Finite Group Databasebeta

← #29658749

Subgroups — #29658749

Order 15616.

986 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456767C211yesmaximal, characteristic, complemented
2#21455827C211yesmaximal, characteristic, complemented
3#21455616C211yesmaximal, characteristic, complemented
4#21445047C214yesmaximal, complemented
5#21444964C214yesmaximal, complemented
6#21444500C211yesmaximal, characteristic, complemented
7#21444244C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic, complemented
9C2×C61×#555C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840141C2212yescomplemented
12#16840112C2212yescomplemented
13#16840092—21no—
14#16840062—22no—
15#16840052C2212yescomplemented
16#16840052C2212yescomplemented
17#16840042C2212yescomplemented
18#16839969—22no—
19#16839967C2212yescomplemented
20#16839918—22no—
21#16839913—21no—
22C4×#11758200—22no—
23#16839815C2211yescharacteristic
24#16839815C2211yescharacteristic
25#16839802—21no—
26#16839786C2211yescharacteristic
27#16839767—21no—
28#16839511C2214yescomplemented
29D4⋊#92418C2214yescomplemented
30D4⋊#92418C2214yescomplemented
31C61×#555—22no—
32C61×#555—22no—
33C61×#555—22no—
34C61×#555C2214yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×#124×C61—22no—
37C2×#121×C61—21no—
38C2×#121×C61—21no—
39C2×C4×D4×C61C2211yescharacteristic
40C2×C4×D4×C61C2211yescharacteristic
41C2×#117×C61—21no—
42C2×#117×C61—21no—
43C2×#11758202—24no—
44C2×#11758200—22no—
45C2×#11758194—21no—
46#14519777C2311yescharacteristic
47#14519777C2311yescharacteristic
48#14519769—21no—
49#14519756—21no—
50#14519756—21no—
51C4×Dic122—21no—
52C4×Dic122—21no—
53#14519736—21no—
54#14519710—22no—
55#14519708—22no—
56#14519708—22no—
57C4⋊#92418—21no—
58C4⋊#92418—21no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519701—24no—
62#14519700—22no—
63#14519698—22no—
64#14519698—22no—
65#14519698—22no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C4×#92418—21no—
69C4×#92418C2312yes—
70C22×Q8×C61D412yes—
71C22×D4×C61—22no—
72C22×D4×C61D411yescharacteristic
73C22×D4×C61D411yescharacteristic
74C23×C4×C61—21no—
75C23×C4×C61—21no—
76C4⋊Q8×C61—21no—
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61D412yes—
79C4⋊Q8×C61C2312yes—
80#124×C61—28no—
81#124×C61—28no—
82#121×C61—24no—
83#121×C61—24no—
84#121×C61—24no—
85#121×C61—24no—
86C4×D4×C61—21no—
87C4×D4×C61—21no—
88C4×D4×C61—21no—
89C4×D4×C61—21no—
90C4×D4×C61D412yes—
91C4×D4×C61D412yes—
92C4×D4×C61C2312yes—
93C4×D4×C61C2312yes—
94#117×C61—24no—
95#117×C61—24no—
96#117×C61—24no—
97#117×C61—24no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C42×C61—22no—
107C2×C42×C61C2311yescharacteristic
108C24×C61—21no—
109C24×C61—21no—
110C2×Q8×C61—22no—
111C2×Q8×C61—82no—
112C2×Q8×C61C2×D414yes—
113C2×D4×C61—28no—
114C2×D4×C61—21no—
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—42no—
119C2×D4×C61—84no—
120C2×D4×C61C2×D412yes—
121C2×D4×C61C2×D412yes—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—44no—
130C22×C4×C61—42no—
131C22×C4×C61—42no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C2×Dic122—21no—
136C2×Dic122—21no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61C2×D412yes—
144C4⋊C4×C61C2×D412yes—
145C22⋊C4×C61—28no—
146C22⋊C4×C61—28no—
147C22⋊C4×C61—22no—
148C22⋊C4×C61—22no—
149C22⋊C4×C61—22no—
150C22⋊C4×C61—22no—
151C42×C61—28no—
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2411yescharacteristic
156C22⋊Dic61—41no—
