Finite Group Databasebeta

← #29654448

Subgroups — #29654448

Order 15616.

946 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
1#21459622C211yesmaximal, characteristic
2#21456640C212yesmaximal, complemented
3#21444826C212yesmaximal, complemented
4#21444809C212yesmaximal, complemented
5#21444792C212yesmaximal, complemented
6#21444775C212yesmaximal, complemented
7#21444351C212yesmaximal, complemented
8#21444157C211yesmaximal, characteristic
9C61×#2638C211yesmaximal, characteristic
10#16840471—24no—
11#16840469—22no—
12D4×Dic122—22no—
13#16840393C2212yes—
14#16840388—21no—
15#16840388C2212yes—
16#16840319—21no—
17C2×#14519756—21no—
18C2×#14519756C2212yes—
19#16840237—21no—
20#16840092C2214yes—
21#16840090—22no—
22#16840018—24no—
23#16839918C2214yes—
24#16839917—22no—
25#16839726C2212yes—
26#16839498C2214yes—
27#16839498C2214yes—
28#16839488C2212yes—
29#16839486C2212yes—
30C2×C22⋊Q8×C61C2211yescharacteristic
31C2×C4×D4×C61C2211yescharacteristic
32C61×#398C2211yescharacteristic
33C61×#397—21no—
34C61×#397C2211yescharacteristic
35C61×#395C2211yescharacteristic
36C61×#394—21no—
37C61×#393C2211yescharacteristic
38C61×#387—21no—
39C61×#385—21no—
40C61×#381C2211yescharacteristic
41C22×Dic122—21no—
42C22×Dic122D412yes—
43C2×#11758202—24no—
44C2×#11758194—22no—
45#14519777—21no—
46#14519777—21no—
47#14519776—24no—
48#14519769—22no—
49#14519768—24no—
50#14519756—22no—
51#14519756—22no—
52#14519756—22no—
53#14519756—22no—
54#14519756—22no—
55#14519756—22no—
56#14519756D414yes—
57#14519756C2314yes—
58#14519754—24no—
59#14519754—24no—
60#14519754—24no—
61#14519754—24no—
62C4×Dic122—22no—
63C4×Dic122—22no—
64D4×Dic61—24no—
65D4×Dic61—24no—
66#14519736—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61D412yes—
69C2×#11758196—22no—
70C2×#11758196—22no—
71C2×C22⋊Dic61—22no—
72C2×C22⋊Dic61—22no—
73#14519708—24no—
74#14519701—24no—
75#14519700—24no—
76C22⋊#92418—22no—
77C22⋊#92418C2314yes—
78C22×Q8×C61C2311yescharacteristic
79C22×D4×C61D411yescharacteristic
80C23×C4×C61D411yescharacteristic
81C23×C4×C61C2311yescharacteristic
82C22⋊Q8×C61—22no—
83C22⋊Q8×C61—22no—
84C4×D4×C61—22no—
85C4×D4×C61—22no—
86C2×C4⋊C4×C61—21no—
87C2×C4⋊C4×C61—21no—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C22⋊C4×C61—21no—
92C2×C22⋊C4×C61—21no—
93C2×C22⋊C4×C61—21no—
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61D411yescharacteristic
96C2×C22⋊C4×C61C2311yescharacteristic
97C2×C42×C61D411yescharacteristic
98#95×C61—21no—
99#95×C61—21no—
100#95×C61—21no—
101#95×C61—21no—
102#95×C61—21no—
103#95×C61C2311yescharacteristic
104#95×C61C2311yescharacteristic
105C24×C61C2×D411yescharacteristic
106C24×C61C2×D411yescharacteristic
107C2×Q8×C61—22no—
108C2×Q8×C61—81no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C22×C4×C61—21no—
113C22×C4×C61—21no—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—81no—
119C22×C4×C61—81no—
120C22×C4×C61C2×D411yescharacteristic
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D412yes—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2411yescharacteristic
127C22×Dic61—22no—
128C22×Dic61—22no—
129C22×Dic61—22no—
130C22×Dic61—22no—
131C2×Dic122—22no—
132C2×Dic122—22no—
133C2×Dic122—22no—
134C2×Dic122—82no—
135C2×Dic122—82no—
136C2×Dic122C2×D414yes—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—41no—
140C4⋊C4×C61—41no—
141C4⋊C4×C61—42no—
142C4⋊C4×C61—81no—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—22no—
145C22⋊C4×C61—42no—
146C22⋊C4×C61—42no—
