Finite Group Databasebeta

← #29658527

Subgroups — #29658527

Order 15616.

986 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456760C211yesmaximal, characteristic
2#21455858C211yesmaximal, characteristic, complemented
3#21455624C211yesmaximal, characteristic, complemented
4#21445005C214yesmaximal, complemented
5#21444940C214yesmaximal, complemented
6#21444487C211yesmaximal, characteristic
7#21444248C211yesmaximal, characteristic, complemented
8#21444214C211yesmaximal, characteristic, complemented
9C2×C61×#559C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840134C2212yes—
12#16840108C2212yes—
13#16840097—21no—
14#16840069—22no—
15#16840057C2212yescomplemented
16#16840057C2212yescomplemented
17#16840047C2212yescomplemented
18#16839972C2212yescomplemented
19#16839959—22no—
20#16839918—21no—
21#16839913—22no—
22C4×#11758202—22no—
23#16839815C2211yescharacteristic
24#16839815C2211yescharacteristic
25#16839802—21no—
26#16839786C2211yescharacteristic
27#16839771—21no—
28#16839506C2214yes—
29Q8⋊#92418C2214yescomplemented
30Q8⋊#92418C2214yescomplemented
31C61×#559—22no—
32C61×#559—22no—
33C61×#559—22no—
34C61×#559C2214yes—
35C2×#127×C61C2211yescharacteristic
36C2×#124×C61—21no—
37C2×#124×C61—21no—
38C2×#123×C61—22no—
39C2×C4×Q8×C61C2211yescharacteristic
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4×D4×C61—21no—
42C2×C4×D4×C61—21no—
43C2×#11758202—22no—
44C2×#11758200—24no—
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#14519710—22no—
56#14519708—22no—
57C4⋊#92418—21no—
58C4⋊#92418—21no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519701—22no—
62#14519701—22no—
63#14519701—22no—
64#14519700—22no—
65#14519698—24no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C4×#92418—21no—
69C4×#92418C2312yes—
70C22×Q8×C61D411yescharacteristic
71C22×Q8×C61D411yescharacteristic
72C22×D4×C61—21no—
73C22×D4×C61—21no—
74C22×D4×C61D412yes—
75C23×C4×C61—22no—
76#127×C61—21no—
77#127×C61—21no—
78#127×C61D412yes—
79#127×C61C2312yes—
80#124×C61—24no—
81#124×C61—24no—
82#124×C61—24no—
83#124×C61—24no—
84#123×C61—28no—
85#123×C61—28no—
86C4×Q8×C61—21no—
87C4×Q8×C61—21no—
88C4×Q8×C61—21no—
89C4×Q8×C61—21no—
90C4×Q8×C61D412yes—
91C4×Q8×C61D412yes—
92C4×Q8×C61C2312yes—
93C4×Q8×C61C2312yes—
94C4×D4×C61—24no—
95C4×D4×C61—24no—
96C4×D4×C61—24no—
97C4×D4×C61—24no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61—22no—
105C2×C42×C61—21no—
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C24×C61—22no—
109C2×Q8×C61—21no—
110C2×Q8×C61—21no—
111C2×Q8×C61—22no—
112C2×Q8×C61—22no—
113C2×Q8×C61—42no—
114C2×Q8×C61C2×D412yes—
115C2×Q8×C61C2×D412yes—
116C2×D4×C61—22no—
117C2×D4×C61—22no—
118C2×D4×C61—22no—
119C2×D4×C61—22no—
120C2×D4×C61—22no—
121C2×D4×C61—44no—
122C2×D4×C61—42no—
123C2×D4×C61—42no—
124C2×D4×C61—82no—
125C2×D4×C61C2×D414yes—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—28no—
131C22×C4×C61—84no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C2×Dic122—21no—
136C2×Dic122—21no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61C2×D412yes—
144C4⋊C4×C61C2×D412yes—
145C22⋊C4×C61—28no—
146C22⋊C4×C61—28no—
147C22⋊C4×C61—28no—
148C42×C61—22no—
149C42×C61—22no—
150C42×C61—22no—
151C42×C61—22no—
152C42×C61C2×D411yescharacteristic
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2411yescharacteristic
