Finite Group Databasebeta

← #29658644

Subgroups — #29658644

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
1C2×#16840486C211yesmaximal, characteristic
2#21455823C211yesmaximal, characteristic
3#21455620C211yesmaximal, characteristic, complemented
4#21445072C214yesmaximal, complemented
5#21444988C214yesmaximal, complemented
6#21444504C211yesmaximal, characteristic, complemented
7#21444245C211yesmaximal, characteristic
8#21444215C211yesmaximal, characteristic, complemented
9C61×#3230C211yesmaximal, characteristic, complemented
10#16840486—22no—
11#16840486—22no—
12#16840486—22no—
13#16840486C2214yes—
14C2×#14519772—22no—
15C2×#14519766—21no—
16C2×#14519757—21no—
17C2×Q8×Dic61—21no—
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×D244C2211yescharacteristic
20C2×#14519735—21no—
21#16840090—21no—
22#16840066—22no—
23#16840054C2212yescomplemented
24#16840049C2212yes—
25#16840044C2212yescomplemented
26#16839964C2212yes—
27#16839912—21no—
28C4×#11758201—22no—
29#16839815C2211yescharacteristic
30#16839805—21no—
31#16839786C2211yescharacteristic
32#16839770—21no—
33#16839508C2214yescomplemented
34#16839494C2214yescomplemented
35#16839491C2214yes—
36C61×#533C2211yescharacteristic
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#500C2212yescomplemented
39C61×#487C2212yescomplemented
40C61×#484—22no—
41C61×#477—22no—
42C61×#431C2211yescharacteristic
43C22×Dic122D411yescharacteristic
44C22×D244D411yescharacteristic
45C22×C4×D61—21no—
46C2×#11758201—22no—
47C2×#11758194—21no—
48#14519772—28no—
49#14519772—28no—
50#14519766—24no—
51#14519766—24no—
52#14519757—24no—
53#14519757—24no—
54Q8×Dic61—24no—
55Q8×Dic61—24no—
56C4×Dic122—21no—
57C4×Dic122—21no—
58C4×Dic122D412yes—
59C4×Dic122C2312yes—
60C4×D244—21no—
61C4×D244—21no—
62C4×D244D412yes—
63C4×D244C2312yes—
64#14519735—24no—
65#14519735—24no—
66C2×C4⋊Dic61—22no—
67C2×C4⋊Dic61D411yescharacteristic
68C2×C4⋊Dic61D411yescharacteristic
69C2×#11758196—21no—
70C2×#11758198—22no—
71C2×#11758198—22no—
72C2×#11758198—21no—
73C2×C4×Dic61—22no—
74C2×C4×Dic61—21no—
75#14519710—22no—
76#14519708—22no—
77#14519708—22no—
78C4⋊#92418—21no—
79C4⋊#92418C2312yes—
80C4⋊#92418C2312yes—
81#14519701—22no—
82#14519700—22no—
83#14519700—22no—
84#14519697—22no—
85C22⋊#92418—21no—
86C4×#92418—21no—
87C22×Q8×C61—21no—
88C4⋊Q8×C61—21no—
89C4⋊Q8×C61—21no—
90C4⋊Q8×C61D412yes—
91C4⋊Q8×C61C2311yescharacteristic
92C4⋊Q8×C61C2311yescharacteristic
93C4⋊Q8×C61C2312yes—
94#125×C61—21no—
95C22⋊Q8×C61—21no—
96C22⋊Q8×C61—21no—
97C4×Q8×C61—21no—
98C4×Q8×C61—21no—
99#117×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61D412yes—
102C2×C42×C61C2311yescharacteristic
103#107×C61—24no—
104C4⋊C8×C61—21no—
105#103×C61—24no—
106C22⋊C8×C61—21no—
107C4×C8×C61C2312yes—
108C23×D61—21no—
109C2×Q8×C61—22no—
110C2×Q8×C61—22no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×Q8×C61—44no—
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—42no—
123C2×D244C2×D412yes—
124C2×C4×D61—22no—
125C2×C4×D61—22no—
126C2×C4×D61—42no—
127C2×C4×D61—42no—
128C2×Dic122—22no—
129C2×Dic122—22no—
130C2×Dic122—21no—
131C2×Dic122C2×D412yes—
132C2×C8×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—41no—
141C4⋊C4×C61—41no—
142C4⋊C4×C61C2×D414yes—
143C22⋊C4×C61—41no—
144C42×C61—41no—
145C42×C61C2×D411yescharacteristic
146C42×C61C2×D411yescharacteristic
147C42×C61C2×D411yescharacteristic
148C42×C61C2411yescharacteristic
