Finite Group Databasebeta

← #29689517

Subgroups — #29689517

Order 15616.

1030 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840318C211yesmaximal, characteristic
2C2×#16839813C211yesmaximal, characteristic
3#21448212C211yesmaximal, characteristic
4#21446785C211yesmaximal, characteristic, complemented
5#21446784C211yesmaximal, characteristic, complemented
6#21446783C211yesmaximal, characteristic, complemented
7#21446769C212yesmaximal, complemented
8#21446765C212yesmaximal, complemented
9#21445913C212yesmaximal
10#21445912C212yesmaximal
11C61×#1991C211yesmaximal, 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#16839813—24no—
25C2×#14519706C2212yes—
26C2×#14519689C2212yes—
27C2×C4×#92418C2211yescharacteristic
28#16839629—21no—
29#16839628C2211yescharacteristic
30#16839624—21no—
31#16839619C2212yes—
32C4⋊C4×Dic61C2212yes—
33#16839609—21no—
34#16839607—21no—
35#16839525C2212yes—
36#16839525C2212yes—
37#16839524C2212yes—
38#16839524C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×#125×C61C2211yescharacteristic
42C2×C4×C8×C61C2211yescharacteristic
43C61×#382C2211yescharacteristic
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—22no—
79C2×C4×Dic61D411yescharacteristic
80C2×C4×Dic61D411yescharacteristic
81C2×C4×Dic61C2311yescharacteristic
82C42×D61C2×C414yes—
83C42×D61D414yes—
84#14519706—24no—
85#14519689—24no—
86C4×#92418—22no—
87#14519684—21no—
88#14519684—21no—
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×C61C2311yescharacteristic
100C4×C8×C61—22no—
101#95×C61—21no—
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—24no—
119C2×C4×D61—42no—
120C2×C4×D61C22⋊C414yes—
121C2×C4×D61C22⋊C414yes—
122C2×C4×D61C22×C412yes—
123C2×C4×D61C2×D412yes—
124C2×Dic122—24no—
125C2×Dic122—24no—
126C2×Dic122—44no—
127C2×Dic122—44no—
128C2×C8×C61—22no—
129C2×C8×C61—22no—
130C2×C8×C61—22no—
131C4⋊C4×C61—24no—
132C4⋊C4×C61—24no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—24no—
136C4⋊C4×C61C22×C412yes—
137C4⋊C4×C61C22×C412yes—
138C42×C61C22×C412yes—
139C42×C61C2×D412yes—
140#11758198—24no—
141#11758198—24no—
142#11758198—44no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145C4⋊Dic61—24no—
146C4⋊Dic61C22×C412yes—
147C4⋊Dic61C22×C412yes—
148#11758196—24no—
149#11758196—24no—
150#11758196—24no—
151#11758196—24no—
152C4×Dic61—24no—
153C4×Dic61—24no—
154C4×Dic61—44no—
155C4×Dic61C22⋊C414yes—
156C4×Dic61C22⋊C414yes—
157C4×Dic61C22×C412yes—
158C4×Dic61C2×D412yes—
159C2×#92418—22no—
160C2×#92418—22no—
161C2×#92418—22no—
162C23×C61C22×D411yescharacteristic
163C22×D61—21no—
164C22×D61—21no—
165C22×D61—22no—
166C22×D61—22no—
167C22×D61C2×C22⋊C412yes—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61—42no—
175C2×C4×C61—42no—
176C2×C4×C61—42no—
177C2×C4×C61C2×C22⋊C412yes—
178C2×C4×C61C2×C22⋊C412yes—
179C2×C4×C61C23×C411yescharacteristic
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C612−1+412yes—
182C2×C4×C612−1+412yes—
183C2×Dic61—21no—
184C2×Dic61—21no—
185C2×Dic61—22no—
186C2×Dic61—22no—
