Finite Group Databasebeta

← #29577099

Subgroups — #29577099

Order 15616.

472 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454839C211yesmaximal, characteristic, complemented
2#21444417C211yesmaximal, characteristic, complemented
3#21444253C211yesmaximal, characteristic, complemented
4#21444171C211yesmaximal, characteristic, complemented
5#21443730C211yesmaximal, characteristic, complemented
6#21443708C211yesmaximal, characteristic, complemented
7C61×#3040C211yesmaximal, characteristic
8Q8×#92418—21no—
9#16839840—22no—
10#16839840—21no—
11#16839819—21no—
12#16839819C2211yescharacteristic
13#16839819C2211yescharacteristic
14#16839815C2211yescharacteristic
15C4×#11758194—21no—
16C4×#11758194C2211yescharacteristic
17#16839502—21no—
18#16839502—21no—
19Q8⋊#92418—21no—
20D4⋊#92418—21no—
21#16839486—22no—
22C2×#137×C61C411yescharacteristic
23C2×AGL(1,ℤ/8)×C61C411yescharacteristic
24C2×#124×C61C2211yescharacteristic
25C4×#41×C61C2211yescharacteristic
26C61×#464—21no—
27C61×#464—21no—
28C61×#463—21no—
29C61×#463—21no—
30C61×#423C2211yescharacteristic
31C61×#419C411yescharacteristic
32C61×#418C411yescharacteristic
33C2×#11758194—22no—
34C2×#11758194—21no—
35C4⋊#92418—22no—
36C4⋊#92418—21no—
37C4⋊#92418—21no—
38C4⋊#92418—21no—
39C22⋊#92418—22no—
40C22⋊#92418—21no—
41C22⋊#92418—21no—
42C22⋊#92418D412yes—
43#14519688—21no—
44#14519688—21no—
45#14519688D411yescharacteristic
46#14519688D411yescharacteristic
47C4×#92418—22no—
48C4×#92418—21no—
49C4×#92418—21no—
50C2×#41×C61C2×C411yescharacteristic
51C22×Q8×C61C2×C411yescharacteristic
52C22×D4×C61C2×C411yescharacteristic
53#137×C61—42no—
54AGL(1,ℤ/8)×C61—42no—
55C2×Q16×C61—21no—
56C2×SD16×C61—21no—
57C2×SD16×C61—21no—
58C2×D8×C61—21no—
59C2×M16×C61C2×C411yescharacteristic
60C2×M16×C61C2×C411yescharacteristic
61#124×C61—21no—
62#124×C61—21no—
63#124×C61—21no—
64#124×C61—21no—
65C4×Q8×C61D411yescharacteristic
66C4×Q8×C61D411yescharacteristic
67C4×D4×C61—21no—
68C4×D4×C61—21no—
69C4×D4×C61D411yescharacteristic
70C4×D4×C61D411yescharacteristic
71#117×C61—21no—
72#117×C61C2×C411yescharacteristic
73C2×C22⋊C4×C61—21no—
74C2×C22⋊C4×C61—21no—
75C2×C42×C61—21no—
76C2×C42×C61C2311yescharacteristic
77C4⋊C8×C61—21no—
78C4⋊C8×C61—21no—
79#103×C61—21no—
80#103×C61—21no—
81#102×C61—21no—
82#102×C61—21no—
83C24×C61—21no—
84#41×C61—22no—
85#41×C61—41no—
86C2×Q8×C61—21no—
87C2×Q8×C61—21no—
88C2×Q8×C61—41no—
89C2×Q8×C61—41no—
90C2×Q8×C61C22⋊C411yescharacteristic
91C2×D4×C61—21no—
92C2×D4×C61—21no—
93C2×D4×C61—21no—
94C2×D4×C61—41no—
95C2×D4×C61—41no—
96C2×D4×C61C22⋊C411yescharacteristic
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—21no—
100C22×C4×C61—21no—
101C22×C4×C61C22×C411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61#4111yescharacteristic
104Q16×C61—42no—
105SD16×C61—42no—
106SD16×C61—42no—
107D8×C61—42no—
108M16×C61—41no—
109M16×C61—41no—
110C2×C8×C61—21no—
111C2×C8×C61—21no—
112C4⋊C4×C61—21no—
113C4⋊C4×C61—21no—
114C4⋊C4×C61C22⋊C411yescharacteristic
115C4⋊C4×C61C22⋊C411yescharacteristic
116C22⋊C4×C61—21no—
