Finite Group Databasebeta

← #29571950

Subgroups — #29571950

Order 15616.

734 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459635C211yesmaximal, characteristic
2#21446768C211yesmaximal, characteristic, complemented
3#21445836C211yesmaximal, characteristic
4#21444510C211yesmaximal, characteristic, complemented
5#21444319C211yesmaximal, characteristic, complemented
6#21444207C211yesmaximal, characteristic
7C61×#1549C211yesmaximal, characteristic, complemented
8#16840492—21no—
9#16840487C412yescomplemented
10#16840473—21no—
11#16840467—22no—
12#16840427C412yescomplemented
13Q8×C22⋊D61—22no—
14#16840393—21no—
15#16840389—22no—
16#16840319C2211yescharacteristic
17#16840317—21no—
18C4⋊Q8×D61—21no—
19#16840243—21no—
20C4×#11758222C2211yescharacteristic
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519703—21no—
23C4×C4⋊Dic61C2211yescharacteristic
24#16839524—22no—
25#16839523—21no—
26#16839510—21no—
27#16839507—21no—
28#16839500—22no—
29#16839494—21no—
30#16839491—21no—
31C2×C4⋊Q8×C61C2211yescharacteristic
32C61×#431C2211yescharacteristic
33C61×#423C2211yescharacteristic
34C61×#332—22no—
35C2×#11758222D411yescharacteristic
36C2×Q8×D61—22no—
37C22×#92418—21no—
38#14519777D411yescharacteristic
39#14519777D411yescharacteristic
40#14519776—22no—
41#14519776—22no—
42#14519776—22no—
43#14519775—24no—
44#14519775—22no—
45#14519769—21no—
46#14519768—22no—
47#14519768—22no—
48#14519767—24no—
49#14519757—22no—
50#14519756—24no—
51#14519756—21no—
52#14519756D412yes—
53#14519755—24no—
54#14519753—24no—
55#14519753—22no—
56Q8×Dic61—22no—
57C4×Dic122—21no—
58C4×Dic122C2×C411yescharacteristic
59C4×Dic122C2×C412yes—
60C4×Dic122D411yescharacteristic
61#14519742—22no—
62#14519742—22no—
63C4×D244C2×C411yescharacteristic
64C4×D244D411yescharacteristic
65#14519739—22no—
66C4×C22⋊D61—24no—
67C4×C22⋊D61—21no—
68C4×C22⋊D61—21no—
69#14519736—21no—
70#14519736D411yescharacteristic
71#14519735—22no—
72#14519734—21no—
73C2×C4⋊Dic61—21no—
74C4⋊C4×D61—22no—
75C4⋊C4×D61—22no—
76C2×C4×Dic61—21no—
77C42×D61—21no—
78C4⋊#92418—21no—
79C4⋊#92418—21no—
80#14519703—42no—
81#14519684—21no—
82C22×Q8×C61—21no—
83C2×M16×C61—21no—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61C2×C412yes—
87C4⋊Q8×C61D412yes—
88C2×C4⋊C4×C61—21no—
89C2×C4⋊C4×C61D412yes—
90C2×C42×C61C2311yescharacteristic
91C4⋊C8×C61—21no—
92C4⋊C8×C61—21no—
93C4⋊C8×C61—21no—
94C22⋊C8×C61—21no—
95C4×C8×C61—21no—
96C2×Q8×C61—22no—
97C2×Q8×C61—22no—
98C2×Q8×C61—82no—
99C22×C4×C61—22no—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C2×C22⋊D61—21no—
104C22×Dic61—41no—
105Q8×D61—84no—
106#11758222—21no—
107#11758222—21no—
108#11758222SD1612yes—
109#11758222Q1612yes—
110C2×D244C22⋊C411yescharacteristic
111C2×C4×D61—21no—
112C2×C4×D61—21no—
113C2×C4×D61—21no—
114C2×C4×D61—24no—
115C2×Dic122—21no—
116C2×Dic122—24no—
117C2×Dic122C22⋊C411yescharacteristic
118C2×Dic122C22⋊C412yes—
119M16×C61—41no—
120M16×C61—41no—
121C2×C8×C61—41no—
122C2×C8×C61—41no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—22no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61C22⋊C414yes—
127C42×C61C22×C411yescharacteristic
128C42×C61C2×D411yescharacteristic
129C42×C61C2×D411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C22⋊Dic61—22no—
132C22⋊Dic61—21no—
133#11758198—24no—
134#11758198—21no—
135#11758198—21no—
136C4⋊Dic61—24no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61—21no—
140C4⋊Dic61—22no—
141C4⋊Dic61—41no—
142C4⋊Dic61C22⋊C411yescharacteristic
143C4⋊Dic61C22⋊C411yescharacteristic
144C4⋊Dic61C22⋊C412yes—
145#11758196—24no—
146#11758196—24no—
147#11758196—24no—
