Finite Group Databasebeta

← #29736168

Subgroups — #29736168

Order 15616.

1094 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459622C211yesmaximal, characteristic
2#21450293C212yesmaximal, complemented
3#21447172C214yesmaximal, complemented
4#21446803C212yesmaximal, complemented
5#21446099C214yesmaximal
6C4×#14519692C211yesmaximal, characteristic
7C61×#2487C211yesmaximal, characteristic
8#16840471—24no—
9#16840469—22no—
10D4×Dic122—22no—
11#16840393—21no—
12#16840388—21no—
13#16840388C2212yes—
14#16840319—21no—
15C2×#14519756—21no—
16C2×#14519756C2212yes—
17#16840237C2212yes—
18C2×#14519692C2212yes—
19#16839693—24no—
20#16839667—22no—
21#16839635C2218yes—
22#16839631—22no—
23C4×C22⋊Dic61C2212yes—
24#16839529C2212yes—
25#16839528C2218yes—
26C2×C4×D4×C61C2211yescharacteristic
27C22×C4⋊C4×C61C2211yescharacteristic
28C61×#400—21no—
29C61×#399—21no—
30C61×#389—21no—
31C61×#386C2212yes—
32C61×#385C2212yes—
33C61×#380—21no—
34C61×#380C2211yescharacteristic
35C22×Dic122—21no—
36C22×Dic122D412yes—
37#14519777—21no—
38#14519777—21no—
39#14519776—24no—
40#14519769—22no—
41#14519768—24no—
42#14519756—22no—
43#14519756—22no—
44#14519756—22no—
45#14519756—22no—
46#14519756—22no—
47#14519756—22no—
48#14519756—22no—
49#14519756—22no—
50#14519754—24no—
51#14519754—24no—
52#14519754—24no—
53#14519754—24no—
54C4×Dic122—22no—
55C4×Dic122—22no—
56D4×Dic61—24no—
57D4×Dic61—24no—
58#14519736—22no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61D412yes—
61C2×#11758196—22no—
62C2×#11758196—28no—
63C2×#11758196D414yes—
64C2×C22⋊Dic61—22no—
65C2×C22⋊Dic61C2314yes—
66C2×C22⋊Dic61C2314yes—
67C2×C4×Dic61—22no—
68#14519692Q818yescomplemented
69#14519684—22no—
70#14519684—28no—
71C22×D4×C61C2311yescharacteristic
72C23×C4×C61D411yescharacteristic
73C23×C4×C61C2312yes—
74C23×C4×C61C2311yescharacteristic
75C4×D4×C61—22no—
76C4×D4×C61—22no—
77C2×C4⋊C4×C61—22no—
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61—22no—
82C2×C4⋊C4×C61—22no—
83C2×C4⋊C4×C61D411yescharacteristic
84C2×C22⋊C4×C61—22no—
85C2×C22⋊C4×C61D411yescharacteristic
86C2×C22⋊C4×C61C2312yes—
87C2×C22⋊C4×C61C2311yescharacteristic
88C2×C42×C61D411yescharacteristic
89#95×C61—22no—
90#95×C61—21no—
91#95×C61—22no—
92C24×C61C2×D411yescharacteristic
93C24×C61C2411yescharacteristic
94C2×D4×C61—22no—
95C2×D4×C61—22no—
96C2×D4×C61C2×Q814yes—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—24no—
102C22×C4×C61—42no—
103C22×C4×C61—42no—
104C22×C4×C61—82no—
105C22×C4×C61C2×D412yes—
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D412yes—
111C22×C4×C61C2×D411yescharacteristic
112C22×Dic61—24no—
113C22×Dic61—22no—
114C22×Dic61—22no—
115C22×Dic61C2×D414yes—
116C22×Dic61C2×D414yes—
117C2×Dic122—22no—
118C2×Dic122—22no—
119C2×Dic122—22no—
120C2×Dic122—22no—
121C2×Dic122—44no—
122C2×Dic122—82no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—42no—
125C4⋊C4×C61—41no—
126C4⋊C4×C61—42no—
127C4⋊C4×C61—41no—
128C4⋊C4×C61—82no—
129C22⋊C4×C61—22no—
130C22⋊C4×C61—22no—
131C22⋊C4×C61—24no—
132C22⋊C4×C61—82no—
133C42×C61—21no—
134C42×C61—21no—
135C22⋊Dic61—24no—
136C22⋊Dic61—24no—
137C22⋊Dic61—24no—
138C22⋊Dic61—24no—
139C22⋊Dic61C2×Q818yescomplemented
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143C4⋊Dic61—22no—
