Finite Group Databasebeta

← #29658545

Subgroups — #29658545

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
1#21456767C211yesmaximal, characteristic, complemented
2#21455827C211yesmaximal, characteristic
3#21455624C211yesmaximal, characteristic, complemented
4#21445026C214yesmaximal, complemented
5#21444953C214yesmaximal, complemented
6#21444500C211yesmaximal, characteristic, complemented
7#21444244C211yesmaximal, characteristic
8#21444214C211yesmaximal, characteristic, complemented
9C2×C61×#554C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840141C2212yescomplemented
12#16840112C2212yescomplemented
13#16840092—21no—
14#16840069—22no—
15#16840057C2212yescomplemented
16#16840052C2212yes—
17#16840047C2212yescomplemented
18#16839969—22no—
19#16839967C2212yes—
20#16839918—22no—
21#16839913—21no—
22C4×#11758200—22no—
23#16839815C2211yescharacteristic
24#16839815C2211yescharacteristic
25#16839802—21no—
26#16839786C2211yescharacteristic
27#16839771—21no—
28#16839511C2214yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2214yes—
31C61×#554—22no—
32C61×#554—22no—
33C61×#554—22no—
34C61×#554C2214yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×#125×C61—21no—
37C2×C22⋊Q8×C61—21no—
38C2×C22⋊Q8×C61—22no—
39C2×C4×Q8×C61C2211yescharacteristic
40C2×C4×D4×C61—21no—
41C2×C4×D4×C61C2211yescharacteristic
42C22×C4⋊C4×C61—21no—
43C2×#11758202—24no—
44C2×#11758200—22no—
45C2×#11758194—21no—
46#14519777C2311yescharacteristic
47#14519777C2311yescharacteristic
48#14519769—21no—
49#14519756—21no—
50#14519756—21no—
51C4×Dic122—21no—
52C4×Dic122—21no—
53#14519736—21no—
54#14519710—22no—
55#14519708—22no—
56#14519708—22no—
57C4⋊#92418—21no—
58C4⋊#92418—21no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519701—24no—
62#14519701—22no—
63#14519701—22no—
64#14519700—22no—
65#14519698—22no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C4×#92418—21no—
69C4×#92418C2312yes—
70C22×Q8×C61D411yescharacteristic
71C22×Q8×C61D412yes—
72C22×D4×C61D411yescharacteristic
73C23×C4×C61—22no—
74C23×C4×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61—21no—
77C4⋊Q8×C61D412yes—
78C4⋊Q8×C61C2312yes—
79#125×C61—24no—
80#125×C61—24no—
81C22⋊Q8×C61—24no—
82C22⋊Q8×C61—28no—
83C22⋊Q8×C61—24no—
84C22⋊Q8×C61—28no—
85C4×Q8×C61—21no—
86C4×Q8×C61—21no—
87C4×Q8×C61D412yes—
88C4×Q8×C61C2312yes—
89C4×D4×C61—24no—
90C4×D4×C61—24no—
91C4×D4×C61—21no—
92C4×D4×C61—21no—
93C4×D4×C61D412yes—
94C4×D4×C61C2312yes—
95C2×C4⋊C4×C61—24no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—24no—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C22⋊C4×C61—22no—
105C2×C22⋊C4×C61—21no—
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—82no—
114C2×Q8×C61C2×D412yes—
115C2×Q8×C61C2×D414yes—
116C2×D4×C61—24no—
117C2×D4×C61—21no—
118C2×D4×C61C2×D412yes—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—28no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—42no—
127C22×C4×C61—42no—
128C22×C4×C61—84no—
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C22×C4×C61C2×D411yescharacteristic
132C2×Dic122—21no—
133C2×Dic122—21no—
134C4⋊C4×C61—28no—
135C4⋊C4×C61—28no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61C2×D412yes—
145C4⋊C4×C61C2×D412yes—
146C22⋊C4×C61—28no—
147C22⋊C4×C61—22no—
148C22⋊C4×C61—22no—
149C42×C61—22no—
