Finite Group Databasebeta

← #29575248

Subgroups — #29575248

Order 15616.

500 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456374C211yesmaximal, characteristic
2#21446763C211yesmaximal, characteristic, complemented
3#21444314C211yesmaximal, characteristic
4#21444221C211yesmaximal, characteristic, complemented
5#21443658C211yesmaximal, characteristic, complemented
6#21443657C211yesmaximal, characteristic, complemented
7C61×#2935C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840158C411yescharacteristic, complemented
10#16840134C411yescharacteristic, complemented
11#16840114C411yescharacteristic, complemented
12#16840108C411yescharacteristic, complemented
13#16839989—22no—
14C4×#11758200—22no—
15#16839815C2211yescharacteristic
16C2×C4×#92418C2211yescharacteristic
17C4×C4⋊Dic61C2211yescharacteristic
18#16839523—21no—
19#16839523—21no—
20#16839496—22no—
21#16839492—21no—
22#16839492—21no—
23C2×#131×C61—21no—
24C61×#532C2211yescharacteristic
25C61×#436—21no—
26C61×#435—22no—
27C8×D4×C61—21no—
28C2×C4⋊C8×C61C2211yescharacteristic
29C61×#408—21no—
30C4×M16×C61C2211yescharacteristic
31C2×#11758200—22no—
32C22×#92418—21no—
33#14519777C2×C411yescharacteristic
34#14519777C2×C411yescharacteristic
35#14519769—21no—
36#14519756—21no—
37#14519756—21no—
38C4×Dic122—21no—
39C4×Dic122—21no—
40#14519736—21no—
41C2×C4⋊Dic61—21no—
42C2×C4×Dic61—21no—
43#14519708—22no—
44C4⋊#92418—21no—
45#14519700—22no—
46#14519698—22no—
47C22⋊#92418—21no—
48C4×#92418—21no—
49C4×#92418—21no—
50C4×#92418C2×C411yescharacteristic
51C4×#92418C2×C411yescharacteristic
52#14519684—21no—
53C2×#41×C61—21no—
54#131×C61—42no—
55#131×C61—42no—
56C2×M16×C61—22no—
57C2×M16×C61D411yescharacteristic
58C2×M16×C61D411yescharacteristic
59C22×C8×C61—22no—
60C22×C8×C61—21no—
61C4⋊Q8×C61C2×C411yescharacteristic
62#127×C61C2×C411yescharacteristic
63#124×C61—21no—
64#121×C61—21no—
65#121×C61D412yes—
66C4×D4×C61—21no—
67C4×D4×C61D412yes—
68C2×C42×C61C2311yescharacteristic
69C4⋊C8×C61—21no—
70C4⋊C8×C61D411yescharacteristic
71C4⋊C8×C61D411yescharacteristic
72C22⋊C8×C61—22no—
73C22⋊C8×C61—22no—
74#97×C61—21no—
75C4×C8×C61—21no—
76#41×C61—41no—
77#41×C61—41no—
78#41×C61—41no—
79#41×C61—41no—
80C2×Q8×C61—21no—
81C2×D4×C61—22no—
82C2×D4×C61—21no—
83C2×D4×C61—21no—
84C2×D4×C61C22⋊C412yes—
85C22×C4×C61—22no—
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61#4111yescharacteristic
89C22×Dic61—41no—
90C2×Dic122—21no—
91C2×Dic122—21no—
92M16×C61—21no—
93M16×C61—21no—
94M16×C61—21no—
95M16×C61—44no—
96M16×C61SD1612yescomplemented
97C2×C8×C61—21no—
98C2×C8×C61—21no—
99C2×C8×C61—21no—
100C2×C8×C61—21no—
101C2×C8×C61—44no—
102C2×C8×C61—41no—
103C2×C8×C61—41no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61C22⋊C412yes—
106C22⋊C4×C61—22no—
107C22⋊C4×C61—22no—
108C42×C61C22×C411yescharacteristic
109C42×C61C2×D411yescharacteristic
110C42×C61C2×D411yescharacteristic
111C42×C61#4111yescharacteristic
112C22⋊Dic61—41no—
113#11758200—44no—
114C4⋊Dic61—22no—
115C4⋊Dic61—21no—
116C4⋊Dic61—21no—
117C4⋊Dic61—21no—
118C4⋊Dic61—21no—
119C4⋊Dic61—41no—
120#11758196—41no—
121#11758196—41no—
122C4×Dic61—41no—
123C4×Dic61—41no—
124C4×Dic61—41no—
125C2×#92418—21no—
126C2×#92418—21no—
