Finite Group Databasebeta

← #30600780

Subgroups — #30600780

Order 15616.

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

normal subgroups only

This group has 442 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#21462314C211yesmaximal, characteristic
2#21462007C211yesmaximal, characteristic
3#21461335C211yesmaximal, characteristic
4#21461251C211yesmaximal, characteristic
5#21460673C211yesmaximal, characteristic
6#21460642C211yesmaximal, characteristic
7#21456966C211yesmaximal, characteristic
8#16840780C2211yescharacteristic
9#16840767C2211yescharacteristic
10(C2×#6923)⋊Q8—22no—
11#16840713—21no—
12#16840711—22no—
13#16840706C2211yescharacteristic
14C4⋊C4×#6923—21no—
15#16840698C2211yescharacteristic
16#16840632—21no—
17#16840632—21no—
18#16840621—22no—
19#16840615—22no—
20#16840317C2211yescharacteristic
21C4⋊Q8×D61C2211yescharacteristic
22#16840167C411yescharacteristic, complemented
23#16840165C411yescharacteristic, complemented
24#16840112C411yescharacteristic, complemented
25#16840096—22no—
26#16840055—22no—
27#16839813C2211yescharacteristic
28C61×#487C411yescharacteristic, complemented
29C2×#11758213—21no—
30C2×C4⋊#6923—21no—
31C2×C4⋊#6923—22no—
32C2×C4×#6923—22no—
33C2×C4×#6923D411yescharacteristic
34C2×C4×#6923D411yescharacteristic
35#14519844—21no—
36#14519844—21no—
37#14519844—21no—
38#14519834—21no—
39C4×#92420—21no—
40(C2×#6923)⋊C4—22no—
41(C2×#6923)⋊C4—22no—
42(C2×#6923)⋊C4—21no—
43(C2×#6923)⋊C4—21no—
44(C2×#6923)⋊C4—21no—
45(C2×#6923)⋊C4D412yes—
46C2×Q8×D61—21no—
47#14519777C2×C411yescharacteristic
48#14519777C2×C411yescharacteristic
49#14519776—21no—
50#14519776D412yes—
51#14519775—21no—
52#14519768—21no—
53#14519757—21no—
54#14519753—21no—
55Q8×Dic61—21no—
56#14519742—21no—
57#14519735—21no—
58C4⋊C4×D61—21no—
59C4⋊C4×D61D412yes—
60C42×D61C2311yescharacteristic
61#14519711—22no—
62#14519710—22no—
63#14519706—21no—
64#14519701—22no—
65#14519689—21no—
66C4×#92418C2×C411yescharacteristic
67C4⋊Q8×C61C2×C411yescharacteristic
68C4⋊Q8×C61C2×C411yescharacteristic
69#103×C61—22no—
70C4×C8×C61C2×C411yescharacteristic
71C22×#6923—21no—
72C22×#6923—21no—
73C22×#6923—22no—
74C2×Q8×C61—21no—
75C2×Q8×C61—21no—
76Q8×D61—82no—
77C2×C4×D61—22no—
78C2×C4×D61C2×D411yescharacteristic
79C2×C4×D61C2×D411yescharacteristic
80C2×C4×D61C2×D411yescharacteristic
81C2×Dic122—21no—
82C2×Dic122—22no—
83C2×Dic122—21no—
84C2×#92420—41no—
85C2×#92420—41no—
86C4⋊#6923—41no—
87C4⋊#6923—41no—
88C4⋊#6923—82no—
89C4×#6923—21no—
90C4×#6923—21no—
91C4×#6923—21no—
92C4×#6923—82no—
93C4×#6923Q1612yes—
94#11758213—41no—
95#11758213—41no—
96C2×C8×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—21no—
100C4⋊C4×C61C22⋊C412yes—
