Finite Group Databasebeta

← #29832821

Subgroups — #29832821

Order 15616.

1188 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
1#21459601C211yesmaximal, characteristic
2#21454281C212yesmaximal, complemented
3#21447494C211yesmaximal, characteristic, complemented
4#21447492C214yesmaximal, complemented
5#21447483C214yesmaximal, complemented
6#21447482C212yesmaximal, complemented
7C61×#2129C211yesmaximal, characteristic, complemented
8C2×#14519825C411yescharacteristic, complemented
9C2×#14519824C2211yescharacteristic
10#16840425—22no—
11#16840414—22no—
12#16840410—24no—
13#16840338C411yescharacteristic, complemented
14#16840337C412yescomplemented
15#16840332C2212yes—
16#16840311C2212yes—
17#16840308C2211yescharacteristic
18C2×#14519775C411yescharacteristic, complemented
19C4⋊Q8×D61C412yescomplemented
20#16840239—22no—
21#16840230C411yescharacteristic, complemented
22#16839736C2212yes—
23#16839735—22no—
24#16839734—24no—
25#16839734C2218yes—
26#16839557C2214yes—
27#16839556C2214yes—
28#16839554C2214yes—
29C61×#536C2211yescharacteristic
30C61×#410C2212yes—
31C61×#354C2214yes—
32#14519825—44no—
33#14519824—44no—
34C2×#11758226—21no—
35C2×#11758226C2×C412yes—
36C2×#11758222—22no—
37C2×#11758222C2×C411yescharacteristic
38C2×#11758222C2×C411yescharacteristic
39C2×#11758222D412yes—
40C2×#11758224D414yes—
41C2×#11758224C2312yes—
42C22×Dic122C2×C411yescharacteristic
43C22×Dic122D412yes—
44C2×Q8×D61—21no—
45C2×Q8×D61C2×C412yes—
46C2×Q8×D61C2×C412yes—
47#14519777C2×C412yes—
48#14519776—24no—
49#14519775—22no—
50#14519775C2×C411yescharacteristic
51#14519775C2×C412yes—
52#14519775C2×C412yes—
53#14519773C2311yescharacteristic
54#14519766C2×C411yescharacteristic
55#14519765C2×C412yes—
56#14519764C2311yescharacteristic
57#14519762—24no—
58#14519761—22no—
59#14519759—24no—
60#14519757—22no—
61#14519755—24no—
62#14519754—24no—
63#14519750—24no—
64#14519749—22no—
65C4×Dic122—44no—
66#14519742—21no—
67#14519742—44no—
68#14519739—21no—
69C4×C22⋊D61—22no—
70#14519737—22no—
71#14519737—24no—
72#14519736—22no—
73#14519734C2×C412yes—
74#14519733—28no—
75C2×#11758196—22no—
76C4⋊C4×D61—24no—
77C2×C4×Dic61C2×C411yescharacteristic
78C42×D61C2×C412yes—
79#14519692—22no—
80#14519692C2314yes—
81#14519691—28no—
82C2×#41×C61C2311yescharacteristic
83C22×Q8×C61C2×C411yescharacteristic
84C4⋊Q8×C61C2×C412yes—
85#124×C61C2312yes—
86#123×C61—22no—
87C22⋊Q8×C61—22no—
88#117×C61—22no—
89#117×C61C2×C411yescharacteristic
90#99×C61—22no—
91#99×C61C2314yes—
92#41×C61—42no—
93C2×Q8×C61—21no—
94C2×Q8×C61—42no—
95C2×Q8×C61C22×C412yes—
96C2×Q8×C61C22×C411yescharacteristic
97C2×D4×C61C2×D412yes—
