Finite Group Databasebeta

← #29576712

Subgroups — #29576712

Order 15616.

594 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456760C211yesmaximal, characteristic
2#21446800C211yesmaximal, characteristic, complemented
3#21444487C211yesmaximal, characteristic
4#21444382C211yesmaximal, characteristic, complemented
5#21443677C211yesmaximal, characteristic, complemented
6#21443648C211yesmaximal, characteristic, complemented
7C61×#2328C211yesmaximal, characteristic
8#16840319C2211yescharacteristic
9#16840134C412yescomplemented
10#16840108C412yescomplemented
11#16839959—22no—
12#16839913—22no—
13#16839815C2211yescharacteristic
14#16839815C2211yescharacteristic
15#16839786C2211yescharacteristic
16#16839522—21no—
17#16839522—21no—
18#16839506—22no—
19#16839498—22no—
20C2×#127×C61C2211yescharacteristic
21C2×C4×D4×C61—21no—
22C61×#390—21no—
23C61×#386—22no—
24C61×#386—22no—
25C61×#382C2211yescharacteristic
26C61×#376C2211yescharacteristic
27C2×#11758200—24no—
28C2×#11758194—21no—
29#14519777C2×C411yescharacteristic, complemented
30#14519777C2×C411yescharacteristic, complemented
31#14519769—21no—
32#14519756—21no—
33#14519756—21no—
34C4×Dic122—21no—
35C4×Dic122—21no—
36#14519736—21no—
37C4⋊#92418—21no—
38C4⋊#92418—21no—
39C4⋊#92418—21no—
40C4⋊#92418—21no—
41#14519698—24no—
42C22⋊#92418—21no—
43C22⋊#92418—21no—
44C4×#92418—21no—
45C4×#92418C2×C412yescomplemented
46C22×D4×C61—21no—
47C22×D4×C61—21no—
48C22×D4×C61D412yes—
49C23×C4×C61—22no—
50#127×C61—21no—
51#127×C61—21no—
52#127×C61C2×C412yes—
53#127×C61D412yes—
54C4×D4×C61—42no—
55C4×D4×C61—42no—
56C2×C4⋊C4×C61—21no—
57C2×C22⋊C4×C61—22no—
58C2×C22⋊C4×C61—22no—
59C2×C22⋊C4×C61—22no—
60C2×C42×C61D411yescharacteristic
61C2×C42×C61D411yescharacteristic
62C2×C42×C61C2311yescharacteristic
63#95×C61—21no—
64#95×C61—21no—
65#95×C61—21no—
66#95×C61D412yes—
67C24×C61—22no—
68C2×D4×C61—22no—
69C2×D4×C61—22no—
70C2×D4×C61—22no—
71C2×D4×C61—22no—
72C2×D4×C61—22no—
73C2×D4×C61—41no—
74C2×D4×C61—41no—
75C2×D4×C61—41no—
76C2×D4×C61—41no—
77C2×D4×C61—82no—
78C2×D4×C61—82no—
79C2×D4×C61C22⋊C414yes—
80C22×C4×C61—21no—
81C22×C4×C61—21no—
82C22×C4×C61—22no—
83C22×C4×C61—44no—
84C22×C4×C61—82no—
85C22×C4×C61—82no—
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D411yescharacteristic
88C22×C4×C61C2×D411yescharacteristic
89C2×Dic122—21no—
90C2×Dic122—21no—
91C4⋊C4×C61—41no—
92C4⋊C4×C61—41no—
93C22⋊C4×C61—44no—
94C22⋊C4×C61—82no—
95C22⋊C4×C61—82no—
96C42×C61—21no—
97C42×C61—21no—
98C42×C61—21no—
99C42×C61SD1612yescomplemented
100C42×C61C22×C411yescharacteristic
101C42×C61C2×D411yescharacteristic
102C42×C61#4111yescharacteristic
103C42×C61#4111yescharacteristic
104C22⋊Dic61—41no—
105#11758200—84no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109C4⋊Dic61—21no—
110#11758196—41no—
111#11758196—41no—
112C4×Dic61—41no—
113#11758194—41no—
114#11758194—41no—
115C2×#92418—22no—
116C2×#92418—41no—
117C2×#92418—41no—
118C23×C61—24no—
119C23×C61—42no—
120C23×C61—42no—
121C23×C61—42no—
122C23×C61—42no—
123C23×C61—42no—
124C23×C61—42no—
125C23×C61#12011yescharacteristic
126D4×C61—44no—
127D4×C61—42no—
128D4×C61—42no—
129D4×C61—42no—
130D4×C61—42no—
131D4×C61—82no—
132D4×C61—82no—
133D4×C61—82no—
134C2×C4×C61—22no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—42no—
142C2×C4×C61—42no—
143C2×C4×C61—42no—
144C2×C4×C61—41no—
145C2×C4×C61—88no—
146C2×C4×C61C2×C22⋊C411yescharacteristic
147C2×C4×C61C4×D411yescharacteristic
148C2×C4×C61C4×D411yescharacteristic
149C2×C4×C61#12111yescharacteristic
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—41no—
157C2×Dic61—41no—
158Dic122—42no—
159Dic122—81no—
160#92418—42no—
161#92418—81no—
162#92418—81no—
163#9295—611nomaximal, Sylow
164C22×C61—21no—
165C22×C61—21no—
166C22×C61—44no—
167C22×C61—44no—
168C22×C61—44no—
169C22×C61—82no—
170C22×C61—82no—
171C22×C61—82no—
172C22×C61—82no—
173C22×C61#38611yescharacteristic
174C22×C61#45011yescharacteristic
175C22×C61#45211yescharacteristic
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—42no—
181C4×C61—81no—
182C4×C61—82no—
183C4×C61C4×SD1611yescharacteristic
184C4×C61#44011yescharacteristic
185C4×C61#46111yescharacteristic
186C4×C61#46311yescharacteristic
187Dic61—81no—
188Dic61—81no—
189#3208—611no—
190#2328D6111yescharacteristic
191#1995—611no—
192#1545—611no—