157#11758202—84no—
158#11758200—44no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163#11758196—41no—
164#11758196—41no—
165C4×Dic61—41no—
166#11758194—42no—
167C2×#92418—22no—
168C2×#92418—22no—
169C2×#92418—22no—
170C23×C61—22no—
171C23×C61—22no—
172C23×C61—42no—
173C23×C61—42no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61—42no—
177C23×C61—42no—
178C23×C61#12011yescharacteristic
179Q8×C61—44no—
180Q8×C61—82no—
181D4×C61—22no—
182D4×C61—22no—
183D4×C61—22no—
184D4×C61—22no—
185D4×C61—42no—
186D4×C61—42no—
187D4×C61—88no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—24no—
192C2×C4×C61—41no—
193C2×C4×C61—44no—
194C2×C4×C61—42no—
195C2×C4×C61—44no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—44no—
200C2×C4×C61—44no—
201C2×C4×C61#12011yescharacteristic
202C2×C4×C61#12011yescharacteristic
203C2×C4×C61#12011yescharacteristic
204C2×C4×C61#13512yes—
205C2×C4×C61#13712yes—
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C22×D411yescharacteristic
209C2×Dic61—41no—
210C2×Dic61—41no—
211Dic122—42no—
212Dic122—42no—
213#92418—42no—
214#92418—42no—
215#92418—42no—
216#18773—611nomaximal, Sylow
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—42no—
220C22×C61—42no—
221C22×C61—42no—
222C22×C61—42no—
223C22×C61—42no—
224C22×C61—42no—
225C22×C61—84no—
226C22×C61—84no—
227C22×C61#45212yes—
228C22×C61C2×#12011yescharacteristic
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—42no—
233C4×C61—42no—
234C4×C61—44no—
235C4×C61#45212yes—
236C4×C61C2×#13511yescharacteristic
237C4×C61C2×#13711yescharacteristic
238C4×C61#57512yes—
239Dic61—81no—
240Dic61—81no—
241C2×#555D6111yescharacteristic
242#3209—611no—
243#3101—611no—
244#3077—611no—
245#1646—614no—
246#1634—614no—
247#1548—611no—
248#1486—611no—
249#1478—611no—
250C2×C61—22no—
251C2×C61—82no—
252C2×C61—82no—
253C2×C61C2×#45211yescharacteristic
254C2×C61#300511yescharacteristic
255C2×C61#301911yescharacteristic
256#555—22no—
257#555—22no—
258#555—22no—
259#555C2×D6114yes—
260#533—611no—
261C2×C4⋊Q8C2×D6111yescharacteristic
262C2×#124—22no—
263C2×#121—21no—
264C2×#121—21no—
265C2×C4×D4C2×D6111yescharacteristic
266C2×C4×D4C2×D6111yescharacteristic
267C2×#117—21no—
268C2×#117—21no—
269#494—612no—
270#487—612no—
271#483—1221no—
272#478—1222no—
273#476—612no—
274#476—612no—
275#474—612no—
276#462—1222no—
277#461—612no—
278#451—1222no—
279#450—1221no—
280C4×SD16—1222no—
281#431—611no—
282#431—611no—
283#428—1221no—
284#423—611no—
285#418—1221no—
286#333—614no—
287D4⋊C8—614no—
288D4⋊C8—614no—
289C61#1877311yescharacteristic, complemented, Sylow
290C22×Q8C22⋊D6112yes—
291C22×D4—22no—
292C22×D4C22⋊D6111yescharacteristic
293C22×D4C22⋊D6111yescharacteristic
294C23×C4—21no—
295C23×C4—21no—
296C2×Q16—1224no—
297C2×SD16—1222no—
298C2×M16—1221no—
299C4⋊Q8—21no—
300C4⋊Q8—21no—
301C4⋊Q8—611no—
302C4⋊Q8—611no—
303C4⋊Q8C22⋊D6112yes—
304C4⋊Q8C22×D6112yes—
305#125—1221no—
306#124—28no—
307#124—28no—
308C22⋊Q8—1221no—
309C22⋊Q8—1221no—
310#121—24no—
311#121—24no—
312#121—24no—
313#121—24no—
314C4×Q8—1221no—
315C4×Q8—1221no—
316C4×D4—21no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4—21no—
320C4×D4C22⋊D6112yes—
321C4×D4C22⋊D6112yes—
322C4×D4C22×D6112yes—
323C4×D4C22×D6112yes—
324#117—24no—
325#117—24no—
326#117—24no—
327#117—24no—
328#117—1221no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4C22⋊D6111yescharacteristic
332C2×C4⋊C4C22⋊D6111yescharacteristic