187C2×Dic61—22no—
188C2×Dic61—42no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61C2×C22⋊C412yes—
194C2×Dic61C2×C22⋊C412yes—
195C2×Dic61C2×C22⋊C412yes—
196C2×Dic61#13514yes—
197C4×D61—24no—
198C4×D61—24no—
199C4×D61—28no—
200C4×D61—44no—
201Dic122—48no—
202Dic122—48no—
203C8×C61—42no—
204#92418—42no—
205#21837—611nomaximal, Sylow
206C22×C61C22×C22⋊C411yescharacteristic
207C22×C61#53611yescharacteristic
208C22×C61#53611yescharacteristic
209C22×C61C2×#13512yes—
210C22×C61#57512yes—
211C2×D61—22no—
212C2×D61—22no—
213C2×D61—22no—
214C2×D61—42no—
215C2×D61—42no—
216C4×C61—22no—
217C4×C61—22no—
218C4×C61—42no—
219C4×C61—42no—
220C4×C61—82no—
221C4×C61#51912yes—
222C4×C61#53612yes—
223Dic61—22no—
224Dic61—22no—
225Dic61—42no—
226Dic61—42no—
227Dic61—82no—
228Dic61#41614yes—
229C2×#533—611no—
230C2×#431—611no—
231#2283—611no—
232#1991—611no—
233#1991—611no—
234#1991—611no—
235#1991D6111yescharacteristic, complemented
236#1988—612no—
237#1987—612no—
238#1825—612no—
239#1825—612no—
240C2×C61#231011yescharacteristic
241C2×C61C2×#41611yescharacteristic
242C2×C61#289611yescharacteristic
243C2×C61#323312yes—
244C2×C61#323312yes—
245D61—42no—
246#533—618no—
247#533—1224no—
248C2×C4⋊Q8—611no—
249C2×C4⋊Q8—611no—
250C2×#125—611no—
251C2×#125C2×D6111yescharacteristic
252C2×C22⋊Q8—1221no—
253C2×C22⋊Q8—1221no—
254C2×C4×Q8—1221no—
255C2×C4×Q8—1221no—
256C2×#117—611no—
257C2×#117—611no—
258C22×C42—611no—
259#431—1224no—
260C2×C4⋊C8—612no—
261C2×C22⋊C8—612no—
262C2×C4×C8—611no—
263C2×C4×C8C2×D6111yescharacteristic
264#383—1221no—
265#382—611no—
266#382C2×D6111yescharacteristic
267#381—1221no—
268#380—612no—
269C4×C4⋊C4—612no—
270C4×C22⋊C4—1221no—
271#376—1221no—
272#340—612no—
273#340—612no—
274#340—612no—
275#340—612no—
276#340—612no—
277#340—612no—
278#340C2×D6112yes—
279#340C2×D6112yes—
280C61#2183711yescharacteristic, complemented, Sylow
281C22×Q8—1221no—
282C22×Q8—1221no—
283C23×C4—611no—
284C23×C4—611no—
285C23×C4—611no—
286C22×C8—612no—
287C22×C8C22×D6112yes—
288C4⋊Q8—614no—
289C4⋊Q8—614no—
290C4⋊Q8—1222no—
291C4⋊Q8—1222no—
292#125—22no—
293#125—614no—
294#125—1222no—
295#125C4×D6114yes—
296C22⋊Q8—1228no—
297C22⋊Q8—1228no—
298C4×Q8—1228no—
299C4×Q8—1228no—
300#117—614no—
301#117—614no—
302#117—1222no—
303#117—1222no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—611no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4—612no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4—1221no—
312C2×C4⋊C4—1221no—
313C2×C4⋊C4C22×D6112yes—
314C2×C4⋊C4C22×D6111yescharacteristic
315C2×C4⋊C4C22×D6111yescharacteristic
316C2×C22⋊C4—1221no—
317C2×C22⋊C4—1221no—
318C2×C22⋊C4—1222no—
319C2×C42—611no—
320C2×C42—611no—
321C2×C42—611no—
322C2×C42—614no—
323C2×C42—614no—
324C2×C42—1221no—
325C2×C42—1221no—
326C2×C42—1222no—
327C2×C42C22×D6111yescharacteristic
328C4⋊C8—1224no—
329C22⋊C8—1224no—
330C4×C8—22no—
331C4×C8—1222no—
332#95—21no—
333#95—21no—
334#95—1222no—
335#95—1222no—
336#95—1221no—
337#95—1221no—
338C24—611no—
