Finite Group Databasebeta

← #29654926

Subgroups — #29654926

Order 15616.

994 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
1#21456960C211yesmaximal, characteristic, complemented
2#21456937C211yesmaximal, characteristic, complemented
3#21450811C211yesmaximal, characteristic, complemented
4#21444833C218yesmaximal, complemented
5#21444318C211yesmaximal, characteristic, complemented
6#21444314C211yesmaximal, characteristic, complemented
7#21444140C211yesmaximal, characteristic, complemented
8C61×#3506C211yesmaximal, characteristic
9C2×#14519777C2211yescharacteristic
10C2×#14519764—21no—
11#16840165—21no—
12#16840158—21no—
13#16840112—21no—
14#16840108—21no—
15#16840090C2214yes—
16#16840073C2214yes—
17#16840047—22no—
18#16840042—22no—
19#16839815C2211yescharacteristic
20#16839815C2211yescharacteristic
21#16839687C2214yes—
22#16839667—21no—
23#16839622—22no—
24C4×C4⋊Dic61C2211yescharacteristic
25#16839499C2214yes—
26#16839496C2214yes—
27#16839486C2214yes—
28C61×#559—22no—
29C61×#556—21no—
30C61×#555—21no—
31C61×#553—22no—
32C61×#550—21no—
33C61×#547—21no—
34C61×#538C2214yes—
35C61×#535—21no—
36C61×#533C2211yescharacteristic
37C61×#532C2211yescharacteristic
38C61×#525—22no—
39C61×#520—21no—
40C2×C4×D4×C61C2211yescharacteristic
41C22×Dic122—21no—
42#14519777—21no—
43#14519777—21no—
44#14519777—21no—
45#14519777—21no—
46#14519764—44no—
47C2×C4⋊Dic61—21no—
48C2×C4⋊Dic61—21no—
49C2×C4⋊Dic61C2312yes—
50C2×C22⋊Dic61—24no—
51C2×C4×Dic61—21no—
52#14519710—24no—
53#14519708—24no—
54C4⋊#92418—21no—
55C4⋊#92418—21no—
56#14519701—24no—
57#14519698—24no—
58C22⋊#92418C2312yes—
59C22⋊#92418C2312yes—
60C4×#92418—21no—
61C4×#92418—21no—
62#14519684—24no—
63#14519684C2312yes—
64C2×#41×C61—22no—
65C2×#41×C61—21no—
66C22×D4×C61D411yescharacteristic
67C23×C4×C61D412yes—
68C4⋊Q8×C61D411yescharacteristic
69C4⋊Q8×C61D411yescharacteristic
70C4⋊Q8×C61D411yescharacteristic
71#127×C61—22no—
72#127×C61D411yescharacteristic
73#126×C61—24no—
74#125×C61—21no—
75#124×C61—22no—
76#124×C61—24no—
77#124×C61—21no—
78#123×C61—24no—
79#123×C61—24no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—21no—
82C22⋊Q8×C61C2312yes—
83#121×C61—22no—
84#121×C61—22no—
85#121×C61—22no—
86#121×C61—24no—
87#121×C61—21no—
88#121×C61C2312yes—
89#120×C61—24no—
90C4×Q8×C61—22no—
91C4×Q8×C61—21no—
92C4×Q8×C61D412yes—
93C4×D4×C61—22no—
94C4×D4×C61—24no—
95C4×D4×C61—24no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C4×D4×C61D412yes—
102#117×C61—22no—
103#117×C61—22no—
104#117×C61—21no—
105C2×C4⋊C4×C61—22no—
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C22⋊C4×C61C2312yes—
108C2×C42×C61C2311yescharacteristic
109C24×C61C2×D412yes—
110#41×C61—42no—
111#41×C61—42no—
112#41×C61—84no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61C2×D412yes—
116C2×D4×C61—22no—
117C2×D4×C61—24no—
118C2×D4×C61—24no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61—21no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61C2×D412yes—
126C22×C4×C61—24no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—22no—
130C22×C4×C61—44no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D412yes—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—22no—
136C22×Dic61—22no—
137C2×Dic122—41no—
138C2×Dic122—41no—
139C2×Dic122—42no—
140C2×Dic122—42no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61C2×D412yes—
151C4⋊C4×C61C2×D412yes—
