Finite Group Databasebeta

← #29735296

Subgroups — #29735296

Order 15616.

1618 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839952C214yesmaximal, complemented
2#21450291C212yesmaximal, complemented
3#21450127C211yesmaximal, characteristic, complemented
4#21446948C214yesmaximal, complemented
5#21446656C212yesmaximal, complemented
6C4×#14519692C211yesmaximal, characteristic, complemented
7C61×#3496C211yesmaximal, characteristic
8C2×#14519758C2212yes—
9C2×#14519758C2218yescomplemented
10C22×C4⋊Dic61C2211yescharacteristic, complemented
11#16839952—216no—
12C2×#14519692C2212yes—
13C2×#14519692C2218yescomplemented
14#16839698—24no—
15#16839692—21no—
16#16839687—21no—
17#16839665—22no—
18#16839660—21no—
19#16839631—22no—
20#16839623C2212yescomplemented
21#16839622—21no—
22#16839622C2211yescharacteristic, complemented
23C4×C22⋊Dic61C2212yescomplemented
24#16839529C2212yescomplemented
25C2×H(5,𝔽2)×C61C2212yes—
26C61×#547—22no—
27C61×#546—24no—
28D42×C61—22no—
29C61×#537—21no—
30C61×#534—21no—
31C61×#534C2212yes—
32C61×#532—21no—
33C2×#121×C61—21no—
34C2×#121×C61C2212yes—
35C2×C4×D4×C61C2211yescharacteristic
36C23×Dic61C2312yes—
37#14519758—28no—
38#14519758—216no—
39#14519758—48no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—22no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—22no—
44C2×C4⋊Dic61—21no—
45C2×#11758196—22no—
46C2×#11758196—28no—
47C2×C22⋊Dic61—22no—
48C2×C22⋊Dic61—28no—
49C2×C22⋊Dic61C2312yes—
50C2×C22⋊Dic61C2314yes—
51C2×C4×Dic61—22no—
52#14519692—216no—
53#14519692D418yescomplemented
54#14519684—21no—
55#14519684—22no—
56#14519684—22no—
57#14519684—22no—
58H(5,𝔽2)×C61—28no—
59H(5,𝔽2)×C61—44no—
60C2×#41×C61—22no—
61C2×#41×C61—24no—
62C22×D4×C61—22no—
63C22×D4×C61—24no—
64C22×D4×C61—21no—
65C22×D4×C61D412yes—
66C22×D4×C61C2314yes—
67C22×D4×C61C2311yescharacteristic
68C23×C4×C61D411yescharacteristic
69C23×C4×C61C2311yescharacteristic
70C4⋊Q8×C61—21no—
71#127×C61—21no—
72#124×C61—22no—
73#123×C61—24no—
74#123×C61—24no—
75C22⋊Q8×C61—24no—
76#121×C61—24no—
77#121×C61—22no—
78#121×C61—22no—
79#121×C61—22no—
80#121×C61—22no—
81#121×C61—22no—
82#121×C61—22no—
83#121×C61—22no—
84#121×C61—22no—
85#120×C61—24no—
86#120×C61—24no—
87C4×D4×C61—22no—
88C4×D4×C61—22no—
89C4×D4×C61—22no—
90C4×D4×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61D411yescharacteristic
93C2×C4⋊C4×C61D412yes—
94C2×C22⋊C4×C61—22no—
95C2×C22⋊C4×C61D411yescharacteristic
96C2×C22⋊C4×C61D414yes—
97C2×C22⋊C4×C61C2311yescharacteristic
98C2×C42×C61D411yescharacteristic
99C24×C61—22no—
100C24×C61C2×D414yes—
101C24×C61C2×D411yescharacteristic
102C24×C61C2411yescharacteristic
103#41×C61—416no—
104#41×C61—44no—
105#41×C61—44no—
106C2×Q8×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—24no—
110C2×D4×C61—22no—
111C2×D4×C61—24no—
112C2×D4×C61—24no—
113C2×D4×C61—28no—
114C2×D4×C61—28no—
115C2×D4×C61—22no—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—44no—
120C2×D4×C61—416no—
121C2×D4×C61—44no—
122C2×D4×C61—44no—
123C2×D4×C61—82no—
124C2×D4×C61C2×D414yes—
125C22×C4×C61—22no—
126C22×C4×C61—24no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—42no—
130C22×C4×C61—42no—
131C22×C4×C61C2×D414yes—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D412yes—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D412yes—
138C22×C4×C61C2×D411yescharacteristic
139C22×Dic61—24no—
140C22×Dic61—22no—
141C22×Dic61—24no—
142C22×Dic61—24no—
143C22×Dic61—22no—
144C22×Dic61—24no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C22⋊C4×C61—24no—
151C22⋊C4×C61—24no—
152C22⋊C4×C61—22no—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—24no—
156C42×C61—21no—
157C42×C61—21no—
158C22⋊Dic61—24no—
159C22⋊Dic61—24no—
160C22⋊Dic61—416no—
161C22⋊Dic61—44no—
162C22⋊Dic61C2×D418yescomplemented
