Finite Group Databasebeta

← #29734231

Subgroups — #29734231

Order 15616.

1020 conjugacy classes of subgroups.

The subgroup lattice has 536 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#21454196C211yesmaximal, characteristic, complemented
4#21447045C214yesmaximal, complemented
5#21446809C211yesmaximal, characteristic, complemented
6#21446648C211yesmaximal, characteristic, complemented
7#21446052C214yesmaximal, complemented
8C4×#14519704C211yesmaximal, characteristic, complemented
9C61×#3511C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840121—22no—
12#16840112—21no—
13#16840108—21no—
14#16840057—21no—
15#16840052—21no—
16#16840047—21no—
17#16840042—21no—
18#16839805C2212yes—
19#16839786C2211yescharacteristic
20C2×#14519704C2212yes—
21C2×#14519704C2214yes—
22#16839763C2212yes—
23#16839763C2214yes—
24C4×#11758194C2211yescharacteristic
25#16839669—22no—
26#16839667—22no—
27#16839627C2214yes—
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—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C4×Dic61—21no—
55C2×C4×Dic61—21no—
56C2×C4×Dic61C2312yes—
57#14519710—22no—
58#14519708—22no—
59C4⋊#92418—21no—
60C4⋊#92418—21no—
61#14519704—24no—
62#14519704—28no—
63#14519701—22no—
64#14519701—24no—
65#14519698—22no—
66#14519698—24no—
67#14519688—21no—
68C4×#92418—21no—
69#14519684—28no—
70C2×#41×C61—22no—
71C2×#41×C61C2312yes—
72C2×#41×C61C2311yescharacteristic
73C22×Q8×C61D412yes—
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61—21no—
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61D411yescharacteristic
79#127×C61D411yescharacteristic
80#126×C61—24no—
81#125×C61—22no—
82#124×C61—22no—
83#124×C61—24no—
84#124×C61D412yes—
85#123×C61—24no—
86C22⋊Q8×C61—22no—
87C22⋊Q8×C61—24no—
88C22⋊Q8×C61—22no—
89C22⋊Q8×C61—22no—
90#121×C61—22no—
91#121×C61—22no—
92#121×C61—22no—
93#121×C61—21no—
94#121×C61—21no—
95#121×C61—24no—
96C4×Q8×C61—22no—
97C4×Q8×C61—22no—
98C4×Q8×C61D411yescharacteristic
99C4×Q8×C61D411yescharacteristic
100C4×D4×C61—22no—
101C4×D4×C61—22no—
102C4×D4×C61—21no—
103C4×D4×C61—24no—
104C4×D4×C61—21no—
105C4×D4×C61—21no—
106C4×D4×C61—21no—
107C4×D4×C61D411yescharacteristic
108C4×D4×C61D411yescharacteristic
109#117×C61—22no—
110#117×C61—22no—
111#117×C61—21no—
112#117×C61C2311yescharacteristic
113C2×C4⋊C4×C61—22no—
114C2×C4⋊C4×C61D412yes—
115C2×C4⋊C4×C61C2312yes—
116C2×C42×C61—21no—
117C2×C42×C61C2311yescharacteristic
118#41×C61—24no—
119#41×C61—22no—
120#41×C61—24no—
121#41×C61—22no—
122#41×C61—84no—
123C2×Q8×C61—22no—
124C2×Q8×C61—22no—
125C2×Q8×C61—44no—
126C2×Q8×C61C2×D412yes—
127C2×Q8×C61C2×D411yescharacteristic
128C2×D4×C61—21no—
129C2×D4×C61—22no—
130C2×D4×C61—24no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61C2×D412yes—
134C2×D4×C61C2×D411yescharacteristic
135C22×C4×C61—24no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—22no—
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D414yes—
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—22no—
145C22×Dic61—22no—
146C2×Dic122—44no—
147C2×Dic122—41no—
148C2×Dic122—41no—
149C4⋊C4×C61—24no—
150C4⋊C4×C61—24no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61—21no—
155C4⋊C4×C61—21no—
156C4⋊C4×C61—21no—
157C4⋊C4×C61—22no—
158C4⋊C4×C61—22no—
159C4⋊C4×C61C2×D411yescharacteristic
160C4⋊C4×C61C2×D411yescharacteristic
161C22⋊C4×C61—24no—
162C22⋊C4×C61—24no—
163C22⋊C4×C61—24no—
164C22⋊C4×C61—21no—
165C22⋊C4×C61—21no—
166C22⋊C4×C61—21no—
167C22⋊C4×C61—22no—
168C22⋊C4×C61—22no—
169C42×C61—24no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61C2×D411yescharacteristic
174C42×C61C2×D411yescharacteristic
175C42×C61C2×D411yescharacteristic
176C42×C61C2×D411yescharacteristic
