Finite Group Databasebeta

← #29572653

Subgroups — #29572653

Order 15616.

1166 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 1166 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×#14519759C211yesmaximal, characteristic
2#21445897C212yesmaximal, complemented
3#21444146C211yesmaximal, characteristic, complemented
4#21444145C212yesmaximal, complemented
5C61×#1462C211yesmaximal, characteristic, complemented
6C23×C22⋊D61C411yescharacteristic
7C23×C4×D61C411yescharacteristic
8C2×#14519759—24no—
9C2×#14519759—24no—
10C2×#14519759—22no—
11C2×#14519759—22no—
12C22×C4⋊Dic61C2211yescharacteristic
13C22×#11758196C411yescharacteristic
14C22×C22⋊Dic61C411yescharacteristic
15C22×#11758198C2211yescharacteristic
16#16839726C2212yes—
17#16839530—24no—
18#16839486—22no—
19#16839485—24no—
20C22×C22⋊C4×C61C2211yescharacteristic
21C61×#407C2212yes—
22C61×#323—22no—
23C24×D61C2×C411yescharacteristic
24C22×C22⋊D61—22no—
25C22×C22⋊D61—22no—
26C22×C22⋊D61—22no—
27C22×C22⋊D61—44no—
28C23×Dic61C2×C411yescharacteristic
29C23×Dic61C2×C411yescharacteristic
30C23×Dic61C2×C411yescharacteristic
31C22×C4×D61—22no—
32C22×C4×D61—22no—
33C22×C4×D61—22no—
34C22×C4×D61—44no—
35C2×#11758194—22no—
36#14519759—216no—
37#14519759—216no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61—22no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61D414yes—
42C2×#11758196—22no—
43C2×#11758196—22no—
44C2×#11758196—22no—
45C2×C22⋊Dic61—22no—
46C2×C22⋊Dic61—22no—
47C2×C22⋊Dic61—22no—
48C2×#11758198—21no—
49C2×#11758198—22no—
50C2×#11758198—21no—
51C2×#11758198D414yes—
52C22⋊#92418—22no—
53C22⋊#92418—22no—
54C25×C61C2×C411yescharacteristic
55C23×C4×C61C2×C411yescharacteristic
56C23×C4×C61C2311yescharacteristic
57C2×M16×C61—22no—
58C2×C22⋊C4×C61—22no—
59C2×C22⋊C4×C61—22no—
60C2×C22⋊C4×C61D412yes—
61C2×C22⋊C4×C61D412yes—
62C22⋊C8×C61—22no—
63C22⋊C8×C61—22no—
64C24×C61—21no—
65C24×C61—21no—
66C24×C61—42no—
67C24×C61—42no—
68C24×C61—42no—
69C24×C61C22⋊C412yes—
70C24×C61C22×C411yescharacteristic
71C23×D61—21no—
72C23×D61—21no—
73C23×D61—24no—
74C23×D61—42no—
75C23×D61—42no—
76C23×D61C22⋊C412yes—
77C22×C4×C61—21no—
78C22×C4×C61—21no—
79C22×C4×C61—42no—
80C22×C4×C61—42no—
81C22×C4×C61C22⋊C412yes—
82C22×C4×C61C2×D412yes—
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D412yes—
85C22×C4×C61C2×D411yescharacteristic
86C2×C22⋊D61—28no—
87C2×C22⋊D61—48no—
88C2×C22⋊D61—44no—
89C2×C22⋊D61—48no—
90C2×C22⋊D61—44no—
91C22×Dic61—21no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—21no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C22×Dic61—42no—
98C22×Dic61—42no—
99C22×Dic61—42no—
100C22×Dic61C22⋊C412yes—
101C22×Dic61C22⋊C412yes—
102C22×Dic61C22⋊C412yes—
103C2×C4×D61—28no—
104C2×C4×D61—48no—
105C2×C4×D61—44no—
106C2×C4×D61—48no—
107C2×C4×D61—44no—
108M16×C61—44no—
109C2×C8×C61—42no—
110C22⋊C4×C61—24no—
111C22⋊C4×C61—24no—
112C22⋊Dic61—28no—
113#11758198—24no—
114#11758198—24no—
115C4⋊Dic61—24no—
116C4⋊Dic61—24no—
