Finite Group Databasebeta

← #29658647

Subgroups — #29658647

Order 15616.

910 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456970C211yesmaximal, characteristic, complemented
2#21455827C211yesmaximal, characteristic
3#21455624C211yesmaximal, characteristic, complemented
4#21445075C214yesmaximal, complemented
5#21444991C214yesmaximal, complemented
6#21444504C211yesmaximal, characteristic, complemented
7#21444244C211yesmaximal, characteristic
8#21444214C211yesmaximal, characteristic, complemented
9C2×C61×#556C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840165C2212yescomplemented
12#16840114C2212yescomplemented
13#16840097—22no—
14#16840092—21no—
15#16840069—22no—
16#16840057C2212yescomplemented
17#16840054—22no—
18#16840052C2212yes—
19#16840047C2212yescomplemented
20#16839967C2212yes—
21#16839913—21no—
22C4×#11758200—22no—
23#16839815C2211yescharacteristic
24#16839815C2211yescharacteristic
25#16839802—21no—
26#16839786C2211yescharacteristic
27#16839771—21no—
28#16839508C2214yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2214yes—
31C61×#556—22no—
32C61×#556—22no—
33C61×#556—22no—
34C61×#556C2214yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×#126×C61—22no—
37C2×#124×C61—21no—
38C2×C22⋊Q8×C61—21no—
39C2×C4×Q8×C61—21no—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4×D4×C61C2211yescharacteristic
42C2×#117×C61—21no—
43C2×#11758200—22no—
44C2×#11758194—21no—
45#14519777C2311yescharacteristic
46#14519777C2311yescharacteristic
47#14519769—21no—
48#14519756—21no—
49#14519756—21no—
50C4×Dic122—21no—
51C4×Dic122—21no—
52#14519736—21no—
53#14519710—24no—
54#14519710—22no—
55#14519708—22no—
56#14519708—22no—
57C4⋊#92418—21no—
58C4⋊#92418—21no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519701—22no—
62#14519701—22no—
63#14519700—22no—
64#14519700—24no—
65#14519698—22no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C4×#92418—21no—
69C4×#92418C2312yes—
70C22×Q8×C61—21no—
71C22×Q8×C61D411yescharacteristic
72C22×D4×C61D411yescharacteristic
73C23×C4×C61—21no—
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61D412yes—
77C4⋊Q8×C61C2312yes—
78#126×C61—28no—
79#126×C61—28no—
80#124×C61—24no—
81#124×C61—24no—
82C22⋊Q8×C61—24no—
83C22⋊Q8×C61—24no—
84C4×Q8×C61—24no—
85C4×Q8×C61—24no—
86C4×Q8×C61—21no—
87C4×Q8×C61—21no—
88C4×Q8×C61D412yes—
89C4×Q8×C61C2312yes—
90C4×D4×C61—21no—
91C4×D4×C61—21no—
92C4×D4×C61D412yes—
93C4×D4×C61C2312yes—
94#117×C61—24no—
95#117×C61—24no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61D412yes—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61—21no—
105C2×C42×C61—22no—
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C24×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—22no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—22no—
114C2×Q8×C61—44no—
115C2×Q8×C61C2×D412yes—
116C2×D4×C61—21no—
117C2×D4×C61—42no—
118C2×D4×C61C2×D412yes—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—42no—
126C22×C4×C61—42no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C2×Dic122—21no—
131C2×Dic122—21no—
132C4⋊C4×C61—28no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61C2×D412yes—
141C4⋊C4×C61C2×D412yes—
142C4⋊C4×C61C2×D414yes—
143C22⋊C4×C61—28no—
144C22⋊C4×C61—28no—
145C22⋊C4×C61—22no—