152C22⋊C4×C61—24no—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—24no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C42×C61—24no—
161C42×C61—22no—
162C42×C61C2×D411yescharacteristic
163C42×C61C2×D411yescharacteristic
164C42×C61C2×D411yescharacteristic
165C42×C61C2×D411yescharacteristic
166C22⋊Dic61—48no—
167C4⋊Dic61—22no—
168C4⋊Dic61—22no—
169C4⋊Dic61—42no—
170C4⋊Dic61—41no—
171C4⋊Dic61—41no—
172C4×Dic61—42no—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—42no—
180C23×C61—42no—
181C23×C61C22×D411yescharacteristic
182Q8×C61—42no—
183Q8×C61—42no—
184Q8×C61—42no—
185D4×C61—42no—
186D4×C61—42no—
187D4×C61—42no—
188D4×C61—42no—
189D4×C61—42no—
190D4×C61—42no—
191D4×C61—42no—
192D4×C61—84no—
193D4×C61—84no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—41no—
200C2×C4×C61—44no—
201C2×C4×C61—44no—
202C2×C4×C61—42no—
203C2×C4×C61—82no—
204C2×C4×C61—84no—
205C2×C4×C61#12011yescharacteristic
206C2×C4×C61#12011yescharacteristic
207C2×C4×C61#12011yescharacteristic
208C2×C4×C61#12011yescharacteristic
209C2×C4×C61#13512yes—
210C2×C4×C61AGL(1,ℤ/8)12yes—
211C2×C4×C61C22×D411yescharacteristic
212C2×C4×C61C22×D411yescharacteristic
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219Dic122—82no—
220Dic122—82no—
221#92418—42no—
222#92418—42no—
223#18412—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—44no—
227C22×C61—42no—
228C22×C61—42no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61—42no—
232C22×C61—42no—
233C22×C61—82no—
234C22×C61C2×#12011yescharacteristic
235C22×C61C2×#13511yescharacteristic
236C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
237C4×C61—21no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—42no—
241C4×C61—42no—
242C4×C61—42no—
243C4×C61—42no—
244C4×C61#44912yes—
245C4×C61#44912yes—
246C4×C61#45512yes—
247Dic61—82no—
248Dic61—82no—
249#3506D6111yescharacteristic
250#3229—611no—
251#3226—611no—
252#2547—611no—
253#1612—618no—
254#1503—611no—
255#1502—611no—
256#1460—611no—
257C2×C61—21no—
258C2×C61—21no—
259C2×C61—82no—
260C2×C61—82no—
261C2×C61C2×#44911yescharacteristic
262C2×C61C2×#45511yescharacteristic
263C2×C61#302811yescharacteristic
264#559—22no—
265#556—21no—
266#555—21no—
267#553—22no—
268#550—21no—
269#547—21no—
270#538C2×D6114yes—
271#535—21no—
272#533C2×D6111yescharacteristic
273#532C2×D6111yescharacteristic
274C2×C4⋊Q8—611no—
275C2×#124—1221no—
276#525—22no—
277#520—21no—
278C2×C4×D4C2×D6111yescharacteristic
279#500—1221no—
280#498—1221no—
281#487—1221no—
282#486—1221no—
283#483—614no—
284#480—614no—
285#475—1222no—
286#474—1222no—
287#431—611no—
288#431—611no—
289#397—614no—
290#391—1221no—
291#380—1222no—
292C4×C4⋊C4—611no—
293#328—614no—
294#327—614no—
295#323—614no—
296C61#1841211yescharacteristic, complemented, Sylow
297C2×#41—22no—
298C2×#41—21no—
299C22×Q8—1221no—
300C22×D4C22⋊D6111yescharacteristic
301C23×C4C22⋊D6112yes—
302C4⋊Q8—1221no—
303C4⋊Q8—1221no—
304C4⋊Q8—1221no—
305C4⋊Q8—1221no—
306C4⋊Q8C22⋊D6111yescharacteristic
307C4⋊Q8C22⋊D6111yescharacteristic
308C4⋊Q8C22⋊D6111yescharacteristic
309#127—22no—
310#127C22⋊D6111yescharacteristic
311#126—24no—
312#125—21no—
313#124—22no—
314#124—24no—
315#124—21no—
316#124—2444no—
317#123—24no—
318#123—24no—
319C22⋊Q8—22no—
320C22⋊Q8—21no—
321C22⋊Q8C22×D6112yes—
322#121—22no—
323#121—22no—
324#121—22no—
325#121—24no—
326#121—21no—
327#121C22×D6112yes—
328#120—24no—
329C4×Q8—22no—
330C4×Q8—21no—
