Finite Group Databasebeta

← #29656122

Subgroups — #29656122

Order 15616.

962 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840473C211yesmaximal, characteristic
2#21455945C211yesmaximal, characteristic, complemented
3#21455914C211yesmaximal, characteristic
4#21445030C214yesmaximal, complemented
5#21445023C214yesmaximal, complemented
6#21444545C211yesmaximal, characteristic, complemented
7#21444267C211yesmaximal, characteristic, complemented
8#21444263C211yesmaximal, characteristic
9C61×#3139C211yesmaximal, characteristic, complemented
10#16840473—22no—
11#16840473—22no—
12#16840473—22no—
13#16840473C2214yes—
14C2×#14519768—21no—
15C2×#14519757—21no—
16C2×#14519753—22no—
17C2×C4×Dic122C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519739—21no—
20C2×C4⋊C4×D61—21no—
21#16840095—21no—
22#16840090—21no—
23#16840044C2212yescomplemented
24#16840039C2212yes—
25#16839969C2212yescomplemented
26#16839964C2212yes—
27#16839917—21no—
28#16839912—21no—
29#16839867—22no—
30#16839860—22no—
31#16839819C2211yescharacteristic
32#16839819C2211yescharacteristic
33#16839511C2214yescomplemented
34#16839494C2214yescomplemented
35#16839491C2214yes—
36C61×#533C2211yescharacteristic
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#491—22no—
39C61×#487C2212yescomplemented
40C61×#487C2212yescomplemented
41C61×#419—22no—
42C4×M16×C61C2211yescharacteristic
43C22×Dic122D411yescharacteristic
44C22×D244D411yescharacteristic
45C22×C4×D61—21no—
46C22×C4×D61—22no—
47C2×#11758202—22no—
48C2×#11758201—22no—
49#14519768—24no—
50#14519768—24no—
51#14519757—24no—
52#14519757—24no—
53#14519753—28no—
54#14519753—28no—
55C4×Dic122—21no—
56C4×Dic122—21no—
57C4×Dic122D412yes—
58C4×Dic122C2312yes—
59C4×D244—21no—
60C4×D244—21no—
61C4×D244D412yes—
62C4×D244C2312yes—
63#14519739—24no—
64#14519739—24no—
65C2×C4⋊Dic61—22no—
66C2×C4⋊Dic61D411yescharacteristic
67C2×C4⋊Dic61D411yescharacteristic
68C2×#11758196—22no—
69C2×#11758196—21no—
70C4⋊C4×D61—24no—
71C4⋊C4×D61—24no—
72C2×#11758198—22no—
73C2×#11758198—21no—
74C2×C4×Dic61—21no—
75#14519710—22no—
76#14519708—22no—
77C4⋊#92418C2312yes—
78C4⋊#92418C2312yes—
79#14519701—22no—
80#14519701—22no—
81#14519700—22no—
82#14519697—22no—
83C22⋊#92418—21no—
84C22⋊#92418—21no—
85#14519688—21no—
86#14519688—21no—
87C22×Q8×C61D412yes—
88C2×M16×C61—21no—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61—21no—
91C4⋊Q8×C61D412yes—
92C4⋊Q8×C61C2311yescharacteristic
93C4⋊Q8×C61C2311yescharacteristic
94C4⋊Q8×C61C2312yes—
95#125×C61—21no—
96C22⋊Q8×C61—21no—
97C22⋊Q8×C61—21no—
98C4×Q8×C61—21no—
99C4×Q8×C61—21no—
100#117×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#103×C61—24no—
105#103×C61—24no—
106#97×C61—21no—
107C4×C8×C61C2312yes—
108C23×D61—21no—
109C2×Q8×C61—22no—
110C2×Q8×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—44no—
113C2×Q8×C61C2×D414yes—
114C22×C4×C61—22no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×Dic61—21no—
119C22×Dic61—21no—
120C22×Dic61—21no—
121C2×D244—21no—
122C2×D244—24no—
123C2×D244C2×D412yes—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—28no—
127C2×C4×D61—44no—
128C2×C4×D61—84no—
129C2×Dic122—21no—
130C2×Dic122—42no—
131C2×Dic122C2×D412yes—
132M16×C61—42no—
133C2×C8×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—41no—
143C4⋊C4×C61—41no—
144C22⋊C4×C61—41no—
145C42×C61—41no—
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C42×C61C2×D411yescharacteristic
