Finite Group Databasebeta

← #29571735

Subgroups — #29571735

Order 15616.

998 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×#16840417C211yesmaximal, characteristic
2#21445808C211yesmaximal, characteristic, complemented
3#21444492C211yesmaximal, characteristic, complemented
4#21444490C211yesmaximal, characteristic, complemented
5#21444244C211yesmaximal, characteristic, complemented
6#21444139C211yesmaximal, characteristic, complemented
7C61×#1486C211yesmaximal, characteristic, complemented
8C22×#11758222—21no—
9#16840417—22no—
10#16840417—22no—
11#16840417—22no—
12#16840417—22no—
13C2×#14519777C2211yescharacteristic
14C2×#14519759—22no—
15C2×#14519756—21no—
16C2×#14519748C412yes—
17C2×C4×D244C2211yescharacteristic
18C2×C4⋊C4×D61C412yes—
19#16839815C2211yescharacteristic
20#16839786C2211yescharacteristic
21C2×#14519693—21no—
22#16839530—21no—
23#16839509—22no—
24#16839507—22no—
25D4⋊#92418—22no—
26#16839485—22no—
27C2×C4×D4×C61C2211yescharacteristic
28C61×#431C2211yescharacteristic
29C61×#423C2211yescharacteristic
30C61×D4⋊C8—22no—
31C2×#11758222—24no—
32C2×#11758222—24no—
33C2×#11758222—42no—
34C2×#11758222—42no—
35C2×#11758222—42no—
36C2×#11758222—42no—
37C22×Dic122—21no—
38C22×D244D411yescharacteristic
39C22×C22⋊D61—22no—
40C22×C4×D61—22no—
41C22×C4×D61C2×C412yes—
42C2×#11758194—21no—
43#14519777—21no—
44#14519777—21no—
45#14519777D412yes—
46#14519777D412yes—
47#14519759—28no—
48#14519759—28no—
49#14519756—24no—
50#14519756—24no—
51#14519748—24no—
52#14519748—24no—
53C4×D244—21no—
54C4×D244—21no—
55C4×D244—21no—
56C4×D244—21no—
57C2×C4⋊Dic61—21no—
58C2×C4⋊Dic61C2×C412yes—
59C2×C4⋊Dic61D411yescharacteristic
60C2×#11758196—22no—
61C4⋊C4×D61—24no—
62C4⋊C4×D61—24no—
63C2×C22⋊Dic61—22no—
64C2×#11758198D412yes—
65C4⋊#92418—21no—
66C4⋊#92418—21no—
67C4⋊#92418—21no—
68#14519693—42no—
69C22⋊#92418—21no—
70C4×#92418—21no—
71C22×D4×C61C2×C411yescharacteristic
72C23×C4×C61—21no—
73C2×M16×C61—21no—
74C4×D4×C61—21no—
75C4×D4×C61—21no—
76C4×D4×C61D412yes—
77C4×D4×C61D412yes—
78C2×C4⋊C4×C61C2×C411yescharacteristic
79C2×C22⋊C4×C61—21no—
80C2×C42×C61C2311yescharacteristic
81C4⋊C8×C61—21no—
82C4⋊C8×C61—21no—
83C4⋊C8×C61—21no—
84C22⋊C8×C61—21no—
85C4×C8×C61—21no—
86C24×C61—21no—
87C23×D61C22⋊C412yes—
88C2×D4×C61—21no—
89C2×D4×C61—42no—
90C2×D4×C61C22⋊C412yes—
91C22×C4×C61—21no—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61C22×C411yescharacteristic
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D411yescharacteristic
101C2×C22⋊D61—28no—
102C2×C22⋊D61—44no—
103C2×C22⋊D61—44no—
104C22×Dic61—22no—
105C22×Dic61C22⋊C412yes—
106#11758222—48no—
107#11758222—48no—
108#11758222—48no—
109#11758222—48no—
110C2×D244—21no—
111C2×D244—21no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244D812yes—
115C2×D244SD1612yes—
116C2×C4×D61—28no—
117C2×C4×D61—22no—
118C2×C4×D61—22no—
119C2×C4×D61—44no—
120C2×C4×D61—44no—
121C2×C4×D61—44no—
122C2×Dic122—22no—
123C2×Dic122—22no—
124C2×Dic122—41no—
125C2×Dic122—41no—
126C2×Dic122—41no—
127C2×Dic122—41no—
128M16×C61—41no—
129M16×C61—41no—
130C2×C8×C61—41no—
131C2×C8×C61—41no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61C22⋊C412yes—
134C22⋊C4×C61—22no—
135C22⋊C4×C61—22no—
136C42×C61C2×D411yescharacteristic
137C42×C61C2×D411yescharacteristic
138C42×C61C2×D411yescharacteristic
139C42×C61C2×D411yescharacteristic
140C22⋊Dic61—28no—
141#11758198—22no—
142#11758198—22no—
143C4⋊Dic61—22no—
144C4⋊Dic61—22no—
145C4⋊Dic61—22no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61C22⋊C414yes—
149#11758196—28no—
