Finite Group Databasebeta

← #30600767

Subgroups — #30600767

Order 15616.

426 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 86 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#21462612C211yesmaximal, characteristic
2#21461950C211yesmaximal, characteristic
3#21461329C211yesmaximal, characteristic
4#21461253C211yesmaximal, characteristic
5#21460673C211yesmaximal, characteristic
6#21460644C211yesmaximal, characteristic
7#21456959C211yesmaximal, characteristic
8#16840814—22no—
9#16840813—22no—
10#16840780C2211yescharacteristic
11#16840769—21no—
12#16840767C2211yescharacteristic
13C2×#14519840—21no—
14C2×#14519840—21no—
15#16840727—21no—
16#16840721C2211yescharacteristic
17#16840706C2211yescharacteristic
18#16840621—22no—
19#16840616—22no—
20#16840317C2211yescharacteristic
21C4⋊Q8×D61C2211yescharacteristic
22#16840160C411yescharacteristic, complemented
23#16840158C411yescharacteristic, complemented
24#16840112C411yescharacteristic, complemented
25#16840091—22no—
26#16840045—22no—
27#16839813C2211yescharacteristic
28C61×#487C411yescharacteristic, complemented
29C2×#11758213—21no—
30C2×D61⋊C8D411yescharacteristic
31C2×D61⋊C8D411yescharacteristic
32C2×C4⋊#6923—21no—
33#14519844—22no—
34#14519844—21no—
35#14519844—21no—
36#14519844D412yes—
37C4⋊#92420—22no—
38#14519840—42no—
39#14519840—42no—
40#14519834—21no—
41#14519832—22no—
42#14519832—21no—
43#14519832—21no—
44C4×#92420—22no—
45C4×#92420—21no—
46(C2×#6923)⋊C4—21no—
47(C2×#6923)⋊C4—21no—
48C2×Q8×D61—21no—
49#14519777C2×C411yescharacteristic
50#14519777C2×C411yescharacteristic
51#14519776—21no—
52#14519776—21no—
53#14519775—21no—
54#14519768—21no—
55#14519757—21no—
56#14519753D412yes—
57Q8×Dic61D412yes—
58#14519742—21no—
59#14519735—21no—
60C4⋊C4×D61—21no—
61C4⋊C4×D61—21no—
62C42×D61C2311yescharacteristic
63#14519709—22no—
64#14519708—22no—
65#14519706—21no—
66#14519701—22no—
67#14519689—21no—
68C4×#92418C2×C411yescharacteristic
69C4⋊Q8×C61C2×C411yescharacteristic
70C4⋊Q8×C61C2×C411yescharacteristic
71#103×C61—22no—
72C4×C8×C61C2×C411yescharacteristic
73C22×#6923—41no—
74C2×Q8×C61—21no—
75C2×Q8×C61C22⋊C412yes—
76Q8×D61—41no—
77Q8×D61—41no—
78Q8×D61—41no—
79Q8×D61—41no—
80C2×C4×D61—41no—
81C2×C4×D61C2×D411yescharacteristic
82C2×C4×D61C2×D411yescharacteristic
83C2×C4×D61#4111yescharacteristic
84C2×Dic122—21no—
85C2×Dic122—21no—
86C2×Dic122—41no—
87C2×#92420—21no—
88C2×#92420—22no—
89C2×#92420—21no—
90C2×#92420—41no—
91C4⋊#6923—41no—
92C4⋊#6923—41no—
93#11758213—41no—
94#11758213—41no—
95D61⋊C8—21no—
96D61⋊C8—21no—
97D61⋊C8—21no—
98D61⋊C8SD1612yes—
99C2×C8×C61—21no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—21no—
104C42×C61C22×C411yescharacteristic
105#11758198—22no—