146C22⋊C4×C61—22no—
147C42×C61—28no—
148C42×C61—22no—
149C42×C61—22no—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61C2411yescharacteristic
154C22⋊Dic61—41no—
155#11758200—44no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160#11758196—41no—
161#11758196—41no—
162C4×Dic61—41no—
163#11758194—42no—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—22no—
167C23×C61—22no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61#12011yescharacteristic
172Q8×C61—22no—
173Q8×C61—22no—
174Q8×C61—44no—
175Q8×C61—44no—
176Q8×C61—42no—
177D4×C61—22no—
178D4×C61—22no—
179D4×C61—42no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—24no—
185C2×C4×C61—41no—
186C2×C4×C61—42no—
187C2×C4×C61—44no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—44no—
193C2×C4×C61#12011yescharacteristic
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#12011yescharacteristic
196C2×C4×C61#13512yes—
197C2×C4×C61#13712yes—
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×Dic61—41no—
202C2×Dic61—41no—
203Dic122—42no—
204Dic122—42no—
205#92418—42no—
206#92418—42no—
207#92418—42no—
208#18763—611nomaximal, Sylow
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—42no—
212C22×C61—42no—
213C22×C61—42no—
214C22×C61—84no—
215C22×C61#45212yes—
216C22×C61C2×#12011yescharacteristic
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—42no—
221C4×C61—42no—
222C4×C61—44no—
223C4×C61—42no—
224C4×C61#45212yes—
225C4×C61C2×#13511yescharacteristic
226C4×C61C2×#13711yescharacteristic
227C4×C61#57512yes—
228Dic61—81no—
229Dic61—81no—
230C2×#556D6111yescharacteristic
231#3230—611no—
232#3101—611no—
233#3078—611no—
234#1650—614no—
235#1638—614no—
236#1549—611no—
237#1486—611no—
238#1479—611no—
239C2×C61—22no—
240C2×C61—82no—
241C2×C61C2×#45211yescharacteristic
242C2×C61#300511yescharacteristic
243C2×C61#302011yescharacteristic
244#556—22no—
245#556—22no—
246#556—22no—
247#556C2×D6114yes—
248#533—611no—
249C2×C4⋊Q8C2×D6111yescharacteristic
250C2×#126—22no—
251C2×#124—21no—
252C2×C22⋊Q8—21no—
253C2×C4×Q8—21no—
254C2×C4×Q8C2×D6111yescharacteristic
255C2×C4×D4C2×D6111yescharacteristic
256C2×#117—21no—
257#500—612no—
258#487—612no—
259#484—1222no—
260#483—1221no—
261#479—1222no—
262#477—612no—
263#477—1222no—
264#476—612no—
265#475—612no—
266#461—612no—
267#450—1221no—
268C4×SD16—1222no—
269#431—611no—
270#431—611no—
271#428—1221no—
272#423—611no—
273#419—1221no—
274#332—614no—
275Q8⋊C8—614no—
276D4⋊C8—614no—
277C61#1876311yescharacteristic, complemented, Sylow
278C22×Q8—21no—
279C22×Q8C22⋊D6111yescharacteristic
280C22×D4C22⋊D6111yescharacteristic
281C23×C4—21no—
282C2×SD16—1222no—
283C2×M16—1221no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—611no—
287C4⋊Q8—611no—
288C4⋊Q8C22⋊D6112yes—
289C4⋊Q8C22×D6112yes—
290#126—28no—
291#126—28no—
292#125—1221no—
293#124—24no—
294#124—24no—
295C22⋊Q8—24no—
296C22⋊Q8—24no—
297C22⋊Q8—1221no—
298C22⋊Q8—1221no—
299C4×Q8—24no—
300C4×Q8—24no—
301C4×Q8—21no—
302C4×Q8—21no—
303C4×Q8—1221no—
304C4×Q8—1221no—
305C4×Q8C22⋊D6112yes—
306C4×Q8C22×D6112yes—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4C22⋊D6112yes—
310C4×D4C22×D6112yes—
311#117—24no—
312#117—24no—
313#117—1221no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4C22⋊D6111yescharacteristic
