Finite Group Databasebeta

← #29662909

Subgroups — #29662909

Order 15616.

680 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456193C211yesmaximal, characteristic
2#21455858C211yesmaximal, characteristic, complemented
3Q8×#11758194C211yesmaximal, characteristic, complemented
4#21445385C212yesmaximal, complemented
5#21445357C212yesmaximal, complemented
6#21445261C212yesmaximal, complemented
7#21445233C212yesmaximal, complemented
8#21444569C211yesmaximal, characteristic, complemented
9#21444507C211yesmaximal, characteristic, complemented
10#21444487C211yesmaximal, characteristic
11C2×C61×#460C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840126—22no—
14#16840108C2212yes—
15#16840108C2212yes—
16#16840097—21no—
17#16840057C2211yescharacteristic, complemented
18#16840057C2211yescharacteristic, complemented
19#16839972C2211yescharacteristic, complemented
20#16839972C2211yescharacteristic, complemented
21#16839918—21no—
22#16839893—22no—
23#16839893C2212yescomplemented
24Q8×#92418C2212yescomplemented
25C4×#11758202—21no—
26C4×#11758202—21no—
27#16839815C2211yescharacteristic
28#16839786—21no—
29#16839786C2211yescharacteristic
30#16839767—21no—
31#16839767—21no—
32C4×#11758194—21no—
33C4×#11758194C2211yescharacteristic
34#16839514C2212yescomplemented
35#16839512C2212yescomplemented
36#16839510C2212yescomplemented
37#16839509C2212yescomplemented
38#16839506C2212yes—
39#16839506C2212yes—
40C22×SD16×C61—21no—
41C2×#127×C61C2211yescharacteristic
42C2×C4×Q8×C61C2211yescharacteristic
43C61×#460—21no—
44C61×#460—21no—
45C61×#460C2212yes—
46C61×#460C2212yes—
47C2×C4⋊C8×C61C2211yescharacteristic
48C2×#102×C61—21no—
49C2×#11758202—21no—
50C2×#11758202—21no—
51C2×#11758194—21no—
52C2×#11758194D411yescharacteristic
53C2×#11758194D411yescharacteristic
54#14519777C2311yescharacteristic
55#14519777C2311yescharacteristic
56#14519769—21no—
57#14519756—21no—
58#14519756—21no—
59C4×Dic122—21no—
60C4×Dic122—21no—
61#14519736—21no—
62#14519710—21no—
63#14519710—21no—
64C4⋊#92418—21no—
65C4⋊#92418—22no—
66C4⋊#92418—22no—
67C4⋊#92418C2311yescharacteristic
68C4⋊#92418C2311yescharacteristic
69C4⋊#92418C2312yes—
70#14519701—21no—
71#14519701—21no—
72#14519700—21no—
73#14519700—21no—
74#14519698—22no—
75#14519698—22no—
76#14519698—22no—
77#14519698—22no—
78C22⋊#92418—21no—
79#14519688—22no—
80#14519688D412yes—
81C4×#92418—21no—
82C4×#92418—22no—
83C4×#92418C2312yes—
84C22×Q8×C61D411yescharacteristic
85C22×D4×C61—21no—
86C22×D4×C61—21no—
87C22×D4×C61—21no—
88C2×SD16×C61—22no—
89C2×SD16×C61—22no—
90C2×SD16×C61—44no—
91C22×C8×C61—21no—
92#127×C61—21no—
93#127×C61—21no—
94#127×C61—21no—
95#127×C61C2312yes—
96C4×Q8×C61—21no—
97C4×Q8×C61—21no—
98C4×Q8×C61D411yescharacteristic
99C4×Q8×C61D411yescharacteristic
100C4×Q8×C61C2311yescharacteristic
101C4×Q8×C61C2311yescharacteristic
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C42×C61—21no—
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—21no—
107C4⋊C8×C61C2312yes—
108#102×C61—22no—
109#102×C61—22no—
110C24×C61—41no—
111C2×Q8×C61—21no—
112C2×Q8×C61—22no—
113C2×Q8×C61—41no—
114C2×Q8×C61C2×D411yescharacteristic
115C2×Q8×C61C2×D411yescharacteristic
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—42no—
120C2×D4×C61—42no—
121C2×D4×C61—41no—
122C2×D4×C61—41no—
123C2×D4×C61—41no—
124C2×D4×C61—42no—
125C2×D4×C61—42no—
126C2×D4×C61—82no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61#4111yescharacteristic
132C2×Dic122—21no—
133C2×Dic122—21no—
134SD16×C61—44no—
135SD16×C61—44no—
136C2×C8×C61—22no—
137C2×C8×C61—42no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61C2×D411yescharacteristic
144C4⋊C4×C61C2×D411yescharacteristic
145C42×C61—21no—
146C42×C61—21no—
147C42×C61—21no—
