Finite Group Databasebeta

← #29656218

Subgroups — #29656218

Order 15616.

1146 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456361C211yesmaximal, characteristic, complemented
2#21455949C211yesmaximal, characteristic, complemented
3#21455918C211yesmaximal, characteristic
4#21445082C214yesmaximal, complemented
5#21445068C214yesmaximal, complemented
6#21444536C211yesmaximal, characteristic, complemented
7#21444266C211yesmaximal, characteristic, complemented
8#21444262C211yesmaximal, characteristic
9C2×D4×Q8×C61C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840168—22no—
12#16840165C2212yescomplemented
13#16840158C2212yescomplemented
14#16840097—21no—
15#16840092—21no—
16#16840047C2212yescomplemented
17#16840042C2212yes—
18#16839972C2212yescomplemented
19#16839967C2212yes—
20#16839918—21no—
21#16839913—21no—
22#16839867—22no—
23#16839862—22no—
24#16839819C2211yescharacteristic
25#16839819C2211yescharacteristic
26#16839802—22no—
27C4×#11758194C2211yescharacteristic
28#16839508C2214yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2214yes—
31C23×Q8×C61—21no—
32D4×Q8×C61—22no—
33D4×Q8×C61—22no—
34D4×Q8×C61—22no—
35D4×Q8×C61C2214yes—
36C2×C4⋊Q8×C61—21no—
37C2×C4⋊Q8×C61C2211yescharacteristic
38C2×C22⋊Q8×C61—21no—
39C2×C22⋊Q8×C61—22no—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4×D4×C61—21no—
42C2×C4×D4×C61C2211yescharacteristic
43C2×#11758202—22no—
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#14519710—24no—
56#14519708—22no—
57#14519708—24no—
58C4⋊#92418C2312yes—
59C4⋊#92418C2312yes—
60#14519701—22no—
61#14519700—22no—
62#14519700—22no—
63#14519698—22no—
64C22⋊#92418—21no—
65C22⋊#92418—21no—
66#14519688—21no—
67#14519688—21no—
68#14519688—21no—
69C4×#92418C2312yes—
70C22×Q8×C61—22no—
71C22×Q8×C61—24no—
72C22×Q8×C61—24no—
73C22×Q8×C61—21no—
74C22×Q8×C61—48no—
75C22×Q8×C61D411yescharacteristic
76C22×D4×C61D411yescharacteristic
77C23×C4×C61—22no—
78C23×C4×C61—21no—
79C4⋊Q8×C61—24no—
80C4⋊Q8×C61—24no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61D412yes—
84C4⋊Q8×C61C2312yes—
85C22⋊Q8×C61—24no—
86C22⋊Q8×C61—28no—
87C22⋊Q8×C61—24no—
88C22⋊Q8×C61—28no—
89C4×Q8×C61—21no—
90C4×Q8×C61—21no—
91C4×Q8×C61D412yes—
92C4×Q8×C61C2312yes—
93C4×D4×C61—24no—
94C4×D4×C61—24no—
95C4×D4×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61D412yes—
98C4×D4×C61C2312yes—
99C2×C4⋊C4×C61—22no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61D412yes—
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C22⋊C4×C61—22no—
106C2×C22⋊C4×C61—21no—
107C2×C42×C61—21no—
108C2×C42×C61C2311yescharacteristic
109C24×C61—21no—
110C2×Q8×C61—24no—
111C2×Q8×C61—28no—
112C2×Q8×C61—21no—
113C2×Q8×C61—22no—
114C2×Q8×C61—22no—
115C2×Q8×C61—48no—
116C2×Q8×C61—44no—
117C2×Q8×C61—416no—
118C2×Q8×C61—48no—
119C2×Q8×C61—48no—
120C2×Q8×C61C2×D412yes—
121C2×D4×C61—24no—
122C2×D4×C61—21no—
123C2×D4×C61C2×D412yes—
124C22×C4×C61—21no—
125C22×C4×C61—28no—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61—21no—
129C22×C4×C61—22no—
130C22×C4×C61—22no—
131C22×C4×C61—48no—
132C22×C4×C61—44no—
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C2×Dic122—21no—
137C2×Dic122—21no—
138C4⋊C4×C61—28no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—21no—
142C4⋊C4×C61—21no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61—22no—
145C4⋊C4×C61—22no—
146C4⋊C4×C61C2×D412yes—
147C4⋊C4×C61C2×D412yes—
148C4⋊C4×C61C2×D414yes—