117C22⋊C4×C61—21no—
118C22⋊C4×C61—21no—
119C22⋊C4×C61—41no—
120C22⋊C4×C61—41no—
121C22⋊C4×C61—41no—
122C22⋊C4×C61—41no—
123C42×C61—21no—
124C42×C61—21no—
125C42×C61—21no—
126C42×C61C2×D411yescharacteristic
127C42×C61C2×D411yescharacteristic
128C42×C61C2×D411yescharacteristic
129C42×C61#4111yescharacteristic
130#11758194—42no—
131#11758194—44no—
132C2×#92418—21no—
133C2×#92418—22no—
134C2×#92418—21no—
135C2×#92418—41no—
136C2×#92418—41no—
137C23×C61—21no—
138C23×C61—42no—
139C23×C61—41no—
140C23×C61—81no—
141C23×C61—81no—
142C23×C61C4×D411yescharacteristic
143Q8×C61—21no—
144Q8×C61—41no—
145Q8×C61—41no—
146Q8×C61—81no—
147Q8×C61#10012yes—
148D4×C61—21no—
149D4×C61—21no—
150D4×C61—42no—
151D4×C61—41no—
152D4×C61—41no—
153D4×C61—81no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61—41no—
164C2×C4×C61—81no—
165C2×C4×C61—81no—
166C2×C4×C61C2×C22⋊C411yescharacteristic
167C2×C4×C61C4×D411yescharacteristic
168C2×C4×C61#12011yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12311yescharacteristic
172C8×C61—41no—
173C8×C61—41no—
174#92418—22no—
175#92418—42no—
176#92418—41no—
177#92418—81no—
178#92418—81no—
179#9350—611nomaximal, Sylow
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—41no—
184C22×C61—41no—
185C22×C61—41no—
186C22×C61—81no—
187C22×C61—81no—
188C22×C61—81no—
189C22×C61—162no—
190C22×C61#38611yescharacteristic
191C4×C61—21no—
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—22no—
195C4×C61—21no—
196C4×C61—21no—
197C4×C61—41no—
198C4×C61—41no—
199C4×C61—81no—
200C4×C61C2×#10011yescharacteristic
201C4×C61#41311yescharacteristic
202#3040D6111yescharacteristic
203#2971—611no—
204#1527—611no—
205#1488—611no—
206#1469—611no—
207#1307—611no—
208#1299—611no—
209C2×C61—41no—
210C2×C61—81no—
211C2×C61—161no—
212C2×C61#188111yescharacteristic
213C2×C61#190011yescharacteristic
214C2×C61#190611yescharacteristic
215C2×#137Dic6111yescharacteristic
216C2×AGL(1,ℤ/8)Dic6111yescharacteristic
217C2×#124C2×D6111yescharacteristic
218C4×#41C2×D6111yescharacteristic
219#464—21no—
220#464—21no—
221#463—21no—
222#463—21no—
223C8×Q8—1221no—
224#436—1222no—
225#436—1221no—
226#432—611no—
227#432—611no—
228#432—1221no—
229#431—611no—
230#423C2×D6111yescharacteristic
231#419Dic6111yescharacteristic
232#418Dic6111yescharacteristic
233C4×M16—611no—
234C4×M16—1221no—
235#329—1221no—
236#329—1221no—
237Q8⋊C8—1221no—
238D4⋊C8—1221no—
239#323—1222no—
240C61#935011yescharacteristic, complemented, Sylow
241C2×#41C2×Dic6111yescharacteristic
242C22×Q8C2×Dic6111yescharacteristic
243C22×D4C2×Dic6111yescharacteristic
244#137—42no—
245AGL(1,ℤ/8)—42no—
246C2×Q16—21no—
247C2×SD16—21no—
248C2×SD16—21no—
249C2×D8—21no—
250C2×M16—1222no—
251C2×M16—1221no—
252C2×M16C2×Dic6111yescharacteristic
253C2×M16C2×Dic6111yescharacteristic
254#124—21no—
255#124—21no—
256#124—21no—
257#124—21no—
258C4×Q8C22⋊D6111yescharacteristic
259C4×Q8C22⋊D6111yescharacteristic
260C4×D4—21no—
261C4×D4—21no—
262C4×D4C22⋊D6111yescharacteristic
263C4×D4C22⋊D6111yescharacteristic
264#117—21no—
