Finite Group Databasebeta

← #29689724

Subgroups — #29689724

Order 15616.

1106 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840314C211yesmaximal, characteristic
2C2×#16839812C211yesmaximal, characteristic
3#21448222C211yesmaximal, characteristic
4#21446778C212yesmaximal, complemented
5#21446619C212yesmaximal, complemented
6#21446132C211yesmaximal, characteristic
7#21446131C211yesmaximal, characteristic
8#21446129C211yesmaximal, characteristic
9#21445833C212yesmaximal, complemented
10#21445832C212yesmaximal, complemented
11C61×#1873C211yesmaximal, characteristic
12#16840314—24no—
13#16840314C418yescomplemented
14C2×#14519776C2211yescharacteristic
15C2×#14519775C2211yescharacteristic
16C2×#14519767—21no—
17C2×#14519756C2211yescharacteristic
18C2×#14519754—21no—
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519742—21no—
21C2×#14519742C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23C2×#14519733—21no—
24C22×C4×Dic61C2211yescharacteristic
25#16839812—24no—
26C2×#14519706C2212yes—
27C2×#14519702C2212yes—
28C2×#14519701C2212yes—
29C2×#14519700C2212yes—
30C2×C22⋊#92418C2211yescharacteristic
31C2×C8×Dic61C2212yes—
32#16839631C2212yes—
33#16839627C2212yes—
34#16839620—21no—
35#16839620—21no—
36C4×#11758196—21no—
37C4×#11758196—21no—
38#16839525C2212yes—
39#16839524C2212yes—
40#16839524C2212yes—
41C2×C22⋊Q8×C61C2211yescharacteristic
42C2×#117×C61C2211yescharacteristic
43C2×#103×C61C2212yes—
44C2×C22⋊C8×C61C2211yescharacteristic
45C61×#340C2212yes—
46C22×Dic122—21no—
47C22×Dic122C2311yescharacteristic
48C23×Dic61D411yescharacteristic
49C23×Dic61D411yescharacteristic
50C22×#92418C2312yes—
51#14519776—22no—
52#14519776C2×C414yes—
53#14519775—22no—
54#14519775C2×C414yes—
55#14519767—28no—
56#14519756—22no—
57#14519756C2×C414yes—
58#14519754—28no—
59Q8×Dic61—22no—
60Q8×Dic61C2×C414yes—
61#14519742—28no—
62#14519742—22no—
63#14519742C2×C414yes—
64#14519736—42no—
65#14519733—28no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61C2312yes—
68C2×C4⋊Dic61C2311yescharacteristic
69C2×#11758196—21no—
70C2×#11758196—21no—
71C2×#11758196—21no—
72C2×#11758196—21no—
73C2×#11758196—21no—
74C2×#11758196—22no—
75C2×#11758196—22no—
76C2×C22⋊Dic61—21no—
77C2×C22⋊Dic61—21no—
78C2×C4×Dic61—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81C2×C4×Dic61C2×C414yes—
82C2×C4×Dic61D411yescharacteristic
83C2×C4×Dic61D411yescharacteristic
84C2×C4×Dic61D414yes—
85C2×C4×Dic61C2311yescharacteristic
86C2×C4×Dic61C2311yescharacteristic
87#14519706—24no—
88#14519702—42no—
89#14519701—42no—
90#14519700—42no—
91C22⋊#92418—22no—
92C8×Dic61—24no—
93#14519684—22no—
94#14519684—22no—
95C22×Q8×C61C2311yescharacteristic
96C23×C4×C61C2311yescharacteristic
97C22×C8×C61C2312yes—
98C22⋊Q8×C61—22no—
99C22⋊Q8×C61C2×C414yes—
100#117×C61—42no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61—21no—
106C2×C42×C61—21no—
107#103×C61—42no—
108C22⋊C8×C61—22no—
109C24×C61C2×D411yescharacteristic
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61C22×C412yes—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61C22×C412yes—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D412yes—
122C22×C4×C61C2411yescharacteristic
123C22×Dic61—24no—
124C22×Dic61—24no—
125C22×Dic61—21no—
126C22×Dic61—21no—
127C22×Dic61—22no—
128C22×Dic61C22⋊C414yes—
129C22×Dic61C22⋊C414yes—
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C2×Dic122—22no—
135C2×Dic122—22no—
136C2×Dic122—21no—
137C2×Dic122—24no—
138C2×Dic122—44no—
139C2×Dic122C22×C412yes—
140C2×C8×C61—22no—
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—21no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61C22×C412yes—
