Finite Group Databasebeta

← #29654371

Subgroups — #29654371

Order 15616.

888 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
1#21457046C212yesmaximal, complemented
2#21451031C211yesmaximal, characteristic, complemented
3#21444861C212yesmaximal, complemented
4#21444856C212yesmaximal
5#21444850C214yesmaximal, complemented
6#21444344C212yesmaximal, complemented
7#21444163C211yesmaximal, characteristic, complemented
8C61×#3461C211yesmaximal, characteristic
9C2×#14519777C2211yescharacteristic
10#16840172—22no—
11#16840121—22no—
12#16840095C2214yes—
13#16840078C2214yes—
14#16840047—22no—
15#16840042—22no—
16#16839819C2212yes—
17#16839701C2212yes—
18#16839701C2212yes—
19#16839669—21no—
20#16839668—21no—
21#16839668—21no—
22#16839627—21no—
23C4×C4⋊Dic61C2211yescharacteristic
24#16839499C2214yes—
25#16839496C2214yes—
26#16839488C2212yes—
27#16839488C2212yes—
28C61×#544C2211yescharacteristic
29C61×#542—22no—
30C61×#541—22no—
31C61×#539—21no—
32C61×#539C2212yes—
33C61×#537C2211yescharacteristic
34C61×#536—21no—
35C61×#533—22no—
36C61×#529—22no—
37C61×#529—21no—
38C61×#529C2212yes—
39C4×#41×C61—21no—
40C2×C4×Q8×C61C2211yescharacteristic
41C22×Dic122—21no—
42#14519777—22no—
43#14519777—22no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61—21no—
46C2×C4⋊Dic61C2312yes—
47C2×C4×Dic61—21no—
48#14519710—24no—
49#14519708—24no—
50C4⋊#92418—22no—
51#14519701—24no—
52#14519698—24no—
53C22⋊#92418C2314yes—
54#14519688—22no—
55#14519684—22no—
56#14519684—22no—
57#14519684—22no—
58#14519684—22no—
59#14519684C2312yes—
60C2×#41×C61—21no—
61C22×Q8×C61D411yescharacteristic
62C4⋊Q8×C61—22no—
63C4⋊Q8×C61—22no—
64#126×C61—22no—
65#126×C61—22no—
66#126×C61—22no—
67#126×C61—22no—
68#125×C61—22no—
69#125×C61—21no—
70#125×C61—22no—
71#125×C61—22no—
72#125×C61D412yes—
73#124×C61—22no—
74#124×C61—21no—
75#124×C61D412yes—
76#123×C61—22no—
77#123×C61—22no—
78#123×C61—22no—
79#123×C61—22no—
80C22⋊Q8×C61—22no—
81C22⋊Q8×C61—22no—
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C22⋊Q8×C61—21no—
85C22⋊Q8×C61—22no—
86C22⋊Q8×C61C2312yes—
87#121×C61—22no—
88#121×C61—21no—
89#121×C61C2312yes—
90C4×Q8×C61—21no—
91C4×Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—22no—
94C4×Q8×C61—22no—
95C4×Q8×C61D412yes—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—22no—
100C4×D4×C61—22no—
101C4×D4×C61D412yes—
102#117×C61—22no—
103#117×C61—22no—
104#117×C61—21no—
105#117×C61—22no—
106C2×C4⋊C4×C61D411yescharacteristic
107C2×C4⋊C4×C61C2311yescharacteristic
108C2×C4⋊C4×C61C2311yescharacteristic
109C2×C42×C61—22no—
110C2×C42×C61D411yescharacteristic
111C2×C42×C61D411yescharacteristic
112C2×C42×C61C2311yescharacteristic
113#41×C61—42no—
114#41×C61—42no—
115C2×Q8×C61—21no—
116C2×Q8×C61—21no—
117C2×Q8×C61—22no—
118C2×Q8×C61—22no—
119C2×Q8×C61—41no—
120C2×Q8×C61C2×D412yes—
121C2×D4×C61—21no—
122C2×D4×C61—22no—
123C2×D4×C61—21no—
124C2×D4×C61C2×D412yes—
125C22×C4×C61—22no—
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×C61C2×D411yescharacteristic
134C22×C4×C61C2411yescharacteristic
135C22×Dic61—22no—
136C22×Dic61—22no—
137C2×Dic122—44no—
138C2×Dic122—41no—
139C2×Dic122—41no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—21no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—22no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—22no—
150C4⋊C4×C61—22no—
151C4⋊C4×C61—22no—
