Finite Group Databasebeta

← #29652728

Subgroups — #29652728

Order 15616.

998 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455739C212yesmaximal, complemented
2C2×#16839701C211yesmaximal, characteristic, complemented
3#21444861C212yesmaximal, complemented
4#21444856C212yesmaximal, complemented
5#21444850C214yesmaximal, complemented
6#21444323C212yesmaximal, complemented
7#21444149C211yesmaximal, characteristic, complemented
8C61×#3492C211yesmaximal, characteristic
9C22×C4⋊Dic61C2211yescharacteristic
10#16840095C2214yes—
11#16840090—22no—
12#16840078C2214yes—
13#16840073—22no—
14#16839805—22no—
15#16839763—22no—
16#16839726C2212yes—
17#16839701—21no—
18#16839701—21no—
19#16839701C2212yes—
20#16839701C2212yes—
21C2×#14519684—21no—
22C2×#14519684—21no—
23C2×#14519684C2211yescharacteristic
24#16839499C2214yes—
25#16839496C2214yes—
26#16839488C2212yes—
27#16839488C2212yes—
28C61×#554—22no—
29C61×#553—22no—
30C61×#552—22no—
31C61×#547—22no—
32C61×#544C2211yescharacteristic
33C61×#539—21no—
34C61×#539C2212yes—
35C61×#537C2211yescharacteristic
36C61×#525—21no—
37C61×#525—21no—
38C61×#525C2212yes—
39C61×#520—21no—
40C22×C4⋊C4×C61C2211yescharacteristic
41C23×Dic61—21no—
42C23×Dic61—21no—
43C2×#11758194—22no—
44C2×C4⋊Dic61—22no—
45C2×C4⋊Dic61—22no—
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61C2312yes—
48#14519710—24no—
49#14519708—24no—
50#14519701—24no—
51#14519698—24no—
52C22⋊#92418—22no—
53C22⋊#92418C2314yes—
54#14519684—21no—
55#14519684—22no—
56#14519684—22no—
57#14519684—22no—
58#14519684—22no—
59#14519684C2312yes—
60C2×#41×C61—21no—
61C2×#41×C61—21no—
62C2×#41×C61D412yes—
63C23×C4×C61D411yescharacteristic
64C23×C4×C61D411yescharacteristic
65C23×C4×C61C2311yescharacteristic
66C4⋊Q8×C61—22no—
67#126×C61—22no—
68#126×C61—22no—
69#125×C61—22no—
70#125×C61—22no—
71#125×C61—22no—
72#123×C61—22no—
73#123×C61—22no—
74#123×C61—22no—
75#123×C61—22no—
76#123×C61—22no—
77C22⋊Q8×C61—22no—
78C22⋊Q8×C61—21no—
79C22⋊Q8×C61—21no—
80C22⋊Q8×C61—21no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—21no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61D412yes—
85C22⋊Q8×C61C2312yes—
86#121×C61—21no—
87#121×C61—21no—
88#121×C61—21no—
89#121×C61—22no—
90#121×C61—21no—
91#121×C61D412yes—
92#121×C61C2312yes—
93#120×C61—22no—
94C4×Q8×C61—22no—
95C4×D4×C61—22no—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102#117×C61—21no—
103#117×C61—22no—
104#117×C61—21no—
105#117×C61D412yes—
106C2×C4⋊C4×C61—22no—
107C2×C4⋊C4×C61—22no—
108C2×C4⋊C4×C61—22no—
109C2×C4⋊C4×C61D411yescharacteristic
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C4⋊C4×C61C2311yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C24×C61C2×D411yescharacteristic
114#41×C61—42no—
115#41×C61—44no—
116#41×C61—42no—
117#41×C61—82no—
118C2×Q8×C61—21no—
119C2×Q8×C61—21no—
120C2×Q8×C61C2×D412yes—
121C2×D4×C61—21no—
122C2×D4×C61—22no—
123C2×D4×C61—22no—
124C2×D4×C61—22no—
125C2×D4×C61—21no—
126C2×D4×C61C2×D412yes—
127C22×C4×C61—22no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—22no—
132C22×C4×C61—21no—
133C22×C4×C61—22no—
134C22×C4×C61—22no—
135C22×C4×C61—41no—
136C22×C4×C61—42no—
137C22×C4×C61—42no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D412yes—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—22no—
147C22×Dic61—22no—
148C22×Dic61—42no—
149C22×Dic61—42no—
150C22×Dic61—44no—
151C22×Dic61—42no—
152C22×Dic61—41no—
153C22×Dic61—41no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61—22no—
