Finite Group Databasebeta

← #29658542

Subgroups — #29658542

Order 15616.

966 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
1C2×#16840473C211yesmaximal, characteristic
2#21455823C211yesmaximal, characteristic
3#21455620C211yesmaximal, characteristic, complemented
4#21445023C214yesmaximal, complemented
5#21444950C214yesmaximal, complemented
6#21444500C211yesmaximal, characteristic, complemented
7#21444245C211yesmaximal, characteristic
8#21444215C211yesmaximal, characteristic, complemented
9C61×#3209C211yesmaximal, 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#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#16839511C2214yescomplemented
34#16839494C2214yescomplemented
35#16839491C2214yes—
36C61×#533C2211yescharacteristic
37C2×C4⋊Q8×C61C2211yescharacteristic
38C61×#494C2212yescomplemented
39C61×#487C2212yescomplemented
40C61×#462—22no—
41C61×#451—22no—
42C61×#431C2211yescharacteristic
43C22×Dic122D411yescharacteristic
44C22×D244D411yescharacteristic
45C22×C4×D61—21no—
46C22×C4×D61—22no—
47C2×#11758201—22no—
48C2×#11758194—21no—
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—
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×C61D412yes—
88C2×Q16×C61—24no—
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
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—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—82no—
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—42no—
128C2×C4×D61—42no—
129C2×C4×D61—84no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C2×Dic122—21no—
133C2×Dic122C2×D412yes—
134Q16×C61—84no—
135C2×C8×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—21no—
144C4⋊C4×C61—41no—
145C4⋊C4×C61—41no—
146C22⋊C4×C61—41no—
147C42×C61—41no—
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61C2411yescharacteristic
152#11758201—44no—
153#11758198—28no—
154#11758198—22no—
155#11758198—22no—
156C4⋊Dic61—28no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61C2×D412yes—
160C4⋊Dic61C2×D412yes—
161#11758196—28no—
162#11758196—22no—
163#11758196—22no—
164C4×Dic61—22no—
165C4×Dic61—22no—
166#11758194—42no—
167C2×#92418—22no—
168C2×#92418—22no—
169C23×C61#12011yescharacteristic
170C22×D61—24no—
171C22×D61—22no—
172C22×D61—42no—
173C22×D61—42no—
174Q8×C61—44no—
175Q8×C61—42no—
176Q8×C61—42no—
177Q8×C61—82no—
178C2×C4×C61—22no—
179C2×C4×C61—24no—
180C2×C4×C61—41no—
181C2×C4×C61—44no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61#12011yescharacteristic
185C2×C4×C61#12011yescharacteristic
186C2×C4×C61#12011yescharacteristic
187C2×C4×C61#13512yes—
188C2×C4×C61#13712yes—
189C2×C4×C61C22×D411yescharacteristic
190C2×C4×C61C22×D411yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×Dic61—22no—
193C2×Dic61—22no—
194C2×Dic61—22no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198D244—24no—
199D244—22no—
200D244#13714yes—
201C4×D61—42no—
202C4×D61—42no—
203C4×D61—44no—
204C4×D61—88no—
205Dic122—22no—
206Dic122—22no—
207Dic122—42no—
208C8×C61—42no—
209#92418—42no—