148C22⋊C4×C61—24no—
149C22⋊C4×C61—42no—
150C42×C61—42no—
151C22⋊Dic61—24no—
152C22⋊Dic61—42no—
153C4⋊Dic61—21no—
154C4⋊Dic61—24no—
155C4⋊Dic61—22no—
156C4⋊Dic61—22no—
157C4⋊Dic61C22×C412yes—
158#11758196—24no—
159#11758196—24no—
160#11758196—24no—
161#11758196—24no—
162#11758196—24no—
163#11758196—44no—
164#11758196—44no—
165C4×Dic61—24no—
166C4×Dic61—24no—
167C4×Dic61—42no—
168C4×Dic61C22⋊C414yes—
169C4×Dic61C22⋊C414yes—
170C4×Dic61C22×C412yes—
171C4×Dic61C22×C412yes—
172C4×Dic61C2×D412yes—
173C4×Dic61C2×D412yes—
174C2×#92418—22no—
175C2×#92418—22no—
176C2×#92418—22no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61C2×C22⋊C412yes—
182C23×C61C22×D411yescharacteristic
183Q8×C61—24no—
184Q8×C61—42no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—24no—
188C2×C4×C61—24no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61C2×C22⋊C412yes—
198C2×C4×C61C23×C411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61H(5,𝔽2)12yes—
201C2×C4×C612−1+412yes—
202C2×Dic61—22no—
203C2×Dic61—22no—
204C2×Dic61—28no—
205C2×Dic61—28no—
206C2×Dic61—22no—
207C2×Dic61—22no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61C2×C22⋊C412yes—
215C2×Dic61C2×C22⋊C412yes—
216C2×Dic61C2×C22⋊C412yes—
217C2×Dic61C2×C22⋊C412yes—
218C2×Dic61#13514yes—
219C2×Dic61#13714yes—
220Dic122—24no—
221Dic122—42no—
222Dic122—48no—
223C8×C61—42no—
224#92418—42no—
225#21864—611nomaximal, Sylow
226C22×C61—22no—
227C22×C61—22no—
228C22×C61—22no—
229C22×C61—42no—
230C22×C61—42no—
231C22×C61C22×C22⋊C411yescharacteristic
232C22×C61#53511yescharacteristic
233C22×C61#53611yescharacteristic
234C22×C61C2×#13512yes—
235C22×C61C2×#13712yes—
236C4×C61—22no—
237C4×C61—42no—
238C4×C61—42no—
239C4×C61—82no—
240C4×C61#52012yes—
241C4×C61#53712yes—
242Dic61—22no—
243Dic61—22no—
244Dic61—42no—
245Dic61—42no—
246Dic61—82no—
247Dic61#41614yes—
248Dic61#41914yes—
249C2×#533—611no—
250C2×#431—611no—
251#2285—611no—
252#1990—612no—
253#1956—612no—
254#1873—611no—
255#1873—611no—
256#1873—611no—
257#1873D6111yescharacteristic
258#1805—612no—
259#1805—612no—
260C2×C61—42no—
261C2×C61#231111yescharacteristic
262C2×C61C2×#41611yescharacteristic
263C2×C61C2×#41911yescharacteristic
264C2×C61#321012yes—
265C2×C61#323112yes—
266#533—618no—
267#533—1224no—
268C2×C4⋊Q8—611no—
269C2×C4⋊Q8—611no—
270C2×#125—1221no—
271C2×C22⋊Q8—611no—
272C2×C22⋊Q8—1221no—
273C2×C22⋊Q8C2×D6111yescharacteristic
274C2×C4×Q8—611no—
275C2×C4×Q8—611no—
276C2×C4×Q8—1221no—
277C2×#117—611no—
278C2×#117—1221no—
279C2×#117C2×D6111yescharacteristic
280C22×C42—611no—
281#431—1224no—
282C2×C4⋊C8—612no—
283C2×#103—612no—
284C2×#103—612no—
285C2×#103—612no—
286C2×#103C2×D6112yes—
287C2×C22⋊C8—611no—
288C2×C22⋊C8C2×D6111yescharacteristic
289C2×C4×C8—612no—
290#384—612no—
291#382—612no—
292#380—1221no—
293#380—1221no—
294C4×C4⋊C4—1221no—
295C4×C4⋊C4—1221no—
296#340—612no—
297#340—612no—
298#340—612no—
299#340C2×D6112yes—
300C61#2186411yescharacteristic, complemented, Sylow
301C22×Q8—611no—
302C22×Q8—1221no—
303C22×Q8C22×D6111yescharacteristic
304C23×C4—611no—
305C23×C4—611no—
306C23×C4C22×D6111yescharacteristic
307C22×C8—612no—
308C22×C8C22×D6112yes—
309C4⋊Q8—614no—
310C4⋊Q8—614no—
311C4⋊Q8—1222no—
312C4⋊Q8—1222no—
313#125—1228no—
314C22⋊Q8—22no—
315C22⋊Q8—614no—
316C22⋊Q8—1228no—
317C22⋊Q8—1222no—
318C22⋊Q8C4×D6114yes—
319C4×Q8—614no—
320C4×Q8—614no—
321C4×Q8—1222no—
322C4×Q8—1228no—
323C4×Q8—1222no—
324#117—42no—
325#117—1228no—
326#117—2442no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—611no—
329C2×C4⋊C4—612no—
