Finite Group Databasebeta

← #29736177

Subgroups — #29736177

Order 15616.

1094 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459622C211yesmaximal, characteristic
2#21451199C212yesmaximal, complemented
3#21447172C212yesmaximal, complemented
4#21447172C212yesmaximal, complemented
5#21446803C212yesmaximal, complemented
6#21446099C212yesmaximal
7#21446099C212yesmaximal
8C4×#14519692C211yesmaximal, characteristic
9C61×#2579C211yesmaximal, characteristic
10#16840471—24no—
11#16840469—22no—
12D4×Dic122—22no—
13#16840393—21no—
14#16840388—21no—
15#16840388C2212yes—
16#16840319—21no—
17C2×#14519756—21no—
18C2×#14519756C2212yes—
19#16840237C2212yes—
20C2×#14519692C2212yes—
21#16839692—22no—
22#16839682—22no—
23#16839657—24no—
24#16839635C2214yes—
25#16839635C2214yes—
26C4×C22⋊Dic61C2212yes—
27#16839529C2212yes—
28#16839528C2214yes—
29#16839528C2214yes—
30C2×C4×D4×C61C2211yescharacteristic
31C22×C4⋊C4×C61C2211yescharacteristic
32C61×#398—21no—
33C61×#396—21no—
34C61×#387—21no—
35C61×#386C2211yescharacteristic
36C61×#386C2211yescharacteristic
37C61×#385—21no—
38C61×#385C2211yescharacteristic
39C61×#385C2211yescharacteristic
40C61×#380C2211yescharacteristic
41C22×Dic122—21no—
42C22×Dic122D412yes—
43#14519777—21no—
44#14519777—21no—
45#14519776—24no—
46#14519769—22no—
47#14519768—24no—
48#14519756—22no—
49#14519756—22no—
50#14519756—22no—
51#14519756—22no—
52#14519756—22no—
53#14519756—22no—
54#14519756—22no—
55#14519756—22no—
56#14519754—24no—
57#14519754—24no—
58#14519754—24no—
59#14519754—24no—
60C4×Dic122—22no—
61C4×Dic122—22no—
62D4×Dic61—24no—
63D4×Dic61—24no—
64#14519736—22no—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61D412yes—
67C2×#11758196—22no—
68C2×#11758196—24no—
69C2×#11758196—24no—
70C2×#11758196D414yes—
71C2×C22⋊Dic61—22no—
72C2×C22⋊Dic61C2314yes—
73C2×C22⋊Dic61C2314yes—
74C2×C4×Dic61—22no—
75#14519692Q818yescomplemented
76#14519684—22no—
77#14519684—24no—
78#14519684—24no—
79C22×D4×C61C2311yescharacteristic
80C23×C4×C61D411yescharacteristic
81C23×C4×C61C2311yescharacteristic
82C23×C4×C61C2311yescharacteristic
83C23×C4×C61C2311yescharacteristic
84C4×D4×C61—22no—
85C4×D4×C61—22no—
86C2×C4⋊C4×C61—22no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61D411yescharacteristic
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61—21no—
96C2×C22⋊C4×C61D411yescharacteristic
97C2×C22⋊C4×C61C2311yescharacteristic
98C2×C22⋊C4×C61C2311yescharacteristic
99C2×C22⋊C4×C61C2311yescharacteristic
100C2×C42×C61D411yescharacteristic
101#95×C61—21no—
102#95×C61—21no—
103#95×C61—21no—
104#95×C61—21no—
105#95×C61—21no—
106C24×C61C2×D411yescharacteristic
107C24×C61C2411yescharacteristic
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61C2×Q814yes—
111C22×C4×C61—21no—
112C22×C4×C61—22no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—42no—
120C22×C4×C61—81no—
121C22×C4×C61—81no—
122C22×C4×C61—81no—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D412yes—
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×Dic61—24no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61C2×D414yes—
134C22×Dic61C2×D414yes—
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—22no—
138C2×Dic122—22no—
139C2×Dic122—82no—
140C2×Dic122—82no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—42no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61—42no—
145C4⋊C4×C61—41no—
146C4⋊C4×C61—81no—
147C4⋊C4×C61—81no—
148C22⋊C4×C61—22no—
149C22⋊C4×C61—22no—
150C22⋊C4×C61—22no—
151C22⋊C4×C61—22no—
152C22⋊C4×C61—42no—
153C22⋊C4×C61—81no—
154C42×C61—21no—
155C42×C61—21no—
156C22⋊Dic61—24no—
157C22⋊Dic61—24no—
