Finite Group Databasebeta

← #29647753

Subgroups — #29647753

Order 15616.

1018 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454843C211yesmaximal, characteristic, complemented
2#21448792C211yesmaximal, characteristic, complemented
3#21444383C211yesmaximal, characteristic, complemented
4#21444379C211yesmaximal, characteristic, complemented
5#21444375C214yesmaximal, complemented
6#21444318C211yesmaximal, characteristic, complemented
7#21444314C211yesmaximal, characteristic, complemented
8#21444310C214yesmaximal, complemented
9C61×#3506C211yesmaximal, characteristic
10D4×#92418—22no—
11D4×#92418—22no—
12#16839815C2211yescharacteristic
13#16839815C2211yescharacteristic
14C2×C22⋊#92418C2214yes—
15#16839665—21no—
16#16839655—22no—
17#16839628C2211yescharacteristic
18#16839625C2212yes—
19C4⋊C4×Dic61—21no—
20C4×C4⋊Dic61C2211yescharacteristic
21C4×C22⋊Dic61C2212yes—
22#16839499C2212yes—
23#16839499C2214yes—
24#16839499C2212yes—
25#16839496C2212yes—
26#16839496C2214yes—
27#16839496C2212yes—
28C61×#559—22no—
29C61×#556C412yescomplemented
30C61×#555C412yescomplemented
31C61×#553—22no—
32C61×#550C412yescomplemented
33C61×#547C412yescomplemented
34C61×#538—22no—
35C61×#535—21no—
36C61×#533C2211yescharacteristic
37C61×#532C2211yescharacteristic
38C61×#525C2214yes—
39C61×#520—21no—
40C2×C4×D4×C61C2211yescharacteristic
41C22×#92418—24no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—21no—
44C2×C22⋊Dic61—22no—
45C2×C22⋊Dic61—22no—
46C2×C4×Dic61—21no—
47C2×C4×Dic61—22no—
48C4⋊#92418—21no—
49C4⋊#92418—21no—
50C22⋊#92418—24no—
51C22⋊#92418C2312yes—
52C22⋊#92418C2312yes—
53C4×#92418—21no—
54C4×#92418—21no—
55#14519684—22no—
56#14519684—21no—
57#14519684C2311yescharacteristic
58#14519684C2312yes—
59#14519684C2311yescharacteristic
60C2×#41×C61—21no—
61C2×#41×C61C2×C414yes—
62C22×D4×C61D411yescharacteristic
63C23×C4×C61C2312yes—
64C4⋊Q8×C61C2×C411yescharacteristic
65C4⋊Q8×C61C2×C411yescharacteristic
66C4⋊Q8×C61C2×C411yescharacteristic
67#127×C61—22no—
68#127×C61C2×C411yescharacteristic
69#126×C61—24no—
70#125×C61—21no—
71#124×C61—22no—
72#124×C61—24no—
73#124×C61—21no—
74#123×C61—24no—
75#123×C61—24no—
76C22⋊Q8×C61—21no—
77C22⋊Q8×C61C2×C414yes—
78C22⋊Q8×C61C2312yes—
79#121×C61—22no—
80#121×C61—22no—
81#121×C61—21no—
82#121×C61—24no—
83#121×C61C2×C414yes—
84#121×C61C2312yes—
85#120×C61—24no—
86C4×Q8×C61—22no—
87C4×Q8×C61—21no—
88C4×Q8×C61C2×C412yes—
89C4×D4×C61—24no—
90C4×D4×C61—24no—
91C4×D4×C61—22no—
92C4×D4×C61—21no—
93C4×D4×C61C2×C412yes—
94C4×D4×C61C2×C412yes—
95C4×D4×C61C2×C412yes—
96C4×D4×C61D412yes—
97C4×D4×C61D412yes—
98#117×C61—21no—
99#117×C61—22no—
100#117×C61C2×C414yes—
101C2×C4⋊C4×C61—22no—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C22⋊C4×C61D412yes—
104C2×C42×C61C2311yescharacteristic
105C24×C61C2×D412yes—
106#41×C61—42no—
107#41×C61—42no—
