Finite Group Databasebeta

← #29576929

Subgroups — #29576929

Order 15616.

574 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 111 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#21455941C211yesmaximal, characteristic
2#21446737C211yesmaximal, characteristic
3#21444326C211yesmaximal, characteristic
4#21444266C211yesmaximal, characteristic, complemented
5#21443712C211yesmaximal, characteristic, complemented
6#21443675C211yesmaximal, characteristic, complemented
7C61×#2276C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840075—21no—
10#16840057C412yescomplemented
11#16839962C412yescomplemented
12#16839901—21no—
13#16839860—22no—
14#16839819C2211yescharacteristic
15#16839819C2211yescharacteristic
16C2×#14519705—22no—
17C4×#11758194C2211yescharacteristic
18#16839495—21no—
19#16839495—21no—
20Q8⋊#92418—22no—
21C2×C4×Q8×C61—21no—
22C2×C4×Q8×C61C2211yescharacteristic
23C4×C4⋊C4×C61—21no—
24C4×C4⋊C4×C61—22no—
25C4×C4⋊C4×C61—21no—
26C4×C4⋊C4×C61C2211yescharacteristic
27C61×#376—21no—
28C61×#376C2211yescharacteristic
29C2×#11758222—21no—
30C2×#11758201—22no—
31C2×#11758194—21no—
32#14519777C2×C411yescharacteristic
33#14519774C2×C411yescharacteristic
34#14519765—21no—
35#14519751—21no—
36#14519751—21no—
37C4×D244—21no—
38C4×D244—21no—
39#14519710—22no—
40C4⋊#92418—21no—
41C4⋊#92418C2×C412yes—
42#14519705—44no—
43#14519701—22no—
44#14519697—22no—
45C22⋊#92418—21no—
46C22⋊#92418—21no—
47#14519688—21no—
48#14519688—21no—
49#14519688—21no—
50C4×#92418—21no—
51C22×Q8×C61D411yescharacteristic
52C4×Q8×C61—28no—
53C4×Q8×C61—21no—
54C4×Q8×C61—21no—
55C4×Q8×C61C2×C412yes—
56C4×Q8×C61D412yes—
57C2×C4⋊C4×C61—22no—
58C2×C4⋊C4×C61—21no—
59C2×C4⋊C4×C61—21no—
60C2×C4⋊C4×C61D411yescharacteristic
61C2×C42×C61—22no—
62C2×C42×C61—21no—
63C2×C42×C61—21no—
64C2×C42×C61—22no—
65C2×C42×C61D412yes—
66C2×C42×C61C2311yescharacteristic
67#95×C61—22no—
68#95×C61—21no—
69#95×C61—21no—
70#95×C61D411yescharacteristic
71#95×C61D411yescharacteristic
72C2×Q8×C61—24no—
73C2×Q8×C61—21no—
74C2×Q8×C61C22⋊C412yes—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61—21no—
78C22×C4×C61—22no—
79C22×C4×C61—21no—
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D411yescharacteristic
83C2×C22⋊D61—41no—
84#11758222—42no—
85#11758222—42no—
86C2×D244—21no—
87C2×D244—21no—
88C2×D244—21no—
89C2×C4×D61—41no—
90C2×Dic122—21no—
91C4⋊C4×C61—28no—
92C4⋊C4×C61—24no—
93C4⋊C4×C61—22no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61C22⋊C412yes—
97C42×C61—28no—
98C42×C61—24no—
99C42×C61—22no—
100C42×C61—22no—
101C42×C61—42no—
102C42×C61—82no—
103C42×C61C22×C411yescharacteristic
104C42×C61C2×D411yescharacteristic
105C42×C61#4111yescharacteristic
106C42×C61#4111yescharacteristic
107#11758201—44no—
108#11758198—41no—
109#11758198—41no—
110C4⋊Dic61—21no—
111C4⋊Dic61—21no—
112#11758194—22no—
113#11758194—41no—
114C2×#92418—22no—
115C2×#92418—41no—
116C2×#92418—41no—
117C23×C61#12011yescharacteristic
118C22×D61—41no—
119Q8×C61—24no—
120Q8×C61—22no—
121Q8×C61#10414yescomplemented
122C2×C4×C61—24no—
123C2×C4×C61—22no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—22no—
127C2×C4×C61—22no—
128C2×C4×C61—24no—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—42no—
134C2×C4×C61—41no—
135C2×C4×C61—42no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61C2×C22⋊C411yescharacteristic
141C2×C4×C61C4×D411yescharacteristic
142C2×C4×C61C4×D411yescharacteristic
143C2×C4×C61#12111yescharacteristic
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61#12311yescharacteristic
146C22⋊D61—81no—
147C22⋊D61—81no—
148C2×Dic61—41no—
149D244—42no—
150D244—42no—
151D244—81no—
152C4×D61—82no—
153Dic122—42no—
154#92418—42no—
155#92418—42no—
156#92418—81no—
157#9326—611nomaximal, Sylow
158C22×C61—21no—
159C22×C61—21no—
160C22×C61#38611yescharacteristic
161C22×C61#45311yescharacteristic
162C22×C61#45411yescharacteristic
163C2×D61—81no—
164C2×D61—81no—
165C2×D61—81no—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—24no—
169C4×C61—21no—
170C4×C61—21no—
171C4×C61—21no—
172C4×C61—44no—
173C4×C61—42no—
174C4×C61—42no—
175C4×C61—42no—
176C4×C61C2×#10412yes—
177Dic61—81no—
