Finite Group Databasebeta

← #28304000

Subgroups — #28304000

Order 13568.

626 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 126 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#20291562C211yesmaximal, characteristic
2#20290605C212yesmaximal, complemented
3#20289951C211yesmaximal, characteristic
4#20289558C212yesmaximal, complemented
5#123×Dic53C211yesmaximal, characteristic
6D4×#88324—24no—
7#16811639—21no—
8C2×C22⋊#88324—21no—
9C2×C22⋊#88324C2212yes—
10C2×C22⋊#88324C2212yes—
11#16811489—24no—
12#16811468—21no—
13#16811466—21no—
14C2×D4×Dic53C2211yescharacteristic
15C22×C4×Dic53C2211yescharacteristic
16#16810516—21no—
17#16810509—21no—
18#16810503C411yescharacteristic
19#16810498C411yescharacteristic
20#16810486C411yescharacteristic
21C4⋊C4×Dic53—21no—
22C22⋊C4×Dic53—21no—
23C22⋊C4×Dic53C2211yescharacteristic
24C2×#123×C53C411yescharacteristic
25C22×#88324—22no—
26C22×#88324—22no—
27C22×#88324D414yes—
28C4⋊#88324—22no—
29C22⋊#88324—22no—
30C22⋊#88324—22no—
31C22⋊#88324—22no—
32C22⋊#88324—22no—
33C22⋊#88324—22no—
34C4×#88324—22no—
35C23×Dic53C2×C411yescharacteristic
36C23×Dic53D411yescharacteristic
37C23×Dic53C2311yescharacteristic
38D4×Dic53—22no—
39D4×Dic53—22no—
40C2×C4⋊Dic53—21no—
41C2×C4⋊Dic53D411yescharacteristic
42C2×C22⋊Dic53—21no—
43C2×C22⋊Dic53C2×C411yescharacteristic
44C2×C22⋊Dic53C2×C411yescharacteristic
45C2×C22⋊Dic53D411yescharacteristic
46C2×C4×Dic53—21no—
47C2×C4×Dic53—22no—
48C2×C4×Dic53—21no—
49C2×C4×Dic53—21no—
50C2×C4×Dic53—22no—
51C2×C4×Dic53D411yescharacteristic
52#13133124—21no—
53#13133124—21no—
54#13133124—21no—
55#13133124—21no—
56#13133124—21no—
57C22×D4×C53C2×C411yescharacteristic
58C23×C4×C53C2×C411yescharacteristic
59#123×C53—22no—
60#123×C53C814yescomplemented
61C2×C4⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53C2×C411yescharacteristic
64C24×C53C22⋊C411yescharacteristic
65C24×C53C22×C411yescharacteristic
66C2×D4×C53—22no—
67C2×D4×C53—22no—
68C2×D4×C53C2×C812yes—
69C2×D4×C53M1612yes—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—22no—
73C22×C4×C53—21no—
74C22×C4×C53—22no—
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C2×C812yescomplemented
77C22×Dic53—21no—
78C22×Dic53—22no—
79C22×Dic53—21no—
80C22×Dic53—21no—
81C22×Dic53—21no—
82C22×Dic53—42no—
83C22×Dic53—42no—
84C22×Dic53—42no—
85C22×Dic53C22⋊C411yescharacteristic
86C22×Dic53C22⋊C411yescharacteristic
87C22×Dic53C2×D412yes—
88C22×Dic53C2×D411yescharacteristic
89C22×Dic53C2×D411yescharacteristic
90C22×Dic53#4112yes—
91C2×#88324—24no—
92C2×#88324—44no—
93C2×#88324—44no—
94C2×#88324—42no—
95C2×#88324—82no—
96C4⋊C4×C53—24no—
97C22⋊C4×C53—24no—
98C22⋊C4×C53—21no—
99C22⋊C4×C53C2×C812yescomplemented
100C22⋊Dic53—22no—
101C22⋊Dic53—22no—
102C22⋊Dic53—22no—
103C22⋊Dic53—42no—
104C4⋊Dic53—22no—
105C4⋊Dic53—42no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108C4×Dic53—42no—
109C4×Dic53—41no—
110C4×Dic53—42no—