108#41×C61—84no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61C22×C412yes—
112C2×D4×C61—22no—
113C2×D4×C61—24no—
114C2×D4×C61—24no—
115C2×D4×C61—21no—
116C2×D4×C61—21no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61C22⋊C412yes—
120C2×D4×C61C22⋊C412yes—
121C2×D4×C61C22×C412yes—
122C22×C4×C61—22no—
123C22×C4×C61—24no—
124C22×C4×C61—82no—
125C22×C4×C61C22×C414yes—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D412yes—
128C22×C4×C61C2×D414yes—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C22×Dic61—21no—
132C22×Dic61—22no—
133C22×Dic61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—24no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61C22⋊C412yes—
142C4⋊C4×C61C22⋊C412yes—
143C4⋊C4×C61C22×C412yes—
144C4⋊C4×C61C22×C412yes—
145C22⋊C4×C61—24no—
146C22⋊C4×C61—24no—
147C22⋊C4×C61—22no—
148C22⋊C4×C61—24no—
149C22⋊C4×C61—22no—
150C22⋊C4×C61—22no—
151C22⋊C4×C61C22⋊C414yes—
152C22⋊C4×C61C22⋊C414yes—
153C42×C61—24no—
154C42×C61—22no—
155C42×C61C22×C411yescharacteristic
156C42×C61C22×C411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C22⋊Dic61—44no—
160C22⋊Dic61—82no—
161C4⋊Dic61—42no—
162C4⋊Dic61—42no—
163C4×Dic61—42no—
164C4×Dic61—82no—
165C2×#92418—22no—
166C2×#92418—22no—
167C2×#92418—84no—
168C23×C61—22no—
169C23×C61—22no—
170C23×C61—22no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61C2×C22⋊C414yes—
174C23×C61C22×D411yescharacteristic
175Q8×C61—42no—
176Q8×C61—42no—
177Q8×C61—81no—
178D4×C61—24no—
179D4×C61—24no—
180D4×C61—42no—
181D4×C61—42no—
182D4×C61—42no—
183D4×C61—42no—
184D4×C61—81no—
185D4×C61—84no—
186D4×C61—84no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—44no—
191C2×C4×C61—42no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—44no—
195C2×C4×C61—44no—
196C2×C4×C61—84no—
197C2×C4×C61C2×C22⋊C414yes—
198C2×C4×C61C2×C22⋊C411yescharacteristic
199C2×C4×C61C2×C22⋊C411yescharacteristic
200C2×C4×C61C2×C22⋊C411yescharacteristic
201C2×C4×C61C2×C22⋊C411yescharacteristic
202C2×C4×C61#13512yes—
203C2×C4×C61#13512yes—
204C2×C4×C61C23×C411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×Dic61—22no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—42no—
210C2×Dic61—41no—
211C2×Dic61—42no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—42no—
215#92418—42no—
216#92418—42no—
217#17617—611nomaximal, Sylow
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—24no—
221C22×C61—24no—
222C22×C61—24no—
223C22×C61—42no—
224C22×C61—42no—
225C22×C61—42no—
226C22×C61—82no—
227C22×C61—82no—
228C22×C61C22×C22⋊C411yescharacteristic
229C22×C61C2×#13511yescharacteristic
230C22×C61C2×#13511yescharacteristic
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—24no—
234C4×C61—21no—
235C4×C61—21no—
236C4×C61—42no—
237C4×C61—42no—
238C4×C61—42no—
239C4×C61#41612yes—
240C4×C61#41612yes—
241Dic61—42no—
242Dic61—81no—
243Dic61—82no—
244#3506D6111yescharacteristic
245#2972—611no—
246#2357—611no—
