Finite Group Databasebeta

← #29576714

Subgroups — #29576714

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#21448545C211yesmaximal, characteristic
2#21446801C211yesmaximal, characteristic, complemented
3#21444487C211yesmaximal, characteristic
4#21444383C211yesmaximal, characteristic, complemented
5#21443678C211yesmaximal, characteristic, complemented
6#21443647C211yesmaximal, characteristic, complemented
7C61×#3208C211yesmaximal, characteristic
8C2×D4×Dic61—21no—
9#16839815C2211yescharacteristic
10#16839786C2211yescharacteristic
11#16839665—21no—
12#16839642—22no—
13#16839642—22no—
14#16839628C2211yescharacteristic
15#16839608C2211yescharacteristic
16#16839521—21no—
17#16839521—21no—
18#16839506—22no—
19#16839499—22no—
20C61×#533C2211yescharacteristic
21C2×#127×C61C2211yescharacteristic
22C61×#492C412yescomplemented
23C61×#486C412yescomplemented
24C61×#460—22no—
25C61×#450—22no—
26C61×#431C2211yescharacteristic
27C23×Dic61—22no—
28C2×#11758194—21no—
29D4×Dic61—42no—
30D4×Dic61—42no—
31C2×C4⋊Dic61—21no—
32C2×C22⋊Dic61—22no—
33C2×C22⋊Dic61—22no—
34C2×C22⋊Dic61—22no—
35C2×C4×Dic61D411yescharacteristic
36C2×C4×Dic61D411yescharacteristic
37C4⋊#92418—21no—
38C4⋊#92418—21no—
39C4⋊#92418—21no—
40C22⋊#92418—21no—
41C4×#92418—21no—
42#14519684—21no—
43#14519684—21no—
44#14519684—21no—
45#14519684D412yes—
46C22×D4×C61—21no—
47C22×D4×C61—21no—
48C22×D4×C61D412yes—
49C2×SD16×C61—24no—
50C4⋊Q8×C61C2×C411yescharacteristic, complemented
51C4⋊Q8×C61C2×C411yescharacteristic, complemented
52#127×C61—21no—
53#127×C61—21no—
54#127×C61C2×C412yes—
55#127×C61D412yes—
56#125×C61—21no—
57C22⋊Q8×C61—21no—
58C22⋊Q8×C61—21no—
59C4×Q8×C61—21no—
60C4×Q8×C61—21no—
61#117×C61—21no—
62C2×C42×C61C2311yescharacteristic
63C4⋊C8×C61—21no—
64#102×C61—24no—
65C22⋊C8×C61—21no—
66C4×C8×C61C2×C412yescomplemented
67C24×C61—22no—
68C2×Q8×C61—21no—
69C2×Q8×C61—21no—
70C2×D4×C61—22no—
71C2×D4×C61—22no—
72C2×D4×C61—22no—
73C2×D4×C61—22no—
74C2×D4×C61—22no—
75C2×D4×C61—41no—
76C2×D4×C61—41no—
77C2×D4×C61—41no—
78C2×D4×C61—41no—
79C2×D4×C61—82no—
80C2×D4×C61—82no—
81C2×D4×C61C22⋊C414yes—
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×Dic61—21no—
86C22×Dic61—22no—
87C22×Dic61—21no—
88C22×Dic61—44no—
89C22×Dic61—82no—
90C22×Dic61—82no—
91SD16×C61—84no—
92C2×C8×C61—22no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—21no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—41no—
98C4⋊C4×C61—41no—
99C22⋊C4×C61—41no—
100C42×C61—41no—
101C42×C61C22×C411yescharacteristic
102C42×C61C2×D411yescharacteristic
103C42×C61#4111yescharacteristic
104C42×C61#4111yescharacteristic
105C22⋊Dic61—44no—
106C22⋊Dic61—82no—
107C22⋊Dic61—82no—
108C4⋊Dic61—41no—
109C4⋊Dic61—41no—
110C4×Dic61—21no—
111C4×Dic61—21no—
112C4×Dic61—21no—
113C4×Dic61SD1612yescomplemented
114#11758194—41no—
115#11758194—41no—
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
126Q8×C61—42no—
127Q8×C61—81no—
128D4×C61—44no—
129D4×C61—42no—
130D4×C61—42no—
131D4×C61—42no—
132D4×C61—42no—
133D4×C61—82no—
134D4×C61—82no—
135D4×C61—82no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61C2×C22⋊C411yescharacteristic
141C2×C4×C61C4×D411yescharacteristic
142C2×C4×C61C4×D411yescharacteristic
143C2×C4×C61#12111yescharacteristic
144C2×C4×C61#12111yescharacteristic
145C2×C4×C61#12311yescharacteristic
146C2×C4×C61C2×SD1611yescharacteristic
147C2×C4×C61#13511yescharacteristic
148C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C61#13711yescharacteristic
150C2×Dic61—22no—
151C2×Dic61—41no—
152C2×Dic61—41no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—42no—
156C2×Dic61—42no—
157C2×Dic61—42no—
158C2×Dic61—41no—
159C2×Dic61—88no—
160C8×C61—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—81no—
181C4×C61—81no—
182C4×C61C4×SD1611yescharacteristic
183C4×C61#44011yescharacteristic
184C4×C61#46111yescharacteristic
185C4×C61#46311yescharacteristic
186Dic61—42no—
187Dic61—81no—
188Dic61—82no—
189#3208D6111yescharacteristic
190#2328—611no—
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#533C2×D6111yescharacteristic
