Finite Group Databasebeta

← #29734438

Subgroups — #29734438

Order 15616.

968 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840172C211yesmaximal, characteristic, complemented
2C2×#16840112C211yesmaximal, characteristic, complemented
3#21451640C211yesmaximal, characteristic
4#21447019C212yesmaximal
5#21447012C212yesmaximal
6#21446785C211yesmaximal, characteristic, complemented
7#21446640C211yesmaximal, characteristic, complemented
8#21446087C212yesmaximal, complemented
9#21446073C212yesmaximal, complemented
10#21445585C211yesmaximal, characteristic
11C2×C61×#541C211yesmaximal, characteristic
12C2×#14519777C2211yescharacteristic
13#16840172—24no—
14#16840112—24no—
15C2×#14519710C2212yes—
16C2×#14519708C2212yes—
17C2×#14519701C2212yes—
18C2×#14519701C2212yes—
19C2×#14519701C2212yes—
20C2×#14519701C2212yes—
21C2×C22⋊#92418C2212yes—
22C2×#14519688C2211yescharacteristic
23C2×C4×#92418C2211yescharacteristic
24#16839701—21no—
25#16839698—21no—
26#16839682—21no—
27#16839674—21no—
28#16839660C2212yes—
29#16839628C2211yescharacteristic
30#16839622C2212yes—
31#16839524C2212yes—
32#16839524C2212yes—
33#16839517C2212yes—
34C61×#541—24no—
35C61×#541—24no—
36C2×C4⋊Q8×C61C2211yescharacteristic
37C2×#126×C61—21no—
38C2×#126×C61—21no—
39C2×#125×C61—21no—
40C2×#125×C61—21no—
41C2×#125×C61C2211yescharacteristic
42C2×C22⋊Q8×C61C2211yescharacteristic
43C2×C22⋊Q8×C61C2211yescharacteristic
44C2×C4×Q8×C61C2211yescharacteristic
45C2×C4×Q8×C61C2211yescharacteristic
46C2×#117×C61C2211yescharacteristic
47C22×Dic122—21no—
48C22×#92418C2312yes—
49C22×#92418C2312yes—
50#14519777—22no—
51#14519777—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2312yes—
54C2×C4⋊Dic61C2312yes—
55C2×#11758196—22no—
56C2×C22⋊Dic61—22no—
57#14519710—24no—
58#14519708—24no—
59#14519701—24no—
60#14519701—24no—
61#14519701—24no—
62#14519701—24no—
63C22⋊#92418—24no—
64#14519688—22no—
65C4×#92418—22no—
66#14519684—22no—
67#14519684—21no—
68#14519684—21no—
69#14519684—22no—
70C22×Q8×C61C2311yescharacteristic
71C22×Q8×C61C2311yescharacteristic
72C23×C4×C61C2311yescharacteristic
73C4⋊Q8×C61—22no—
74C4⋊Q8×C61D414yes—
75#126×C61—28no—
76#126×C61—28no—
77#125×C61—28no—
78#125×C61—28no—
79#125×C61—22no—
80#125×C61D414yes—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—22no—
85C4×Q8×C61—22no—
86C4×Q8×C61—22no—
87C4×Q8×C61—22no—
88C4×Q8×C61—22no—
89#117×C61—22no—
90#117×C61—22no—
91C2×C4⋊C4×C61—21no—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61C2311yescharacteristic
102C2×C4⋊C4×C61C2311yescharacteristic
103C2×C22⋊C4×C61—21no—
104C2×C22⋊C4×C61—21no—
105C2×C22⋊C4×C61D411yescharacteristic
106C2×C22⋊C4×C61D411yescharacteristic
107C2×C42×C61—21no—
108C2×C42×C61—21no—
109C2×C42×C61C2311yescharacteristic
110C2×C42×C61C2311yescharacteristic
111C24×C61C2×D411yescharacteristic
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—22no—
115C2×Q8×C61—22no—
116C2×Q8×C61—22no—
117C2×Q8×C61—22no—
118C2×Q8×C61C2×D412yes—
119C2×Q8×C61C2×D412yes—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61—22no—
127C22×C4×C61—22no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—22no—
135C22×Dic61—22no—
136C2×Dic122—42no—
137C2×Dic122—82no—
138C4⋊C4×C61—24no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—24no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—24no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—24no—
