Finite Group Databasebeta

← #30600776

Subgroups — #30600776

Order 15616.

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

normal subgroups only

This group has 470 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#21462310C211yesmaximal, characteristic
2#21462007C211yesmaximal, characteristic
3#21461329C211yesmaximal, characteristic
4#21461251C211yesmaximal, characteristic
5#21460673C211yesmaximal, characteristic
6#21460642C211yesmaximal, characteristic
7#21456959C211yesmaximal, characteristic
8C2×Q8×#6923—21no—
9#16840780C2211yescharacteristic
10#16840767C2211yescharacteristic
11(C2×#6923)⋊Q8—21no—
12#16840711—22no—
13#16840711—22no—
14#16840706C2211yescharacteristic
15#16840698C2211yescharacteristic
16#16840632—21no—
17#16840632—21no—
18#16840621—22no—
19#16840615—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—
30Q8×#6923—42no—
31Q8×#6923—42no—
32C2×C4⋊#6923—22no—
33C2×C4⋊#6923—21no—
34C2×C4×#6923—22no—
35C2×C4×#6923D411yescharacteristic
36C2×C4×#6923D411yescharacteristic
37#14519844—21no—
38#14519844—21no—
39#14519844—21no—
40#14519834—21no—
41C4×#92420—21no—
42(C2×#6923)⋊C4—24no—
43(C2×#6923)⋊C4—21no—
44(C2×#6923)⋊C4—21no—
45(C2×#6923)⋊C4—21no—
46(C2×#6923)⋊C4D412yes—
47C2×Q8×D61—21no—
48#14519777C2×C411yescharacteristic
49#14519777C2×C411yescharacteristic
50#14519776—21no—
51#14519776D412yes—
52#14519775—21no—
53#14519768—21no—
54#14519757—21no—
55#14519753—21no—
56Q8×Dic61—21no—
57#14519742—21no—
58#14519735—21no—
59C4⋊C4×D61—21no—
60C4⋊C4×D61D412yes—
61C42×D61C2311yescharacteristic
62#14519709—22no—
63#14519708—22no—
64#14519706—21no—
65#14519701—22no—
66#14519689—21no—
67C4×#92418C2×C411yescharacteristic
68C4⋊Q8×C61C2×C411yescharacteristic
69C4⋊Q8×C61C2×C411yescharacteristic
70#103×C61—22no—
71C4×C8×C61C2×C411yescharacteristic
72C22×#6923—21no—
73C22×#6923—21no—
74C22×#6923—22no—
75C2×Q8×C61—21no—
76C2×Q8×C61—21no—
77Q8×D61—41no—
78Q8×D61—41no—
79Q8×D61—41no—
80Q8×D61—41no—
81C2×C4×D61—22no—
82C2×C4×D61C2×D411yescharacteristic
83C2×C4×D61C2×D411yescharacteristic
84C2×C4×D61C2×D411yescharacteristic
85C2×Dic122—21no—
86C2×Dic122—22no—
87C2×Dic122—21no—
88C2×#92420—41no—
89C2×#92420—41no—
90C4⋊#6923—44no—
91C4⋊#6923—41no—
92C4⋊#6923—41no—
93C4×#6923—21no—
94C4×#6923—21no—
95C4×#6923—21no—
96C4×#6923—44no—
97C4×#6923SD1612yes—
98#11758213—41no—
99#11758213—41no—
100C2×C8×C61—21no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61C22⋊C412yes—
105C42×C61C22×C411yescharacteristic
106#11758198—41no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109C4⋊Dic61—21no—
110C4⋊Dic61—41no—
111C4⋊Dic61C22⋊C412yes—
112#11758196—22no—
113#11758196—22no—
114#11758196—41no—
115C4×Dic61—41no—
116C4×Dic61C2×D411yescharacteristic
117C4×Dic61#4111yescharacteristic
118C4×Dic61#4111yescharacteristic
119C2×#92418—21no—
120C22×D61#12011yescharacteristic
121C2×#6923—22no—
122C2×#6923—41no—
123C2×#6923—41no—
124C2×#6923—41no—
125C2×#6923—42no—
126C2×#6923—41no—
127C2×#6923—42no—
128C2×#6923—42no—
129C2×#6923—41no—
130Q8×C61—42no—
131Q8×C61—41no—
132Q8×C61—41no—
133C2×C4×C61—22no—
134C2×C4×C61—41no—
135C2×C4×C61C2×C22⋊C411yescharacteristic
136C2×C4×C61C4×D411yescharacteristic
137C2×C4×C61C4×D411yescharacteristic
138C2×Dic61—22no—
139C2×Dic61—41no—
140C2×Dic61#12111yescharacteristic
141C2×Dic61#12111yescharacteristic
142C2×Dic61#12311yescharacteristic
143C4×D61—41no—
144C4×D61—41no—
145C4×D61—42no—
146C4×D61—42no—
147C4×D61C2×SD1611yescharacteristic
148C4×D61#13511yescharacteristic
149C4×D61AGL(1,ℤ/8)11yescharacteristic
150C4×D61#13711yescharacteristic
151Dic122—41no—
152Dic122—41no—
153Dic122—42no—
154Dic122—42no—
155Dic122—81no—
156#92420—81no—
157#92420—81no—
158C8×C61—41no—
159#92418—41no—
160#9296—611nomaximal, Sylow
161C22×C61#38611yescharacteristic
162C2×D61—21no—
163C2×D61—21no—
164C2×D61#45011yescharacteristic
165C2×D61#45211yescharacteristic
166#6923—42no—
167#6923—82no—
168#6923—81no—
169C4×C61—21no—
170C4×C61—21no—
171C4×C61—42no—
172C4×C61—81no—
173C4×C61C4×SD1611yescharacteristic