149#11758201—44no—
150#11758198—28no—
151#11758198—28no—
152#11758198—22no—
153#11758198—22no—
154C4⋊Dic61—28no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61C2×D412yes—
158C4⋊Dic61C2×D412yes—
159#11758196—22no—
160#11758196—22no—
161C4×Dic61—28no—
162C4×Dic61—22no—
163C4×Dic61—22no—
164#11758194—42no—
165C2×#92418—22no—
166C2×#92418—22no—
167C23×C61#12011yescharacteristic
168C22×D61—22no—
169C22×D61—42no—
170C22×D61—42no—
171C22×D61—42no—
172Q8×C61—44no—
173Q8×C61—42no—
174Q8×C61—44no—
175Q8×C61—42no—
176C2×C4×C61—22no—
177C2×C4×C61—24no—
178C2×C4×C61—41no—
179C2×C4×C61—44no—
180C2×C4×C61—41no—
181C2×C4×C61—41no—
182C2×C4×C61#12011yescharacteristic
183C2×C4×C61#12011yescharacteristic
184C2×C4×C61#12011yescharacteristic
185C2×C4×C61#13512yes—
186C2×C4×C61#13712yes—
187C2×C4×C61C22×D411yescharacteristic
188C2×C4×C61C22×D411yescharacteristic
189C2×C4×C61C22×D411yescharacteristic
190C2×Dic61—22no—
191C2×Dic61—22no—
192C2×Dic61—22no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195C2×Dic61—42no—
196D244—22no—
197D244—22no—
198D244—42no—
199C4×D61—42no—
200C4×D61—42no—
201C4×D61—44no—
202Dic122—22no—
203Dic122—22no—
204Dic122—42no—
205C8×C61—42no—
206#92418—42no—
207#92418—42no—
208#18763—611nomaximal, Sylow
209C22×C61—21no—
210C22×C61—21no—
211C22×C61#45212yes—
212C22×C61C2×#12011yescharacteristic
213C2×D61—42no—
214C2×D61—42no—
215C2×D61—42no—
216C2×D61—84no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—44no—
221C4×C61—81no—
222C4×C61—81no—
223C4×C61#45212yes—
224C4×C61C2×#13511yescharacteristic
225C4×C61C2×#13711yescharacteristic
226C4×C61#57512yes—
227Dic61—42no—
228Dic61—42no—
229Dic61—42no—
230C2×#556—611no—
231#3230D6111yescharacteristic, complemented
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×C61C2×#45211yescharacteristic
241C2×C61#300511yescharacteristic
242C2×C61#302011yescharacteristic
243D61—82no—
244#556—614no—
245#556—1222no—
246#556—1222no—
247#556—1222no—
248#533C2×D6111yescharacteristic
249C2×C4⋊Q8C2×D6111yescharacteristic
250C2×#126—1222no—
251C2×#124—1221no—
252C2×C22⋊Q8—1221no—
253C2×C4×Q8—611no—
254C2×C4×Q8—1221no—
255C2×C4×D4—611no—
256C2×#117—1221no—
257#500C2×D6112yescomplemented
258#487C2×D6112yescomplemented
259#484—22no—
260#483—1221no—
261#479—1222no—
262#477—22no—
263#477—612no—
264#476—612no—
265#475—612no—
266#461—612no—
267#450—1221no—
268C4×SD16—1222no—
269#431—611no—
270#431C2×D6111yescharacteristic
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×Q8—611no—
280C22×D4—611no—
281C23×C4—1221no—
282C2×SD16—1222no—
283C2×M16—1221no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8C22⋊D6112yes—
287C4⋊Q8C22×D6111yescharacteristic
288C4⋊Q8C22×D6111yescharacteristic
289C4⋊Q8C22×D6112yes—
290#126—1228no—
291#126—1228no—
292#125—21no—
293#124—1224no—
294#124—1224no—
295C22⋊Q8—21no—
296C22⋊Q8—21no—
297C22⋊Q8—1224no—
298C22⋊Q8—1224no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—612no—
302C4×Q8—612no—
303C4×Q8—1224no—
304C4×Q8—1224no—
305C4×Q8—1221no—
306C4×Q8—1221no—
307C4×D4—612no—
308C4×D4—612no—
309C4×D4—1221no—
310C4×D4—1221no—
311#117—21no—
312#117—1224no—
313#117—1224no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—611no—
317C2×C4⋊C4—1222no—
318C2×C4⋊C4—1221no—
319C2×C4⋊C4C22⋊D6112yes—
320C2×C22⋊C4—1222no—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1221no—
323C2×C42—1222no—
324C2×C42—1221no—
