Finite Group Databasebeta

← #29653138

Subgroups — #29653138

Order 15616.

958 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 187 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#21459629C211yesmaximal, characteristic, complemented
2#21455827C212yesmaximal
3#21444826C214yesmaximal, complemented
4#21444758C214yesmaximal
5#21444382C212yesmaximal, complemented
6#21444143C211yesmaximal, characteristic, complemented
7C61×#2536C211yesmaximal, characteristic
8Q8×Dic122—22no—
9#16840491—22no—
10#16840488—24no—
11#16840393—21no—
12#16840393C2214yes—
13#16840319—22no—
14#16840319C2212yes—
15#16840240—21no—
16#16840092C2214yes—
17#16840052—22no—
18#16839967—22no—
19#16839913C2214yes—
20C4×#11758200—24no—
21#16839815C2212yes—
22#16839498C2218yes—
23#16839488C2214yes—
24C2×C22⋊Q8×C61—21no—
25C2×C4×Q8×C61C2211yescharacteristic
26C2×C4×D4×C61C2211yescharacteristic
27C61×#398C2212yes—
28C61×#394C2212yes—
29C61×#388—22no—
30C61×#382C2211yescharacteristic
31C61×#380—21no—
32C2×#11758200—24no—
33#14519777—22no—
34#14519777—22no—
35#14519777D412yes—
36#14519777D412yes—
37#14519776—24no—
38#14519776—28no—
39#14519769—24no—
40#14519769—22no—
41#14519768—24no—
42#14519768—28no—
43#14519756—22no—
44#14519756—24no—
45#14519756C2314yes—
46Q8×Dic61—24no—
47Q8×Dic61—24no—
48C4×Dic122—24no—
49C4×Dic122—24no—
50C4×Dic122—22no—
51C4×Dic122D414yes—
52#14519736—24no—
53#14519736—22no—
54#14519708—24no—
55C4⋊#92418—22no—
56#14519700—24no—
57#14519698—24no—
58C22⋊#92418C2314yes—
59C4×#92418—22no—
60C22×Q8×C61D411yescharacteristic
61C22×D4×C61C2311yescharacteristic
62C23×C4×C61—21no—
63C22⋊Q8×C61—44no—
64C4×Q8×C61—22no—
65C4×Q8×C61—22no—
66C4×D4×C61—22no—
67C4×D4×C61—22no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—22no—
70C2×C4⋊C4×C61D411yescharacteristic
71C2×C4⋊C4×C61C2311yescharacteristic
72C2×C4⋊C4×C61C2312yes—
73C2×C22⋊C4×C61—22no—
74C2×C22⋊C4×C61—21no—
75C2×C22⋊C4×C61—22no—
76C2×C42×C61D412yes—
77C2×C42×C61C2311yescharacteristic
78#95×C61—22no—
79#95×C61—21no—
80#95×C61C2312yes—
81C24×C61—21no—
82C2×Q8×C61—22no—
83C2×Q8×C61—22no—
84C2×Q8×C61SD1614yescomplemented
85C2×D4×C61—22no—
86C2×D4×C61—42no—
87C22×C4×C61—22no—
88C22×C4×C61—21no—
89C22×C4×C61—42no—
90C22×C4×C61—42no—
91C22×C4×C61—42no—
92C22×C4×C61C2×D412yes—
93C22×C4×C61C2×D412yes—
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2411yescharacteristic
97C2×Dic122—22no—
98C2×Dic122—24no—
99C2×Dic122C2×D414yes—
100C4⋊C4×C61—22no—
101C4⋊C4×C61—24no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—42no—
104C4⋊C4×C61—44no—
105C22⋊C4×C61—44no—
106C22⋊C4×C61—41no—
107C22⋊C4×C61—41no—
108C22⋊C4×C61—44no—
109C42×C61—22no—
110C42×C61—22no—
111C42×C61C2×D412yes—
112C42×C61C2×D412yes—
113C22⋊Dic61—24no—
114#11758200—48no—
115C4⋊Dic61—24no—
116C4⋊Dic61—24no—
117C4⋊Dic61—24no—
118C4⋊Dic61—22no—
119C4⋊Dic61—22no—
120C4⋊Dic61—22no—
121C4⋊Dic61C2×D414yes—
122#11758196—28no—
123#11758196—24no—
124#11758196—24no—
125C4×Dic61—28no—
126C4×Dic61—24no—
127C2×#92418—24no—
128C23×C61—22no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61—41no—
135C23×C61C22×D411yescharacteristic
136Q8×C61—28no—
137D4×C61—24no—
138D4×C61—42no—
139C2×C4×C61—22no—
140C2×C4×C61—22no—
141C2×C4×C61—22no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61—42no—
147C2×C4×C61—41no—
148C2×C4×C61—42no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—84no—
152C2×C4×C61#12011yescharacteristic
153C2×C4×C61#12011yescharacteristic
154C2×C4×C61#12012yes—
155C2×C4×C61C2×SD1614yes—
156C2×C4×C61C2×SD1612yes—
157C2×C4×C61#13712yes—
158C2×C4×C61C22×D412yes—
159C2×Dic61—24no—
160C2×Dic61—24no—
161Dic122—44no—
162Dic122—44no—
163Dic122—84no—
164#92418—44no—
165#18220—611nomaximal, Sylow
166C22×C61—21no—
167C22×C61—21no—
168C22×C61—42no—
169C22×C61—42no—
170C22×C61—42no—
171C22×C61—81no—
172C22×C61—81no—
173C22×C61—81no—
174C22×C61—81no—
175C22×C61C2×#12011yescharacteristic
176C22×C61C22×SD1611yescharacteristic
177C22×C61C2×#13711yescharacteristic
178C4×C61—24no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—22no—
182C4×C61—42no—
183C4×C61—82no—
184C4×C61—81no—
