Finite Group Databasebeta

← #29655238

Subgroups — #29655238

Order 15616.

888 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457046C211yesmaximal, characteristic, complemented
2#21456944C211yesmaximal, characteristic, complemented
3#21451770C211yesmaximal, characteristic, complemented
4#21444856C214yesmaximal, complemented
5#21444850C214yesmaximal, complemented
6#21444327C211yesmaximal, characteristic, complemented
7#21444305C211yesmaximal, characteristic, complemented
8#21444175C211yesmaximal, characteristic, complemented
9C61×#3559C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840172—21no—
12#16840158—21no—
13#16840121—21no—
14#16840108—21no—
15#16840095C2212yes—
16#16840078C2214yes—
17#16840078C2212yes—
18#16840052—22no—
19#16840047—21no—
20#16840042—21no—
21#16839819C2211yescharacteristic
22#16839815C2211yescharacteristic
23#16839701—22no—
24#16839701C2214yes—
25#16839667—21no—
26#16839667—21no—
27#16839608C2211yescharacteristic
28#16839499C2212yes—
29#16839496C2212yes—
30#16839496C2214yes—
31#16839488C2214yes—
32C61×#563—21no—
33C61×#562—21no—
34C61×#554—21no—
35C61×#553—21no—
36C61×#553—21no—
37C61×#547—22no—
38C61×#547—21no—
39C61×#544—21no—
40C61×#540—21no—
41C61×#539—22no—
42C61×#539C2212yes—
43C61×#537C2212yes—
44C61×#532C2211yescharacteristic
45C61×#529C2211yescharacteristic
46C2×#125×C61C2211yescharacteristic
47C22×Dic122—21no—
48#14519777—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C4×Dic61—21no—
55#14519710—22no—
56#14519708—22no—
57#14519708—24no—
58C4⋊#92418—21no—
59C4⋊#92418—21no—
60#14519701—22no—
61#14519698—22no—
62#14519698—24no—
63C22⋊#92418C2312yes—
64C22⋊#92418C2312yes—
65#14519688—21no—
66C4×#92418—21no—
67#14519684—21no—
68#14519684—24no—
69#14519684—24no—
70#14519684C2312yes—
71C2×#41×C61—21no—
72C2×#41×C61—22no—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61—22no—
75C4⋊Q8×C61D411yescharacteristic
76#127×C61D411yescharacteristic
77#126×C61—22no—
78#126×C61—22no—
79#126×C61—21no—
80#125×C61—21no—
81#125×C61—22no—
82#125×C61—21no—
83#125×C61—21no—
84#125×C61—21no—
85#125×C61—21no—
86#125×C61D411yescharacteristic
87#124×C61—21no—
88#124×C61D411yescharacteristic
89#123×C61—22no—
90#123×C61—22no—
91#123×C61—22no—
92#123×C61—22no—
93#123×C61—22no—
94#123×C61—22no—
95#123×C61—21no—
96C22⋊Q8×C61—22no—
97C22⋊Q8×C61—22no—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61—22no—
100C22⋊Q8×C61—22no—
101C22⋊Q8×C61—22no—
102C22⋊Q8×C61C2311yescharacteristic
103#121×C61—21no—
104#121×C61—22no—
105#121×C61—21no—
106#121×C61—22no—
107#121×C61—22no—
108#121×C61—21no—
109#121×C61—22no—
110#121×C61C2312yes—
111#121×C61C2311yescharacteristic
112C4×Q8×C61D411yescharacteristic
113C4×D4×C61—22no—
114C4×D4×C61—21no—
115C4×D4×C61—21no—
116C4×D4×C61—22no—
117C4×D4×C61—21no—
118C4×D4×C61—21no—
119C4×D4×C61D412yes—
120C4×D4×C61D411yescharacteristic
121#117×C61—21no—
122#117×C61—21no—
123C2×C4⋊C4×C61—22no—
124C2×C4⋊C4×C61—21no—
125C2×C4⋊C4×C61D412yes—
126C2×C4⋊C4×C61D411yescharacteristic
127C2×C4⋊C4×C61D411yescharacteristic
128C2×C4⋊C4×C61C2312yes—
129C2×C42×C61C2311yescharacteristic
130#41×C61—82no—
131#41×C61—84no—
132C2×Q8×C61—22no—
133C2×Q8×C61—21no—
134C2×Q8×C61C2×D411yescharacteristic
135C2×D4×C61—22no—
136C2×D4×C61—21no—
137C2×D4×C61—22no—
138C2×D4×C61—22no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61C2×D411yescharacteristic
142C2×D4×C61C2×D412yes—
143C22×C4×C61—22no—
144C22×C4×C61—22no—
145C22×C4×C61—22no—
146C22×C4×C61—21no—
147C22×C4×C61—22no—
148C22×C4×C61C2×D412yes—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D412yes—
152C22×C4×C61C2411yescharacteristic
153C22×Dic61—22no—
154C22×Dic61—22no—