177C4⋊Dic61—22no—
178C4⋊Dic61—22no—
179C4⋊Dic61—41no—
180C4⋊Dic61—41no—
181C4×Dic61—22no—
182C4×Dic61—42no—
183C4×Dic61—42no—
184C4×Dic61#4114yescomplemented
185#11758194—24no—
186#11758194—22no—
187#11758194—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C23×C61—21no—
191C23×C61—22no—
192C23×C61C22×D411yescharacteristic
193Q8×C61—21no—
194Q8×C61—21no—
195Q8×C61—24no—
196Q8×C61—44no—
197Q8×C61—82no—
198D4×C61—21no—
199D4×C61—21no—
200D4×C61—24no—
201D4×C61—44no—
202D4×C61—42no—
203D4×C61—42no—
204D4×C61—42no—
205D4×C61—84no—
206C2×C4×C61—22no—
207C2×C4×C61—22no—
208C2×C4×C61—21no—
209C2×C4×C61—22no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—24no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—42no—
216C2×C4×C61—42no—
217C2×C4×C61—42no—
218C2×C4×C61—44no—
219C2×C4×C61—84no—
220C2×C4×C61#12011yescharacteristic
221C2×C4×C61#12011yescharacteristic
222C2×C4×C61#12011yescharacteristic
223C2×C4×C61#12011yescharacteristic
224C2×C4×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61C2×#4112yes—
227C2×C4×C612−1+412yes—
228C2×Dic61—24no—
229C2×Dic61—42no—
230C2×Dic61—42no—
231C2×Dic61—44no—
232C2×Dic61—42no—
233Dic122—82no—
234Dic122—82no—
235#92418—42no—
236#92418—42no—
237#25667—611nomaximal, Sylow
238C22×C61—21no—
239C22×C61—21no—
240C22×C61—42no—
241C22×C61—41no—
242C22×C61—41no—
243C22×C61—42no—
244C22×C61—41no—
245C22×C61—42no—
246C22×C61C2×#12011yescharacteristic
247C22×C61#54812yes—
248C4×C61—21no—
249C4×C61—21no—
250C4×C61—42no—
251C4×C61—41no—
252C4×C61—41no—
253C4×C61—44no—
254C4×C61—41no—
255C4×C61#45412yes—
256C4×C61#45512yes—
257C4×C61#53211yescharacteristic
258C4×C61#53611yescharacteristic
259C4×C61#54712yes—
260Dic61—44no—
261Dic61—82no—
262#3511D6111yescharacteristic
263#3141—611no—
264#3079—611no—
265#2862—611no—
266#2040—614no—
267#1997—611no—
268#1962—611no—
269#1860—614no—
270C4×#104—611no—
271C2×C61—22no—
272C2×C61—81no—
273C2×C61—82no—
274C2×C61#243211yescharacteristic
275C2×C61C2×#45511yescharacteristic
276C2×C61C2×#45411yescharacteristic
277#556—21no—
278#555—21no—
279#554—21no—
280#553—22no—
281#548—22no—
282#547—21no—
283#542—22no—
284#537C2×D6112yes—
285#533—21no—
286#532—21no—
287#532C2×D6111yescharacteristic
288C2×C4⋊Q8—611no—
289C2×C4⋊Q8C2×D6111yescharacteristic
290#520C2×D6112yes—
291C4×#41C2×D6111yescharacteristic
292#489—1222no—
293#487—1221no—
294#486—1221no—
295#477—1221no—
296#476—1221no—
297#475—1221no—
298#474—1221no—
299#428—612no—
300#423—611no—
301C2×#104—614no—
302C2×#104—612no—
303#417—612no—
304#417—614no—
305C4×M16—611no—
306#391—1222no—
307#391—1222no—
308#382—614no—
309C43—611no—
310#343—614no—
311#339—612no—
312C61#2566711yescharacteristic, complemented, Sylow
313C2×#41—22no—
314C2×#41C22×D6111yescharacteristic
315C2×#41C22×D6112yes—
316C22×Q8—1221no—
317C22×Q8C22⋊D6112yes—
318C2×M16—612no—
319C2×M16—612no—
320C4⋊Q8—21no—
321C4⋊Q8—21no—
322C4⋊Q8—21no—
323C4⋊Q8—21no—
324C4⋊Q8—1221no—
325C4⋊Q8—1221no—
326C4⋊Q8—1221no—
327C4⋊Q8—1221no—
328C4⋊Q8C22⋊D6111yescharacteristic
329#127C22⋊D6111yescharacteristic
330#126—24no—
331#125—22no—
332#124—22no—
333#124—24no—
334#124C22⋊D6112yes—
335#123—24no—
336C22⋊Q8—22no—
337C22⋊Q8—24no—
338C22⋊Q8—22no—
339C22⋊Q8—22no—
340#121—22no—
341#121—22no—
342#121—22no—
343#121—21no—
344#121—21no—
345#121—24no—
346C4×Q8—22no—
347C4×Q8—22no—
348C4×Q8C22⋊D6111yescharacteristic
349C4×Q8C22⋊D6111yescharacteristic
350C4×D4—22no—
351C4×D4—22no—
352C4×D4—21no—
353C4×D4—21no—
354C4×D4—24no—
355C4×D4—21no—
356C4×D4—21no—
357C4×D4C22⋊D6111yescharacteristic
358C4×D4C22⋊D6111yescharacteristic
359#117—22no—
360#117—22no—
361#117—21no—
362#117C22×D6111yescharacteristic
