Finite Group Databasebeta

← #29828740

Subgroups — #29828740

Order 15616.

3580 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 3580 conjugacy classes of subgroups — too many to draw. Only its 255 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×#14519825C211yesmaximal, characteristic
2C2×#16839944C214yesmaximal, complemented
3#21455077C212yesmaximal, complemented
4#21445777C214yesmaximal, complemented
5#21445764C212yesmaximal, complemented
6#21445743C211yesmaximal, characteristic, complemented
7C61×#2434C211yesmaximal, characteristic, complemented
8C2×#14519825—22no—
9C2×#14519825—216no—
10C2×#14519825—24no—
11C2×#14519825C2214yes—
12C22×#11758226—22no—
13C22×#11758222—22no—
14C22×#11758222C2212yes—
15C22×Q8×D61—22no—
16C2×#14519808C2214yescomplemented
17#16839944—216no—
18#16839922—24no—
19#16839917—22no—
20#16839912—22no—
21C2×#14519705C2218yescomplemented
22#16839761C2214yescomplemented
23C2×#14519696C2214yescomplemented
24#16839726C2212yes—
25C23×Q8×C61C2211yescharacteristic
26C2×C4⋊Q8×C61C2212yescomplemented
27C2×C22⋊Q8×C61C2211yescharacteristic
28C61×#395—22no—
29C61×#391C2212yescomplemented
30C61×#385—22no—
31C4×C22⋊C4×C61C2211yescharacteristic
32#14519825—232no—
33#14519825—232no—
34#14519825—232no—
35#14519825—28no—
36#14519825—216no—
37#14519825D4116yes—
38C2×#11758226—232no—
39C2×#11758226—28no—
40C2×#11758226—28no—
41C2×#11758222—28no—
42C2×#11758222—232no—
43C2×#11758222—28no—
44C2×#11758222—22no—
45C2×#11758222—22no—
46C2×#11758222—216no—
47C2×#11758222D414yes—
48C2×#11758222C2314yes—
49C22×Dic122—22no—
50C22×Dic122D412yes—
51C2×Q8×D61—28no—
52C2×Q8×D61—232no—
53C2×Q8×D61—28no—
54C22×D244—22no—
55C22×D244D412yes—
56C22×C22⋊D61—22no—
57C22×C4×D61—24no—
58C22×C4×D61—22no—
59#14519808—216no—
60C2×#11758202—24no—
61C2×#11758201—24no—
62C2×#11758194C2314yes—
63#14519705—216no—
64#14519700—24no—
65#14519697—24no—
66#14519696—28no—
67C22⋊#92418—22no—
68C22⋊#92418—22no—
69C22×Q8×C61—28no—
70C22×Q8×C61—22no—
71C22×Q8×C61—22no—
72C22×Q8×C61D412yes—
73C22×Q8×C61C2311yescharacteristic
74C22×Q8×C61C2312yes—
75C23×C4×C61D412yes—
76C23×C4×C61C2311yescharacteristic
77C4⋊Q8×C61—28no—
78C22⋊Q8×C61—22no—
79C22⋊Q8×C61—22no—
80C2×C4⋊C4×C61—24no—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61C2311yescharacteristic
83C2×C22⋊C4×C61—21no—
84C2×C22⋊C4×C61—21no—
85C2×C42×C61C2312yes—
86#95×C61—21no—
87#95×C61—24no—
88C24×C61C2×D411yescharacteristic
89C23×D61—22no—
90C2×Q8×C61—216no—
91C2×Q8×C61—216no—
92C2×Q8×C61—24no—
93C2×Q8×C61—28no—
94C2×Q8×C61—24no—
95C2×Q8×C61—24no—
96C2×Q8×C61—24no—
97C2×Q8×C61—22no—
98C2×Q8×C61C2×D418yes—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—28no—
102C22×C4×C61—24no—
103C22×C4×C61—22no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61C2×D414yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D412yes—
109C22×C4×C61C2×D412yes—
110C22×C4×C61C2411yescharacteristic
111C2×C22⋊D61—216no—
112C2×C22⋊D61—24no—
113C2×C22⋊D61—24no—
114C22×Dic61—22no—
115#11758226—264no—
116#11758226—232no—
117#11758226—232no—
118Q8×D61—264no—
119Q8×D61—232no—
120Q8×D61—232no—
121#11758222—264no—
122#11758222—232no—
123#11758222—216no—
124#11758222—216no—
125#11758222—232no—
126#11758222—28no—
127#11758222—28no—
128#11758222—28no—
129#11758222C2×D4116yes—
130C2×D244—216no—
131C2×D244—28no—
132C2×D244—22no—
133C2×D244—28no—
134C2×D244C2×D414yes—
135C2×C4×D61—232no—
136C2×C4×D61—216no—
