Finite Group Databasebeta

← #29572663

Subgroups — #29572663

Order 15616.

842 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C22×#14519772C211yesmaximal, characteristic
2#21445899C211yesmaximal, characteristic, complemented
3#21444148C212yesmaximal, complemented
4#21444145C212yesmaximal
5C61×#1821C211yesmaximal, characteristic, complemented
6C2×#14519772—24no—
7C2×#14519772—24no—
8C2×#14519772—22no—
9C2×#14519772—22no—
10C22×C4⋊Dic61C411yescharacteristic
11C22×#11758196C2211yescharacteristic
12C22×#11758198C411yescharacteristic
13C22×#11758198C411yescharacteristic
14C22×#11758198C2211yescharacteristic
15C22×C4×Dic61C411yescharacteristic
16#16839726C2212yes—
17#16839531—22no—
18#16839487—24no—
19#16839485—24no—
20C22×C4⋊C4×C61C2211yescharacteristic
21C61×#407C2212yes—
22C61×#343—22no—
23C24×D61C2×C411yescharacteristic
24C23×Dic61C2×C411yescharacteristic
25C23×Dic61C2×C411yescharacteristic
26C23×Dic61C2×C411yescharacteristic
27C2×#11758194—22no—
28#14519772—216no—
29#14519772—216no—
30C2×C4⋊Dic61—22no—
31C2×C4⋊Dic61—22no—
32C2×C4⋊Dic61—22no—
33C2×#11758196—21no—
34C2×#11758196—22no—
35C2×#11758196—21no—
36C2×#11758196D414yes—
37C2×#11758198—22no—
38C2×#11758198—22no—
39C2×#11758198—22no—
40C2×#11758198—22no—
41C2×#11758198—21no—
42C2×#11758198—22no—
43C2×#11758198—21no—
44C2×#11758198—22no—
45C2×#11758198—22no—
46C2×#11758198D414yes—
47C2×C4×Dic61—22no—
48C2×C4×Dic61—22no—
49C2×C4×Dic61—22no—
50C22⋊#92418—22no—
51C22⋊#92418—22no—
52C23×C4×C61C2×C411yescharacteristic
53C23×C4×C61C2×C411yescharacteristic
54C23×C4×C61C2311yescharacteristic
55C2×M16×C61—22no—
56C2×C4⋊C4×C61—22no—
57C2×C4⋊C4×C61—22no—
58C2×C4⋊C4×C61D412yes—
59C2×C4⋊C4×C61D412yes—
60C22⋊C8×C61—22no—
61C22⋊C8×C61—22no—
62C24×C61C22×C411yescharacteristic
63C23×D61—21no—
64C23×D61—21no—
65C23×D61—42no—
66C23×D61—42no—
67C23×D61—42no—
68C23×D61C22⋊C412yes—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—21no—
73C22×C4×C61—42no—
74C22×C4×C61—42no—
75C22×C4×C61—42no—
76C22×C4×C61C22⋊C412yes—
77C22×C4×C61C22⋊C412yes—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C22×C4×C61C2×D412yes—
81C22×C4×C61C2×D412yes—
82C22×Dic61—21no—
83C22×Dic61—21no—
84C22×Dic61—21no—
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—42no—
89C22×Dic61—42no—
90C22×Dic61—42no—
91C22×Dic61C22⋊C412yes—
92C22×Dic61C22⋊C412yes—
93C22×Dic61C22⋊C412yes—
94M16×C61—44no—
95C2×C8×C61—42no—
96C4⋊C4×C61—24no—
97C4⋊C4×C61—24no—
98#11758198—28no—
99#11758198—28no—
100#11758198—24no—
101#11758198—24no—
102C4⋊Dic61—28no—
103#11758196—24no—
104#11758196—24no—
105C4×Dic61—28no—
106#11758194—44no—
107C2×#92418—42no—
108C23×C61—22no—
109C23×C61—22no—
110C23×C61—22no—
111C23×C61C2×C22⋊C411yescharacteristic
112C23×C61C2×C22⋊C411yescharacteristic
113C23×C61C2×C22⋊C411yescharacteristic
114C22×D61—22no—
115C22×D61—24no—
116C22×D61—42no—
117C22×D61—42no—
118C22×D61—42no—
119C22×D61—42no—
120C22×D61—42no—
121C22×D61—42no—
122C22×D61—42no—
123C22×D61—42no—
124C22×D61—88no—
125C2×C4×C61—22no—
126C2×C4×C61—22no—
127C2×C4×C61—22no—
128C2×C4×C61—22no—
129C2×C4×C61—42no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61—42no—
133C2×C4×C61—42no—
134C2×C4×C61—42no—
135C2×C4×C61—42no—
136C2×C4×C61—42no—
137C2×C4×C61#12012yes—
138C2×C4×C61#12012yes—
139C2×Dic61—22no—
140C2×Dic61—22no—
141C2×Dic61—22no—
142C2×Dic61—24no—
143C2×Dic61—42no—
144C2×Dic61—42no—
145C2×Dic61—42no—
146C2×Dic61—42no—
147C2×Dic61—42no—
148C2×Dic61—42no—
149C2×Dic61—42no—
150C2×Dic61—42no—
151C8×C61—82no—
152#92418—82no—
153#8556—611nomaximal, Sylow
154C22×C61—28no—
155C22×C61—21no—
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—21no—
161C22×C61#37911yescharacteristic
162C22×C61#41012yes—
163C22×C61#41312yes—
164C22×C61#41312yes—
165C2×D61—42no—
166C2×D61—42no—
167C2×D61—42no—
168C2×D61—84no—
169C2×D61—84no—
170C2×D61—84no—
171C4×C61—22no—
172C4×C61—22no—
173C4×C61—42no—
174C4×C61—42no—
175Dic61—42no—
176Dic61—42no—