106C4⋊Dic61—21no—
107C4⋊Dic61—22no—
108C4⋊Dic61—21no—
109C4⋊Dic61—21no—
110C4⋊Dic61C22⋊C412yes—
111#11758196—22no—
112#11758196—41no—
113#11758196—41no—
114C4×Dic61—22no—
115C4×Dic61C2×D411yescharacteristic
116C4×Dic61C2×D411yescharacteristic
117C4×Dic61#4111yescharacteristic
118C2×#92418—21no—
119C22×D61#12111yescharacteristic
120C2×#6923—81no—
121C2×#6923—81no—
122C2×#6923—81no—
123Q8×C61—42no—
124Q8×C61—41no—
125Q8×C61—41no—
126C2×C4×C61—22no—
127C2×C4×C61—41no—
128C2×C4×C61C2×C22⋊C411yescharacteristic
129C2×C4×C61C4×D411yescharacteristic
130C2×C4×C61C4×D411yescharacteristic
131C2×Dic61—22no—
132C2×Dic61—41no—
133C2×Dic61#12011yescharacteristic
134C2×Dic61#12111yescharacteristic
135C2×Dic61#12311yescharacteristic
136C4×D61—41no—
137C4×D61—41no—
138C4×D61—81no—
139C4×D61—81no—
140C4×D61C2×SD1611yescharacteristic
141C4×D61#13511yescharacteristic
142C4×D61AGL(1,ℤ/8)11yescharacteristic
143C4×D61#13711yescharacteristic
144Dic122—41no—
145Dic122—41no—
146Dic122—81no—
147Dic122—81no—
148Dic122—81no—
149#92420—22no—
150#92420—41no—
151#92420—42no—
152#92420—81no—
153C8×C61—41no—
154#92418—41no—
155#9292—611nomaximal, Sylow
156C22×C61#38611yescharacteristic
157C2×D61—21no—
158C2×D61—21no—
159C2×D61#46011yescharacteristic
160C2×D61#46411yescharacteristic
161#6923—161no—
162C4×C61—21no—
163C4×C61—21no—
164C4×C61—42no—
165C4×C61—81no—
166C4×C61C4×SD1611yescharacteristic
167C4×C61#44011yescharacteristic
168Dic61—21no—
169Dic61—21no—
170Dic61—42no—
171Dic61—81no—
172Dic61#44811yescharacteristic
173Dic61#44911yescharacteristic
174#3229—611no—
175#2961—611no—
176#1960—611no—
177#1546—611no—
178#1518—611no—
179#1286—611no—
180#1275—611no—
181C2×C61#188011yescharacteristic
182C2×C61#190211yescharacteristic
183C2×C61#190911yescharacteristic
184D61—21no—
185D61—21no—
186#533—611no—
187C2×C4⋊Q8—611no—
188#498—611no—
189#498—611no—
190#487—611no—
191#487#692311yescharacteristic, complemented
192#483—1222no—
193#475—1222no—
194#446—1222no—
195#446—1222no—
196#431—611no—
197#431—611no—
198#424—1221no—
199#423—611no—
200C2×#103—1221no—
201C2×#103—1221no—
202#405—1221no—
203C2×#97—611no—
204#382—611no—
205#332—1222no—
206#328—1222no—
207C61#929211yescharacteristic, complemented, Sylow
208C22×Q8—1221no—
209C2×M16—1221no—
210C22×C8—611no—
211C22×C8—611no—
212C4⋊Q8—611no—
213C4⋊Q8—611no—
214C4⋊Q8—1221no—
215C4⋊Q8—1221no—
216C4⋊Q8—1221no—
217C4⋊Q8C2×#692311yescharacteristic
218C4⋊Q8C2×#692311yescharacteristic
219#125—1221no—
220C22⋊Q8—612no—
221C22⋊Q8—1221no—
222C4×Q8—612no—
223C4×Q8—1221no—
224#117—1221no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4—1221no—
227C2×C4⋊C4—1221no—
228C2×C42—611no—
229#106—1222no—
230#106—1222no—
231C4⋊C8—612no—
