Finite Group Databasebeta

← #29580192

Subgroups — #29580192

Order 15616.

640 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21445799C211yesmaximal, characteristic
2C4×#14519684C211yesmaximal, characteristic
3#21443697C212yesmaximal
4#21443697C212yesmaximal
5C61×#2851C211yesmaximal, characteristic
6C22×C4×Dic61—21no—
7C2×C4⋊#92418—21no—
8C2×C22⋊#92418C2212yes—
9C2×#14519684—21no—
10C2×#14519684C2212yes—
11#16839517—21no—
12C2×#125×C61C411yescharacteristic, complemented
13C22×C42×C61C2211yescharacteristic
14C61×#389C411yescharacteristic, complemented
15C61×#389C411yescharacteristic, complemented
16C61×#387—21no—
17C61×#387—21no—
18C61×#385—21no—
19C61×#385—21no—
20C61×#382C411yescharacteristic, complemented
21C61×#380C2211yescharacteristic
22C61×#380C2211yescharacteristic
23C23×Dic61—22no—
24C22×#92418—22no—
25C2×C4×Dic61—24no—
26C2×C4×Dic61—42no—
27C2×C4×Dic61—42no—
28C4⋊#92418—42no—
29C22⋊#92418—24no—
30#14519684—24no—
31#14519684—22no—
32#14519684Q814yes—
33C23×C4×C61D411yescharacteristic
34C23×C4×C61D411yescharacteristic
35C23×C4×C61C2311yescharacteristic
36#125×C61—42no—
37C2×C4⋊C4×C61—21no—
38C2×C4⋊C4×C61—21no—
39C2×C4⋊C4×C61—21no—
40C2×C4⋊C4×C61—21no—
41C2×C4⋊C4×C61C2×C411yescharacteristic
42C2×C4⋊C4×C61C2×C411yescharacteristic
43C2×C4⋊C4×C61C2×C411yescharacteristic
44C2×C4⋊C4×C61C2×C411yescharacteristic
45C2×C22⋊C4×C61—21no—
46C2×C22⋊C4×C61—21no—
47C2×C42×C61—42no—
48C2×C42×C61C2×C411yescharacteristic
49C2×C42×C61C2×C411yescharacteristic
50C2×C42×C61D411yescharacteristic
51C2×C42×C61D411yescharacteristic
52#95×C61—21no—
53#95×C61—21no—
54#95×C61—21no—
55#95×C61—21no—
56#95×C61—21no—
57#95×C61—21no—
58C24×C61C2×D411yescharacteristic
59C22×C4×C61—21no—
60C22×C4×C61—21no—
61C22×C4×C61—22no—
62C22×C4×C61—22no—
63C22×C4×C61—22no—
64C22×C4×C61—22no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—41no—
70C22×C4×C61—41no—
71C22×C4×C61—41no—
72C22×C4×C61C22⋊C411yescharacteristic
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22×C411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×Q812yes—
79C22×C4×C61#4112yes—
80C22×Dic61—24no—
81C22×Dic61—24no—
82C22×Dic61—24no—
83C22×Dic61—42no—
84C22×Dic61—42no—
85C22×Dic61—42no—
86C22×Dic61—42no—
87C4⋊C4×C61—42no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—41no—
91C4⋊C4×C61—41no—
92C4⋊C4×C61—42no—
93C4⋊C4×C61—42no—
94C4⋊C4×C61—42no—
95C22⋊C4×C61—42no—
96C22⋊C4×C61—41no—
97C22⋊C4×C61—41no—
98C42×C61—22no—
99C42×C61—22no—
100C42×C61—41no—
101C42×C61—41no—
102C4×Dic61—48no—
103C2×#92418—42no—
104C2×#92418—42no—
105C2×#92418—42no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—21no—
110C23×C61C2×C22⋊C411yescharacteristic
111C23×C61C22⋊Q812yes—
112C23×C61#13512yes—
113C23×C61AGL(1,ℤ/8)11yescharacteristic
114C23×C61#13711yescharacteristic
115C2×C4×C61—24no—
116C2×C4×C61—24no—
117C2×C4×C61—21no—
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—21no—
128C2×C4×C61—21no—
129C2×C4×C61—21no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61—44no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—41no—
145C2×C4×C61#9912yes—
146C2×C4×C61#10412yes—
147C2×C4×C61#10412yes—
148C2×C4×C61C4×Q812yes—
149C2×C4×C61#12411yescharacteristic
150C2×C4×C61C4⋊Q811yescharacteristic
151C2×Dic61—28no—
152C2×Dic61—44no—
153C2×Dic61—44no—
154C2×Dic61—44no—
155C2×Dic61—44no—
156C2×Dic61—44no—
157C2×Dic61—44no—
158#92418—82no—
159#9910—611nomaximal, Sylow
160C22×C61—22no—
161C22×C61—22no—
162C22×C61—22no—
163C22×C61—22no—
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—21no—
167C22×C61—21no—
168C22×C61#39111yescharacteristic
169C22×C61C2×#9911yescharacteristic
170C22×C61#41011yescharacteristic
171C22×C61#41611yescharacteristic
172C22×C61#41711yescharacteristic
173C22×C61C2×#10411yescharacteristic
174C22×C61C2×#10411yescharacteristic
175C22×C61#47812yes—
176C22×C61#47912yes—
177C4×C61—21no—
178C4×C61—22no—
179C4×C61—21no—
180C4×C61—22no—
181C4×C61—22no—
182C4×C61—22no—
183C4×C61—41no—
184C4×C61—81no—
185C4×C61—81no—
186C4×C61—81no—
187C4×C61—81no—
188Dic61—48no—
189#2851D6111yescharacteristic
190#1797—611no—