155C2×Dic122—44no—
156C2×Dic122—41no—
157C2×Dic122—41no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61C2×D412yes—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C22⋊C4×C61—22no—
179C22⋊C4×C61—22no—
180C22⋊C4×C61—22no—
181C22⋊C4×C61—22no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—21no—
186C22⋊C4×C61—21no—
187C22⋊C4×C61—22no—
188C22⋊C4×C61—21no—
189C22⋊C4×C61—22no—
190C42×C61—22no—
191C42×C61—21no—
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D411yescharacteristic
194C42×C61C2×D411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198C4⋊Dic61—41no—
199C4⋊Dic61—41no—
200C4×Dic61—42no—
201C2×#92418—22no—
202C2×#92418—22no—
203C23×C61—21no—
204C23×C61—22no—
205C23×C61C22×D411yescharacteristic
206Q8×C61—41no—
207Q8×C61—81no—
208Q8×C61—82no—
209D4×C61—42no—
210D4×C61—41no—
211D4×C61—44no—
212D4×C61—42no—
213D4×C61—42no—
214D4×C61—81no—
215D4×C61—82no—
216D4×C61—82no—
217C2×C4×C61—22no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—21no—
224C2×C4×C61—22no—
225C2×C4×C61—24no—
226C2×C4×C61—41no—
227C2×C4×C61—42no—
228C2×C4×C61—42no—
229C2×C4×C61—82no—
230C2×C4×C61—82no—
231C2×C4×C61—82no—
232C2×C4×C61—82no—
233C2×C4×C61#12011yescharacteristic
234C2×C4×C61#12011yescharacteristic
235C2×C4×C61#12011yescharacteristic
236C2×C4×C61#12011yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×Dic61—42no—
240C2×Dic61—44no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244Dic122—82no—
245Dic122—82no—
246#92418—42no—
247#92418—42no—
248#18441—611nomaximal, Sylow
249C22×C61—21no—
250C22×C61—21no—
251C22×C61—42no—
252C22×C61—41no—
253C22×C61—41no—
254C22×C61—42no—
255C22×C61—42no—
256C22×C61—41no—
257C22×C61C2×#12011yescharacteristic
258C22×C61#57712yes—
259C4×C61—21no—
260C4×C61—21no—
261C4×C61—21no—
262C4×C61—21no—
263C4×C61—21no—
264C4×C61—42no—
265C4×C61—42no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61—42no—
269C4×C61—41no—
270C4×C61#45412yes—
271Dic61—82no—
272Dic61—82no—
273#3559D6111yescharacteristic
274#3239—611no—
275#3227—611no—
276#2642—611no—
277#1616—614no—
278#1615—614no—
279#1505—611no—
280#1500—611no—
281#1470—611no—
282C2×C61—22no—
283C2×C61—81no—
284C2×C61—82no—
285C2×C61#300911yescharacteristic
286C2×C61C2×#45411yescharacteristic
287C2×C61#302811yescharacteristic
288#563—21no—
289#562—21no—
290#554—21no—
291#553—21no—
292#553—21no—
293#547—22no—
294#547—21no—
295#544—21no—
296#540—21no—
297#539—22no—
298#539C2×D6112yes—
299#537C2×D6112yes—
300#532C2×D6111yescharacteristic
301C2×C4⋊Q8—611no—
302#529C2×D6111yescharacteristic
303C2×#125C2×D6111yescharacteristic
304#501—1221no—
305#498—1221no—
306#489—1221no—
307#486—1221no—
308#484—612no—
309#481—614no—
310#481—612no—
311#476—1222no—
312#475—1221no—
313#474—1221no—
314#432—611no—
315#431—611no—
316#400—614no—
317#400—1222no—
318#391—1221no—
319#391—1221no—
320#376—611no—
321#328—612no—
322#327—614no—
323#327—612no—
324#324—614no—
325C61#1844111yescharacteristic, complemented, Sylow
326C2×#41—21no—
327C2×#41—22no—
328C22×Q8—1221no—
329C4⋊Q8—21no—
330C4⋊Q8—22no—
331C4⋊Q8—1221no—
332C4⋊Q8—1221no—
333C4⋊Q8—1221no—
334C4⋊Q8—1221no—
335C4⋊Q8C22⋊D6111yescharacteristic
336#127C22⋊D6111yescharacteristic
337#126—22no—
338#126—22no—
339#126—21no—
340#125—21no—
341#125—22no—
342#125—21no—
343#125—21no—
344#125—21no—
345#125—21no—
346#125C22⋊D6111yescharacteristic
347#124—21no—
348#124C22⋊D6111yescharacteristic
349#123—22no—
350#123—22no—
351#123—22no—
352#123—22no—
353#123—22no—
354#123—21no—
355#123—22no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8C22×D6111yescharacteristic
363#121—21no—
364#121—22no—
365#121—21no—
366#121—22no—
367#121—22no—
368#121—21no—