174C4×C61#44011yescharacteristic
175Dic61—21no—
176Dic61—21no—
177Dic61—42no—
178Dic61—81no—
179Dic61#46111yescharacteristic
180Dic61#46311yescharacteristic
181#3229—611no—
182#2327—611no—
183#1995—611no—
184#1546—611no—
185#1518—611no—
186#1286—611no—
187#1274—611no—
188C2×C61#187711yescharacteristic
189C2×C61#190511yescharacteristic
190C2×C61#191211yescharacteristic
191D61—21no—
192D61—21no—
193#533—611no—
194C2×C4⋊Q8—611no—
195C2×C4×Q8—1221no—
196#498—611no—
197#498—611no—
198#487—611no—
199#487#692311yescharacteristic, complemented
200#483—1222no—
201#475—1222no—
202#431—611no—
203#431—611no—
204#423—611no—
205#391—1221no—
206#387—1222no—
207#387—1222no—
208#382—611no—
209#376—611no—
210#339—1221no—
211#339—1221no—
212#332—1222no—
213#328—1222no—
214C61#929611yescharacteristic, complemented, Sylow
215C22×Q8—1221no—
216C2×M16—1221no—
217C4⋊Q8—612no—
218C4⋊Q8—611no—
219C4⋊Q8—611no—
220C4⋊Q8—1221no—
221C4⋊Q8—1221no—
222C4⋊Q8C2×#692311yescharacteristic
223C4⋊Q8C2×#692311yescharacteristic
224#125—1221no—
225C22⋊Q8—1221no—
226C22⋊Q8—1221no—
227C4×Q8—1221no—
228C4×Q8—1221no—
229C4×Q8—2442no—
230C4×Q8—2442no—
231#117—1221no—
232C2×C4⋊C4—612no—
233C2×C4⋊C4—1221no—
234C2×C4⋊C4—1221no—
235C2×C4⋊C4—1222no—
236C2×C42—611no—
237C2×C42—611no—
238C2×C42—611no—
239C2×C42—1222no—
240#106—1222no—
241#106—1222no—
242C4⋊C8—1221no—
243C4⋊C8—1221no—
244C4⋊C8—1221no—
245C4⋊C8—1221no—
246#103—22no—
247#103—1222no—
248C22⋊C8—1221no—
249C22⋊C8—1221no—
250C4×C8—611no—
251C4×C8—1221no—
252C4×C8C2×#692311yescharacteristic
253#95—612no—
254#95—1221no—
255#95—1221no—
256#95—1221no—
257#95—1224no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—1222no—
261C2×Q8—1221no—
262C2×Q8—1221no—
263C2×Q8—2441no—
264C2×Q8—2441no—
265C2×Q8—2441no—
266C2×Q8—2441no—
267C22×C4—611no—
268C22×C4—611no—
269C22×C4—611no—
270C22×C4—1222no—
271C22×C4—1221no—
272C22×C4—1222no—
273C22×C4—1221no—
274M16—2441no—
275M16—2441no—
276C2×C8—21no—
277C2×C8—1221no—
278C2×C8—2441no—
279C2×C8—2441no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—612no—
284C4⋊C4—1222no—
285C4⋊C4—1222no—
286C4⋊C4—1221no—
287C4⋊C4—1221no—
288C4⋊C4—1221no—
289C4⋊C4—2441no—
290C4⋊C4—2441no—
291C4⋊C4—2441no—
292C4⋊C4—2441no—
293C4⋊C4—2444no—
294C4⋊C4C22⋊#692312yes—
295C22⋊C4—2441no—
296C42—611no—
297C42—611no—
298C42—611no—
299C42—612no—
300C42—1221no—
301C42—1221no—
302C42—1221no—
303C42—2441no—
304C42—2444no—
305C42C22×#692311yescharacteristic
306C23—611no—
307Q8—42no—
308Q8—41no—
309Q8—41no—
310Q8—4881no—
311Q8—2442no—
312Q8—2442no—
313Q8—2441no—
314Q8—2441no—
315C2×C4—22no—
316C2×C4—41no—
317C2×C4—611no—
318C2×C4—611no—
319C2×C4—611no—
320C2×C4—611no—
321C2×C4—611no—
322C2×C4—611no—
323C2×C4—611no—
324C2×C4—1222no—
325C2×C4—1222no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4—2442no—
329C2×C4—2442no—
330C2×C4—2441no—
331C2×C4—2442no—
332C2×C4—2442no—
333C2×C4—2441no—
334C2×C4—2442no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4—2441no—
338C2×C4—2441no—
339C2×C4C2×C22⋊#692311yescharacteristic
340C2×C4D4×#692311yescharacteristic
341C2×C4D4×#692311yescharacteristic
342C8—41no—
343C8—4881no—
344C8—4881no—
345C8—2441no—
346C22—611no—
347C22—611no—
348C22—1221no—
349C22—1221no—
350C22#1684071011yescharacteristic
351C4—21no—
352C4—21no—
353C4—42no—
354C4—81no—
355C4—611no—
356C4—611no—
357C4—1221no—
358C4—1221no—
359C4—4882no—
360C4—4881no—
361C4—4881no—
362C4—2442no—
363C4—2442no—
364C4SD16×#692311yescharacteristic
365C4#1684080111yescharacteristic
366C2—1221no—
367C2—1221no—
368C2#2146181211yescharacteristic
369C2#2146185411yescharacteristic
370C2#2146186711yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#487C4
socleC22×C61#386
prime radicalC22×D61#120
cosocleC42×D61C23
hypercenter#487#6923
hypocenterC61#9296
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
199. #487#6923semidirect
214. C61#9296semidirect

Maximal subgroups

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

subgroupindex
#214623102
#214620072
#214613292
#214612512
#214606732
#214606422
#214569592
#929661