144C4⋊Dic61—22no—
145C4⋊Dic61—22no—
146#11758196—24no—
147#11758196—24no—
148#11758196—24no—
149#11758196—24no—
150#11758196—88no—
151C4×Dic61—24no—
152C4×Dic61—44no—
153C23×C61—21no—
154C23×C61—21no—
155C23×C61—41no—
156C23×C61—41no—
157C23×C61—41no—
158C23×C61—41no—
159C23×C61C4⋊Q812yes—
160C23×C61C22×D411yescharacteristic
161C23×C61C22×D412yes—
162C23×C61C22×Q812yes—
163C23×C612−1+412yes—
164D4×C61—28no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—24no—
168C2×C4×C61—22no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—41no—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—48no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—81no—
184C2×C4×C61#12011yescharacteristic
185C2×C4×C61#12012yes—
186C2×C4×C61#12011yescharacteristic
187C2×C4×C61C4⋊Q812yes—
188C2×Dic61—24no—
189C2×Dic61—24no—
190C2×Dic61—28no—
191C2×Dic61—44no—
192C2×Dic61—44no—
193C2×Dic61—44no—
194C2×Dic61—44no—
195C2×Dic61—82no—
196C2×Dic61—82no—
197Dic122—44no—
198Dic122—82no—
199Dic122—82no—
200Dic122—82no—
201#25853—611nomaximal, Sylow
202C22×C61—22no—
203C22×C61—21no—
204C22×C61—22no—
205C22×C61—21no—
206C22×C61—22no—
207C22×C61—42no—
208C22×C61—42no—
209C22×C61—42no—
210C22×C61—41no—
211C22×C61—41no—
212C22×C61—81no—
213C22×C61C2×#12011yescharacteristic
214C22×C61C2×C4⋊Q811yescharacteristic
215C22×C61#53611yescharacteristic
216C22×C61D4×Q814yes—
217C4×C61—22no—
218C4×C61—22no—
219C4×C61—21no—
220C4×C61—41no—
221C4×C61—41no—
222C4×C61—82no—
223C4×C61—82no—
224C4×C61#45812yes—
225Dic61—48no—
226Dic61—44no—
227Dic61—44no—
228#3501—611no—
229#2502—612no—
230#2487D6111yescharacteristic
231#2062—614no—
232#1996—612no—
233#1867—614no—
234C4×#99—611no—
235C2×C61—24no—
236C2×C61—21no—
237C2×C61—21no—
238C2×C61—42no—
239C2×C61#243411yescharacteristic
240C2×C61C2×#45811yescharacteristic
241C2×C61#302711yescharacteristic
242#554—1222no—
243#554—1224no—
244D4×Q8—1222no—
245#544—1221no—
246#543—612no—
247#543—1221no—
248#533—1221no—
249C2×C22⋊Q8—612no—
250C2×C22⋊Q8—1221no—
251#518—612no—
252C2×C4×D4C2×D6111yescharacteristic
253C22×C4⋊C4C2×D6111yescharacteristic
254C2×#99—612no—
255#400—21no—
256#399—21no—
257#398—1224no—
258#391—1222no—
259#389—21no—
260#386C2×D6112yes—
261#385—618no—
262#385C2×D6112yes—
263#384—1222no—
264#380—21no—
265#380C2×D6111yescharacteristic
266C4×C22⋊C4—612no—
267#342—612no—
268#342—618no—
269C61#2585311yescharacteristic, complemented, Sylow
270C22×Q8—612no—
271C22×Q8—1221no—
272C22×D4C22×D6111yescharacteristic
273C23×C4C22⋊D6111yescharacteristic
274C23×C4C22×D6112yes—
275C23×C4C22×D6111yescharacteristic
276C4⋊Q8—1221no—
277C4⋊Q8—1221no—
278C4⋊Q8—1224no—
279#125—1222no—
280#125—1224no—
281C22⋊Q8—1224no—
282C22⋊Q8—1224no—
283C22⋊Q8—1224no—
284C22⋊Q8—1224no—
285C22⋊Q8—1222no—
286C22⋊Q8—1222no—
287C22⋊Q8—1222no—
288C22⋊Q8—1222no—
289C22⋊Q8—1222no—
290C22⋊Q8—1222no—
291C22⋊Q8—1222no—
292C22⋊Q8—1222no—
293C4×Q8—1222no—
294C4×Q8—1222no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—1224no—
298C4×D4—1224no—
299#117—1222no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—612no—
307C2×C4⋊C4—614no—
308C2×C4⋊C4—1222no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1228no—