177Dic61—42no—
178C22×#126—611no—
179#1821—611no—
180#1821D6111yescharacteristic, complemented
181#1463—612no—
182#1462—612no—
183C2×C61—22no—
184C2×C61—22no—
185C2×C61—22no—
186C2×C61#178211yescharacteristic
187C2×C61#178511yescharacteristic
188C2×C61#178511yescharacteristic
189D61—82no—
190C2×#126—1224no—
191C2×#126—1224no—
192C2×#126—1222no—
193C2×#126—1222no—
194C22×C4⋊C4—611no—
195C22×C4⋊C4—611no—
196C22×C4⋊C4C2×D6111yescharacteristic
197C22×C22⋊C4—611no—
198C22×C22⋊C4—611no—
199C22×C22⋊C4—611no—
200C22×C42—611no—
201#407—612no—
202#407C2×D6112yes—
203#343—22no—
204#343—1222no—
205#324—1224no—
206#323—1224no—
207C61#855611yescharacteristic, complemented, Sylow
208C25—611no—
209C23×C4—611no—
210C23×C4—611no—
211C23×C4—611no—
212C23×C4C4×D6111yescharacteristic
213C23×C4C4×D6111yescharacteristic
214C23×C4C22×D6111yescharacteristic
215C2×M16—22no—
216C2×M16—1222no—
217#126—12216no—
218#126—12216no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—614no—
222C2×C4⋊C4—1222no—
223C2×C4⋊C4—1222no—
224C2×C4⋊C4—1222no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4—1222no—
227C2×C4⋊C4—1221no—
228C2×C4⋊C4D24412yes—
229C2×C4⋊C4C22⋊D6112yes—
230C2×C22⋊C4—614no—
231C2×C22⋊C4—1222no—
232C2×C22⋊C4—1222no—
233C2×C22⋊C4—1222no—
234C2×C22⋊C4—1222no—
235C2×C22⋊C4—1222no—
236C2×C22⋊C4—1222no—
237C2×C22⋊C4—1221no—
238C2×C22⋊C4—1222no—
239C2×C22⋊C4—1221no—
240C2×C42—1222no—
241C2×C42—1222no—
242C2×C42—1222no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—1222no—
246C22⋊C8—1222no—
247C24—612no—
248C24—1221no—
249C24—1221no—
250C24—2442no—
251C24—2442no—
252C24—2442no—
253C24C2×C4×D6111yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—612no—
262C22×C4—612no—
263C22×C4—612no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4—1221no—
268C22×C4—1221no—
269C22×C4—1221no—
270C22×C4—2442no—
271C22×C4—2442no—
272C22×C4—2442no—
273C22×C4#1175819812yes—
274C22×C4#1175819812yes—
275C22×C4C2×D24411yescharacteristic
276C22×C4D4×D6112yes—
277C22×C4D4×D6112yes—
278C22×C4C2×C22⋊D6111yescharacteristic
279M16—44no—
280M16—2444no—
281C2×C8—42no—
282C2×C8—2442no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—1228no—
286C4⋊C4—1224no—
287C4⋊C4—1224no—
288C22⋊C4—1228no—
289C22⋊C4—1228no—
290C22⋊C4—1224no—
291C22⋊C4—1224no—
292C42—1228no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—1224no—
297C23—1222no—
298C23—4888no—
299C23—2442no—
300C23—2442no—
301C23—2442no—
302C23—2442no—
303C23—2442no—
304C23—2442no—
305C23—2442no—
306C23—2442no—
307C23C22⋊C4×D6111yescharacteristic
308C23C22⋊C4×D6111yescharacteristic
309C23C2×#1175819811yescharacteristic
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—1224no—
323C2×C4—1222no—
324C2×C4—1222no—
325C2×C4—1222no—
326C2×C4—2442no—
327C2×C4—2442no—
328C2×C4—2442no—
329C2×C4—2442no—
330C2×C4—2442no—
331C2×C4—2442no—
332C2×C4—2442no—
333C2×C4—2442no—
334C2×C4#1451974512yes—
335C2×C4#1451974712yes—
336C8—82no—
337C8—4882no—
338C22—28no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—4884no—
346C22—4884no—
347C22—4884no—
348C22—2442no—
349C22—2442no—
350C22—2442no—
351C22#1683961711yescharacteristic
352C22#1683973512yes—
353C22#1683974312yes—
354C22#1683974512yes—
355C4—22no—
356C4—22no—
357C4—42no—
358C4—42no—
359C4—2442no—
360C4—2442no—
361C4—2442no—
362C2—22no—
363C2—22no—
364C2—22no—
365C2—4882no—
366C2#2144573611yescharacteristic
367C2#2144574411yescharacteristic
368C2#2144574611yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1821C2
socleC22×C61#379
prime radicalC24×D61C2×C4
cosocleC23×C4×C61C23
hypercenter#1821D61
hypocenterC61#8556
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
2. #21445899C2semidirect
3. #21444148C2semidirect
5. C61×#1821C2semidirect
180. #1821D61semidirect
207. C61#8556semidirect

Maximal subgroups

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

subgroupindex
C22×#145197722
#214458992
#214441482
#214441452
C61×#18212
#855661