Finite Group Databasebeta

← #29652744

Subgroups — #29652744

Order 15616.

962 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455763C212yesmaximal, complemented
2#21449078C211yesmaximal, characteristic
3#21444861C212yesmaximal, complemented
4#21444844C212yesmaximal, complemented
5#21444827C212yesmaximal, complemented
6#21444810C212yesmaximal, complemented
7#21444339C212yesmaximal, complemented
8#21444157C211yesmaximal, characteristic
9C61×#3501C211yesmaximal, characteristic
10C2×#14519736C2211yescharacteristic
11#16840100—24no—
12#16840095C2214yes—
13#16840090C2214yes—
14#16839802—22no—
15#16839771—22no—
16#16839726C2212yes—
17#16839701C2211yescharacteristic
18#16839698C2211yescharacteristic
19#16839692C2211yescharacteristic
20#16839690—21no—
21#16839687C2211yescharacteristic
22#16839676—21no—
23C4×#11758196—21no—
24C4×C22⋊Dic61—21no—
25C4×C22⋊Dic61C2211yescharacteristic
26#16839499C2214yes—
27#16839499C2214yes—
28#16839488C2212yes—
29#16839486C2212yes—
30C61×#554—24no—
31C61×#554—22no—
32D4×Q8×C61—22no—
33C61×#544C2212yes—
34C61×#543—21no—
35C61×#543C2212yes—
36C61×#533—21no—
37C2×C22⋊Q8×C61—21no—
38C2×C22⋊Q8×C61C2212yes—
39C61×#518—21no—
40C2×C4×D4×C61C2211yescharacteristic
41C2×#11758194—22no—
42#14519736—22no—
43#14519736—22no—
44C2×C4⋊Dic61C2311yescharacteristic
45C2×C4⋊Dic61C2311yescharacteristic
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61—21no—
50C2×C22⋊Dic61—21no—
51C2×C22⋊Dic61—21no—
52C2×C4×Dic61—21no—
53C2×C4×Dic61—21no—
54#14519710—24no—
55#14519708—24no—
56#14519701—24no—
57#14519701—24no—
58C22⋊#92418—22no—
59C22⋊#92418C2314yes—
60#14519684—21no—
61#14519684—21no—
62#14519684—21no—
63#14519684—21no—
64#14519684C2311yescharacteristic
65#14519684C2311yescharacteristic
66C22×Q8×C61—21no—
67C22×Q8×C61D412yes—
68C22×D4×C61D411yescharacteristic
69C23×C4×C61D411yescharacteristic
70C23×C4×C61C2311yescharacteristic
71C4⋊Q8×C61—21no—
72C4⋊Q8×C61—21no—
73C4⋊Q8×C61—24no—
74#125×C61—22no—
75#125×C61—24no—
76C22⋊Q8×C61—24no—
77C22⋊Q8×C61—24no—
78C22⋊Q8×C61—22no—
79C22⋊Q8×C61—22no—
80C22⋊Q8×C61—24no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—22no—
85C22⋊Q8×C61—24no—
86C22⋊Q8×C61D414yes—
87C22⋊Q8×C61C2314yes—
88C4×Q8×C61—22no—
89C4×Q8×C61—22no—
90C4×D4×C61—24no—
91C4×D4×C61—24no—
92C4×D4×C61—22no—
93C4×D4×C61—22no—
94#117×C61—22no—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61D412yes—
99C2×C4⋊C4×C61C2311yescharacteristic
100C2×C22⋊C4×C61—22no—
101C2×C22⋊C4×C61—22no—
102C2×C22⋊C4×C61D411yescharacteristic
103C2×C22⋊C4×C61C2311yescharacteristic
104C2×C42×C61D411yescharacteristic
105C24×C61C2×D411yescharacteristic
106C24×C61C2×D411yescharacteristic
107C2×Q8×C61—22no—
108C2×Q8×C61—22no—
109C2×Q8×C61—22no—
110C2×Q8×C61—44no—
111C2×Q8×C61—82no—
112C2×Q8×C61C2×D414yes—
113C2×D4×C61—22no—
114C2×D4×C61—22no—
115C2×D4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—42no—
123C22×C4×C61—42no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D412yes—
130C22×C4×C61C2411yescharacteristic
131C22×Dic61—21no—
132C22×Dic61—21no—
133C22×Dic61—21no—
134C22×Dic61—21no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—24no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—24no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—24no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—24no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61C2×D414yes—
