Finite Group Databasebeta

← #29658178

Subgroups — #29658178

Order 15616.

984 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456967C211yesmaximal, characteristic, complemented
2#21456944C211yesmaximal, characteristic, complemented
3#21456279C211yesmaximal, characteristic, complemented
4#21444953C214yesmaximal, complemented
5#21444946C214yesmaximal, complemented
6#21444481C211yesmaximal, characteristic, complemented
7#21444240C211yesmaximal, characteristic, complemented
8#21444236C211yesmaximal, characteristic, complemented
9C61×#3548C211yesmaximal, characteristic
10C2×#14519807—22no—
11C2×#14519777C2211yescharacteristic
12#16840165—21no—
13#16840158—21no—
14#16840157—22no—
15#16840141C2212yes—
16#16840134C2212yes—
17#16840112—21no—
18#16840108—21no—
19#16840095—22no—
20#16840078—22no—
21#16840057C2214yes—
22#16840052C2214yes—
23#16839815C2211yescharacteristic
24#16839815C2211yescharacteristic
25C4×#11758194C2211yescharacteristic
26#16839511C2212yes—
27#16839506C2212yes—
28Q8⋊#92418C2214yes—
29D4⋊#92418C2214yes—
30C61×#562—22no—
31C61×#554—21no—
32C61×#554C2212yes—
33C61×#550—21no—
34C61×#550C2212yes—
35C61×#547—21no—
36C61×#547—21no—
37C61×#540—22no—
38C61×#537—22no—
39C61×#537—22no—
40C61×#533C2211yescharacteristic
41C61×#532C2211yescharacteristic
42C61×#532C2211yescharacteristic
43C22×Dic122—21no—
44#14519807—48no—
45C2×#11758202—24no—
46C2×#11758200—24no—
47C2×#11758194—21no—
48#14519777—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777C2312yes—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54#14519710—24no—
55#14519708—24no—
56C4⋊#92418C2312yes—
57C4⋊#92418C2312yes—
58#14519701—24no—
59#14519698—24no—
60C22⋊#92418—21no—
61C22⋊#92418—21no—
62#14519688—21no—
63C4×#92418—21no—
64C4×#92418—21no—
65C4×#92418C2312yes—
66C2×#41×C61—22no—
67C2×#41×C61—22no—
68C2×#41×C61—21no—
69C2×#41×C61D412yes—
70C4⋊Q8×C61D411yescharacteristic
71C4⋊Q8×C61D411yescharacteristic
72C4⋊Q8×C61D411yescharacteristic
73C4⋊Q8×C61C2311yescharacteristic
74#127×C61—22no—
75#127×C61D411yescharacteristic
76#127×C61C2311yescharacteristic
77#125×C61—22no—
78#125×C61—21no—
79#124×C61—21no—
80#124×C61D412yes—
81#123×C61—24no—
82#123×C61—24no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—24no—
85C22⋊Q8×C61—22no—
86C22⋊Q8×C61—24no—
87C22⋊Q8×C61—21no—
88C22⋊Q8×C61D412yes—
89#121×C61—24no—
90#121×C61—24no—
91#121×C61—22no—
92#121×C61—24no—
93#121×C61—24no—
94#121×C61—21no—
95#121×C61—21no—
96#121×C61—21no—
97#121×C61—22no—
98#121×C61D412yes—
99C4×Q8×C61—21no—
100C4×Q8×C61C2312yes—
101C4×D4×C61—22no—
102C4×D4×C61—22no—
103C4×D4×C61—21no—
104C4×D4×C61—21no—
105C4×D4×C61—21no—
106C4×D4×C61C2312yes—
107#117×C61—21no—
108C2×C4⋊C4×C61—22no—
109C2×C4⋊C4×C61—22no—
110C2×C42×C61C2311yescharacteristic
111#41×C61—44no—
112#41×C61—82no—
113#41×C61—84no—
114#41×C61—84no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61C2×D412yes—
118C2×Q8×C61C2×D412yes—
119C2×D4×C61—24no—
120C2×D4×C61—24no—
121C2×D4×C61—22no—
122C2×D4×C61—22no—
123C2×D4×C61—21no—
124C2×D4×C61—21no—
125C2×D4×C61—21no—
126C2×D4×C61—21no—
127C2×D4×C61C2×D412yes—
128C2×D4×C61C2×D412yes—
129C22×C4×C61—24no—
130C22×C4×C61—24no—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×Dic61—41no—
137C2×Dic122—22no—
138C2×Dic122—44no—
