Finite Group Databasebeta

← #29734222

Subgroups — #29734222

Order 15616.

1092 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456212C211yesmaximal, characteristic, complemented
2#21455634C211yesmaximal, characteristic, complemented
3#21454215C211yesmaximal, characteristic, complemented
4#21447040C214yesmaximal, complemented
5#21446809C211yesmaximal, characteristic, complemented
6#21446648C211yesmaximal, characteristic, complemented
7#21446045C214yesmaximal, complemented
8C4×#14519704C211yesmaximal, characteristic, complemented
9C61×#3511C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11C2×#14519775—22no—
12#16840121—22no—
13#16840112—21no—
14#16840108—21no—
15#16840057—21no—
16#16840052—21no—
17#16840047—21no—
18#16840042—21no—
19#16839805C2212yes—
20#16839786C2211yescharacteristic
21C2×#14519704C2212yes—
22C2×#14519704C2214yes—
23#16839763C2212yes—
24#16839763C2214yes—
25C4×#11758194C2211yescharacteristic
26#16839631—22no—
27#16839631C2214yes—
28C42×Dic61C2211yescharacteristic
29#16839531C2214yes—
30#16839521C2212yes—
31C61×#556—21no—
32C61×#555—21no—
33C61×#554—21no—
34C61×#553—22no—
35C61×#548—22no—
36C61×#547—21no—
37C61×#542—22no—
38C61×#537C2212yes—
39C61×#533—21no—
40C61×#532—21no—
41C61×#532C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#520C2212yes—
44C4×#41×C61C2211yescharacteristic
45C22×Dic122—21no—
46C2×#11758194C2312yes—
47C2×#11758194C2312yes—
48#14519777—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52#14519775—48no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2312yes—
55C2×#11758196—28no—
56C2×C4×Dic61—21no—
57C2×C4×Dic61—21no—
58C2×C4×Dic61C2312yes—
59#14519710—22no—
60#14519708—22no—
61C4⋊#92418—21no—
62C4⋊#92418—21no—
63#14519704—24no—
64#14519704—28no—
65#14519701—22no—
66#14519701—24no—
67#14519698—22no—
68#14519698—24no—
69#14519688—21no—
70C4×#92418—21no—
71C2×#41×C61—22no—
72C2×#41×C61C2312yes—
73C2×#41×C61C2311yescharacteristic
74C22×Q8×C61D412yes—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61—21no—
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61—21no—
79C4⋊Q8×C61D411yescharacteristic
80#127×C61D411yescharacteristic
81#126×C61—24no—
82#125×C61—22no—
83#124×C61—22no—
84#124×C61—24no—
85#124×C61D412yes—
86#123×C61—24no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—24no—
89C22⋊Q8×C61—22no—
90C22⋊Q8×C61—22no—
91#121×C61—22no—
92#121×C61—22no—
93#121×C61—22no—
94#121×C61—21no—
95#121×C61—21no—
96#121×C61—24no—
97C4×Q8×C61—22no—
98C4×Q8×C61—22no—
99C4×Q8×C61D411yescharacteristic
100C4×Q8×C61D411yescharacteristic
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61—21no—
104C4×D4×C61—24no—
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107C4×D4×C61—21no—
108C4×D4×C61D411yescharacteristic
109C4×D4×C61D411yescharacteristic
110#117×C61—22no—
111#117×C61—22no—
112#117×C61—21no—
113#117×C61C2311yescharacteristic
114C2×C4⋊C4×C61—22no—
115C2×C4⋊C4×C61D412yes—
116C2×C4⋊C4×C61C2312yes—
117C2×C42×C61—21no—
118C2×C42×C61C2311yescharacteristic
119#41×C61—24no—
120#41×C61—22no—
121#41×C61—24no—
122#41×C61—22no—
123#41×C61—84no—
124C2×Q8×C61—22no—
125C2×Q8×C61—22no—
126C2×Q8×C61—44no—
127C2×Q8×C61C2×D412yes—
128C2×Q8×C61C2×D411yescharacteristic
129C2×D4×C61—21no—
130C2×D4×C61—22no—
131C2×D4×C61—24no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61C2×D412yes—
135C2×D4×C61C2×D411yescharacteristic
136C22×C4×C61—24no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—22no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D414yes—
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—22no—
146C22×Dic61—22no—
147C2×Dic122—44no—
148C2×Dic122—41no—
149C2×Dic122—41no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—24no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61—22no—
160C4⋊C4×C61C2×D411yescharacteristic
