Finite Group Databasebeta

← #29689554

Subgroups — #29689554

Order 15616.

1010 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840318C211yesmaximal, characteristic
2C2×#16839818C211yesmaximal, characteristic
3#21448215C211yesmaximal, characteristic
4#21446790C212yesmaximal, complemented
5#21446790C212yesmaximal, complemented
6#21446777C211yesmaximal, characteristic, complemented
7#21446776C211yesmaximal, characteristic, complemented
8#21446774C211yesmaximal, characteristic, complemented
9#21445913C212yesmaximal
10#21445912C212yesmaximal
11C61×#1989C211yesmaximal, characteristic, complemented
12#16840318—24no—
13#16840318C418yescomplemented
14C2×#14519776C2211yescharacteristic
15C2×#14519776C2211yescharacteristic
16C2×#14519769C2211yescharacteristic
17C2×#14519755—21no—
18C2×#14519755—21no—
19C2×#14519742—21no—
20C2×#14519742—21no—
21C2×#14519735C2211yescharacteristic
22C2×#14519735C2211yescharacteristic
23C2×C42×D61C2211yescharacteristic
24#16839818—24no—
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519689C2212yes—
27C2×#14519687C2212yes—
28#16839629—21no—
29#16839627C2212yes—
30#16839624—21no—
31#16839619C2212yes—
32C4×C4⋊Dic61C2211yescharacteristic
33C4×#11758198—21no—
34#16839608—21no—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#125×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C4×C4⋊C4×C61C2211yescharacteristic
44C61×#340C2212yes—
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122—21no—
48C22×C4×D61D411yescharacteristic
49C22×C4×D61D411yescharacteristic
50C22×C4×D61C2311yescharacteristic
51C22×#92418C2312yes—
52#14519776—22no—
53#14519776—22no—
54#14519776C2×C414yes—
55#14519776C2×C414yes—
56#14519769—22no—
57#14519769C2×C414yes—
58#14519755—28no—
59#14519755—28no—
60#14519742—28no—
61#14519742—28no—
62#14519735—22no—
63#14519735—22no—
64#14519735C2×C414yes—
65#14519735C2×C414yes—
66C2×C4⋊Dic61C2312yes—
67C2×C4⋊Dic61C2311yescharacteristic
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—21no—
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758198—21no—
74C2×#11758198—21no—
75C2×#11758198—22no—
76C2×C4×Dic61—21no—
77C2×C4×Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61C2311yescharacteristic
82C42×D61C2×C414yes—
83C42×D61D414yes—
84C4⋊#92418—22no—
85#14519689—24no—
86#14519687—24no—
87#14519684—21no—
88#14519684—22no—
89#14519684—22no—
90#14519684—22no—
91C22×C8×C61C2312yes—
92#125×C61—22no—
93#125×C61C2×C414yes—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61C2312yes—
97C2×C4⋊C4×C61C2311yescharacteristic
98C2×C4⋊C4×C61C2311yescharacteristic
99C2×C42×C61—21no—
100C2×C42×C61C2311yescharacteristic
101C4⋊C8×C61—22no—
102#95×C61—21no—
103C23×D61C2×D411yescharacteristic
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—22no—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2411yescharacteristic
110C22×Dic61—21no—
111C22×Dic61—21no—
112C22×Dic61—22no—
113C22×Dic61C2×D411yescharacteristic
114C22×Dic61C2×D411yescharacteristic
115C22×Dic61C2×D411yescharacteristic
116C2×C4×D61—22no—
117C2×C4×D61—22no—
118C2×C4×D61—41no—
119C2×C4×D61—41no—
120C2×C4×D61—42no—
121C2×C4×D61C22⋊C414yes—
122C2×C4×D61C22⋊C414yes—
123C2×C4×D61C22×C412yes—
124C2×C4×D61C2×D412yes—
125C2×Dic122—24no—
126C2×Dic122—24no—
127C2×Dic122—42no—
128C2×Dic122—44no—
129C2×Dic122—42no—
130C2×C8×C61—22no—
131C2×C8×C61—24no—
132C4⋊C4×C61—24no—
133C4⋊C4×C61—24no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61C22×C412yes—
139C4⋊C4×C61C22×C412yes—
140C42×C61—42no—
141C42×C61C22×C412yes—
142C42×C61C2×D412yes—
143#11758198—24no—
144#11758198—24no—