156C22⋊Dic61—41no—
157#11758202—44no—
158#11758200—84no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163#11758196—41no—
164#11758196—41no—
165C4×Dic61—41no—
166#11758194—42no—
167C2×#92418—22no—
168C2×#92418—22no—
169C2×#92418—22no—
170C23×C61—24no—
171C23×C61—44no—
172C23×C61—44no—
173C23×C61—44no—
174C23×C61#12011yescharacteristic
175Q8×C61—22no—
176Q8×C61—22no—
177Q8×C61—22no—
178Q8×C61—22no—
179Q8×C61—42no—
180Q8×C61—42no—
181D4×C61—44no—
182D4×C61—44no—
183D4×C61—42no—
184D4×C61—42no—
185D4×C61—44no—
186D4×C61—44no—
187D4×C61—82no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—41no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—88no—
199C2×C4×C61#12011yescharacteristic
200C2×C4×C61#12011yescharacteristic
201C2×C4×C61#12011yescharacteristic
202C2×C4×C61#13512yes—
203C2×C4×C61#13712yes—
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×Dic61—41no—
208C2×Dic61—41no—
209Dic122—42no—
210Dic122—42no—
211#92418—42no—
212#92418—42no—
213#92418—42no—
214#18751—611nomaximal, Sylow
215C22×C61—21no—
216C22×C61—21no—
217C22×C61—44no—
218C22×C61—44no—
219C22×C61—44no—
220C22×C61—88no—
221C22×C61#45212yes—
222C22×C61C2×#12011yescharacteristic
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—42no—
227C4×C61—42no—
228C4×C61—42no—
229C4×C61—42no—
230C4×C61#45212yes—
231C4×C61C2×#13511yescharacteristic
232C4×C61C2×#13711yescharacteristic
233C4×C61#57512yes—
234Dic61—81no—
235Dic61—81no—
236C2×#559D6111yescharacteristic
237#3208—611no—
238#3104—611no—
239#3078—611no—
240#1640—614no—
241#1630—614no—
242#1545—611no—
243#1487—611no—
244#1479—611no—
245C2×C61—22no—
246C2×C61—84no—
247C2×C61C2×#45211yescharacteristic
248C2×C61#300511yescharacteristic
249C2×C61#301911yescharacteristic
250#559—22no—
251#559—22no—
252#559—22no—
253#559C2×D6114yes—
254#533—611no—
255C2×#127C2×D6111yescharacteristic
256C2×#124—21no—
257C2×#124—21no—
258C2×#123—22no—
259C2×C4×Q8C2×D6111yescharacteristic
260C2×C4×Q8C2×D6111yescharacteristic
261C2×C4×D4—21no—
262C2×C4×D4—21no—
263#492—612no—
264#486—612no—
265#484—1221no—
266#479—1222no—
267#477—612no—
268#477—612no—
269#475—612no—
270#462—612no—
271#460—1222no—
272#451—1221no—
273#450—1222no—
274C4×Q16—1222no—
275#431—611no—
276#431—611no—
277#428—1221no—
278#423—611no—
279#419—1221no—
280#331—614no—
281Q8⋊C8—614no—
282Q8⋊C8—614no—
283C61#1875111yescharacteristic, complemented, Sylow
284C22×Q8C22⋊D6111yescharacteristic
285C22×Q8C22⋊D6111yescharacteristic
286C22×D4—21no—
287C22×D4—21no—
288C22×D4C22⋊D6112yes—
289C23×C4—22no—
290C2×Q16—1222no—
291C2×SD16—1224no—
292C2×M16—1221no—
293C4⋊Q8—611no—
294C4⋊Q8—611no—
295#127—21no—
296#127—21no—
297#127C22⋊D6112yes—
298#127C22×D6112yes—
299#125—1221no—
300#124—24no—
301#124—24no—
302#124—24no—
303#124—24no—
304#123—28no—
305#123—28no—
306C22⋊Q8—1221no—
307C22⋊Q8—1221no—
308C4×Q8—21no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×Q8—1221no—
313C4×Q8—1221no—
314C4×Q8C22⋊D6112yes—
315C4×Q8C22⋊D6112yes—
316C4×Q8C22×D6112yes—
317C4×Q8C22×D6112yes—
318C4×D4—24no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—24no—
322#117—1221no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4C22⋊D6111yescharacteristic
326C2×C4⋊C4C22⋊D6111yescharacteristic
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C42—21no—
