Finite Group Databasebeta

← #29658794

Subgroups — #29658794

Order 15616.

1138 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840427C211yesmaximal, characteristic
2#21455823C211yesmaximal, characteristic
3#21455620C211yesmaximal, characteristic, complemented
4#21445079C214yesmaximal, complemented
5#21444981C214yesmaximal, complemented
6#21444491C211yesmaximal, characteristic, complemented
7#21444245C211yesmaximal, characteristic
8#21444215C211yesmaximal, characteristic, complemented
9C61×#3229C211yesmaximal, 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#16840090—21no—
22#16840066—22no—
23#16840054C2212yescomplemented
24#16840049C2212yes—
25#16840044C2212yescomplemented
26#16839964C2212yes—
27#16839912—21no—
28C4×#11758201—22no—
29#16839815C2211yescharacteristic
30#16839805—21no—
31#16839786C2211yescharacteristic
32#16839770—21no—
33#16839508C2214yescomplemented
34#16839494C2214yescomplemented
35#16839491C2214yes—
36C61×#533C2211yescharacteristic
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#498C2212yescomplemented
39C61×#487C2212yescomplemented
40C61×#483—22no—
41C61×#475—22no—
42C61×#431C2211yescharacteristic
43C22×Dic122—22no—
44C22×Dic122D411yescharacteristic
45C2×Q8×D61—24no—
46C2×Q8×D61—24no—
47C2×Q8×D61—44no—
48C2×Q8×D61—44no—
49C22×D244D411yescharacteristic
50C22×C4×D61—22no—
51C22×C4×D61—21no—
52C2×#11758201—22no—
53C2×#11758194—21no—
54#14519776—24no—
55#14519776—24no—
56#14519755—28no—
57#14519755—28no—
58#14519753—24no—
59#14519753—24no—
60C4×Dic122—21no—
61C4×Dic122—21no—
62C4×Dic122D412yes—
63C4×Dic122C2312yes—
64C4×D244—21no—
65C4×D244—21no—
66C4×D244D412yes—
67C4×D244C2312yes—
68#14519739—24no—
69#14519739—24no—
70C2×C4⋊Dic61D411yescharacteristic
71C2×C4⋊Dic61D411yescharacteristic
72C2×#11758196—22no—
73C2×#11758196—21no—
74C2×#11758198—22no—
75C2×#11758198—21no—
76C2×C4×Dic61—21no—
77#14519710—22no—
78#14519708—22no—
79#14519708—22no—
80C4⋊#92418—21no—
81C4⋊#92418C2312yes—
82C4⋊#92418C2312yes—
83#14519701—22no—
84#14519700—22no—
85#14519700—22no—
86#14519697—22no—
87C22⋊#92418—21no—
88C4×#92418—21no—
89C22×Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61D412yes—
93C4⋊Q8×C61C2311yescharacteristic
94C4⋊Q8×C61C2311yescharacteristic
95C4⋊Q8×C61C2312yes—
96#125×C61—21no—
97C22⋊Q8×C61—21no—
98C22⋊Q8×C61—21no—
99C4×Q8×C61—21no—
100C4×Q8×C61—21no—
101#117×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61D412yes—
104C2×C42×C61C2311yescharacteristic
105#106×C61—24no—
106C4⋊C8×C61—21no—
107#103×C61—24no—
108C22⋊C8×C61—21no—
109C4×C8×C61C2312yes—
110C23×D61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—21no—
114C2×Q8×C61—21no—
115C2×Q8×C61—42no—
116C2×Q8×C61—42no—
117C22×C4×C61—22no—
118C22×C4×C61C2×D411yescharacteristic
119C22×C4×C61C2×D411yescharacteristic
120C22×C4×C61C2×D411yescharacteristic
121C22×Dic61—21no—
122C22×Dic61—21no—
123C22×Dic61—21no—
124Q8×D61—48no—
125Q8×D61—416no—
126Q8×D61—48no—
127C2×D244—21no—
128C2×D244—24no—
129C2×D244C2×D412yes—
130C2×C4×D61—28no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—48no—
134C2×C4×D61—42no—
135C2×C4×D61—42no—
136C2×Dic122—21no—
137C2×Dic122—22no—
138C2×Dic122—28no—
139C2×Dic122—48no—
140C2×Dic122SD1614yescomplemented
141C2×Dic122C2×D412yes—
142C2×C8×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61—41no—
151C4⋊C4×C61—41no—
152C4⋊C4×C61C2×D414yes—
153C22⋊C4×C61—41no—
154C42×C61—41no—
155C42×C61C2×D411yescharacteristic
156C42×C61C2×D411yescharacteristic
157C42×C61C2×D411yescharacteristic
158C42×C61C2411yescharacteristic
159#11758201—44no—
160#11758198—28no—
161#11758198—22no—
