Finite Group Databasebeta

← #29656125

Subgroups — #29656125

Order 15616.

962 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456199C211yesmaximal, characteristic, complemented
2#21455949C211yesmaximal, characteristic, complemented
3#21455918C211yesmaximal, characteristic
4#21445033C214yesmaximal, complemented
5#21445026C214yesmaximal, complemented
6#21444545C211yesmaximal, characteristic, complemented
7#21444266C211yesmaximal, characteristic, complemented
8#21444262C211yesmaximal, characteristic
9C2×C61×#554C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840130—22no—
12#16840112C2212yescomplemented
13#16840112C2212yescomplemented
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#16839771—22no—
27C4×#11758194C2211yescharacteristic
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—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#14519708—22no—
56C4⋊#92418C2312yes—
57C4⋊#92418C2312yes—
58#14519701—24no—
59#14519701—24no—
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×C61D412yes—
71C22×Q8×C61D411yescharacteristic
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—42no—
112C2×Q8×C61—44no—
113C2×Q8×C61C2×D412yes—
114C2×Q8×C61C2×D414yes—
115C2×D4×C61—24no—
116C2×D4×C61—21no—
117C2×D4×C61C2×D412yes—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—22no—
122C22×C4×C61—28no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—44no—
126C22×C4×C61—84no—
127C22×C4×C61C2×D411yescharacteristic
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C2×Dic122—21no—
131C2×Dic122—21no—
132C4⋊C4×C61—28no—
133C4⋊C4×C61—28no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61C2×D412yes—
143C4⋊C4×C61C2×D412yes—
144C22⋊C4×C61—28no—
145C22⋊C4×C61—22no—
146C22⋊C4×C61—22no—
147C42×C61—22no—
148C42×C61—22no—
149C42×C61C2×D411yescharacteristic
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61C2411yescharacteristic
153C22⋊Dic61—41no—
154#11758202—44no—
155#11758200—44no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160#11758196—41no—
161#11758196—41no—
162C4×Dic61—41no—
163#11758194—42no—
164C2×#92418—22no—
165C2×#92418—22no—
166C2×#92418—22no—
167C23×C61—24no—
168C23×C61—22no—
169C23×C61—42no—
170C23×C61—42no—
171C23×C61#12011yescharacteristic
172Q8×C61—22no—
173Q8×C61—22no—
174Q8×C61—42no—
175Q8×C61—44no—
176Q8×C61—82no—
177D4×C61—24no—
178D4×C61—22no—
179D4×C61#13714yes—
180C2×C4×C61—22no—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—24no—
187C2×C4×C61—44no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—42no—
191C2×C4×C61—44no—
192C2×C4×C61—44no—
193C2×C4×C61—88no—
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#12011yescharacteristic
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61C22×D411yescharacteristic
198C2×C4×C61C22×D411yescharacteristic
199C2×C4×C61C22×D411yescharacteristic
200C2×Dic61—41no—
201C2×Dic61—41no—
202Dic122—42no—
203Dic122—42no—
204#92418—42no—
205#92418—42no—
206#92418—42no—
207#18530—611nomaximal, Sylow
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—24no—
211C22×C61—48no—
212C22×C61—42no—
213C22×C61—42no—
214C22×C61#45512yes—
215C22×C61C2×#12011yescharacteristic
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—42no—
221C4×C61—42no—
222C4×C61—42no—
223C4×C61—44no—
224C4×C61C2×#13712yes—
225C4×C61#57512yes—
226Dic61—81no—
227Dic61—81no—
228C2×#554D6111yescharacteristic
229#3139—611no—
230#3113—611no—
231#3110—611no—
232#1644—614no—
233#1643—614no—
234#1558—611no—
235#1491—611no—
236#1490—611no—
237C2×C61—22no—
238C2×C61—44no—
239C2×C61#300411yescharacteristic
240C2×C61#300511yescharacteristic
241C2×C61C2×#45511yescharacteristic
242#554—22no—
243#554—22no—
244#554—22no—
245#554C2×D6114yes—
246#533—611no—
247C2×C4⋊Q8C2×D6111yescharacteristic
248C2×#125—21no—
249C2×C22⋊Q8—21no—
250C2×C22⋊Q8—22no—
251C2×C4×Q8C2×D6111yescharacteristic
252C2×C4×D4—21no—
253C2×C4×D4C2×D6111yescharacteristic
254C22×C4⋊C4—21no—
255#491—1222no—
256#487—612no—
257#487—612no—
258#484—1221no—
259#483—1221no—
260#475—612no—
261#474—612no—
262#462—612no—
263#461—612no—
264#451—1221no—
265#450—1221no—
266#441—1222no—
267#440—1222no—