210#92418—42no—
211#18753—611nomaximal, Sylow
212C22×C61—21no—
213C22×C61—21no—
214C22×C61#45212yes—
215C22×C61C2×#12011yescharacteristic
216C2×D61—24no—
217C2×D61—48no—
218C2×D61—42no—
219C2×D61—42no—
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—21no—
223C4×C61—44no—
224C4×C61—81no—
225C4×C61—81no—
226C4×C61#45212yes—
227C4×C61C2×#13511yescharacteristic
228C4×C61C2×#13712yes—
229C4×C61C2×#13711yescharacteristic
230Dic61—42no—
231Dic61—42no—
232Dic61—42no—
233C2×#554—611no—
234#3209D6111yescharacteristic, complemented
235#3101—611no—
236#3078—611no—
237#1643—614no—
238#1632—614no—
239#1548—611no—
240#1486—611no—
241#1479—611no—
242C2×C61—22no—
243C2×C61C2×#45211yescharacteristic
244C2×C61#300411yescharacteristic
245C2×C61#301911yescharacteristic
246D61—44no—
247#554—614no—
248#554—1222no—
249#554—1222no—
250#554—1222no—
251#533C2×D6111yescharacteristic
252C2×C4⋊Q8C2×D6111yescharacteristic
253C2×#125—1221no—
254C2×C22⋊Q8—1221no—
255C2×C22⋊Q8—1222no—
256C2×C4×Q8—611no—
257C2×C4×D4—611no—
258C2×C4×D4—1221no—
259C22×C4⋊C4—1221no—
260#494C2×D6112yescomplemented
261#487C2×D6112yescomplemented
262#483—1221no—
263#479—1222no—
264#477—612no—
265#476—612no—
266#475—612no—
267#462—22no—
268#461—612no—
269#451—22no—
270#450—1221no—
271C4×SD16—1222no—
272#431—611no—
273#431C2×D6111yescharacteristic
274#428—1221no—
275#423—611no—
276#419—1221no—
277#333—614no—
278Q8⋊C8—614no—
279D4⋊C8—614no—
280C61#1875311yescharacteristic, complemented, Sylow
281C22×Q8—611no—
282C22×Q8C22⋊D6112yes—
283C22×D4—611no—
284C23×C4—1222no—
285C23×C4—1221no—
286C2×Q16—24no—
287C2×SD16—1222no—
288C2×M16—1221no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8C22⋊D6112yes—
292C4⋊Q8C22×D6111yescharacteristic
293C4⋊Q8C22×D6111yescharacteristic
294C4⋊Q8C22×D6112yes—
295#125—21no—
296#125—1224no—
297#125—1224no—
298C22⋊Q8—21no—
299C22⋊Q8—21no—
300C22⋊Q8—1224no—
301C22⋊Q8—1224no—
302C22⋊Q8—1228no—
303C22⋊Q8—1228no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8—612no—
307C4×Q8—612no—
308C4×Q8—1221no—
309C4×Q8—1221no—
310C4×D4—612no—
311C4×D4—612no—
312C4×D4—1224no—
313C4×D4—1224no—
314C4×D4—1221no—
315C4×D4—1221no—
316#117—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—611no—
320C2×C4⋊C4—611no—
321C2×C4⋊C4—1224no—
322C2×C4⋊C4—1224no—
323C2×C4⋊C4—1222no—
324C2×C4⋊C4—1222no—
325C2×C4⋊C4—1221no—
326C2×C22⋊C4—1222no—
327C2×C22⋊C4—1221no—
328C2×C42—1221no—
329C2×C42C22×D6111yescharacteristic
330#107—1222no—
331#106—1222no—
332#106—1222no—
333C4⋊C8—21no—
334C4⋊C8—612no—
335C4⋊C8—612no—
336C4⋊C8—1221no—
337#103—24no—
338#103—1222no—
339#103—1222no—
340#103—1222no—
341#102—1222no—
342C22⋊C8—21no—
343C22⋊C8—1221no—
344C4×C8—1221no—
345C4×C8C22×D6112yes—
346C24—1221no—
347C2×Q8—22no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—82no—
351C2×Q8—612no—
352C2×Q8—1221no—
353C2×Q8—1222no—
354C2×Q8—1222no—
355C2×Q8C2×C22⋊D6114yes—
356C2×D4—612no—
357C2×D4—1224no—
358C2×D4—1221no—
359C22×C4—22no—
360C22×C4—1228no—
361C22×C4—1221no—
362C22×C4—1221no—
363C22×C4—1221no—