247#1518—611no—
248#1517—611no—
249#1516—614no—
250#1503—611no—
251#1502—611no—
252#1501—614no—
253C2×C61—21no—
254C2×C61—21no—
255C2×C61—44no—
256C2×C61—82no—
257C2×C61#288311yescharacteristic
258C2×C61C2×#41611yescharacteristic
259C2×C61#290111yescharacteristic
260#559—22no—
261#556Dic6112yescomplemented
262#555Dic6112yescomplemented
263#553—22no—
264#550Dic6112yescomplemented
265#547Dic6112yescomplemented
266#538—22no—
267#535—21no—
268#533C2×D6111yescharacteristic
269#532C2×D6111yescharacteristic
270#525C2×D6114yes—
271#520—21no—
272C2×C4×D4C2×D6111yescharacteristic
273C8×D4—1222no—
274C8×D4—1222no—
275#431—611no—
276#431—611no—
277C2×C22⋊C8—614no—
278#390—1221no—
279#388—1222no—
280#382—611no—
281#381—612no—
282C4×C4⋊C4—611no—
283C4×C4⋊C4—1221no—
284C4×C22⋊C4—612no—
285#328—612no—
286#328—614no—
287#328—612no—
288#327—612no—
289#327—614no—
290#327—612no—
291C61#1761711yescharacteristic, complemented, Sylow
292C2×#41—21no—
293C2×#41C2×Dic6114yes—
294C22×D4C22⋊D6111yescharacteristic
295C23×C4C22×D6112yes—
296C22×C8—1224no—
297C4⋊Q8C2×Dic6111yescharacteristic
298C4⋊Q8C2×Dic6111yescharacteristic
299C4⋊Q8C2×Dic6111yescharacteristic
300#127—22no—
301#127C2×Dic6111yescharacteristic
302#126—24no—
303#125—21no—
304#124—22no—
305#124—24no—
306#124—21no—
307#123—24no—
308#123—24no—
309C22⋊Q8—21no—
310C22⋊Q8C2×Dic6114yes—
311C22⋊Q8C22×D6112yes—
312#121—22no—
313#121—22no—
314#121—21no—
315#121—24no—
316#121C2×Dic6114yes—
317#121C22×D6112yes—
318#120—24no—
319C4×Q8—22no—
320C4×Q8—21no—
321C4×Q8C2×Dic6112yes—
322C4×D4—24no—
323C4×D4—24no—
324C4×D4—22no—
325C4×D4—21no—
326C4×D4C2×Dic6112yes—
327C4×D4C2×Dic6112yes—
328C4×D4C2×Dic6112yes—
329C4×D4C22⋊D6112yes—
330C4×D4C22⋊D6112yes—
331#117—21no—
332#117—22no—
333#117C2×Dic6114yes—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1221no—
337C2×C4⋊C4C22⋊D6111yescharacteristic
338C2×C22⋊C4—1222no—
339C2×C22⋊C4—1222no—
340C2×C22⋊C4C22⋊D6112yes—
341C2×C42—1222no—
342C2×C42—1221no—
343C2×C42C22×D6111yescharacteristic
344C4⋊C8—1221no—
345C4⋊C8—1221no—
346C22⋊C8—612no—
347C22⋊C8—612no—
348C22⋊C8—1224no—
349C4×C8—1221no—
350C4×C8—1221no—
351#95—611no—
352#95—612no—
353#95—611no—
354#95—1222no—
355#95—1221no—
356C24C2×C22⋊D6112yes—
357#41—42no—
358#41—42no—
359#41—84no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8C22×Dic6112yes—
363C2×D4—22no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4C22⋊Dic6112yes—
371C2×D4C22⋊Dic6112yes—
372C2×D4C22×Dic6112yes—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—82no—
376C22×C4—1221no—
377C22×C4—1222no—
378C22×C4—1221no—
379C22×C4C22×Dic6114yes—
380C22×C4C2×C22⋊D6114yes—
381C22×C4C2×C22⋊D6112yes—
382C22×C4C2×C22⋊D6111yescharacteristic
383C22×C4C2×C22⋊D6111yescharacteristic
384C22×C4C23×D6111yescharacteristic
385C2×C8—1222no—
386C2×C8—1222no—