149C42×C61C2411yescharacteristic
150#11758202—44no—
151#11758201—44no—
152#11758198—28no—
153#11758198—22no—
154#11758198—22no—
155C4⋊Dic61—28no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61C2×D412yes—
159C4⋊Dic61C2×D412yes—
160#11758196—28no—
161#11758196—22no—
162#11758196—22no—
163C4×Dic61—22no—
164C4×Dic61—22no—
165C2×#92418—22no—
166C2×#92418—22no—
167C23×C61#12011yescharacteristic
168C22×D61—24no—
169C22×D61—22no—
170C22×D61—42no—
171C22×D61—42no—
172Q8×C61—44no—
173Q8×C61—42no—
174Q8×C61—42no—
175Q8×C61—82no—
176C2×C4×C61—22no—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—24no—
180C2×C4×C61—44no—
181C2×C4×C61—41no—
182C2×C4×C61—41no—
183C2×C4×C61#12011yescharacteristic
184C2×C4×C61#12011yescharacteristic
185C2×C4×C61#12011yescharacteristic
186C2×C4×C61C22×D411yescharacteristic
187C2×C4×C61C22×D411yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C2×Dic61—22no—
190C2×Dic61—22no—
191C2×Dic61—22no—
192C2×Dic61—42no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195D244—24no—
196D244—22no—
197D244#13714yes—
198C4×D61—44no—
199C4×D61—44no—
200C4×D61—88no—
201Dic122—22no—
202Dic122—22no—
203Dic122—42no—
204C8×C61—42no—
205#92418—42no—
206#92418—42no—
207#18530—611nomaximal, Sylow
208C22×C61—21no—
209C22×C61—21no—
210C22×C61#45512yes—
211C22×C61C2×#12011yescharacteristic
212C2×D61—24no—
213C2×D61—48no—
214C2×D61—42no—
215C2×D61—42no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—44no—
221C4×C61—81no—
222C4×C61—81no—
223C4×C61C2×#13712yes—
224C4×C61#57512yes—
225Dic61—42no—
226Dic61—42no—
227Dic61—42no—
228C2×#554—611no—
229#3139D6111yescharacteristic, complemented
230#3113—611no—
231#3110—611no—
232#1644—614no—
233#1643—614no—
234#1558—611no—
235#1491—611no—
236#1490—611no—
237C2×C61—22no—
238C2×C61#300411yescharacteristic
239C2×C61#300511yescharacteristic
240C2×C61C2×#45511yescharacteristic
241D61—44no—
242#554—614no—
243#554—1222no—
244#554—1222no—
245#554—1222no—
246#533C2×D6111yescharacteristic
247C2×C4⋊Q8C2×D6111yescharacteristic
248C2×#125—1221no—
249C2×C22⋊Q8—1221no—
250C2×C22⋊Q8—1222no—
251C2×C4×Q8—611no—
252C2×C4×D4—611no—
253C2×C4×D4—1221no—
254C22×C4⋊C4—1221no—
255#491—22no—
256#487C2×D6112yescomplemented
257#487C2×D6112yescomplemented
258#484—1221no—
259#483—1221no—
260#475—612no—
261#474—612no—
262#462—612no—
263#461—612no—
264#451—1221no—
265#450—1221no—
266#441—1222no—
267#440—1222no—
268#432—611no—
269#432—611no—
270#419—22no—
271C4×M16C2×D6111yescharacteristic
272#333—614no—
273Q8⋊C8—614no—
274D4⋊C8—614no—
275C61#1853011yescharacteristic, complemented, Sylow
276C22×Q8—611no—
277C22×Q8C22⋊D6112yes—
278C22×D4—611no—
279C23×C4—1222no—
280C23×C4—1221no—
281C2×Q16—1222no—
282C2×SD16—1222no—
283C2×M16—21no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8C22⋊D6112yes—
287C4⋊Q8C22×D6111yescharacteristic
288C4⋊Q8C22×D6111yescharacteristic
289C4⋊Q8C22×D6112yes—
290#125—21no—
291#125—1224no—
292#125—1224no—
293C22⋊Q8—21no—
294C22⋊Q8—21no—
295C22⋊Q8—1224no—
296C22⋊Q8—1224no—
297C22⋊Q8—1228no—
298C22⋊Q8—1228no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—612no—
302C4×Q8—612no—
303C4×Q8—1221no—
304C4×Q8—1221no—
305C4×D4—612no—
306C4×D4—612no—
307C4×D4—1224no—
308C4×D4—1224no—
309C4×D4—1221no—
310C4×D4—1221no—
311#117—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—611no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—1224no—
317C2×C4⋊C4—1224no—
