Finite Group Databasebeta

← #29658508

Subgroups — #29658508

Order 15616.

966 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840472C211yesmaximal, characteristic
2#21456764C211yesmaximal, characteristic, complemented
3#21455612C211yesmaximal, characteristic
4#21445031C214yesmaximal, complemented
5#21444952C214yesmaximal, complemented
6#21444499C211yesmaximal, characteristic, complemented
7#21444248C211yesmaximal, characteristic, complemented
8#21444209C211yesmaximal, characteristic
9C61×#3104C211yesmaximal, characteristic, complemented
10#16840472—22no—
11#16840472—22no—
12#16840472—22no—
13#16840472C2214yes—
14C2×#14519777C2211yescharacteristic
15C2×#14519769—21no—
16C2×#14519755—22no—
17C2×#14519753—21no—
18C2×C4×D244—21no—
19C2×C4×D244C2211yescharacteristic
20C2×C4⋊C4×D61—21no—
21#16840141C2212yescomplemented
22#16840114C2212yescomplemented
23#16840059—22no—
24#16840049C2212yes—
25#16840039C2212yes—
26#16839972—22no—
27#16839917—22no—
28#16839815C2211yescharacteristic
29#16839805—21no—
30#16839786C2211yescharacteristic
31#16839766—21no—
32#16839510C2214yescomplemented
33Q8⋊#92418C2214yescomplemented
34#16839489C2214yes—
35C61×#533C2211yescharacteristic
36C2×C4×Q8×C61C2211yescharacteristic
37C61×#484—21no—
38C61×#477C2212yescomplemented
39C61×#462C2212yescomplemented
40C61×#451—21no—
41C4×Q16×C61—22no—
42C61×#431C2211yescharacteristic
43C22×Dic122D412yes—
44C22×D244D411yescharacteristic
45C22×C4×D61—21no—
46C22×C4×D61—22no—
47C2×#11758202—24no—
48C2×#11758194—21no—
49#14519777—21no—
50#14519777—21no—
51#14519777D412yes—
52#14519777C2312yes—
53#14519769—24no—
54#14519769—24no—
55#14519755—28no—
56#14519755—28no—
57#14519753—24no—
58#14519753—24no—
59C4×D244—24no—
60C4×D244—24no—
61C4×D244—21no—
62C4×D244—21no—
63C4×D244D412yes—
64C4×D244C2312yes—
65C2×C4⋊Dic61—21no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×#11758196—22no—
69C2×#11758196—22no—
70C4⋊C4×D61—24no—
71C4⋊C4×D61—24no—
72C2×#11758198—22no—
73C2×#11758198—21no—
74#14519710—22no—
75#14519708—22no—
76C4⋊#92418—21no—
77C4⋊#92418—21no—
78C4⋊#92418C2312yes—
79#14519700—24no—
80#14519697—22no—
81#14519697—22no—
82C22⋊#92418—21no—
83C4×#92418C2312yes—
84C22×Q8×C61D411yescharacteristic
85C2×Q16×C61—22no—
86C4⋊Q8×C61C2311yescharacteristic
87C4⋊Q8×C61C2311yescharacteristic
88#125×C61—21no—
89C22⋊Q8×C61—21no—
90C22⋊Q8×C61—21no—
91C4×Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61D412yes—
96C4×Q8×C61C2312yes—
97#117×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C42×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#107×C61—22no—
103C4⋊C8×C61C2312yes—
104#103×C61—22no—
105#103×C61—22no—
106C22⋊C8×C61—21no—
107C4×C8×C61—21no—
108C23×D61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—42no—
113C2×Q8×C61C2×D412yes—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×Dic61—22no—
120C22×Dic61—21no—
121C2×D244—21no—
122C2×D244—22no—
123C2×D244Q1614yescomplemented
124C2×D244C2×D412yes—
125C2×C4×D61—22no—
126C2×C4×D61—22no—
127C2×C4×D61—28no—
128C2×C4×D61—44no—
129C2×C4×D61—84no—
130C2×Dic122—22no—
131C2×Dic122—82no—
132C2×Dic122C2×D414yes—
133Q16×C61—44no—
134C2×C8×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61—41no—
144C4⋊C4×C61C2×D412yes—
145C22⋊C4×C61—41no—
146C42×C61—22no—
147C42×C61—22no—
148C42×C61—41no—
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61C2411yescharacteristic
153#11758202—84no—