158C22⋊Dic61—24no—
159C22⋊Dic61—24no—
160C22⋊Dic61C2×Q818yescomplemented
161C4⋊Dic61—22no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167#11758196—24no—
168#11758196—24no—
169#11758196—24no—
170#11758196—24no—
171#11758196—48no—
172#11758196—84no—
173C4×Dic61—24no—
174C4×Dic61—82no—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—41no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61C4⋊Q812yes—
182C23×C61C22×D411yescharacteristic
183C23×C61C22×D412yes—
184C23×C61C22×Q812yes—
185C23×C612−1+412yes—
186D4×C61—28no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—24no—
190C2×C4×C61—22no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—42no—
194C2×C4×C61—41no—
195C2×C4×C61—42no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—44no—
204C2×C4×C61—41no—
205C2×C4×C61—41no—
206C2×C4×C61—41no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—41no—
210C2×C4×C61—41no—
211C2×C4×C61—41no—
212C2×C4×C61—41no—
213C2×C4×C61—81no—
214C2×C4×C61#12011yescharacteristic
215C2×C4×C61#12012yes—
216C2×C4×C61#12011yescharacteristic
217C2×C4×C61C4⋊Q812yes—
218C2×Dic61—24no—
219C2×Dic61—24no—
220C2×Dic61—28no—
221C2×Dic61—44no—
222C2×Dic61—44no—
223C2×Dic61—44no—
224C2×Dic61—44no—
225C2×Dic61—82no—
226C2×Dic61—82no—
227Dic122—44no—
228Dic122—82no—
229Dic122—82no—
230Dic122—82no—
231#25854—611nomaximal, Sylow
232C22×C61—22no—
233C22×C61—21no—
234C22×C61—22no—
235C22×C61—21no—
236C22×C61—22no—
237C22×C61—42no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—41no—
241C22×C61—41no—
242C22×C61—81no—
243C22×C61C2×#12011yescharacteristic
244C22×C61C2×C4⋊Q811yescharacteristic
245C22×C61#53611yescharacteristic
246C22×C61D4×Q814yes—
247C4×C61—22no—
248C4×C61—21no—
249C4×C61—22no—
250C4×C61—21no—
251C4×C61—41no—
252C4×C61—42no—
253C4×C61—41no—
254C4×C61—81no—
255C4×C61—81no—
256C4×C61—81no—
257Dic61—48no—
258Dic61—44no—
259Dic61—44no—
260#3501—611no—
261#2583—612no—
262#2579D6111yescharacteristic
263#2062—612no—
264#2062—612no—
265#1996—612no—
266#1867—612no—
267#1867—612no—
268C4×#99—611no—
269C2×C61—24no—
270C2×C61—21no—
271C2×C61—21no—
272C2×C61—42no—
273C2×C61#243411yescharacteristic
274C2×C61#302511yescharacteristic
275C2×C61#302611yescharacteristic
276#554—1222no—
277#554—1224no—
278D4×Q8—1222no—
279#544—1221no—
280#543—612no—
281#543—1221no—
282#533—1221no—
283C2×C22⋊Q8—612no—
284C2×C22⋊Q8—1221no—
285#518—612no—
286C2×C4×D4C2×D6111yescharacteristic
287C22×C4⋊C4C2×D6111yescharacteristic
288C2×#99—612no—
289#398—21no—
290#398—1222no—
291#396—21no—
292#395—1222no—
293#389—1224no—
294#387—21no—
295#386C2×D6111yescharacteristic
296#386C2×D6111yescharacteristic
297#385—21no—
298#385—614no—
299#385—614no—
300#385C2×D6111yescharacteristic
301#385C2×D6111yescharacteristic
302#380C2×D6111yescharacteristic
303C4×C22⋊C4—612no—
304#342—612no—
305#342—614no—
306#342—614no—
307C61#2585411yescharacteristic, complemented, Sylow
308C22×Q8—612no—
309C22×Q8—1221no—
310C22×D4C22×D6111yescharacteristic
311C23×C4C22⋊D6111yescharacteristic
312C23×C4C22×D6111yescharacteristic
313C23×C4C22×D6111yescharacteristic
314C23×C4C22×D6111yescharacteristic
315C4⋊Q8—1221no—
316C4⋊Q8—1221no—
317C4⋊Q8—1224no—
318#125—1222no—
319#125—1224no—
320C22⋊Q8—1224no—
321C22⋊Q8—1224no—
322C22⋊Q8—1224no—
323C22⋊Q8—1224no—
324C22⋊Q8—1222no—
325C22⋊Q8—1222no—
326C22⋊Q8—1222no—
327C22⋊Q8—1222no—
328C22⋊Q8—1222no—
329C22⋊Q8—1222no—
330C22⋊Q8—1222no—
331C22⋊Q8—1222no—
332C4×Q8—1222no—
333C4×Q8—1222no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—1224no—
337C4×D4—1224no—
338#117—1222no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—614no—
