Finite Group Databasebeta

← #29577629

Subgroups — #29577629

Order 15616.

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

normal subgroups only

This group has 490 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#21456208C211yesmaximal, characteristic
2#21448929C211yesmaximal, characteristic
3#21444442C211yesmaximal, characteristic
4#21444331C211yesmaximal, characteristic, complemented
5#21443712C211yesmaximal, characteristic, complemented
6#21443711C211yesmaximal, characteristic, complemented
7C61×#1506C211yesmaximal, characteristic, complemented
8C2×#14519765C2211yescharacteristic
9#16840124C412yescomplemented
10#16840124C412yescomplemented
11#16840119—22no—
12#16839819C2211yescharacteristic
13#16839770—21no—
14#16839766—21no—
15C4×#11758194C2211yescharacteristic
16#16839666—21no—
17#16839653—21no—
18#16839651—21no—
19#16839640—21no—
20C4×#11758198—21no—
21#16839609—21no—
22#16839608C2211yescharacteristic
23#16839501—22no—
24#16839499—21no—
25#16839499—21no—
26C61×#533C2211yescharacteristic
27C61×#432C2211yescharacteristic
28C61×#376C2211yescharacteristic
29C61×#328—21no—
30C61×#328—21no—
31C22×Dic122D411yescharacteristic
32C22×D244D411yescharacteristic
33C22×C4×D61—21no—
34C2×#11758194—21no—
35#14519765—21no—
36#14519765—21no—
37#14519765C2×C412yes—
38#14519765D412yes—
39C2×#11758196—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×C4×Dic61—21no—
46C2×C4×Dic61—21no—
47C4⋊#92418—21no—
48#14519700—22no—
49#14519700—22no—
50#14519697—22no—
51#14519697—22no—
52C22⋊#92418—21no—
53#14519688—21no—
54#14519688C2×C412yescomplemented
55C4×#92418—21no—
56#14519684—21no—
57#14519684—21no—
58#14519684—21no—
59#14519684D411yescharacteristic
60#14519684D411yescharacteristic
61C4⋊Q8×C61C2×C411yescharacteristic, complemented
62C4⋊Q8×C61C2×C411yescharacteristic, complemented
63#125×C61—21no—
64C22⋊Q8×C61—21no—
65C22⋊Q8×C61—21no—
66C4×Q8×C61—21no—
67C4×Q8×C61—21no—
68#117×C61—21no—
69C2×C42×C61—21no—
70C2×C42×C61C2311yescharacteristic
71C4⋊C8×C61—21no—
72C22⋊C8×C61—21no—
73#97×C61—21no—
74#95×C61—21no—
75#95×C61D411yescharacteristic
76#95×C61D411yescharacteristic
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—41no—
90C2×D244—41no—
91C2×D244C22⋊C412yes—
92C2×C4×D61—42no—
93C2×C4×D61—42no—
94C2×C4×D61—81no—
95C2×Dic122—21no—
96C2×Dic122—41no—
97C2×Dic122—41no—
98C2×Dic122C22⋊C412yes—
99C2×C8×C61—41no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—41no—
105C4⋊C4×C61—41no—
106C22⋊C4×C61—41no—
107C42×C61—42no—
108C42×C61—41no—
109C42×C61C22×C411yescharacteristic
110C42×C61C2×D411yescharacteristic
111C42×C61#4111yescharacteristic
112C42×C61#4111yescharacteristic
113#11758198—22no—
114#11758198—22no—
115#11758198—22no—
116#11758198—22no—
117#11758198—42no—
118#11758198—42no—
119#11758198—81no—
120#11758196—81no—
121C4×Dic61—42no—
122C4×Dic61—81no—
123#11758194—41no—
124#11758194—41no—
125C2×#92418—22no—
126C2×#92418—41no—
127C23×C61#12011yescharacteristic
128C22×D61—22no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C22×D61—41no—
135Q8×C61—81no—
136Q8×C61—81no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—22no—
140C2×C4×C61—41no—
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×C61C2×C22⋊C411yescharacteristic
150C2×C4×C61C4×D411yescharacteristic
151C2×C4×C61C4×D411yescharacteristic
152C2×C4×C61#12111yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12311yescharacteristic
155C2×Dic61—22no—
156C2×Dic61—22no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—41no—
161C2×Dic61—41no—
162C2×Dic61—41no—
163C2×Dic61—41no—
164D244—42no—
165D244—81no—
166C4×D61—84no—
167Dic122—42no—
168Dic122—81no—
169C8×C61—81no—
170#92418—42no—
171#92418—81no—
172#9439—611nomaximal, Sylow
173C22×C61—21no—
174C22×C61—21no—
175C22×C61#38611yescharacteristic
176C22×C61#45511yescharacteristic
177C22×C61#45511yescharacteristic
178C2×D61—42no—
179C2×D61—42no—
180C2×D61—42no—
181C2×D61—81no—
182C2×D61—81no—
183C2×D61—81no—
184C2×D61—81no—
185C4×C61—21no—
186C4×C61—21no—
187C4×C61—21no—
188C4×C61—21no—
189C4×C61—21no—
190C4×C61—21no—
191C4×C61—42no—
192C4×C61—81no—
193C4×C61—81no—
194C4×C61—81no—
195Dic61—42no—
196Dic61—42no—
197Dic61—81no—
198#3140—611no—
199#2372—611no—
200#1533—611no—
201#1506—611no—
202#1506D6111yescharacteristic, complemented
203#1300—611no—
204#1300—611no—
205C2×C61—21no—