148C42×C61—21no—
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61#4111yescharacteristic
152C42×C61C2411yescharacteristic
153C22⋊Dic61—41no—
154#11758202—42no—
155#11758202—42no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160#11758196—41no—
161#11758196—41no—
162C4×Dic61—41no—
163#11758194—22no—
164#11758194—22no—
165#11758194—42no—
166C2×#92418—22no—
167C2×#92418—21no—
168C2×#92418—21no—
169C2×#92418C2×D412yes—
170C2×#92418C2×D412yes—
171C23×C61—42no—
172C23×C61—82no—
173C23×C61—82no—
174C23×C61—82no—
175C23×C61#12111yescharacteristic
176Q8×C61—21no—
177Q8×C61—21no—
178Q8×C61—21no—
179Q8×C61—42no—
180Q8×C61AGL(1,ℤ/8)12yes—
181D4×C61—42no—
182D4×C61—42no—
183D4×C61—42no—
184D4×C61—42no—
185D4×C61—82no—
186D4×C61—82no—
187D4×C61—82no—
188D4×C61—82no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—41no—
196C2×C4×C61—42no—
197C2×C4×C61—81no—
198C2×C4×C61#12111yescharacteristic
199C2×C4×C61#12111yescharacteristic
200C2×C4×C61#12111yescharacteristic
201C2×C4×C61#13512yes—
202C2×C4×C61#13712yes—
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C2×#4111yescharacteristic
206C2×Dic61—41no—
207C2×Dic61—41no—
208Dic122—41no—
209Dic122—41no—
210Dic122—81no—
211C8×C61—42no—
212#92418—22no—
213#92418—22no—
214#92418—42no—
215#92418—41no—
216#92418—41no—
217#19174—611nomaximal, Sylow
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—82no—
221C22×C61—82no—
222C22×C61—82no—
223C22×C61—164no—
224C22×C61#46412yes—
225C22×C61C2×#12111yescharacteristic
226C4×C61—21no—
227C4×C61—22no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—41no—
231C4×C61—41no—
232C4×C61—41no—
233C4×C61#46412yes—
234C4×C61C2×#13511yescharacteristic
235C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
236C4×C61C2×#13711yescharacteristic
237C4×C61#57511yescharacteristic
238Dic61—81no—
239Dic61—81no—
240#3138—611no—
241#3104—611no—
242C2×#460D6111yescharacteristic
243Q8×M16—611no—
244#1702—612no—
245#1698—612no—
246#1682—612no—
247#1678—612no—
248#1565—611no—
249#1550—611no—
250#1545—611no—
251C2×C61—22no—
252C2×C61—162no—
253C2×C61C2×#46411yescharacteristic
254C2×C61#305511yescharacteristic
255C2×C61#306811yescharacteristic
256C22×SD16—21no—
257#533—611no—
258C2×#127C2×D6111yescharacteristic
259C2×C4×Q8C2×D6111yescharacteristic
260#490—1222no—
261#486—612no—
262#486—612no—
263#484—1221no—
264#477—611no—
265#477—611no—
266#462—611no—
267#462—611no—
268#460—21no—
269#460—21no—
270#460C2×D6112yes—
271#460C2×D6112yes—
272#451—1221no—
273#446—612no—
274#446—1222no—
275C8×Q8—612no—
276C4×Q16—1221no—
277C4×Q16—1221no—
278#431—611no—
279#423—611no—
280#423—1221no—
281C2×C4⋊C8C2×D6111yescharacteristic
282#418—1221no—
283#418—1221no—
284C2×#102—21no—
285C4×M16—611no—
286C4×M16—1221no—
287#335—612no—
288#334—612no—
289#333—612no—
290#332—612no—
291#331—612no—
292#331—612no—
293C61#1917411yescharacteristic, complemented, Sylow
294C22×Q8C22⋊D6111yescharacteristic
295C22×D4—21no—
296C22×D4—21no—
297C22×D4—21no—
298C2×Q16—1221no—
299C2×Q16—1221no—
300C2×SD16—22no—
301C2×SD16—22no—
302C2×SD16—44no—
303C2×M16—611no—
304C2×M16—611no—
305C2×M16—1221no—
306C22×C8—21no—
307C4⋊Q8—611no—
308C4⋊Q8—611no—
309#127—21no—
310#127—21no—
311#127—21no—
312#127C22×D6112yes—
313#125—1221no—
314C22⋊Q8—1221no—
315C22⋊Q8—1221no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8—1221no—
319C4×Q8—1221no—
320C4×Q8C22⋊D6111yescharacteristic
321C4×Q8C22⋊D6111yescharacteristic
322C4×Q8C22×D6111yescharacteristic
323C4×Q8C22×D6111yescharacteristic
324#117—1221no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4C22⋊D6111yescharacteristic
327C2×C42—21no—
328C2×C42C22×D6111yescharacteristic
329#107—1221no—
330#107—1221no—
331C4⋊C8—21no—
332C4⋊C8—612no—