139C2×Dic122—41no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61C2×D412yes—
151C4⋊C4×C61C2×D412yes—
152C22⋊C4×C61—24no—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C22⋊C4×C61—22no—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—22no—
159C42×C61—22no—
160C42×C61C2×D411yescharacteristic
161C42×C61C2×D411yescharacteristic
162C42×C61C2×D411yescharacteristic
163C42×C61C2411yescharacteristic
164#11758202—48no—
165#11758200—48no—
166C4⋊Dic61—22no—
167C4⋊Dic61—22no—
168C4⋊Dic61—41no—
169C4⋊Dic61—41no—
170#11758194—24no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—22no—
174C23×C61—22no—
175C23×C61—22no—
176C23×C61#12011yescharacteristic
177Q8×C61—24no—
178Q8×C61—42no—
179Q8×C61—42no—
180Q8×C61—81no—
181D4×C61—24no—
182D4×C61—42no—
183D4×C61—42no—
184D4×C61—42no—
185D4×C61—42no—
186D4×C61—81no—
187D4×C61—82no—
188D4×C61—82no—
189D4×C61—84no—
190D4×C61—84no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—44no—
199C2×C4×C61—44no—
200C2×C4×C61—84no—
201C2×C4×C61—84no—
202C2×C4×C61#12011yescharacteristic
203C2×C4×C61#12011yescharacteristic
204C2×C4×C61#12011yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×Dic61—42no—
209C2×Dic61—82no—
210Dic122—44no—
211Dic122—82no—
212#92418—24no—
213#92418—42no—
214#92418—42no—
215#18717—611nomaximal, Sylow
216C22×C61—21no—
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—42no—
220C22×C61—42no—
221C22×C61—42no—
222C22×C61—42no—
223C22×C61—42no—
224C22×C61—42no—
225C22×C61C2×#12011yescharacteristic
226C4×C61—21no—
227C4×C61—21no—
228C4×C61—21no—
229C4×C61—42no—
230C4×C61—42no—
231C4×C61—42no—
232C4×C61—42no—
233C4×C61#45312yes—
234C4×C61#57712yes—
235C4×C61#57812yes—
236Dic61—82no—
237#3548D6111yescharacteristic
238#3230—611no—
239#3227—611no—
240#3152—611no—
241#1632—614no—
242#1631—614no—
243#1543—611no—
244#1485—611no—
245#1484—611no—
246C2×C61—41no—
247C2×C61—82no—
248C2×C61—82no—
249C2×C61#300911yescharacteristic
250C2×C61#301111yescharacteristic
251C2×C61C2×#45311yescharacteristic
252C2×#137—1222no—
253#562—22no—
254#554—21no—
255#554C2×D6112yes—
256#550—21no—
257#550C2×D6112yes—
258#547—21no—
259#547—21no—
260#540—22no—
261#537—22no—
262#537—22no—
263#533C2×D6111yescharacteristic
264#532C2×D6111yescharacteristic
265#532C2×D6111yescharacteristic
266C2×C4⋊Q8—611no—
267#500—1221no—
268#498—1221no—
269#497—1222no—
270#494—612no—
271#492—612no—
272#487—1221no—
273#486—1221no—
274#484—1222no—
275#481—1222no—
276#477—614no—
277#476—614no—
278#431—611no—
279#431—611no—
280C4×M16—611no—
281#333—612no—
282#331—612no—
283Q8⋊C8—614no—
284D4⋊C8—614no—
285C61#1871711yescharacteristic, complemented, Sylow
286C2×#41—22no—
287C2×#41—22no—
288C2×#41—21no—
289C2×#41C22⋊D6112yes—
290C22×Q8—1221no—
291#137—2448no—
292C2×Q16—1224no—
293C2×SD16—1224no—
294C2×M16—1221no—
295C4⋊Q8—612no—
296C4⋊Q8—1221no—
297C4⋊Q8—1221no—
298C4⋊Q8—1221no—
299C4⋊Q8C22⋊D6111yescharacteristic
300C4⋊Q8C22⋊D6111yescharacteristic
301C4⋊Q8C22⋊D6111yescharacteristic
302C4⋊Q8C22×D6111yescharacteristic
303#127—22no—
304#127C22⋊D6111yescharacteristic
305#127C22×D6111yescharacteristic
306#125—22no—
307#125—21no—
308#124—21no—
309#124C22⋊D6112yes—
310#123—24no—
311#123—24no—
312C22⋊Q8—22no—
313C22⋊Q8—24no—
314C22⋊Q8—22no—
315C22⋊Q8—24no—
316C22⋊Q8—21no—
