Finite Group Databasebeta

← #29658608

Subgroups — #29658608

Order 15616.

1138 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840427C211yesmaximal, characteristic
2#21455854C211yesmaximal, characteristic, complemented
3#21455612C211yesmaximal, characteristic
4#21445065C214yesmaximal, complemented
5#21444981C214yesmaximal, complemented
6#21444504C211yesmaximal, characteristic, complemented
7#21444249C211yesmaximal, characteristic, complemented
8#21444211C211yesmaximal, characteristic
9C61×#3230C211yesmaximal, characteristic, complemented
10C22×Q8×D61—21no—
11#16840427—22no—
12#16840427—22no—
13#16840427—22no—
14#16840427C2214yes—
15C2×#14519776—21no—
16C2×#14519755—22no—
17C2×#14519753—21no—
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×#14519739—21no—
21#16840095—21no—
22#16840059—22no—
23#16840054C2212yescomplemented
24#16840049C2212yes—
25#16840039C2212yes—
26#16839969C2212yescomplemented
27#16839917—21no—
28C4×#11758202—22no—
29#16839815C2211yescharacteristic
30#16839805—21no—
31#16839786C2211yescharacteristic
32#16839766—21no—
33#16839508C2214yescomplemented
34#16839494C2214yescomplemented
35#16839491C2214yes—
36C61×#533C2211yescharacteristic
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#500C2212yescomplemented
39C61×#487C2212yescomplemented
40C61×#484—22no—
41C61×#477—22no—
42C61×#431C2211yescharacteristic
43C22×Dic122—22no—
44C22×Dic122D411yescharacteristic
45C2×Q8×D61—24no—
46C2×Q8×D61—24no—
47C2×Q8×D61—48no—
48C22×D244D411yescharacteristic
49C22×C4×D61—22no—
50C22×C4×D61—21no—
51C2×#11758202—22no—
52C2×#11758194—21no—
53#14519776—24no—
54#14519776—24no—
55#14519755—28no—
56#14519755—28no—
57#14519753—24no—
58#14519753—24no—
59C4×Dic122—21no—
60C4×Dic122—21no—
61C4×Dic122D412yes—
62C4×Dic122C2312yes—
63C4×D244—21no—
64C4×D244—21no—
65C4×D244D412yes—
66C4×D244C2312yes—
67#14519739—24no—
68#14519739—24no—
69C2×C4⋊Dic61D411yescharacteristic
70C2×C4⋊Dic61D411yescharacteristic
71C2×#11758196—22no—
72C2×#11758196—21no—
73C2×#11758198—22no—
74C2×#11758198—21no—
75C2×C4×Dic61—21no—
76#14519710—22no—
77#14519710—22no—
78#14519708—22no—
79C4⋊#92418—21no—
80C4⋊#92418C2312yes—
81C4⋊#92418C2312yes—
82#14519701—22no—
83#14519700—22no—
84#14519697—22no—
85#14519697—22no—
86C22⋊#92418—21no—
87C4×#92418—21no—
88C22×Q8×C61—21no—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61D412yes—
92C4⋊Q8×C61C2311yescharacteristic
93C4⋊Q8×C61C2311yescharacteristic
94C4⋊Q8×C61C2312yes—
95#125×C61—21no—
96C22⋊Q8×C61—21no—
97C22⋊Q8×C61—21no—
98C4×Q8×C61—21no—
99C4×Q8×C61—21no—
100#117×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61D412yes—
103C2×C42×C61C2311yescharacteristic
104#107×C61—24no—
105C4⋊C8×C61—21no—
106#103×C61—24no—
107C22⋊C8×C61—21no—
108C4×C8×C61C2312yes—
109C23×D61—21no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—44no—
115C22×C4×C61—22no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×Dic61—21no—
120C22×Dic61—21no—
121C22×Dic61—21no—
122Q8×D61—416no—
123Q8×D61—48no—
124Q8×D61—48no—
125C2×D244—21no—
126C2×D244—22no—
127C2×D244Q1614yescomplemented
128C2×D244C2×D412yes—
129C2×C4×D61—28no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—48no—
133C2×C4×D61—44no—
134C2×Dic122—21no—
135C2×Dic122—24no—
136C2×Dic122—28no—
137C2×Dic122—48no—
138C2×Dic122C2×D412yes—
139C2×C8×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—41no—
148C4⋊C4×C61—41no—
149C4⋊C4×C61C2×D414yes—
150C22⋊C4×C61—41no—
151C42×C61—41no—
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2411yescharacteristic
156#11758202—44no—