331C4×Q8C22⋊D6112yes—
332C4×D4—22no—
333C4×D4—24no—
334C4×D4—24no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4—21no—
340C4×D4C22⋊D6112yes—
341#117—22no—
342#117—22no—
343#117—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—612no—
346C2×C4⋊C4—1221no—
347C2×C4⋊C4—1221no—
348C2×C4⋊C4C22⋊D6111yescharacteristic
349C2×C22⋊C4—1224no—
350C2×C22⋊C4C22×D6112yes—
351C2×C42—1221no—
352C2×C42C22×D6111yescharacteristic
353#107—1224no—
354#106—1224no—
355C4⋊C8—1221no—
356C4⋊C8—1221no—
357#103—1224no—
358#102—1224no—
359C22⋊C8—612no—
360C22⋊C8—612no—
361C4×C8—1221no—
362C4×C8—1221no—
363#95—612no—
364#95—1224no—
365C24C2×C22⋊D6112yes—
366#41—42no—
367#41—42no—
368#41—84no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—2441no—
372C2×Q8—2441no—
373C2×Q8—2442no—
374C2×Q8—2442no—
375C2×Q8C2×C22⋊D6112yes—
376C2×D4—22no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4C2×C22⋊D6112yes—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—44no—
391C22×C4—1222no—
392C22×C4—1222no—
393C22×C4C2×C22⋊D6111yescharacteristic
394C22×C4C2×C22⋊D6112yes—
395C22×C4C2×C22⋊D6111yescharacteristic
396C22×C4C23×D6111yescharacteristic
397C2×C8—1222no—
398C2×C8—1222no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C4⋊C4—2442no—
411C4⋊C4—2441no—
412C4⋊C4—2441no—
413C4⋊C4C2×C22⋊D6112yes—
414C4⋊C4C2×C22⋊D6112yes—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—2448no—
424C42—24no—
425C42—22no—
426C42—2442no—
427C42C2×C22⋊D6111yescharacteristic
428C42C2×C22⋊D6111yescharacteristic
429C42C2×C22⋊D6111yescharacteristic
430C42C2×C22⋊D6111yescharacteristic
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—42no—
436C23—42no—
437C23C22×C22⋊D6111yescharacteristic
438Q8—42no—
439Q8—42no—
440Q8—42no—
441Q8—4882no—
442Q8—4882no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—42no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—84no—
451D4—84no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—82no—
462C2×C4—84no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2442no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4#1451974611yescharacteristic
470C2×C4#1451974611yescharacteristic
471C2×C4#1451974611yescharacteristic
472C2×C4#1451974611yescharacteristic
473C2×C4#1451979712yes—
474C2×C4#1451980212yes—
475C2×C4C22×C22⋊D6111yescharacteristic
476C2×C4C22×C22⋊D6111yescharacteristic
477C8—2442no—
478C8—2442no—
479C22—21no—
480C22—21no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—82no—
489C22C2×#1451974611yescharacteristic
490C22C2×#1451979711yescharacteristic
491C22C2×#1451980211yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—4882no—
500C4—4882no—
501C4#1683990712yes—
502C4#1683990712yes—
503C4#1683994012yes—
504C2—21no—
505C2—21no—
506C2—82no—
507C2—82no—
508C2C2×#1683990711yescharacteristic
509C2C2×#1683994011yescharacteristic
510C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3506C2
socleC22×C61C2×#120
hypercenter#3506D61
hypocenterC61#18412
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #21456960C2semidirect
2. #21456937C2semidirect
3. #21450811C2semidirect
4. #21444833C2semidirect
5. #21444318C2semidirect
6. #21444314C2semidirect
7. #21444140C2semidirect
296. C61#18412semidirect

Maximal subgroups

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

subgroupindex
#214569602
#214569372
#214508112
#214448332
#214443182
#214443142
#214441402
C61×#35062
#1841261