333C4⋊C8—611no—
334C4⋊C8—611no—
335C4⋊C8—1222no—
336C4⋊C8—1222no—
337C4⋊C8—1221no—
338C4⋊C8C22×D6112yes—
339#103—1221no—
340#103—1221no—
341#103—1221no—
342#103—1221no—
343#102—22no—
344#102—22no—
345#102—1222no—
346#102—1222no—
347#102—1222no—
348#102—1222no—
349C22⋊C8—1221no—
350#97—612no—
351#97—1222no—
352C4×C8—612no—
353C4×C8—1222no—
354C4×C8—1221no—
355C24—41no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—41no—
359C2×Q8—1221no—
360C2×Q8—1221no—
361C2×Q8C2×C22⋊D6111yescharacteristic
362C2×Q8C2×C22⋊D6111yescharacteristic
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—82no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4D4×D6111yescharacteristic
377C22×C4#1175822411yescharacteristic
378C22×C4C2×C22⋊D6111yescharacteristic
379Q16—2442no—
380Q16—2442no—
381SD16—44no—
382SD16—44no—
383M16—1222no—
384M16—1222no—
385M16—2442no—
386C2×C8—22no—
387C2×C8—42no—
388C2×C8—612no—
389C2×C8—612no—
390C2×C8—1222no—
391C2×C8—1221no—
392C2×C8—1221no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4—1221no—
401C4⋊C4—1221no—
402C4⋊C4—2441no—
403C4⋊C4—2441no—
404C4⋊C4C2×C22⋊D6111yescharacteristic
405C4⋊C4C2×C22⋊D6111yescharacteristic
406C22⋊C4—2441no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—2441no—
412C42D4×D6111yescharacteristic
413C42#1175822411yescharacteristic
414C42C2×C22⋊D6111yescharacteristic
415C42C23×D6111yescharacteristic
416C23—42no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23#1451974911yescharacteristic
421Q8—21no—
422Q8—21no—
423Q8—21no—
424Q8—42no—
425Q8—4881no—
426Q8—2441no—
427Q8—2441no—
428Q8#1451980312yes—
429D4—42no—
430D4—42no—
431D4—42no—
432D4—42no—
433D4—82no—
434D4—82no—
435D4—82no—
436D4—82no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—81no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4#1451974911yescharacteristic
449C2×C4#1451974911yescharacteristic
450C2×C4#1451974911yescharacteristic
451C2×C4#1451979812yes—
452C2×C4#1451980612yes—
453C2×C4C2×D4×D6111yescharacteristic
454C2×C4C22×C22⋊D6111yescharacteristic
455C2×C4C2×#1175822411yescharacteristic
456C8—42no—
457C8—1222no—
458C8—1222no—
459C8—2442no—
460C8—2441no—
461C8—2441no—
462C22—21no—
463C22—21no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—164no—
468C22#1683998212yes—
469C22C2×#1451974911yescharacteristic
470C4—21no—
471C4—22no—
472C4—21no—
473C4—21no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—4881no—
478C4—4881no—
479C4#1683998212yes—
480C4C2×#1451979811yescharacteristic
481C4C2×#1451980311yescharacteristic
482C4C2×#1451980611yescharacteristic
483C4#1684056011yescharacteristic
484C2—22no—
485C2—162no—
486C2C2×#1683998211yescharacteristic
487C2#2145543411yescharacteristic
488C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#460C2
socleC22×C61C2×#121
prime radicalC2×#127×C61C22
hypercenterC2×#460D61
hypocenterC61#19174
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#460

Complements

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

NH ≅ G/Ntype
2. #21455858C2semidirect
3. Q8×#11758194C2semidirect
4. #21445385C2semidirect
5. #21445357C2semidirect
6. #21445261C2semidirect
7. #21445233C2semidirect
8. #21444569C2semidirect
9. #21444507C2semidirect
17. #16840057C22semidirect
18. #16840057C22semidirect
19. #16839972C22semidirect
20. #16839972C22semidirect
23. #16839893C22semidirect
24. Q8×#92418C22semidirect
34. #16839514C22semidirect
35. #16839512C22semidirect
36. #16839510C22semidirect
37. #16839509C22semidirect
293. C61#19174semidirect

Maximal subgroups

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

subgroupindex
#214561932
#214558582
Q8×#117581942
#214453852
#214453572
#214452612
#214452332
#214445692
#214445072
#214444872
C2×C61×#4602
#1917461