Finite Group Databasebeta

← #29657404

Subgroups — #29657404

Order 15616.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456944C211yesmaximal, characteristic, complemented
2C2×#16840114C211yesmaximal, characteristic, complemented
3#21455634C211yesmaximal, characteristic, complemented
4#21444991C214yesmaximal, complemented
5#21444970C214yesmaximal, complemented
6#21444504C211yesmaximal, characteristic, complemented
7#21444270C211yesmaximal, characteristic, complemented
8#21444201C211yesmaximal, characteristic, complemented
9C61×#3511C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840158—21no—
12#16840114—22no—
13#16840114C2214yes—
14#16840108—21no—
15#16840078—22no—
16#16840057C2212yes—
17#16840052C2212yes—
18#16840052C2214yes—
19#16840047—21no—
20#16840042—21no—
21#16839815C2211yescharacteristic
22#16839805—21no—
23#16839786C2211yescharacteristic
24#16839763—21no—
25C2×#14519700—22no—
26C2×C4×#92418C2211yescharacteristic
27#16839508C2214yes—
28Q8⋊#92418C2212yes—
29D4⋊#92418C2212yes—
30D4⋊#92418C2214yes—
31C61×#556C2212yes—
32C61×#555—21no—
33C61×#554—21no—
34C61×#553—22no—
35C61×#548—22no—
36C61×#547C2212yes—
37C61×#542—22no—
38C61×#537—21no—
39C61×#533—21no—
40C61×#532—21no—
41C61×#532C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C61×#520—21no—
44C4×#41×C61C2211yescharacteristic
45C22×Dic122—21no—
46C2×#11758194—21no—
47C22×#92418—21no—
48#14519777—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777C2312yes—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54#14519710—22no—
55#14519708—22no—
56#14519708—24no—
57C4⋊#92418—21no—
58C4⋊#92418C2312yes—
59C4⋊#92418C2312yes—
60#14519701—22no—
61#14519700—28no—
62#14519698—22no—
63#14519698—24no—
64C22⋊#92418—21no—
65C4×#92418—21no—
66C4×#92418—21no—
67C4×#92418C2312yes—
68C2×#41×C61—22no—
69C2×#41×C61—21no—
70C2×#41×C61D411yescharacteristic
71C22×Q8×C61—21no—
72C4⋊Q8×C61—21no—
73C4⋊Q8×C61—21no—
74C4⋊Q8×C61D412yes—
75C4⋊Q8×C61D411yescharacteristic
76C4⋊Q8×C61C2312yes—
77#127×C61D411yescharacteristic
78#126×C61—24no—
79#125×C61—22no—
80#124×C61—24no—
81#124×C61—22no—
82#124×C61—21no—
83#123×C61—24no—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61—24no—
86C22⋊Q8×C61—22no—
87C22⋊Q8×C61—22no—
88#121×C61—22no—
89#121×C61—22no—
90#121×C61—24no—
91#121×C61—22no—
92#121×C61—21no—
93#121×C61D412yes—
94C4×Q8×C61—22no—
95C4×Q8×C61—22no—
96C4×Q8×C61D411yescharacteristic
97C4×Q8×C61C2311yescharacteristic
98C4×D4×C61—22no—
99C4×D4×C61—22no—
100C4×D4×C61—24no—
101C4×D4×C61—21no—
102C4×D4×C61—21no—
103C4×D4×C61—21no—
104C4×D4×C61D411yescharacteristic
105C4×D4×C61C2312yes—
106C4×D4×C61C2311yescharacteristic
107#117×C61—22no—
108#117×C61—22no—
109#117×C61—21no—
110#117×C61D411yescharacteristic
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—22no—
113C2×C4⋊C4×C61D412yes—