331C2×C42—21no—
332C2×C42C22×D6111yescharacteristic
333#107—1222no—
334#107—1222no—
335#106—1222no—
336C4⋊C8—612no—
337C4⋊C8—612no—
338C4⋊C8—1221no—
339C4⋊C8—1221no—
340#103—1222no—
341#103—1222no—
342#103—1222no—
343#103—1222no—
344#102—1224no—
345C22⋊C8—1221no—
346C22⋊C8—1221no—
347C4×C8—612no—
348C4×C8—1221no—
349C24—22no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—42no—
355C2×Q8—1221no—
356C2×Q8—1221no—
357C2×Q8C2×C22⋊D6112yes—
358C2×Q8C2×C22⋊D6112yes—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—44no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—82no—
368C2×D4C2×C22⋊D6114yes—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—28no—
374C22×C4—84no—
375C22×C4C2×C22⋊D6111yescharacteristic
376C22×C4C2×C22⋊D6111yescharacteristic
377C22×C4C2×C22⋊D6111yescharacteristic
378Q16—2444no—
379SD16—4884no—
380M16—2442no—
381C2×C8—1222no—
382C2×C8—1222no—
383C2×C8—1222no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—1221no—
391C4⋊C4—1221no—
392C4⋊C4—1221no—
393C4⋊C4—1221no—
394C4⋊C4—2441no—
395C4⋊C4—2441no—
396C4⋊C4C2×C22⋊D6112yes—
397C4⋊C4C2×C22⋊D6112yes—
398C22⋊C4—28no—
399C22⋊C4—28no—
400C22⋊C4—28no—
401C22⋊C4—2441no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—2441no—
407C42C2×C22⋊D6111yescharacteristic
408C42C2×C22⋊D6111yescharacteristic
409C42C2×C22⋊D6111yescharacteristic
410C42C23×D6111yescharacteristic
411C23—24no—
412C23—44no—
413C23—44no—
414C23—44no—
415C23#1451974611yescharacteristic
416Q8—22no—
417Q8—22no—
418Q8—22no—
419Q8—22no—
420Q8—42no—
421Q8—42no—
422Q8—2442no—
423Q8—2442no—
424D4—44no—
425D4—44no—
426D4—42no—
427D4—42no—
428D4—44no—
429D4—44no—
430D4—82no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—88no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4#1451974611yescharacteristic
445C2×C4#1451974611yescharacteristic
446C2×C4#1451974611yescharacteristic
447C2×C4#1451979712yes—
448C2×C4#1451980712yes—
449C2×C4C22×C22⋊D6111yescharacteristic
450C2×C4C22×C22⋊D6111yescharacteristic
451C2×C4C22×C22⋊D6111yescharacteristic
452C8—2442no—
453C8—2442no—
454C8—2442no—
455C22—21no—
456C22—21no—
457C22—44no—
458C22—44no—
459C22—44no—
460C22—88no—
461C22#1683992412yes—
462C22C2×#1451974611yescharacteristic
463C4—21no—
464C4—21no—
465C4—21no—
466C4—42no—
467C4—42no—
468C4—42no—
469C4—42no—
470C4—4881no—
471C4—4881no—
472C4#1683992412yes—
473C4C2×#1451979711yescharacteristic
474C4C2×#1451980711yescharacteristic
475C4#1684056012yes—
476C2—22no—
477C2—84no—
478C2C2×#1683992411yescharacteristic
479C2#2145506111yescharacteristic
480C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#559C2
socleC22×C61C2×#120
prime radicalC2×#127×C61C22
hypercenterC2×#559D61
hypocenterC61#18751
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. #21455858C2semidirect
3. #21455624C2semidirect
4. #21445005C2semidirect
5. #21444940C2semidirect
7. #21444248C2semidirect
8. #21444214C2semidirect
15. #16840057C22semidirect
16. #16840057C22semidirect
17. #16840047C22semidirect
18. #16839972C22semidirect
29. Q8⋊#92418C22semidirect
30. Q8⋊#92418C22semidirect
283. C61#18751semidirect

Maximal subgroups

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

subgroupindex
#214567602
#214558582
#214556242
#214450052
#214449402
#214444872
#214442482
#214442142
C2×C61×#5592
#1875161