162#11758198—22no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165C4⋊Dic61C2×D412yes—
166C4⋊Dic61C2×D412yes—
167#11758196—28no—
168#11758196—22no—
169#11758196—22no—
170C4×Dic61—22no—
171C4×Dic61—22no—
172#11758194—42no—
173C2×#92418—22no—
174C2×#92418—22no—
175C23×C61#12011yescharacteristic
176C22×D61—24no—
177C22×D61—22no—
178C22×D61—42no—
179C22×D61—42no—
180Q8×C61—42no—
181Q8×C61—42no—
182Q8×C61—42no—
183Q8×C61—44no—
184Q8×C61—42no—
185C2×C4×C61—22no—
186C2×C4×C61—24no—
187C2×C4×C61—41no—
188C2×C4×C61—44no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61#12011yescharacteristic
192C2×C4×C61#12011yescharacteristic
193C2×C4×C61#12011yescharacteristic
194C2×C4×C61C2×SD1612yes—
195C2×C4×C61AGL(1,ℤ/8)12yes—
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×Dic61—22no—
200C2×Dic61—24no—
201C2×Dic61—22no—
202C2×Dic61—22no—
203C2×Dic61—42no—
204C2×Dic61—42no—
205D244—24no—
206D244—22no—
207D244#13714yes—
208C4×D61—416no—
209C4×D61—42no—
210C4×D61—42no—
211C4×D61—44no—
212Dic122—22no—
213Dic122—24no—
214Dic122—416no—
215Dic122C2×SD1614yes—
216C8×C61—42no—
217#92418—42no—
218#92418—42no—
219#18777—611nomaximal, Sylow
220C22×C61—21no—
221C22×C61—21no—
222C22×C61#45012yes—
223C22×C61C2×#12011yescharacteristic
224C2×D61—24no—
225C2×D61—48no—
226C2×D61—42no—
227C2×D61—42no—
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—44no—
232C4×C61—81no—
233C4×C61—81no—
234C4×C61#45012yes—
235C4×C61C22×SD1611yescharacteristic
236C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
237C4×C61C2×#13712yes—
238Dic61—24no—
239Dic61—42no—
240Dic61—42no—
241C2×D4×Q8—611no—
242#3229D6111yescharacteristic, complemented
243#3101—611no—
244#3078—611no—
245#1651—614no—
246#1637—614no—
247#1546—611no—
248#1486—611no—
249#1479—611no—
250C2×C61—22no—
251C2×C61C2×#45011yescharacteristic
252C2×C61#300411yescharacteristic
253C2×C61#302011yescharacteristic
254D61—44no—
255C23×Q8—1221no—
256D4×Q8—614no—
257D4×Q8—1222no—
258D4×Q8—1222no—
259D4×Q8—1222no—
260#533C2×D6111yescharacteristic
261C2×C4⋊Q8—1221no—
262C2×C4⋊Q8C2×D6111yescharacteristic
263C2×C22⋊Q8—1221no—
264C2×C22⋊Q8—1222no—
265C2×C4×Q8—611no—
266C2×C4×D4—611no—
267C2×C4×D4—1221no—
268#498C2×D6112yescomplemented
269#487C2×D6112yescomplemented
270#483—22no—
271#483—1221no—
272#479—1222no—
273#477—612no—
274#476—612no—
275#475—22no—
276#475—612no—
277#461—612no—
278#450—1221no—
279C4×SD16—1222no—
280#431—611no—
281#431C2×D6111yescharacteristic
282#428—1221no—
283#423—611no—
284#419—1221no—
285#332—614no—
286Q8⋊C8—614no—
287D4⋊C8—614no—
288C61#1877711yescharacteristic, complemented, Sylow
289C22×Q8—21no—
290C22×Q8—611no—
291C22×Q8—1222no—
292C22×Q8—1224no—
293C22×Q8—1224no—
294C22×Q8—2444no—
295C22×Q8—2444no—
296C22×D4—611no—
297C23×C4—1222no—
298C23×C4—1221no—
299C2×SD16—1222no—
300C2×M16—1221no—
301C4⋊Q8—21no—
302C4⋊Q8—21no—
303C4⋊Q8—1224no—
304C4⋊Q8—1224no—
305C4⋊Q8C22⋊D6112yes—
306C4⋊Q8C22×D6111yescharacteristic
307C4⋊Q8C22×D6111yescharacteristic
308C4⋊Q8C22×D6112yes—
309#125—21no—
310C22⋊Q8—21no—
311C22⋊Q8—21no—
312C22⋊Q8—1228no—
313C22⋊Q8—1224no—
314C22⋊Q8—1224no—
315C22⋊Q8—1228no—
316C4×Q8—21no—
317C4×Q8—21no—
318C4×Q8—612no—
319C4×Q8—612no—
320C4×Q8—1221no—
321C4×Q8—1221no—
322C4×D4—612no—
323C4×D4—612no—
324C4×D4—1224no—
325C4×D4—1224no—
326C4×D4—1221no—
327C4×D4—1221no—
328#117—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—611no—
331C2×C4⋊C4—611no—
332C2×C4⋊C4—1222no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4C22⋊D6112yes—
335C2×C22⋊C4—1222no—
336C2×C22⋊C4—1221no—
337C2×C42—1221no—
338C2×C42C22×D6111yescharacteristic
339#107—1222no—
340#106—24no—