147C22⋊C4×C61—81no—
148C22⋊C4×C61—81no—
149C42×C61—21no—
150C42×C61—21no—
151C22⋊Dic61—24no—
152C22⋊Dic61—24no—
153C22⋊Dic61—24no—
154#11758202—84no—
155C4⋊Dic61—22no—
156C4⋊Dic61—22no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61—22no—
160C4⋊Dic61C2×D414yes—
161#11758196—24no—
162#11758196—24no—
163#11758196—24no—
164#11758196—24no—
165C4×Dic61—24no—
166#11758194—82no—
167C2×#92418—24no—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—21no—
172C23×C61—41no—
173C23×C61—41no—
174C23×C61—41no—
175C23×C61—41no—
176C23×C61#12012yes—
177C23×C61#13712yes—
178C23×C61C22×D411yescharacteristic
179Q8×C61—44no—
180D4×C61—28no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—21no—
184C2×C4×C61—21no—
185C2×C4×C61—42no—
186C2×C4×C61—42no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—42no—
190C2×C4×C61—41no—
191C2×C4×C61—41no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—41no—
198C2×C4×C61—41no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—41no—
202C2×C4×C61—41no—
203C2×C4×C61—41no—
204C2×C4×C61—81no—
205C2×C4×C61#12011yescharacteristic
206C2×C4×C61#12011yescharacteristic
207C2×C4×C61#13712yes—
208C2×C4×C61C22×D412yes—
209C2×Dic61—24no—
210C2×Dic61—24no—
211C2×Dic61—82no—
212C2×Dic61—82no—
213C2×Dic61—82no—
214C2×Dic61—82no—
215Dic122—44no—
216Dic122—44no—
217Dic122—82no—
218Dic122—82no—
219#92418—44no—
220#18353—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—22no—
224C22×C61—22no—
225C22×C61—22no—
226C22×C61—22no—
227C22×C61—42no—
228C22×C61—41no—
229C22×C61—41no—
230C22×C61—81no—
231C22×C61—81no—
232C22×C61—81no—
233C22×C61C2×#12011yescharacteristic
234C22×C61C2×#13711yescharacteristic
235C22×C61C2×#13711yescharacteristic
236C4×C61—22no—
237C4×C61—22no—
238C4×C61—21no—
239C4×C61—21no—
240C4×C61—41no—
241C4×C61—41no—
242C4×C61—81no—
243C4×C61—81no—
244C4×C61—81no—
245C4×C61—81no—
246Dic61—44no—
247Dic61—44no—
248#3501—611no—
249#3195—612no—
250#2638D6111yescharacteristic
251#1611—612no—
252#1608—612no—
253#1605—612no—
254#1602—612no—
255#1511—612no—
256#1465—611no—
257C2×C61—21no—
258C2×C61—21no—
259C2×C61—42no—
260C2×C61—42no—
261C2×C61#300411yescharacteristic
262C2×C61#302211yescharacteristic
263C2×C61#302611yescharacteristic
264#554—1222no—
265#554—1224no—
266D4×Q8—1222no—
267#544—612no—
268#543—612no—
269#543—1221no—
270#533—1221no—
271C2×C22⋊Q8—612no—
272C2×C22⋊Q8—1221no—
273C2×C22⋊Q8C2×D6111yescharacteristic
274#518—1221no—
275C2×C4×D4C2×D6111yescharacteristic
276#483—614no—
277#483—1222no—
278#470—1224no—
279#451—614no—
280#451—1222no—
281#407—612no—
282#398C2×D6111yescharacteristic
283#397—21no—
284#397C2×D6111yescharacteristic
285#395C2×D6111yescharacteristic
286#394—21no—
287#393C2×D6111yescharacteristic
288#387—21no—
289#385—21no—
290#381C2×D6111yescharacteristic
291#328—614no—
292#328—614no—
293#324—612no—
294#323—612no—
295C61#1835311yescharacteristic, complemented, Sylow
296C22×Q8—612no—
297C22×Q8—1221no—
298C22×Q8C22×D6111yescharacteristic
299C22×D4C22⋊D6111yescharacteristic
300C23×C4C22⋊D6111yescharacteristic
301C23×C4C22×D6111yescharacteristic
302C2×Q16—1224no—
303C2×M16—1222no—
304C4⋊Q8—1221no—
305C4⋊Q8—1221no—
306C4⋊Q8—1224no—
307#125—1222no—
308#125—1224no—
309C22⋊Q8—22no—
310C22⋊Q8—22no—
311C22⋊Q8—614no—
312C22⋊Q8—614no—
313C22⋊Q8—1224no—
314C22⋊Q8—1224no—
315C22⋊Q8—1224no—
316C22⋊Q8—1224no—
317C22⋊Q8—1222no—
318C22⋊Q8—1222no—
319C22⋊Q8—1222no—
320C22⋊Q8—1222no—
321C22⋊Q8—1222no—
322C22⋊Q8—1222no—
323C4×Q8—1222no—
324C4×Q8—1222no—