117#11758196—28no—
118#11758194—44no—
119C2×#92418—42no—
120C23×C61—22no—
121C23×C61—22no—
122C23×C61—22no—
123C23×C61—22no—
124C23×C61—24no—
125C23×C61—42no—
126C23×C61—42no—
127C23×C61—42no—
128C23×C61—42no—
129C23×C61—42no—
130C23×C61—42no—
131C23×C61—42no—
132C23×C61—42no—
133C23×C61—88no—
134C23×C61C2×C22⋊C411yescharacteristic
135C23×C61C2×C22⋊C411yescharacteristic
136C23×C61C2×C22⋊C411yescharacteristic
137C22×D61—24no—
138C22×D61—24no—
139C22×D61—22no—
140C22×D61—22no—
141C22×D61—416no—
142C22×D61—42no—
143C22×D61—42no—
144C22×D61—42no—
145C22×D61—42no—
146C22×D61—42no—
147C22×D61—42no—
148C22×D61#10014yes—
149C2×C4×C61—24no—
150C2×C4×C61—22no—
151C2×C4×C61—41no—
152C2×C4×C61—42no—
153C2×C4×C61—41no—
154C2×C4×C61—42no—
155C2×C4×C61—42no—
156C2×C4×C61—42no—
157C2×C4×C61#12012yes—
158C2×C4×C61#12012yes—
159C22⋊D61—416no—
160C2×Dic61—22no—
161C2×Dic61—22no—
162C2×Dic61—22no—
163C2×Dic61—42no—
164C2×Dic61—42no—
165C2×Dic61—42no—
166C2×Dic61—42no—
167C2×Dic61—42no—
168C2×Dic61—42no—
169C2×Dic61—42no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C4×D61—416no—
173C8×C61—82no—
174#92418—82no—
175#8553—611nomaximal, Sylow
176C22×C61—28no—
177C22×C61—21no—
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—42no—
184C22×C61—42no—
185C22×C61—42no—
186C22×C61—84no—
187C22×C61—84no—
188C22×C61—84no—
189C22×C61#37911yescharacteristic
190C22×C61#41012yes—
191C22×C61C2×#10012yes—
192C22×C61#41312yes—
193C2×D61—24no—
194C2×D61—48no—
195C2×D61—44no—
196C2×D61—48no—
197C2×D61—42no—
198C2×D61—42no—
199C2×D61—44no—
200C4×C61—22no—
201C4×C61—22no—
202C4×C61—42no—
203Dic61—42no—
204Dic61—42no—
205Dic61—42no—
206C22×#123—611no—
207#1821—612no—
208#1462—612no—
209#1462—611no—
210#1462D6111yescharacteristic, complemented
211C2×C61—22no—
212C2×C61—22no—
213C2×C61—22no—
214C2×C61—82no—
215C2×C61#178211yescharacteristic
216C2×C61#178311yescharacteristic
217C2×C61#178511yescharacteristic
218D61—44no—
219C23×D4—611no—
220C24×C4—611no—
221C2×#123—1224no—
222C2×#123—1224no—
223C2×#123—1222no—
224C2×#123—1222no—
225C22×C4⋊C4—611no—
226C22×C4⋊C4—611no—
227C22×C22⋊C4—611no—
228C22×C22⋊C4—611no—
229C22×C22⋊C4C2×D6111yescharacteristic
230#407—612no—
231#407C2×D6112yes—
232#343—1224no—
233#323—22no—
234#323—1224no—
235#323—1222no—
236C61#855311yescharacteristic, complemented, Sylow
237C25—611no—
238C25C4×D6111yescharacteristic
239C22×D4—1222no—
240C22×D4—1222no—
241C22×D4—1222no—
242C22×D4—2444no—
243C23×C4—611no—
244C23×C4—611no—
245C23×C4—611no—
246C23×C4—1222no—
247C23×C4—1222no—
248C23×C4—1222no—
249C23×C4—2444no—
250C23×C4C4×D6111yescharacteristic
251C23×C4C22×D6111yescharacteristic
252C2×M16—22no—
253C2×M16—1222no—
254#123—12216no—
255#123—12216no—
256C2×C4⋊C4—614no—
257C2×C4⋊C4—1222no—
258C2×C4⋊C4—1222no—
259C2×C4⋊C4—1222no—
260C2×C4⋊C4—1221no—
261C2×C4⋊C4—1222no—
262C2×C4⋊C4—1221no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—614no—
266C2×C22⋊C4—1222no—
267C2×C22⋊C4—1222no—