265#117C2×Dic6111yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C42—21no—
269C2×C42C22×D6111yescharacteristic
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—1222no—
273C4⋊C8—1221no—
274C4⋊C8—1221no—
275C4⋊C8—1221no—
276#103—21no—
277#103—21no—
278#102—21no—
279#102—21no—
280C22⋊C8—612no—
281C22⋊C8—1222no—
282C22⋊C8—1221no—
283C22⋊C8—1221no—
284#97—611no—
285#97—611no—
286#97—1221no—
287#97—1221no—
288C4×C8—1222no—
289C4×C8—1221no—
290C4×C8—1221no—
291C24—21no—
292#41—22no—
293#41—41no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8C22⋊Dic6111yescharacteristic
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4C22⋊Dic6111yescharacteristic
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4D4×D6111yescharacteristic
310C22×C4#1175822411yescharacteristic
311C22×C4C22×Dic6111yescharacteristic
312Q16—42no—
313SD16—42no—
314SD16—42no—
315D8—42no—
316M16—41no—
317M16—41no—
318M16—2442no—
319M16—2444no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—1221no—
323C2×C8—1221no—
324C2×C8—1222no—
325C2×C8—2441no—
326C2×C8—2441no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4C22⋊Dic6111yescharacteristic
330C4⋊C4C22⋊Dic6111yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42D4×D6111yescharacteristic
342C42#1175822411yescharacteristic
343C42C2×C22⋊D6111yescharacteristic
344C42C2×C22⋊D6111yescharacteristic
345C23—21no—
346C23—42no—
347C23—41no—
348C23—81no—
349C23—81no—
350C23D4×Dic6111yescharacteristic
351Q8—21no—
352Q8—41no—
353Q8—41no—
354Q8—81no—
355Q8#1451969412yes—
356D4—21no—
357D4—21no—
358D4—42no—
359D4—41no—
360D4—41no—
361D4—81no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4C2×C22⋊Dic6111yescharacteristic
375C2×C4D4×Dic6111yescharacteristic
376C2×C4#1451974511yescharacteristic
377C2×C4#1451974911yescharacteristic
378C2×C4#1451974911yescharacteristic
379C2×C4#1451975811yescharacteristic
380C8—41no—
381C8—41no—
382C8—1222no—
383C8—4881no—
384C8—4881no—
385C8—2441no—
386C8—2442no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—162no—
397C22#1683964211yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—22no—
402C4—21no—
403C4—21no—
404C4—41no—
405C4—41no—
406C4—81no—
407C4C2×#1451969411yescharacteristic
408C4#1683974611yescharacteristic
409C2—41no—
410C2—81no—
411C2—161no—
412C2#2144618311yescharacteristic
413C2#2144628311yescharacteristic
414C2#2144632211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#3040C2
socleC22×C61#386
prime radicalC2×AGL(1,ℤ/8)×C61C4
cosocleC2×C42×C61C23
hypercenter#3040D61
hypocenterC61#9350
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3040

Complements

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

NH ≅ G/Ntype
1. #21454839C2semidirect
2. #21444417C2semidirect
3. #21444253C2semidirect
4. #21444171C2semidirect
5. #21443730C2semidirect
6. #21443708C2semidirect
240. C61#9350semidirect

Maximal subgroups

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

subgroupindex
#214548392
#214444172
#214442532
#214441712
#214437302
#214437082
C61×#30402
#935061