Finite Group Databasebeta

← #30601032

Subgroups — #30601032

Order 15616.

618 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 84 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#21462257C211yesmaximal, characteristic
2#21461218C212yesmaximal
3#21461068C211yesmaximal, characteristic
4#21460700C212yesmaximal
5#21451193C211yesmaximal, characteristic
6C2×#14519856—21no—
7#16840733C2212yes—
8C4⋊C4×#6923—24no—
9C4×C22⋊#6923—21no—
10C4×C22⋊#6923C2212yes—
11#16840615—24no—
12#16840608—21no—
13#16840607—21no—
14C2×#14519757C2211yescharacteristic
15C2×C4⋊C4×D61C2211yescharacteristic
16#16839693C411yescharacteristic, complemented
17#16839683C411yescharacteristic, complemented
18#16839682C411yescharacteristic, complemented
19#16839661—21no—
20#16839659—21no—
21#16839636—21no—
22#16839633—21no—
23#16839609C2211yescharacteristic
24C61×#398C411yescharacteristic, complemented
25C2×#11758213—22no—
26#14519856—24no—
27#14519856—24no—
28C2×C4⋊#6923—22no—
29C2×C22⋊#6923—22no—
30C2×C4×#6923—22no—
31C2×C4×#6923—22no—
32C2×C4×#6923—22no—
33#14519834—22no—
34C22⋊#92420—22no—
35(C2×#6923)⋊C4—22no—
36(C2×#6923)⋊C4D414yes—
37C22×Dic122C2×C411yescharacteristic
38C22×C4×D61D411yescharacteristic
39C22×C4×D61D411yescharacteristic
40C22×C4×D61C2311yescharacteristic
41#14519757—22no—
42#14519757—22no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61D411yescharacteristic
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196C2×C411yescharacteristic
49C2×#11758196D411yescharacteristic
50C4⋊C4×D61—22no—
51C4⋊C4×D61—22no—
52C2×#11758198—21no—
53C2×#11758198—21no—
54C2×C4×Dic61—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58#14519684C2×C411yescharacteristic
59C2×C4⋊C4×C61—21no—
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C4⋊C4×C61C2×C411yescharacteristic
62#95×C61—21no—
63#95×C61C2×C411yescharacteristic
64C23×D61C2×D411yescharacteristic
65C22×#6923—24no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22×C411yescharacteristic
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C22⋊C411yescharacteristic
75C22×Dic61C2×D411yescharacteristic
76C2×C4×D61—21no—
77C2×C4×D61—21no—
78C2×C4×D61—22no—
79C2×C4×D61—22no—
80C2×C4×D61—21no—
81C2×C4×D61—21no—
82C2×C4×D61—81no—
83C2×C4×D61—81no—
84C2×C4×D61C2×D412yes—
85C2×C4×D61#4112yes—
86C2×Dic122—22no—
87C2×Dic122—81no—
88C22⋊#6923—24no—
89C22⋊#6923—24no—
90C2×#92420—42no—
91C4⋊#6923—28no—
92C4×#6923—28no—
93C4×#6923—82no—
94C4×#6923—82no—
95#11758213—82no—
96C4⋊C4×C61—22no—
97C4⋊C4×C61—22no—
98C4⋊C4×C61—42no—
99#11758198—24no—
100#11758198—41no—
101#11758198—41no—
102C4⋊Dic61—22no—
103C4⋊Dic61—42no—
104C4⋊Dic61—81no—
105#11758196—22no—
106#11758196—22no—
107#11758196—42no—
108#11758196—81no—
109C4×Dic61—81no—
110C23×C61C2×C22⋊C411yescharacteristic
111C22×D61—21no—
112C22×D61—21no—
113C22×D61—22no—
114C22×D61—41no—
115C22×D61#12011yescharacteristic
116C22×D61#12311yescharacteristic
117C2×#6923—28no—
118C2×#6923—44no—
119C2×#6923—44no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—22no—
123C2×C4×C61—21no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61#10412yes—
132C2×C4×C61C4×D412yes—
133C2×Dic61—21no—
134C2×Dic61—21no—
135C2×Dic61—22no—
136C2×Dic61—21no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61#10412yes—
145C2×Dic61#12112yes—
146C4×D61—22no—
147C4×D61—22no—
148C4×D61—24no—
149C4×D61—42no—
150C4×D61—42no—
151C4×D61—42no—
152C4×D61—81no—
153Dic122—82no—
154#92420—82no—
155#9408—611nomaximal, Sylow
156C22×C61—21no—
157C22×C61—21no—
158C22×C61#38611yescharacteristic
159C22×C61C2×#10411yescharacteristic
160C22×C61C2×#10411yescharacteristic
161C2×D61—21no—
162C2×D61—21no—
163C2×D61—21no—
164C2×D61—21no—
165C2×D61—21no—
166C2×D61—42no—
167C2×D61—42no—
168C2×D61#45612yes—
169#6923—48no—
170C4×C61—22no—
171C4×C61—22no—
172C4×C61—42no—
173C4×C61—41no—
174C4×C61—81no—
175Dic61—22no—
176Dic61—22no—
177Dic61—42no—
178Dic61—41no—
179Dic61—81no—
180#2583—611no—
181#2274—611no—
182#1508—612no—
183#1463—611no—