163C4⋊Dic61—42no—
164C4⋊Dic61—41no—
165C4⋊Dic61—42no—
166C4⋊Dic61—41no—
167#11758196—416no—
168#11758196—44no—
169C4×Dic61—44no—
170C23×C61—21no—
171C23×C61—21no—
172C23×C61—24no—
173C23×C61—28no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61—41no—
177C23×C61—41no—
178C23×C61—44no—
179C23×C61—44no—
180C23×C61—44no—
181C23×C61—82no—
182C23×C61—82no—
183C23×C61#12712yes—
184C23×C61C22×D411yescharacteristic
185C23×C61C22×D412yes—
186C23×C61C22×D412yes—
187C23×C61H(5,𝔽2)12yes—
188Q8×C61—44no—
189D4×C61—28no—
190D4×C61—44no—
191D4×C61—48no—
192D4×C61—44no—
193D4×C61—44no—
194D4×C61—48no—
195D4×C61—44no—
196D4×C61—82no—
197D4×C61—82no—
198D4×C61—82no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—24no—
202C2×C4×C61—22no—
203C2×C4×C61—21no—
204C2×C4×C61—21no—
205C2×C4×C61—24no—
206C2×C4×C61—28no—
207C2×C4×C61—42no—
208C2×C4×C61—41no—
209C2×C4×C61—48no—
210C2×C4×C61—44no—
211C2×C4×C61—42no—
212C2×C4×C61—41no—
213C2×C4×C61—41no—
214C2×C4×C61—81no—
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61#12012yes—
217C2×C4×C61#12011yescharacteristic
218C2×C4×C61#12712yes—
219C2×Dic61—28no—
220C2×Dic61—44no—
221C2×Dic61—44no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—48no—
229#25772—611nomaximal, Sylow
230C22×C61—21no—
231C22×C61—22no—
232C22×C61—21no—
233C22×C61—22no—
234C22×C61—22no—
235C22×C61—42no—
236C22×C61—42no—
237C22×C61—42no—
238C22×C61—41no—
239C22×C61—41no—
240C22×C61—44no—
241C22×C61—44no—
242C22×C61—44no—
243C22×C61—81no—
244C22×C61—88no—
245C22×C61—164no—
246C22×C61C2×#12011yescharacteristic
247C22×C61C2×#12711yescharacteristic
248C22×C61#53511yescharacteristic
249C22×C61D4214yes—
250C4×C61—22no—
251C4×C61—21no—
252C4×C61—22no—
253C4×C61—44no—
254C4×C61—41no—
255C4×C61—41no—
256C4×C61#45812yescomplemented
257Dic61—48no—
258Dic61—82no—
259Dic61—82no—
260#3496D6111yescharacteristic
261C2×#458—614no—
262#2501—612no—
263#2486—611no—
264#2022—614no—
265#1964—612no—
266C4×#99—611no—
267C2×C61—24no—
268C2×C61—21no—
269C2×C61—21no—
270C2×C61—42no—
271C2×C61—84no—
272C2×C61#243311yescharacteristic
273C2×C61C2×#45811yescharacteristic
274C2×C61#302811yescharacteristic
275C2×H(5,𝔽2)C2×D6112yes—
276#547—22no—
277#546—24no—
278D42—22no—
279#537—21no—
280#534—21no—
281#534C2×D6112yes—
282#532—21no—
283C2×#123—612no—
284C2×#123—618no—
285C2×#121—21no—
286C2×#121C2×D6112yes—
287C2×C4×D4C2×D6111yescharacteristic
288C22×C4⋊C4—611no—
289#458—12216no—
290C2×#99—612no—
291C2×#99—618no—
292#399—1224no—
293#398—1221no—
294#397—1221no—
295#390—1222no—
296#389—1221no—
297#384—1222no—
298#381—612no—
299#380—611no—
300#380—1221no—
301C4×C22⋊C4—612no—
302#342—612no—
303C61#2577211yescharacteristic, complemented, Sylow
304H(5,𝔽2)—28no—
305H(5,𝔽2)—44no—
306C2×#41—22no—
307C2×#41—24no—
308C22×D4—22no—
309C22×D4—24no—
310C22×D4—21no—
311C22×D4C22⋊D6112yes—
312C22×D4C22×D6114yes—
313C22×D4C22×D6111yescharacteristic
314C23×C4—612no—
315C23×C4C22⋊D6111yescharacteristic
316C23×C4C22×D6111yescharacteristic
317C4⋊Q8—21no—
318#127—21no—
319#124—22no—
320#123—24no—
321#123—24no—
322#123—12216no—
323#123—1228no—
324#123—2448no—
325C22⋊Q8—24no—
326#121—22no—
327#121—22no—
328#121—24no—
329#121—22no—
330#121—22no—
331#121—22no—
332#121—22no—
333#121—22no—
334#121—22no—
335#120—24no—
336#120—24no—
337C4×D4—22no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1222no—
344C2×C4⋊C4—1221no—
345C2×C4⋊C4—1222no—
346C2×C4⋊C4—1228no—
347C2×C4⋊C4—1222no—
348C2×C4⋊C4—1221no—
349C2×C4⋊C4C22⋊D6111yescharacteristic
350C2×C4⋊C4C22⋊D6112yes—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—614no—
353C2×C22⋊C4—612no—
354C2×C22⋊C4—1222no—
355C2×C22⋊C4—1228no—
356C2×C22⋊C4C22⋊D6111yescharacteristic
357C2×C22⋊C4C22⋊D6114yes—
358C2×C22⋊C4C22×D6111yescharacteristic
359C2×C42—1222no—
