Finite Group Databasebeta

← #29663168

Subgroups — #29663168

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456741C211yesmaximal, characteristic, complemented
2C2×#16840108C211yesmaximal, characteristic, complemented
3#21454531C211yesmaximal, characteristic, complemented
4#21445378C214yesmaximal, complemented
5#21445254C214yesmaximal, complemented
6#21444565C211yesmaximal, characteristic, complemented
7#21444503C211yesmaximal, characteristic, complemented
8#21444487C211yesmaximal, characteristic, complemented
9C61×#3205C211yesmaximal, characteristic
10C2×#14519784—21no—
11C2×#14519777C2211yescharacteristic
12#16840134—21no—
13#16840114—21no—
14#16840108—21no—
15#16840108—21no—
16#16840108C2212yes—
17#16840108C2212yes—
18#16839967C2214yescomplemented
19#16839900—22no—
20#16839840—21no—
21#16839840C2212yescomplemented
22D4×#92418—21no—
23D4×#92418C2212yescomplemented
24#16839815C2211yescharacteristic
25#16839786C2211yescharacteristic
26C2×#14519698—21no—
27C2×#14519698—21no—
28#16839729—21no—
29C2×C4×#92418C2211yescharacteristic
30#16839514C2212yescomplemented
31#16839512C2212yescomplemented
32#16839509C2214yescomplemented
33#16839506C2214yes—
34C61×#532C2211yescharacteristic
35C2×#127×C61C2211yescharacteristic
36C61×#493—21no—
37C61×#486—21no—
38C61×#459C2214yes—
39C61×#447—22no—
40C61×#431C2211yescharacteristic
41C22×Dic122—21no—
42C2×#11758200—24no—
43#14519784—44no—
44C2×#11758194—21no—
45C2×#11758194—22no—
46C22×#92418—22no—
47C22×#92418D411yescharacteristic
48C22×#92418D411yescharacteristic
49#14519777—21no—
50#14519777—21no—
51#14519777—21no—
52#14519777C2312yes—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61—21no—
55C4⋊#92418—21no—
56C4⋊#92418C2312yes—
57C4⋊#92418C2312yes—
58#14519700—24no—
59#14519698—22no—
60#14519698—22no—
61#14519698—22no—
62#14519698—22no—
63C22⋊#92418—21no—
64C22⋊#92418—22no—
65C22⋊#92418—22no—
66C4×#92418—21no—
67C4×#92418D411yescharacteristic
68C4×#92418D411yescharacteristic
69C4×#92418C2311yescharacteristic
70C4×#92418C2311yescharacteristic
71C2×#41×C61—21no—
72C22×D4×C61—21no—
73C22×D4×C61—21no—
74C22×D4×C61—21no—
75C2×D8×C61—24no—
76C4⋊Q8×C61D411yescharacteristic
77#127×C61—21no—
78#127×C61—21no—
79#127×C61—21no—
80#127×C61D411yescharacteristic
81#127×C61C2312yes—
82#124×C61—21no—
83#121×C61—21no—
84#121×C61D412yes—
85C4×D4×C61—21no—
86C4×D4×C61C2312yes—
87C2×C42×C61C2311yescharacteristic
88C4⋊C8×C61C2312yes—
89#102×C61—24no—
90C22⋊C8×C61—21no—
91C4×C8×C61—21no—
92C24×C61—41no—
93#41×C61—42no—
94#41×C61—42no—
95C2×Q8×C61—21no—
96C2×D4×C61—22no—
97C2×D4×C61—21no—
98C2×D4×C61—21no—
99C2×D4×C61—22no—
100C2×D4×C61—22no—
101C2×D4×C61—22no—
102C2×D4×C61—41no—
103C2×D4×C61—41no—
104C2×D4×C61—41no—
105C2×D4×C61—42no—
106C2×D4×C61—42no—
107C2×D4×C61—82no—
108C2×D4×C61—82no—
109C2×D4×C61C2×D412yes—
110C22×C4×C61—22no—
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61#4111yescharacteristic
114C22×Dic61—41no—
115C2×Dic122—22no—
116C2×Dic122—41no—
117C2×Dic122—82no—
118D8×C61—84no—
119C2×C8×C61—22no—
120C4⋊C4×C61—21no—
121C4⋊C4×C61C2×D412yes—
122C22⋊C4×C61—22no—
123C22⋊C4×C61—22no—
124C42×C61C2×D411yescharacteristic
125C42×C61C2×D411yescharacteristic
126C42×C61#4111yescharacteristic