111C4×Dic53—41no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53C22⋊C812yes—
119C23×C53C2×C22⋊C411yescharacteristic
120C23×C53C4×D412yes—
121C23×C53C22×C811yescharacteristic
122C23×C53C2×M1611yescharacteristic
123C23×C53#13112yes—
124D4×C53—24no—
125D4×C53—24no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—22no—
129C2×C4×C53—22no—
130C2×C4×C53—24no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—42no—
136C2×C4×C53C22⋊C812yes—
137C2×Dic53—24no—
138C2×Dic53—22no—
139C2×Dic53—24no—
140C2×Dic53—22no—
141C2×Dic53—21no—
142C2×Dic53—21no—
143C2×Dic53—41no—
144C2×Dic53—42no—
145C2×Dic53—41no—
146C2×Dic53—42no—
147C2×Dic53—41no—
148C2×Dic53—42no—
149C2×Dic53—41no—
150C2×Dic53—41no—
151C2×Dic53—41no—
152C2×Dic53—41no—
153C2×Dic53—41no—
154C2×Dic53—81no—
155C2×Dic53#12011yescharacteristic
156C2×Dic53#12112yes—
157C2×Dic53#12311yescharacteristic
158#88324—44no—
159#88324—82no—
160#8966—531nomaximal, Sylow
161C22×C53—21no—
162C22×C53—22no—
163C22×C53—21no—
164C22×C53—22no—
165C22×C53—22no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—42no—
169C22×C53—42no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—81no—
173C22×C53#38611yescharacteristic
174C22×C53C2×C22⋊C811yescharacteristic
175C22×C53#40811yescharacteristic
176C22×C53C8×D412yes—
177C22×C53#43512yes—
178C4×C53—22no—
179C4×C53—42no—
180C4×C53—42no—
181Dic53—22no—
182Dic53—21no—
183Dic53—41no—
184Dic53—42no—
185Dic53—41no—
186Dic53—81no—
187Dic53#45812yes—
188#2972—531no—
189C4×#123—531no—
190#1817—532no—
191#1459—531no—
192#1282—532no—
193C2×C53—21no—
194C2×C53—21no—
195C2×C53—22no—
196C2×C53—22no—
197C2×C53—42no—
198C2×C53#186911yescharacteristic
199C2×C53#189011yescharacteristic
200C2×C53#189111yescharacteristic
201C2×#123#615711yescharacteristic
202C2×C4×D4—531no—
203C22×C42—531no—
204C8×D4—1064no—
205#431—1061no—
206C2×C22⋊C8—532no—
207C2×C22⋊C8—532no—
208C2×C22⋊C8—1061no—
209#388—1061no—
210#387—1061no—
211#386—531no—
212#385—531no—
213#381—531no—
214C4×C4⋊C4—1061no—
215C4×C22⋊C4—531no—
216C4×C22⋊C4—1061no—
217#336—1064no—
218#324—1061no—
219#323—1061no—
220C53#896611yescharacteristic, complemented, Sylow
221C22×D4C2×#615711yescharacteristic
222C23×C4—531no—
223C23×C4—531no—
224C23×C4—531no—
225C23×C4C2×#615711yescharacteristic
226C22×C8—534no—
227C22×C8—1062no—
228C22×C8—1062no—
229#123—22no—
230#123#8832414yescomplemented
231C4×D4—1062no—
232C4×D4—1062no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—531no—
235C2×C4⋊C4—1061no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—531no—
238C2×C22⋊C4—531no—
239C2×C22⋊C4—531no—
240C2×C22⋊C4—1061no—
241C2×C22⋊C4C2×#615711yescharacteristic
242C2×C42—531no—
243C2×C42—1062no—
244C2×C42—1061no—
245C2×C42—1061no—
246C2×C42—1061no—
247C2×C42—1062no—
248C4⋊C8—1062no—
249C22⋊C8—1062no—
250C22⋊C8—1062no—
251C22⋊C8—1062no—
252C22⋊C8—1062no—
253C22⋊C8—1062no—
254C4×C8—1062no—
255#95—1061no—
256#95—1061no—
257#95—1061no—
258#95—1061no—