161C4⋊C4×C61—22no—
162C4⋊C4×C61—22no—
163C4⋊C4×C61—22no—
164C4⋊C4×C61—22no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61C2×D412yes—
169C4⋊C4×C61C2×D412yes—
170C22⋊C4×C61—22no—
171C22⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—22no—
178C22⋊C4×C61—22no—
179C42×C61—22no—
180C42×C61—22no—
181C42×C61—22no—
182C4⋊Dic61—24no—
183C4⋊Dic61—42no—
184#11758194—44no—
185C2×#92418—24no—
186C23×C61—21no—
187C23×C61—21no—
188C23×C61—22no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61#12011yescharacteristic
192C23×C61#12011yescharacteristic
193C23×C61C22×D411yescharacteristic
194Q8×C61—42no—
195Q8×C61—42no—
196Q8×C61—81no—
197D4×C61—42no—
198D4×C61—42no—
199D4×C61—42no—
200D4×C61—42no—
201D4×C61—44no—
202D4×C61—44no—
203D4×C61—81no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—24no—
207C2×C4×C61—22no—
208C2×C4×C61—22no—
209C2×C4×C61—21no—
210C2×C4×C61—22no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—42no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—44no—
221C2×C4×C61—42no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—82no—
225C2×C4×C61—81no—
226C2×C4×C61—82no—
227C2×C4×C61#12012yes—
228C2×C4×C61#13712yes—
229C2×C4×C61C22×D412yes—
230C2×Dic61—42no—
231C2×Dic61—42no—
232C2×Dic61—44no—
233C2×Dic61—42no—
234C2×Dic61—42no—
235C2×Dic61—84no—
236C2×Dic61—82no—
237C2×Dic61—82no—
238#92418—44no—
239#18172—611nomaximal, Sylow
240C22×C61—24no—
241C22×C61—24no—
242C22×C61—21no—
243C22×C61—21no—
244C22×C61—21no—
245C22×C61—21no—
246C22×C61—21no—
247C22×C61—21no—
248C22×C61—42no—
249C22×C61—42no—
250C22×C61—42no—
251C22×C61#45412yes—
252C22×C61#45812yes—
253C22×C61C2×#12011yescharacteristic
254C22×C61C2×#13711yescharacteristic
255C22×C61#57511yescharacteristic
256C4×C61—22no—
257C4×C61—22no—
258C4×C61—22no—
259C4×C61—41no—
260C4×C61—42no—
261C4×C61—42no—
262C4×C61—41no—
263C4×C61—41no—
264C4×C61—42no—
265Dic61—82no—
266Dic61—82no—
267#3492D6111yescharacteristic
268#3092—612no—
269C2×#400—611no—
270#1617—612no—
271#1616—612no—
272#1615—614no—
273#1504—612no—
274#1463—611no—
275C2×C61—21no—
276C2×C61—21no—
277C2×C61—22no—
278C2×C61—22no—
279C2×C61—82no—
280C2×C61#300711yescharacteristic
281C2×C61C2×#45411yescharacteristic
282C2×C61C2×#45811yescharacteristic
283#554—22no—
284#553—22no—
285#552—22no—
286#547—22no—
287#544C2×D6111yescharacteristic
288#539—21no—
289#539C2×D6112yes—
290#537C2×D6111yescharacteristic
291#525—21no—
292#525—21no—
293#525C2×D6112yes—
294#520—21no—
295C22×C4⋊C4—611no—
296C22×C4⋊C4C2×D6111yescharacteristic
297#484—614no—
298#483—1222no—
299#481—614no—
300#480—1222no—
301#428—1222no—
302#417—1222no—
303#407—612no—
304#400—612no—
305#400—612no—
306#400—1221no—
307#400—1221no—
308C2×#95—611no—
309C2×#95—1221no—
310C2×#95—1221no—
311#328—614no—
312#327—614no—
313#324—612no—
314#324—612no—
315C61#1817211yescharacteristic, complemented, Sylow
316C2×#41—21no—
317C2×#41—21no—
318C2×#41C22⋊D6112yes—
319C23×C4—1221no—
320C23×C4—1221no—
321C23×C4C22⋊D6111yescharacteristic
322C23×C4C22⋊D6111yescharacteristic
323C23×C4C22×D6111yescharacteristic
324C2×M16—1222no—
325C4⋊Q8—22no—
326#126—22no—
327#126—22no—
328#125—22no—
329#125—22no—
330#125—22no—
331#123—22no—
332#123—22no—
333#123—22no—
334#123—22no—
335#123—22no—
336C22⋊Q8—21no—
337C22⋊Q8—21no—
338C22⋊Q8—21no—
339C22⋊Q8—22no—
340C22⋊Q8—22no—
341C22⋊Q8—21no—
342C22⋊Q8—22no—
343C22⋊Q8C22⋊D6112yes—
344C22⋊Q8C22×D6112yes—
345#121—21no—
346#121—21no—