206C2×C61—21no—
207C2×C61#188511yescharacteristic
208C2×C61#190611yescharacteristic
209C2×C61#190611yescharacteristic
210D61—82no—
211#533C2×D6111yescharacteristic
212C2×#124—611no—
213#489—612no—
214#489—612no—
215#488—1222no—
216#432—611no—
217#432C2×D6111yescharacteristic
218#419—1221no—
219#418—1221no—
220C4×M16—611no—
221#391—1221no—
222#388—1221no—
223#388—1221no—
224#386—1221no—
225C4×C22⋊C4—1221no—
226C4×C22⋊C4—1221no—
227#376—611no—
228#376C2×D6111yescharacteristic
229#329—1222no—
230#328—21no—
231#328—21no—
232#328—1221no—
233#328—1221no—
234C61#943911yescharacteristic, complemented, Sylow
235C22×Q8—611no—
236C22×D4—611no—
237C23×C4—1221no—
238C2×M16—1221no—
239C4⋊Q8C4×D6111yescharacteristic, complemented
240C4⋊Q8C4×D6111yescharacteristic, complemented
241#125—21no—
242#124—612no—
243#124—612no—
244#124—1221no—
245#124—1221no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248C4×Q8—21no—
249C4×Q8—21no—
250#117—21no—
251C2×C4⋊C4—1221no—
252C2×C22⋊C4—1221no—
253C2×C22⋊C4—1221no—
254C2×C22⋊C4—1221no—
255C2×C22⋊C4—1221no—
256C2×C22⋊C4—1221no—
257C2×C42—21no—
258C2×C42—1221no—
259C2×C42—1221no—
260C2×C42C22×D6111yescharacteristic
261C4⋊C8—21no—
262C4⋊C8—1221no—
263#103—1222no—
264#103—1222no—
265#102—1222no—
266#102—1222no—
267C22⋊C8—21no—
268C22⋊C8—1221no—
269#97—21no—
270#97—612no—
271#97—1221no—
272C4×C8—1221no—
273#95—21no—
274#95—611no—
275#95—611no—
276#95—1221no—
277#95—1221no—
278#95—1221no—
279#95C22⋊D6111yescharacteristic
280#95C22⋊D6111yescharacteristic
281C24—1221no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—612no—
285C2×Q8—1221no—
286C2×Q8—2441no—
287C2×Q8—2441no—
288C2×D4—612no—
289C2×D4—1221no—
290C2×D4—2441no—
291C2×D4—2441no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4—1221no—
297C22×C4—4881no—
298C22×C4—2442no—
299C22×C4—2442no—
300C22×C4D4×D6111yescharacteristic
301C22×C4D4×D6111yescharacteristic
302C22×C4C2×C22⋊D6111yescharacteristic
303M16—2441no—
304M16—2441no—
305C2×C8—41no—
306C2×C8—1222no—
307C2×C8—2441no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—4881no—
315C22⋊C4—41no—
316C22⋊C4—1222no—
317C22⋊C4—1222no—
318C22⋊C4—1222no—
319C22⋊C4—1222no—
320C22⋊C4—4881no—
321C22⋊C4—2442no—
322C22⋊C4—2442no—
323C42—42no—
324C42—41no—
325C42—4881no—
326C42—2442no—
327C42C2×C4×D6111yescharacteristic
328C42#1175822211yescharacteristic
329C42D4×D6111yescharacteristic
330C42#1175822411yescharacteristic
331C23—1222no—
332C23—2441no—
333C23—2441no—
334C23—2441no—
335C23—2441no—
336C23—2441no—
337C23—2441no—
338C23#1451974511yescharacteristic
339Q8—81no—
340Q8—81no—
341Q8—4881no—
342Q8—2442no—
343D4—4881no—
344D4—2442no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—1222no—
358C2×C4—1222no—
359C2×C4—4884no—
360C2×C4—2441no—
361C2×C4—2441no—
362C2×C4—2441no—
363C2×C4—2441no—
364C2×C4—2441no—
365C2×C4—2441no—
366C2×C4—2441no—
367C2×C4C22⋊C4×D6111yescharacteristic
368C2×C4#1451973711yescharacteristic
369C2×C4C4×C22⋊D6111yescharacteristic
370C2×C4#1451974811yescharacteristic
371C2×C4#1451975011yescharacteristic
372C2×C4#1451975911yescharacteristic
373C8—81no—
374C8—4881no—
375C8—2442no—
376C22—21no—
377C22—21no—
378C22—4881no—
379C22—4881no—
380C22—4881no—
381C22—4881no—
382C22—2442no—
383C22—2442no—
384C22—2442no—
385C22#1683963811yescharacteristic
386C22#1683993811yescharacteristic
387C22#1683993811yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—4881no—
399C4—2442no—
400C4—2442no—
401C2—21no—
402C2—21no—
403C2—4882no—
404C2#2144620411yescharacteristic
405C2#2144631711yescharacteristic
406C2#2144632311yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1506C2
socleC22×C61#386
prime radicalC22×D244D4
cosocleC2×C42×C61C23
hypercenter#1506D61
hypocenterC61#9439
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1506

Complements

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

NH ≅ G/Ntype
4. #21444331C2semidirect
5. #21443712C2semidirect
6. #21443711C2semidirect
7. C61×#1506C2semidirect
9. #16840124C4semidirect
10. #16840124C4semidirect
54. #14519688C2×C4semidirect
61. C4⋊Q8×C61C2×C4semidirect
62. C4⋊Q8×C61C2×C4semidirect
202. #1506D61semidirect
234. C61#9439semidirect
239. C4⋊Q8C4×D61semidirect
240. C4⋊Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214562082
#214489292
#214444422
#214443312
#214437122
#214437112
C61×#15062
#943961