330C2×C4⋊C4—1221no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4—1221no—
333C2×C4⋊C4—1221no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4—1222no—
338C2×C4⋊C4C22×D6112yes—
339C2×C4⋊C4C22×D6111yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—1221no—
343C2×C22⋊C4—1221no—
344C2×C42—21no—
345C2×C42—614no—
346C2×C42—611no—
347C2×C42—611no—
348C2×C42—614no—
349C2×C42—611no—
350C2×C42—611no—
351C2×C42—1221no—
352C2×C42—1221no—
353C2×C42—1221no—
354C4⋊C8—1224no—
355#103—42no—
356#103—2442no—
357#103—2442no—
358#103—2442no—
359C22⋊C8—22no—
360C22⋊C8—1222no—
361C4×C8—1224no—
362#95—1222no—
363#95—1222no—
364C24D4×D6111yescharacteristic
365C2×Q8—21no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—612no—
369C2×Q8—1224no—
370C2×Q8—1221no—
371C2×Q8—1222no—
372C2×Q8—1222no—
373C2×Q8—2444no—
374C2×Q8C2×C4×D6112yes—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—611no—
381C22×C4—611no—
382C22×C4—611no—
383C22×C4—611no—
384C22×C4—614no—
385C22×C4—614no—
386C22×C4—1224no—
387C22×C4—1224no—
388C22×C4—1221no—
389C22×C4—1221no—
390C22×C4—1222no—
391C22×C4C2×C4×D6112yes—
392C22×C4D4×D6112yes—
393C22×C4D4×D6111yescharacteristic
394C22×C4C23×D6111yescharacteristic
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—1222no—
399C2×C8—1222no—
400C2×C8—1222no—
401C4⋊C4—21no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—612no—
406C4⋊C4—1224no—
407C4⋊C4—1224no—
408C4⋊C4—1224no—
409C4⋊C4—1224no—
410C4⋊C4—1224no—
411C4⋊C4—1221no—
412C4⋊C4—1224no—
413C4⋊C4—1222no—
414C4⋊C4—1222no—
415C4⋊C4—2444no—
416C4⋊C4—2444no—
417C4⋊C4C2×C4×D6112yes—
418C22⋊C4—24no—
419C22⋊C4—42no—
420C22⋊C4—1224no—
421C22⋊C4—2442no—
422C42—42no—
423C42—612no—
424C42—612no—
425C42—612no—
426C42—612no—
427C42—614no—
428C42—614no—
429C42—1224no—
430C42—1224no—
431C42—2442no—
432C23—22no—
433C23—22no—
434C23—21no—
435C23—21no—
436C23C22⋊C4×D6112yes—
437C23C2×D4×D6111yescharacteristic
438Q8—24no—
439Q8—42no—
440Q8—1224no—
441Q8—2448no—
442Q8—2442no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—612no—
456C2×C4—612no—
457C2×C4—614no—
458C2×C4—614no—
459C2×C4—612no—
460C2×C4—612no—
461C2×C4—1228no—
462C2×C4—1228no—
463C2×C4—1222no—
464C2×C4—1222no—
465C2×C4—1222no—
466C2×C4—1222no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2441no—
472C2×C4—2441no—
473C2×C4C22⋊C4×D6112yes—
474C2×C4C22×C4×D6111yescharacteristic
475C2×C4C2×D4×D6111yescharacteristic
476C2×C4#1451982212yes—
477C2×C4#1451982412yes—
478C8—42no—
479C8—2442no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—42no—
484C22—42no—
485C22C2×C22⋊C4×D6111yescharacteristic
486C22#1684032711yescharacteristic
487C22#1684033411yescharacteristic
488C22#135×D6112yes—
489C22#137×D6112yes—
490C4—22no—
491C4—42no—
492C4—42no—
493C4—82no—
494C4—614no—
495C4—614no—
496C4—1222no—
497C4—1222no—
498C4—4882no—
499C4—2442no—
500C4—2442no—
501C4#1684024112yes—
502C4#1684033912yes—
503C2—42no—
504C2#2144840611yescharacteristic
505C2#416×D6111yescharacteristic
506C2#419×D6111yescharacteristic
507C2#2145677312yes—
508C2#2145697612yes—
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1873C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1873D61
hypocenterC61#21864
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
4. #21446778C2semidirect
5. #21446619C2semidirect
9. #21445833C2semidirect
10. #21445832C2semidirect
13. #16840314C4semidirect
300. C61#21864semidirect

Maximal subgroups

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

subgroupindex
C2×#168403142
C2×#168398122
#214482222
#214467782
#214466192
#214461322
#214461312
#214461292
#214458332
#214458322
C61×#18732
#2186461