259#95—1061no—
260C24C22⋊#615711yescharacteristic
261C24C22×#615711yescharacteristic
262C2×D4—22no—
263C2×D4—22no—
264C2×D4C2×#8832412yes—
265C2×D4#1171994012yes—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—531no—
272C22×C4—532no—
273C22×C4—531no—
274C22×C4—532no—
275C22×C4—531no—
276C22×C4—531no—
277C22×C4—1061no—
278C22×C4—1061no—
279C22×C4—1061no—
280C22×C4—1062no—
281C22×C4—1061no—
282C22×C4—2122no—
283C22×C4—2122no—
284C22×C4—2122no—
285C22×C4C2×#8832412yescomplemented
286C22×C4C22⋊#615711yescharacteristic
287C2×C8—1064no—
288C2×C8—4242no—
289C2×C8—2122no—
290C2×C8—2124no—
291C2×C8—2124no—
292C4⋊C4—24no—
293C4⋊C4—1062no—
294C4⋊C4—2122no—
295C22⋊C4—24no—
296C22⋊C4—21no—
297C22⋊C4—1062no—
298C22⋊C4—1062no—
299C22⋊C4—1062no—
300C22⋊C4—2122no—
301C22⋊C4C2×#8832412yescomplemented
302C42—1061no—
303C42—1061no—
304C42—2121no—
305C42—2122no—
306C42—2121no—
307C42—2122no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23C22⋊#8832412yes—
315C23C2×C22⋊#615711yescharacteristic
316C23D4×#615712yes—
317C23C22×#8832411yescharacteristic
318C23C2×#1171994011yescharacteristic
319C23#1313330412yes—
320D4—24no—
321D4—24no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—24no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—531no—
333C2×C4—532no—
334C2×C4—531no—
335C2×C4—1062no—
336C2×C4—1064no—
337C2×C4—1062no—
338C2×C4—1064no—
339C2×C4—1061no—
340C2×C4—1061no—
341C2×C4—4241no—
342C2×C4—2122no—
343C2×C4—2121no—
344C2×C4—2122no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4—2121no—
349C2×C4—2122no—
350C2×C4—2121no—
351C2×C4—2121no—
352C2×C4—2121no—
353C2×C4C22⋊#8832412yes—
354C8—4242no—
355C8—2124no—
356C22—21no—
357C22—22no—
358C22—21no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—42no—
364C22—42no—
365C22—41no—
366C22—41no—
367C22—81no—
368C22#1681157111yescharacteristic
369C22C2×C22⋊#8832411yescharacteristic
370C22#1681159611yescharacteristic
371C22D4×#8832412yes—
372C22#1681165412yes—
373C4—22no—
374C4—42no—
375C4—42no—
376C4—532no—
377C4—1062no—
378C4—1061no—
379C4—4241no—
380C4—2121no—
381C4—2122no—
382C4—2121no—
383C2—21no—
384C2—21no—
385C2—22no—
386C2—22no—
387C2—42no—
388C2#2029068811yescharacteristic
389C2#2029072511yescharacteristic
390C2#2029072711yescharacteristic
subgroupG/H
centerC22#16811571
commutatorC23×C53C22×C8
FrattiniC24C22×#6157
FittingC2×#123×C53C4
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC23×Dic53C23
hypercenterC2×#123#6157
hypocenterC53#8966
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. #20290605C2semidirect
4. #20289558C2semidirect
60. #123×C53C8semidirect
76. C22×C4×C53C2×C8semidirect
99. C22⋊C4×C53C2×C8semidirect
220. C53#8966semidirect
230. #123#88324semidirect
285. C22×C4C2×#88324semidirect
301. C22⋊C4C2×#88324semidirect

Maximal subgroups

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

subgroupindex
#202915622
#202906052
#202899512
#202895582
#123×Dic532
#896653