317C22⋊Q8C22⋊D6112yes—
318#121—24no—
319#121—24no—
320#121—22no—
321#121—24no—
322#121—24no—
323#121—21no—
324#121—21no—
325#121—21no—
326#121—22no—
327#121C22⋊D6112yes—
328C4×Q8—21no—
329C4×Q8C22×D6112yes—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4C22×D6112yes—
336#117—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—1221no—
340C2×C4⋊C4—1221no—
341C2×C42C22×D6111yescharacteristic
342#107—1224no—
343#106—1224no—
344C4⋊C8—612no—
345C4⋊C8—612no—
346#103—1224no—
347#102—1224no—
348C22⋊C8—1221no—
349C22⋊C8—1221no—
350#97—1221no—
351C4×C8—612no—
352C4×C8—1221no—
353C4×C8—1221no—
354#41—44no—
355#41—82no—
356#41—84no—
357#41—84no—
358C2×Q8—21no—
359C2×Q8—21no—
360C2×Q8—1222no—
361C2×Q8—2444no—
362C2×Q8—2441no—
363C2×Q8C2×C22⋊D6112yes—
364C2×Q8C2×C22⋊D6112yes—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—21no—
372C2×D4—21no—
373C2×D4C2×C22⋊D6112yes—
374C2×D4C2×C22⋊D6112yes—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—2441no—
380C22×C4C2×C22⋊D6111yescharacteristic
381C22×C4C2×C22⋊D6111yescharacteristic
382C22×C4C2×C22⋊D6111yescharacteristic
383Q16—2448no—
384SD16—2448no—
385M16—1224no—
386C2×C8—1222no—
387C2×C8—1222no—
388C2×C8—1222no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—1222no—
400C4⋊C4—1222no—
401C4⋊C4—2441no—
402C4⋊C4—2441no—
403C4⋊C4C2×C22⋊D6112yes—
404C4⋊C4C2×C22⋊D6112yes—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C42—22no—
413C42C2×C22⋊D6111yescharacteristic
414C42C2×C22⋊D6111yescharacteristic
415C42C2×C22⋊D6111yescharacteristic
416C42C23×D6111yescharacteristic
417C23—22no—
418C23—22no—
419C23#1451974611yescharacteristic
420Q8—24no—
421Q8—42no—
422Q8—42no—
423Q8—81no—
424Q8—4882no—
425Q8—2444no—
426D4—24no—
427D4—42no—
428D4—42no—
429D4—42no—
430D4—42no—
431D4—81no—
432D4—82no—
433D4—82no—
434D4—84no—
435D4—84no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—84no—
446C2×C4—84no—
447C2×C4—4882no—
448C2×C4—2442no—
449C2×C4#1451974611yescharacteristic
450C2×C4#1451974611yescharacteristic
451C2×C4#1451974611yescharacteristic
452C2×C4C22×C22⋊D6111yescharacteristic
453C2×C4C22×C22⋊D6111yescharacteristic
454C2×C4C22×C22⋊D6111yescharacteristic
455C8—1224no—
456C8—2442no—
457C8—2442no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—42no—
467C22C2×#1451974611yescharacteristic
468C4—21no—
469C4—21no—
470C4—21no—
471C4—42no—
472C4—42no—
473C4—42no—
474C4—42no—
475C4—4882no—
476C4#1683993012yes—
477C4#1684057312yes—
478C4#1684057712yes—
479C2—41no—
480C2—82no—
481C2—82no—
482C2#2145509611yescharacteristic
483C2#2145511411yescharacteristic
484C2C2×#1683993011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3548C2
socleC22×C61C2×#120
hypercenter#3548D61
hypocenterC61#18717
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3548

Complements

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

NH ≅ G/Ntype
1. #21456967C2semidirect
2. #21456944C2semidirect
3. #21456279C2semidirect
4. #21444953C2semidirect
5. #21444946C2semidirect
6. #21444481C2semidirect
7. #21444240C2semidirect
8. #21444236C2semidirect
285. C61#18717semidirect

Maximal subgroups

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

subgroupindex
#214569672
#214569442
#214562792
#214449532
#214449462
#214444812
#214442402
#214442362
C61×#35482
#1871761