Finite Group Databasebeta

← #29572649

Subgroups — #29572649

Order 15616.

1154 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1154 conjugacy classes of subgroups — too many to draw. Only its 119 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×#14519763C211yesmaximal, characteristic
2#21445742C212yesmaximal, complemented
3#21444146C211yesmaximal, characteristic, complemented
4#21444145C212yesmaximal, complemented
5C61×#1462C211yesmaximal, characteristic, complemented
6C23×Dic122C2211yescharacteristic
7C23×C22⋊D61C411yescharacteristic
8C2×#14519763—24no—
9C2×#14519763—22no—
10C2×#14519763—22no—
11C2×#14519763—24no—
12C22×C22⋊Dic61C411yescharacteristic
13C22×#11758198C411yescharacteristic
14C22×#11758198C2211yescharacteristic
15C22×C4×Dic61C411yescharacteristic
16C2×#14519695—24no—
17#16839726C2212yes—
18#16839486—22no—
19#16839485—24no—
20C22×C22⋊C4×C61C2211yescharacteristic
21C61×#407C2212yes—
22C61×#323—22no—
23C24×D61C2×C411yescharacteristic
24C22×Dic122—21no—
25C22×Dic122—22no—
26C22×Dic122—21no—
27C22×Dic122—44no—
28C22×Dic122D414yes—
29C22×C22⋊D61—22no—
30C22×C22⋊D61—22no—
31C22×C22⋊D61—22no—
32C22×C22⋊D61—44no—
33C23×Dic61C2×C411yescharacteristic
34C23×Dic61C2×C411yescharacteristic
35C23×Dic61C2×C411yescharacteristic
36C2×#11758194—22no—
37#14519763—216no—
38#14519763—216no—
39C2×C22⋊Dic61—22no—
40C2×C22⋊Dic61—22no—
41C2×C22⋊Dic61—22no—
42C2×#11758198—22no—
43C2×#11758198—22no—
44C2×#11758198—21no—
45C2×#11758198—22no—
46C2×#11758198—21no—
47C2×#11758198—22no—
48C2×#11758198D414yes—
49C2×C4×Dic61—22no—
50C2×C4×Dic61—22no—
51C2×C4×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—21no—
66C24×C61—21no—
67C24×C61—42no—
68C24×C61—42no—
69C24×C61—42no—
70C24×C61C22⋊C412yes—
71C24×C61C22×C411yescharacteristic
72C23×D61—21no—
73C23×D61—21no—
74C23×D61—42no—
75C23×D61—42no—
76C23×D61—42no—
77C23×D61C22⋊C412yes—
78C22×C4×C61—21no—
79C22×C4×C61—21no—
80C22×C4×C61—42no—
81C22×C4×C61—42no—
82C22×C4×C61C22⋊C412yes—
83C22×C4×C61C2×D412yes—
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D412yes—
86C22×C4×C61C2×D411yescharacteristic
87C2×C22⋊D61—28no—
88C2×C22⋊D61—48no—
89C2×C22⋊D61—48no—
90C2×C22⋊D61—44no—
91C2×C22⋊D61—44no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—24no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C22×Dic61—21no—
98C22×Dic61—21no—
99C22×Dic61—42no—
100C22×Dic61—42no—
101C22×Dic61C22⋊C412yes—
102C22×Dic61C22⋊C412yes—
103C22×Dic61C22⋊C412yes—
104C2×Dic122—24no—
105C2×Dic122—24no—
106C2×Dic122—28no—
107C2×Dic122—44no—
108C2×Dic122—44no—
109C2×Dic122—48no—
110C2×Dic122—44no—
111M16×C61—44no—
112C2×C8×C61—42no—
113C22⋊C4×C61—24no—
114C22⋊C4×C61—24no—
115C22⋊Dic61—28no—
116#11758198—28no—
117#11758198—24no—
118#11758198—24no—
119C4×Dic61—28no—
120#11758194—44no—
121C2×#92418—42no—
122C23×C61—22no—
123C23×C61—22no—
124C23×C61—22no—
125C23×C61—22no—
126C23×C61—24no—
127C23×C61—42no—