150C42×C61—22no—
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2411yescharacteristic
155C22⋊Dic61—41no—
156#11758202—84no—
157#11758200—44no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162#11758196—41no—
163#11758196—41no—
164C4×Dic61—41no—
165#11758194—42no—
166C2×#92418—22no—
167C2×#92418—22no—
168C2×#92418—22no—
169C23×C61—24no—
170C23×C61—22no—
171C23×C61—42no—
172C23×C61—42no—
173C23×C61#12011yescharacteristic
174Q8×C61—22no—
175Q8×C61—22no—
176Q8×C61—44no—
177Q8×C61—42no—
178Q8×C61—82no—
179D4×C61—24no—
180D4×C61—22no—
181D4×C61#13714yes—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—24no—
187C2×C4×C61—41no—
188C2×C4×C61—42no—
189C2×C4×C61—44no—
190C2×C4×C61—42no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—44no—
195C2×C4×C61—88no—
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61#12011yescharacteristic
198C2×C4×C61#12011yescharacteristic
199C2×C4×C61#13512yes—
200C2×C4×C61#13712yes—
201C2×C4×C61C22×D411yescharacteristic
202C2×C4×C61C22×D411yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×Dic61—41no—
205C2×Dic61—41no—
206Dic122—42no—
207Dic122—42no—
208#92418—42no—
209#92418—42no—
210#92418—42no—
211#18753—611nomaximal, Sylow
212C22×C61—21no—
213C22×C61—21no—
214C22×C61—24no—
215C22×C61—48no—
216C22×C61—42no—
217C22×C61—42no—
218C22×C61#45212yes—
219C22×C61C2×#12011yescharacteristic
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—21no—
223C4×C61—42no—
224C4×C61—42no—
225C4×C61—42no—
226C4×C61—44no—
227C4×C61#45212yes—
228C4×C61C2×#13511yescharacteristic
229C4×C61C2×#13711yescharacteristic
230C4×C61C2×#13712yes—
231Dic61—81no—
232Dic61—81no—
233C2×#554D6111yescharacteristic
234#3209—611no—
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×C61—44no—
244C2×C61C2×#45211yescharacteristic
245C2×C61#300411yescharacteristic
246C2×C61#301911yescharacteristic
247#554—22no—
248#554—22no—
249#554—22no—
250#554C2×D6114yes—
251#533—611no—
252C2×C4⋊Q8C2×D6111yescharacteristic
253C2×#125—21no—
254C2×C22⋊Q8—21no—
255C2×C22⋊Q8—22no—
256C2×C4×Q8C2×D6111yescharacteristic
257C2×C4×D4—21no—
258C2×C4×D4C2×D6111yescharacteristic
259C22×C4⋊C4—21no—
260#494—612no—
261#487—612no—
262#483—1221no—
263#479—1222no—
264#477—612no—
265#476—612no—
266#475—612no—
267#462—1222no—
268#461—612no—
269#451—1222no—
270#450—1221no—
271C4×SD16—1222no—
272#431—611no—
273#431—611no—
274#428—1221no—
275#423—611no—
276#419—1221no—
277#333—614no—
278Q8⋊C8—614no—
279D4⋊C8—614no—
280C61#1875311yescharacteristic, complemented, Sylow
281C22×Q8C22⋊D6111yescharacteristic
282C22×Q8C22⋊D6112yes—
283C22×D4C22⋊D6111yescharacteristic
284C23×C4—22no—
285C23×C4—21no—
286C2×Q16—1224no—
287C2×SD16—1222no—
288C2×M16—1221no—
289C4⋊Q8—21no—
290C4⋊Q8—21no—
291C4⋊Q8—611no—
292C4⋊Q8—611no—
293C4⋊Q8C22⋊D6112yes—
294C4⋊Q8C22×D6112yes—
295#125—24no—
296#125—24no—
297#125—1221no—
298C22⋊Q8—24no—
299C22⋊Q8—28no—
300C22⋊Q8—24no—
301C22⋊Q8—28no—
302C22⋊Q8—1221no—
303C22⋊Q8—1221no—
304C4×Q8—21no—
305C4×Q8—21no—
306C4×Q8—1221no—
307C4×Q8—1221no—
308C4×Q8C22⋊D6112yes—
309C4×Q8C22×D6112yes—
310C4×D4—24no—
311C4×D4—24no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4C22⋊D6112yes—
315C4×D4C22×D6112yes—
316#117—1221no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4C22⋊D6111yescharacteristic
325C2×C4⋊C4C22⋊D6111yescharacteristic