114C2×C42×C61—21no—
115C2×C42×C61C2311yescharacteristic
116#41×C61—22no—
117#41×C61—22no—
118#41×C61—42no—
119#41×C61—42no—
120#41×C61—84no—
121C2×Q8×C61—21no—
122C2×Q8×C61—22no—
123C2×Q8×C61—22no—
124C2×Q8×C61—82no—
125C2×Q8×C61C2×D411yescharacteristic
126C2×D4×C61—21no—
127C2×D4×C61—22no—
128C2×D4×C61—24no—
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61C2×D411yescharacteristic
132C2×D4×C61C2×D412yes—
133C22×C4×C61—24no—
134C22×C4×C61—22no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—22no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×Dic61—41no—
143C2×Dic122—22no—
144C2×Dic122—44no—
145C2×Dic122—41no—
146C4⋊C4×C61—24no—
147C4⋊C4×C61—24no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—21no—
151C4⋊C4×C61—21no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—21no—
154C4⋊C4×C61C2×D411yescharacteristic
155C4⋊C4×C61C2×D414yes—
156C4⋊C4×C61C2×D411yescharacteristic
157C4⋊C4×C61C2×D412yes—
158C22⋊C4×C61—24no—
159C22⋊C4×C61—24no—
160C22⋊C4×C61—24no—
161C22⋊C4×C61—21no—
162C22⋊C4×C61—21no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—21no—
165C22⋊C4×C61—22no—
166C42×C61—24no—
167C42×C61—21no—
168C42×C61—21no—
169C42×C61—21no—
170C42×C61C2×D411yescharacteristic
171C42×C61C2×D411yescharacteristic
172C42×C61C2×D411yescharacteristic
173C42×C61C2411yescharacteristic
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61—41no—
177C4⋊Dic61—41no—
178#11758194—42no—
179C2×#92418—22no—
180C2×#92418—22no—
181C2×#92418—22no—
182C2×#92418—42no—
183C23×C61—21no—
184C23×C61—22no—
185C23×C61#12011yescharacteristic
186Q8×C61—21no—
187Q8×C61—21no—
188Q8×C61—44no—
189Q8×C61—42no—
190Q8×C61—82no—
191D4×C61—21no—
192D4×C61—21no—
193D4×C61—24no—
194D4×C61—42no—
195D4×C61—44no—
196D4×C61—42no—
197D4×C61—42no—
198D4×C61—84no—
199C2×C4×C61—22no—
200C2×C4×C61—22no—
201C2×C4×C61—21no—
202C2×C4×C61—21no—
203C2×C4×C61—22no—
204C2×C4×C61—21no—
205C2×C4×C61—24no—
206C2×C4×C61—42no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—82no—
212C2×C4×C61—84no—
213C2×C4×C61#12011yescharacteristic
214C2×C4×C61#12011yescharacteristic
215C2×C4×C61#12011yescharacteristic
216C2×C4×C61#13512yes—
217C2×C4×C61#13712yes—
218C2×C4×C61C22×D411yescharacteristic
219C2×C4×C61C22×D411yescharacteristic
220C2×C4×C61C22×D411yescharacteristic
221C2×Dic61—42no—
222C2×Dic61—82no—
223Dic122—44no—
224Dic122—82no—
225#92418—42no—
226#92418—42no—
227#92418—42no—
228#18647—611nomaximal, Sylow
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—41no—
232C22×C61—42no—
233C22×C61—41no—
234C22×C61—42no—
235C22×C61—41no—
236C22×C61—42no—
237C22×C61#45212yes—
238C22×C61C2×#12011yescharacteristic
239C4×C61—21no—
240C4×C61—21no—
241C4×C61—41no—
242C4×C61—42no—
243C4×C61—41no—
244C4×C61—44no—
245C4×C61—41no—
246C4×C61#45212yes—
247C4×C61#45412yes—