145#11758198—44no—
146C4⋊Dic61—21no—
147C4⋊Dic61—21no—
148C4⋊Dic61—22no—
149C4⋊Dic61—22no—
150C4⋊Dic61C22×C412yes—
151C4⋊Dic61C22×C412yes—
152#11758196—24no—
153#11758196—24no—
154#11758196—24no—
155#11758196—24no—
156C4×Dic61—24no—
157C4×Dic61—24no—
158C4×Dic61—42no—
159C4×Dic61C22⋊C414yes—
160C4×Dic61C22⋊C414yes—
161C4×Dic61C22×C412yes—
162C4×Dic61C2×D412yes—
163C2×#92418—22no—
164C2×#92418—24no—
165C23×C61C22×D411yescharacteristic
166C22×D61—22no—
167C22×D61—22no—
168C22×D61—21no—
169C22×D61—21no—
170C22×D61C2×C22⋊C412yes—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—22no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—41no—
177C2×C4×C61—42no—
178C2×C4×C61—41no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61C2×C22⋊C412yes—
183C2×C4×C61C2×C22⋊C412yes—
184C2×C4×C61C23×C411yescharacteristic
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C612−1+412yes—
187C2×C4×C612−1+412yes—
188C2×Dic61—22no—
189C2×Dic61—21no—
190C2×Dic61—22no—
191C2×Dic61—21no—
192C2×Dic61—22no—
193C2×Dic61—22no—
194C2×Dic61—41no—
195C2×Dic61—42no—
196C2×Dic61—41no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61C2×C22⋊C412yes—
201C2×Dic61C2×C22⋊C412yes—
202C2×Dic61C2×C22⋊C412yes—
203C4×D61—24no—
204C4×D61—22no—
205C4×D61—22no—
206C4×D61—44no—
207C4×D61—44no—
208Dic122—48no—
209Dic122—48no—
210C8×C61—42no—
211#92418—42no—
212#21842—611nomaximal, Sylow
213C22×C61C22×C22⋊C411yescharacteristic
214C22×C61#53612yes—
215C22×C61#57511yescharacteristic
216C22×C61#57512yes—
217C22×C61#57511yescharacteristic
218C2×D61—22no—
219C2×D61—22no—
220C2×D61—22no—
221C2×D61—42no—
222C2×D61—42no—
223C4×C61—22no—
224C4×C61—22no—
225C4×C61—42no—
226C4×C61—42no—
227C4×C61—82no—
228C4×C61#51912yes—
229C4×C61#53611yescharacteristic
230C4×C61#53611yescharacteristic
231Dic61—22no—
232Dic61—22no—
233Dic61—22no—
234Dic61—42no—
235Dic61—42no—
236Dic61—82no—
237C2×#533—611no—
238C2×#432—611no—
239#2283—611no—
240#1993—612no—
241#1993—612no—
242#1989—611no—
243#1989—611no—
244#1989—611no—
245#1989D6111yescharacteristic, complemented
246#1825—612no—
247#1825—612no—
248C2×C61#231011yescharacteristic
249C2×C61#289611yescharacteristic
250C2×C61#289611yescharacteristic
251C2×C61#324012yes—
252C2×C61#324012yes—
253D61—42no—
254#533—618no—
255#533—1224no—
256C2×C4⋊Q8—611no—
257C2×C4⋊Q8—611no—
258C2×#125—611no—
259C2×#125C2×D6111yescharacteristic
260C2×C22⋊Q8—1221no—
261C2×C22⋊Q8—1221no—
262C2×C4×Q8—1221no—
263C2×C4×Q8—1221no—
264C2×#117—611no—
265C2×#117—611no—
266C22×C42—611no—
267#432—1224no—
268C2×C4⋊C8—611no—
269C2×C4⋊C8C2×D6111yescharacteristic
270C2×C22⋊C8—612no—
271C2×#97—612no—
272#383—1221no—
273#382—612no—
274#381—1221no—
275#380—612no—
276C4×C4⋊C4—611no—
277C4×C4⋊C4C2×D6111yescharacteristic
278C4×C22⋊C4—1221no—
279#376—1221no—
280#340—612no—
281#340—612no—
282#340—612no—
283#340—612no—
284#340—612no—
285#340—612no—
286#340C2×D6112yes—
287#340C2×D6112yes—
288C61#2184211yescharacteristic, complemented, Sylow
289C22×Q8—1221no—
290C22×Q8—1221no—
291C23×C4—611no—
292C23×C4—611no—
293C23×C4—611no—
294C22×C8—612no—
295C22×C8C22×D6112yes—
296C4⋊Q8—614no—
297C4⋊Q8—614no—
298C4⋊Q8—1222no—
299C4⋊Q8—1222no—
300#125—22no—
301#125—614no—
302#125—1222no—
303#125C4×D6114yes—
304C22⋊Q8—1228no—
305C22⋊Q8—1228no—
306C4×Q8—1228no—
307C4×Q8—1228no—
308#117—614no—
309#117—614no—
310#117—1222no—
311#117—1222no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—611no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—612no—
317C2×C4⋊C4—1221no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4—1221no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4C22×D6112yes—