325C2×C42C22×D6111yescharacteristic
326#107—24no—
327#107—1222no—
328#106—1222no—
329#106—1222no—
330C4⋊C8—21no—
331C4⋊C8—612no—
332C4⋊C8—612no—
333C4⋊C8—1221no—
334#103—24no—
335#103—1222no—
336#103—1222no—
337#103—1222no—
338#102—1222no—
339C22⋊C8—21no—
340C22⋊C8—1221no—
341C4×C8—1221no—
342C4×C8C22×D6112yes—
343C24—1221no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—44no—
349C2×Q8—612no—
350C2×Q8—1221no—
351C2×Q8—1222no—
352C2×Q8—1222no—
353C2×D4—612no—
354C2×D4—1221no—
355C2×D4—2442no—
356C22×C4—22no—
357C22×C4—1221no—
358C22×C4—1221no—
359C22×C4—1221no—
360C22×C4—1222no—
361C22×C4—1222no—
362C22×C4—2442no—
363C22×C4—2442no—
364C22×C4D4×D6111yescharacteristic
365C22×C4D4×D6111yescharacteristic
366C22×C4C2×C22⋊D6111yescharacteristic
367SD16—2444no—
368M16—2442no—
369C2×C8—22no—
370C2×C8—1222no—
371C2×C8—1222no—
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—1222no—
385C4⋊C4—1221no—
386C4⋊C4—1221no—
387C4⋊C4—1222no—
388C4⋊C4C2×C22⋊D6114yes—
389C22⋊C4—41no—
390C22⋊C4—1228no—
391C22⋊C4—1228no—
392C22⋊C4—1222no—
393C22⋊C4—1222no—
394C42—41no—
395C42—1228no—
396C42—1222no—
397C42—1222no—
398C42D4×D6111yescharacteristic
399C42D4×D6111yescharacteristic
400C42C2×C22⋊D6111yescharacteristic
401C42C23×D6111yescharacteristic
402C23—1222no—
403C23—2442no—
404C23—2442no—
405C23—2442no—
406C23#1451974511yescharacteristic
407Q8—44no—
408Q8—42no—
409Q8—44no—
410Q8—42no—
411Q8—1222no—
412Q8—1222no—
413Q8—2442no—
414D4—1222no—
415D4—1222no—
416D4—2442no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—1222no—
424C2×C4—1222no—
425C2×C4—1222no—
426C2×C4—2442no—
427C2×C4—2442no—
428C2×C4—2442no—
429C2×C4—2442no—
430C2×C4—2442no—
431C2×C4—2444no—
432C2×C4#1451974511yescharacteristic
433C2×C4#1451974511yescharacteristic
434C2×C4#1451974511yescharacteristic
435C2×C4#1451980012yes—
436C2×C4#1451980912yes—
437C2×C4C2×D4×D6111yescharacteristic
438C2×C4C2×D4×D6111yescharacteristic
439C2×C4C22×C22⋊D6111yescharacteristic
440C8—42no—
441C8—2442no—
442C8—2442no—
443C22—21no—
444C22—21no—
445C22—4884no—
446C22—2442no—
447C22—2442no—
448C22—2442no—
449C22#1683992312yes—
450C22C2×#1451974511yescharacteristic
451C4—21no—
452C4—21no—
453C4—21no—
454C4—44no—
455C4—81no—
456C4—81no—
457C4—2442no—
458C4—2442no—
459C4—2442no—
460C4#1683992312yes—
461C4C2×#1451980011yescharacteristic
462C4C2×#1451980911yescharacteristic
463C4#1684055812yes—
464C2—22no—
465C2—4882no—
466C2C2×#1683992311yescharacteristic
467C2#2145505811yescharacteristic
468C2#2145518211yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3230C2
socleC22×C61C2×#120
prime radicalC22×D244D4
hypercenter#3230D61
hypocenterC61#18763
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3230

Complements

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

NH ≅ G/Ntype
3. #21455620C2semidirect
4. #21445072C2semidirect
5. #21444988C2semidirect
6. #21444504C2semidirect
8. #21444215C2semidirect
9. C61×#3230C2semidirect
23. #16840054C22semidirect
25. #16840044C22semidirect
33. #16839508C22semidirect
34. #16839494C22semidirect
38. C61×#500C22semidirect
39. C61×#487C22semidirect
231. #3230D61semidirect
257. #500C2×D61semidirect
258. #487C2×D61semidirect
277. C61#18763semidirect

Maximal subgroups

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

subgroupindex
C2×#168404862
#214558232
#214556202
#214450722
#214449882
#214445042
#214442452
#214442152
C61×#32302
#1876361