Finite Group Databasebeta

← #29736140

Subgroups — #29736140

Order 15616.

1056 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840471C211yesmaximal, characteristic
2C2×#16840168C211yesmaximal, characteristic, complemented
3#21451028C211yesmaximal, characteristic, complemented
4#21447155C212yesmaximal, complemented
5#21447145C212yesmaximal, complemented
6#21446774C211yesmaximal, characteristic, complemented
7#21446758C211yesmaximal, characteristic, complemented
8#21446078C212yesmaximal
9#21446064C212yesmaximal
10#21445644C211yesmaximal, characteristic
11C2×C61×#474C211yesmaximal, characteristic
12#16840471—24no—
13#16840471—24no—
14C2×#14519776C2211yescharacteristic
15C2×#14519769C2211yescharacteristic
16C2×#14519768C2211yescharacteristic
17C2×#14519756—21no—
18C2×#14519754—21no—
19C2×#14519754—21no—
20C2×C4×Dic122C2211yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22C2×D4×Dic61C2211yescharacteristic
23C22×#11758196—21no—
24#16840168—24no—
25C2×#14519710C2212yes—
26C2×#14519708C2212yes—
27C2×#14519698C2212yes—
28C2×#14519688C2211yescharacteristic
29#16839681—21no—
30#16839681—21no—
31#16839667—21no—
32#16839658C2212yes—
33#16839658C2212yes—
34#16839631—21no—
35C4×C4⋊Dic61C2211yescharacteristic
36#16839525C2212yes—
37#16839525C2212yes—
38#16839524C2212yes—
39#16839516C2212yes—
40#16839516C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C61×#474—24no—
44C2×#107×C61C2212yes—
45C2×C4⋊C8×C61C2211yescharacteristic
46C2×#102×C61C2212yes—
47C22×Dic122—21no—
48C22×Dic122D411yescharacteristic
49C23×Dic61—21no—
50C23×Dic61—21no—
51C22×#92418C2312yes—
52#14519776—22no—
53#14519776—22no—
54#14519769—22no—
55#14519769D414yes—
56#14519768—22no—
57#14519768—22no—
58#14519756—28no—
59#14519754—28no—
60#14519754—28no—
61C4×Dic122—22no—
62C4×Dic122—22no—
63D4×Dic61—22no—
64D4×Dic61—22no—
65D4×Dic61—22no—
66D4×Dic61—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61D411yescharacteristic
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2312yes—
72C2×#11758196—28no—
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×#11758196—22no—
78C2×#11758196—22no—
79C2×#11758196D411yescharacteristic
80C2×#11758196D411yescharacteristic
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519710—24no—
87#14519708—24no—
88#14519698Q818yescomplemented
89#14519688—22no—
90#14519684—21no—
91#14519684—22no—
92#14519684—22no—
93C22×Q8×C61—21no—
94C22×D4×C61C2311yescharacteristic
95C23×C4×C61—21no—
96C22×C8×C61C2312yes—
97C4⋊Q8×C61—22no—
98C4⋊Q8×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61D414yes—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C42×C61C2311yescharacteristic
106#107×C61—24no—
107C4⋊C8×C61—22no—
108#102×C61—24no—
109C24×C61—21no—
110C2×Q8×C61—42no—
111C2×Q8×C61—82no—
112C2×D4×C61—21no—
113C2×D4×C61—22no—
114C2×D4×C61C2×D412yes—
115C2×D4×C61C2×Q814yes—
116C22×C4×C61—21no—
117C22×C4×C61—24no—
118C22×C4×C61—22no—
119C22×C4×C61—82no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—24no—
124C22×Dic61—21no—
125C22×Dic61—21no—
126C22×Dic61—22no—
127C22×Dic61—24no—
128C22×Dic61—42no—
129C22×Dic61—42no—
130C22×Dic61—82no—
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C22×Dic61C2×D411yescharacteristic
135C2×Dic122—22no—
136C2×Dic122—22no—
137C2×Dic122—24no—
138C2×Dic122—41no—
139C2×Dic122—82no—
140C2×C8×C61—22no—
141C2×C8×C61—24no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—42no—
146C4⋊C4×C61C2×D412yes—
147C22⋊C4×C61—24no—
148C42×C61C2×D412yes—
149C42×C61C2×D412yes—
150C22⋊Dic61—24no—
151C22⋊Dic61—24no—
152C4⋊Dic61—24no—
153C4⋊Dic61—22no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—22no—
157C4⋊Dic61C2×D412yes—
