Finite Group Databasebeta

← #29656234

Subgroups — #29656234

Order 15616.

918 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840485C211yesmaximal, characteristic
2#21456366C211yesmaximal, characteristic, complemented
3#21455914C211yesmaximal, characteristic
4#21445087C214yesmaximal, complemented
5#21445074C214yesmaximal, complemented
6#21444535C211yesmaximal, characteristic, complemented
7#21444266C211yesmaximal, characteristic, complemented
8#21444261C211yesmaximal, characteristic
9C61×#3113C211yesmaximal, characteristic, complemented
10#16840485—22no—
11#16840485—22no—
12#16840485—22no—
13#16840485C2214yes—
14C2×#14519777C2211yescharacteristic
15C2×#14519772—22no—
16C2×#14519765—21no—
17C2×#14519753—21no—
18C2×C4×Dic122—21no—
19C2×C4×D244C2211yescharacteristic
20C2×#14519735—21no—
21#16840172—22no—
22#16840165C2212yescomplemented
23#16840158C2212yescomplemented
24#16840095—21no—
25#16840039C2212yes—
26#16839964C2212yes—
27#16839912—21no—
28#16839860—22no—
29#16839819C2211yescharacteristic
30#16839805—22no—
31C4×#11758194C2211yescharacteristic
32#16839507C2214yescomplemented
33Q8⋊#92418C2214yescomplemented
34#16839489C2214yes—
35C61×#533C2211yescharacteristic
36C2×C4×Q8×C61C2211yescharacteristic
37C61×#483—21no—
38C61×#475C2212yescomplemented
39C61×#462C2212yescomplemented
40C61×#451—21no—
41C61×#441—22no—
42C61×#432C2211yescharacteristic
43C22×Dic122—21no—
44C22×D244D411yescharacteristic
45C22×C4×D61—21no—
46C2×#11758201—22no—
47C2×#11758194—21no—
48#14519777—21no—
49#14519777—21no—
50#14519777D412yes—
51#14519777C2312yes—
52#14519772—28no—
53#14519772—28no—
54#14519765—24no—
55#14519765—24no—
56#14519753—24no—
57#14519753—24no—
58C4×Dic122—24no—
59C4×Dic122—24no—
60C4×D244—21no—
61C4×D244—21no—
62C4×D244D412yes—
63C4×D244C2312yes—
64#14519735—24no—
65#14519735—24no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61D412yes—
69C2×#11758196—22no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×#11758198—21no—
73C2×C4×Dic61—22no—
74#14519710—22no—
75#14519710—24no—
76#14519708—24no—
77C4⋊#92418C2312yes—
78#14519701—22no—
79#14519697—22no—
80C22⋊#92418—21no—
81#14519688—21no—
82#14519688—21no—
83C4×#92418C2312yes—
84C22×Q8×C61D411yescharacteristic
85C2×Q16×C61—22no—
86C4⋊Q8×C61C2311yescharacteristic
87C4⋊Q8×C61C2311yescharacteristic
88#125×C61—21no—
89C22⋊Q8×C61—21no—
90C22⋊Q8×C61—21no—
91C4×Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61D412yes—
96C4×Q8×C61C2312yes—
97#117×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61D411yescharacteristic
100C2×C42×C61—21no—
101C2×C42×C61C2311yescharacteristic
102#106×C61—22no—
103C4⋊C8×C61C2312yes—
104#103×C61—22no—
105#103×C61—22no—
106C22⋊C8×C61—21no—
107#97×C61—21no—
108C23×D61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—42no—
113C2×Q8×C61C2×D412yes—
114C22×C4×C61—21no—
115C22×C4×C61—21no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61C2×D411yescharacteristic
119C22×Dic61—21no—
120C22×Dic61—22no—
121C2×D244—21no—
122C2×D244—42no—
123C2×D244C2×D412yes—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—44no—
127C2×Dic122—22no—
128C2×Dic122—22no—
129C2×Dic122—44no—
130Q16×C61—44no—
131C2×C8×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—41no—
140C4⋊C4×C61—41no—
141C4⋊C4×C61C2×D412yes—
142C22⋊C4×C61—41no—
143C42×C61—22no—
144C42×C61—22no—