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C42—22no—
338C2×C42C22×D6111yescharacteristic
339#107—1222no—
340#106—1222no—
341#106—1222no—
342C4⋊C8—612no—
343C4⋊C8—612no—
344C4⋊C8—1221no—
345C4⋊C8—1221no—
346#103—1224no—
347#103—1222no—
348#102—1222no—
349#102—1222no—
350#102—1222no—
351C22⋊C8—1221no—
352C22⋊C8—1221no—
353C4×C8—612no—
354C4×C8—1221no—
355C24—21no—
356C24—21no—
357C2×Q8—22no—
358C2×Q8—82no—
359C2×Q8—1221no—
360C2×Q8—1221no—
361C2×Q8C2×C22⋊D6114yes—
362C2×D4—28no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—42no—
368C2×D4—84no—
369C2×D4C2×C22⋊D6112yes—
370C2×D4C2×C22⋊D6112yes—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4C2×C22⋊D6111yescharacteristic
382C22×C4C2×C22⋊D6111yescharacteristic
383C22×C4C2×C22⋊D6111yescharacteristic
384Q16—4884no—
385SD16—2444no—
386M16—2442no—
387C2×C8—1222no—
388C2×C8—1222no—
389C2×C8—1222no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—1221no—
397C4⋊C4—1221no—
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4—2441no—
401C4⋊C4—2441no—
402C4⋊C4C2×C22⋊D6112yes—
403C4⋊C4C2×C22⋊D6112yes—
404C22⋊C4—28no—
405C22⋊C4—28no—
406C22⋊C4—22no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—2441no—
411C42—28no—
412C42—2441no—
413C42C2×C22⋊D6111yescharacteristic
414C42C2×C22⋊D6111yescharacteristic
415C42C2×C22⋊D6111yescharacteristic
416C42C23×D6111yescharacteristic
417C23—22no—
418C23—22no—
419C23—42no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23#1451974611yescharacteristic
426Q8—44no—
427Q8—82no—
428Q8—2442no—
429Q8—2442no—
430D4—22no—
431D4—22no—
432D4—22no—
433D4—22no—
434D4—42no—
435D4—42no—
436D4—88no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—41no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—2441no—
451C2×C4—2441no—
452C2×C4#1451974611yescharacteristic
453C2×C4#1451974611yescharacteristic
454C2×C4#1451974611yescharacteristic
455C2×C4#1451979712yes—
456C2×C4#1451980712yes—
457C2×C4C22×C22⋊D6111yescharacteristic
458C2×C4C22×C22⋊D6111yescharacteristic
459C2×C4C22×C22⋊D6111yescharacteristic
460C8—2442no—
461C8—2442no—
462C8—2442no—
463C22—21no—
464C22—21no—
465C22—42no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—84no—
472C22—84no—
473C22#1683992412yes—
474C22C2×#1451974611yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—44no—
479C4—42no—
480C4—42no—
481C4—4881no—
482C4—4881no—
483C4#1683992412yes—
484C4C2×#1451979711yescharacteristic
485C4C2×#1451980711yescharacteristic
486C4#1684056012yes—
487C2—22no—
488C2—82no—
489C2—82no—
490C2C2×#1683992411yescharacteristic
491C2#2145506111yescharacteristic
492C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC2×C61×#555C2
socleC22×C61C2×#120
hypercenterC2×#555D61
hypocenterC61#18773
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. #21456767C2semidirect
2. #21455827C2semidirect
3. #21455616C2semidirect
4. #21445047C2semidirect
5. #21444964C2semidirect
6. #21444500C2semidirect
7. #21444244C2semidirect
8. #21444210C2semidirect
11. #16840141C22semidirect
12. #16840112C22semidirect
15. #16840052C22semidirect
16. #16840052C22semidirect
17. #16840042C22semidirect
19. #16839967C22semidirect
28. #16839511C22semidirect
29. D4⋊#92418C22semidirect
30. D4⋊#92418C22semidirect
289. C61#18773semidirect

Maximal subgroups

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

subgroupindex
#214567672
#214558272
#214556162
#214450472
#214449642
#214445002
#214442442
#214442102
C2×C61×#5552
#1877361