Finite Group Databasebeta

← #29572642

Subgroups — #29572642

Order 15616.

1134 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 127 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×#14519754C211yesmaximal, characteristic
2#21445742C212yesmaximal, complemented
3#21444148C212yesmaximal, complemented
4#21444146C211yesmaximal, characteristic
5C61×#1462C211yesmaximal, characteristic
6C23×Dic122C2211yescharacteristic
7C24×Dic61C411yescharacteristic
8C2×#14519754—24no—
9C2×#14519754—22no—
10C2×#14519754—22no—
11C2×#14519754—24no—
12C22×C4⋊Dic61C411yescharacteristic
13C22×#11758196C411yescharacteristic
14C22×#11758196C2211yescharacteristic
15C22×C22⋊Dic61C411yescharacteristic
16C2×#14519695—24no—
17#16839726C2212yes—
18#16839487—24no—
19#16839486—22no—
20C22×C22⋊C4×C61C2211yescharacteristic
21C61×#407C2212yes—
22C61×#323—22no—
23C22×Dic122—21no—
24C22×Dic122—22no—
25C22×Dic122—21no—
26C22×Dic122—44no—
27C22×Dic122D414yes—
28C23×Dic61—22no—
29C23×Dic61—22no—
30C23×Dic61—22no—
31C23×Dic61—44no—
32C23×Dic61C2×C411yescharacteristic
33C23×Dic61C2×C411yescharacteristic
34C23×Dic61C2×C411yescharacteristic
35C23×Dic61C2×C411yescharacteristic
36C2×#11758194—22no—
37#14519754—216no—
38#14519754—216no—
39C2×C4⋊Dic61—22no—
40C2×C4⋊Dic61—22no—
41C2×C4⋊Dic61—22no—
42C2×#11758196—22no—
43C2×#11758196—21no—
44C2×#11758196—22no—
45C2×#11758196—21no—
46C2×#11758196—22no—
47C2×#11758196—22no—
48C2×#11758196D414yes—
49C2×C22⋊Dic61—22no—
50C2×C22⋊Dic61—22no—
51C2×C22⋊Dic61—22no—
52#14519695—48no—
53C22⋊#92418—22no—
54C22⋊#92418—22no—
55C25×C61C2×C411yescharacteristic
56C23×C4×C61C2×C411yescharacteristic
57C23×C4×C61C2311yescharacteristic
58C2×M16×C61—22no—
59C2×C22⋊C4×C61—22no—
60C2×C22⋊C4×C61—22no—
61C2×C22⋊C4×C61D412yes—
62C2×C22⋊C4×C61D412yes—
63C22⋊C8×C61—22no—
64C22⋊C8×C61—22no—
65C24×C61—24no—
66C24×C61—21no—
67C24×C61—21no—
68C24×C61—42no—
69C24×C61—42no—
70C24×C61C22⋊C412yes—
71C24×C61C22×C411yescharacteristic
72C22×C4×C61—21no—
73C22×C4×C61—21no—
74C22×C4×C61—42no—
75C22×C4×C61—42no—
76C22×C4×C61C22⋊C412yes—
77C22×C4×C61C2×D412yes—
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D412yes—
80C22×C4×C61C2×D411yescharacteristic
81C22×Dic61—21no—
82C22×Dic61—21no—
83C22×Dic61—24no—
84C22×Dic61—28no—
85C22×Dic61—21no—
86C22×Dic61—21no—
87C22×Dic61—21no—
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—42no—
92C22×Dic61—42no—
93C22×Dic61—48no—
94C22×Dic61—48no—
95C22×Dic61—44no—
96C22×Dic61—44no—
97C22×Dic61—42no—
98C22×Dic61C22⋊C412yes—
99C22×Dic61C22⋊C412yes—
100C22×Dic61C22⋊C412yes—
101C22×Dic61C22⋊C412yes—
102C2×Dic122—24no—
103C2×Dic122—24no—
104C2×Dic122—28no—
105C2×Dic122—44no—
106C2×Dic122—44no—
107C2×Dic122—48no—
108C2×Dic122—44no—
109M16×C61—44no—
110C2×C8×C61—42no—
111C22⋊C4×C61—24no—
112C22⋊C4×C61—24no—
113C22⋊Dic61—28no—
114C4⋊Dic61—28no—
115#11758196—28no—
116#11758196—24no—
117#11758196—24no—
118#11758194—44no—
119C2×#92418—42no—
120C23×C61—22no—