191C4×#95—611no—
192#1294—612no—
193#1294—612no—
194C2×C61—22no—
195C2×C61—22no—
196C2×C61#150011yescharacteristic
197C2×C61#150111yescharacteristic
198C2×C61#150411yescharacteristic
199C2×C61#150911yescharacteristic
200C2×C61#198911yescharacteristic
201C2×C61#199411yescharacteristic
202C2×C61#199611yescharacteristic
203C2×#125Dic6111yescharacteristic, complemented
204C22×C42—1221no—
205C22×C42C2×D6111yescharacteristic
206C2×C4⋊C8—1221no—
207C2×C22⋊C8—612no—
208#389Dic6111yescharacteristic, complemented
209#389Dic6111yescharacteristic, complemented
210#387—21no—
211#387—21no—
212#385—21no—
213#385—21no—
214#382Dic6111yescharacteristic, complemented
215#380C2×D6111yescharacteristic
216#380C2×D6111yescharacteristic
217C2×#95—612no—
218C2×#95—1221no—
219#336—1221no—
220C61#991011yescharacteristic, complemented, Sylow
221C23×C4—1222no—
222C23×C4C22⋊D6111yescharacteristic
223C23×C4C22⋊D6111yescharacteristic
224C23×C4C22×D6111yescharacteristic
225C22×C8—1222no—
226#125—42no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4C2×Dic6111yescharacteristic
232C2×C4⋊C4C2×Dic6111yescharacteristic
233C2×C4⋊C4C2×Dic6111yescharacteristic
234C2×C4⋊C4C2×Dic6111yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C42—42no—
238C2×C42—1224no—
239C2×C42—2442no—
240C2×C42—2442no—
241C2×C42C2×Dic6111yescharacteristic
242C2×C42C2×Dic6111yescharacteristic
243C2×C42C22⋊D6111yescharacteristic
244C2×C42C22⋊D6111yescharacteristic
245C4⋊C8—2442no—
246C22⋊C8—1224no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—614no—
254#95—1224no—
255#95—1222no—
256C24C2×C22⋊D6111yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—1224no—
271C22×C4—1224no—
272C22×C4—1224no—
273C22×C4—2442no—
274C22×C4—2442no—
275C22×C4—2442no—
276C22×C4—2442no—
277C22×C4C22⋊Dic6111yescharacteristic
278C22×C4C22⋊Dic6111yescharacteristic
279C22×C4C22⋊Dic6111yescharacteristic
280C22×C4C22⋊Dic6111yescharacteristic
281C22×C4Q8×D6112yes—
282C22×C4#1175822612yes—
283C22×C4C22×Dic6111yescharacteristic
284C22×C4C2×C22⋊D6111yescharacteristic
285C2×C8—2442no—
286C2×C8—2442no—
287C2×C8—2442no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C42—22no—
300C42—22no—
301C42—41no—
302C42—41no—
303C42—2448no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23C2×C22⋊Dic6111yescharacteristic
309C23#1451975312yes—
310C23#1451979712yes—
311C23#1451980211yescharacteristic
312C23#1451980711yescharacteristic
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—1228no—
344C2×C4—2444no—
345C2×C4—2444no—
346C2×C4—2444no—
347C2×C4—2444no—
348C2×C4—2444no—
349C2×C4—2444no—
350C2×C4#1451969212yes—
351C2×C4#1451970412yes—
352C2×C4#1451970412yes—
353C2×C4Q8×Dic6112yes—
354C2×C4#1451976611yescharacteristic
355C2×C4#1451977511yescharacteristic
356C8—4882no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22#1683966911yescharacteristic
366C22C2×#1451969211yescharacteristic
367C22#1683973611yescharacteristic
368C22#1683975811yescharacteristic
369C22#1683976311yescharacteristic
370C22C2×#1451970411yescharacteristic
371C22C2×#1451970411yescharacteristic
372C22#1684005912yes—
373C22#1684006612yes—
374C4—21no—
375C4—22no—
376C4—21no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—2448no—
386C2—22no—
387C2—22no—
388C2#2144430511yescharacteristic
389C2#2144431011yescharacteristic
390C2#2144432311yescharacteristic
391C2#2144434411yescharacteristic
392C2#2144677711yescharacteristic
393C2#2144679711yescharacteristic
394C2#2144680511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic61
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2851C2
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#2851D61
hypocenterC61#9910
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
12. C2×#125×C61C4semidirect
14. C61×#389C4semidirect
15. C61×#389C4semidirect
20. C61×#382C4semidirect
203. C2×#125Dic61semidirect
208. #389Dic61semidirect
209. #389Dic61semidirect
214. #382Dic61semidirect
220. C61#9910semidirect

Maximal subgroups

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

subgroupindex
#214457992
C4×#145196842
#214436972
C61×#28512
#991061