184#1297—612no—
185C2×C61—21no—
186C2×C61—21no—
187C2×C61#188611yescharacteristic
188C2×C61#189611yescharacteristic
189C2×C61#191311yescharacteristic
190D61—22no—
191D61—41no—
192C2×C22⋊Q8—611no—
193C2×#117—1221no—
194C22×C4⋊C4—611no—
195#407—612no—
196#398—611no—
197#398#692311yescharacteristic, complemented
198#395—611no—
199#395—611no—
200#389—1221no—
201#389—1221no—
202#385—1221no—
203#385—1221no—
204C4×C4⋊C4—1224no—
205C4×C22⋊C4—612no—
206C4×C22⋊C4—611no—
207C4×C22⋊C4—1221no—
208#328—1224no—
209#324—1221no—
210#324—1221no—
211C61#940811yescharacteristic, complemented, Sylow
212C22×Q8—611no—
213C23×C4—611no—
214C23×C4—611no—
215C23×C4—611no—
216C2×M16—1222no—
217C22⋊Q8—1222no—
218C22⋊Q8—1222no—
219#117—1224no—
220#117—1224no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—611no—
223C2×C4⋊C4—611no—
224C2×C4⋊C4—611no—
225C2×C4⋊C4—1222no—
226C2×C4⋊C4—1222no—
227C2×C4⋊C4—1222no—
228C2×C4⋊C4—1221no—
229C2×C4⋊C4—1221no—
230C2×C4⋊C4—1221no—
231C2×C4⋊C4—1221no—
232C2×C4⋊C4C2×#692311yescharacteristic
233C2×C4⋊C4C2×#692311yescharacteristic
234C2×C22⋊C4—1221no—
235C2×C22⋊C4—1222no—
236C2×C22⋊C4—1221no—
237C2×C42—1222no—
238C2×C42—1221no—
239C2×C42—1222no—
240C2×C42—1222no—
241C22⋊C8—1222no—
242C22⋊C8—1222no—
243#95—21no—
244#95—614no—
245#95—611no—
246#95—1221no—
247#95—1221no—
248#95—1222no—
249#95—1221no—
250#95C2×#692311yescharacteristic
251C24—611no—
252C2×Q8—1222no—
253C2×Q8—4881no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—611no—
257C22×C4—611no—
258C22×C4—612no—
259C22×C4—612no—
260C22×C4—611no—
261C22×C4—1222no—
262C22×C4—1221no—
263C22×C4—1221no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—1222no—
267C22×C4—1224no—
268C22×C4—1221no—
269C22×C4—1221no—
270C22×C4—4881no—
271C22×C4—4881no—
272C22×C4C22⋊#692311yescharacteristic
273C22×C4C22⋊#692311yescharacteristic
274C22×C4C22×#692311yescharacteristic
275M16—4882no—
276C2×C8—2442no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—42no—
280C4⋊C4—1222no—
281C4⋊C4—1222no—
282C4⋊C4—1228no—
283C4⋊C4—1222no—
284C4⋊C4—4881no—
285C4⋊C4—4881no—
286C4⋊C4—2442no—
287C4⋊C4—2442no—
288C22⋊C4—1224no—
289C22⋊C4—1224no—
290C22⋊C4—1224no—
291C22⋊C4—2441no—
292C22⋊C4—2441no—
293C42—1228no—
294C42—4881no—
295C42—4882no—
296C42—4882no—
297C23—611no—
298C23—611no—
299C23—1222no—
300C23—1221no—
301C23—1221no—
302C23—2441no—
303C23C2×C22⋊#692311yescharacteristic
304Q8—4882no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—612no—
317C2×C4—612no—
318C2×C4—1222no—
319C2×C4—1222no—
320C2×C4—1224no—
321C2×C4—1221no—
322C2×C4—1221no—
323C2×C4—1222no—
324C2×C4—1228no—
325C2×C4—1221no—
326C2×C4—4881no—
327C2×C4—2442no—
328C2×C4—2442no—
329C2×C4—2442no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4—2441no—
333C2×C4—2441no—
334C2×C4—2441no—
335C2×C4—2444no—
336C2×C4—2444no—
337C2×C4—2441no—
338C2×C4—2441no—
339C2×C4#1451984112yes—
340C2×C4D4×#692312yes—
341C8—4882no—
342C22—21no—
343C22—21no—
344C22—612no—
345C22—1221no—
346C22—1221no—
347C22—1221no—
348C22—1221no—
349C22—1221no—
350C22—2442no—
351C22—2442no—
352C22#1684071011yescharacteristic
353C22C2×#1451984111yescharacteristic
354C22D61⋊#10411yescharacteristic
355C4—22no—
356C4—22no—
357C4—42no—
358C4—41no—
359C4—81no—
360C4—1222no—
361C4—1222no—
362C4—4881no—
363C4—2442no—
364C4—2448no—
365C4—2441no—
366C2—21no—
367C2—21no—
368C2—1222no—
369C2—2441no—
370C2#2146182511yescharacteristic
371C2#2146183711yescharacteristic
372C2#2146186911yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#398C4
socleC22×C61#386
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#398#6923
hypocenterC61#9408
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
16. #16839693C4semidirect
17. #16839683C4semidirect
18. #16839682C4semidirect
24. C61×#398C4semidirect
197. #398#6923semidirect
211. C61#9408semidirect

Maximal subgroups

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

subgroupindex
#214622572
#214612182
#214610682
#214607002
#214511932
#940861