150#11758194—41no—
151#11758194—41no—
152C2×#92418—41no—
153C2×#92418—41no—
154C23×C61—22no—
155C23×C61—42no—
156C23×C61—42no—
157C23×C61—41no—
158C23×C61—41no—
159C23×C61C2×C22⋊C411yescharacteristic
160C22×D61—22no—
161C22×D61—22no—
162C22×D61—42no—
163C22×D61—42no—
164C22×D61#10214yes—
165D4×C61—24no—
166D4×C61—42no—
167C2×C4×C61—22no—
168C2×C4×C61—22no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61—42no—
175C2×C4×C61—42no—
176C2×C4×C61C2×C22⋊C411yescharacteristic
177C2×C4×C61C2×C22⋊C411yescharacteristic
178C2×C4×C61#12011yescharacteristic
179C2×C4×C61#12011yescharacteristic
180C2×C4×C61#12011yescharacteristic
181C2×C4×C61#12011yescharacteristic
182C2×C4×C61C2×D811yescharacteristic
183C2×C4×C61C2×SD1611yescharacteristic
184C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
185C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
186C22⋊D61—416no—
187C2×Dic61—22no—
188C2×Dic61—24no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191C2×Dic61—42no—
192D244—22no—
193D244—22no—
194D244—22no—
195D244—22no—
196C4×D61—44no—
197C4×D61—416no—
198C4×D61—44no—
199Dic122—42no—
200Dic122—42no—
201Dic122—42no—
202Dic122—42no—
203C8×C61—81no—
204C8×C61—81no—
205#92418—81no—
206#92418—81no—
207#8387—611nomaximal, Sylow
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—42no—
211C22×C61—42no—
212C22×C61—42no—
213C22×C61—82no—
214C22×C61—82no—
215C22×C61#37911yescharacteristic
216C22×C61C2×#10211yescharacteristic
217C22×C61#41811yescharacteristic
218C2×D61—24no—
219C2×D61—44no—
220C2×D61—42no—
221C2×D61—42no—
222C2×D61—44no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—42no—
227C4×C61#41312yes—
228C4×C61#44711yescharacteristic
229C4×C61#44711yescharacteristic
230C4×C61#45011yescharacteristic
231C4×C61#45011yescharacteristic
232Dic61—42no—
233Dic61—44no—
234C2×#547—611no—
235#1800—611no—
236#1546—611no—
237#1546—611no—
238#1486—611no—
239#1486D6111yescharacteristic, complemented
240#1460—611no—
241C2×C61—21no—
242C2×C61—21no—
243C2×C61—82no—
244C2×C61#178511yescharacteristic
245C2×C61#178611yescharacteristic
246C2×C61#179611yescharacteristic
247D61—44no—
248C22×#41—1221no—
249#547—1222no—
250#547—1222no—
251#547—1222no—
252#547—1222no—
253C2×C4⋊Q8—611no—
254C2×#123—1222no—
255C2×C22⋊Q8—1221no—
256C2×#121—612no—
257C2×C4×D4—611no—
258C2×C4×D4C2×D6111yescharacteristic
259C22×C4⋊C4—612no—
260#431—611no—
261#431C2×D6111yescharacteristic
262#423—611no—
263#423C2×D6111yescharacteristic
264C2×#100—1221no—
265#343—1221no—
266#332—1222no—
267#332—1222no—
268D4⋊C8—22no—
269D4⋊C8—1222no—
270#323—1222no—
271C61#838711yescharacteristic, complemented, Sylow
272C2×#41—1224no—
273C2×#41—1224no—
274C2×#41—2442no—
275C2×#41—2442no—
276C2×#41—2442no—
277C2×#41—2442no—
278C22×Q8—1221no—
279C22×D4—611no—
280C22×D4—1222no—
281C22×D4C4×D6111yescharacteristic
282C23×C4—21no—
283C23×C4—612no—
284C23×C4—1222no—
285C2×M16—21no—
286C2×M16—1221no—
287C4⋊Q8—612no—
288C4⋊Q8—612no—
289C4⋊Q8—1221no—
290C4⋊Q8—1221no—
291#123—1228no—
292#123—1228no—
293C22⋊Q8—1224no—
294C22⋊Q8—1224no—
295#121—1224no—
296#121—1224no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4—1221no—
300C4×D4—1221no—
301C4×D4—1221no—
302C4×D4—1221no—
303C4×D4D24412yes—
304C4×D4C22⋊D6112yes—
305C2×C4⋊C4—612no—
306C2×C4⋊C4—611no—
307C2×C4⋊C4—1224no—
308C2×C4⋊C4—1222no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1224no—
311C2×C4⋊C4C4×D6111yescharacteristic
312C2×C22⋊C4—21no—
313C2×C22⋊C4—612no—
314C2×C22⋊C4—1222no—
315C2×C42C22×D6111yescharacteristic
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—21no—
