Finite Group Databasebeta

← #29575744

Subgroups — #29575744

Order 15616.

1010 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 1010 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C2×C4×C22⋊Dic61C211yesmaximal, characteristic
2#21445729C211yesmaximal, characteristic
3C2×#16839499C212yesmaximal, complemented
4#21443697C212yesmaximal, complemented
5C61×#2848C211yesmaximal, characteristic
6C22×C22⋊Dic61—21no—
7C22×C4×Dic61—21no—
8C2×C22⋊#92418—22no—
9C2×C22⋊#92418C2212yes—
10C4×C22⋊Dic61—24no—
11C2×#14519684C2212yes—
12#16839499—24no—
13C24×C4×C61C2211yescharacteristic
14C2×C22⋊Q8×C61C412yescomplemented
15C2×C22⋊Q8×C61C2211yescharacteristic
16C61×#385—21no—
17C61×#385—22no—
18C61×#380C412yescomplemented
19C61×#380C2211yescharacteristic
20C61×#379—21no—
21C23×Dic61—22no—
22C22×#92418—22no—
23C2×C22⋊Dic61—28no—
24C2×C22⋊Dic61—24no—
25C2×C4×Dic61—24no—
26C2×C4×Dic61—44no—
27C22⋊#92418—22no—
28C22⋊#92418—22no—
29C22⋊#92418—44no—
30#14519684—22no—
31#14519684D414yes—
32C25×C61D411yescharacteristic
33C22×Q8×C61C2×C411yescharacteristic
34C23×C4×C61—22no—
35C23×C4×C61—22no—
36C23×C4×C61—44no—
37C23×C4×C61C2×C412yes—
38C23×C4×C61D411yescharacteristic
39C23×C4×C61D412yes—
40C23×C4×C61C2311yescharacteristic
41C22⋊Q8×C61—22no—
42C22⋊Q8×C61—22no—
43C22⋊Q8×C61—44no—
44C2×C4⋊C4×C61—22no—
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61C2×C411yescharacteristic
47C2×C4⋊C4×C61C2×C412yes—
48C2×C22⋊C4×C61—22no—
49C2×C22⋊C4×C61—22no—
50C2×C22⋊C4×C61—21no—
51C2×C22⋊C4×C61—21no—
52#95×C61—22no—
53#95×C61—21no—
54C24×C61—24no—
55C24×C61—21no—
56C24×C61—21no—
57C24×C61—22no—
58C24×C61—22no—
59C24×C61—42no—
60C24×C61C22⋊C412yes—
61C24×C61C2×D411yescharacteristic
62C2×Q8×C61—22no—
63C2×Q8×C61—42no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—24no—
67C22×C4×C61—24no—
68C22×C4×C61—24no—
69C22×C4×C61—24no—
70C22×C4×C61—24no—
71C22×C4×C61—21no—
72C22×C4×C61—22no—
73C22×C4×C61—21no—
74C22×C4×C61—42no—
75C22×C4×C61—42no—
76C22×C4×C61—42no—
77C22×C4×C61—48no—
78C22×C4×C61—48no—
79C22×C4×C61C22⋊C412yes—
80C22×C4×C61C22×C411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61C2×D412yes—
83C22×C4×C61#4112yes—
84C22×Dic61—28no—
85C22×Dic61—24no—
86C22×Dic61—44no—
87C22×Dic61—42no—
88C22×Dic61—42no—
89C4⋊C4×C61—22no—
90C4⋊C4×C61—42no—
91C4⋊C4×C61—44no—
92C4⋊C4×C61—42no—
93C22⋊C4×C61—44no—
94C22⋊C4×C61—44no—
95C22⋊C4×C61—41no—
96C22⋊C4×C61—41no—
97C22⋊C4×C61—42no—
98C22⋊Dic61—216no—
99C4×Dic61—48no—
100C2×#92418—44no—
101C2×#92418—42no—
102C23×C61—22no—
103C23×C61—22no—
104C23×C61—24no—
105C23×C61—24no—
106C23×C61—22no—
107C23×C61—22no—
108C23×C61—22no—
109C23×C61—22no—
110C23×C61—22no—
111C23×C61—22no—
112C23×C61—48no—
113C23×C61—48no—
114C23×C61—42no—
115C23×C61—42no—
116C23×C61—42no—
117C23×C61—42no—
118C23×C61#10414yes—
119C23×C61C2×C22⋊C411yescharacteristic
120C23×C61#12011yescharacteristic
121C23×C61#12311yescharacteristic
122C23×C61#13512yes—
123C23×C61#13712yes—
124Q8×C61—82no—
125C2×C4×C61—24no—
126C2×C4×C61—24no—
127C2×C4×C61—22no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61—42no—
132C2×C4×C61—42no—
133C2×C4×C61—48no—
134C2×C4×C61—48no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—42no—
138C2×C4×C61—42no—
139C2×C4×C61—42no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61#9914yes—
143C2×C4×C61C4×D412yes—
144C2×C4×C61#12112yes—
145C2×Dic61—28no—
146C2×Dic61—44no—
147C2×Dic61—48no—
148C2×Dic61—48no—
149C2×Dic61—44no—
150#92418—82no—
151#9146—611nomaximal, Sylow
152C22×C61—21no—
153C22×C61—21no—
154C22×C61—21no—
155C22×C61—21no—
156C22×C61—24no—
157C22×C61—24no—
158C22×C61—24no—
159C22×C61—24no—
160C22×C61—24no—
161C22×C61—48no—
162C22×C61—48no—
163C22×C61—42no—
164C22×C61—42no—
165C22×C61—42no—
166C22×C61—42no—
167C22×C61#38611yescharacteristic
168C22×C61C2×#9912yes—
169C22×C61#41611yescharacteristic
170C22×C61#41911yescharacteristic
171C22×C61C2×#10412yes—
172C22×C61#45212yes—
173C22×C61#48512yes—
174C4×C61—22no—
175C4×C61—24no—
176C4×C61—22no—