369#121—22no—
370#121C22×D6111yescharacteristic
371#121C22×D6112yes—
372C4×Q8C22⋊D6111yescharacteristic
373C4×D4—22no—
374C4×D4—21no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—22no—
378C4×D4—21no—
379C4×D4C22⋊D6111yescharacteristic
380C4×D4C22⋊D6112yes—
381#117—21no—
382#117—21no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—612no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4C22⋊D6111yescharacteristic
388C2×C4⋊C4C22⋊D6112yes—
389C2×C4⋊C4C22⋊D6111yescharacteristic
390C2×C4⋊C4C22×D6112yes—
391C2×C42—1221no—
392C2×C42C22×D6111yescharacteristic
393#107—1222no—
394#106—1222no—
395#106—1224no—
396C4⋊C8—1221no—
397C4⋊C8—1221no—
398#103—1222no—
399#102—1222no—
400#102—1224no—
401C22⋊C8—612no—
402C22⋊C8—612no—
403#97—1221no—
404C4×C8—1221no—
405#95—612no—
406#95—1221no—
407#95—1224no—
408#95—1224no—
409#41—82no—
410#41—84no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—2444no—
414C2×Q8—2441no—
415C2×Q8—2441no—
416C2×Q8C2×C22⋊D6111yescharacteristic
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4C2×C22⋊D6111yescharacteristic
424C2×D4C2×C22⋊D6112yes—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—1222no—
431C22×C4—1222no—
432C22×C4C2×C22⋊D6112yes—
433C22×C4C2×C22⋊D6111yescharacteristic
434C22×C4C2×C22⋊D6111yescharacteristic
435C22×C4C2×C22⋊D6112yes—
436C22×C4C23×D6111yescharacteristic
437C2×C8—1222no—
438C2×C8—1222no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—1222no—
457C4⋊C4—1222no—
458C4⋊C4—2441no—
459C4⋊C4—2441no—
460C4⋊C4C2×C22⋊D6112yes—
461C4⋊C4C2×C22⋊D6111yescharacteristic
462C4⋊C4C2×C22⋊D6111yescharacteristic
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—22no—
471C22⋊C4—21no—
472C22⋊C4—21no—
473C22⋊C4—21no—
474C22⋊C4—22no—
475C42—22no—
476C42—21no—
477C42—2442no—
478C42C2×C22⋊D6111yescharacteristic
479C42C2×C22⋊D6111yescharacteristic
480C42C2×C22⋊D6111yescharacteristic
481C42C2×C22⋊D6111yescharacteristic
482C23—21no—
483C23—22no—
484C23C22×C22⋊D6111yescharacteristic
485Q8—41no—
486Q8—81no—
487Q8—82no—
488Q8—4882no—
489Q8—4882no—
490D4—42no—
491D4—41no—
492D4—44no—
493D4—42no—
494D4—42no—
495D4—81no—
496D4—82no—
497D4—82no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—2442no—
515C2×C4—2442no—
516C2×C4—2444no—
517C2×C4—2442no—
518C2×C4—2442no—
519C2×C4#1451974611yescharacteristic
520C2×C4#1451974611yescharacteristic
521C2×C4#1451974611yescharacteristic
522C2×C4#1451974611yescharacteristic
523C2×C4C22×C22⋊D6111yescharacteristic
524C2×C4C22×C22⋊D6111yescharacteristic
525C8—2442no—
526C8—2442no—
527C22—21no—
528C22—21no—
529C22—41no—
530C22—42no—
531C22—42no—
532C22—41no—
533C22—42no—
534C22—41no—
535C22C2×#1451974611yescharacteristic
536C22#1684057312yes—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—41no—
543C4—42no—
544C4—41no—
545C4—42no—
546C4—42no—
547C4—41no—
548C4—4882no—
549C4—4882no—
550C4#1683993512yes—
551C2—22no—
552C2—81no—
553C2—82no—
554C2#2145509611yescharacteristic
555C2C2×#1683993511yescharacteristic
556C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3559C2
socleC22×C61C2×#120
hypercenter#3559D61
hypocenterC61#18441
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3559

Complements

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

NH ≅ G/Ntype
1. #21457046C2semidirect
2. #21456944C2semidirect
3. #21451770C2semidirect
4. #21444856C2semidirect
5. #21444850C2semidirect
6. #21444327C2semidirect
7. #21444305C2semidirect
8. #21444175C2semidirect
325. C61#18441semidirect

Maximal subgroups

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

subgroupindex
#214570462
#214569442
#214517702
#214448562
#214448502
#214443272
#214443052
#214441752
C61×#35592
#1844161