137C2×C4×D61—216no—
138C2×C4×D61—24no—
139C2×C4×D61—24no—
140C2×Dic122—216no—
141C2×Dic122—28no—
142C2×Dic122—22no—
143C2×Dic122—28no—
144C2×Dic122C2×D414yes—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—42no—
147C4⋊C4×C61—48no—
148C22⋊C4×C61—24no—
149C22⋊C4×C61—41no—
150C22⋊C4×C61—41no—
151C42×C61—24no—
152#11758202—48no—
153#11758201—48no—
154#11758194—28no—
155C2×#92418—24no—
156C23×C61—22no—
157C23×C61—22no—
158C23×C61—41no—
159C23×C61#12011yescharacteristic
160C23×C61#12011yescharacteristic
161C23×C61C22×D411yescharacteristic
162C22×D61—28no—
163C22×D61—24no—
164C22×D61—44no—
165C22×D61—44no—
166Q8×C61—216no—
167Q8×C61—216no—
168Q8×C61—44no—
169C2×C4×C61—28no—
170C2×C4×C61—28no—
171C2×C4×C61—24no—
172C2×C4×C61—24no—
173C2×C4×C61—22no—
174C2×C4×C61—24no—
175C2×C4×C61—24no—
176C2×C4×C61—24no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61—42no—
183C2×C4×C61#12012yes—
184C2×C4×C61#12018yes—
185C2×C4×C61C22×D412yes—
186C2×C4×C61C22×D414yes—
187C22⋊D61—216no—
188C22⋊D61—216no—
189C2×Dic61—28no—
190C2×Dic61—24no—
191D244—232no—
192D244—28no—
193D244—24no—
194D244#12018yes—
195C4×D61—264no—
196C4×D61—216no—
197C4×D61—216no—
198Dic122—232no—
199Dic122—28no—
200Dic122—24no—
201Dic122#12018yes—
202#92418—44no—
203#32892—611nomaximal, Sylow
204C22×C61—21no—
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—44no—
212C22×C61#45612yescomplemented
213C22×C61C2×#12012yes—
214C22×C61C2×#12011yescharacteristic
215C2×D61—28no—
216C2×D61—416no—
217C2×D61—44no—
218C2×D61—44no—
219C4×C61—28no—
220C4×C61—22no—
221C4×C61—44no—
222C4×C61—81no—
223C4×C61—81no—
224C4×C61C2×#12014yes—
225Dic61—28no—
226C22×2−1+4—611no—
227C2×#456—614no—
228#3007—612no—
229#2434D6111yescharacteristic, complemented
230#1793—614no—
231#1789—612no—
232#1784—611no—
233C2×C61—22no—
234C2×C61—22no—
235C2×C61—41no—
236C2×C61#284611yescharacteristic
237C2×C61C2×#45611yescharacteristic
238C2×C61#302011yescharacteristic
239D61—48no—
240C2×2−1+4—614no—
241C2×2−1+4—1224no—
242C2×2−1+4—12216no—
243C2×2−1+4—1222no—
244C22×#41—612no—
245C22×#41—1222no—
246C22×#41—1222no—
247C23×Q8—1222no—
248C23×Q8C2×D6111yescharacteristic
249C2×#137—614no—
250C2×C4⋊Q8C2×D6112yescomplemented
251C2×C22⋊Q8C2×D6111yescharacteristic
252#456—12216no—
253#452—1224no—
254#451—1222no—
255#450—1222no—
256C2×#104—618no—
257#417—614no—
258C2×#101—614no—
259#407—612no—
260#395—22no—
261#391C2×D6112yescomplemented
262#385—22no—
263C4×C22⋊C4C2×D6111yescharacteristic
264C61#3289211yescharacteristic, complemented, Sylow
2652−1+4—6116no—
2662−1+4—12216no—
2672−1+4—12232no—
2682−1+4—12232no—
2692−1+4—12232no—
2702−1+4—1228no—
271C2×#41—614no—
272C2×#41—614no—
273C2×#41—12216no—
274C2×#41—1228no—
275C2×#41—12232no—
276C2×#41—1228no—
277C2×#41—12232no—
278C2×#41—1228no—
279C2×#41—1222no—
280C2×#41—1222no—
281C2×#41—1228no—
282C22×Q8—28no—
283C22×Q8—22no—
284C22×Q8—22no—
285C22×Q8—612no—
286C22×Q8—1222no—
287C22×Q8—1228no—
288C22×Q8—12232no—
289C22×Q8—1228no—
290C22×Q8C22⋊D6112yes—
291C22×Q8C22×D6111yescharacteristic
292C22×Q8C22×D6112yes—
293C22×D4—612no—
294C22×D4—1222no—
295C22×D4—1222no—
296C23×C4—1224no—
297C23×C4—1222no—
298C23×C4C22⋊D6112yes—
299C23×C4C22×D6111yescharacteristic
300#137—12216no—
301C2×Q16—1224no—
302C2×SD16—1224no—
303C2×M16—614no—
304C4⋊Q8—28no—
305C22⋊Q8—22no—
306C22⋊Q8—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4C22×D6111yescharacteristic