268C2×C22⋊C4—1222no—
269C2×C22⋊C4—1221no—
270C2×C22⋊C4—1222no—
271C2×C22⋊C4—1221no—
272C2×C22⋊C4D24412yes—
273C2×C22⋊C4C22⋊D6112yes—
274C22⋊C8—22no—
275C22⋊C8—22no—
276C22⋊C8—1222no—
277C22⋊C8—1222no—
278C24—21no—
279C24—21no—
280C24—42no—
281C24—42no—
282C24—42no—
283C24—612no—
284C24—1224no—
285C24—1221no—
286C24—1221no—
287C24—2442no—
288C24—2442no—
289C24#1175819812yes—
290C24C2×C4×D6111yescharacteristic
291C2×D4—1228no—
292C2×D4—2444no—
293C2×D4—2448no—
294C2×D4—2444no—
295C2×D4—2448no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—612no—
301C22×C4—612no—
302C22×C4—612no—
303C22×C4—1228no—
304C22×C4—1221no—
305C22×C4—1221no—
306C22×C4—1221no—
307C22×C4—1221no—
308C22×C4—1221no—
309C22×C4—1221no—
310C22×C4—2444no—
311C22×C4—2448no—
312C22×C4—2444no—
313C22×C4—2448no—
314C22×C4—2442no—
315C22×C4—2442no—
316C22×C4—2442no—
317C22×C4#1175819812yes—
318C22×C4C2×D24411yescharacteristic
319C22×C4D4×D6112yes—
320C22×C4D4×D6112yes—
321C22×C4C2×C22⋊D6111yescharacteristic
322M16—44no—
323M16—2444no—
324C2×C8—42no—
325C2×C8—2442no—
326C4⋊C4—1228no—
327C4⋊C4—1224no—
328C4⋊C4—1224no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—1228no—
332C22⋊C4—1224no—
333C22⋊C4—1224no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—24no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—88no—
348C23—614no—
349C23—1224no—
350C23—1224no—
351C23—1222no—
352C23—1222no—
353C23—2442no—
354C23—2442no—
355C23—24416no—
356C23—2442no—
357C23—2442no—
358C23—2442no—
359C23—2442no—
360C23C22⋊C4×D6111yescharacteristic
361C23C22⋊C4×D6111yescharacteristic
362C23C2×#1175819811yescharacteristic
363D4—24416no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—1222no—
373C2×C4—1222no—
374C2×C4—1222no—
375C2×C4—24416no—
376C2×C4—2442no—
377C2×C4—2442no—
378C2×C4—2442no—
379C2×C4—2442no—
380C2×C4—2442no—
381C2×C4—2442no—
382C2×C4—2442no—
383C2×C4—2442no—
384C2×C4—2442no—
385C2×C4#1451974512yes—
386C2×C4#1451974712yes—
387C8—82no—
388C8—4882no—
389C22—28no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—42no—
397C22—42no—
398C22—42no—
399C22—84no—
400C22—84no—
401C22—84no—
402C22—1224no—
403C22—2444no—
404C22—2448no—
405C22—2444no—
406C22—2448no—
407C22—2442no—
408C22—2442no—
409C22#1683961711yescharacteristic
410C22#1683973412yes—
411C22#100×D6112yes—
412C22#1683974412yes—
413C4—22no—
414C4—22no—
415C4—42no—
416C4—2442no—
417C4—2442no—
418C4—2442no—
419C2—22no—
420C2—22no—
421C2—22no—
422C2—82no—
423C2—2444no—
424C2#2144573511yescharacteristic
425C2#2144573811yescharacteristic
426C2#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#8553
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. #21445897C2semidirect
3. #21444146C2semidirect
4. #21444145C2semidirect
5. C61×#1462C2semidirect
210. #1462D61semidirect
236. C61#8553semidirect

Maximal subgroups

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

subgroupindex
C22×#145197592
#214458972
#214441462
#214441452
C61×#14622
#855361