161C4⋊C4×C61C2×D411yescharacteristic
162C22⋊C4×C61—24no—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—24no—
165C22⋊C4×C61—21no—
166C22⋊C4×C61—21no—
167C22⋊C4×C61—21no—
168C22⋊C4×C61—22no—
169C22⋊C4×C61—22no—
170C42×C61—24no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61C2×D411yescharacteristic
175C42×C61C2×D411yescharacteristic
176C42×C61C2×D411yescharacteristic
177C42×C61C2×D411yescharacteristic
178C4⋊Dic61—22no—
179C4⋊Dic61—22no—
180C4⋊Dic61—41no—
181C4⋊Dic61—41no—
182#11758196—416no—
183C4×Dic61—24no—
184C4×Dic61—24no—
185C4×Dic61—22no—
186C4×Dic61#4114yescomplemented
187#11758194—24no—
188#11758194—22no—
189#11758194—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C23×C61—21no—
193C23×C61—22no—
194C23×C61C22×D411yescharacteristic
195Q8×C61—21no—
196Q8×C61—21no—
197Q8×C61—24no—
198Q8×C61—44no—
199Q8×C61—82no—
200D4×C61—21no—
201D4×C61—21no—
202D4×C61—24no—
203D4×C61—44no—
204D4×C61—42no—
205D4×C61—42no—
206D4×C61—42no—
207D4×C61—84no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—22no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—24no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—42no—
219C2×C4×C61—42no—
220C2×C4×C61—44no—
221C2×C4×C61—84no—
222C2×C4×C61#12011yescharacteristic
223C2×C4×C61#12011yescharacteristic
224C2×C4×C61#12011yescharacteristic
225C2×C4×C61#12011yescharacteristic
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C2×#4112yes—
229C2×C4×C612−1+412yes—
230C2×Dic61—24no—
231C2×Dic61—24no—
232C2×Dic61—24no—
233C2×Dic61—44no—
234C2×Dic61—42no—
235Dic122—82no—
236Dic122—82no—
237#92418—42no—
238#92418—42no—
239#25666—611nomaximal, Sylow
240C22×C61—21no—
241C22×C61—21no—
242C22×C61—42no—
243C22×C61—41no—
244C22×C61—41no—
245C22×C61—42no—
246C22×C61—41no—
247C22×C61—42no—
248C22×C61C2×#12011yescharacteristic
249C22×C61#54812yes—
250C4×C61—21no—
251C4×C61—21no—
252C4×C61—42no—
253C4×C61—41no—
254C4×C61—41no—
255C4×C61—44no—
256C4×C61—41no—
257C4×C61#45312yes—
258C4×C61#45612yes—
259C4×C61#53211yescharacteristic
260C4×C61#53611yescharacteristic
261C4×C61#54712yes—
262Dic61—28no—
263Dic61—82no—
264#3511D6111yescharacteristic
265#3141—611no—
266#3079—611no—
267#2868—611no—
268#2039—614no—
269#1997—611no—
270#1962—611no—
271#1859—614no—
272C4×#104—611no—
273C2×C61—22no—
274C2×C61—81no—
275C2×C61—82no—
276C2×C61#243211yescharacteristic
277C2×C61C2×#45311yescharacteristic
278C2×C61C2×#45611yescharacteristic
279#556—21no—
280#555—21no—
281#554—21no—
282#553—22no—
283#548—22no—
284#547—21no—
285#542—22no—
286#537C2×D6112yes—
287#533—21no—
288#532—21no—
289#532C2×D6111yescharacteristic
290C2×C4⋊Q8—611no—
291C2×C4⋊Q8—1222no—
292C2×C4⋊Q8C2×D6111yescharacteristic
293#520C2×D6112yes—
294C4×#41C2×D6111yescharacteristic
295#489—1222no—
296#487—1221no—
297#486—1221no—
298#477—1221no—
299#476—1221no—
300#475—1221no—
301#474—1221no—
302#428—612no—
303#423—611no—
304C2×#104—614no—
305C2×#104—612no—
306#417—612no—
307#417—614no—
308C4×M16—611no—
309#384—614no—
310#384—1222no—
311C43—611no—
312#343—614no—
313#339—612no—
314C61#2566611yescharacteristic, complemented, Sylow
315C2×#41—22no—
316C2×#41C22×D6111yescharacteristic
317C2×#41C22×D6112yes—
318C22×Q8—1221no—
319C22×Q8C22⋊D6112yes—
320C2×M16—612no—
321C2×M16—612no—
322C4⋊Q8—21no—
323C4⋊Q8—21no—
324C4⋊Q8—21no—
325C4⋊Q8—21no—
326C4⋊Q8—1221no—
327C4⋊Q8—1221no—
328C4⋊Q8—1221no—
329C4⋊Q8—1221no—
330C4⋊Q8—2448no—
331C4⋊Q8C22⋊D6111yescharacteristic
332#127C22⋊D6111yescharacteristic
333#126—24no—
334#125—22no—
335#124—22no—
336#124—24no—
337#124C22⋊D6112yes—
338#123—24no—
339C22⋊Q8—22no—
340C22⋊Q8—24no—
341C22⋊Q8—22no—
342C22⋊Q8—22no—
343#121—22no—
344#121—22no—
345#121—22no—
346#121—21no—
347#121—21no—
348#121—24no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8C22⋊D6111yescharacteristic