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C42C22×D6112yes—
313#104—12216no—
314#103—1224no—
315#102—1224no—
316#101—1228no—
317C22⋊C8—1222no—
318C22⋊C8—1222no—
319#95—21no—
320#95—24no—
321C24—1222no—
322C24C2×C22⋊D6111yescharacteristic
323#41—6116no—
324#41—1228no—
325#41—1228no—
326#41—12264no—
327#41—12216no—
328#41—12232no—
329#41—12216no—
330#41—12232no—
331#41—12264no—
332#41—12232no—
333#41—1228no—
334#41—12232no—
335C2×Q8—216no—
336C2×Q8—216no—
337C2×Q8—24no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—28no—
341C2×Q8—22no—
342C2×Q8—24no—
343C2×Q8—614no—
344C2×Q8—1228no—
345C2×Q8—12216no—
346C2×Q8—1228no—
347C2×Q8—12232no—
348C2×Q8—12264no—
349C2×Q8—12232no—
350C2×Q8—1222no—
351C2×Q8D4×D6118yes—
352C2×D4—614no—
353C2×D4—1228no—
354C2×D4—12216no—
355C2×D4—1228no—
356C2×D4—12216no—
357C2×D4—1224no—
358C2×D4—1224no—
359C2×D4—1222no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—12232no—
368C22×C4—12216no—
369C22×C4—12216no—
370C22×C4—1222no—
371C22×C4—1224no—
372C22×C4—1224no—
373C22×C4D4×D6112yes—
374C22×C4D4×D6112yes—
375C22×C4C2×C22⋊D6114yes—
376C22×C4C2×C22⋊D6111yescharacteristic
377C22×C4C23×D6111yescharacteristic
378Q16—2448no—
379SD16—2448no—
380M16—1228no—
381C2×C8—1224no—
382C4⋊C4—22no—
383C4⋊C4—42no—
384C4⋊C4—48no—
385C22⋊C4—24no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C42—24no—
389C23—22no—
390C23—22no—
391C23—41no—
392C23—1228no—
393C23—1224no—
394C23—2444no—
395C23—2444no—
396C23#1451974511yescharacteristic
397C23#1451974511yescharacteristic
398C23C22×C22⋊D6111yescharacteristic
399Q8—216no—
400Q8—216no—
401Q8—44no—
402Q8—618no—
403Q8—1228no—
404Q8—12232no—
405Q8—1224no—
406D4—618no—
407D4—1228no—
408D4—12232no—
409D4—12216no—
410D4—12216no—
411D4—1224no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—12264no—
427C2×C4—12216no—
428C2×C4—1228no—
429C2×C4—1224no—
430C2×C4—12216no—
431C2×C4#1451974518yes—
432C2×C4#1451974512yes—
433C2×C4C2×D4×D6114yes—
434C2×C4C2×D4×D6112yes—
435C8—2444no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—44no—
444C22—1228no—
445C22—24416no—
446C22—2444no—
447C22—2444no—
448C22#1683994412yescomplemented
449C22C2×#1451974512yes—
450C22C2×#1451974511yescharacteristic
451C4—28no—
452C4—22no—
453C4—44no—
454C4—81no—
455C4—81no—
456C4—1228no—
457C4#120×D6114yes—
458C2—22no—
459C2—22no—
460C2—41no—
461C2—2448no—
462C2#2145409511yescharacteristic
463C2C2×#1683994411yescharacteristic
464C2#2145518211yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2434C2
socleC22×C61C2×#120
prime radicalC22×#14519825C2
hypercenter#2434D61
hypocenterC61#32892
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2434

Complements

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

NH ≅ G/Ntype
2. C2×#16839944C2semidirect
3. #21455077C2semidirect
4. #21445777C2semidirect
5. #21445764C2semidirect
6. #21445743C2semidirect
7. C61×#2434C2semidirect
16. C2×#14519808C22semidirect
21. C2×#14519705C22semidirect
22. #16839761C22semidirect
23. C2×#14519696C22semidirect
26. C2×C4⋊Q8×C61C22semidirect
29. C61×#391C22semidirect
212. C22×C61#456semidirect
229. #2434D61semidirect
250. C2×C4⋊Q8C2×D61semidirect
261. #391C2×D61semidirect
264. C61#32892semidirect
448. C22#16839944semidirect

Maximal subgroups

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

subgroupindex
C22×#145198252
C2×#168399442
#214550772
#214457772
#214457642
#214457432
C61×#24342
#3289261