145C42×C61—41no—
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C42×C61C2411yescharacteristic
150#11758201—44no—
151#11758198—28no—
152#11758198—28no—
153#11758198—22no—
154#11758198—22no—
155C4⋊Dic61—22no—
156C4⋊Dic61—21no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4⋊Dic61C2×D412yes—
160C4⋊Dic61C2×D414yes—
161#11758196—28no—
162C4×Dic61—28no—
163#11758194—24no—
164C2×#92418—22no—
165C2×#92418—22no—
166C23×C61#12011yescharacteristic
167C22×D61—22no—
168C22×D61—42no—
169C22×D61—42no—
170C22×D61—42no—
171Q8×C61—22no—
172Q8×C61—22no—
173Q8×C61—42no—
174Q8×C61—42no—
175Q8×C61—42no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—42no—
182C2×C4×C61—42no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61#12011yescharacteristic
186C2×C4×C61#12011yescharacteristic
187C2×C4×C61#12011yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C2×Dic61—22no—
192C2×Dic61—24no—
193C2×Dic61—44no—
194C2×Dic61—42no—
195D244—22no—
196D244—22no—
197D244—42no—
198C4×D61—44no—
199C4×D61—44no—
200Dic122—44no—
201Dic122—44no—
202C8×C61—42no—
203#92418—24no—
204#92418—42no—
205#18542—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61#45612yes—
209C22×C61C2×#12011yescharacteristic
210C2×D61—42no—
211C2×D61—42no—
212C2×D61—42no—
213C2×D61—84no—
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—42no—
219C4×C61—42no—
220C4×C61—81no—
221C4×C61—81no—
222C4×C61#57512yes—
223C4×C61#57512yes—
224Dic61—44no—
225Dic61—42no—
226C2×#556—611no—
227#3165—611no—
228#3113D6111yescharacteristic, complemented
229#3110—611no—
230#1652—614no—
231#1650—614no—
232#1556—611no—
233#1491—611no—
234#1490—611no—
235C2×C61—22no—
236C2×C61#300511yescharacteristic
237C2×C61#300511yescharacteristic
238C2×C61C2×#45611yescharacteristic
239D61—82no—
240#556—614no—
241#556—1222no—
242#556—1222no—
243#556—1222no—
244#533C2×D6111yescharacteristic
245C2×C4⋊Q8—611no—
246C2×#126—1222no—
247C2×#124—1221no—
248C2×C22⋊Q8—1221no—
249C2×C4×Q8—1221no—
250C2×C4×Q8C2×D6111yescharacteristic
251C2×C4×D4—611no—
252C2×#117—1221no—
253#501—1222no—
254#500—612no—
255#498—612no—
256#484—1221no—
257#483—21no—
258#475C2×D6112yescomplemented
259#474—612no—
260#462C2×D6112yescomplemented
261#461—612no—
262#451—21no—
263#450—1221no—
264#441—22no—
265#440—1222no—
266#432—611no—
267#432C2×D6111yescharacteristic
268#428—1222no—
269C4×M16—611no—
270#332—614no—
271Q8⋊C8—614no—
272D4⋊C8—614no—
273C61#1854211yescharacteristic, complemented, Sylow
274C22×Q8—1221no—
275C22×Q8C22⋊D6111yescharacteristic
276C22×D4—611no—
277C23×C4—1221no—
278C2×Q16—22no—
279C2×SD16—1222no—
280C2×M16—1221no—
281C4⋊Q8—612no—
282C4⋊Q8—612no—
283C4⋊Q8—1221no—
284C4⋊Q8—1221no—
285C4⋊Q8C22×D6111yescharacteristic
286C4⋊Q8C22×D6111yescharacteristic
287#126—1228no—
288#126—1228no—
289#125—21no—
290#124—1224no—
291#124—1224no—
292C22⋊Q8—21no—
293C22⋊Q8—21no—
294C22⋊Q8—1224no—
295C22⋊Q8—1224no—
296C4×Q8—21no—
297C4×Q8—21no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8—1224no—
301C4×Q8—1224no—
302C4×Q8C22⋊D6112yes—
303C4×Q8C22×D6112yes—
304C4×D4—612no—
305C4×D4—612no—
306C4×D4—1221no—
307C4×D4—1221no—
308#117—21no—
309#117—1224no—
310#117—1224no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—611no—
313C2×C4⋊C4—612no—
314C2×C4⋊C4—1222no—
315C2×C4⋊C4—1221no—
316C2×C4⋊C4C22⋊D6111yescharacteristic
317C2×C22⋊C4—1222no—