347#121—21no—
348#121—22no—
349#121—21no—
350#121C22⋊D6112yes—
351#121C22×D6112yes—
352#120—22no—
353C4×Q8—22no—
354C4×D4—22no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4—22no—
359C4×D4—22no—
360C4×D4—22no—
361#117—22no—
362#117—21no—
363#117—21no—
364#117C22⋊D6112yes—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4—22no—
368C2×C4⋊C4—612no—
369C2×C4⋊C4—1222no—
370C2×C4⋊C4—1222no—
371C2×C4⋊C4—1221no—
372C2×C4⋊C4C22⋊D6111yescharacteristic
373C2×C4⋊C4C22⋊D6111yescharacteristic
374C2×C4⋊C4C22×D6111yescharacteristic
375C2×C4⋊C4C22×D6111yescharacteristic
376#107—1224no—
377#106—1224no—
378#103—1224no—
379#102—1224no—
380C22⋊C8—614no—
381C22⋊C8—1222no—
382#95—612no—
383#95—1221no—
384#95—1222no—
385#95—1222no—
386#95—1222no—
387#95—1222no—
388C24C2×C22⋊D6111yescharacteristic
389#41—42no—
390#41—44no—
391#41—42no—
392#41—82no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8C2×C22⋊D6112yes—
396C2×D4—21no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4C2×C22⋊D6112yes—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—1222no—
414C22×C4—1222no—
415C22×C4—2442no—
416C22×C4—2442no—
417C22×C4—2442no—
418C22×C4—2444no—
419C22×C4—2441no—
420C22×C4—2441no—
421C22×C4C2×C22⋊D6111yescharacteristic
422C22×C4C2×C22⋊D6112yes—
423C22×C4C2×C22⋊D6111yescharacteristic
424C22×C4C2×C22⋊D6111yescharacteristic
425C22×C4C2×C22⋊D6111yescharacteristic
426C22×C4C2×C22⋊D6112yes—
427C22×C4C2×C22⋊D6111yescharacteristic
428C22×C4C23×D6111yescharacteristic
429M16—2444no—
430C2×C8—1224no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—1224no—
446C4⋊C4—2442no—
447C4⋊C4C2×C22⋊D6112yes—
448C4⋊C4C2×C22⋊D6112yes—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C23—22no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23—22no—
466C23#1451974611yescharacteristic
467C23#1451974611yescharacteristic
468C23C22×C22⋊D6111yescharacteristic
469Q8—42no—
470Q8—42no—
471Q8—81no—
472D4—42no—
473D4—42no—
474D4—44no—
475D4—42no—
476D4—42no—
477D4—44no—
478D4—81no—
479C2×C4—24no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—4884no—
503C2×C4—4882no—
504C2×C4—4882no—
505C2×C4—2442no—
506C2×C4—2442no—
507C2×C4—2444no—
508C2×C4—2442no—
509C2×C4—2442no—
510C2×C4#1451974612yes—
511C2×C4#1451980712yes—
512C2×C4C22×C22⋊D6112yes—
513C8—2444no—
514C22—24no—
515C22—24no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22#1683993512yes—
526C22#1683995212yes—
527C22C2×#1451974611yescharacteristic
528C22C2×#1451980711yescharacteristic
529C22#1684056011yescharacteristic
530C4—22no—
531C4—22no—
532C4—22no—
533C4—42no—
534C4—41no—
535C4—41no—
536C4—42no—
537C4—41no—
538C4—42no—
539C4—4882no—
540C4—4882no—
541C2—21no—
542C2—22no—
543C2—21no—
544C2—22no—
545C2—82no—
546C2#2145507911yescharacteristic
547C2C2×#1683993511yescharacteristic
548C2C2×#1683995211yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3492C2
socleC22×C61C2×#120
hypercenter#3492D61
hypocenterC61#18172
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3492

Complements

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

NH ≅ G/Ntype
1. #21455739C2semidirect
2. C2×#16839701C2semidirect
3. #21444861C2semidirect
4. #21444856C2semidirect
5. #21444850C2semidirect
6. #21444323C2semidirect
7. #21444149C2semidirect
315. C61#18172semidirect

Maximal subgroups

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

subgroupindex
#214557392
C2×#168397012
#214448612
#214448562
#214448502
#214443232
#214441492
C61×#34922
#1817261