364C22×C4—1222no—
365C22×C4—1222no—
366C22×C4—4884no—
367C22×C4—2442no—
368C22×C4—2442no—
369C22×C4D4×D6111yescharacteristic
370C22×C4D4×D6111yescharacteristic
371C22×C4C2×C22⋊D6111yescharacteristic
372Q16—84no—
373SD16—2444no—
374M16—2442no—
375C2×C8—22no—
376C2×C8—1222no—
377C2×C8—1222no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—612no—
389C4⋊C4—612no—
390C4⋊C4—1228no—
391C4⋊C4—1228no—
392C4⋊C4—1222no—
393C4⋊C4—1221no—
394C4⋊C4—1221no—
395C4⋊C4—1222no—
396C22⋊C4—41no—
397C22⋊C4—1228no—
398C22⋊C4—1222no—
399C22⋊C4—1222no—
400C42—41no—
401C42—1222no—
402C42—1222no—
403C42D4×D6111yescharacteristic
404C42D4×D6111yescharacteristic
405C42C2×C22⋊D6111yescharacteristic
406C42C23×D6111yescharacteristic
407C23—1224no—
408C23—1222no—
409C23—2442no—
410C23—2442no—
411C23#1451974511yescharacteristic
412Q8—44no—
413Q8—42no—
414Q8—42no—
415Q8—82no—
416Q8—1222no—
417Q8—1222no—
418Q8—2442no—
419D4—614no—
420D4—1224no—
421D4—1222no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—41no—
425C2×C4—44no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—1222no—
429C2×C4—1222no—
430C2×C4—1222no—
431C2×C4—4888no—
432C2×C4—2442no—
433C2×C4—2442no—
434C2×C4—2442no—
435C2×C4—2442no—
436C2×C4—2442no—
437C2×C4—2444no—
438C2×C4#1451974511yescharacteristic
439C2×C4#1451974511yescharacteristic
440C2×C4#1451974511yescharacteristic
441C2×C4#1451980012yes—
442C2×C4#1451980912yes—
443C2×C4C2×D4×D6111yescharacteristic
444C2×C4C2×D4×D6111yescharacteristic
445C2×C4C22×C22⋊D6111yescharacteristic
446C8—42no—
447C8—2442no—
448C8—2442no—
449C22—21no—
450C22—21no—
451C22—1224no—
452C22—2448no—
453C22—2442no—
454C22—2442no—
455C22#1683992312yes—
456C22C2×#1451974511yescharacteristic
457C4—21no—
458C4—21no—
459C4—21no—
460C4—44no—
461C4—81no—
462C4—81no—
463C4—2442no—
464C4—2442no—
465C4—2442no—
466C4#1683992312yes—
467C4C2×#1451980011yescharacteristic
468C4#137×D6112yes—
469C4C2×#1451980911yescharacteristic
470C2—22no—
471C2—2444no—
472C2C2×#1683992311yescharacteristic
473C2#2145505111yescharacteristic
474C2#2145517511yescharacteristic
subgroupG/H
centerC22C2×#14519745
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3209C2
socleC22×C61C2×#120
prime radicalC22×D244D4
hypercenter#3209D61
hypocenterC61#18753
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3209

Complements

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

NH ≅ G/Ntype
3. #21455620C2semidirect
4. #21445023C2semidirect
5. #21444950C2semidirect
6. #21444500C2semidirect
8. #21444215C2semidirect
9. C61×#3209C2semidirect
23. #16840054C22semidirect
25. #16840044C22semidirect
33. #16839511C22semidirect
34. #16839494C22semidirect
38. C61×#494C22semidirect
39. C61×#487C22semidirect
234. #3209D61semidirect
260. #494C2×D61semidirect
261. #487C2×D61semidirect
280. C61#18753semidirect

Maximal subgroups

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

subgroupindex
C2×#168404732
#214558232
#214556202
#214450232
#214449502
#214445002
#214442452
#214442152
C61×#32092
#1875361