232C4⋊C8—1221no—
233C4⋊C8—1221no—
234C4⋊C8—1221no—
235C4⋊C8—1222no—
236C4⋊C8—1222no—
237#103—22no—
238#103—1222no—
239#103—2442no—
240#103—2442no—
241C22⋊C8—1221no—
242C22⋊C8—1221no—
243#97—1221no—
244#97—1221no—
245#97—1222no—
246C4×C8—611no—
247C4×C8—1221no—
248C4×C8—1222no—
249C4×C8C2×#692311yescharacteristic
250#95—1221no—
251#95—1221no—
252C2×Q8—21no—
253C2×Q8—1221no—
254C2×Q8—1221no—
255C2×Q8—2441no—
256C2×Q8—2441no—
257C2×Q8—2441no—
258C2×Q8—2441no—
259C2×Q8—2441no—
260C2×Q8C22⋊#692312yes—
261C22×C4—611no—
262C22×C4—611no—
263C22×C4—611no—
264C22×C4—2441no—
265C22×C4—2441no—
266M16—2441no—
267M16—2441no—
268C2×C8—21no—
269C2×C8—612no—
270C2×C8—1221no—
271C2×C8—1221no—
272C2×C8—1222no—
273C2×C8—1221no—
274C2×C8—1221no—
275C2×C8—1221no—
276C2×C8—1221no—
277C2×C8—2441no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—612no—
283C4⋊C4—1222no—
284C4⋊C4—1222no—
285C4⋊C4—1221no—
286C4⋊C4—1221no—
287C4⋊C4—1221no—
288C4⋊C4—2441no—
289C4⋊C4—2441no—
290C4⋊C4—2441no—
291C4⋊C4—2441no—
292C22⋊C4—1222no—
293C42—611no—
294C42—611no—
295C42—611no—
296C42—1222no—
297C42C22×#692311yescharacteristic
298C23—611no—
299Q8—42no—
300Q8—41no—
301Q8—41no—
302Q8—4881no—
303Q8—4881no—
304Q8—4881no—
305Q8—2441no—
306Q8—2441no—
307C2×C4—22no—
308C2×C4—41no—
309C2×C4—611no—
310C2×C4—611no—
311C2×C4—611no—
312C2×C4—611no—
313C2×C4—611no—
314C2×C4—611no—
315C2×C4—611no—
316C2×C4—1222no—
317C2×C4—4881no—
318C2×C4—4881no—
319C2×C4—4881no—
320C2×C4—4881no—
321C2×C4—4881no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4—2441no—
325C2×C4C2×C22⋊#692311yescharacteristic
326C2×C4D4×#692311yescharacteristic
327C2×C4D4×#692311yescharacteristic
328C8—41no—
329C8—1222no—
330C8—4881no—
331C8—2441no—
332C8—2441no—
333C8—2442no—
334C22—611no—
335C22—611no—
336C22—1221no—
337C22—1221no—
338C22#1684071011yescharacteristic
339C4—21no—
340C4—21no—
341C4—42no—
342C4—81no—
343C4—611no—
344C4—611no—
345C4—1221no—
346C4—1221no—
347C4—9761no—
348C4—4881no—
349C4—2442no—
350C4SD16×#692311yescharacteristic
351C4#1684080111yescharacteristic
352C2—1221no—
353C2—1221no—
354C2#2146181511yescharacteristic
355C2#2146184911yescharacteristic
356C2#2146186011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#487C4
socleC22×C61#386
prime radicalC22×D61#121
cosocleC42×D61C23
hypercenter#487#6923
hypocenterC61#9292
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
22. #16840160C4semidirect
23. #16840158C4semidirect
24. #16840112C4semidirect
28. C61×#487C4semidirect
191. #487#6923semidirect
207. C61#9292semidirect

Maximal subgroups

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

subgroupindex
#214626122
#214619502
#214613292
#214612532
#214606732
#214606442
#214569592
#929261