193#1518—611no—
194#1286—611no—
195#1274—611no—
196C2×C61—21no—
197C2×C61—21no—
198C2×C61—84no—
199C2×C61#187711yescharacteristic
200C2×C61#190411yescharacteristic
201C2×C61#191111yescharacteristic
202#533—611no—
203C2×#127C2×D6111yescharacteristic
204C2×C4×D4—21no—
205#492—612no—
206#486—612no—
207#460—1222no—
208#450—1222no—
209#431—611no—
210#431—611no—
211#423—611no—
212#390—21no—
213#386—22no—
214#386—22no—
215#382C2×D6111yescharacteristic
216#376C2×D6111yescharacteristic
217#339—1221no—
218#339—1221no—
219#331—1222no—
220#328—1222no—
221C61#929511yescharacteristic, complemented, Sylow
222C22×D4—21no—
223C22×D4—21no—
224C22×D4C22⋊D6112yes—
225C23×C4—22no—
226C2×SD16—1224no—
227C2×M16—1221no—
228C4⋊Q8—611no—
229C4⋊Q8—611no—
230#127—21no—
231#127—21no—
232#127C4×D6112yes—
233#127D24412yes—
234#125—1221no—
235C22⋊Q8—1221no—
236C22⋊Q8—1221no—
237C4×Q8—1221no—
238C4×Q8—1221no—
239C4×D4—42no—
240C4×D4—42no—
241#117—1221no—
242C2×C4⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C42C22⋊D6111yescharacteristic
247C2×C42C22⋊D6111yescharacteristic
248C2×C42C22×D6111yescharacteristic
249C4⋊C8—1221no—
250C4⋊C8—1221no—
251C4⋊C8—1221no—
252C4⋊C8—1221no—
253#102—1224no—
254C22⋊C8—1221no—
255C22⋊C8—1221no—
256C4×C8—612no—
257C4×C8—1221no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95C22⋊D6112yes—
262C24—22no—
263C2×Q8—1221no—
264C2×Q8—1221no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—82no—
275C2×D4—82no—
276C2×D4#1175819814yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—44no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4C2×C22⋊D6111yescharacteristic
284C22×C4C2×C22⋊D6111yescharacteristic
285C22×C4C2×C22⋊D6111yescharacteristic
286SD16—4884no—
287M16—2441no—
288M16—2441no—
289C2×C8—1222no—
290C2×C8—2441no—
291C2×C8—2441no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—1221no—
295C4⋊C4—1221no—
296C4⋊C4—1221no—
297C4⋊C4—1221no—
298C4⋊C4—2441no—
299C4⋊C4—2441no—
300C22⋊C4—44no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C22⋊C4—2441no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—2441no—
308C42#1175820012yescomplemented
309C42C2×C4×D6111yescharacteristic
310C42C2×D24411yescharacteristic
311C42#1175822211yescharacteristic
312C42#1175822211yescharacteristic
313C23—24no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23#1451974611yescharacteristic
321Q8—4881no—
322Q8—2442no—
323D4—44no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—82no—
329D4—82no—
330D4—82no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—88no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4C2×#1175819811yescharacteristic
346C2×C4C4×C22⋊D6111yescharacteristic
347C2×C4C4×C22⋊D6111yescharacteristic
348C2×C4#1451975111yescharacteristic
349C2×C4#1451975111yescharacteristic
350C2×C4#1451976011yescharacteristic
351C2×C4C2×#1175820011yescharacteristic
352C2×C4#1451979711yescharacteristic
353C2×C4#1451980311yescharacteristic
354C2×C4#1451980711yescharacteristic
355C8—4881no—
356C8—4881no—
357C8—2442no—
358C22—21no—
359C22—21no—
360C22—44no—
361C22—44no—
362C22—44no—
363C22—82no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22#1683964111yescharacteristic
368C22#1683991311yescharacteristic
369C22#1683992411yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—42no—
375C4—81no—
376C4—82no—
377C4—4881no—
378C4—4881no—
379C4C4×#1175820011yescharacteristic
380C4#1683986211yescharacteristic
381C4#1683996711yescharacteristic
382C4#1683997711yescharacteristic
383C2—21no—
384C2—21no—
385C2—84no—
386C2#2144615311yescharacteristic
387C2#2144630611yescharacteristic
388C2#2144635111yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2328C2
socleC22×C61#386
prime radicalC2×#127×C61C22
cosocleC2×C42×C61C23
hypercenter#2328D61
hypocenterC61#9295
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. #21446800C2semidirect
4. #21444382C2semidirect
5. #21443677C2semidirect
6. #21443648C2semidirect
9. #16840134C4semidirect
10. #16840108C4semidirect
29. #14519777C2×C4semidirect
30. #14519777C2×C4semidirect
45. C4×#92418C2×C4semidirect
99. C42×C61SD16semidirect
221. C61#9295semidirect
308. C42#11758200semidirect

Maximal subgroups

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

subgroupindex
#214567602
#214468002
#214444872
#214443822
#214436772
#214436482
C61×#23282
#929561