Finite Group Databasebeta

← #29658246

Subgroups — #29658246

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456960C211yesmaximal, characteristic, complemented
2#21456764C211yesmaximal, characteristic, complemented
3#21456212C211yesmaximal, characteristic
4#21445040C214yesmaximal, complemented
5#21445005C214yesmaximal, complemented
6#21444481C211yesmaximal, characteristic
7#21444240C211yesmaximal, characteristic, complemented
8#21444240C211yesmaximal, characteristic, complemented
9C61×#3550C211yesmaximal, characteristic
10C2×#14519777C2211yescharacteristic
11#16840158—21no—
12#16840141—21no—
13#16840121—22no—
14#16840114—21no—
15#16840112—21no—
16#16840112C2212yes—
17#16840108C2212yes—
18#16840090—22no—
19#16840047C2214yes—
20#16839972C2214yes—
21#16839917—22no—
22#16839815C2211yescharacteristic
23#16839815C2211yescharacteristic
24#16839763—22no—
25C4×#11758194C2211yescharacteristic
26#16839511C2212yes—
27#16839506C2212yes—
28Q8⋊#92418C2214yes—
29Q8⋊#92418C2214yes—
30C61×#563—22no—
31C61×#559C2212yes—
32Q82×C61C2212yes—
33C61×#556—21no—
34C61×#556—21no—
35D4×Q8×C61—21no—
36D4×Q8×C61—21no—
37C61×#542—22no—
38C61×#542—22no—
39C61×#536—22no—
40C61×#533C2211yescharacteristic
41C61×#533C2211yescharacteristic
42C61×#532C2211yescharacteristic
43C22×Dic122—21no—
44C2×#11758202—24no—
45C2×#11758194—21no—
46#14519777—21no—
47#14519777—21no—
48#14519777—21no—
49#14519777C2312yes—
50C2×C4⋊Dic61—21no—
51C2×C4⋊Dic61—21no—
52#14519708—24no—
53C4⋊#92418C2312yes—
54C4⋊#92418C2312yes—
55#14519701—24no—
56#14519701—24no—
57#14519700—24no—
58#14519698—24no—
59C22⋊#92418—21no—
60C22⋊#92418—21no—
61#14519688—21no—
62C4×#92418—21no—
63C4×#92418—21no—
64C4×#92418C2312yes—
65C2×#41×C61D412yes—
66C22×Q8×C61—22no—
67C4⋊Q8×C61—24no—
68C4⋊Q8×C61—22no—
69C4⋊Q8×C61—22no—
70C4⋊Q8×C61D411yescharacteristic
71C4⋊Q8×C61D411yescharacteristic
72C4⋊Q8×C61D411yescharacteristic
73C4⋊Q8×C61D411yescharacteristic
74C4⋊Q8×C61C2311yescharacteristic
75#127×C61C2311yescharacteristic
76#126×C61—24no—
77#126×C61—24no—
78#125×C61—21no—
79#125×C61—21no—
80#124×C61—24no—
81#124×C61—22no—
82#124×C61—22no—
83#124×C61D412yes—
84#123×C61—24no—
85#123×C61—24no—
86C22⋊Q8×C61—22no—
87C22⋊Q8×C61—22no—
88C22⋊Q8×C61—24no—
89C22⋊Q8×C61—24no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—21no—
92C22⋊Q8×C61D412yes—
93C22⋊Q8×C61D412yes—
94#121×C61—21no—
95#121×C61—21no—
96C4×Q8×C61—22no—
97C4×Q8×C61—22no—
98C4×Q8×C61—21no—
99C4×Q8×C61—21no—
100C4×Q8×C61C2312yes—
101C4×Q8×C61C2312yes—
102C4×D4×C61—22no—
103C4×D4×C61—22no—
104C4×D4×C61—21no—
105C4×D4×C61—21no—
106#117×C61—22no—
107#117×C61—21no—
108#117×C61—21no—
109C2×C42×C61C2311yescharacteristic
110#41×C61—44no—
111C2×Q8×C61—24no—
112C2×Q8×C61—21no—
113C2×Q8×C61—21no—
114C2×Q8×C61—84no—
115C2×Q8×C61C2×D412yes—
116C2×Q8×C61C2×D412yes—
117C2×Q8×C61C2×D412yes—
118C2×D4×C61—22no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61C2×D412yes—
122C22×C4×C61—24no—
123C22×C4×C61—22no—
124C22×C4×C61C2×D411yescharacteristic
125C22×C4×C61C2×D411yescharacteristic
126C22×C4×C61C2×D411yescharacteristic
127C22×Dic61—41no—
128C2×Dic122—22no—
129C2×Dic122—41no—
130C2×Dic122—82no—
131C4⋊C4×C61—24no—
132C4⋊C4×C61—24no—