145C4⋊C4×C61—24no—
146C4⋊C4×C61—22no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—21no—
149C4⋊C4×C61—21no—
150C4⋊C4×C61C2×D412yes—
151C4⋊C4×C61C2×D412yes—
152C22⋊C4×C61—24no—
153C22⋊C4×C61—24no—
154C22⋊C4×C61—22no—
155C22⋊C4×C61—22no—
156C42×C61—24no—
157C42×C61—24no—
158C42×C61—21no—
159C42×C61—21no—
160C42×C61C2×D412yes—
161C42×C61C2×D412yes—
162C22⋊Dic61—82no—
163C4⋊Dic61—22no—
164C4⋊Dic61—22no—
165C4⋊Dic61—22no—
166C4⋊Dic61—22no—
167C4⋊Dic61—42no—
168#11758196—82no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418#4114yes—
175C23×C61—21no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61C22×D411yescharacteristic
181Q8×C61—24no—
182Q8×C61—24no—
183Q8×C61—42no—
184Q8×C61—42no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—21no—
196C2×C4×C61—22no—
197C2×C4×C61—22no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61#12012yes—
207C2×C4×C61#12012yes—
208C2×C4×C61C22×D411yescharacteristic
209C2×C4×C61C22×D411yescharacteristic
210C2×C4×C61C2×#4112yes—
211C2×C4×C612−1+412yes—
212C2×Dic61—42no—
213C2×Dic61—42no—
214C2×Dic61—42no—
215C2×Dic61—42no—
216C2×Dic61—42no—
217C2×Dic61—42no—
218Dic122—84no—
219#92418—24no—
220#92418—42no—
221#25686—611nomaximal, Sylow
222C22×C61—42no—
223C22×C61—41no—
224C22×C61—42no—
225C22×C61—41no—
226C22×C61—41no—
227C22×C61C2×#12011yescharacteristic
228C22×C61#53211yescharacteristic
229C22×C61#53611yescharacteristic
230C22×C61#57712yes—
231C22×C61#57812yes—
232C4×C61—22no—
233C4×C61—22no—
234C4×C61—41no—
235C4×C61—42no—
236C4×C61—42no—
237C4×C61—41no—
238C4×C61—42no—
239C4×C61—41no—
240C4×C61—42no—
241C4×C61#54712yes—
242C4×C61#54812yes—
243Dic61—82no—
244Dic61—82no—
245C2×#541D6111yescharacteristic
246C2×#501—611no—
247C2×#487—611no—
248#2628—611no—
249#2036—612no—
250#2035—612no—
251#1991—611no—
252#1960—611no—
253#1865—612no—
254#1863—612no—
255#1750—611no—
256C2×C61—81no—
257C2×C61#243211yescharacteristic
258C2×C61#301011yescharacteristic
259C2×C61#301111yescharacteristic
260C2×C61#317812yes—
261C2×C61#324212yes—
262#541—24no—
263#541—24no—
264C2×C4⋊Q8—611no—
265C2×C4⋊Q8C2×D6111yescharacteristic
266C2×#126—21no—
267C2×#126—21no—
268C2×#125—21no—
269C2×#125—21no—
270C2×#125C2×D6111yescharacteristic
271C2×C22⋊Q8C2×D6111yescharacteristic
272C2×C22⋊Q8C2×D6111yescharacteristic
273C2×C4×Q8C2×D6111yescharacteristic
274C2×C4×Q8C2×D6111yescharacteristic
275C2×#117C2×D6111yescharacteristic
276#501—1224no—
277#487—1224no—
278C2×#107—612no—
279C2×#106—612no—
280C2×#103—612no—
281C2×#103—612no—
282C2×#103—612no—
283C2×#103—612no—
284C2×C22⋊C8—612no—
285C2×#97—611no—
286C2×C4×C8—611no—
287#400—1221no—
288#399—1221no—
289#395—1221no—
290#393—1221no—
291#389—612no—
292#382—611no—
293#380—612no—
294#340—612no—
295#340—612no—
296#336—612no—
297C61#2568611yescharacteristic, complemented, Sylow
298C22×Q8—1221no—
299C22×Q8C22×D6111yescharacteristic
300C22×Q8C22×D6111yescharacteristic
301C23×C4C22×D6111yescharacteristic
302C22×C8—612no—
303C22×C8—612no—
304C4⋊Q8—22no—
305C4⋊Q8—1222no—
306C4⋊Q8—1222no—
307C4⋊Q8C22⋊D6114yes—
308#126—28no—
309#126—28no—
310#125—28no—
311#125—28no—
312#125—22no—
313#125C22⋊D6114yes—
314C22⋊Q8—22no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C4×Q8—22no—
319C4×Q8—22no—
320C4×Q8—22no—
321C4×Q8—22no—
322#117—22no—
323#117—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—612no—
333C2×C4⋊C4—612no—