127C42×C61C2411yescharacteristic
128#11758200—84no—
129C4⋊Dic61—22no—
130C4⋊Dic61—22no—
131C4⋊Dic61—41no—
132C4⋊Dic61—41no—
133#11758194—44no—
134#11758194—42no—
135C2×#92418—22no—
136C2×#92418—22no—
137C2×#92418—22no—
138C2×#92418—22no—
139C2×#92418—44no—
140C2×#92418C2×D411yescharacteristic
141C2×#92418C2×D411yescharacteristic
142C2×#92418C2×D411yescharacteristic
143C2×#92418C2×D411yescharacteristic
144C23×C61—22no—
145C23×C61—42no—
146C23×C61—82no—
147C23×C61—82no—
148C23×C61—82no—
149C23×C61#12111yescharacteristic
150Q8×C61—42no—
151D4×C61—24no—
152D4×C61—44no—
153D4×C61—42no—
154D4×C61—42no—
155D4×C61—44no—
156D4×C61—42no—
157D4×C61—42no—
158D4×C61—82no—
159D4×C61—82no—
160D4×C61—82no—
161D4×C61—82no—
162C2×C4×C61—22no—
163C2×C4×C61—22no—
164C2×C4×C61—22no—
165C2×C4×C61—42no—
166C2×C4×C61—42no—
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#13512yes—
171C2×C4×C61AGL(1,ℤ/8)12yes—
172C2×C4×C61C22×D411yescharacteristic
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C2×#4111yescharacteristic
175C2×Dic61—42no—
176C2×Dic61—82no—
177Dic122—44no—
178Dic122—82no—
179C8×C61—42no—
180#92418—21no—
181#92418—21no—
182#92418—21no—
183#92418—21no—
184#92418—42no—
185#92418—42no—
186#19198—611nomaximal, Sylow
187C22×C61—21no—
188C22×C61—21no—
189C22×C61—42no—
190C22×C61—42no—
191C22×C61—42no—
192C22×C61—82no—
193C22×C61—82no—
194C22×C61—82no—
195C22×C61—164no—
196C22×C61#46312yes—
197C22×C61C2×#12111yescharacteristic
198C4×C61—21no—
199C4×C61—21no—
200C4×C61—42no—
201C4×C61#46312yes—
202C4×C61#47112yes—
203C4×C61C2×#13511yescharacteristic
204C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
205C4×C61#57612yes—
206Dic61—82no—
207#3206—611no—
208#3205D6111yescharacteristic
209C2×#486—611no—
210#2931—611no—
211#1701—614no—
212#1681—614no—
213#1564—611no—
214#1549—611no—
215#1545—611no—
216C2×C61—22no—
217C2×C61—82no—
218C2×C61—162no—
219C2×C61C2×#46311yescharacteristic
220C2×C61#305811yescharacteristic
221C2×C61C2×#47111yescharacteristic
222C2×#131—1221no—
223#532C2×D6111yescharacteristic
224C2×C4⋊Q8—611no—
225C2×#127C2×D6111yescharacteristic
226#493—21no—
227#492—1221no—
228#487—1221no—
229#486—21no—
230#486—612no—
231#486—612no—
232#486—1221no—
233#486—1221no—
234#461—614no—
235#459C2×D6114yes—
236#448—1222no—
237#447—22no—
238#436—612no—
239#436—1221no—
240C8×D4—612no—
241C8×D4—1221no—
242#431—611no—
243#431C2×D6111yescharacteristic
244#423—611no—
245C2×#102—1221no—
246C2×#102—1221no—
247#408—1221no—
248C2×C4×C8—611no—
249#335—612no—
250#334—612no—
251#332—614no—
252#331—614no—
253C61#1919811yescharacteristic, complemented, Sylow
254C2×#41—21no—
255C22×Q8—1221no—
256C22×D4—21no—
257C22×D4—21no—
258C22×D4—21no—
259C2×SD16—1224no—
260C2×D8—24no—
261#131—2444no—
262C2×M16—1222no—
263C2×M16—1221no—
264C22×C8—611no—
265C22×C8—611no—
266C22×C8—1222no—
267C4⋊Q8—612no—
268C4⋊Q8—1221no—
269C4⋊Q8—1221no—
270C4⋊Q8—1221no—
271C4⋊Q8C22⋊D6111yescharacteristic
272#127—21no—
273#127—21no—
274#127—21no—
275#127C22⋊D6111yescharacteristic
276#127C22×D6112yes—
277#124—21no—
278#121—21no—
279#121C22⋊D6112yes—
280C4×D4—21no—
281C4×D4C22×D6112yes—
282C2×C4⋊C4—1221no—
283C2×C4⋊C4—1221no—
284C2×C42C22×D6111yescharacteristic