387C2×C8—4884no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—24no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—2442no—
396C4⋊C4—2442no—
397C4⋊C4C22⋊Dic6112yes—
398C4⋊C4C22⋊Dic6112yes—
399C4⋊C4C22×Dic6112yes—
400C4⋊C4C22×Dic6112yes—
401C22⋊C4—22no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—22no—
405C22⋊C4—24no—
406C22⋊C4—22no—
407C22⋊C4—4882no—
408C22⋊C4—2444no—
409C22⋊C4C22⋊Dic6114yes—
410C22⋊C4C22⋊Dic6114yes—
411C42—24no—
412C42—22no—
413C42—4882no—
414C42—2442no—
415C42C22×Dic6111yescharacteristic
416C42C22×Dic6111yescharacteristic
417C42C2×C22⋊D6111yescharacteristic
418C42C2×C22⋊D6111yescharacteristic
419C23—22no—
420C23—22no—
421C23—22no—
422C23—42no—
423C23—42no—
424C23C2×C22⋊Dic6114yes—
425C23C22×C22⋊D6111yescharacteristic
426Q8—42no—
427Q8—42no—
428Q8—81no—
429D4—24no—
430D4—24no—
431D4—42no—
432D4—42no—
433D4—42no—
434D4—42no—
435D4—81no—
436D4—84no—
437D4—84no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—44no—
444C2×C4—41no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—84no—
448C2×C4—1222no—
449C2×C4—2442no—
450C2×C4—2441no—
451C2×C4—2442no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4—2442no—
455C2×C4—2441no—
456C2×C4—2441no—
457C2×C4C2×C22⋊Dic6114yes—
458C2×C4C2×C22⋊Dic6111yescharacteristic
459C2×C4C2×C22⋊Dic6111yescharacteristic
460C2×C4C2×C22⋊Dic6111yescharacteristic
461C2×C4C2×C22⋊Dic6111yescharacteristic
462C2×C4#1451979712yes—
463C2×C4#1451979712yes—
464C2×C4C23×Dic6111yescharacteristic
465C2×C4C22×C22⋊D6111yescharacteristic
466C8—2442no—
467C8—2442no—
468C22—21no—
469C22—21no—
470C22—24no—
471C22—24no—
472C22—24no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—82no—
477C22—82no—
478C22C22×C22⋊Dic6111yescharacteristic
479C22C2×#1451979711yescharacteristic
480C22C2×#1451979711yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—24no—
486C4—42no—
487C4—42no—
488C4—42no—
489C4—4882no—
490C4—4881no—
491C4—2442no—
492C4#1683975812yes—
493C4#1683975812yes—
494C2—21no—
495C2—21no—
496C2—44no—
497C2—82no—
498C2#2145428011yescharacteristic
499C2C2×#1683975811yescharacteristic
500C2#2145436611yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3506C2
socleC22×C61C22×C22⋊C4
cosocleC22×C4×C61C24
hypercenter#3506D61
hypocenterC61#17617
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #21454843C2semidirect
2. #21448792C2semidirect
3. #21444383C2semidirect
4. #21444379C2semidirect
5. #21444375C2semidirect
6. #21444318C2semidirect
7. #21444314C2semidirect
8. #21444310C2semidirect
29. C61×#556C4semidirect
30. C61×#555C4semidirect
32. C61×#550C4semidirect
33. C61×#547C4semidirect
261. #556Dic61semidirect
262. #555Dic61semidirect
264. #550Dic61semidirect
265. #547Dic61semidirect
291. C61#17617semidirect

Maximal subgroups

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

subgroupindex
#214548432
#214487922
#214443832
#214443792
#214443752
#214443182
#214443142
#214443102
C61×#35062
#1761761