154#11758198—28no—
155#11758198—22no—
156#11758198—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—21no—
160C4⋊Dic61—22no—
161C4⋊Dic61—22no—
162C4⋊Dic61C2×D412yes—
163#11758196—28no—
164#11758196—28no—
165#11758194—42no—
166C2×#92418—22no—
167C2×#92418—22no—
168C23×C61#12011yescharacteristic
169C22×D61—22no—
170C22×D61—24no—
171C22×D61—42no—
172C22×D61—42no—
173Q8×C61—22no—
174Q8×C61—22no—
175Q8×C61—42no—
176Q8×C61—42no—
177Q8×C61—42no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—41no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61#12011yescharacteristic
187C2×C4×C61#12011yescharacteristic
188C2×C4×C61#12011yescharacteristic
189C2×C4×C61C2×Q1612yes—
190C2×C4×C61#13712yes—
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C22×D411yescharacteristic
194C2×Dic61—22no—
195C2×Dic61—24no—
196C2×Dic61—44no—
197C2×Dic61—42no—
198D244—22no—
199D244—24no—
200D244C2×Q1614yes—
201C4×D61—44no—
202C4×D61—44no—
203C4×D61—88no—
204Dic122—44no—
205Dic122—82no—
206C8×C61—42no—
207#92418—42no—
208#92418—42no—
209#18749—611nomaximal, Sylow
210C22×C61—21no—
211C22×C61—21no—
212C22×C61#45112yes—
213C22×C61C2×#12011yescharacteristic
214C2×D61—24no—
215C2×D61—48no—
216C2×D61—42no—
217C2×D61—42no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—21no—
221C4×C61—42no—
222C4×C61—42no—
223C4×C61—81no—
224C4×C61—81no—
225C4×C61#45112yes—
226C4×C61C22×Q1611yescharacteristic
227C4×C61C2×#13711yescharacteristic
228C4×C61#57512yes—
229Dic61—42no—
230Dic61—44no—
231C2×#554—611no—
232#3209—611no—
233#3104D6111yescharacteristic, complemented
234#3077—611no—
235#1644—614no—
236#1632—614no—
237#1548—611no—
238#1487—611no—
239#1478—611no—
240C2×C61—22no—
241C2×C61C2×#45111yescharacteristic
242C2×C61#300511yescharacteristic
243C2×C61#301911yescharacteristic
244D61—44no—
245#554—614no—
246#554—1222no—
247#554—1222no—
248#554—1222no—
249#533C2×D6111yescharacteristic
250C2×C4⋊Q8—611no—
251C2×#125—1221no—
252C2×C22⋊Q8—1222no—
253C2×C22⋊Q8—1221no—
254C2×C4×Q8C2×D6111yescharacteristic
255C2×C4×D4—611no—
256C2×C4×D4—1221no—
257C22×C4⋊C4—1221no—
258#494—612no—
259#487—612no—
260#484—21no—
261#478—1222no—
262#477C2×D6112yescomplemented
263#476—612no—
264#474—612no—
265#462—1222no—
266#462C2×D6112yescomplemented
267#451—21no—
268#451—1222no—
269C4×Q16—22no—
270#431—611no—
271#431C2×D6111yescharacteristic
272#428—1221no—
273#423—611no—
274#418—1221no—
275#333—614no—
276Q8⋊C8—614no—
277D4⋊C8—614no—
278C61#1874911yescharacteristic, complemented, Sylow
279C22×Q8—612no—
280C22×Q8C22⋊D6111yescharacteristic
281C22×D4—611no—
282C23×C4—1222no—
283C23×C4—1221no—
284C2×Q16—22no—
285C2×Q16—1224no—
286C2×M16—1221no—
287C4⋊Q8—612no—
288C4⋊Q8—612no—
289C4⋊Q8—1221no—
290C4⋊Q8—1221no—
291C4⋊Q8C22×D6111yescharacteristic
292C4⋊Q8C22×D6111yescharacteristic
293#125—21no—
294#125—1224no—
295#125—1224no—
296C22⋊Q8—21no—
297C22⋊Q8—21no—
298C22⋊Q8—1228no—
299C22⋊Q8—1228no—
300C22⋊Q8—1224no—
301C22⋊Q8—1224no—
302C4×Q8—21no—
303C4×Q8—21no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8C22⋊D6112yes—
307C4×Q8C22×D6112yes—
308C4×D4—612no—
309C4×D4—612no—
310C4×D4—1224no—
311C4×D4—1224no—
312C4×D4—1221no—
313C4×D4—1221no—
314#117—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—611no—
317C2×C4⋊C4—1224no—
318C2×C4⋊C4—1222no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4—1221no—
322C2×C4⋊C4—1224no—
323C2×C4⋊C4C22⋊D6111yescharacteristic