318C2×C22⋊C4—1222no—
319C2×C22⋊C4—1221no—
320C2×C42—21no—
321C2×C42—1222no—
322C2×C42C22×D6111yescharacteristic
323#107—1224no—
324#107—1222no—
325#106—22no—
326#106—1224no—
327C4⋊C8—612no—
328C4⋊C8C22×D6112yes—
329#103—22no—
330#103—22no—
331#103—1222no—
332#102—1222no—
333C22⋊C8—21no—
334C22⋊C8—1221no—
335#97—21no—
336#97—1221no—
337#97—1221no—
338C4×C8—612no—
339C24—1221no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—42no—
344C2×Q8—1222no—
345C2×Q8—1222no—
346C2×Q8—2444no—
347C2×Q8C2×C22⋊D6112yes—
348C2×D4—612no—
349C2×D4—1221no—
350C2×D4—2442no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—1221no—
354C22×C4—1222no—
355C22×C4—1222no—
356C22×C4—1222no—
357C22×C4—2444no—
358C22×C4D4×D6111yescharacteristic
359C22×C4D4×D6111yescharacteristic
360C22×C4C2×C22⋊D6111yescharacteristic
361Q16—44no—
362SD16—2444no—
363M16—1224no—
364C2×C8—22no—
365C2×C8—1222no—
366C2×C8—1222no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—612no—
377C4⋊C4—614no—
378C4⋊C4—1228no—
379C4⋊C4—1222no—
380C4⋊C4—1221no—
381C4⋊C4—1222no—
382C4⋊C4—1222no—
383C4⋊C4C2×C22⋊D6112yes—
384C22⋊C4—41no—
385C22⋊C4—1228no—
386C22⋊C4—1228no—
387C22⋊C4—1222no—
388C22⋊C4—1222no—
389C42—22no—
390C42—22no—
391C42—41no—
392C42—1228no—
393C42D4×D6111yescharacteristic
394C42D4×D6111yescharacteristic
395C42C2×C22⋊D6111yescharacteristic
396C42C23×D6111yescharacteristic
397C23—1222no—
398C23—2442no—
399C23—2442no—
400C23—2442no—
401C23#1451974511yescharacteristic
402Q8—22no—
403Q8—22no—
404Q8—42no—
405Q8—42no—
406Q8—42no—
407Q8—2444no—
408Q8—2444no—
409D4—1222no—
410D4—1222no—
411D4—2442no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—1222no—
422C2×C4—1224no—
423C2×C4—2444no—
424C2×C4—2442no—
425C2×C4—2444no—
426C2×C4—2444no—
427C2×C4#1451974511yescharacteristic
428C2×C4#1451974511yescharacteristic
429C2×C4#1451974511yescharacteristic
430C2×C4C2×D4×D6111yescharacteristic
431C2×C4C2×D4×D6111yescharacteristic
432C2×C4C22×C22⋊D6111yescharacteristic
433C8—42no—
434C8—1224no—
435C8—2442no—
436C22—21no—
437C22—21no—
438C22—4884no—
439C22—2442no—
440C22—2442no—
441C22—2442no—
442C22#1683994312yes—
443C22C2×#1451974511yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—42no—
449C4—42no—
450C4—81no—
451C4—81no—
452C4—2442no—
453C4—2444no—
454C4#1684056012yes—
455C4#1684056112yes—
456C2—22no—
457C2—4882no—
458C2#2145505911yescharacteristic
459C2#2145506011yescharacteristic
460C2C2×#1683994311yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3113C2
socleC22×C61C2×#120
prime radicalC22×D244D4
hypercenter#3113D61
hypocenterC61#18542
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3113

Complements

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

NH ≅ G/Ntype
2. #21456366C2semidirect
4. #21445087C2semidirect
5. #21445074C2semidirect
6. #21444535C2semidirect
7. #21444266C2semidirect
9. C61×#3113C2semidirect
22. #16840165C22semidirect
23. #16840158C22semidirect
32. #16839507C22semidirect
33. Q8⋊#92418C22semidirect
38. C61×#475C22semidirect
39. C61×#462C22semidirect
228. #3113D61semidirect
258. #475C2×D61semidirect
260. #462C2×D61semidirect
273. C61#18542semidirect

Maximal subgroups

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

subgroupindex
C2×#168404852
#214563662
#214559142
#214450872
#214450742
#214445352
#214442662
#214442612
C61×#31132
#1854261