152C4⋊C4×C61—22no—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61C2×D412yes—
156C4⋊C4×C61C2×D412yes—
157C22⋊C4×C61—22no—
158C22⋊C4×C61—22no—
159C22⋊C4×C61—22no—
160C22⋊C4×C61—22no—
161C22⋊C4×C61—22no—
162C22⋊C4×C61—22no—
163C22⋊C4×C61—22no—
164C22⋊C4×C61—22no—
165C22⋊C4×C61—22no—
166C42×C61—22no—
167C42×C61—22no—
168C42×C61—21no—
169C42×C61—22no—
170C42×C61—21no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61C2×D412yes—
174C42×C61C2×D412yes—
175C4⋊Dic61—24no—
176C4⋊Dic61—42no—
177C4⋊Dic61—42no—
178C4×Dic61—42no—
179C2×#92418—24no—
180C23×C61—22no—
181C23×C61C22×D411yescharacteristic
182Q8×C61—44no—
183Q8×C61—42no—
184Q8×C61—42no—
185Q8×C61—81no—
186D4×C61—42no—
187D4×C61—42no—
188D4×C61—42no—
189D4×C61—42no—
190D4×C61—81no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—22no—
196C2×C4×C61—21no—
197C2×C4×C61—22no—
198C2×C4×C61—22no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—22no—
202C2×C4×C61—21no—
203C2×C4×C61—21no—
204C2×C4×C61—41no—
205C2×C4×C61—44no—
206C2×C4×C61—44no—
207C2×C4×C61—41no—
208C2×C4×C61—41no—
209C2×C4×C61—82no—
210C2×C4×C61#12012yes—
211C2×C4×C61#12011yescharacteristic
212C2×C4×C61#12011yescharacteristic
213C2×C4×C61#13712yes—
214C2×C4×C61C22×D412yes—
215C2×Dic61—42no—
216C2×Dic61—44no—
217C2×Dic61—42no—
218C2×Dic61—42no—
219C2×Dic61—42no—
220Dic122—82no—
221Dic122—82no—
222#92418—44no—
223#18345—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—42no—
227C22×C61—42no—
228C22×C61—42no—
229C22×C61C2×#12011yescharacteristic
230C22×C61C2×#13711yescharacteristic
231C22×C61#57511yescharacteristic
232C4×C61—22no—
233C4×C61—22no—
234C4×C61—21no—
235C4×C61—22no—
236C4×C61—41no—
237C4×C61—42no—
238C4×C61—42no—
239C4×C61—41no—
240C4×C61—41no—
241C4×C61—42no—
242C4×C61#45412yes—
243Dic61—82no—
244Dic61—82no—
245#3461D6111yescharacteristic
246#3239—612no—
247#2568—611no—
248#1617—612no—
249#1616—612no—
250#1615—614no—
251#1509—612no—
252#1467—611no—
253C2×C61—21no—
254C2×C61—21no—
255C2×C61—82no—
256C2×C61#300711yescharacteristic
257C2×C61C2×#45411yescharacteristic
258C2×C61#302811yescharacteristic
259#544C2×D6111yescharacteristic
260#542—22no—
261#541—22no—
262#539—21no—
263#539C2×D6112yes—
264#537C2×D6111yescharacteristic
265#536—21no—
266#533—22no—
267C2×C4⋊Q8—611no—
268#529—22no—
269#529—21no—
270#529C2×D6112yes—
271C4×#41—21no—
272C2×C4×Q8C2×D6111yescharacteristic
273#501—1222no—
274#489—1222no—
275#484—614no—
276#481—614no—
277#475—1222no—
278#474—1222no—
279#432—612no—
280#400—612no—
281#400—612no—
282#391—1221no—
283#391—1221no—
284#391—1221no—
285#382—1221no—
286C4×C4⋊C4—611no—
287#328—614no—
288#327—614no—
289#324—612no—
290#324—612no—
291C61#1834511yescharacteristic, complemented, Sylow
292C2×#41—21no—
293C22×Q8—1221no—
294C22×Q8C22⋊D6111yescharacteristic
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—1222no—
298C4⋊Q8—1222no—
299#126—22no—
300#126—22no—
301#126—22no—
302#126—22no—
303#125—21no—
304#125—22no—
305#125—22no—
306#125—22no—
307#125C22⋊D6112yes—
308#124—22no—
309#124—21no—
310#124C22⋊D6112yes—
311#123—22no—
312#123—22no—
313#123—22no—
314#123—22no—
315C22⋊Q8—22no—
316C22⋊Q8—22no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319C22⋊Q8—21no—
320C22⋊Q8—22no—
321C22⋊Q8C22×D6112yes—
322#121—22no—
323#121—21no—
324#121C22×D6112yes—
325C4×Q8—21no—
326C4×Q8—21no—