133C4⋊C4×C61—21no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—21no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
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×C61C2×D412yes—
146C4⋊C4×C61C2×D412yes—
147C22⋊C4×C61—24no—
148C22⋊C4×C61—24no—
149C22⋊C4×C61—24no—
150C22⋊C4×C61—22no—
151C22⋊C4×C61—22no—
152C22⋊C4×C61—22no—
153C22⋊C4×C61—22no—
154C42×C61—24no—
155C42×C61—22no—
156C42×C61—22no—
157C42×C61C2×D411yescharacteristic
158C42×C61C2×D411yescharacteristic
159C42×C61C2×D411yescharacteristic
160C42×C61C2411yescharacteristic
161#11758202—84no—
162C4⋊Dic61—22no—
163C4⋊Dic61—22no—
164C4⋊Dic61—41no—
165C4⋊Dic61—41no—
166#11758194—42no—
167C2×#92418—22no—
168C2×#92418—22no—
169C2×#92418—22no—
170C23×C61—22no—
171C23×C61#12011yescharacteristic
172Q8×C61—24no—
173Q8×C61—24no—
174Q8×C61—42no—
175Q8×C61—42no—
176Q8×C61—42no—
177Q8×C61—84no—
178D4×C61—42no—
179D4×C61—42no—
180D4×C61—42no—
181D4×C61—82no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—22no—
190C2×C4×C61—44no—
191C2×C4×C61—84no—
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—42no—
199C2×Dic61—82no—
200Dic122—44no—
201Dic122—82no—
202#92418—42no—
203#92418—42no—
204#92418—42no—
205#18725—611nomaximal, Sylow
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—42no—
210C22×C61—42no—
211C22×C61—42no—
212C22×C61C2×#12011yescharacteristic
213C4×C61—21no—
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—42no—
217C4×C61—42no—
218C4×C61—42no—
219C4×C61—42no—
220C4×C61—42no—
221C4×C61#45512yes—
222C4×C61#57712yes—
223C4×C61#57812yes—
224Dic61—82no—
225#3550D6111yescharacteristic
226#3229—611no—
227#3209—611no—
228#3141—611no—
229#1645—614no—
230#1640—614no—
231#1543—611no—
232#1485—611no—
233#1485—611no—
234C2×C61—41no—
235C2×C61—82no—
236C2×C61#301011yescharacteristic
237C2×C61#301111yescharacteristic
238C2×C61C2×#45511yescharacteristic
239#563—22no—
240#559C2×D6112yes—
241Q82C2×D6112yes—
242#556—21no—
243#556—21no—
244D4×Q8—21no—
245D4×Q8—21no—
246#542—22no—
247#542—22no—
248#536—22no—
249#533C2×D6111yescharacteristic
250#533C2×D6111yescharacteristic
251#532C2×D6111yescharacteristic
252C2×C4⋊Q8—611no—
253#498—1221no—
254#494—1221no—
255#489—1222no—
256#487—612no—
257#487—1221no—
258#487—1221no—
259#486—612no—
260#483—1222no—
261#475—614no—
262#462—614no—
263#451—1222no—
264#431—611no—
265#431—611no—
266#417—1222no—
267C4×M16—611no—
268#333—612no—
269#331—612no—
270Q8⋊C8—614no—
271Q8⋊C8—614no—
272C61#1872511yescharacteristic, complemented, Sylow
273C2×#41C22⋊D6112yes—
274C22×Q8—22no—
275C22×Q8—1221no—
276C2×Q16—1224no—
277C2×M16—1221no—
278C4⋊Q8—24no—
279C4⋊Q8—22no—
280C4⋊Q8—22no—
281C4⋊Q8—612no—
282C4⋊Q8—1221no—
283C4⋊Q8—1221no—
284C4⋊Q8—1221no—
285C4⋊Q8C22⋊D6111yescharacteristic
286C4⋊Q8C22⋊D6111yescharacteristic
287C4⋊Q8C22⋊D6111yescharacteristic
288C4⋊Q8C22⋊D6111yescharacteristic
289C4⋊Q8C22×D6111yescharacteristic
290#127C22×D6111yescharacteristic
291#126—24no—
292#126—24no—
293#125—21no—
294#125—21no—
295#124—24no—
296#124—22no—
297#124—22no—
298#124C22⋊D6112yes—
299#123—24no—
300#123—24no—
301C22⋊Q8—22no—
302C22⋊Q8—22no—
303C22⋊Q8—24no—
304C22⋊Q8—24no—
305C22⋊Q8—21no—
306C22⋊Q8—21no—
307C22⋊Q8C22⋊D6112yes—