128C23×C61—42no—
129C23×C61—42no—
130C23×C61—42no—
131C23×C61—42no—
132C23×C61—42no—
133C23×C61—42no—
134C23×C61—42no—
135C23×C61—88no—
136C23×C61C2×C22⋊C411yescharacteristic
137C23×C61C2×C22⋊C411yescharacteristic
138C23×C61C2×C22⋊C411yescharacteristic
139C22×D61—22no—
140C22×D61—24no—
141C22×D61—42no—
142C22×D61—42no—
143C22×D61—42no—
144C22×D61—42no—
145C22×D61—42no—
146C22×D61—42no—
147C22×D61—42no—
148C22×D61—42no—
149C22×D61—88no—
150C2×C4×C61—24no—
151C2×C4×C61—22no—
152C2×C4×C61—41no—
153C2×C4×C61—42no—
154C2×C4×C61—41no—
155C2×C4×C61—42no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61#12012yes—
159C2×C4×C61#12012yes—
160C22⋊D61—416no—
161C2×Dic61—24no—
162C2×Dic61—24no—
163C2×Dic61—22no—
164C2×Dic61—22no—
165C2×Dic61—22no—
166C2×Dic61—22no—
167C2×Dic61—24no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61—42no—
173C2×Dic61#9914yes—
174Dic122—48no—
175Dic122—48no—
176C8×C61—82no—
177#92418—82no—
178#8552—611nomaximal, Sylow
179C22×C61—28no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—21no—
185C22×C61—21no—
186C22×C61—42no—
187C22×C61—42no—
188C22×C61—42no—
189C22×C61—84no—
190C22×C61—84no—
191C22×C61—84no—
192C22×C61#37911yescharacteristic
193C22×C61C2×#9912yes—
194C22×C61#41312yes—
195C22×C61#41312yes—
196C2×D61—42no—
197C2×D61—42no—
198C2×D61—42no—
199C2×D61—84no—
200C2×D61—84no—
201C2×D61—84no—
202C4×C61—22no—
203C4×C61—22no—
204C4×C61—42no—
205Dic61—24no—
206Dic61—42no—
207Dic61—42no—
208C22×#124—611no—
209#1784—612no—
210#1462—612no—
211#1462—611no—
212#1462D6111yescharacteristic, complemented
213C2×C61—22no—
214C2×C61—22no—
215C2×C61—22no—
216C2×C61—82no—
217C2×C61#178111yescharacteristic
218C2×C61#178511yescharacteristic
219C2×C61#178511yescharacteristic
220D61—82no—
221C23×Q8—611no—
222C23×D4—611no—
223C2×#124—1224no—
224C2×#124—1224no—
225C2×#124—1222no—
226C2×#124—1222no—
227C22×C22⋊C4—611no—
228C22×C22⋊C4—611no—
229C22×C22⋊C4—611no—
230C22×C22⋊C4C2×D6111yescharacteristic
231C22×C42—611no—
232C2×#101—1224no—
233#407—612no—
234#407C2×D6112yes—
235#323—22no—
236#323—1224no—
237#323—1222no—
238C61#855211yescharacteristic, complemented, Sylow
239C25—611no—
240C25C4×D6111yescharacteristic
241C22×Q8—614no—
242C22×Q8—1221no—
243C22×Q8—1222no—
244C22×Q8—1221no—
245C22×Q8—2444no—
246C22×D4—1222no—
247C22×D4—1222no—
248C22×D4—1222no—
249C22×D4—2444no—
250C23×C4—611no—
251C23×C4—611no—
252C23×C4—611no—
253C23×C4C4×D6111yescharacteristic
254C23×C4C22×D6111yescharacteristic
255C2×M16—22no—
256C2×M16—1222no—
257#124—12216no—
258#124—12216no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—614no—
262C2×C22⋊C4—1222no—
263C2×C22⋊C4—1222no—
264C2×C22⋊C4—1222no—
265C2×C22⋊C4—1222no—
266C2×C22⋊C4—1222no—
267C2×C22⋊C4—1222no—
268C2×C22⋊C4—1221no—
269C2×C22⋊C4—1222no—
270C2×C22⋊C4—1221no—
271C2×C22⋊C4D24412yes—
272C2×C22⋊C4C22⋊D6112yes—
273C2×C42—1222no—
274C2×C42—1222no—
275C2×C42—1222no—
276#101—2448no—