360C2×C42C22⋊D6111yescharacteristic
361#99—618no—
362#99—12216no—
363#95—1222no—
364#95—1221no—
365#95—1222no—
366#95—1222no—
367C24—22no—
368C24C2×C22⋊D6114yes—
369C24C2×C22⋊D6111yescharacteristic
370C24C23×D6111yescharacteristic
371#41—416no—
372#41—44no—
373#41—44no—
374C2×Q8—22no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4—24no—
378C2×D4—22no—
379C2×D4—28no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—28no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—44no—
388C2×D4—416no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—82no—
392C2×D4D4×D6114yes—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—1224no—
400C22×C4—1224no—
401C22×C4—1222no—
402C22×C4—1222no—
403C22×C4—1224no—
404C22×C4—1224no—
405C22×C4C2×C22⋊D6114yes—
406C22×C4C2×C22⋊D6112yes—
407C22×C4C2×C22⋊D6111yescharacteristic
408C22×C4C2×C22⋊D6111yescharacteristic
409C22×C4C2×C22⋊D6111yescharacteristic
410C22×C4C2×C22⋊D6111yescharacteristic
411C22×C4C2×C22⋊D6112yes—
412C22×C4C2×C22⋊D6111yescharacteristic
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24416no—
419C4⋊C4—2442no—
420C4⋊C4—2441no—
421C4⋊C4—2442no—
422C4⋊C4—2441no—
423C4⋊C4—2444no—
424C22⋊C4—24no—
425C22⋊C4—24no—
426C22⋊C4—22no—
427C22⋊C4—24no—
428C22⋊C4—22no—
429C22⋊C4—24no—
430C22⋊C4—618no—
431C22⋊C4—1224no—
432C22⋊C4—1224no—
433C22⋊C4—24416no—
434C22⋊C4—2444no—
435C42—21no—
436C42—21no—
437C42—2444no—
438C23—21no—
439C23—21no—
440C23—24no—
441C23—28no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—82no—
450C23—82no—
451C23#1451977312yes—
452C23C2×D4×D6112yes—
453C23C22×C22⋊D6111yescharacteristic
454C23C22×C22⋊D6112yes—
455C23#1451982212yes—
456Q8—44no—
457D4—28no—
458D4—44no—
459D4—48no—
460D4—44no—
461D4—44no—
462D4—48no—
463D4—44no—
464D4—82no—
465D4—82no—
466D4—82no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—24no—
474C2×C4—28no—
475C2×C4—42no—
476C2×C4—41no—
477C2×C4—48no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—81no—
483C2×C4—1228no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4—2442no—
487C2×C4—2442no—
488C2×C4—2442no—
489C2×C4—2442no—
490C2×C4—2448no—
491C2×C4—2444no—
492C2×C4—2444no—
493C2×C4#1451974611yescharacteristic
494C2×C4#1451974612yes—
495C2×C4#1451974611yescharacteristic
496C2×C4#1451977312yes—
497C22—21no—
498C22—22no—
499C22—21no—
500C22—22no—
501C22—22no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—41no—
506C22—41no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22—81no—
511C22—88no—
512C22—164no—
513C22C2×#1451974611yescharacteristic
514C22C2×#1451977311yescharacteristic
515C22#1684032911yescharacteristic
516C22D4×C22⋊D6114yes—
517C4—21no—
518C4—22no—
519C4—22no—
520C4—41no—
521C4—41no—
522C4—44no—
523C4—4882no—
524C4—4882no—
525C4—2448no—
526C4#1683995212yescomplemented
527C2—21no—
528C2—24no—
529C2—21no—
530C2—42no—
531C2—84no—
532C2#2144945211yescharacteristic
533C2C2×#1683995211yescharacteristic
534C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
Fitting, prime radicalC61×#3496C2
socleC22×C61C2×#120
hypercenter#3496D61
hypocenterC61#25772
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. C2×#16839952C2semidirect
2. #21450291C2semidirect
3. #21450127C2semidirect
4. #21446948C2semidirect
5. #21446656C2semidirect
6. C4×#14519692C2semidirect
9. C2×#14519758C22semidirect
10. C22×C4⋊Dic61C22semidirect
13. C2×#14519692C22semidirect
20. #16839623C22semidirect
22. #16839622C22semidirect
23. C4×C22⋊Dic61C22semidirect
24. #16839529C22semidirect
53. #14519692D4semidirect
162. C22⋊Dic61C2×D4semidirect
256. C4×C61#458semidirect
303. C61#25772semidirect
526. C4#16839952semidirect

Maximal subgroups

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

subgroupindex
C2×#168399522
#214502912
#214501272
#214469482
#214466562
C4×#145196922
C61×#34962
#2577261