146C22⋊C4×C61—24no—
147C22⋊C4×C61—24no—
148C22⋊C4×C61—22no—
149C22⋊C4×C61—24no—
150C22⋊C4×C61—22no—
151C42×C61—24no—
152C42×C61—21no—
153C42×C61—21no—
154C22⋊Dic61—42no—
155C22⋊Dic61—42no—
156C22⋊Dic61—42no—
157C22⋊Dic61—42no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160#11758196—42no—
161#11758196—42no—
162C4×Dic61—42no—
163C4×Dic61—41no—
164C4×Dic61—41no—
165#11758194—44no—
166C2×#92418—24no—
167C23×C61—21no—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61#12012yes—
176C23×C61#13712yes—
177C23×C61C22×D411yescharacteristic
178Q8×C61—44no—
179Q8×C61—44no—
180Q8×C61—82no—
181Q8×C61—82no—
182D4×C61—28no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—24no—
186C2×C4×C61—22no—
187C2×C4×C61—21no—
188C2×C4×C61—24no—
189C2×C4×C61—24no—
190C2×C4×C61—21no—
191C2×C4×C61—42no—
192C2×C4×C61—41no—
193C2×C4×C61—42no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—81no—
197C2×C4×C61—82no—
198C2×C4×C61—82no—
199C2×C4×C61—82no—
200C2×C4×C61—82no—
201C2×C4×C61#12011yescharacteristic
202C2×C4×C61#12011yescharacteristic
203C2×C4×C61#13712yes—
204C2×C4×C61C22×D412yes—
205C2×Dic61—22no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216C2×Dic61—41no—
217#92418—44no—
218#18174—611nomaximal, Sylow
219C22×C61—21no—
220C22×C61—21no—
221C22×C61—22no—
222C22×C61—22no—
223C22×C61—22no—
224C22×C61—22no—
225C22×C61—42no—
226C22×C61—41no—
227C22×C61—41no—
228C22×C61—81no—
229C22×C61—81no—
230C22×C61—81no—
231C22×C61C2×#12011yescharacteristic
232C22×C61C2×#13711yescharacteristic
233C22×C61C2×#13711yescharacteristic
234C4×C61—22no—
235C4×C61—22no—
236C4×C61—21no—
237C4×C61—41no—
238C4×C61—44no—
239C4×C61—41no—
240C4×C61—44no—
241C4×C61#45812yes—
242Dic61—42no—
243Dic61—81no—
244Dic61—81no—
245Dic61—81no—
246#3501D6111yescharacteristic
247#3095—612no—
248#2395—611no—
249#1617—612no—
250#1614—612no—
251#1611—612no—
252#1608—612no—
253#1508—612no—
254#1465—611no—
255C2×C61—21no—
256C2×C61—21no—
257C2×C61—42no—
258C2×C61—42no—
259C2×C61#300411yescharacteristic
260C2×C61#301811yescharacteristic
261C2×C61C2×#45811yescharacteristic
262#554—24no—
263#554—22no—
264D4×Q8—22no—
265#544C2×D6112yes—
266#543—21no—
267#543C2×D6112yes—
268#533—21no—
269C2×C22⋊Q8—21no—
270C2×C22⋊Q8C2×D6112yes—
271#518—21no—
272C2×C4×D4C2×D6111yescharacteristic
273C2×#117—611no—
274#485—1224no—
275#484—614no—
276#483—614no—
277#428—1222no—
278#419—1222no—
279#407—612no—
280#400—611no—
281#399—611no—
282#398—611no—
283#397—611no—
284#397—1221no—
285#394—1221no—
286C4×C4⋊C4—1221no—
287C4×C22⋊C4—611no—
288C4×C22⋊C4—1221no—
289#328—614no—
290#328—614no—
291#324—612no—
292#323—612no—
293C61#1817411yescharacteristic, complemented, Sylow
294C22×Q8—21no—
295C22×Q8C22⋊D6112yes—
296C22×D4C22⋊D6111yescharacteristic
297C23×C4C22⋊D6111yescharacteristic
298C23×C4C22×D6111yescharacteristic
299C2×M16—1222no—
300C4⋊Q8—21no—
301C4⋊Q8—21no—
302C4⋊Q8—24no—
303#125—22no—
304#125—24no—
305C22⋊Q8—22no—
306C22⋊Q8—22no—
307C22⋊Q8—24no—
308C22⋊Q8—24no—
309C22⋊Q8—24no—
310C22⋊Q8—22no—
311C22⋊Q8—22no—
312C22⋊Q8—22no—
313C22⋊Q8—22no—
314C22⋊Q8—24no—
315C22⋊Q8C22⋊D6114yes—
316C22⋊Q8C22×D6114yes—
317C4×Q8—22no—
318C4×Q8—22no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—22no—
322C4×D4—22no—
323#117—22no—
324#117—1222no—