268#432—611no—
269#432—611no—
270#419—1222no—
271C4×M16—611no—
272#333—614no—
273Q8⋊C8—614no—
274D4⋊C8—614no—
275C61#1853011yescharacteristic, complemented, Sylow
276C22×Q8C22⋊D6111yescharacteristic
277C22×Q8C22⋊D6112yes—
278C22×D4C22⋊D6111yescharacteristic
279C23×C4—22no—
280C23×C4—21no—
281C2×Q16—1222no—
282C2×SD16—1222no—
283C2×M16—1221no—
284C4⋊Q8—21no—
285C4⋊Q8—21no—
286C4⋊Q8—611no—
287C4⋊Q8—611no—
288C4⋊Q8C22⋊D6112yes—
289C4⋊Q8C22×D6112yes—
290#125—24no—
291#125—24no—
292#125—1221no—
293C22⋊Q8—24no—
294C22⋊Q8—28no—
295C22⋊Q8—24no—
296C22⋊Q8—28no—
297C22⋊Q8—1221no—
298C22⋊Q8—1221no—
299C4×Q8—21no—
300C4×Q8—21no—
301C4×Q8—1221no—
302C4×Q8—1221no—
303C4×Q8C22⋊D6112yes—
304C4×Q8C22×D6112yes—
305C4×D4—24no—
306C4×D4—24no—
307C4×D4—21no—
308C4×D4—21no—
309C4×D4C22⋊D6112yes—
310C4×D4C22×D6112yes—
311#117—1221no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4C22⋊D6111yescharacteristic
320C2×C4⋊C4C22⋊D6111yescharacteristic
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C42—21no—
324C2×C42C22×D6111yescharacteristic
325#107—1222no—
326#106—1222no—
327C4⋊C8—612no—
328C4⋊C8—612no—
329#103—1224no—
330#103—1224no—
331#103—1222no—
332#103—1222no—
333#103—1222no—
334#102—1222no—
335C22⋊C8—1221no—
336C22⋊C8—1221no—
337#97—1221no—
338#97—1221no—
339#97—1221no—
340C4×C8—612no—
341C24—21no—
342C2×Q8—21no—
343C2×Q8—22no—
344C2×Q8—42no—
345C2×Q8—44no—
346C2×Q8—1221no—
347C2×Q8—1221no—
348C2×Q8C2×C22⋊D6112yes—
349C2×Q8C2×C22⋊D6114yes—
350C2×D4—24no—
351C2×D4—21no—
352C2×D4C2×C22⋊D6112yes—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—28no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—44no—
361C22×C4—84no—
362C22×C4C2×C22⋊D6111yescharacteristic
363C22×C4C2×C22⋊D6111yescharacteristic
364C22×C4C2×C22⋊D6111yescharacteristic
365Q16—2444no—
366SD16—2444no—
367M16—2442no—
368C2×C8—1222no—
369C2×C8—1222no—
370C2×C8—1222no—
371C4⋊C4—28no—
372C4⋊C4—28no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—1221no—
382C4⋊C4—1221no—
383C4⋊C4—1221no—
384C4⋊C4—1221no—
385C4⋊C4—2441no—
386C4⋊C4—2441no—
387C4⋊C4C2×C22⋊D6112yes—
388C4⋊C4C2×C22⋊D6112yes—
389C22⋊C4—28no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—2441no—
393C42—22no—
394C42—22no—
395C42—2441no—
396C42C2×C22⋊D6111yescharacteristic
397C42C2×C22⋊D6111yescharacteristic
398C42C2×C22⋊D6111yescharacteristic
399C42C23×D6111yescharacteristic
400C23—24no—
401C23—22no—
402C23—42no—
403C23—42no—
404C23#1451974611yescharacteristic
405Q8—22no—
406Q8—22no—
407Q8—44no—
408Q8—42no—
409Q8—82no—
410Q8—2442no—
411Q8—2442no—
412D4—24no—
413D4—22no—
414D4#1451980814yes—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—88no—
429C2×C4—2441no—
430C2×C4—2441no—
431C2×C4#1451974611yescharacteristic
432C2×C4#1451974611yescharacteristic
433C2×C4#1451974611yescharacteristic
434C2×C4C22×C22⋊D6111yescharacteristic
435C2×C4C22×C22⋊D6111yescharacteristic
436C2×C4C22×C22⋊D6111yescharacteristic
437C8—2442no—
438C8—2442no—
439C8—2442no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—48no—
444C22—42no—
445C22—42no—
446C22#1683994012yes—
447C22C2×#1451974611yescharacteristic
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—42no—
453C4—44no—
454C4—42no—
455C4—42no—
456C4—4881no—
457C4—4881no—
458C4C2×#1451980812yes—
459C4#1684056012yes—
460C2—22no—
461C2—44no—
462C2#2145505311yescharacteristic
463C2#2145506111yescharacteristic
464C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#554C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenterC2×#554D61
hypocenterC61#18530
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. #21456199C2semidirect
2. #21455949C2semidirect
4. #21445033C2semidirect
5. #21445026C2semidirect
6. #21444545C2semidirect
7. #21444266C2semidirect
12. #16840112C22semidirect
13. #16840112C22semidirect
16. #16840047C22semidirect
18. #16839972C22semidirect
28. #16839511C22semidirect
29. Q8⋊#92418C22semidirect
275. C61#18530semidirect

Maximal subgroups

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

subgroupindex
#214561992
#214559492
#214559182
#214450332
#214450262
#214445452
#214442662
#214442622
C2×C61×#5542
#1853061