311C2×C4⋊C4C22⋊D6111yescharacteristic
312C2×C22⋊C4—22no—
313C2×C22⋊C4—614no—
314C2×C22⋊C4—614no—
315C2×C22⋊C4—1222no—
316C2×C22⋊C4C22⋊D6111yescharacteristic
317C2×C22⋊C4C22×D6112yes—
318C2×C22⋊C4C22×D6111yescharacteristic
319C2×C42—1222no—
320C2×C42C22⋊D6111yescharacteristic
321#99—618no—
322#95—22no—
323#95—21no—
324#95—22no—
325#95—1228no—
326#95—1222no—
327C24C2×C22⋊D6111yescharacteristic
328C24C23×D6111yescharacteristic
329C2×Q8—1222no—
330C2×Q8—1222no—
331C2×Q8—1222no—
332C2×Q8—1222no—
333C2×Q8—4882no—
334C2×Q8—2444no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4Q8×D6114yes—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—82no—
346C22×C4—614no—
347C22×C4—614no—
348C22×C4—1222no—
349C22×C4—1222no—
350C22×C4—1224no—
351C22×C4D4×D6112yes—
352C22×C4D4×D6112yes—
353C22×C4C2×C22⋊D6111yescharacteristic
354C22×C4C2×C22⋊D6111yescharacteristic
355C22×C4C2×C22⋊D6111yescharacteristic
356C22×C4C2×C22⋊D6111yescharacteristic
357C22×C4C2×C22⋊D6111yescharacteristic
358C4⋊C4—22no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—82no—
364C4⋊C4—1224no—
365C4⋊C4—1222no—
366C4⋊C4—1224no—
367C4⋊C4—1222no—
368C4⋊C4—1222no—
369C4⋊C4—1222no—
370C4⋊C4—1224no—
371C4⋊C4—1222no—
372C4⋊C4—1224no—
373C4⋊C4—1222no—
374C4⋊C4—4888no—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—24no—
378C22⋊C4—82no—
379C22⋊C4—618no—
380C22⋊C4—1224no—
381C22⋊C4—1224no—
382C22⋊C4—1224no—
383C22⋊C4—1224no—
384C42—21no—
385C42—21no—
386C42—1224no—
387C42—2444no—
388C23—21no—
389C23—21no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23#1451977512yes—
395C23C2×D4×D6112yes—
396C23C22×C22⋊D6111yescharacteristic
397C23C2×Q8×D6112yes—
398C23#1451982412yes—
399Q8—4882no—
400Q8—4882no—
401Q8—4882no—
402Q8—2444no—
403D4—28no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—81no—
423C2×C4—1224no—
424C2×C4—1224no—
425C2×C4—1228no—
426C2×C4—4882no—
427C2×C4—4882no—
428C2×C4—2444no—
429C2×C4—2444no—
430C2×C4—2444no—
431C2×C4—2444no—
432C2×C4#1451974511yescharacteristic
433C2×C4#1451974512yes—
434C2×C4#1451974511yescharacteristic
435C2×C4#1451977512yes—
436C22—21no—
437C22—22no—
438C22—21no—
439C22—22no—
440C22—22no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—41no—
445C22—41no—
446C22—81no—
447C22C2×#1451974511yescharacteristic
448C22C2×#1451977511yescharacteristic
449C22#1684033311yescharacteristic
450C22#1684042514yes—
451C4—21no—
452C4—22no—
453C4—22no—
454C4—41no—
455C4—41no—
456C4—82no—
457C4—82no—
458C4—2448no—
459C4—2444no—
460C4—2444no—
461C4#1683995112yes—
462C2—21no—
463C2—24no—
464C2—21no—
465C2—42no—
466C2#2144946411yescharacteristic
467C2C2×#1683995111yescharacteristic
468C2#2145522011yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#2487C2
socleC22×C61C2×#120
prime radicalC22×D4×C61C23
hypercenter#2487D61
hypocenterC61#25853
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2487

Complements

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

NH ≅ G/Ntype
2. #21450293C2semidirect
3. #21447172C2semidirect
4. #21446803C2semidirect
68. #14519692Q8semidirect
139. C22⋊Dic61C2×Q8semidirect
269. C61#25853semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214502932
#214471722
#214468032
#214460992
C4×#145196922
C61×#24872
#2585361