341#106—1222no—
342#106—1222no—
343C4⋊C8—21no—
344C4⋊C8—612no—
345C4⋊C8—612no—
346C4⋊C8—1221no—
347#103—24no—
348#103—1222no—
349#103—1222no—
350#103—1222no—
351#102—1222no—
352C22⋊C8—21no—
353C22⋊C8—1221no—
354C4×C8—1221no—
355C4×C8C22×D6112yes—
356C24—1221no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—612no—
364C2×Q8—614no—
365C2×Q8—1228no—
366C2×Q8—1221no—
367C2×Q8—1222no—
368C2×Q8—2448no—
369C2×Q8—2448no—
370C2×Q8—2448no—
371C2×Q8—24416no—
372C2×D4—612no—
373C2×D4—1224no—
374C2×D4—1221no—
375C22×C4—22no—
376C22×C4—1228no—
377C22×C4—1221no—
378C22×C4—1221no—
379C22×C4—1221no—
380C22×C4—1222no—
381C22×C4—1222no—
382C22×C4—2448no—
383C22×C4—2442no—
384C22×C4—2442no—
385C22×C4D4×D6111yescharacteristic
386C22×C4D4×D6111yescharacteristic
387C22×C4C2×C22⋊D6111yescharacteristic
388SD16—2444no—
389M16—2442no—
390C2×C8—22no—
391C2×C8—1222no—
392C2×C8—1222no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—612no—
403C4⋊C4—612no—
404C4⋊C4—1228no—
405C4⋊C4—1222no—
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C4⋊C4—1222no—
409C4⋊C4C2×C22⋊D6114yes—
410C22⋊C4—41no—
411C22⋊C4—1228no—
412C22⋊C4—1222no—
413C22⋊C4—1222no—
414C42—41no—
415C42—1222no—
416C42—1222no—
417C42D4×D6111yescharacteristic
418C42D4×D6111yescharacteristic
419C42C2×C22⋊D6111yescharacteristic
420C42C23×D6111yescharacteristic
421C23—1224no—
422C23—1222no—
423C23—2442no—
424C23—2442no—
425C23#1451974511yescharacteristic
426Q8—42no—
427Q8—42no—
428Q8—42no—
429Q8—44no—
430Q8—42no—
431Q8—614no—
432Q8—1222no—
433Q8—1224no—
434Q8—24416no—
435D4—614no—
436D4—1224no—
437D4—1222no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—41no—
441C2×C4—44no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—1222no—
445C2×C4—1224no—
446C2×C4—1222no—
447C2×C4—1222no—
448C2×C4—24416no—
449C2×C4—2442no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4—2444no—
454C2×C4#1451974511yescharacteristic
455C2×C4#1451974511yescharacteristic
456C2×C4#1451974511yescharacteristic
457C2×C4SD16×D6112yes—
458C2×C4#1451980412yes—
459C2×C4C2×D4×D6111yescharacteristic
460C2×C4C2×D4×D6111yescharacteristic
461C2×C4C22×C22⋊D6111yescharacteristic
462C8—42no—
463C8—2442no—
464C8—2442no—
465C22—21no—
466C22—21no—
467C22—1224no—
468C22—2448no—
469C22—2442no—
470C22—2442no—
471C22#1683991112yes—
472C22C2×#1451974511yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—44no—
477C4—81no—
478C4—81no—
479C4—1224no—
480C4—2442no—
481C4—2442no—
482C4#1683991112yes—
483C4C2×SD16×D6111yescharacteristic
484C4C2×#1451980411yescharacteristic
485C4#137×D6112yes—
486C2—22no—
487C2—2444no—
488C2C2×#1683991111yescharacteristic
489C2#2145505111yescharacteristic
490C2#2145518211yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3229C2
socleC22×C61C2×#120
prime radicalC22×D244D4
hypercenter#3229D61
hypocenterC61#18777
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3229

Complements

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

NH ≅ G/Ntype
3. #21455620C2semidirect
4. #21445079C2semidirect
5. #21444981C2semidirect
6. #21444491C2semidirect
8. #21444215C2semidirect
9. C61×#3229C2semidirect
23. #16840054C22semidirect
25. #16840044C22semidirect
33. #16839508C22semidirect
34. #16839494C22semidirect
38. C61×#498C22semidirect
39. C61×#487C22semidirect
140. C2×Dic122SD16semidirect
242. #3229D61semidirect
268. #498C2×D61semidirect
269. #487C2×D61semidirect
288. C61#18777semidirect

Maximal subgroups

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

subgroupindex
C2×#168404272
#214558232
#214556202
#214450792
#214449812
#214444912
#214442452
#214442152
C61×#32292
#1877761