121C23×C61—22no—
122C23×C61—22no—
123C23×C61—24no—
124C23×C61—24no—
125C23×C61—22no—
126C23×C61—22no—
127C23×C61—42no—
128C23×C61—416no—
129C23×C61—42no—
130C23×C61—42no—
131C23×C61—42no—
132C23×C61—42no—
133C23×C61—42no—
134C23×C61#10114yescomplemented
135C23×C61C2×C22⋊C411yescharacteristic
136C23×C61C2×C22⋊C411yescharacteristic
137C23×C61C2×C22⋊C411yescharacteristic
138C2×C4×C61—24no—
139C2×C4×C61—22no—
140C2×C4×C61—41no—
141C2×C4×C61—42no—
142C2×C4×C61—41no—
143C2×C4×C61—42no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61#12012yes—
147C2×C4×C61#12012yes—
148C2×Dic61—24no—
149C2×Dic61—24no—
150C2×Dic61—22no—
151C2×Dic61—22no—
152C2×Dic61—22no—
153C2×Dic61—22no—
154C2×Dic61—22no—
155C2×Dic61—24no—
156C2×Dic61—24no—
157C2×Dic61—416no—
158C2×Dic61—42no—
159C2×Dic61—42no—
160C2×Dic61—42no—
161C2×Dic61—42no—
162C2×Dic61—42no—
163C2×Dic61—42no—
164C2×Dic61—42no—
165C2×Dic61#9914yes—
166Dic122—48no—
167Dic122—48no—
168C8×C61—82no—
169#92418—82no—
170#8550—611nomaximal, Sylow
171C22×C61—28no—
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—21no—
175C22×C61—21no—
176C22×C61—21no—
177C22×C61—21no—
178C22×C61—24no—
179C22×C61—44no—
180C22×C61—48no—
181C22×C61—44no—
182C22×C61—48no—
183C22×C61—42no—
184C22×C61—42no—
185C22×C61#37911yescharacteristic
186C22×C61C2×#9912yes—
187C22×C61C2×#10112yes—
188C22×C61#41312yes—
189C4×C61—22no—
190C4×C61—22no—
191C4×C61—42no—
192Dic61—24no—
193Dic61—42no—
194Dic61—42no—
195Dic61—42no—
196C22×C22⋊Q8—611no—
197#1784—612no—
198#1463—612no—
199#1462—611no—
200#1462D6111yescharacteristic
201C2×C61—22no—
202C2×C61—22no—
203C2×C61—22no—
204C2×C61—44no—
205C2×C61#178111yescharacteristic
206C2×C61#178411yescharacteristic
207C2×C61#178511yescharacteristic
208C23×Q8—611no—
209C24×C4—611no—
210C2×C22⋊Q8—1224no—
211C2×C22⋊Q8—1224no—
212C2×C22⋊Q8—1222no—
213C2×C22⋊Q8—1222no—
214C22×C4⋊C4—611no—
215C22×C4⋊C4—611no—
216C22×C4⋊C4—611no—
217C22×C22⋊C4—611no—
218C22×C22⋊C4C2×D6111yescharacteristic
219C2×#101—1224no—
220#407—612no—
221#407C2×D6112yes—
222#324—1224no—
223#323—22no—
224#323—1222no—
225C61#855011yescharacteristic, complemented, Sylow
226C25C4×D6111yescharacteristic
227C22×Q8—614no—
228C22×Q8—1221no—
229C22×Q8—1222no—
230C22×Q8—1221no—
231C22×Q8—2444no—
232C23×C4—611no—
233C23×C4—611no—
234C23×C4—611no—
235C23×C4—611no—
236C23×C4—1222no—
237C23×C4—1222no—
238C23×C4—1222no—
239C23×C4—2444no—
240C23×C4C4×D6111yescharacteristic
241C23×C4C22×D6111yescharacteristic
242C2×M16—22no—
243C2×M16—1222no—
244C22⋊Q8—12216no—
245C22⋊Q8—12216no—
246C2×C4⋊C4—614no—
247C2×C4⋊C4—1222no—
248C2×C4⋊C4—1222no—
249C2×C4⋊C4—1222no—
250C2×C4⋊C4—1222no—
251C2×C4⋊C4—1222no—
252C2×C4⋊C4—1222no—
253C2×C4⋊C4—1221no—
254C2×C4⋊C4—1222no—
255C2×C4⋊C4—1221no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—1222no—
259C2×C22⋊C4—1222no—
260C2×C22⋊C4—1222no—
261C2×C22⋊C4D24412yes—
262C2×C22⋊C4C22⋊D6112yes—
263#101—2448no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—1222no—