318C2×C4⋊C4—1222no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1221no—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1221no—
323C2×C42—1221no—
324C2×C42C22×D6111yescharacteristic
325#107—1222no—
326#106—1222no—
327C4⋊C8—612no—
328C4⋊C8—612no—
329#103—24no—
330#103—24no—
331#103—1222no—
332#103—1222no—
333#103—1222no—
334#102—1222no—
335C22⋊C8—1221no—
336C22⋊C8—1221no—
337#97—21no—
338#97—1221no—
339#97—1221no—
340C4×C8C22×D6112yes—
341C24—1221no—
342C2×Q8—22no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—44no—
346C2×Q8—612no—
347C2×Q8—1221no—
348C2×Q8—2442no—
349C2×Q8C2×C22⋊D6114yes—
350C2×D4—612no—
351C2×D4—1224no—
352C2×D4—1221no—
353C22×C4—22no—
354C22×C4—1228no—
355C22×C4—1221no—
356C22×C4—1221no—
357C22×C4—1221no—
358C22×C4—1222no—
359C22×C4—1222no—
360C22×C4—4884no—
361C22×C4—2444no—
362C22×C4D4×D6111yescharacteristic
363C22×C4D4×D6111yescharacteristic
364C22×C4C2×C22⋊D6111yescharacteristic
365Q16—2444no—
366SD16—2444no—
367M16—42no—
368C2×C8—22no—
369C2×C8—1222no—
370C2×C8—1222no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—612no—
382C4⋊C4—612no—
383C4⋊C4—1228no—
384C4⋊C4—1228no—
385C4⋊C4—1222no—
386C4⋊C4—1221no—
387C4⋊C4—1221no—
388C4⋊C4—1222no—
389C22⋊C4—41no—
390C22⋊C4—1228no—
391C22⋊C4—1222no—
392C22⋊C4—1222no—
393C42—41no—
394C42—1222no—
395C42—1222no—
396C42D4×D6111yescharacteristic
397C42D4×D6111yescharacteristic
398C42C2×C22⋊D6111yescharacteristic
399C42C23×D6111yescharacteristic
400C23—1224no—
401C23—1222no—
402C23—2442no—
403C23—2442no—
404C23#1451974511yescharacteristic
405Q8—44no—
406Q8—42no—
407Q8—42no—
408Q8—82no—
409Q8—1222no—
410Q8—1222no—
411Q8—2442no—
412D4—614no—
413D4—1224no—
414D4—1222no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—44no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—1222no—
423C2×C4—1222no—
424C2×C4—1222no—
425C2×C4—4888no—
426C2×C4—2444no—
427C2×C4—2442no—
428C2×C4—2442no—
429C2×C4—2442no—
430C2×C4—2444no—
431C2×C4#1451974511yescharacteristic
432C2×C4#1451974511yescharacteristic
433C2×C4#1451974511yescharacteristic
434C2×C4C2×D4×D6111yescharacteristic
435C2×C4C2×D4×D6111yescharacteristic
436C2×C4C22×C22⋊D6111yescharacteristic
437C8—42no—
438C8—2442no—
439C8—2442no—
440C22—21no—
441C22—21no—
442C22—1224no—
443C22—2448no—
444C22—2442no—
445C22—2442no—
446C22#1683993912yes—
447C22C2×#1451974511yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—44no—
453C4—81no—
454C4—81no—
455C4—2442no—
456C4—2442no—
457C4—2442no—
458C4#137×D6112yes—
459C4#1684056112yes—
460C2—22no—
461C2—2444no—
462C2#2145505111yescharacteristic
463C2#2145506011yescharacteristic
464C2C2×#1683993911yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3139C2
socleC22×C61C2×#120
prime radicalC22×D244D4
hypercenter#3139D61
hypocenterC61#18530
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
2. #21455945C2semidirect
4. #21445030C2semidirect
5. #21445023C2semidirect
6. #21444545C2semidirect
7. #21444267C2semidirect
9. C61×#3139C2semidirect
23. #16840044C22semidirect
25. #16839969C22semidirect
33. #16839511C22semidirect
34. #16839494C22semidirect
39. C61×#487C22semidirect
40. C61×#487C22semidirect
229. #3139D61semidirect
256. #487C2×D61semidirect
257. #487C2×D61semidirect
275. C61#18530semidirect

Maximal subgroups

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

subgroupindex
C2×#168404732
#214559452
#214559142
#214450302
#214450232
#214445452
#214442672
#214442632
C61×#31392
#1853061