101C42×C61C22×C411yescharacteristic
102#11758198—41no—
103C4⋊Dic61—21no—
104C4⋊Dic61—21no—
105C4⋊Dic61—21no—
106C4⋊Dic61—41no—
107C4⋊Dic61C22⋊C412yes—
108#11758196—22no—
109#11758196—22no—
110#11758196—41no—
111C4×Dic61—41no—
112C4×Dic61C2×D411yescharacteristic
113C4×Dic61#4111yescharacteristic
114C4×Dic61#4111yescharacteristic
115C2×#92418—21no—
116C22×D61#12011yescharacteristic
117C2×#6923—22no—
118C2×#6923—41no—
119C2×#6923—41no—
120C2×#6923—42no—
121C2×#6923—41no—
122C2×#6923—41no—
123C2×#6923—42no—
124C2×#6923—41no—
125C2×#6923—42no—
126Q8×C61—42no—
127Q8×C61—81no—
128C2×C4×C61—22no—
129C2×C4×C61—41no—
130C2×C4×C61C2×C22⋊C411yescharacteristic
131C2×C4×C61C4×D411yescharacteristic
132C2×C4×C61C4×D411yescharacteristic
133C2×Dic61—22no—
134C2×Dic61—41no—
135C2×Dic61#12111yescharacteristic
136C2×Dic61#12111yescharacteristic
137C2×Dic61#12311yescharacteristic
138C4×D61—41no—
139C4×D61—41no—
140C4×D61—42no—
141C4×D61—42no—
142C4×D61C2×Q1611yescharacteristic
143C4×D61#13511yescharacteristic
144C4×D61#13711yescharacteristic
145C4×D61#13711yescharacteristic
146Dic122—82no—
147Dic122—81no—
148Dic122—81no—
149#92420—81no—
150#92420—81no—
151C8×C61—41no—
152#92418—41no—
153#9298—611nomaximal, Sylow
154C22×C61#38611yescharacteristic
155C2×D61—21no—
156C2×D61—21no—
157C2×D61#45111yescharacteristic
158C2×D61#45211yescharacteristic
159#6923—42no—
160#6923—82no—
161#6923—81no—
162C4×C61—21no—
163C4×C61—21no—
164C4×C61—42no—
165C4×C61—81no—
166C4×C61C4×Q1611yescharacteristic
167C4×C61#44111yescharacteristic
168Dic61—21no—
169Dic61—21no—
170Dic61—42no—
171Dic61—81no—
172Dic61#46211yescharacteristic
173Dic61#46411yescharacteristic
174#3230—611no—
175#2331—611no—
176#1995—611no—
177#1549—611no—
178#1518—611no—
179#1286—611no—
180#1274—611no—
181C2×C61#187811yescharacteristic
182C2×C61#190411yescharacteristic
183C2×C61#191211yescharacteristic
184D61—21no—
185D61—21no—
186#533—611no—
187C2×C4⋊Q8—611no—
188#500—611no—
189#500—611no—
190#487—611no—
191#487#692311yescharacteristic, complemented
192#484—1222no—
193#477—1222no—
194#431—611no—
195#431—611no—
196#423—611no—
197#391—1222no—
198#389—1221no—
199#387—1222no—
200#382—611no—
201C4×C4⋊C4—1221no—
202#376—611no—
203#339—1221no—
204#339—1221no—
205#332—1222no—
206#328—1222no—
207C61#929811yescharacteristic, complemented, Sylow
208C22×Q8—1221no—
209C2×M16—1221no—
210C4⋊Q8—612no—
211C4⋊Q8—611no—
212C4⋊Q8—611no—
213C4⋊Q8—1221no—
214C4⋊Q8—1221no—
215C4⋊Q8C2×#692311yescharacteristic
216C4⋊Q8C2×#692311yescharacteristic
217#125—1221no—
218C22⋊Q8—1221no—
219C22⋊Q8—1221no—
220C4×Q8—1221no—
221C4×Q8—1221no—
222#117—1221no—
223C2×C4⋊C4—612no—
224C2×C4⋊C4—1221no—
225C2×C4⋊C4—1222no—
226C2×C4⋊C4—1221no—