177C4×C61—81no—
178C4×C61—81no—
179C4×C61—82no—
180Dic61—48no—
181#2848D6111yescharacteristic
182C2×C4×C22⋊C4—611no—
183#1779—611no—
184C2×#328—612no—
185#1294—612no—
186C2×C61—22no—
187C2×C61—22no—
188C2×C61—44no—
189C2×C61#150112yes—
190C2×C61#150812yes—
191C2×C61#187311yescharacteristic
192C2×C61#189611yescharacteristic
193C2×C61#189811yescharacteristic
194C24×C4C2×D6111yescharacteristic
195C2×C22⋊Q8Dic6112yescomplemented
196C2×C22⋊Q8C2×D6111yescharacteristic
197C22×C22⋊C4—1221no—
198C22×C42—1221no—
199C2×C22⋊C8—612no—
200C2×C22⋊C8—1222no—
201#385—21no—
202#385—22no—
203#380Dic6112yescomplemented
204#380C2×D6111yescharacteristic
205#379—21no—
206C4×C22⋊C4—1224no—
207C2×#95—612no—
208#328—1224no—
209C61#914611yescharacteristic, complemented, Sylow
210C25C22⋊D6111yescharacteristic
211C22×Q8C2×Dic6111yescharacteristic
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—44no—
215C23×C4—1222no—
216C23×C4C2×Dic6112yes—
217C23×C4C22⋊D6111yescharacteristic
218C23×C4C22⋊D6112yes—
219C23×C4C22×D6111yescharacteristic
220C22×C8—1222no—
221C22⋊Q8—22no—
222C22⋊Q8—22no—
223C22⋊Q8—44no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4C2×Dic6111yescharacteristic
227C2×C4⋊C4C2×Dic6112yes—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—1224no—
233C2×C22⋊C4—1228no—
234C2×C42—1224no—
235C2×C42—2444no—
236C22⋊C8—1222no—
237C22⋊C8—1222no—
238C22⋊C8—2444no—
239#95—22no—
240#95—21no—
241#95—614no—
242#95—1222no—
243C24—24no—
244C24—21no—
245C24—21no—
246C24—22no—
247C24—22no—
248C24—42no—
249C24C22⋊Dic6112yes—
250C24C2×C22⋊D6111yescharacteristic
251C2×Q8—22no—
252C2×Q8—42no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—48no—
267C22×C4—48no—
268C22×C4—1228no—
269C22×C4—1224no—
270C22×C4—2444no—
271C22×C4—2442no—
272C22×C4—2442no—
273C22×C4C22⋊Dic6112yes—
274C22×C4D4×D6112yes—
275C22×C4#1175822412yes—
276C22×C4C22×Dic6111yescharacteristic
277C22×C4C2×C22⋊D6111yescharacteristic
278C2×C8—2444no—
279C2×C8—2442no—
280C4⋊C4—22no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—42no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—42no—
289C22⋊C4—12216no—
290C42—2448no—
291C23—22no—
292C23—22no—
293C23—24no—
294C23—24no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—48no—
302C23—48no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23#1451970414yes—
308C23C2×C22⋊Dic6111yescharacteristic
309C23#1451974511yescharacteristic
310C23#1451975811yescharacteristic
311C23#1451979712yes—
312C23#1451980812yes—
313Q8—82no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—48no—
323C2×C4—48no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—1228no—
332C2×C4—2444no—
333C2×C4—2448no—
334C2×C4—2448no—
335C2×C4—2444no—
336C2×C4#1451969214yes—
337C2×C4D4×Dic6112yes—
338C2×C4#1451974912yes—
339C8—4882no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—24no—
345C22—24no—
346C22—24no—
347C22—24no—
348C22—24no—
349C22—48no—
350C22—48no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22#1683964211yescharacteristic
356C22C2×#1451969212yes—
357C22#1683975811yescharacteristic
358C22#1683977111yescharacteristic
359C22C2×#1451970412yes—
360C22#1683992212yes—
361C22#1684010012yes—
362C4—22no—
363C4—24no—
364C4—22no—
365C4—81no—
366C4—81no—
367C4—82no—
368C4—2448no—
369C2—22no—
370C2—22no—
371C2—44no—
372C2#2144431012yes—
373C2#2144433912yes—
374C2#2144613211yescharacteristic
375C2#2144626311yescharacteristic
376C2#2144627311yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic61
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2848C2
socleC23×C61C2×C22⋊C4
prime radicalC25×C61D4
cosocleC23×C4×C61C23
hypercenter#2848D61
hypocenterC61#9146
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
3. C2×#16839499C2semidirect
4. #21443697C2semidirect
14. C2×C22⋊Q8×C61C4semidirect
18. C61×#380C4semidirect
195. C2×C22⋊Q8Dic61semidirect
203. #380Dic61semidirect
209. C61#9146semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Dic612
#214457292
C2×#168394992
#214436972
C61×#28482
#914661