352C4×Q8C22⋊D6111yescharacteristic
353C4×D4—22no—
354C4×D4—22no—
355C4×D4—21no—
356C4×D4—21no—
357C4×D4—24no—
358C4×D4—21no—
359C4×D4—21no—
360C4×D4C22⋊D6111yescharacteristic
361C4×D4C22⋊D6111yescharacteristic
362#117—22no—
363#117—22no—
364#117—21no—
365#117C22×D6111yescharacteristic
366C2×C4⋊C4—22no—
367C2×C4⋊C4—612no—
368C2×C4⋊C4—1221no—
369C2×C4⋊C4—1228no—
370C2×C4⋊C4C22⋊D6112yes—
371C2×C4⋊C4C22×D6112yes—
372C2×C42—21no—
373C2×C42—612no—
374C2×C42—1221no—
375C2×C42—1221no—
376C2×C42C22×D6111yescharacteristic
377#107—1222no—
378#106—1222no—
379C4⋊C8—1221no—
380C4⋊C8—1221no—
381#104—1228no—
382#104—1224no—
383#103—1222no—
384#103—1224no—
385#102—1222no—
386#102—1224no—
387#97—1221no—
388C4×C8—1221no—
389#41—24no—
390#41—22no—
391#41—24no—
392#41—22no—
393#41—84no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—44no—
397C2×Q8—2444no—
398C2×Q8—2441no—
399C2×Q8—2441no—
400C2×Q8C2×C22⋊D6112yes—
401C2×Q8C2×C22⋊D6111yescharacteristic
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4C2×C22⋊D6112yes—
408C2×D4C2×C22⋊D6111yescharacteristic
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—1222no—
415C22×C4—1222no—
416C22×C4C2×C22⋊D6111yescharacteristic
417C22×C4C2×C22⋊D6111yescharacteristic
418C22×C4C2×C22⋊D6114yes—
419C22×C4C23×D6111yescharacteristic
420M16—1224no—
421M16—1222no—
422M16—1222no—
423C2×C8—1222no—
424C2×C8—1222no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—1222no—
436C4⋊C4—1222no—
437C4⋊C4—24416no—
438C4⋊C4—2441no—
439C4⋊C4—2441no—
440C4⋊C4C2×C22⋊D6111yescharacteristic
441C4⋊C4C2×C22⋊D6111yescharacteristic
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C42—24no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—614no—
455C42—1224no—
456C42—1224no—
457C42—1222no—
458C42C2×C22⋊D6111yescharacteristic
459C42C2×C22⋊D6111yescharacteristic
460C42C2×C22⋊D6111yescharacteristic
461C42C2×C22⋊D6111yescharacteristic
462C23—21no—
463C23—22no—
464C23C22×C22⋊D6111yescharacteristic
465Q8—21no—
466Q8—21no—
467Q8—24no—
468Q8—44no—
469Q8—82no—
470Q8—4882no—
471Q8—4882no—
472D4—21no—
473D4—21no—
474D4—24no—
475D4—44no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—84no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—84no—
494C2×C4—1224no—
495C2×C4—1224no—
496C2×C4—1224no—
497C2×C4—2444no—
498C2×C4—2442no—
499C2×C4#1451974611yescharacteristic
500C2×C4#1451974611yescharacteristic
501C2×C4#1451974611yescharacteristic
502C2×C4#1451974611yescharacteristic
503C2×C4C22×C22⋊D6111yescharacteristic
504C2×C4C22×C22⋊D6111yescharacteristic
505C2×C4#41×D6112yes—
506C2×C4#1451982412yes—
507C8—2442no—
508C8—2442no—
509C22—21no—
510C22—21no—
511C22—41no—
512C22—42no—
513C22—41no—
514C22—42no—
515C22—42no—
516C22—41no—
517C22C2×#1451974611yescharacteristic
518C22#1684041912yes—
519C4—21no—
520C4—21no—
521C4—41no—
522C4—44no—
523C4—41no—
524C4—42no—
525C4—41no—
526C4—1228no—
527C4—4882no—
528C4#1683993012yes—
529C4#1683994512yes—
530C4#1684030811yescharacteristic
531C4#1684033311yescharacteristic
532C4#1684041112yes—
533C2—22no—
534C2—81no—
535C2—82no—
536C2#2144944311yescharacteristic
537C2C2×#1683993011yescharacteristic
538C2C2×#1683994511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3511C2
socleC22×C61C2×#120
hypercenter#3511D61
hypocenterC61#25666
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. #21456212C2semidirect
2. #21455634C2semidirect
3. #21454215C2semidirect
4. #21447040C2semidirect
5. #21446809C2semidirect
6. #21446648C2semidirect
7. #21446045C2semidirect
8. C4×#14519704C2semidirect
186. C4×Dic61#41semidirect
314. C61#25666semidirect

Maximal subgroups

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

subgroupindex
#214562122
#214556342
#214542152
#214470402
#214468092
#214466482
#214460452
C4×#145197042
C61×#35112
#2566661