324C2×C22⋊C4—1222no—
325C2×C22⋊C4—1221no—
326C2×C42—21no—
327C2×C42C22×D6111yescharacteristic
328#107—22no—
329#107—1222no—
330#106—1222no—
331C4⋊C8—612no—
332C4⋊C8—1221no—
333C4⋊C8—1221no—
334C4⋊C8C22×D6112yes—
335#103—22no—
336#103—22no—
337#103—1224no—
338#102—1222no—
339#102—1222no—
340C22⋊C8—21no—
341C22⋊C8—1221no—
342C4×C8—21no—
343C4×C8—612no—
344C24—1221no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—42no—
349C2×Q8—614no—
350C2×Q8—1222no—
351C2×Q8—4882no—
352C2×Q8C2×C22⋊D6112yes—
353C2×D4—612no—
354C2×D4—614no—
355C2×D4—1222no—
356C2×D4—1221no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—1228no—
360C22×C4—1221no—
361C22×C4—1222no—
362C22×C4—1222no—
363C22×C4—1222no—
364C22×C4—4884no—
365C22×C4—2444no—
366C22×C4D4×D6111yescharacteristic
367C22×C4D4×D6111yescharacteristic
368C22×C4C2×C22⋊D6111yescharacteristic
369Q16—44no—
370Q16—4884no—
371M16—2442no—
372C2×C8—22no—
373C2×C8—1222no—
374C2×C8—1222no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—612no—
385C4⋊C4—1228no—
386C4⋊C4—1228no—
387C4⋊C4—1222no—
388C4⋊C4—1222no—
389C4⋊C4—1221no—
390C4⋊C4—1222no—
391C4⋊C4—1222no—
392C4⋊C4C2×C22⋊D6112yes—
393C22⋊C4—41no—
394C22⋊C4—1228no—
395C22⋊C4—1222no—
396C22⋊C4—1222no—
397C42—22no—
398C42—22no—
399C42—41no—
400C42D4×D6111yescharacteristic
401C42D4×D6111yescharacteristic
402C42C2×C22⋊D6111yescharacteristic
403C42C23×D6111yescharacteristic
404C23—1222no—
405C23—1224no—
406C23—2442no—
407C23—2442no—
408C23#1451974511yescharacteristic
409Q8—22no—
410Q8—22no—
411Q8—42no—
412Q8—42no—
413Q8—42no—
414Q8—4882no—
415Q8—2444no—
416D4—614no—
417D4—1222no—
418D4—1224no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—1222no—
428C2×C4—1224no—
429C2×C4—4888no—
430C2×C4—2444no—
431C2×C4—2444no—
432C2×C4—2442no—
433C2×C4—2444no—
434C2×C4#1451974511yescharacteristic
435C2×C4#1451974511yescharacteristic
436C2×C4#1451974511yescharacteristic
437C2×C4Q16×D6112yes—
438C2×C4#1451980912yes—
439C2×C4C2×D4×D6111yescharacteristic
440C2×C4C2×D4×D6111yescharacteristic
441C2×C4C22×C22⋊D6111yescharacteristic
442C8—42no—
443C8—2442no—
444C8—2442no—
445C22—21no—
446C22—21no—
447C22—1224no—
448C22—2448no—
449C22—2442no—
450C22—2442no—
451C22#1683991612yes—
452C22C2×#1451974511yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—42no—
457C4—42no—
458C4—81no—
459C4—81no—
460C4—2444no—
461C4—2442no—
462C4#1683991612yes—
463C4C2×Q16×D6111yescharacteristic
464C4C2×#1451980911yescharacteristic
465C4#1684056012yes—
466C2—22no—
467C2—2444no—
468C2C2×#1683991611yescharacteristic
469C2#2145505911yescharacteristic
470C2#2145517411yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3104C2
socleC22×C61C2×#120
prime radicalC22×D244D4
hypercenter#3104D61
hypocenterC61#18749
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
2. #21456764C2semidirect
4. #21445031C2semidirect
5. #21444952C2semidirect
6. #21444499C2semidirect
7. #21444248C2semidirect
9. C61×#3104C2semidirect
21. #16840141C22semidirect
22. #16840114C22semidirect
32. #16839510C22semidirect
33. Q8⋊#92418C22semidirect
38. C61×#477C22semidirect
39. C61×#462C22semidirect
123. C2×D244Q16semidirect
233. #3104D61semidirect
262. #477C2×D61semidirect
266. #462C2×D61semidirect
278. C61#18749semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
#214567642
#214556122
#214450312
#214449522
#214444992
#214442482
#214442092
C61×#31042
#1874961