158C4⋊Dic61C2×D412yes—
159C4⋊Dic61C2×Q814yes—
160#11758196—24no—
161#11758196—24no—
162#11758196—22no—
163#11758196—24no—
164#11758196—24no—
165#11758196—22no—
166#11758196—82no—
167#11758196—82no—
168C4×Dic61—21no—
169C4×Dic61—21no—
170C4×Dic61—21no—
171C4×Dic61—21no—
172C4×Dic61—42no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418C2×Q814yes—
176C23×C61—22no—
177C23×C61—24no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61—42no—
181C23×C61C22×D411yescharacteristic
182Q8×C61—84no—
183D4×C61—22no—
184D4×C61—24no—
185D4×C61C4⋊Q814yes—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—84no—
195C2×C4×C61#12012yes—
196C2×C4×C61#12012yes—
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×Q812yes—
200C2×C4×C612−1+412yes—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—21no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—24no—
209C2×Dic61—21no—
210C2×Dic61—21no—
211C2×Dic61—22no—
212C2×Dic61—22no—
213C2×Dic61—48no—
214C2×Dic61—41no—
215C2×Dic61—42no—
216C2×Dic61—41no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—84no—
220C2×Dic61#13712yes—
221Dic122—44no—
222Dic122—84no—
223C8×C61—42no—
224#92418—24no—
225#25850—611nomaximal, Sylow
226C22×C61—24no—
227C22×C61—44no—
228C22×C61—44no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61C2×#12011yescharacteristic
232C22×C61D4×Q812yes—
233C22×C61C2×#13711yescharacteristic
234C22×C61#57511yescharacteristic
235C22×C61#57612yes—
236C4×C61—22no—
237C4×C61—22no—
238C4×C61—42no—
239C4×C61—82no—
240C4×C61C2×C4⋊Q811yescharacteristic
241C4×C61#53611yescharacteristic
242C4×C61D4×Q812yes—
243Dic61—22no—
244Dic61—44no—
245Dic61—41no—
246Dic61—42no—
247Dic61—41no—
248Dic61—42no—
249Dic61—41no—
250C2×#554—611no—
251C2×#501—611no—
252C2×#474D6111yescharacteristic
253#2567—611no—
254#2059—612no—
255#2057—612no—
256#1989—611no—
257#1985—611no—
258#1864—612no—
259#1862—612no—
260#1762—611no—
261C2×C61—44no—
262C2×C61#243411yescharacteristic
263C2×C61#300711yescharacteristic
264C2×C61#300811yescharacteristic
265C2×C61#338512yes—
266C2×C61#338912yes—
267#554—1224no—
268#554—1224no—
269C2×C4⋊Q8—611no—
270C2×C4⋊Q8C2×D6111yescharacteristic
271C2×#125—611no—
272C2×#125—611no—
273C2×C22⋊Q8—1221no—
274C2×C22⋊Q8—1221no—
275C2×C22⋊Q8—1221no—
276C2×C4×Q8—611no—
277C2×C4×D4—611no—
278C2×C4×D4—611no—
279C2×C4×D4C2×D6111yescharacteristic
280C22×C4⋊C4—1221no—
281#501—1224no—
282#474—24no—
283C2×#107—612no—
284C2×#107C2×D6112yes—
285C2×#106—612no—
286C2×C4⋊C8C2×D6111yescharacteristic
287C2×#102—612no—
288C2×#102C2×D6112yes—
289C2×#97—611no—
290#395—1221no—
291#395—1221no—
292#391—1221no—
293#389—612no—
294#389—612no—
295#384—1221no—
296C4×C4⋊C4—611no—
297#340—612no—
298#340—612no—
299#340—612no—
300#336—612no—
301#336—612no—
302C61#2585011yescharacteristic, complemented, Sylow
303C22×Q8—21no—
304C22×Q8—611no—
305C22×Q8—1221no—
306C22×D4C22×D6111yescharacteristic
307C23×C4—21no—
308C23×C4—1221no—
309C23×C4—1221no—
310C22×C8—612no—
311C22×C8C22×D6112yes—
312C4⋊Q8—22no—
313C4⋊Q8—22no—
314C4⋊Q8—1222no—
315C4⋊Q8—1222no—
316#125—614no—
317#125—1222no—
318#125—1222no—
319#125—1222no—
320C22⋊Q8—1228no—
321C22⋊Q8—1228no—
322C22⋊Q8—1228no—
323C4×Q8—1222no—
324C4×Q8—1222no—
325C4×D4—22no—
326C4×D4—1222no—
327C4×D4—1222no—
328C4×D4—1222no—
329C4×D4—1222no—
330C4×D4C22⋊D6114yes—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—611no—
334C2×C4⋊C4—612no—
335C2×C4⋊C4—611no—
336C2×C4⋊C4—611no—
337C2×C4⋊C4—611no—
338C2×C4⋊C4—1221no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C4⋊C4—1228no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4—1222no—
345C2×C4⋊C4—1222no—
346C2×C4⋊C4C22×D6112yes—