318C2×C4⋊C4C22⋊D6111yescharacteristic
319C2×C4⋊C4C22⋊D6112yes—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42C22×D6111yescharacteristic
326#107—1222no—
327#107—1224no—
328#106—1222no—
329#106—1222no—
330C4⋊C8—612no—
331C4⋊C8—612no—
332C4⋊C8—1221no—
333C4⋊C8—1221no—
334#103—1222no—
335#103—1222no—
336#103—1222no—
337#103—1224no—
338#102—1222no—
339C22⋊C8—1221no—
340C22⋊C8—1221no—
341C4×C8—612no—
342C4×C8—1221no—
343C24—21no—
344C2×Q8—22no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—22no—
348C2×Q8—22no—
349C2×Q8—44no—
350C2×Q8—1221no—
351C2×Q8—1221no—
352C2×Q8C2×C22⋊D6112yes—
353C2×D4—21no—
354C2×D4—42no—
355C2×D4C2×C22⋊D6112yes—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4C2×C22⋊D6111yescharacteristic
365C22×C4C2×C22⋊D6111yescharacteristic
366C22×C4C2×C22⋊D6111yescharacteristic
367SD16—2444no—
368M16—2442no—
369C2×C8—1222no—
370C2×C8—1222no—
371C2×C8—1222no—
372C4⋊C4—28no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—1221no—
381C4⋊C4—1221no—
382C4⋊C4—1221no—
383C4⋊C4—1221no—
384C4⋊C4—2441no—
385C4⋊C4—2441no—
386C4⋊C4C2×C22⋊D6112yes—
387C4⋊C4C2×C22⋊D6114yes—
388C4⋊C4C2×C22⋊D6112yes—
389C22⋊C4—28no—
390C22⋊C4—28no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—2441no—
394C42—28no—
395C42—22no—
396C42—22no—
397C42—2441no—
398C42C2×C22⋊D6111yescharacteristic
399C42C2×C22⋊D6111yescharacteristic
400C42C2×C22⋊D6111yescharacteristic
401C42C23×D6111yescharacteristic
402C23—22no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23#1451974611yescharacteristic
407Q8—22no—
408Q8—22no—
409Q8—44no—
410Q8—44no—
411Q8—42no—
412Q8—2442no—
413Q8—2442no—
414D4—22no—
415D4—22no—
416D4—42no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—2441no—
431C2×C4—2441no—
432C2×C4#1451974611yescharacteristic
433C2×C4#1451974611yescharacteristic
434C2×C4#1451974611yescharacteristic
435C2×C4#1451979712yes—
436C2×C4#1451980712yes—
437C2×C4C22×C22⋊D6111yescharacteristic
438C2×C4C22×C22⋊D6111yescharacteristic
439C2×C4C22×C22⋊D6111yescharacteristic
440C8—2442no—
441C8—2442no—
442C8—2442no—
443C22—21no—
444C22—21no—
445C22—42no—
446C22—42no—
447C22—42no—
448C22—84no—
449C22#1683992412yes—
450C22C2×#1451974611yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—42no—
455C4—42no—
456C4—42no—
457C4—44no—
458C4—4881no—
459C4—4881no—
460C4#1683992412yes—
461C4C2×#1451979711yescharacteristic
462C4C2×#1451980711yescharacteristic
463C4#1684056012yes—
464C2—22no—
465C2—82no—
466C2C2×#1683992411yescharacteristic
467C2#2145506111yescharacteristic
468C2#2145518311yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#556C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenterC2×#556D61
hypocenterC61#18763
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. #21456970C2semidirect
3. #21455624C2semidirect
4. #21445075C2semidirect
5. #21444991C2semidirect
6. #21444504C2semidirect
8. #21444214C2semidirect
11. #16840165C22semidirect
12. #16840114C22semidirect
16. #16840057C22semidirect
19. #16840047C22semidirect
28. #16839508C22semidirect
29. Q8⋊#92418C22semidirect
277. C61#18763semidirect

Maximal subgroups

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

subgroupindex
#214569702
#214558272
#214556242
#214450752
#214449912
#214445042
#214442442
#214442142
C2×C61×#5562
#1876361