277C22⋊C8—22no—
278C22⋊C8—22no—
279C22⋊C8—1222no—
280C22⋊C8—1222no—
281C24—21no—
282C24—21no—
283C24—42no—
284C24—42no—
285C24—42no—
286C24—612no—
287C24—1221no—
288C24—1221no—
289C24—2442no—
290C24—2442no—
291C24—2442no—
292C24#1175819812yes—
293C24C2×C4×D6111yescharacteristic
294C2×Q8—1228no—
295C2×Q8—1224no—
296C2×Q8—1224no—
297C2×Q8—2444no—
298C2×Q8—2444no—
299C2×Q8—2448no—
300C2×Q8—2444no—
301C2×D4—1228no—
302C2×D4—2444no—
303C2×D4—2448no—
304C2×D4—2448no—
305C2×D4—2444no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—612no—
311C22×C4—612no—
312C22×C4—612no—
313C22×C4—1224no—
314C22×C4—1221no—
315C22×C4—1221no—
316C22×C4—1221no—
317C22×C4—1221no—
318C22×C4—1221no—
319C22×C4—1221no—
320C22×C4—2442no—
321C22×C4—2442no—
322C22×C4#1175819812yes—
323C22×C4C2×D24411yescharacteristic
324C22×C4D4×D6112yes—
325C22×C4D4×D6112yes—
326C22×C4C2×C22⋊D6111yescharacteristic
327M16—44no—
328M16—2444no—
329C2×C8—42no—
330C2×C8—2442no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—1228no—
334C22⋊C4—1228no—
335C22⋊C4—1224no—
336C22⋊C4—1224no—
337C42—1228no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—24no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—88no—
352C23—1224no—
353C23—1222no—
354C23—4888no—
355C23—2442no—
356C23—2442no—
357C23—2442no—
358C23—2442no—
359C23—2442no—
360C23—2442no—
361C23—2442no—
362C23—2442no—
363C23C22⋊C4×D6111yescharacteristic
364C23C22⋊C4×D6111yescharacteristic
365C23C2×#1175819811yescharacteristic
366Q8—2448no—
367Q8—2448no—
368D4—24416no—
369C2×C4—24no—
370C2×C4—22no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—614no—
378C2×C4—1224no—
379C2×C4—1224no—
380C2×C4—1222no—
381C2×C4—1222no—
382C2×C4—1224no—
383C2×C4—1222no—
384C2×C4—1222no—
385C2×C4—2442no—
386C2×C4—2442no—
387C2×C4—2442no—
388C2×C4—2442no—
389C2×C4—2442no—
390C2×C4#1451974512yes—
391C2×C4#1451974712yes—
392C8—82no—
393C8—4882no—
394C22—28no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—84no—
405C22—84no—
406C22—84no—
407C22—4884no—
408C22—4884no—
409C22—4884no—
410C22—2442no—
411C22—2442no—
412C22—2442no—
413C22#1683961711yescharacteristic
414C22#99×D6112yes—
415C22#1683974312yes—
416C22#1683974412yes—
417C4—22no—
418C4—22no—
419C4—42no—
420C4—1224no—
421C4—2442no—
422C4—2442no—
423C2—22no—
424C2—22no—
425C2—22no—
426C2—82no—
427C2—4882no—
428C2#2144573211yescharacteristic
429C2#2144574411yescharacteristic
430C2#2144574511yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1462C2
socleC22×C61#379
prime radicalC23×C22⋊D61C4
cosocleC23×C4×C61C23
hypercenter#1462D61
hypocenterC61#8552
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. #21444146C2semidirect
4. #21444145C2semidirect
5. C61×#1462C2semidirect
212. #1462D61semidirect
238. C61#8552semidirect

Maximal subgroups

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

subgroupindex
C22×#145197632
#214457422
#214441462
#214441452
C61×#14622
#855261