127C2×#92418—42no—
128C2×#92418—41no—
129C23×C61—22no—
130C23×C61#12111yescharacteristic
131Q8×C61—41no—
132Q8×C61—41no—
133D4×C61—42no—
134D4×C61—42no—
135D4×C61—42no—
136D4×C61—41no—
137D4×C61—41no—
138D4×C61—41no—
139D4×C61—41no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—22no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61—42no—
146C2×C4×C61C2×C22⋊C411yescharacteristic
147C2×C4×C61C4×D411yescharacteristic
148C2×C4×C61C4×D411yescharacteristic
149C2×C4×C61#12011yescharacteristic
150C2×C4×C61#12111yescharacteristic
151C2×C4×C61#12311yescharacteristic
152C2×C4×C61C2×SD1611yescharacteristic
153C2×C4×C61#13511yescharacteristic
154C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C61#13711yescharacteristic
156C2×Dic61—42no—
157C2×Dic61—41no—
158C2×Dic61—41no—
159C2×Dic61—81no—
160C2×Dic61—81no—
161Dic122—42no—
162Dic122—41no—
163Dic122—41no—
164C8×C61—22no—
165C8×C61—41no—
166C8×C61—41no—
167C8×C61—41no—
168#92418—22no—
169#92418—41no—
170#92418—81no—
171#9055—611nomaximal, Sylow
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—42no—
175C22×C61—42no—
176C22×C61—42no—
177C22×C61#38611yescharacteristic
178C22×C61#46111yescharacteristic
179C22×C61#46311yescharacteristic
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—21no—
183C4×C61—42no—
184C4×C61C4×SD1611yescharacteristic
185C4×C61#44011yescharacteristic
186C4×C61#44311yescharacteristic
187C4×C61#44411yescharacteristic
188C4×C61#45011yescharacteristic
189C4×C61#45211yescharacteristic
190Dic61—82no—
191Dic61—81no—
192Dic61—81no—
193#3167—611no—
194#2935D6111yescharacteristic
195#1986—611no—
196#1502—611no—
197#1481—611no—
198#1279—611no—
199#1278—611no—
200C2×C61—21no—
201C2×C61—21no—
202C2×C61—82no—
203C2×C61#187611yescharacteristic
204C2×C61#187711yescharacteristic
205C2×C61#188411yescharacteristic
206C2×#131—21no—
207#533—611no—
208#532C2×D6111yescharacteristic
209#498—611no—
210#492—611no—
211#487—611no—
212#486—611no—
213#465—1222no—
214C4×SD16—1222no—
215#436—21no—
216#435—22no—
217C8×D4—21no—
218#431—611no—
219C2×C4⋊C8C2×D6111yescharacteristic
220#408—21no—
221C4×M16C2×D6111yescharacteristic
222C2×C4×C8—611no—
223C4×C4⋊C4—611no—
224#339—1221no—
225#339—1221no—
226#327—1222no—
227Q8⋊C8—1221no—
228Q8⋊C8—1221no—
229C61#905511yescharacteristic, complemented, Sylow
230C2×#41—21no—
231C2×SD16—1222no—
232#131—42no—
233#131—42no—
234C2×M16—22no—
235C2×M16C22⋊D6111yescharacteristic
236C2×M16C22⋊D6111yescharacteristic
237C22×C8—22no—
238C22×C8—21no—
239C22×C8—1221no—
240C4⋊Q8—611no—
241C4⋊Q8—611no—
242C4⋊Q8C4×D6111yescharacteristic
243#127C4×D6111yescharacteristic
244#125—1221no—
245#124—21no—
246C22⋊Q8—1221no—
247C22⋊Q8—1221no—
248#121—21no—
249#121D24412yes—
250C4×Q8—1221no—
251C4×Q8—1221no—
252C4×D4—21no—
253C4×D4C22⋊D6112yes—
254#117—1221no—
255C2×C4⋊C4—1221no—
256C2×C42—1221no—
257C2×C42C22×D6111yescharacteristic
258#106—1222no—
259C4⋊C8—21no—
260C4⋊C8—1221no—
261C4⋊C8C22⋊D6111yescharacteristic
262C4⋊C8C22⋊D6111yescharacteristic
263#103—1222no—
264#102—1222no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267C22⋊C8—1221no—
268#97—21no—
269C4×C8—21no—
270C4×C8—611no—
271C4×C8—611no—
272C4×C8—1221no—
273C4×C8—1221no—
