Finite Group Databasebeta

← #29575553

Subgroups — #29575553

Order 15616.

586 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448484C211yesmaximal, characteristic
2#21444137C211yesmaximal, characteristic
3C2×#16839487C212yesmaximal, complemented
4#21443665C212yesmaximal, complemented
5C61×#2972C211yesmaximal, characteristic
6C2×#14519758C411yescharacteristic
7C22×#11758196C2211yescharacteristic
8C2×C22⋊#92418C2212yes—
9#16839657—21no—
10#16839656—21no—
11#16839646—21no—
12#16839642C411yescharacteristic
13#16839635—21no—
14#16839634C411yescharacteristic
15#16839623C411yescharacteristic
16C4×C22⋊Dic61C2211yescharacteristic
17#16839488—21no—
18#16839487—24no—
19#16839486—21no—
20C2×C4×D4×C61C2211yescharacteristic
21C8×D4×C61—24no—
22C61×#431—21no—
23C2×C22⋊C8×C61—21no—
24C2×C22⋊C8×C61C2212yes—
25C23×Dic61C2×C411yescharacteristic
26C23×Dic61C2×C411yescharacteristic
27C22×#92418—22no—
28#14519758—22no—
29#14519758C814yescomplemented
30C2×#11758196—21no—
31C2×#11758196—22no—
32C2×#11758196—21no—
33C2×#11758196—21no—
34C2×#11758196—22no—
35C2×#11758196—21no—
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61C2×C411yescharacteristic
39C2×C22⋊Dic61C2×C411yescharacteristic
40C2×C4×Dic61—21no—
41C22⋊#92418—22no—
42C22⋊#92418—22no—
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48C22×D4×C61C2×C411yescharacteristic
49C23×C4×C61D411yescharacteristic
50C23×C4×C61C2311yescharacteristic
51C22×C8×C61—22no—
52C22×C8×C61—22no—
53C4×D4×C61—22no—
54C4×D4×C61—22no—
55C2×C4⋊C4×C61D411yescharacteristic
56C2×C22⋊C4×C61C2×C411yescharacteristic
57C2×C22⋊C4×C61D411yescharacteristic
58C2×C42×C61D411yescharacteristic
59C4⋊C8×C61—22no—
60C22⋊C8×C61—22no—
61C22⋊C8×C61—22no—
62C22⋊C8×C61D414yes—
63C4×C8×C61—22no—
64C24×C61C22⋊C411yescharacteristic
65C24×C61C22×C411yescharacteristic
66C2×D4×C61—22no—
67C2×D4×C61—22no—
68C2×D4×C61C2×C812yes—
69C2×D4×C61M1612yes—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—81no—
73C22×C4×C61—81no—
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22⋊C411yescharacteristic
77C22×C4×C61C2×D412yes—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61#4112yes—
81C22×Dic61—21no—
82C22×Dic61—21no—
83C22×Dic61—22no—
84C22×Dic61—21no—
85C22×Dic61—22no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—22no—
89C22×Dic61—81no—
90C22×Dic61C2×C812yescomplemented
91C2×C8×C61—24no—
92C2×C8×C61—82no—
93C2×C8×C61—82no—
94C4⋊C4×C61—22no—
95C22⋊C4×C61—22no—
96C22⋊C4×C61—22no—
97C42×C61—21no—
98C42×C61—21no—
99C22⋊Dic61—24no—
100C22⋊Dic61—21no—
101C22⋊Dic61—22no—
102C22⋊Dic61—81no—
103C22⋊Dic61C2×C812yescomplemented
104#11758196—24no—
105#11758196—41no—
106#11758196—42no—
107#11758196—41no—
108#11758196—81no—
109C4×Dic61—81no—
110C2×#92418—42no—
111C2×#92418—82no—
112C23×C61—21no—
113C23×C61—21no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61—41no—
118C23×C61C22⋊C812yes—
119C23×C61C2×C22⋊C411yescharacteristic
120C23×C61C4×D412yes—
121C23×C61C22×C811yescharacteristic
122C23×C61C2×M1611yescharacteristic
123C23×C61#13112yes—
124D4×C61—24no—
125D4×C61—24no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—42no—
131C2×C4×C61—41no—
132C2×C4×C61—42no—
133C2×C4×C61—41no—
134C2×C4×C61—42no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—81no—
138C2×C4×C61C22⋊C812yes—
139C2×C4×C61#12011yescharacteristic
140C2×C4×C61#12112yes—
141C2×C4×C61#12311yescharacteristic
142C2×Dic61—22no—
143C2×Dic61—22no—
144C2×Dic61—24no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—42no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—82no—
157C8×C61—44no—
158#92418—82no—
159#9108—611nomaximal, Sylow
160C22×C61—21no—
161C22×C61—22no—
162C22×C61—21no—
163C22×C61—22no—
164C22×C61—22no—
165C22×C61—41no—
166C22×C61—41no—
167C22×C61—42no—
168C22×C61—42no—
169C22×C61—41no—
170C22×C61—41no—
171C22×C61—81no—
172C22×C61#38611yescharacteristic
173C22×C61C2×C22⋊C811yescharacteristic
174C22×C61#40811yescharacteristic
175C22×C61C8×D412yes—
176C22×C61#43512yes—
177C4×C61—22no—
178C4×C61—22no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—41no—
182C4×C61—41no—
183Dic61—42no—
184Dic61—42no—
185Dic61—81no—
186Dic61—81no—
187#2972D6111yescharacteristic
188#2322—611no—
189#1459—611no—