157#11758198—28no—
158#11758198—22no—
159#11758198—22no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61C2×D412yes—
163C4⋊Dic61C2×D412yes—
164#11758196—28no—
165#11758196—22no—
166#11758196—22no—
167C4×Dic61—22no—
168C4×Dic61—22no—
169#11758194—42no—
170C2×#92418—22no—
171C2×#92418—22no—
172C23×C61#12011yescharacteristic
173C22×D61—22no—
174C22×D61—24no—
175C22×D61—42no—
176C22×D61—42no—
177Q8×C61—44no—
178Q8×C61—42no—
179Q8×C61—44no—
180Q8×C61—42no—
181C2×C4×C61—22no—
182C2×C4×C61—24no—
183C2×C4×C61—41no—
184C2×C4×C61—44no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61#12011yescharacteristic
188C2×C4×C61#12011yescharacteristic
189C2×C4×C61#12011yescharacteristic
190C2×C4×C61C2×Q1612yes—
191C2×C4×C61#13712yes—
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×C4×C61C22×D411yescharacteristic
195C2×Dic61—24no—
196C2×Dic61—22no—
197C2×Dic61—22no—
198C2×Dic61—22no—
199C2×Dic61—42no—
200C2×Dic61—42no—
201D244—22no—
202D244—24no—
203D244C2×Q1614yes—
204C4×D61—416no—
205C4×D61—44no—
206C4×D61—44no—
207Dic122—24no—
208Dic122—22no—
209Dic122—416no—
210Dic122AGL(1,ℤ/8)14yes—
211C8×C61—42no—
212#92418—42no—
213#92418—42no—
214#18759—611nomaximal, Sylow
215C22×C61—21no—
216C22×C61—21no—
217C22×C61#45112yes—
218C22×C61C2×#12011yescharacteristic
219C2×D61—24no—
220C2×D61—48no—
221C2×D61—42no—
222C2×D61—42no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—44no—
227C4×C61—81no—
228C4×C61—81no—
229C4×C61#45112yes—
230C4×C61C22×Q1611yescharacteristic
231C4×C61C2×AGL(1,ℤ/8)12yes—
232C4×C61C2×#13711yescharacteristic
233Dic61—24no—
234Dic61—42no—
235Dic61—42no—
236C2×D4×Q8—611no—
237#3230D6111yescharacteristic, complemented
238#3104—611no—
239#3077—611no—
240#1649—614no—
241#1637—614no—
242#1549—611no—
243#1487—611no—
244#1478—611no—
245C2×C61—22no—
246C2×C61C2×#45111yescharacteristic
247C2×C61#300311yescharacteristic
248C2×C61#302011yescharacteristic
249D61—44no—
250C23×Q8—1221no—
251D4×Q8—614no—
252D4×Q8—1222no—
253D4×Q8—1222no—
254D4×Q8—1222no—
255#533C2×D6111yescharacteristic
256C2×C4⋊Q8—1221no—
257C2×C4⋊Q8C2×D6111yescharacteristic
258C2×C22⋊Q8—1221no—
259C2×C22⋊Q8—1222no—
260C2×C4×Q8—611no—
261C2×C4×D4—611no—
262C2×C4×D4—1221no—
263#500C2×D6112yescomplemented
264#487C2×D6112yescomplemented
265#484—22no—
266#484—1221no—
267#478—1222no—
268#477—22no—
269#477—612no—
270#476—612no—
271#474—612no—
272#462—612no—
273#451—1221no—
274C4×Q16—1222no—
275#431—611no—
276#431C2×D6111yescharacteristic
277#428—1221no—
278#423—611no—
279#418—1221no—
280#332—614no—
281Q8⋊C8—614no—
282D4⋊C8—614no—
283C61#1875911yescharacteristic, complemented, Sylow
284C22×Q8—21no—
285C22×Q8—611no—
286C22×Q8—1222no—
287C22×Q8—1224no—
288C22×Q8—1224no—
289C22×Q8—2448no—
290C22×D4—611no—
291C23×C4—1222no—
292C23×C4—1221no—
293C2×Q16—1222no—
294C2×M16—1221no—
295C4⋊Q8—21no—
296C4⋊Q8—21no—
297C4⋊Q8—1224no—
298C4⋊Q8—1224no—
299C4⋊Q8C22⋊D6112yes—
300C4⋊Q8C22×D6111yescharacteristic
301C4⋊Q8C22×D6111yescharacteristic
302C4⋊Q8C22×D6112yes—
303#125—21no—
304C22⋊Q8—21no—
305C22⋊Q8—21no—
306C22⋊Q8—1228no—
307C22⋊Q8—1224no—
308C22⋊Q8—1224no—
309C22⋊Q8—1228no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—612no—
313C4×Q8—612no—
314C4×Q8—1221no—
315C4×Q8—1221no—
316C4×D4—612no—
317C4×D4—612no—
318C4×D4—1224no—
319C4×D4—1224no—
320C4×D4—1221no—
321C4×D4—1221no—
322#117—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—611no—
325C2×C4⋊C4—611no—
326C2×C4⋊C4—1222no—
327C2×C4⋊C4—1221no—
328C2×C4⋊C4C22⋊D6112yes—
329C2×C22⋊C4—1222no—
330C2×C22⋊C4—1221no—
331C2×C42—1221no—
332C2×C42C22×D6111yescharacteristic