185C4×C61#44814yes—
186Dic61—44no—
187Dic61—44no—
188#3502—611no—
189#3101—612no—
190#2536D6111yescharacteristic
191#1611—614no—
192#1599—614no—
193#1518—612no—
194#1461—611no—
195C2×C61—21no—
196C2×C61—21no—
197C2×C61—82no—
198C2×C61C2×#44811yescharacteristic
199C2×C61#302011yescharacteristic
200C2×C61#302511yescharacteristic
201Q82—1222no—
202#557—1224no—
203#557—1222no—
204#544—614no—
205#544—1221no—
206#533—612no—
207#533—1222no—
208C2×C22⋊Q8—21no—
209#519—1221no—
210C2×C4×Q8C2×D6111yescharacteristic
211C2×C4×D4C2×D6111yescharacteristic
212#483—614no—
213#476—1222no—
214#461—1222no—
215#450—614no—
216C4×SD16—1224no—
217#431—612no—
218#398C2×D6112yes—
219#394C2×D6112yes—
220#388—22no—
221#382C2×D6111yescharacteristic
222#380—21no—
223#328—618no—
224#324—614no—
225C61#1822011yescharacteristic, complemented, Sylow
226C22×Q8C22⋊D6111yescharacteristic
227C22×D4C22×D6111yescharacteristic
228C23×C4—21no—
229C2×SD16—1224no—
230C4⋊Q8—612no—
231C4⋊Q8—612no—
232C4⋊Q8—1222no—
233C4⋊Q8—1222no—
234C4⋊Q8—1228no—
235C4⋊Q8—1224no—
236#125—1224no—
237#125—1228no—
238#125—1224no—
239#125—1222no—
240C22⋊Q8—44no—
241C22⋊Q8—614no—
242C22⋊Q8—1222no—
243C22⋊Q8—1224no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—614no—
247C4×Q8—1224no—
248C4×Q8—1224no—
249C4×Q8—1224no—
250C4×Q8—1224no—
251C4×Q8—1222no—
252C4×D4—22no—
253C4×D4—22no—
254#117—1222no—
255#117—1224no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4C22⋊D6111yescharacteristic
259C2×C4⋊C4C22×D6111yescharacteristic
260C2×C4⋊C4C22×D6112yes—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C42C22⋊D6112yes—
265C2×C42C22×D6111yescharacteristic
266#106—1224no—
267C4⋊C8—1222no—
268#103—1224no—
269#102—1224no—
270C22⋊C8—614no—
271C4×C8—1222no—
272#95—22no—
273#95—21no—
274#95C22×D6112yes—
275C24—21no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—614no—
279C2×Q8—1224no—
280C2×Q8—1222no—
281C2×Q8#1175820014yescomplemented
282C2×D4—22no—
283C2×D4—42no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4C2×C22⋊D6112yes—
290C22×C4C2×C22⋊D6112yes—
291C22×C4C2×C22⋊D6111yescharacteristic
292C22×C4C2×C22⋊D6111yescharacteristic
293C22×C4C23×D6111yescharacteristic
294SD16—2448no—
295C2×C8—1224no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—614no—
302C4⋊C4—1228no—
303C4⋊C4—1224no—
304C4⋊C4—1224no—
305C4⋊C4—1224no—
306C4⋊C4—1224no—
307C4⋊C4—1224no—
308C4⋊C4—1222no—
309C4⋊C4—1222no—
310C4⋊C4—1222no—
311C22⋊C4—44no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—44no—
315C22⋊C4—1224no—
316C42—22no—
317C42—22no—
318C42—1228no—
319C42—1224no—
320C42D4×D6112yes—
321C42D4×D6112yes—
322C23—22no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23C22×C22⋊D6111yescharacteristic
330Q8—28no—
331Q8—4884no—
332Q8—2444no—
333Q8—2444no—
334D4—24no—
335D4—42no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—84no—
349C2×C4—1224no—
350C2×C4—1224no—
351C2×C4#1451974511yescharacteristic
352C2×C4#1451974511yescharacteristic
353C2×C4#1451974512yes—
354C2×C4C2×#1175820014yes—
355C2×C4C2×#1175820012yes—
356C2×C4#1451980712yes—
357C2×C4C2×D4×D6112yes—
358C8—2444no—
359C22—21no—
360C22—21no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22C2×#1451974511yescharacteristic
369C22C22×#1175820011yescharacteristic
370C22C2×#1451980711yescharacteristic
371C4—21no—
372C4—21no—
373C4—22no—
374C4—24no—
375C4—42no—
376C4—82no—
377C4—81no—
378C4—2444no—
379C4—2444no—
380C4#1683990014yes—
381C2—21no—
382C2—21no—
383C2—82no—
384C2C2×#1683990011yescharacteristic
385C2#2145518211yescharacteristic
386C2#2145521011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2536C2
socleC22×C61C2×#120
prime radicalC22×D4×C61C23
hypercenter#2536D61
hypocenterC61#18220
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. #21459629C2semidirect
3. #21444826C2semidirect
5. #21444382C2semidirect
6. #21444143C2semidirect
84. C2×Q8×C61SD16semidirect
225. C61#18220semidirect
281. C2×Q8#11758200semidirect

Maximal subgroups

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

subgroupindex
#214596292
#214558272
#214448262
#214447582
#214443822
#214441432
C61×#25362
#1822061