190C2×#324—612no—
191#1282—612no—
192C2×C61—21no—
193C2×C61—21no—
194C2×C61—22no—
195C2×C61—22no—
196C2×C61—42no—
197C2×C61#186911yescharacteristic
198C2×C61#191311yescharacteristic
199C2×C61#191511yescharacteristic
200C2×#123—611no—
201C2×C4×D4C2×D6111yescharacteristic
202C22×C4⋊C4—611no—
203C8×D4—24no—
204#431—21no—
205C2×C22⋊C8—21no—
206C2×C22⋊C8—612no—
207C2×C22⋊C8C2×D6112yes—
208#389—1221no—
209#388—1221no—
210#387—1221no—
211#386—611no—
212#385—611no—
213#385—1221no—
214#381—611no—
215C4×C22⋊C4—611no—
216#324—1221no—
217#324—1224no—
218#323—1221no—
219C61#910811yescharacteristic, complemented, Sylow
220C22×D4C4×D6111yescharacteristic
221C23×C4—611no—
222C23×C4—611no—
223C23×C4C22⋊D6111yescharacteristic
224C23×C4C22×D6111yescharacteristic
225C22×C8—22no—
226C22×C8—22no—
227C22×C8—1222no—
228#123—614no—
229#123—1222no—
230C4×D4—22no—
231C4×D4—22no—
232C2×C4⋊C4—1221no—
233C2×C4⋊C4—1221no—
234C2×C4⋊C4—1221no—
235C2×C4⋊C4—1222no—
236C2×C4⋊C4—1222no—
237C2×C4⋊C4—1221no—
238C2×C4⋊C4D24411yescharacteristic
239C2×C22⋊C4—611no—
240C2×C22⋊C4—611no—
241C2×C22⋊C4—1221no—
242C2×C22⋊C4—1221no—
243C2×C22⋊C4C4×D6111yescharacteristic
244C2×C22⋊C4D24411yescharacteristic
245C2×C42—1221no—
246C2×C42C22⋊D6111yescharacteristic
247C4⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—1222no—
251C22⋊C8—1222no—
252C22⋊C8C22⋊D6114yes—
253C4×C8—22no—
254#95—1221no—
255#95—1221no—
256#95—1221no—
257#95—1221no—
258#95—1221no—
259C24#1175819811yescharacteristic
260C24C2×C4×D6111yescharacteristic
261C2×D4—22no—
262C2×D4—22no—
263C2×D4C8×D6112yes—
264C2×D4#1175818912yes—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—612no—
270C22×C4—1221no—
271C22×C4—1221no—
272C22×C4—1222no—
273C22×C4—1221no—
274C22×C4—1222no—
275C22×C4—1222no—
276C22×C4—1221no—
277C22×C4—1221no—
278C22×C4—4881no—
279C22×C4#1175819811yescharacteristic
280C22×C4#1175819811yescharacteristic
281C22×C4#1175819811yescharacteristic
282C22×C4C2×D24411yescharacteristic
283C22×C4#1175822212yes—
284C22×C4C2×C22⋊D6112yes—
285C22×C4C2×C22⋊D6111yescharacteristic
286C2×C8—24no—
287C2×C8—82no—
288C2×C8—82no—
289C2×C8—4882no—
290C2×C8—2442no—
291C4⋊C4—22no—
292C4⋊C4—1224no—
293C4⋊C4—4881no—
294C4⋊C4—2441no—
295C4⋊C4—2442no—
296C4⋊C4—2441no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—612no—
300C22⋊C4—1224no—
301C22⋊C4—1221no—
302C22⋊C4—1222no—
303C22⋊C4—4881no—
304C42—21no—
305C42—21no—
306C42—4881no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23#1451968912yes—
314C23C2×#1175819811yescharacteristic
315C23C4×C22⋊D6112yes—
316C23C2×C8×D6111yescharacteristic
317C23C2×#1175818911yescharacteristic
318C23#1451978312yes—
319D4—24no—
320D4—24no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—81no—
333C2×C4—1222no—
334C2×C4—1222no—
335C2×C4—1224no—
336C2×C4—4882no—
337C2×C4—2441no—
338C2×C4—2441no—
339C2×C4—2441no—
340C2×C4—2441no—
341C2×C4—2441no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2442no—
347C2×C4—2441no—
348C2×C4#1451968912yes—
349C2×C4#1451974611yescharacteristic
350C2×C4#1451975112yes—
351C2×C4#1451976011yescharacteristic
352C8—44no—
353C8—4882no—
354C22—21no—
355C22—22no—
356C22—21no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22#1683964111yescharacteristic
367C22C2×#1451968911yescharacteristic
368C22#1683972811yescharacteristic
369C22C8×C22⋊D6112yes—
370C22#1683983312yes—
371C4—21no—
372C4—22no—
373C4—21no—
374C4—22no—
375C4—41no—
376C4—41no—
377C4—4881no—
378C4—4881no—
379C4—2442no—
380C4—2442no—
381C2—21no—
382C2—21no—
383C2—22no—
384C2—22no—
385C2—42no—
386C2#2144611111yescharacteristic
387C2#2144636111yescharacteristic
388C2#2144637111yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC23×C61C22×C8
FrattiniC23×C4C22×D61
FittingC61×#2972C2
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC23×C4×C61C23
hypercenter#2972D61
hypocenterC61#9108
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
3. C2×#16839487C2semidirect
4. #21443665C2semidirect
29. #14519758C8semidirect
90. C22×Dic61C2×C8semidirect
103. C22⋊Dic61C2×C8semidirect
219. C61#9108semidirect

Maximal subgroups

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

subgroupindex
#214484842
#214441372
C2×#168394872
#214436652
C61×#29722
#910861