248C4×C61C2×#13511yescharacteristic
249C4×C61C2×#13711yescharacteristic
250C4×C61#57712yes—
251Dic61—82no—
252#3511D6111yescharacteristic
253#3227—611no—
254C2×#487—611no—
255#3079—611no—
256#1638—614no—
257#1635—614no—
258#1549—611no—
259#1492—611no—
260#1476—611no—
261C2×C61—22no—
262C2×C61—81no—
263C2×C61—82no—
264C2×C61C2×#45211yescharacteristic
265C2×C61#300911yescharacteristic
266C2×C61C2×#45411yescharacteristic
267#556C2×D6112yes—
268#555—21no—
269#554—21no—
270#553—22no—
271#548—22no—
272#547C2×D6112yes—
273#542—22no—
274#537—21no—
275#533—21no—
276#532—21no—
277#532C2×D6111yescharacteristic
278C2×C4⋊Q8—611no—
279C2×C4⋊Q8C2×D6111yescharacteristic
280#520—21no—
281C4×#41C2×D6111yescharacteristic
282#498—1221no—
283#487—614no—
284#487—1222no—
285#486—1221no—
286#481—1222no—
287#477—612no—
288#476—612no—
289#476—614no—
290#475—1221no—
291#474—1221no—
292#431—611no—
293#428—1221no—
294#423—611no—
295#417—1221no—
296C2×#103—1222no—
297C2×C4×C8—611no—
298#332—614no—
299Q8⋊C8—612no—
300D4⋊C8—612no—
301D4⋊C8—614no—
302C61#1864711yescharacteristic, complemented, Sylow
303C2×#41—22no—
304C2×#41—21no—
305C2×#41C22⋊D6111yescharacteristic
306C22×Q8—21no—
307C22×Q8—1221no—
308C2×M16—1221no—
309C22×C8—1221no—
310C4⋊Q8—21no—
311C4⋊Q8—21no—
312C4⋊Q8—612no—
313C4⋊Q8—1221no—
314C4⋊Q8—1221no—
315C4⋊Q8—1221no—
316C4⋊Q8C22⋊D6112yes—
317C4⋊Q8C22⋊D6111yescharacteristic
318C4⋊Q8C22×D6112yes—
319#127C22⋊D6111yescharacteristic
320#126—24no—
321#125—22no—
322#124—24no—
323#124—22no—
324#124—21no—
325#123—24no—
326C22⋊Q8—22no—
327C22⋊Q8—24no—
328C22⋊Q8—22no—
329C22⋊Q8—22no—
330#121—22no—
331#121—22no—
332#121—24no—
333#121—22no—
334#121—21no—
335#121C22⋊D6112yes—
336C4×Q8—22no—
337C4×Q8—22no—
338C4×Q8C22⋊D6111yescharacteristic
339C4×Q8C22×D6111yescharacteristic
340C4×D4—22no—
341C4×D4—24no—
342C4×D4—22no—
343C4×D4—21no—
344C4×D4—21no—
345C4×D4—21no—
346C4×D4C22⋊D6111yescharacteristic
347C4×D4C22×D6111yescharacteristic
348C4×D4C22×D6112yes—
349#117—22no—
350#117—22no—
351#117—21no—
352#117C22⋊D6111yescharacteristic
353C2×C4⋊C4—21no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—1221no—
356C2×C4⋊C4—1221no—
357C2×C4⋊C4C22⋊D6112yes—
358C2×C42—21no—
359C2×C42C22×D6111yescharacteristic
360#107—1222no—
361#106—1222no—
362#106—1224no—
363C4⋊C8—612no—
364C4⋊C8—612no—
365C4⋊C8—1221no—
366#103—1222no—
367#103—1228no—
368#102—1222no—
369#102—1224no—
370C22⋊C8—1221no—
371C4×C8—612no—
372C4×C8—1221no—
373C4×C8—1221no—
374#41—22no—
375#41—22no—
376#41—42no—
377#41—42no—
378#41—84no—
379C2×Q8—22no—
380C2×Q8—21no—
381C2×Q8—22no—
382C2×Q8—82no—
383C2×Q8—1222no—
384C2×Q8—2444no—
385C2×Q8—2441no—
386C2×Q8C2×C22⋊D6111yescharacteristic
387C2×D4—21no—
388C2×D4—22no—
389C2×D4—24no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4C2×C22⋊D6111yescharacteristic