148#11758196—21no—
149#11758196—22no—
150#11758196—21no—
151#11758196—22no—
152C4×Dic61—24no—
153C4×Dic61—21no—
154C4×Dic61—22no—
155C4×Dic61—21no—
156C4×Dic61—21no—
157C4×Dic61—41no—
158C4×Dic61—41no—
159C2×#92418—42no—
160C2×#92418—41no—
161C23×C61#12011yescharacteristic
162C22×D61—21no—
163Q8×C61—44no—
164Q8×C61—82no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—24no—
168C2×C4×C61—41no—
169C2×C4×C61—82no—
170C2×C4×C61C2×C22⋊C411yescharacteristic
171C2×C4×C61C2×C22⋊C411yescharacteristic
172C2×C4×C61C2×C22⋊C411yescharacteristic
173C2×C4×C61#12011yescharacteristic
174C2×C4×C61#12011yescharacteristic
175C2×C4×C61#12011yescharacteristic
176C2×C4×C61C2×SD1611yescharacteristic
177C2×C4×C61C2×Q1611yescharacteristic
178C2×C4×C61#13711yescharacteristic
179C2×C4×C61#13711yescharacteristic
180C22⋊D61—22no—
181C22⋊D61—22no—
182C2×Dic61—21no—
183C2×Dic61—22no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—22no—
187C2×Dic61—42no—
188C2×Dic61—81no—
189C2×Dic61—81no—
190D244—21no—
191D244#10312yes—
192C4×D61—22no—
193C4×D61—22no—
194C4×D61—42no—
195C4×D61—41no—
196C4×D61—41no—
197C4×D61—84no—
198Dic122—21no—
199Dic122—41no—
200Dic122—41no—
201Dic122—42no—
202Dic122—84no—
203Dic122#10312yes—
204C8×C61—81no—
205C8×C61—81no—
206#92418—81no—
207#8424—611nomaximal, Sylow
208C22×C61—21no—
209C22×C61—21no—
210C22×C61#37911yescharacteristic
211C22×C61#44811yescharacteristic
212C22×C61#45111yescharacteristic
213C2×D61—22no—
214C2×D61—41no—
215C2×D61—41no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—44no—
220C4×C61C2×#10311yescharacteristic
221C4×C61#41911yescharacteristic
222C4×C61#44811yescharacteristic
223C4×C61#45111yescharacteristic
224C4×C61#45411yescharacteristic
225C4×C61#45611yescharacteristic
226Dic61—22no—
227Dic61—42no—
228Dic61—41no—
229Dic61—42no—
230Dic61—41no—
231Dic61—82no—
232#3503—611no—
233#1987—611no—
234#1806—611no—
235#1550—611no—
236#1549D6111yescharacteristic, complemented
237#1503—611no—
238#1477—611no—
239C2×C61—21no—
240C2×C61—21no—
241C2×C61#178711yescharacteristic
242C2×C61#179311yescharacteristic
243C2×C61#179511yescharacteristic
244D61—42no—
245Q82—1221no—
246#557—612no—
247#554—1221no—
248#554—1222no—
249D4×Q8—612no—
250D4×Q8—1222no—
251#544—1221no—
252#544—1222no—
253#533—611no—
254#533—1221no—
255C2×C4⋊Q8—1221no—
256C2×C4⋊Q8C2×D6111yescharacteristic
257#520—1221no—
258C4×#41—611no—
259#431C2×D6111yescharacteristic
260#423C2×D6111yescharacteristic
261C2×C4⋊C8—611no—
262C2×#104—1221no—
263C4×C4⋊C4—611no—
264#340—1222no—
265#339—1221no—
266#333—1221no—
267#332—22no—
268#332—1221no—
269#328—1222no—
270Q8⋊C8—1221no—
271D4⋊C8—1221no—
272C61#842411yescharacteristic, complemented, Sylow
273C2×#41—611no—
274C22×Q8—21no—
275C22×Q8—1222no—
276C2×M16—21no—
277C22×C8—1221no—
278C4⋊Q8—21no—
279C4⋊Q8—21no—
280C4⋊Q8—611no—
281C4⋊Q8—611no—
282C4⋊Q8—1222no—
283C4⋊Q8—1222no—
284C4⋊Q8—1222no—
285C4⋊Q8—1222no—
286C4⋊Q8—1224no—
287C4⋊Q8C4×D6112yes—
288C4⋊Q8C22⋊D6112yes—
289#125—1222no—
290#125—1222no—
291#125—1224no—
292#125—1221no—
293C22⋊Q8—612no—
294C22⋊Q8—1222no—
295C22⋊Q8—1222no—
296C22⋊Q8—1224no—
297C22⋊Q8—1224no—
298C22⋊Q8—1221no—
299C22⋊Q8—1224no—
300C4×Q8—611no—
301C4×Q8—612no—
302C4×Q8—611no—
303C4×Q8—1222no—
304C4×Q8—1222no—
305C4×Q8—1222no—
306C4×Q8—1221no—
307C4×D4—611no—
308C4×D4—611no—
309C4×D4—1222no—
310C4×D4—1224no—
311C4×D4—1221no—
312C4×D4—1221no—
313#117—611no—
314#117—1222no—
315#117—1221no—
316#117—1221no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—1222no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4D24412yes—
322C2×C42—1221no—
323C2×C42—1221no—