203C2×#127C2×D6111yescharacteristic
204C2×C4×D4—1221no—
205#492Dic6112yescomplemented
206#486Dic6112yescomplemented
207#460—22no—
208#450—22no—
209#431—611no—
210#431C2×D6111yescharacteristic
211#423—611no—
212#390—1221no—
213#386—1222no—
214#386—1222no—
215#382—611no—
216#376—611no—
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—1222no—
226C2×SD16—24no—
227C2×M16—1221no—
228C4⋊Q8C2×Dic6111yescharacteristic, complemented
229C4⋊Q8C2×Dic6111yescharacteristic, complemented
230#127—21no—
231#127—21no—
232#127C2×Dic6112yes—
233#127C22⋊D6112yes—
234#125—21no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×D4—2442no—
240C4×D4—2442no—
241#117—21no—
242C2×C4⋊C4—1221no—
243C2×C22⋊C4—1222no—
244C2×C22⋊C4—1222no—
245C2×C22⋊C4—1222no—
246C2×C42—611no—
247C2×C42—611no—
248C2×C42C22×D6111yescharacteristic
249C4⋊C8—21no—
250C4⋊C8—1221no—
251C4⋊C8—1221no—
252C4⋊C8—1221no—
253#102—24no—
254C22⋊C8—21no—
255C22⋊C8—1221no—
256C4×C8—1221no—
257C4×C8C2×Dic6112yescomplemented
258#95—612no—
259#95—1221no—
260#95—1221no—
261#95—1221no—
262C24—22no—
263C2×Q8—21no—
264C2×Q8—21no—
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×D4C22⋊Dic6114yes—
277C22×C4—1221no—
278C22×C4—1222no—
279C22×C4—1221no—
280C22×C4—4882no—
281C22×C4—4882no—
282C22×C4—2444no—
283C22×C4D4×D6111yescharacteristic
284C22×C4D4×D6111yescharacteristic
285C22×C4C2×C22⋊D6111yescharacteristic
286SD16—84no—
287M16—2441no—
288M16—2441no—
289C2×C8—22no—
290C2×C8—2441no—
291C2×C8—2441no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—2441no—
299C4⋊C4—2441no—
300C22⋊C4—41no—
301C22⋊C4—4882no—
302C22⋊C4—4882no—
303C22⋊C4—2444no—
304C42—41no—
305C42—612no—
306C42—1221no—
307C42—1221no—
308C42—1221no—
309C42#1175822411yescharacteristic
310C42#1175822411yescharacteristic
311C42C22×Dic6111yescharacteristic
312C42C2×C22⋊D6111yescharacteristic
313C23—24no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23#1451974511yescharacteristic
321Q8—42no—
322Q8—81no—
323D4—44no—
324D4—42no—
325D4—42no—
326D4—42no—
327D4—42no—
328D4—82no—
329D4—82no—
330D4—82no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—1222no—
336C2×C4—4888no—
337C2×C4—2441no—
338C2×C4—2441no—
339C2×C4—2442no—
340C2×C4—2442no—
341C2×C4—2442no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4C2×C22⋊Dic6111yescharacteristic
346C2×C4D4×Dic6111yescharacteristic
347C2×C4D4×Dic6111yescharacteristic
348C2×C4#1451974911yescharacteristic
349C2×C4#1451974911yescharacteristic
350C2×C4#1451975811yescharacteristic
351C2×C4SD16×D6111yescharacteristic
352C2×C4#1451979811yescharacteristic
353C2×C4#1451980411yescharacteristic
354C2×C4#1451980611yescharacteristic
355C8—42no—
356C8—4881no—
357C8—4881no—
358C22—21no—
359C22—21no—
360C22—44no—
361C22—44no—
362C22—44no—
363C22—82no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22#1683964211yescharacteristic
368C22#1683991111yescharacteristic
369C22#1683992111yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—81no—
375C4—81no—
376C4—4882no—
377C4—4881no—
378C4—2442no—
379C4SD16×Dic6111yescharacteristic
380C4#1683986311yescharacteristic
381C4#1683996511yescharacteristic
382C4#1683997911yescharacteristic
383C2—21no—
384C2—21no—
385C2—84no—
386C2#2144615511yescharacteristic
387C2#2144630811yescharacteristic
388C2#2144635211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#3208C2
socleC22×C61#386
prime radicalC2×#127×C61C22
cosocleC2×C42×C61C23
hypercenter#3208D61
hypocenterC61#9295
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
2. #21446801C2semidirect
4. #21444383C2semidirect
5. #21443678C2semidirect
6. #21443647C2semidirect
22. C61×#492C4semidirect
23. C61×#486C4semidirect
50. C4⋊Q8×C61C2×C4semidirect
51. C4⋊Q8×C61C2×C4semidirect
66. C4×C8×C61C2×C4semidirect
113. C4×Dic61SD16semidirect
205. #492Dic61semidirect
206. #486Dic61semidirect
221. C61#9295semidirect
228. C4⋊Q8C2×Dic61semidirect
229. C4⋊Q8C2×Dic61semidirect
257. C4×C8C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214485452
#214468012
#214444872
#214443832
#214436782
#214436472
C61×#32082
#929561