333#107—24no—
334#107—1222no—
335#107—1222no—
336#106—1222no—
337C4⋊C8—21no—
338C4⋊C8—612no—
339C4⋊C8—612no—
340C4⋊C8—1221no—
341#103—24no—
342#103—1222no—
343#103—1222no—
344#102—1222no—
345#102—1222no—
346C22⋊C8—21no—
347C22⋊C8—1221no—
348C4×C8—1221no—
349C4×C8C22×D6112yes—
350C24—1221no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—44no—
356C2×Q8—612no—
357C2×Q8—1224no—
358C2×Q8—1228no—
359C2×Q8—1221no—
360C2×Q8—2448no—
361C2×Q8—2448no—
362C2×Q8—24416no—
363C2×Q8—2448no—
364C2×D4—612no—
365C2×D4—614no—
366C2×D4—1222no—
367C2×D4—1221no—
368C22×C4—22no—
369C22×C4—1228no—
370C22×C4—1221no—
371C22×C4—1221no—
372C22×C4—1221no—
373C22×C4—1222no—
374C22×C4—1222no—
375C22×C4—2444no—
376C22×C4—2448no—
377C22×C4D4×D6111yescharacteristic
378C22×C4D4×D6111yescharacteristic
379C22×C4C2×C22⋊D6111yescharacteristic
380Q16—2444no—
381M16—2442no—
382C2×C8—22no—
383C2×C8—1222no—
384C2×C8—1222no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—612no—
395C4⋊C4—612no—
396C4⋊C4—1228no—
397C4⋊C4—1222no—
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4—1222no—
401C4⋊C4C2×C22⋊D6114yes—
402C22⋊C4—41no—
403C22⋊C4—1228no—
404C22⋊C4—1222no—
405C22⋊C4—1222no—
406C42—41no—
407C42—1222no—
408C42—1222no—
409C42D4×D6111yescharacteristic
410C42D4×D6111yescharacteristic
411C42C2×C22⋊D6111yescharacteristic
412C42C23×D6111yescharacteristic
413C23—1222no—
414C23—1224no—
415C23—2442no—
416C23—2442no—
417C23#1451974511yescharacteristic
418Q8—44no—
419Q8—42no—
420Q8—44no—
421Q8—42no—
422Q8—614no—
423Q8—1224no—
424Q8—1222no—
425Q8—24416no—
426D4—614no—
427D4—1222no—
428D4—1224no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—41no—
432C2×C4—44no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—1224no—
436C2×C4—1222no—
437C2×C4—1222no—
438C2×C4—1222no—
439C2×C4—2444no—
440C2×C4—24416no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4—2444no—
444C2×C4#1451974511yescharacteristic
445C2×C4#1451974511yescharacteristic
446C2×C4#1451974511yescharacteristic
447C2×C4Q16×D6112yes—
448C2×C4#1451980912yes—
449C2×C4C2×D4×D6111yescharacteristic
450C2×C4C2×D4×D6111yescharacteristic
451C2×C4C22×C22⋊D6111yescharacteristic
452C8—42no—
453C8—2442no—
454C8—2442no—
455C22—21no—
456C22—21no—
457C22—1224no—
458C22—2448no—
459C22—2442no—
460C22—2442no—
461C22#1683991612yes—
462C22C2×#1451974511yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—44no—
467C4—81no—
468C4—81no—
469C4—1224no—
470C4—2442no—
471C4—2442no—
472C4#1683991612yes—
473C4C2×Q16×D6111yescharacteristic
474C4AGL(1,ℤ/8)×D6112yes—
475C4C2×#1451980911yescharacteristic
476C2—22no—
477C2—2444no—
478C2C2×#1683991611yescharacteristic
479C2#2145504411yescharacteristic
480C2#2145518211yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3230C2
socleC22×C61C2×#120
prime radicalC22×D244D4
hypercenter#3230D61
hypocenterC61#18759
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
2. #21455854C2semidirect
4. #21445065C2semidirect
5. #21444981C2semidirect
6. #21444504C2semidirect
7. #21444249C2semidirect
9. C61×#3230C2semidirect
23. #16840054C22semidirect
26. #16839969C22semidirect
33. #16839508C22semidirect
34. #16839494C22semidirect
38. C61×#500C22semidirect
39. C61×#487C22semidirect
127. C2×D244Q16semidirect
237. #3230D61semidirect
263. #500C2×D61semidirect
264. #487C2×D61semidirect
283. C61#18759semidirect

Maximal subgroups

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

subgroupindex
C2×#168404272
#214558542
#214556122
#214450652
#214449812
#214445042
#214442492
#214442112
C61×#32302
#1875961