327C4×Q8—21no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8C22⋊D6112yes—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4C22⋊D6112yes—
337#117—22no—
338#117—22no—
339#117—21no—
340#117—22no—
341C2×C4⋊C4—612no—
342C2×C4⋊C4—1221no—
343C2×C4⋊C4—1221no—
344C2×C4⋊C4C22⋊D6111yescharacteristic
345C2×C4⋊C4C22×D6111yescharacteristic
346C2×C4⋊C4C22×D6111yescharacteristic
347C2×C42—22no—
348C2×C42—1221no—
349C2×C42C22⋊D6111yescharacteristic
350C2×C42C22⋊D6111yescharacteristic
351C2×C42C22×D6111yescharacteristic
352#107—1224no—
353#106—1224no—
354C4⋊C8—1222no—
355#103—1224no—
356#102—1224no—
357C22⋊C8—614no—
358#97—1222no—
359#95—612no—
360#95—1222no—
361#95—1222no—
362#95—1222no—
363#95—1222no—
364#41—42no—
365#41—42no—
366C2×Q8—21no—
367C2×Q8—21no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—41no—
371C2×Q8—2444no—
372C2×Q8—2441no—
373C2×Q8—2441no—
374C2×Q8C2×C22⋊D6112yes—
375C2×D4—21no—
376C2×D4—22no—
377C2×D4—21no—
378C2×D4C2×C22⋊D6112yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4C2×C22⋊D6111yescharacteristic
385C22×C4C2×C22⋊D6111yescharacteristic
386C22×C4C2×C22⋊D6111yescharacteristic
387C22×C4C2×C22⋊D6111yescharacteristic
388C22×C4C2×C22⋊D6111yescharacteristic
389C22×C4C2×C22⋊D6111yescharacteristic
390C22×C4C23×D6111yescharacteristic
391C2×C8—1224no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—21no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—1224no—
408C4⋊C4—2442no—
409C4⋊C4—2442no—
410C4⋊C4C2×C22⋊D6112yes—
411C4⋊C4C2×C22⋊D6112yes—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C42—22no—
422C42—22no—
423C42—21no—
424C42—22no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—2442no—
429C42C2×C22⋊D6112yes—
430C42C2×C22⋊D6112yes—
431C23—22no—
432C23C22×C22⋊D6111yescharacteristic
433Q8—44no—
434Q8—42no—
435Q8—42no—
436Q8—81no—
437Q8—4882no—
438Q8—4882no—
439D4—42no—
440D4—42no—
441D4—42no—
442D4—42no—
443D4—81no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—82no—
463C2×C4—2442no—
464C2×C4—2442no—
465C2×C4—2444no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4#1451974611yescharacteristic
469C2×C4#1451974612yes—
470C2×C4#1451974611yescharacteristic
471C2×C4#1451980712yes—
472C2×C4C22×C22⋊D6112yes—
473C8—2444no—
474C22—21no—
475C22—21no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22C2×#1451974611yescharacteristic
480C22C2×#1451980711yescharacteristic
481C22#1684056011yescharacteristic
482C4—21no—
483C4—22no—
484C4—22no—
485C4—22no—
486C4—42no—
487C4—41no—
488C4—41no—
489C4—42no—
490C4—41no—
491C4—42no—
492C4—4882no—
493C4—4882no—
494C4#1683993512yes—
495C2—21no—
496C2—21no—
497C2—82no—
498C2#2145507911yescharacteristic
499C2C2×#1683993511yescharacteristic
500C2#2145522511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3461C2
socleC22×C61C2×#120
prime radicalC61×#537C22
hypercenter#3461D61
hypocenterC61#18345
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3461

Complements

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

NH ≅ G/Ntype
1. #21457046C2semidirect
2. #21451031C2semidirect
3. #21444861C2semidirect
5. #21444850C2semidirect
6. #21444344C2semidirect
7. #21444163C2semidirect
291. C61#18345semidirect

Maximal subgroups

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

subgroupindex
#214570462
#214510312
#214448612
#214448562
#214448502
#214443442
#214441632
C61×#34612
#1834561