347C2×C4⋊C4C22×D6111yescharacteristic
348C2×C22⋊C4—21no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1221no—
351C2×C42—611no—
352C2×C42—611no—
353C2×C42—1221no—
354C2×C42C22×D6111yescharacteristic
355#107—24no—
356#107—1224no—
357#106—1224no—
358C4⋊C8—22no—
359#102—24no—
360#102—618no—
361#97—1222no—
362#95—1222no—
363#95—1222no—
364#95—1221no—
365C24—21no—
366C2×Q8—42no—
367C2×Q8—82no—
368C2×Q8—1222no—
369C2×Q8—1224no—
370C2×Q8—1222no—
371C2×Q8—4882no—
372C2×Q8—2441no—
373C2×D4—21no—
374C2×D4—22no—
375C2×D4Q8×D6114yes—
376C2×D4C2×C22⋊D6112yes—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—82no—
381C22×C4—611no—
382C22×C4—611no—
383C22×C4—611no—
384C22×C4—611no—
385C22×C4—1224no—
386C22×C4—1224no—
387C22×C4—1222no—
388C22×C4—1221no—
389C22×C4—1221no—
390C22×C4—4882no—
391C22×C4—2442no—
392C22×C4—2442no—
393C22×C4D4×D6111yescharacteristic
394C22×C4D4×D6111yescharacteristic
395C22×C4C23×D6111yescharacteristic
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—614no—
399C2×C8—1222no—
400C2×C8—1222no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—42no—
405C4⋊C4—612no—
406C4⋊C4—612no—
407C4⋊C4—614no—
408C4⋊C4—1224no—
409C4⋊C4—1224no—
410C4⋊C4—1224no—
411C4⋊C4—1221no—
412C4⋊C4—1221no—
413C4⋊C4—1222no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C4⋊C4—1222no—
417C4⋊C4—1224no—
418C4⋊C4—1224no—
419C4⋊C4—4882no—
420C4⋊C4—4882no—
421C4⋊C4C2×C22⋊D6112yes—
422C22⋊C4—24no—
423C22⋊C4—1224no—
424C22⋊C4—1224no—
425C42—1221no—
426C42—1221no—
427C42—1221no—
428C42—1221no—
429C42—2442no—
430C42D4×D6112yes—
431C42C2×C22⋊D6112yes—
432C23—22no—
433C23—24no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23C2×D4×D6111yescharacteristic
438Q8—84no—
439Q8—4884no—
440Q8—2444no—
441D4—22no—
442D4—24no—
443D4#1451977514yes—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—84no—
453C2×C4—612no—
454C2×C4—1221no—
455C2×C4—1221no—
456C2×C4—1221no—
457C2×C4—1221no—
458C2×C4—1221no—
459C2×C4—1222no—
460C2×C4—1222no—
461C2×C4—1221no—
462C2×C4—1221no—
463C2×C4—1222no—
464C2×C4—1222no—
465C2×C4—1224no—
466C2×C4—4884no—
467C2×C4—2448no—
468C2×C4—2441no—
469C2×C4—2442no—
470C2×C4—2441no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4#1451974512yes—
474C2×C4#1451974512yes—
475C2×C4C2×D4×D6111yescharacteristic
476C2×C4C22×C22⋊D6111yescharacteristic
477C2×C4C2×Q8×D6112yes—
478C2×C4#1451982412yes—
479C8—42no—
480C8—1224no—
481C22—24no—
482C22—44no—
483C22—44no—
484C22—42no—
485C22—42no—
486C22C2×#1451974511yescharacteristic
487C22#1684042512yes—
488C22#137×D6111yescharacteristic
489C22#1684056111yescharacteristic
490C22#1684056312yes—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—82no—
495C4—1222no—
496C4—2444no—
497C4—2441no—
498C4—2442no—
499C4—2441no—
500C4—2441no—
501C4—2442no—
502C4C2×#1451977511yescharacteristic
503C4#1684033311yescharacteristic
504C4#1684042512yes—
505C2—44no—
506C2#2144946411yescharacteristic
507C2#2145507511yescharacteristic
508C2#2145508311yescharacteristic
509C2#2145872412yes—
510C2#2145877212yes—
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#474C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenterC2×#474D61
hypocenterC61#25850
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
2. C2×#16840168C2semidirect
3. #21451028C2semidirect
4. #21447155C2semidirect
5. #21447145C2semidirect
6. #21446774C2semidirect
7. #21446758C2semidirect
88. #14519698Q8semidirect
302. C61#25850semidirect

Maximal subgroups

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

subgroupindex
C2×#168404712
C2×#168401682
#214510282
#214471552
#214471452
#214467742
#214467582
#214460782
#214460642
#214456442
C2×C61×#4742
#2585061