98C2×D4×C61C2411yescharacteristic
99C22×C4×C61—21no—
100C22×C4×C61C22×C411yescharacteristic
101C22×C4×C61C2×D412yes—
102C2×C22⋊D61—21no—
103C2×C22⋊D61—22no—
104C2×C22⋊D61C22⋊C412yes—
105C2×C22⋊D61C2×D414yes—
106C22×Dic61—21no—
107C22×Dic61C22⋊C412yes—
108C22×Dic61C2×D414yes—
109#11758226—44no—
110#11758226—44no—
111Q8×D61—44no—
112Q8×D61—44no—
113Q8×D61—44no—
114#11758224—48no—
115#11758224—44no—
116#11758222—24no—
117#11758222—42no—
118#11758222—42no—
119#11758222—48no—
120#11758222—42no—
121C2×D244—22no—
122C2×D244C22⋊C411yescharacteristic
123C2×D244C22×C411yescharacteristic
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—21no—
127C2×C4×D61—22no—
128C2×C4×D61C22⋊C414yes—
129C2×C4×D61C22×C412yes—
130C2×Dic122—24no—
131C2×Dic122—24no—
132C2×Dic122—22no—
133C2×Dic122—42no—
134C2×Dic122C22⋊C414yes—
135C2×Dic122C22⋊C412yes—
136C2×Dic122C22⋊C411yescharacteristic
137C2×Dic122C22×C411yescharacteristic
138C2×Dic122C22×C412yes—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—42no—
142C22⋊C4×C61—21no—
143C22⋊C4×C61—24no—
144C22⋊C4×C61—24no—
145C42×C61—42no—
146C42×C61C22×C412yes—
147C22⋊Dic61—22no—
148C22⋊Dic61—24no—
149#11758198—22no—
150#11758198—22no—
151#11758198—24no—
152C4⋊Dic61—24no—
153C4⋊Dic61—42no—
154#11758196—22no—
155#11758196—24no—
156#11758196—24no—
157#11758196—22no—
158#11758196—48no—
159C4×Dic61—21no—
160C4×Dic61—22no—
161C4×Dic61—48no—
162C4×Dic61—42no—
163C4×Dic61C22×C411yescharacteristic
164C4×Dic61C22×C411yescharacteristic
165C23×C61C2×C22⋊C411yescharacteristic
166C23×C61C22×D412yes—
167C22×D61—21no—
168C22×D61C2×C22⋊C412yes—
169Q8×C61—41no—
170Q8×C61—42no—
171Q8×C61—42no—
172D4×C61—42no—
173D4×C61—41no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—42no—
180C2×C4×C61—44no—
181C2×C4×C61C2×C22⋊C412yes—
182C2×C4×C61C2×C22⋊C411yescharacteristic
183C2×C4×C61C23×C411yescharacteristic
184C22⋊D61—24no—
185C22⋊D61—44no—
186C22⋊D61—44no—
187C22⋊D61—42no—
188C2×Dic61—21no—
189C2×Dic61—28no—
190C2×Dic61—21no—
191C2×Dic61—21no—
192C2×Dic61—42no—
193C2×Dic61—42no—
194C2×Dic61—44no—
195C2×Dic61C2×C22⋊C414yes—
196C2×Dic61C2×C22⋊C412yes—
197D244—22no—
198D244—41no—
199D244—44no—
200C4×D61—24no—
201C4×D61—44no—
202C4×D61—44no—
203C4×D61—44no—
204C4×D61—42no—
205C4×D61—44no—
206Dic122—22no—
207Dic122—22no—
208Dic122—28no—
209Dic122—42no—
210Dic122—44no—
211Dic122—41no—
212Dic122#9914yes—
213#33360—611nomaximal, Sylow
214C22×C61—22no—
215C22×C61—21no—
216C22×C61—41no—
217C22×C61—42no—
218C22×C61C22×C22⋊C411yescharacteristic
219C2×D61—22no—
220C2×D61—42no—
221C2×D61—42no—
222C2×D61—41no—
223C4×C61—22no—
224C4×C61—21no—
225C4×C61—42no—
226C4×C61—42no—