325#117—1222no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—611no—
330C2×C4⋊C4—611no—
331C2×C4⋊C4—1221no—
332C2×C4⋊C4—1221no—
333C2×C4⋊C4C22⋊D6112yes—
334C2×C4⋊C4C22×D6111yescharacteristic
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—1221no—
338C2×C22⋊C4—1221no—
339C2×C22⋊C4—1221no—
340C2×C22⋊C4—1221no—
341C2×C22⋊C4C22⋊D6111yescharacteristic
342C2×C22⋊C4C22×D6111yescharacteristic
343C2×C42—1221no—
344C2×C42—1221no—
345C2×C42C22⋊D6111yescharacteristic
346#107—1224no—
347#106—1224no—
348#103—1224no—
349#103—1224no—
350C22⋊C8—614no—
351C22⋊C8—1222no—
352#95—611no—
353#95—611no—
354#95—1221no—
355#95—1221no—
356#95—1221no—
357#95—1221no—
358C24C2×C22⋊D6111yescharacteristic
359C24C2×C22⋊D6111yescharacteristic
360C2×Q8—22no—
361C2×Q8—22no—
362C2×Q8—22no—
363C2×Q8—44no—
364C2×Q8—82no—
365C2×Q8C2×C22⋊D6114yes—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—1221no—
378C22×C4—1221no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4C2×C22⋊D6111yescharacteristic
382C22×C4C2×C22⋊D6112yes—
383C22×C4C2×C22⋊D6111yescharacteristic
384C22×C4C2×C22⋊D6111yescharacteristic
385C22×C4C2×C22⋊D6111yescharacteristic
386C22×C4C2×C22⋊D6112yes—
387C22×C4C23×D6111yescharacteristic
388M16—2444no—
389C2×C8—1224no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—1222no—
401C4⋊C4—1222no—
402C4⋊C4—2442no—
403C4⋊C4—2442no—
404C4⋊C4C2×C22⋊D6114yes—
405C22⋊C4—24no—
406C22⋊C4—24no—
407C22⋊C4—24no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—2442no—
411C22⋊C4—2442no—
412C22⋊C4—2442no—
413C22⋊C4—2442no—
414C42—24no—
415C42—21no—
416C42—21no—
417C42—2442no—
418C42—2441no—
419C42—2441no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23#1451974612yes—
429C23#1451980812yes—
430C23C22×C22⋊D6111yescharacteristic
431Q8—44no—
432Q8—44no—
433Q8—82no—
434Q8—82no—
435D4—28no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—21no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—1222no—
455C2×C4—2441no—
456C2×C4—2441no—
457C2×C4—2441no—
458C2×C4—2441no—
459C2×C4—2441no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4#1451974611yescharacteristic
467C2×C4#1451974611yescharacteristic
468C2×C4#1451980812yes—
469C2×C4C22×C22⋊D6112yes—
470C8—2444no—
471C22—21no—
472C22—21no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—42no—
478C22—41no—
479C22—41no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22C2×#1451974611yescharacteristic
484C22C2×#1451980811yescharacteristic
485C22C2×#1451980811yescharacteristic
486C4—22no—
487C4—21no—
488C4—22no—
489C4—41no—
490C4—44no—
491C4—41no—
492C4—44no—
493C4—4881no—
494C4—4881no—
495C4—4881no—
496C4—2442no—
497C4#1683995212yes—
498C2—21no—
499C2—21no—
500C2—42no—
501C2—42no—
502C2#2145505311yescharacteristic
503C2#2145516811yescharacteristic
504C2C2×#1683995211yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3501C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenter#3501D61
hypocenterC61#18174
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. #21455763C2semidirect
3. #21444861C2semidirect
4. #21444844C2semidirect
5. #21444827C2semidirect
6. #21444810C2semidirect
7. #21444339C2semidirect
293. C61#18174semidirect

Maximal subgroups

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

subgroupindex
#214557632
#214490782
#214448612
#214448442
#214448272
#214448102
#214443392
#214441572
C61×#35012
#1817461