274#95—1221no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—1221no—
281C2×Q8—1221no—
282C2×D4—22no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4#1175819812yes—
286C22×C4—22no—
287C22×C4—2441no—
288C22×C4C2×D24411yescharacteristic
289C22×C4#1175822211yescharacteristic
290C22×C4C2×C22⋊D6111yescharacteristic
291SD16—2444no—
292M16—21no—
293M16—21no—
294M16—21no—
295M16—44no—
296M16#1175820012yescomplemented
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—44no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—1221no—
305C2×C8—1221no—
306C2×C8—2442no—
307C2×C8—2441no—
308C4⋊C4—21no—
309C4⋊C4—1222no—
310C4⋊C4—1221no—
311C4⋊C4—1221no—
312C4⋊C4—1221no—
313C4⋊C4—1221no—
314C4⋊C4—2441no—
315C4⋊C4—2441no—
316C4⋊C4—2441no—
317C4⋊C4#1175819812yes—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—2441no—
321C42—2441no—
322C42—2441no—
323C42—2441no—
324C42C2×C4×D6111yescharacteristic
325C42#1175822211yescharacteristic
326C42C2×C22⋊D6111yescharacteristic
327C42C2×C22⋊D6111yescharacteristic
328C23—22no—
329C23#1451975111yescharacteristic
330Q8—41no—
331Q8—41no—
332Q8—2442no—
333Q8—2441no—
334Q8—2441no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—41no—
339D4—41no—
340D4—41no—
341D4—41no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—4881no—
349C2×C4—4881no—
350C2×C4—2442no—
351C2×C4—2441no—
352C2×C4—2441no—
353C2×C4C2×#1175819811yescharacteristic
354C2×C4C4×C22⋊D6111yescharacteristic
355C2×C4C4×C22⋊D6111yescharacteristic
356C2×C4#1451974611yescharacteristic
357C2×C4#1451975111yescharacteristic
358C2×C4#1451976011yescharacteristic
359C2×C4C2×#1175820011yescharacteristic
360C2×C4#1451979711yescharacteristic
361C2×C4#1451980311yescharacteristic
362C2×C4#1451980711yescharacteristic
363C8—22no—
364C8—41no—
365C8—41no—
366C8—41no—
367C8—1222no—
368C8—4881no—
369C8—2441no—
370C22—21no—
371C22—21no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22#1683964111yescharacteristic
376C22#1683996711yescharacteristic
377C22#1683997711yescharacteristic
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—4882no—
383C4—4881no—
384C4—4881no—
385C4C4×#1175820011yescharacteristic
386C4#1683986211yescharacteristic
387C4#1683987711yescharacteristic
388C4#1683988211yescharacteristic
389C4#1683991311yescharacteristic
390C4#1683992411yescharacteristic
391C2—21no—
392C2—21no—
393C2—82no—
394C2#2144614611yescharacteristic
395C2#2144615311yescharacteristic
396C2#2144620211yescharacteristic
subgroupG/H
centerC2×C4#14519746
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2935C2
socleC22×C61#386
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#2935D61
hypocenterC61#9055
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2935

Complements

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

NH ≅ G/Ntype
2. #21446763C2semidirect
4. #21444221C2semidirect
5. #21443658C2semidirect
6. #21443657C2semidirect
9. #16840158C4semidirect
10. #16840134C4semidirect
11. #16840114C4semidirect
12. #16840108C4semidirect
96. M16×C61SD16semidirect
229. C61#9055semidirect
296. M16#11758200semidirect

Maximal subgroups

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

subgroupindex
#214563742
#214467632
#214443142
#214442212
#214436582
#214436572
C61×#29352
#905561