Finite Group Databasebeta

← #29656236

Subgroups — #29656236

Order 15616.

918 conjugacy classes of subgroups.

The subgroup lattice has 462 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#21445089C214yesmaximal, complemented
5#21445075C214yesmaximal, complemented
6#21444536C211yesmaximal, characteristic, complemented
7#21444266C211yesmaximal, characteristic, complemented
8#21444262C211yesmaximal, characteristic
9C2×C61×#556C211yesmaximal, 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—
31C61×#556—22no—
32C61×#556—22no—
33C61×#556—22no—
34C61×#556C2214yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×#126×C61—22no—
37C2×#124×C61—21no—
38C2×C22⋊Q8×C61—21no—
39C2×C4×Q8×C61—21no—
40C2×C4×Q8×C61C2211yescharacteristic
41C2×C4×D4×C61C2211yescharacteristic
42C2×#117×C61—21no—
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—21no—
71C22×Q8×C61D411yescharacteristic
72C22×D4×C61D411yescharacteristic
73C23×C4×C61—21no—
74C4⋊Q8×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61D412yes—
77C4⋊Q8×C61C2312yes—
78#126×C61—28no—
79#126×C61—28no—
80#124×C61—24no—
81#124×C61—24no—
82C22⋊Q8×C61—24no—
83C22⋊Q8×C61—24no—
84C4×Q8×C61—24no—
85C4×Q8×C61—24no—
86C4×Q8×C61—21no—
87C4×Q8×C61—21no—
88C4×Q8×C61D412yes—
89C4×Q8×C61C2312yes—
90C4×D4×C61—21no—
91C4×D4×C61—21no—
92C4×D4×C61D412yes—
93C4×D4×C61C2312yes—
94#117×C61—24no—
95#117×C61—24no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—21no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61D412yes—
100C2×C4⋊C4×C61D411yescharacteristic
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61—22no—
103C2×C22⋊C4×C61—22no—
104C2×C22⋊C4×C61—21no—
105C2×C42×C61—22no—
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—42no—
113C2×Q8×C61—44no—
114C2×Q8×C61C2×D412yes—
115C2×D4×C61—21no—
116C2×D4×C61—42no—
117C2×D4×C61C2×D412yes—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61—22no—
123C22×C4×C61—22no—
124C22×C4×C61—44no—
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D411yescharacteristic
128C2×Dic122—21no—
129C2×Dic122—21no—
130C4⋊C4×C61—28no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61C2×D412yes—
139C4⋊C4×C61C2×D412yes—
140C4⋊C4×C61C2×D414yes—
141C22⋊C4×C61—28no—
142C22⋊C4×C61—28no—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—22no—
145C42×C61—28no—
146C42×C61—22no—
147C42×C61—22no—
148C42×C61C2×D411yescharacteristic
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61C2411yescharacteristic
152C22⋊Dic61—41no—
153#11758202—44no—
154#11758200—44no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159#11758196—41no—
160#11758196—41no—
161C4×Dic61—41no—
162#11758194—24no—
163C2×#92418—22no—
164C2×#92418—22no—
165C2×#92418—22no—
166C23×C61—22no—
167C23×C61—42no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61#12011yescharacteristic
171Q8×C61—22no—
172Q8×C61—22no—
173Q8×C61—42no—
174Q8×C61—44no—
175Q8×C61—44no—
176D4×C61—22no—
177D4×C61—22no—
178D4×C61—42no—
179C2×C4×C61—22no—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—24no—
186C2×C4×C61—44no—
187C2×C4×C61—42no—
188C2×C4×C61—42no—
189C2×C4×C61—44no—
190C2×C4×C61—42no—
191C2×C4×C61—44no—
192C2×C4×C61#12011yescharacteristic
193C2×C4×C61#12011yescharacteristic
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×Dic61—41no—
199C2×Dic61—41no—
200Dic122—42no—
201Dic122—42no—
202#92418—24no—
203#92418—42no—
204#92418—42no—
205#18542—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—42no—
209C22×C61—42no—
210C22×C61—42no—
211C22×C61—84no—
212C22×C61#45612yes—
213C22×C61C2×#12011yescharacteristic
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—42no—
219C4×C61—42no—
220C4×C61—44no—
221C4×C61—42no—
222C4×C61#57512yes—
223C4×C61#57512yes—
224Dic61—81no—
225Dic61—81no—
226C2×#556D6111yescharacteristic
227#3165—611no—
228#3113—611no—
229#3110—611no—
230#1652—614no—
231#1650—614no—
232#1556—611no—
233#1491—611no—
234#1490—611no—
235C2×C61—22no—
236C2×C61—82no—
237C2×C61#300511yescharacteristic
238C2×C61#300511yescharacteristic
239C2×C61C2×#45611yescharacteristic
240#556—22no—
241#556—22no—
242#556—22no—
243#556C2×D6114yes—
244#533—611no—
245C2×C4⋊Q8C2×D6111yescharacteristic
246C2×#126—22no—
247C2×#124—21no—
248C2×C22⋊Q8—21no—
249C2×C4×Q8—21no—
250C2×C4×Q8C2×D6111yescharacteristic
251C2×C4×D4C2×D6111yescharacteristic
252C2×#117—21no—
253#501—1222no—