363C2×C4⋊C4—22no—
364C2×C4⋊C4—612no—
365C2×C4⋊C4—1221no—
366C2×C4⋊C4C22⋊D6112yes—
367C2×C4⋊C4C22×D6112yes—
368C2×C42—21no—
369C2×C42—612no—
370C2×C42—1221no—
371C2×C42—1221no—
372C2×C42C22×D6111yescharacteristic
373#107—1222no—
374#106—1222no—
375C4⋊C8—1221no—
376C4⋊C8—1221no—
377#104—1228no—
378#104—1224no—
379#103—1222no—
380#103—1224no—
381#102—1222no—
382#102—1224no—
383#97—1221no—
384C4×C8—1221no—
385#95—1228no—
386#41—24no—
387#41—22no—
388#41—24no—
389#41—22no—
390#41—84no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—44no—
394C2×Q8—2444no—
395C2×Q8—2441no—
396C2×Q8—2441no—
397C2×Q8C2×C22⋊D6112yes—
398C2×Q8C2×C22⋊D6111yescharacteristic
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—24no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4C2×C22⋊D6112yes—
405C2×D4C2×C22⋊D6111yescharacteristic
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—1222no—
412C22×C4—1222no—
413C22×C4C2×C22⋊D6111yescharacteristic
414C22×C4C2×C22⋊D6111yescharacteristic
415C22×C4C2×C22⋊D6114yes—
416C22×C4C23×D6111yescharacteristic
417M16—1224no—
418M16—1222no—
419M16—1222no—
420C2×C8—1222no—
421C2×C8—1222no—
422C4⋊C4—24no—
423C4⋊C4—24no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—1222no—
433C4⋊C4—1222no—
434C4⋊C4—2441no—
435C4⋊C4—2441no—
436C4⋊C4C2×C22⋊D6111yescharacteristic
437C4⋊C4C2×C22⋊D6111yescharacteristic
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—21no—
442C22⋊C4—21no—
443C22⋊C4—21no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C42—24no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—614no—
451C42—1222no—
452C42—2442no—
453C42—2442no—
454C42C2×C22⋊D6111yescharacteristic
455C42C2×C22⋊D6111yescharacteristic
456C42C2×C22⋊D6111yescharacteristic
457C42C2×C22⋊D6111yescharacteristic
458C23—21no—
459C23—22no—
460C23C22×C22⋊D6111yescharacteristic
461Q8—21no—
462Q8—21no—
463Q8—24no—
464Q8—44no—
465Q8—82no—
466Q8—4882no—
467Q8—4882no—
468D4—21no—
469D4—21no—
470D4—24no—
471D4—44no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—84no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—24no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—84no—
490C2×C4—1224no—
491C2×C4—2444no—
492C2×C4—2442no—
493C2×C4—2442no—
494C2×C4—2442no—
495C2×C4#1451974611yescharacteristic
496C2×C4#1451974611yescharacteristic
497C2×C4#1451974611yescharacteristic
498C2×C4#1451974611yescharacteristic
499C2×C4C22×C22⋊D6111yescharacteristic
500C2×C4C22×C22⋊D6111yescharacteristic
501C2×C4#41×D6112yes—
502C2×C4#1451982412yes—
503C8—2442no—
504C8—2442no—
505C22—21no—
506C22—21no—
507C22—41no—
508C22—42no—
509C22—41no—
510C22—42no—
511C22—42no—
512C22—41no—
513C22C2×#1451974611yescharacteristic
514C22#1684041912yes—
515C4—21no—
516C4—21no—
517C4—41no—
518C4—44no—
519C4—41no—
520C4—42no—
521C4—41no—
522C4—4882no—
523C4—2444no—
524C4#1683993512yes—
525C4#1683994012yes—
526C4#1684030811yescharacteristic
527C4#1684033311yescharacteristic
528C4#1684041112yes—
529C2—22no—
530C2—81no—
531C2—82no—
532C2#2144944311yescharacteristic
533C2C2×#1683994011yescharacteristic
534C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
Fitting, prime radicalC61×#3511C2
socleC22×C61C2×#120
hypercenter#3511D61
hypocenterC61#25667
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. #21454196C2semidirect
4. #21447045C2semidirect
5. #21446809C2semidirect
6. #21446648C2semidirect
7. #21446052C2semidirect
8. C4×#14519704C2semidirect
184. C4×Dic61#41semidirect
312. C61#25667semidirect

Maximal subgroups

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

subgroupindex
#214562122
#214556342
#214541962
#214470452
#214468092
#214466482
#214460522
C4×#145197042
C61×#35112
#2566761