227C4×C61—84no—
228C4×C61C2×#9912yes—
229Dic61—22no—
230Dic61—24no—
231Dic61—41no—
232Dic61—41no—
233Dic61—84no—
234Dic61C2×#9914yes—
235#3499—611no—
236#2884—612no—
237#2129—614no—
238#2129—611no—
239#2129D6111yescharacteristic, complemented
240#2126—612no—
241#2126—614no—
242C2×C61—21no—
243C2×C61—42no—
244C2×C61—41no—
245C2×C61C22×#9911yescharacteristic
246D61—42no—
247D61—81no—
248C2×2−1+4—611no—
249C2×2−1+4—611no—
250D4×Q8—1222no—
251#547—1222no—
252#547—1224no—
253#536—612no—
254#536—612no—
255#536—611no—
256#536C2×D6111yescharacteristic
257#532—611no—
258#532—612no—
259C2×C4⋊Q8—611no—
260C2×C4⋊Q8—612no—
261#519—1222no—
262C4×#41—611no—
263#410—612no—
264#410—618no—
265#410—1224no—
266#410—1222no—
267#410C2×D6112yes—
268#354—614no—
269#354—614no—
270#354C2×D6114yes—
271#353—614no—
272C61#3336011yescharacteristic, complemented, Sylow
2732−1+4—2444no—
2742−1+4—2444no—
275C2×#41—614no—
276C2×#41—612no—
277C2×#41—611no—
278C2×#41—612no—
279C2×#41—612no—
280C2×#41—611no—
281C2×#41—1222no—
282C2×#41—1221no—
283C2×#41C22×D6111yescharacteristic
284C22×Q8—612no—
285C22×Q8—611no—
286C22×Q8—612no—
287C22×Q8—612no—
288C22×Q8—1221no—
289C22×Q8C4×D6111yescharacteristic
290C4⋊Q8—612no—
291C4⋊Q8—611no—
292C4⋊Q8—612no—
293C4⋊Q8—612no—
294C4⋊Q8—1224no—
295C4⋊Q8—1222no—
296C4⋊Q8C4×D6112yes—
297#127—611no—
298#124—611no—
299#124—611no—
300#124—612no—
301#124C22×D6112yes—
302#123—22no—
303#123—1222no—
304#123—1224no—
305#123—1224no—
306C22⋊Q8—22no—
307C22⋊Q8—1224no—
308C22⋊Q8—1224no—
309C22⋊Q8—1222no—
310#121—1224no—
311#121—1222no—
312C4×Q8—1221no—
313C4×Q8—2444no—
314C4×Q8—2444no—
315C4×D4—1224no—
316C4×D4—1222no—
317C4×D4—1221no—
318C4×D4—1222no—
319#117—22no—
320#117—612no—
321#117—1228no—
322#117—1222no—
323#117C4×D6111yescharacteristic
324C2×C4⋊C4—1224no—
325C2×C4⋊C4—1222no—
326C2×C42—611no—
327C2×C42—612no—
328#99—22no—
329#99—614no—
330#99—1228no—
331#99—1222no—
332#99C22×D6114yes—
333#41—42no—
334#41—1224no—
335#41—2442no—
336#41—2442no—
337#41—2442no—
338#41—2448no—
339#41—2444no—
340#41—2448no—
341#41—2444no—
342#41—2444no—
343C2×Q8—21no—
344C2×Q8—42no—
345C2×Q8—612no—
346C2×Q8—614no—
347C2×Q8—611no—
348C2×Q8—612no—
349C2×Q8—611no—
350C2×Q8—1222no—
351C2×Q8—1224no—
352C2×Q8—1224no—
353C2×Q8—2442no—
354C2×Q8—2444no—
355C2×Q8—2444no—
356C2×Q8—2444no—
357C2×Q8C2×C4×D6112yes—
358C2×Q8C2×C4×D6111yescharacteristic
359C2×D4—612no—
360C2×D4—614no—
361C2×D4—611no—
362C2×D4—611no—
363C2×D4—1222no—
364C2×D4—1222no—
365C2×D4—1221no—
366C2×D4D4×D6112yes—
367C2×D4C23×D6111yescharacteristic
368C22×C4—21no—
369C22×C4—614no—
370C22×C4—612no—
371C22×C4—612no—
372C22×C4—614no—
