—
366C2×C4—2441no—
367C2×C4—2441no—
368C2×C4—2441no—
369C2×C4C22⋊C4×D6111yescharacteristic
370C2×C4#1451973711yescharacteristic
371C2×C4C4×C22⋊D6111yescharacteristic
372C2×C4#1451974811yescharacteristic
373C2×C4#1451975011yescharacteristic
374C2×C4#1451975911yescharacteristic
375C8—42no—
376C8—4881no—
377C8—2442no—
378C22—21no—
379C22—21no—
380C22—4881no—
381C22—4881no—
382C22—4881no—
383C22—4881no—
384C22—2442no—
385C22—2442no—
386C22—2442no—
387C22#1683963811yescharacteristic
388C22#1683993911yescharacteristic
389C22#1683994411yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4—4881no—
401C4—4881no—
402C4—2442no—
403C2—21no—
404C2—21no—
405C2—4882no—
406C2#2144619111yescharacteristic
407C2#2144632311yescharacteristic
408C2#2144634811yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1986C2
socleC22×C61#386
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1986D61
hypocenterC61#9379
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
4. #21444370C2semidirect
5. #21443684C2semidirect
6. #21443675C2semidirect
7. C61×#1986C2semidirect
9. #16840154C4semidirect
10. #16840124C4semidirect
54. #14519688C2×C4semidirect
62. C4⋊Q8×C61C2×C4semidirect
63. C4⋊Q8×C61C2×C4semidirect
201. #1986D61semidirect
235. C61#9379semidirect
242. C4⋊Q8C4×D61semidirect
243. C4⋊Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214568182
#214486322
#214444162
#214443702
#214436842
#214436752
C61×#19862
#937961