308C22⋊Q8C22⋊D6112yes—
309#121—21no—
310#121—21no—
311C4×Q8—22no—
312C4×Q8—22no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8C22×D6112yes—
316C4×Q8C22×D6112yes—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—21no—
320C4×D4—21no—
321#117—22no—
322#117—21no—
323#117—21no—
324C2×C4⋊C4—1221no—
325C2×C4⋊C4—1221no—
326C2×C42C22×D6111yescharacteristic
327#106—1224no—
328C4⋊C8—612no—
329C4⋊C8—612no—
330#103—1224no—
331#103—1224no—
332#103—1224no—
333#102—1224no—
334C22⋊C8—1221no—
335C22⋊C8—1221no—
336#97—1221no—
337C4×C8—612no—
338C4×C8—1221no—
339C4×C8—1221no—
340#41—44no—
341C2×Q8—24no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—84no—
345C2×Q8—1222no—
346C2×Q8—4882no—
347C2×Q8—2441no—
348C2×Q8C2×C22⋊D6112yes—
349C2×Q8C2×C22⋊D6112yes—
350C2×Q8C2×C22⋊D6112yes—
351C2×D4—22no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4C2×C22⋊D6112yes—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—2441no—
358C22×C4C2×C22⋊D6111yescharacteristic
359C22×C4C2×C22⋊D6111yescharacteristic
360C22×C4C2×C22⋊D6111yescharacteristic
361Q16—4884no—
362M16—2442no—
363C2×C8—1222no—
364C2×C8—1222no—
365C2×C8—1222no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—1222no—
381C4⋊C4—1222no—
382C4⋊C4—2441no—
383C4⋊C4—2441no—
384C4⋊C4C2×C22⋊D6112yes—
385C4⋊C4C2×C22⋊D6112yes—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—24no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C42—24no—
394C42—22no—
395C42—22no—
396C42C2×C22⋊D6111yescharacteristic
397C42C2×C22⋊D6111yescharacteristic
398C42C2×C22⋊D6111yescharacteristic
399C42C23×D6111yescharacteristic
400C23—22no—
401C23#1451974611yescharacteristic
402Q8—24no—
403Q8—24no—
404Q8—42no—
405Q8—42no—
406Q8—42no—
407Q8—84no—
408Q8—4882no—
409Q8—2444no—
410D4—42no—
411D4—42no—
412D4—42no—
413D4—82no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—44no—
423C2×C4—84no—
424C2×C4—4882no—
425C2×C4—2442no—
426C2×C4#1451974611yescharacteristic
427C2×C4#1451974611yescharacteristic
428C2×C4#1451974611yescharacteristic
429C2×C4C22×C22⋊D6111yescharacteristic
430C2×C4C22×C22⋊D6111yescharacteristic
431C2×C4C22×C22⋊D6111yescharacteristic
432C8—2442no—
433C8—2442no—
434C8—2442no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22C2×#1451974611yescharacteristic
442C4—21no—
443C4—21no—
444C4—21no—
445C4—42no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—4882no—
451C4#1683994012yes—
452C4#1684057012yes—
453C4#1684057712yes—
454C2—41no—
455C2—82no—
456C2#2145510511yescharacteristic
457C2#2145511411yescharacteristic
458C2C2×#1683994011yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3550C2
socleC22×C61C2×#120
prime radicalC61×#532C22
hypercenter#3550D61
hypocenterC61#18725
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3550

Complements

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

NH ≅ G/Ntype
1. #21456960C2semidirect
2. #21456764C2semidirect
4. #21445040C2semidirect
5. #21445005C2semidirect
7. #21444240C2semidirect
8. #21444240C2semidirect
272. C61#18725semidirect

Maximal subgroups

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

subgroupindex
#214569602
#214567642
#214562122
#214450402
#214450052
#214444812
#214442402
C61×#35502
#1872561