178#3112—611no—
179#2276D6111yescharacteristic, complemented
180#1980—611no—
181#1505—611no—
182#1491—611no—
183#1300—611no—
184#1285—611no—
185C2×C61—21no—
186C2×C61—21no—
187C2×C61#188311yescharacteristic
188C2×C61#189611yescharacteristic
189C2×C61#190511yescharacteristic
190D61—161no—
191#532—611no—
192C2×C4×Q8—21no—
193C2×C4×Q8C2×D6111yescharacteristic
194#480—1221no—
195#477—612no—
196#460—612no—
197#448—1221no—
198#440—1222no—
199#432—611no—
200#432—611no—
201C2×#104—1222no—
202C4×M16—611no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4C2×D6111yescharacteristic
207#376—21no—
208#376C2×D6111yescharacteristic
209#327—1221no—
210#327—1221no—
211Q8⋊C8—1222no—
212C61#932611yescharacteristic, complemented, Sylow
213C2×#41—1221no—
214C22×Q8C22⋊D6111yescharacteristic
215C2×SD16—1222no—
216C2×M16—1221no—
217C4⋊Q8—611no—
218#127—611no—
219#124—1221no—
220#121—1221no—
221#121—1221no—
222C4×Q8—28no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8C4×D6112yes—
226C4×Q8D24412yes—
227C4×D4—1221no—
228C4×D4—1221no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4C22⋊D6111yescharacteristic
233C2×C42—22no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—22no—
237C2×C42C22⋊D6112yes—
238C2×C42C22×D6111yescharacteristic
239#107—1222no—
240C4⋊C8—612no—
241C4⋊C8—1221no—
242#104—2444no—
243#103—1222no—
244#102—1222no—
245C22⋊C8—1221no—
246C22⋊C8—1221no—
247#97—1221no—
248#97—1221no—
249#97—1221no—
250C4×C8—1221no—
251#95—22no—
252#95—21no—
253#95—21no—
254#95C22⋊D6111yescharacteristic
255#95C22⋊D6111yescharacteristic
256#41—2442no—
257#41—2442no—
258C2×Q8—24no—
259C2×Q8—21no—
260C2×Q8—1221no—
261C2×Q8#1175819812yes—
262C2×D4—1221no—
263C2×D4—1221no—
264C2×D4—1221no—
265C2×D4—2441no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—2441no—
272C22×C4C2×C22⋊D6111yescharacteristic
273C22×C4C2×C22⋊D6111yescharacteristic
274C22×C4C2×C22⋊D6111yescharacteristic
275SD16—2444no—
276M16—1222no—
277M16—2441no—
278C2×C8—1222no—
279C2×C8—2441no—
280C2×C8—2441no—
281C4⋊C4—28no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—21no—
286C4⋊C4—1221no—
287C4⋊C4—1221no—
288C4⋊C4#1175819812yes—
289C22⋊C4—2441no—
290C22⋊C4—2441no—
291C42—28no—
292C42—24no—
293C42—22no—
294C42—22no—
295C42—42no—
296C42—82no—
297C42C2×C4×D6111yescharacteristic
298C42C2×D24411yescharacteristic
299C42#1175822211yescharacteristic
300C42#1175822211yescharacteristic
301C23—2441no—
302C23#1451974611yescharacteristic
303Q8—24no—
304Q8—22no—
305Q8—2442no—
306Q8#1451970514yescomplemented
307D4—4881no—
308D4—4881no—
309D4—4881no—
310D4—2442no—
311D4—2442no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—4882no—
331C2×C4—2441no—
332C2×C4C2×#1175819811yescharacteristic
333C2×C4C4×C22⋊D6111yescharacteristic
334C2×C4C4×C22⋊D6111yescharacteristic
335C2×C4#1451975111yescharacteristic
336C2×C4#1451975111yescharacteristic
337C2×C4#1451976011yescharacteristic
338C8—4881no—
339C8—2442no—
340C8—2442no—
341C22—21no—
342C22—21no—
343C22—4881no—
344C22—4881no—
345C22—4881no—
346C22#1683964111yescharacteristic
347C22#1683993011yescharacteristic
348C22#1683993511yescharacteristic
349C4—21no—
350C4—21no—
351C4—24no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—44no—
356C4—42no—
357C4—42no—
358C4—42no—
359C4—4881no—
360C4C2×#1451970512yes—
361C2—21no—
362C2—21no—
363C2—9761no—
364C2#2144619511yescharacteristic
365C2#2144626211yescharacteristic
366C2#2144631311yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2276C2
socleC22×C61#386
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#2276D61
hypocenterC61#9326
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2276

Complements

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

NH ≅ G/Ntype
4. #21444266C2semidirect
5. #21443712C2semidirect
6. #21443675C2semidirect
7. C61×#2276C2semidirect
10. #16840057C4semidirect
11. #16839962C4semidirect
121. Q8×C61#104semidirect
179. #2276D61semidirect
212. C61#9326semidirect
306. Q8#14519705semidirect

Maximal subgroups

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

subgroupindex
#214559412
#214467372
#214443262
#214442662
#214437122
#214436752
C61×#22762
#932661