285C4⋊C8—612no—
286C4⋊C8—612no—
287C4⋊C8—1221no—
288C4⋊C8C22×D6112yes—
289#103—1224no—
290#102—24no—
291#102—1222no—
292#102—1222no—
293#102—1222no—
294#102—1222no—
295C22⋊C8—21no—
296C22⋊C8—1222no—
297C22⋊C8—1222no—
298C22⋊C8—1221no—
299C4×C8—21no—
300C4×C8—611no—
301C4×C8—611no—
302C4×C8—611no—
303C4×C8—611no—
304C4×C8—1221no—
305C24—41no—
306#41—42no—
307#41—42no—
308C2×Q8—21no—
309C2×Q8—1222no—
310C2×Q8—4882no—
311C2×Q8—2441no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—82no—
324C2×D4—82no—
325C2×D4C2×C22⋊D6112yes—
326C22×C4—22no—
327C22×C4—2441no—
328C22×C4D4×D6111yescharacteristic
329C22×C4#1175822411yescharacteristic
330C22×C4C2×C22⋊D6111yescharacteristic
331SD16—4884no—
332D8—84no—
333M16—2444no—
334M16—2442no—
335C2×C8—22no—
336C2×C8—611no—
337C2×C8—611no—
338C2×C8—611no—
339C2×C8—611no—
340C2×C8—1222no—
341C2×C8—1222no—
342C2×C8—1222no—
343C2×C8—1222no—
344C2×C8—2444no—
345C4⋊C4—21no—
346C4⋊C4—1222no—
347C4⋊C4—1222no—
348C4⋊C4—2441no—
349C4⋊C4—2441no—
350C4⋊C4C2×C22⋊D6112yes—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C42D4×D6111yescharacteristic
354C42#1175822411yescharacteristic
355C42C2×C22⋊D6111yescharacteristic
356C42C23×D6111yescharacteristic
357C23—22no—
358C23—42no—
359C23—82no—
360C23—82no—
361C23—82no—
362C23#1451974911yescharacteristic
363Q8—42no—
364Q8—4882no—
365Q8—2444no—
366D4—24no—
367D4—44no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—42no—
372D4—44no—
373D4—82no—
374D4—82no—
375D4—82no—
376D4—82no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—4882no—
383C2×C4—2442no—
384C2×C4#1451974911yescharacteristic
385C2×C4#1451974911yescharacteristic
386C2×C4#1451974911yescharacteristic
387C2×C4#1451979612yes—
388C2×C4#1451980112yes—
389C2×C4C2×D4×D6111yescharacteristic
390C2×C4C22×C22⋊D6111yescharacteristic
391C2×C4C2×#1175822411yescharacteristic
392C8—42no—
393C8—1221no—
394C8—1221no—
395C8—1221no—
396C8—1221no—
397C8—2442no—
398C8—2442no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—82no—
405C22—82no—
406C22—82no—
407C22—164no—
408C22#1683997512yes—
409C22C2×#1451974911yescharacteristic
410C4—21no—
411C4—21no—
412C4—42no—
413C4—4882no—
414C4#1683997512yes—
415C4#1684002312yes—
416C4C2×#1451979611yescharacteristic
417C4C2×#1451980111yescharacteristic
418C4#1684056512yes—
419C2—22no—
420C2—82no—
421C2—162no—
422C2C2×#1683997511yescharacteristic
423C2#2145546411yescharacteristic
424C2C2×#1684002311yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3205C2
socleC22×C61C2×#121
hypercenter#3205D61
hypocenterC61#19198
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #21456741C2semidirect
2. C2×#16840108C2semidirect
3. #21454531C2semidirect
4. #21445378C2semidirect
5. #21445254C2semidirect
6. #21444565C2semidirect
7. #21444503C2semidirect
8. #21444487C2semidirect
18. #16839967C22semidirect
21. #16839840C22semidirect
23. D4×#92418C22semidirect
30. #16839514C22semidirect
31. #16839512C22semidirect
32. #16839509C22semidirect
253. C61#19198semidirect

Maximal subgroups

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

subgroupindex
#214567412
C2×#168401082
#214545312
#214453782
#214452542
#214445652
#214445032
#214444872
C61×#32052
#1919861