227C2×C42—611no—
228C2×C42—611no—
229C2×C42—611no—
230C2×C42—1222no—
231#107—1222no—
232#107—1222no—
233C4⋊C8—1221no—
234C4⋊C8—1221no—
235C4⋊C8—1221no—
236C4⋊C8—1221no—
237#103—22no—
238#103—1222no—
239C22⋊C8—1221no—
240C22⋊C8—1221no—
241C4×C8—611no—
242C4×C8—1221no—
243C4×C8C2×#692311yescharacteristic
244#95—612no—
245#95—1221no—
246#95—1221no—
247#95—1221no—
248#95—1222no—
249#95—1222no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—1222no—
253C2×Q8—1221no—
254C2×Q8—1221no—
255C2×Q8—4882no—
256C22×C4—611no—
257C22×C4—611no—
258C22×C4—611no—
259C22×C4—1222no—
260C22×C4—1221no—
261C22×C4—1222no—
262C22×C4—1221no—
263M16—2441no—
264M16—2441no—
265C2×C8—21no—
266C2×C8—1221no—
267C2×C8—2441no—
268C2×C8—2441no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—612no—
273C4⋊C4—1222no—
274C4⋊C4—1222no—
275C4⋊C4—1221no—
276C4⋊C4—1221no—
277C4⋊C4—1221no—
278C4⋊C4—4882no—
279C4⋊C4—2441no—
280C4⋊C4—2441no—
281C4⋊C4—2441no—
282C4⋊C4—2441no—
283C4⋊C4C22⋊#692312yes—
284C22⋊C4—2441no—
285C42—611no—
286C42—611no—
287C42—611no—
288C42—612no—
289C42—1221no—
290C42—1221no—
291C42—1221no—
292C42—4882no—
293C42—2441no—
294C42C22×#692311yescharacteristic
295C23—611no—
296Q8—42no—
297Q8—81no—
298Q8—4881no—
299Q8—4881no—
300Q8—4882no—
301C2×C4—22no—
302C2×C4—41no—
303C2×C4—611no—
304C2×C4—611no—
305C2×C4—611no—
306C2×C4—611no—
307C2×C4—611no—
308C2×C4—611no—
309C2×C4—611no—
310C2×C4—1222no—
311C2×C4—1222no—
312C2×C4—2441no—
313C2×C4—2441no—
314C2×C4—2442no—
315C2×C4—2442no—
316C2×C4—2441no—
317C2×C4—2442no—
318C2×C4—2442no—
319C2×C4—2441no—
320C2×C4—2442no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4—2441no—
325C2×C4C2×C22⋊#692311yescharacteristic
326C2×C4D4×#692311yescharacteristic
327C2×C4D4×#692311yescharacteristic
328C8—41no—
329C8—4881no—
330C8—4881no—
331C8—2441no—
332C22—611no—
333C22—611no—
334C22—1221no—
335C22—1221no—
336C22#1684071011yescharacteristic
337C4—21no—
338C4—21no—
339C4—42no—
340C4—81no—
341C4—611no—
342C4—611no—
343C4—1221no—
344C4—1221no—
345C4—4881no—
346C4—4882no—
347C4—4881no—
348C4—2442no—
349C4—2442no—
350C4Q16×#692311yescharacteristic
351C4#1684080211yescharacteristic
352C2—1221no—
353C2—1221no—
354C2#2146181311yescharacteristic
355C2#2146185211yescharacteristic
356C2#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#9298
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. #16840167C4semidirect
23. #16840165C4semidirect
24. #16840112C4semidirect
28. C61×#487C4semidirect
191. #487#6923semidirect
207. C61#9298semidirect

Maximal subgroups

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

subgroupindex
#214623142
#214620072
#214613352
#214612512
#214606732
#214606422
#214569662
#929861