373C22×C4—1222no—
374C22×C4—1222no—
375C22×C4—1222no—
376C22×C4—1221no—
377C22×C4—1221no—
378C22×C4C2×C4×D6111yescharacteristic
379C22×C4D4×D6112yes—
380C4⋊C4—24no—
381C4⋊C4—21no—
382C4⋊C4—42no—
383C4⋊C4—1224no—
384C4⋊C4—1224no—
385C4⋊C4—1224no—
386C4⋊C4—1222no—
387C4⋊C4—1222no—
388C4⋊C4—2442no—
389C4⋊C4—2448no—
390C22⋊C4—24no—
391C22⋊C4—21no—
392C22⋊C4—24no—
393C22⋊C4—1224no—
394C22⋊C4—1222no—
395C22⋊C4—1222no—
396C22⋊C4—1222no—
397C22⋊C4—1224no—
398C42—42no—
399C42—611no—
400C42—611no—
401C42—1222no—
402C42—1221no—
403C42—2442no—
404C42—2448no—
405C42C2×C4×D6112yes—
406C23—612no—
407C23—1221no—
408C23C22⋊C4×D6111yescharacteristic
409C23C2×D4×D6112yes—
410Q8—41no—
411Q8—42no—
412Q8—42no—
413Q8—614no—
414Q8—1228no—
415Q8—1222no—
416Q8—1222no—
417Q8—2442no—
418Q8—2441no—
419Q8—2444no—
420D4—42no—
421D4—41no—
422D4—1224no—
423D4—1222no—
424D4—2441no—
425D4—2444no—
426D4—2444no—
427D4—2444no—
428D4—2442no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—612no—
437C2×C4—614no—
438C2×C4—1224no—
439C2×C4—1228no—
440C2×C4—1221no—
441C2×C4—1221no—
442C2×C4—1221no—
443C2×C4—2444no—
444C2×C4—2444no—
445C2×C4—2444no—
446C2×C4—2444no—
447C2×C4—2442no—
448C2×C4—2442no—
449C2×C4—2444no—
450C2×C4—2442no—
451C2×C4C22⋊C4×D6112yes—
452C2×C4C22⋊C4×D6111yescharacteristic
453C2×C4C22×C4×D6111yescharacteristic
454C22—22no—
455C22—21no—
456C22—41no—
457C22—42no—
458C22—1222no—
459C22—2442no—
460C22—2442no—
461C22—2441no—
462C22C2×C22⋊C4×D6111yescharacteristic
463C4—21no—
464C4—22no—
465C4—42no—
466C4—42no—
467C4—84no—
468C4—614no—
469C4—1222no—
470C4—1224no—
471C4—4884no—
472C4—2441no—
473C4—2441no—
474C4#99×D6112yes—
475C2—21no—
476C2—42no—
477C2—41no—
478C2—4881no—
479C2—2442no—
480C2C2×#99×D6111yescharacteristic
subgroupG/H
centerC2C2×#99×D61
commutatorC2×C4×C61C23×C4
FrattiniC2×D4C23×D61
FittingC61×#2129C2
socleC2×C61C22×#99
prime radical#21459601C2
cosocleC2×D4×C61C24
hypercenter#2129D61
hypocenterC61#33360
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #2129

Complements

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

NH ≅ G/Ntype
2. #21454281C2semidirect
3. #21447494C2semidirect
4. #21447492C2semidirect
5. #21447483C2semidirect
6. #21447482C2semidirect
7. C61×#2129C2semidirect
8. C2×#14519825C4semidirect
13. #16840338C4semidirect
14. #16840337C4semidirect
18. C2×#14519775C4semidirect
19. C4⋊Q8×D61C4semidirect
21. #16840230C4semidirect
239. #2129D61semidirect
272. C61#33360semidirect

Maximal subgroups

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

subgroupindex
#214596012
#214542812
#214474942
#214474922
#214474832
#214474822
C61×#21292
#3336061