254#500—612no—
255#498—612no—
256#484—1221no—
257#483—1221no—
258#475—612no—
259#474—612no—
260#462—612no—
261#461—612no—
262#451—1221no—
263#450—1221no—
264#441—1222no—
265#440—1222no—
266#432—611no—
267#432—611no—
268#428—1222no—
269C4×M16—611no—
270#332—614no—
271Q8⋊C8—614no—
272D4⋊C8—614no—
273C61#1854211yescharacteristic, complemented, Sylow
274C22×Q8—21no—
275C22×Q8C22⋊D6111yescharacteristic
276C22×D4C22⋊D6111yescharacteristic
277C23×C4—21no—
278C2×Q16—1222no—
279C2×SD16—1222no—
280C2×M16—1221no—
281C4⋊Q8—21no—
282C4⋊Q8—21no—
283C4⋊Q8—611no—
284C4⋊Q8—611no—
285C4⋊Q8C22⋊D6112yes—
286C4⋊Q8C22×D6112yes—
287#126—28no—
288#126—28no—
289#125—1221no—
290#124—24no—
291#124—24no—
292C22⋊Q8—24no—
293C22⋊Q8—24no—
294C22⋊Q8—1221no—
295C22⋊Q8—1221no—
296C4×Q8—24no—
297C4×Q8—24no—
298C4×Q8—21no—
299C4×Q8—21no—
300C4×Q8—1221no—
301C4×Q8—1221no—
302C4×Q8C22⋊D6112yes—
303C4×Q8C22×D6112yes—
304C4×D4—21no—
305C4×D4—21no—
306C4×D4C22⋊D6112yes—
307C4×D4C22×D6112yes—
308#117—24no—
309#117—24no—
310#117—1221no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4C22⋊D6111yescharacteristic
315C2×C4⋊C4C22⋊D6111yescharacteristic
316C2×C4⋊C4C22⋊D6112yes—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—21no—
320C2×C42—22no—
321C2×C42—21no—
322C2×C42C22×D6111yescharacteristic
323#107—1224no—
324#107—1222no—
325#106—1224no—
326#106—1222no—
327C4⋊C8—612no—
328C4⋊C8—612no—
329#103—1222no—
330#103—1222no—
331#103—1222no—
332#102—1222no—
333C22⋊C8—1221no—
334C22⋊C8—1221no—
335#97—1221no—
336#97—1221no—
337#97—1221no—
338C4×C8—612no—
339C24—21no—
340C2×Q8—22no—
341C2×Q8—21no—
342C2×Q8—22no—
343C2×Q8—42no—
344C2×Q8—44no—
345C2×Q8—1221no—
346C2×Q8—1221no—
347C2×Q8C2×C22⋊D6112yes—
348C2×D4—21no—
349C2×D4—42no—
350C2×D4C2×C22⋊D6112yes—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—44no—
358C22×C4C2×C22⋊D6111yescharacteristic
359C22×C4C2×C22⋊D6111yescharacteristic
360C22×C4C2×C22⋊D6111yescharacteristic
361Q16—2444no—
362SD16—2444no—
363M16—1224no—
364C2×C8—1222no—
365C2×C8—1222no—
366C2×C8—1222no—
367C4⋊C4—28no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4—1221no—
376C4⋊C4—1221no—
377C4⋊C4—1221no—
378C4⋊C4—1221no—
379C4⋊C4—2441no—
380C4⋊C4—2441no—
381C4⋊C4C2×C22⋊D6112yes—
382C4⋊C4C2×C22⋊D6112yes—
383C4⋊C4C2×C22⋊D6114yes—
384C22⋊C4—28no—
385C22⋊C4—28no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—2441no—
389C42—28no—
390C42—22no—
391C42—22no—
392C42—2441no—
393C42C2×C22⋊D6111yescharacteristic
394C42C2×C22⋊D6111yescharacteristic
395C42C2×C22⋊D6111yescharacteristic
396C42C23×D6111yescharacteristic
397C23—22no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23#1451974611yescharacteristic
402Q8—22no—
403Q8—22no—
404Q8—42no—
405Q8—44no—
406Q8—44no—
407Q8—2442no—
408Q8—2442no—
409D4—22no—
410D4—22no—
411D4—42no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—2441no—
426C2×C4—2441no—
427C2×C4#1451974611yescharacteristic
428C2×C4#1451974611yescharacteristic
429C2×C4#1451974611yescharacteristic
430C2×C4C22×C22⋊D6111yescharacteristic
431C2×C4C22×C22⋊D6111yescharacteristic
432C2×C4C22×C22⋊D6111yescharacteristic
433C8—1224no—
434C8—2442no—
435C8—2442no—
436C22—21no—
437C22—21no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—84no—
442C22#1683994512yes—
443C22C2×#1451974611yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—44no—
452C4—4881no—
453C4—4881no—
454C4#1684056012yes—
455C4#1684056012yes—
456C2—22no—
457C2—82no—
458C2#2145506111yescharacteristic
459C2#2145506111yescharacteristic
460C2C2×#1683994511yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#556C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenterC2×#556D61
hypocenterC61#18542
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#556

Complements

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

NH ≅ G/Ntype
1. #21456361C2semidirect
2. #21455949C2semidirect
4. #21445089C2semidirect
5. #21445075C2semidirect
6. #21444536C2semidirect
7. #21444266C2semidirect
12. #16840165C22semidirect
13. #16840158C22semidirect
16. #16840047C22semidirect
18. #16839972C22semidirect
28. #16839508C22semidirect
29. Q8⋊#92418C22semidirect
273. C61#18542semidirect

Maximal subgroups

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

subgroupindex
#214563612
#214559492
#214559182
#214450892
#214450752
#214445362
#214442662
#214442622
C2×C61×#5562
#1854261