322C2×C4⋊C4C22×D6111yescharacteristic
323C2×C4⋊C4C22×D6111yescharacteristic
324C2×C22⋊C4—1221no—
325C2×C22⋊C4—1221no—
326C2×C22⋊C4—1222no—
327C2×C42—21no—
328C2×C42—611no—
329C2×C42—611no—
330C2×C42—611no—
331C2×C42—614no—
332C2×C42—614no—
333C2×C42—1221no—
334C2×C42—1221no—
335C2×C42—1221no—
336C2×C42C22×D6111yescharacteristic
337C4⋊C8—22no—
338C4⋊C8—1222no—
339C22⋊C8—1224no—
340#97—1224no—
341#95—21no—
342#95—1222no—
343#95—1222no—
344#95—1222no—
345#95—1221no—
346C24—611no—
347C2×Q8—1224no—
348C2×Q8—1224no—
349C2×Q8—2442no—
350C2×Q8—2444no—
351C2×Q8—2442no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—611no—
356C22×C4—611no—
357C22×C4—611no—
358C22×C4—612no—
359C22×C4—612no—
360C22×C4—614no—
361C22×C4—614no—
362C22×C4—1222no—
363C22×C4—1222no—
364C22×C4—1221no—
365C22×C4—1221no—
366C22×C4—1222no—
367C22×C4—2441no—
368C22×C4—2442no—
369C22×C4—2441no—
370C22×C4D4×D6111yescharacteristic
371C22×C4D4×D6111yescharacteristic
372C22×C4C23×D6111yescharacteristic
373C2×C8—22no—
374C2×C8—24no—
375C2×C8—1222no—
376C2×C8—1224no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—612no—
384C4⋊C4—612no—
385C4⋊C4—1224no—
386C4⋊C4—1224no—
387C4⋊C4—1221no—
388C4⋊C4—1221no—
389C4⋊C4—1224no—
390C4⋊C4—1224no—
391C4⋊C4—1222no—
392C4⋊C4—1222no—
393C4⋊C4C2×C4×D6112yes—
394C4⋊C4C2×C4×D6112yes—
395C22⋊C4—1224no—
396C22⋊C4—1224no—
397C22⋊C4—2444no—
398C42—42no—
399C42—612no—
400C42—612no—
401C42—614no—
402C42—614no—
403C42—1224no—
404C42—1224no—
405C42—2442no—
406C42C2×C4×D6112yes—
407C42D4×D6112yes—
408C23—612no—
409C23—1221no—
410C23—1222no—
411C23—1221no—
412C23—1222no—
413C23C2×D4×D6111yescharacteristic
414Q8—2448no—
415Q8—2448no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—612no—
428C2×C4—612no—
429C2×C4—612no—
430C2×C4—1221no—
431C2×C4—1222no—
432C2×C4—1221no—
433C2×C4—1222no—
434C2×C4—1224no—
435C2×C4—1222no—
436C2×C4—1222no—
437C2×C4—1222no—
438C2×C4—1222no—
439C2×C4—2444no—
440C2×C4—2444no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4—2442no—
444C2×C4—2441no—
445C2×C4—2442no—
446C2×C4—2441no—
447C2×C4C22⋊C4×D6112yes—
448C2×C4C22⋊C4×D6112yes—
449C2×C4C22×C4×D6111yescharacteristic
450C2×C4C2×D4×D6111yescharacteristic
451C2×C4#1451982412yes—
452C2×C4#1451982512yes—
453C8—42no—
454C8—2442no—
455C22—1222no—
456C22—1222no—
457C22—1222no—
458C22—2442no—
459C22—2442no—
460C22C2×C22⋊C4×D6111yescharacteristic
461C22#1684033512yes—
462C22#1684055811yescharacteristic
463C22#1684055912yes—
464C22#1684056111yescharacteristic
465C4—22no—
466C4—22no—
467C4—42no—
468C4—42no—
469C4—82no—
470C4—1222no—
471C4—1222no—
472C4—1222no—
473C4—4882no—
474C4—2442no—
475C4—2442no—
476C4#1684023812yes—
477C4#1684033211yescharacteristic
478C4#1684033711yescharacteristic
479C2—2442no—
480C2#2144840211yescharacteristic
481C2#2145433411yescharacteristic
482C2#2145433511yescharacteristic
483C2#2145705812yes—
484C2#2145706412yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1989C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1989D61
hypocenterC61#21842
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
4. #21446790C2semidirect
5. #21446790C2semidirect
6. #21446777C2semidirect
7. #21446776C2semidirect
8. #21446774C2semidirect
11. C61×#1989C2semidirect
13. #16840318C4semidirect
245. #1989D61semidirect
288. C61#21842semidirect

Maximal subgroups

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

subgroupindex
C2×#168403182
C2×#168398182
#214482152
#214467902
#214467772
#214467762
#214467742
#214459132
#214459122
C61×#19892
#2184261