325C4×D4—22no—
326C4×D4—22no—
327C4×D4—1224no—
328C4×D4—1224no—
329#117—1222no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—612no—
334C2×C4⋊C4—1222no—
335C2×C4⋊C4—1221no—
336C2×C4⋊C4—1222no—
337C2×C4⋊C4C22×D6111yescharacteristic
338C2×C4⋊C4C22×D6111yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—1222no—
344C2×C22⋊C4—1222no—
345C2×C22⋊C4C22⋊D6111yescharacteristic
346C2×C22⋊C4C22×D6111yescharacteristic
347C2×C42C22⋊D6111yescharacteristic
348#106—1224no—
349#103—1224no—
350#103—1224no—
351C22⋊C8—614no—
352C22⋊C8—1222no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95C22×D6111yescharacteristic
359#95C22×D6111yescharacteristic
360C24C2×C22⋊D6111yescharacteristic
361C24C2×C22⋊D6111yescharacteristic
362C2×Q8—22no—
363C2×Q8—81no—
364C2×Q8—614no—
365C2×Q8—1222no—
366C2×Q8—1222no—
367C2×Q8—1222no—
368C2×Q8—4882no—
369C2×Q8—4882no—
370C2×D4—22no—
371C2×D4—22no—
372C2×D4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—1222no—
382C22×C4—1222no—
383C22×C4—1222no—
384C22×C4—1222no—
385C22×C4D4×D6112yes—
386C22×C4D4×D6112yes—
387C22×C4C2×C22⋊D6111yescharacteristic
388C22×C4C2×C22⋊D6111yescharacteristic
389C22×C4C2×C22⋊D6111yescharacteristic
390C22×C4C2×C22⋊D6111yescharacteristic
391C22×C4C23×D6111yescharacteristic
392Q16—4884no—
393M16—4882no—
394C2×C8—1224no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—42no—
400C4⋊C4—81no—
401C4⋊C4—614no—
402C4⋊C4—1224no—
403C4⋊C4—1222no—
404C4⋊C4—1224no—
405C4⋊C4—1222no—
406C4⋊C4—1222no—
407C4⋊C4—1222no—
408C4⋊C4—1224no—
409C4⋊C4—1222no—
410C4⋊C4—1224no—
411C22⋊C4—22no—
412C22⋊C4—22no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—81no—
416C22⋊C4—81no—
417C22⋊C4—1224no—
418C22⋊C4—1224no—
419C22⋊C4—1224no—
420C42—21no—
421C42—21no—
422C42—1224no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23#1451974512yes—
432C23#1451980712yes—
433C23C22×C22⋊D6111yescharacteristic
434Q8—44no—
435Q8—4882no—
436Q8—4882no—
437Q8—2444no—
438Q8—2444no—
439D4—28no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—81no—
464C2×C4—1224no—
465C2×C4—1224no—
466C2×C4—4882no—
467C2×C4—4882no—
468C2×C4—4882no—
469C2×C4—4882no—
470C2×C4#1451974511yescharacteristic
471C2×C4#1451974511yescharacteristic
472C2×C4#1451980712yes—
473C2×C4C2×D4×D6112yes—
474C8—2444no—
475C22—21no—
476C22—21no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—42no—
482C22—41no—
483C22—41no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22C2×#1451974511yescharacteristic
488C22C2×#1451980711yescharacteristic
489C22C2×#1451980711yescharacteristic
490C4—22no—
491C4—21no—
492C4—21no—
493C4—22no—
494C4—41no—
495C4—41no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—2444no—
501C4—2444no—
502C2—21no—
503C2—21no—
504C2—42no—
505C2—42no—
506C2#2145505211yescharacteristic
507C2#2145519711yescharacteristic
508C2#2145521511yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2638C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenter#2638D61
hypocenterC61#18353
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2638

Complements

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

NH ≅ G/Ntype
2. #21456640C2semidirect
3. #21444826C2semidirect
4. #21444809C2semidirect
5. #21444792C2semidirect
6. #21444775C2semidirect
7. #21444351C2semidirect
295. C61#18353semidirect

Maximal subgroups

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

subgroupindex
#214596222
#214566402
#214448262
#214448092
#214447922
#214447752
#214443512
#214441572
C61×#26382
#1835361