393C2×D4C2×C22⋊D6112yes—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—2441no—
401C22×C4C2×C22⋊D6111yescharacteristic
402C22×C4C2×C22⋊D6111yescharacteristic
403C22×C4C2×C22⋊D6111yescharacteristic
404M16—2442no—
405C2×C8—1222no—
406C2×C8—1222no—
407C2×C8—1222no—
408C2×C8—2442no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—1222no—
418C4⋊C4—1222no—
419C4⋊C4—2441no—
420C4⋊C4—2441no—
421C4⋊C4C2×C22⋊D6114yes—
422C4⋊C4C2×C22⋊D6111yescharacteristic
423C4⋊C4C2×C22⋊D6112yes—
424C4⋊C4C2×C22⋊D6111yescharacteristic
425C22⋊C4—24no—
426C22⋊C4—24no—
427C22⋊C4—24no—
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—22no—
431C22⋊C4—21no—
432C22⋊C4—22no—
433C42—24no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42C2×C22⋊D6111yescharacteristic
438C42C2×C22⋊D6111yescharacteristic
439C42C2×C22⋊D6111yescharacteristic
440C42C23×D6111yescharacteristic
441C23—21no—
442C23—22no—
443C23#1451974611yescharacteristic
444Q8—21no—
445Q8—21no—
446Q8—44no—
447Q8—42no—
448Q8—82no—
449Q8—4882no—
450Q8—2444no—
451D4—21no—
452D4—21no—
453D4—24no—
454D4—42no—
455D4—44no—
456D4—42no—
457D4—42no—
458D4—84no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—24no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—82no—
472C2×C4—84no—
473C2×C4—4882no—
474C2×C4—2442no—
475C2×C4#1451974611yescharacteristic
476C2×C4#1451974611yescharacteristic
477C2×C4#1451974611yescharacteristic
478C2×C4#1451979712yes—
479C2×C4#1451980712yes—
480C2×C4C22×C22⋊D6111yescharacteristic
481C2×C4C22×C22⋊D6111yescharacteristic
482C2×C4C22×C22⋊D6111yescharacteristic
483C8—2442no—
484C8—2442no—
485C8—2442no—
486C22—21no—
487C22—21no—
488C22—41no—
489C22—42no—
490C22—41no—
491C22—42no—
492C22—41no—
493C22—42no—
494C22#1683992412yes—
495C22C2×#1451974611yescharacteristic
496C4—21no—
497C4—21no—
498C4—41no—
499C4—41no—
500C4—42no—
501C4—41no—
502C4—44no—
503C4—4882no—
504C4#1683992412yes—
505C4#1683993512yes—
506C4C2×#1451979711yescharacteristic
507C4C2×#1451980711yescharacteristic
508C4#1684057312yes—
509C2—22no—
510C2—81no—
511C2—82no—
512C2C2×#1683992411yescharacteristic
513C2#2145509611yescharacteristic
514C2C2×#1683993511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3511C2
socleC22×C61C2×#120
hypercenter#3511D61
hypocenterC61#18647
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3511

Complements

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

NH ≅ G/Ntype
1. #21456944C2semidirect
2. C2×#16840114C2semidirect
3. #21455634C2semidirect
4. #21444991C2semidirect
5. #21444970C2semidirect
6. #21444504C2semidirect
7. #21444270C2semidirect
8. #21444201C2semidirect
302. C61#18647semidirect

Maximal subgroups

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

subgroupindex
#214569442
C2×#168401142
#214556342
#214449912
#214449702
#214445042
#214442702
#214442012
C61×#35112
#1864761