319C4⋊C8—1221no—
320C4⋊C8—1221no—
321C4⋊C8—1221no—
322#100—2442no—
323C22⋊C8—21no—
324C22⋊C8—1221no—
325C4×C8—21no—
326C4×C8—1221no—
327C24—21no—
328C24—612no—
329#41—2448no—
330#41—2448no—
331#41—2448no—
332#41—2448no—
333C2×Q8—1222no—
334C2×Q8—1222no—
335C2×Q8—2441no—
336C2×Q8—2441no—
337C2×Q8—2441no—
338C2×Q8—2441no—
339C2×D4—21no—
340C2×D4—42no—
341C2×D4—612no—
342C2×D4—612no—
343C2×D4—1228no—
344C2×D4—1221no—
345C2×D4—1221no—
346C2×D4—1221no—
347C2×D4—1221no—
348C2×D4—2444no—
349C2×D4—2444no—
350C2×D4#1175819812yes—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—612no—
359C22×C4—1228no—
360C22×C4—1222no—
361C22×C4—1222no—
362C22×C4—1222no—
363C22×C4—2444no—
364C22×C4—2444no—
365C22×C4—2444no—
366C22×C4C2×C4×D6111yescharacteristic
367C22×C4D4×D6111yescharacteristic
368C22×C4D4×D6111yescharacteristic
369M16—41no—
370M16—41no—
371M16—2441no—
372M16—2441no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—2441no—
376C2×C8—2441no—
377C4⋊C4—21no—
378C4⋊C4—614no—
379C4⋊C4—1228no—
380C4⋊C4—1222no—
381C4⋊C4—1221no—
382C4⋊C4—1222no—
383C4⋊C4—1221no—
384C4⋊C4—1222no—
385C4⋊C4#1175819812yes—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—1228no—
389C22⋊C4—1222no—
390C22⋊C4—1222no—
391C42C2×D24411yescharacteristic
392C42D4×D6111yescharacteristic
393C42D4×D6111yescharacteristic
394C42C2×C22⋊D6111yescharacteristic
395C23—22no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—41no—
400C23—614no—
401C23—1222no—
402C23—1222no—
403C23—2442no—
404C23—2442no—
405C23C22⋊C4×D6111yescharacteristic
406Q8—2442no—
407Q8—2442no—
408Q8—2442no—
409Q8—2442no—
410D4—24no—
411D4—42no—
412D4—1222no—
413D4—1222no—
414D4—1222no—
415D4—1222no—
416D4—24416no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—1222no—
427C2×C4—1224no—
428C2×C4—2444no—
429C2×C4—24416no—
430C2×C4—2442no—
431C2×C4—2442no—
432C2×C4—2444no—
433C2×C4—2442no—
434C2×C4C22⋊C4×D6111yescharacteristic
435C2×C4C2×#1175819811yescharacteristic
436C2×C4#1451974511yescharacteristic
437C2×C4#1451974511yescharacteristic
438C2×C4#1451974711yescharacteristic
439C2×C4#1451974711yescharacteristic
440C2×C4D8×D6111yescharacteristic
441C2×C4SD16×D6111yescharacteristic
442C2×C4#1451980111yescharacteristic
443C2×C4#1451980411yescharacteristic
444C8—81no—
445C8—81no—
446C8—4881no—
447C8—4881no—
448C22—21no—
449C22—21no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—82no—
454C22—82no—
455C22—1224no—
456C22—2444no—
457C22—2444no—
458C22—2442no—
459C22—2442no—
460C22#1683961711yescharacteristic
461C22#102×D6111yescharacteristic
462C22#1683976511yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—2442no—
468C4—2444no—
469C4#1683974412yes—
470C4#1683989411yescharacteristic
471C4#1683989711yescharacteristic
472C4#1683991111yescharacteristic
473C4#1683991411yescharacteristic
474C2—21no—
475C2—21no—
476C2—82no—
477C2—2444no—
478C2#2144574511yescharacteristic
479C2#2144574811yescharacteristic
480C2#2144579311yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1486C2
socleC22×C61#379
prime radicalC2×#16840417C2
cosocleC2×C42×C61C23
hypercenter#1486D61
hypocenterC61#8387
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
2. #21445808C2semidirect
3. #21444492C2semidirect
4. #21444490C2semidirect
5. #21444244C2semidirect
6. #21444139C2semidirect
7. C61×#1486C2semidirect
239. #1486D61semidirect
271. C61#8387semidirect

Maximal subgroups

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

subgroupindex
C2×#168404172
#214458082
#214444922
#214444902
#214442442
#214441392
C61×#14862
#838761