149C22⋊C4×C61—28no—
150C22⋊C4×C61—22no—
151C22⋊C4×C61—22no—
152C42×C61—22no—
153C42×C61—22no—
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C42×C61C2×D411yescharacteristic
157C42×C61C2411yescharacteristic
158C22⋊Dic61—41no—
159#11758202—44no—
160#11758200—44no—
161C4⋊Dic61—21no—
162C4⋊Dic61—21no—
163C4⋊Dic61—21no—
164C4⋊Dic61—21no—
165#11758196—41no—
166#11758196—41no—
167C4×Dic61—41no—
168#11758194—24no—
169C2×#92418—22no—
170C2×#92418—22no—
171C2×#92418—22no—
172C23×C61—24no—
173C23×C61—22no—
174C23×C61—42no—
175C23×C61—42no—
176C23×C61#12011yescharacteristic
177Q8×C61—24no—
178Q8×C61—22no—
179Q8×C61—416no—
180Q8×C61—44no—
181Q8×C61—44no—
182Q8×C61AGL(1,ℤ/8)14yes—
183D4×C61—24no—
184D4×C61—22no—
185D4×C61#13714yes—
186C2×C4×C61—24no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—24no—
193C2×C4×C61—22no—
194C2×C4×C61—416no—
195C2×C4×C61—44no—
196C2×C4×C61—44no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61—44no—
200C2×C4×C61#12011yescharacteristic
201C2×C4×C61#12011yescharacteristic
202C2×C4×C61#12011yescharacteristic
203C2×C4×C61C22×D411yescharacteristic
204C2×C4×C61C22×D411yescharacteristic
205C2×C4×C61C22×D411yescharacteristic
206C2×Dic61—41no—
207C2×Dic61—41no—
208Dic122—42no—
209Dic122—42no—
210#92418—24no—
211#92418—42no—
212#92418—42no—
213#18540—611nomaximal, Sylow
214C22×C61—21no—
215C22×C61—21no—
216C22×C61—24no—
217C22×C61—48no—
218C22×C61—42no—
219C22×C61—42no—
220C22×C61#45612yes—
221C22×C61C2×#12011yescharacteristic
222C4×C61—24no—
223C4×C61—21no—
224C4×C61—21no—
225C4×C61—21no—
226C4×C61—21no—
227C4×C61—44no—
228C4×C61—42no—
229C4×C61—42no—
230C4×C61C2×AGL(1,ℤ/8)12yes—
231C4×C61C2×#13712yes—
232Dic61—81no—
233Dic61—81no—
234C2×D4×Q8D6111yescharacteristic
235#3165—611no—
236#3113—611no—
237#3110—611no—
238#1651—614no—
239#1649—614no—
240#1556—611no—
241#1491—611no—
242#1490—611no—
243C2×C61—22no—
244C2×C61—44no—
245C2×C61#300311yescharacteristic
246C2×C61#300411yescharacteristic
247C2×C61C2×#45611yescharacteristic
248C23×Q8—21no—
249D4×Q8—22no—
250D4×Q8—22no—
251D4×Q8—22no—
252D4×Q8C2×D6114yes—
253#533—611no—
254C2×C4⋊Q8—21no—
255C2×C4⋊Q8C2×D6111yescharacteristic
256C2×C22⋊Q8—21no—
257C2×C22⋊Q8—22no—
258C2×C4×Q8C2×D6111yescharacteristic
259C2×C4×D4—21no—
260C2×C4×D4C2×D6111yescharacteristic
261#501—1222no—
262#500—612no—
263#498—612no—
264#484—1221no—
265#483—1221no—
266#475—612no—
267#474—612no—
268#462—612no—
269#461—612no—
270#451—1221no—
271#450—1221no—
272#441—1222no—
273#440—1222no—
274#432—611no—
275#432—611no—
276#428—1222no—
277C4×M16—611no—
278#332—614no—
279Q8⋊C8—614no—
280D4⋊C8—614no—
281C61#1854011yescharacteristic, complemented, Sylow
282C22×Q8—22no—
283C22×Q8—24no—
284C22×Q8—24no—
285C22×Q8—21no—
286C22×Q8—48no—
287C22×Q8C22⋊D6111yescharacteristic
288C22×D4C22⋊D6111yescharacteristic
289C23×C4—22no—
290C23×C4—21no—
291C2×Q16—1222no—
292C2×SD16—1222no—
293C2×M16—1221no—
294C4⋊Q8—24no—
295C4⋊Q8—24no—
296C4⋊Q8—21no—
297C4⋊Q8—21no—
298C4⋊Q8—611no—
299C4⋊Q8—611no—
300C4⋊Q8C22⋊D6112yes—
301C4⋊Q8C22×D6112yes—
302#125—1221no—
303C22⋊Q8—24no—
304C22⋊Q8—28no—
305C22⋊Q8—24no—
306C22⋊Q8—28no—
307C22⋊Q8—1221no—
308C22⋊Q8—1221no—
309C4×Q8—21no—
310C4×Q8—21no—
311C4×Q8—1221no—
312C4×Q8—1221no—
313C4×Q8C22⋊D6112yes—
314C4×Q8C22×D6112yes—
315C4×D4—24no—
316C4×D4—24no—
317C4×D4—21no—
318C4×D4—21no—
319C4×D4C22⋊D6112yes—
320C4×D4C22×D6112yes—