324C2×C42C22×D6111yescharacteristic
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—1221no—
329C4⋊C8—1221no—
330#104—2442no—
331C22⋊C8—21no—
332C4×C8—21no—
333#95—1221no—
334#41—612no—
335#41—612no—
336#41—1221no—
337#41—1221no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—82no—
341C2×Q8—611no—
342C2×Q8—612no—
343C2×Q8—1224no—
344C2×Q8—1221no—
345C2×Q8—4884no—
346C2×D4—611no—
347C2×D4—1221no—
348C22×C4—22no—
349C22×C4—1224no—
350C22×C4—1221no—
351C22×C4—1221no—
352C22×C4—1221no—
353C22×C4—2441no—
354C22×C4C2×D24411yescharacteristic
355C22×C4D4×D6111yescharacteristic
356C22×C4D4×D6111yescharacteristic
357M16—41no—
358M16—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—2442no—
362C2×C8—2441no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—611no—
367C4⋊C4—611no—
368C4⋊C4—612no—
369C4⋊C4—1224no—
370C4⋊C4—1224no—
371C4⋊C4—1224no—
372C4⋊C4—1224no—
373C4⋊C4—1222no—
374C4⋊C4—1221no—
375C4⋊C4—1222no—
376C4⋊C4—1221no—
377C4⋊C4—1221no—
378C4⋊C4—1222no—
379C4⋊C4—1221no—
380C4⋊C4—1221no—
381C4⋊C4—2441no—
382C4⋊C4#1175819814yes—
383C22⋊C4—1224no—
384C22⋊C4—1221no—
385C22⋊C4—1222no—
386C22⋊C4—1221no—
387C22⋊C4—1221no—
388C42—1224no—
389C42—1221no—
390C42—1222no—
391C42—1221no—
392C42—1221no—
393C42—2441no—
394C42—2441no—
395C42C2×C4×D6111yescharacteristic
396C42D4×D6111yescharacteristic
397C42D4×D6111yescharacteristic
398C42C2×C22⋊D6111yescharacteristic
399C23—1221no—
400C23#1451974711yescharacteristic
401Q8—44no—
402Q8—82no—
403Q8—612no—
404Q8—1221no—
405Q8—4884no—
406Q8—2442no—
407Q8—2441no—
408Q8—2441no—
409D4—612no—
410D4—1222no—
411D4—1222no—
412D4—1221no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—24no—
416C2×C4—41no—
417C2×C4—82no—
418C2×C4—1221no—
419C2×C4—1222no—
420C2×C4—1222no—
421C2×C4—1222no—
422C2×C4—1221no—
423C2×C4—1221no—
424C2×C4—1222no—
425C2×C4—4884no—
426C2×C4—4881no—
427C2×C4—4881no—
428C2×C4—2442no—
429C2×C4—2442no—
430C2×C4—2441no—
431C2×C4—2441no—
432C2×C4C22⋊C4×D6111yescharacteristic
433C2×C4C22⋊C4×D6111yescharacteristic
434C2×C4C2×#1175819811yescharacteristic
435C2×C4#1451974511yescharacteristic
436C2×C4#1451974511yescharacteristic
437C2×C4#1451974711yescharacteristic
438C2×C4SD16×D6111yescharacteristic
439C2×C4Q16×D6111yescharacteristic
440C2×C4#1451980611yescharacteristic
441C2×C4#1451980911yescharacteristic
442C8—81no—
443C8—81no—
444C8—4881no—
445C22—21no—
446C22—21no—
447C22—1222no—
448C22—2441no—
449C22—2441no—
450C22#1683961711yescharacteristic
451C22#1683990211yescharacteristic
452C22#1683991911yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—44no—
457C4—1222no—
458C4—4882no—
459C4—2441no—
460C4—2442no—
461C4—2442no—
462C4—2441no—
463C4#103×D6111yescharacteristic
464C4#1683976911yescharacteristic
465C4#1683989911yescharacteristic
466C4#1683991611yescharacteristic
467C4#1683993711yescharacteristic
468C4#1683994711yescharacteristic
469C2—21no—
470C2—21no—
471C2—2442no—
472C2#2144575211yescharacteristic
473C2#2144578011yescharacteristic
474C2#2144578611yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1549C2
socleC22×C61#379
prime radicalC2×#11758222D4
cosocleC2×C42×C61C23
hypercenter#1549D61
hypocenterC61#8424
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
2. #21446768C2semidirect
4. #21444510C2semidirect
5. #21444319C2semidirect
7. C61×#1549C2semidirect
9. #16840487C4semidirect
12. #16840427C4semidirect
236. #1549D61semidirect
272. C61#8424semidirect

Maximal subgroups

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

subgroupindex
#214596352
#214467682
#214458362
#214445102
#214443192
#214442072
C61×#15492
#842461