334C2×C4⋊C4—1222no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4C22⋊D6111yescharacteristic
337C2×C4⋊C4C22⋊D6111yescharacteristic
338C2×C4⋊C4C22×D6111yescharacteristic
339C2×C4⋊C4C22×D6111yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—1222no—
343C2×C22⋊C4C22⋊D6111yescharacteristic
344C2×C22⋊C4C22⋊D6111yescharacteristic
345C2×C42—21no—
346C2×C42—21no—
347C2×C42C22×D6111yescharacteristic
348C2×C42C22×D6111yescharacteristic
349#107—1224no—
350#106—1224no—
351#103—1224no—
352#103—1224no—
353#103—1224no—
354#103—1224no—
355C22⋊C8—1224no—
356#97—1222no—
357C4×C8—1222no—
358#95—1222no—
359#95—1221no—
360#95—1221no—
361#95—1222no—
362C24C2×C22⋊D6111yescharacteristic
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—4882no—
370C2×Q8—2442no—
371C2×Q8C2×C22⋊D6112yes—
372C2×Q8C2×C22⋊D6112yes—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—1222no—
382C22×C4—1222no—
383C22×C4C2×C22⋊D6111yescharacteristic
384C22×C4C2×C22⋊D6111yescharacteristic
385C22×C4C2×C22⋊D6111yescharacteristic
386C22×C4C2×C22⋊D6111yescharacteristic
387C22×C4C2×C22⋊D6111yescharacteristic
388C22×C4C23×D6111yescharacteristic
389C2×C8—614no—
390C2×C8—1222no—
391C2×C8—1222no—
392C2×C8—1222no—
393C2×C8—1222no—
394C2×C8—1222no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—1222no—
408C4⋊C4—1222no—
409C4⋊C4—1222no—
410C4⋊C4—1222no—
411C4⋊C4—4882no—
412C4⋊C4—2442no—
413C4⋊C4C2×C22⋊D6112yes—
414C4⋊C4C2×C22⋊D6112yes—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—4882no—
420C42—24no—
421C42—24no—
422C42—21no—
423C42—21no—
424C42C2×C22⋊D6112yes—
425C42C2×C22⋊D6112yes—
426C23—21no—
427C23—21no—
428C23—22no—
429C23—22no—
430C23—21no—
431C23C22×C22⋊D6111yescharacteristic
432Q8—24no—
433Q8—24no—
434Q8—42no—
435Q8—42no—
436Q8—4884no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—2442no—
459C2×C4—2442no—
460C2×C4—2442no—
461C2×C4—2442no—
462C2×C4—2442no—
463C2×C4—2442no—
464C2×C4#1451974612yes—
465C2×C4#1451974612yes—
466C2×C4C22×C22⋊D6111yescharacteristic
467C2×C4C22×C22⋊D6111yescharacteristic
468C2×C4#41×D6112yes—
469C2×C4#1451982412yes—
470C8—1224no—
471C8—2442no—
472C22—41no—
473C22—41no—
474C22—42no—
475C22—42no—
476C22—41no—
477C22C2×#1451974611yescharacteristic
478C22#1684030811yescharacteristic
479C22#1684033311yescharacteristic
480C22#1684057012yes—
481C22#1684057712yes—
482C4—22no—
483C4—22no—
484C4—41no—
485C4—42no—
486C4—41no—
487C4—42no—
488C4—41no—
489C4—42no—
490C4—42no—
491C4—4882no—
492C4—4882no—
493C4#1684041112yes—
494C4#1684041912yes—
495C2—81no—
496C2#2144944311yescharacteristic
497C2#2145510511yescharacteristic
498C2#2145511411yescharacteristic
499C2#2145648912yes—
500C2#2145708112yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#541C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterC2×#541D61
hypocenterC61#25686
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#541

Complements

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

NH ≅ G/Ntype
1. C2×#16840172C2semidirect
2. C2×#16840112C2semidirect
6. #21446785C2semidirect
7. #21446640C2semidirect
8. #21446087C2semidirect
9. #21446073C2semidirect
297. C61#25686semidirect

Maximal subgroups

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

subgroupindex
C2×#168401722
C2×#168401122
#214516402
#214470192
#214470122
#214467852
#214466402
#214460872
#214460732
#214455852
C2×C61×#5412
#2568661