326C2×C22⋊C4—22no—
327C2×C22⋊C4—21no—
328C2×C42—21no—
329C2×C42C22×D6111yescharacteristic
330#107—1222no—
331#106—1222no—
332#106—1222no—
333C4⋊C8—612no—
334C4⋊C8—612no—
335C4⋊C8—1221no—
336C4⋊C8—1221no—
337#103—1222no—
338#103—1222no—
339#103—1224no—
340#103—1222no—
341#102—1222no—
342C22⋊C8—1221no—
343C22⋊C8—1221no—
344C4×C8—612no—
345C4×C8—1221no—
346C24—21no—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—22no—
351C2×Q8—82no—
352C2×Q8—1221no—
353C2×Q8—1221no—
354C2×Q8C2×C22⋊D6112yes—
355C2×Q8C2×C22⋊D6114yes—
356C2×D4—24no—
357C2×D4—21no—
358C2×D4C2×C22⋊D6112yes—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—28no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—84no—
369C22×C4C2×C22⋊D6111yescharacteristic
370C22×C4C2×C22⋊D6111yescharacteristic
371C22×C4C2×C22⋊D6111yescharacteristic
372Q16—4884no—
373SD16—2444no—
374M16—2442no—
375C2×C8—1222no—
376C2×C8—1222no—
377C2×C8—1222no—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—1221no—
389C4⋊C4—1221no—
390C4⋊C4—1221no—
391C4⋊C4—1221no—
392C4⋊C4—2441no—
393C4⋊C4—2441no—
394C4⋊C4C2×C22⋊D6112yes—
395C4⋊C4C2×C22⋊D6112yes—
396C22⋊C4—28no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—2441no—
400C42—22no—
401C42—22no—
402C42—2441no—
403C42C2×C22⋊D6111yescharacteristic
404C42C2×C22⋊D6111yescharacteristic
405C42C2×C22⋊D6111yescharacteristic
406C42C23×D6111yescharacteristic
407C23—24no—
408C23—22no—
409C23—42no—
410C23—42no—
411C23#1451974611yescharacteristic
412Q8—22no—
413Q8—22no—
414Q8—44no—
415Q8—42no—
416Q8—82no—
417Q8—2442no—
418Q8—2442no—
419D4—24no—
420D4—22no—
421D4#1451980814yes—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—24no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—88no—
436C2×C4—2441no—
437C2×C4—2441no—
438C2×C4#1451974611yescharacteristic
439C2×C4#1451974611yescharacteristic
440C2×C4#1451974611yescharacteristic
441C2×C4#1451979712yes—
442C2×C4#1451980712yes—
443C2×C4C22×C22⋊D6111yescharacteristic
444C2×C4C22×C22⋊D6111yescharacteristic
445C2×C4C22×C22⋊D6111yescharacteristic
446C8—2442no—
447C8—2442no—
448C8—2442no—
449C22—21no—
450C22—21no—
451C22—24no—
452C22—48no—
453C22—42no—
454C22—42no—
455C22#1683992412yes—
456C22C2×#1451974611yescharacteristic
457C4—21no—
458C4—21no—
459C4—21no—
460C4—42no—
461C4—44no—
462C4—42no—
463C4—42no—
464C4—4881no—
465C4—4881no—
466C4#1683992412yes—
467C4C2×#1451979711yescharacteristic
468C4C2×#1451980711yescharacteristic
469C4C2×#1451980812yes—
470C2—22no—
471C2—44no—
472C2C2×#1683992411yescharacteristic
473C2#2145505311yescharacteristic
474C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#554C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenterC2×#554D61
hypocenterC61#18753
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. #21456767C2semidirect
3. #21455624C2semidirect
4. #21445026C2semidirect
5. #21444953C2semidirect
6. #21444500C2semidirect
8. #21444214C2semidirect
11. #16840141C22semidirect
12. #16840112C22semidirect
15. #16840057C22semidirect
17. #16840047C22semidirect
28. #16839511C22semidirect
29. Q8⋊#92418C22semidirect
280. C61#18753semidirect

Maximal subgroups

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

subgroupindex
#214567672
#214558272
#214556242
#214450262
#214449532
#214445002
#214442442
#214442142
C2×C61×#5542
#1875361