339C2×Q8—1224no—
340C2×Q8—1224no—
341C2×Q8—2444no—
342C2×Q8—2444no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—611no—
347C22×C4—611no—
348C22×C4—611no—
349C22×C4—612no—
350C22×C4—612no—
351C22×C4—614no—
352C22×C4—614no—
353C22×C4—1224no—
354C22×C4—1222no—
355C22×C4—1222no—
356C22×C4—1221no—
357C22×C4—1221no—
358C22×C4—1222no—
359C22×C4—2442no—
360C22×C4D4×D6111yescharacteristic
361C22×C4D4×D6111yescharacteristic
362C22×C4C23×D6111yescharacteristic
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—1222no—
367C2×C8—1222no—
368C2×C8—1222no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—612no—
375C4⋊C4—612no—
376C4⋊C4—1224no—
377C4⋊C4—1224no—
378C4⋊C4—1221no—
379C4⋊C4—1221no—
380C4⋊C4—1224no—
381C4⋊C4—1224no—
382C4⋊C4—1224no—
383C4⋊C4C2×C4×D6112yes—
384C4⋊C4C2×C4×D6112yes—
385C22⋊C4—1224no—
386C22⋊C4—1224no—
387C22⋊C4—2444no—
388C42—612no—
389C42—612no—
390C42—614no—
391C42—614no—
392C42—1224no—
393C42—1224no—
394C42—2444no—
395C42C2×C4×D6112yes—
396C42D4×D6112yes—
397C23—612no—
398C23—1222no—
399C23—1222no—
400C23—1221no—
401C23—1221no—
402C23C2×D4×D6111yescharacteristic
403Q8—2448no—
404Q8—2448no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—612no—
415C2×C4—614no—
416C2×C4—612no—
417C2×C4—612no—
418C2×C4—1228no—
419C2×C4—1222no—
420C2×C4—1221no—
421C2×C4—1221no—
422C2×C4—1224no—
423C2×C4—1224no—
424C2×C4—1222no—
425C2×C4—1222no—
426C2×C4—2444no—
427C2×C4—2442no—
428C2×C4—2442no—
429C2×C4—2442no—
430C2×C4—2442no—
431C2×C4—2442no—
432C2×C4C22⋊C4×D6112yes—
433C2×C4C22⋊C4×D6112yes—
434C2×C4C22×C4×D6111yescharacteristic
435C2×C4C2×D4×D6111yescharacteristic
436C2×C4#1451982412yes—
437C2×C4#1451982512yes—
438C8—42no—
439C8—2442no—
440C22—1222no—
441C22—1222no—
442C22—1222no—
443C22—2442no—
444C22—2442no—
445C22C2×C22⋊C4×D6111yescharacteristic
446C22#1684033211yescharacteristic
447C22#1684033711yescharacteristic
448C22#135×D6112yes—
449C22#1684055912yes—
450C4—22no—
451C4—22no—
452C4—42no—
453C4—42no—
454C4—82no—
455C4—614no—
456C4—1222no—
457C4—1222no—
458C4—4882no—
459C4—2442no—
460C4—2442no—
461C4#1684023812yes—
462C4#1684033512yes—
463C2—2442no—
464C2#2144840211yescharacteristic
465C2#416×D6111yescharacteristic
466C2#2145433411yescharacteristic
467C2#2145699312yes—
468C2#2145699912yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1991C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1991D61
hypocenterC61#21837
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1991

Complements

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

NH ≅ G/Ntype
4. #21446785C2semidirect
5. #21446784C2semidirect
6. #21446783C2semidirect
7. #21446769C2semidirect
8. #21446765C2semidirect
11. C61×#1991C2semidirect
13. #16840318C4semidirect
235. #1991D61semidirect
280. C61#21837semidirect

Maximal subgroups

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

subgroupindex
C2×#168403182
C2×#168398132
#214482122
#214467852
#214467842
#214467832
#214467692
#214467652
#214459132
#214459122
C61×#19912
#2183761