321#117—1221no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4C22⋊D6111yescharacteristic
326C2×C4⋊C4C22⋊D6111yescharacteristic
327C2×C4⋊C4C22⋊D6112yes—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C42—21no—
331C2×C42C22×D6111yescharacteristic
332#107—1224no—
333#107—1222no—
334#106—1224no—
335#106—1222no—
336C4⋊C8—612no—
337C4⋊C8—612no—
338#103—1222no—
339#103—1222no—
340#103—1222no—
341#102—1222no—
342C22⋊C8—1221no—
343C22⋊C8—1221no—
344#97—1221no—
345#97—1221no—
346#97—1221no—
347C4×C8—612no—
348C24—21no—
349C2×Q8—24no—
350C2×Q8—28no—
351C2×Q8—22no—
352C2×Q8—21no—
353C2×Q8—22no—
354C2×Q8—48no—
355C2×Q8—44no—
356C2×Q8—416no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—1221no—
360C2×Q8—1221no—
361C2×Q8C2×C22⋊D6112yes—
362C2×D4—24no—
363C2×D4—21no—
364C2×D4C2×C22⋊D6112yes—
365C22×C4—21no—
366C22×C4—28no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—48no—
373C22×C4—44no—
374C22×C4C2×C22⋊D6111yescharacteristic
375C22×C4C2×C22⋊D6111yescharacteristic
376C22×C4C2×C22⋊D6111yescharacteristic
377Q16—2444no—
378SD16—2444no—
379M16—1224no—
380C2×C8—1222no—
381C2×C8—1222no—
382C2×C8—1222no—
383C4⋊C4—28no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—1221no—
392C4⋊C4—1221no—
393C4⋊C4—1221no—
394C4⋊C4—1221no—
395C4⋊C4—2441no—
396C4⋊C4—2441no—
397C4⋊C4C2×C22⋊D6112yes—
398C4⋊C4C2×C22⋊D6112yes—
399C4⋊C4C2×C22⋊D6114yes—
400C22⋊C4—28no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—2441no—
404C42—22no—
405C42—22no—
406C42—2441no—
407C42C2×C22⋊D6111yescharacteristic
408C42C2×C22⋊D6111yescharacteristic
409C42C2×C22⋊D6111yescharacteristic
410C42C23×D6111yescharacteristic
411C23—24no—
412C23—22no—
413C23—42no—
414C23—42no—
415C23#1451974611yescharacteristic
416Q8—24no—
417Q8—22no—
418Q8—416no—
419Q8—44no—
420Q8—44no—
421Q8—2442no—
422Q8—2442no—
423Q8#1451980314yes—
424D4—24no—
425D4—22no—
426D4#1451980814yes—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—416no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4#1451974611yescharacteristic
444C2×C4#1451974611yescharacteristic
445C2×C4#1451974611yescharacteristic
446C2×C4C22×C22⋊D6111yescharacteristic
447C2×C4C22×C22⋊D6111yescharacteristic
448C2×C4C22×C22⋊D6111yescharacteristic
449C8—1224no—
450C8—2442no—
451C8—2442no—
452C22—21no—
453C22—21no—
454C22—24no—
455C22—48no—
456C22—42no—
457C22—42no—
458C22#1683994512yes—
459C22C2×#1451974611yescharacteristic
460C4—21no—
461C4—21no—
462C4—24no—
463C4—21no—
464C4—21no—
465C4—42no—
466C4—44no—
467C4—42no—
468C4—4881no—
469C4—4881no—
470C4C2×#1451980312yes—
471C4C2×#1451980812yes—
472C2—22no—
473C2—44no—
474C2#2145504611yescharacteristic
475C2#2145505311yescharacteristic
476C2C2×#1683994511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×D4×Q8×C61C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenterC2×D4×Q8D61
hypocenterC61#18540
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. #21456361C2semidirect
2. #21455949C2semidirect
4. #21445082C2semidirect
5. #21445068C2semidirect
6. #21444536C2semidirect
7. #21444266C2semidirect
12. #16840165C22semidirect
13. #16840158C22semidirect
16. #16840047C22semidirect
18. #16839972C22semidirect
28. #16839508C22semidirect
29. Q8⋊#92418C22semidirect
281. C61#18540semidirect

Maximal subgroups

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

subgroupindex
#214563612
#214559492
#214559182
#214450822
#214450682
#214445362
#214442662
#214442622
C2×D4×Q8×C612
#1854061