347C2×C4⋊C4—612no—
348C2×C4⋊C4—1222no—
349C2×C4⋊C4—1221no—
350C2×C4⋊C4—1224no—
351C2×C4⋊C4—1224no—
352C2×C4⋊C4C22⋊D6111yescharacteristic
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—614no—
356C2×C22⋊C4—614no—
357C2×C22⋊C4—1222no—
358C2×C22⋊C4C22⋊D6111yescharacteristic
359C2×C22⋊C4C22×D6111yescharacteristic
360C2×C22⋊C4C22×D6111yescharacteristic
361C2×C22⋊C4C22×D6111yescharacteristic
362C2×C42—1222no—
363C2×C42C22⋊D6111yescharacteristic
364#99—618no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95—21no—
370#95—1224no—
371#95—1224no—
372#95—1222no—
373C24C2×C22⋊D6111yescharacteristic
374C24C23×D6111yescharacteristic
375C2×Q8—1222no—
376C2×Q8—1222no—
377C2×Q8—1222no—
378C2×Q8—1222no—
379C2×Q8—4882no—
380C2×Q8—4882no—
381C2×D4—22no—
382C2×D4—22no—
383C2×D4Q8×D6114yes—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—42no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4—614no—
397C22×C4—614no—
398C22×C4—1222no—
399C22×C4—1222no—
400C22×C4—1224no—
401C22×C4D4×D6112yes—
402C22×C4D4×D6112yes—
403C22×C4C2×C22⋊D6111yescharacteristic
404C22×C4C2×C22⋊D6111yescharacteristic
405C22×C4C2×C22⋊D6111yescharacteristic
406C22×C4C2×C22⋊D6111yescharacteristic
407C22×C4C2×C22⋊D6111yescharacteristic
408C4⋊C4—22no—
409C4⋊C4—42no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—42no—
413C4⋊C4—81no—
414C4⋊C4—81no—
415C4⋊C4—1224no—
416C4⋊C4—1222no—
417C4⋊C4—1224no—
418C4⋊C4—1222no—
419C4⋊C4—1222no—
420C4⋊C4—1222no—
421C4⋊C4—1224no—
422C4⋊C4—1222no—
423C4⋊C4—1224no—
424C4⋊C4—1222no—
425C4⋊C4—4884no—
426C4⋊C4—2448no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—42no—
432C22⋊C4—81no—
433C22⋊C4—618no—
434C22⋊C4—1224no—
435C22⋊C4—1224no—
436C22⋊C4—1224no—
437C22⋊C4—1224no—
438C42—21no—
439C42—21no—
440C42—1224no—
441C42—4882no—
442C23—21no—
443C23—21no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23#1451977512yes—
449C23C2×D4×D6112yes—
450C23C22×C22⋊D6111yescharacteristic
451C23C2×Q8×D6112yes—
452C23#1451982412yes—
453Q8—4882no—
454Q8—4882no—
455Q8—4882no—
456Q8—2444no—
457D4—28no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—41no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—81no—
485C2×C4—1224no—
486C2×C4—1224no—
487C2×C4—1228no—
488C2×C4—4882no—
489C2×C4—4882no—
490C2×C4—2444no—
491C2×C4—2444no—
492C2×C4—2444no—
493C2×C4—2444no—
494C2×C4#1451974511yescharacteristic
495C2×C4#1451974511yescharacteristic
496C2×C4#1451974512yes—
497C2×C4#1451977512yes—
498C22—21no—
499C22—22no—
500C22—21no—
501C22—22no—
502C22—22no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—81no—
509C22C2×#1451974511yescharacteristic
510C22C2×#1451977511yescharacteristic
511C22#1684033311yescharacteristic
512C22#1684042514yes—
513C4—21no—
514C4—21no—
515C4—22no—
516C4—22no—
517C4—42no—
518C4—41no—
519C4—41no—
520C4—81no—
521C4—81no—
522C4—81no—
523C4—2448no—
524C4—2444no—
525C4—2444no—
526C2—21no—
527C2—24no—
528C2—21no—
529C2—42no—
530C2#2144946411yescharacteristic
531C2#2145521011yescharacteristic
532C2#2145521511yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#2579C2
socleC22×C61C2×#120
prime radicalC22×D4×C61C23
hypercenter#2579D61
hypocenterC61#25854
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2579

Complements

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

NH ≅ G/Ntype
2. #21451199C2semidirect
3. #21447172C2semidirect
4. #21447172C2semidirect
5. #21446803C2semidirect
75. #14519692Q8semidirect
160. C22⋊Dic61C2×Q8semidirect
307. C61#25854semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214511992
#214471722
#214468032
#214460992
C4×#145196922
C61×#25792
#2585461