267C22⋊C8—1222no—
268C24—24no—
269C24—21no—
270C24—21no—
271C24—42no—
272C24—42no—
273C24#1175819812yes—
274C24C2×C4×D6111yescharacteristic
275C2×Q8—1228no—
276C2×Q8—1224no—
277C2×Q8—1224no—
278C2×Q8—2444no—
279C2×Q8—2444no—
280C2×Q8—2448no—
281C2×Q8—2444no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—612no—
287C22×C4—612no—
288C22×C4—612no—
289C22×C4—612no—
290C22×C4—1224no—
291C22×C4—1221no—
292C22×C4—1221no—
293C22×C4—1228no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4—1221no—
297C22×C4—1221no—
298C22×C4—1221no—
299C22×C4—1221no—
300C22×C4—2444no—
301C22×C4—2442no—
302C22×C4—2442no—
303C22×C4—2448no—
304C22×C4—2448no—
305C22×C4—2444no—
306C22×C4—2442no—
307C22×C4#1175819812yes—
308C22×C4C2×D24411yescharacteristic
309C22×C4D4×D6112yes—
310C22×C4D4×D6112yes—
311C22×C4C2×C22⋊D6111yescharacteristic
312M16—44no—
313M16—2444no—
314C2×C8—42no—
315C2×C8—2442no—
316C4⋊C4—1228no—
317C4⋊C4—1228no—
318C4⋊C4—1224no—
319C4⋊C4—1224no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—1228no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—24no—
327C23—24no—
328C23—22no—
329C23—22no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—416no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23#1451969514yescomplemented
338C23C22⋊C4×D6111yescharacteristic
339C23C22⋊C4×D6111yescharacteristic
340C23C2×#1175819811yescharacteristic
341Q8—2448no—
342Q8—2448no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—614no—
352C2×C4—1224no—
353C2×C4—1224no—
354C2×C4—1222no—
355C2×C4—1222no—
356C2×C4—1224no—
357C2×C4—1224no—
358C2×C4—1222no—
359C2×C4—1222no—
360C2×C4—1222no—
361C2×C4—24416no—
362C2×C4—2442no—
363C2×C4—2442no—
364C2×C4—2442no—
365C2×C4—2442no—
366C2×C4—2442no—
367C2×C4—2442no—
368C2×C4—2442no—
369C2×C4#1451974512yes—
370C2×C4#1451974712yes—
371C8—82no—
372C8—4882no—
373C22—28no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—24no—
381C22—44no—
382C22—48no—
383C22—44no—
384C22—48no—
385C22—42no—
386C22—42no—
387C22#1683961711yescharacteristic
388C22#99×D6112yes—
389C22C2×#1451969512yes—
390C22#1683974312yes—
391C4—22no—
392C4—22no—
393C4—42no—
394C4—1224no—
395C4—2442no—
396C4—2442no—
397C4—2442no—
398C2—22no—
399C2—22no—
400C2—22no—
401C2—44no—
402C2#2144573211yescharacteristic
403C2#2144574211yescharacteristic
404C2#2144574411yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1462C2
socleC22×C61#379
prime radicalC25×C61C2×C4
cosocleC23×C4×C61C23
hypercenter#1462D61
hypocenterC61#8550
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1462

Complements

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

NH ≅ G/Ntype
2. #21445742C2semidirect
3. #21444148C2semidirect
134. C23×C61#101semidirect
225. C61#8550semidirect
337. C23#14519695semidirect

Maximal subgroups

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

subgroupindex
C22×#145197542
#214457422
#214441482
#214441462
C61×#14622
#855061