Finite Group Databasebeta

← #29579416

Subgroups — #29579416

Order 15616.

440 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21445505C211yesmaximal, characteristic, complemented
2#21444582C211yesmaximal, characteristic, complemented
3#21444434C211yesmaximal, characteristic, complemented
4#21444430C211yesmaximal, characteristic
5#21444331C211yesmaximal, characteristic, complemented
6#21444327C211yesmaximal, characteristic
7C61×#3167C211yesmaximal, characteristic
8#16839819C2211yescharacteristic
9#16839819C2211yescharacteristic
10C2×#14519688C2211yescharacteristic
11C2×#14519687—21no—
12C2×C8×Dic61—21no—
13#16839608C2211yescharacteristic
14#16839515—21no—
15#16839513—21no—
16#16839504—22no—
17#16839502—22no—
18#16839499—22no—
19#16839496—22no—
20C61×#533C2211yescharacteristic
21C61×#532C2211yescharacteristic
22C61×#498C411yescharacteristic, complemented
23C61×#492C411yescharacteristic, complemented
24C61×#487C411yescharacteristic, complemented
25C61×#486C411yescharacteristic, complemented
26C61×#465—21no—
27C61×#465—21no—
28C4×SD16×C61—21no—
29C4×SD16×C61—21no—
30C2×C4×C8×C61C2211yescharacteristic
31C22×#92418—21no—
32C2×C4×Dic61—21no—
33C4⋊#92418—21no—
34C4⋊#92418—21no—
35C22⋊#92418—21no—
36C22⋊#92418—21no—
37#14519688—21no—
38#14519688—21no—
39#14519688—21no—
40#14519688D412yes—
41#14519687—24no—
42C8×Dic61—24no—
43#14519684—21no—
44#14519684D412yes—
45C2×#41×C61—21no—
46C2×SD16×C61—21no—
47C2×SD16×C61—21no—
48C22×C8×C61D411yescharacteristic
49C22×C8×C61D411yescharacteristic
50C4⋊Q8×C61C2×C411yescharacteristic
51C4⋊Q8×C61C2×C411yescharacteristic
52C4⋊Q8×C61C2×C411yescharacteristic
53#127×C61C2×C411yescharacteristic
54#125×C61—21no—
55#124×C61—21no—
56C22⋊Q8×C61—21no—
57C22⋊Q8×C61—21no—
58#121×C61—21no—
59#121×C61—21no—
60C4×Q8×C61—21no—
61C4×Q8×C61—21no—
62C4×D4×C61—21no—
63C4×D4×C61—21no—
64#117×C61—21no—
65C2×C42×C61C2311yescharacteristic
66#106×C61—21no—
67#106×C61—21no—
68#103×C61—21no—
69#103×C61—21no—
70#102×C61—21no—
71#102×C61—21no—
72C4×C8×C61C2×C411yescharacteristic
73C4×C8×C61C2×C411yescharacteristic
74C4×C8×C61D411yescharacteristic
75C4×C8×C61D411yescharacteristic
76#41×C61—82no—
77C2×Q8×C61—21no—
78C2×Q8×C61—21no—
79C2×Q8×C61—21no—
80C2×D4×C61—21no—
81C2×D4×C61—21no—
82C2×D4×C61—21no—
83C2×D4×C61—41no—
84C22×C4×C61—41no—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61#4111yescharacteristic
88C22×Dic61—22no—
89SD16×C61—42no—
90SD16×C61—42no—
91C2×C8×C61—22no—
92C2×C8×C61—22no—
93C2×C8×C61C22⋊C411yescharacteristic
94C2×C8×C61C22⋊C411yescharacteristic
95C2×C8×C61C22⋊C411yescharacteristic
96C2×C8×C61C22⋊C411yescharacteristic
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—21no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—41no—
104C4⋊C4×C61—41no—
105C22⋊C4×C61—41no—
106C22⋊C4×C61—41no—
107C22⋊C4×C61—41no—
108C42×C61—41no—
109C42×C61C22×C411yescharacteristic
110C42×C61C2×D411yescharacteristic
111C42×C61C2×D411yescharacteristic
112C42×C61#4111yescharacteristic
113C4×Dic61—22no—
114C4×Dic61—22no—
115C2×#92418—22no—
116C2×#92418—22no—
117C2×#92418—22no—
118C2×#92418—41no—
119C2×#92418—41no—
120C23×C61—41no—
121C23×C61#12111yescharacteristic
122Q8×C61—41no—
123Q8×C61—41no—
124Q8×C61—81no—
125Q8×C61—81no—
126D4×C61—41no—
127D4×C61—41no—
128D4×C61—81no—
129D4×C61—81no—
130D4×C61—82no—
131C2×C4×C61—22no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—82no—
139C2×C4×C61C2×C22⋊C411yescharacteristic
140C2×C4×C61C4×D411yescharacteristic
141C2×C4×C61C4×D411yescharacteristic
142C2×C4×C61#12111yescharacteristic
143C2×C4×C61#12411yescharacteristic
144C2×C4×C61#12711yescharacteristic
145C2×Dic61—24no—
146C2×Dic61—42no—
147C2×Dic61—42no—
148C8×C61—21no—
149C8×C61—21no—
150C8×C61#10412yes—
151C8×C61#10412yes—
152#92418—24no—
153#92418—81no—
154#92418—81no—
155#9761—611nomaximal, Sylow
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—81no—
159C22×C61—81no—
160C22×C61—81no—
161C22×C61#39011yescharacteristic
162C22×C61#47112yes—
163C4×C61—21no—
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—81no—
169C4×C61—81no—
170C4×C61—81no—
171C4×C61C2×#10411yescharacteristic
172C4×C61C2×#10411yescharacteristic
173Dic61—44no—
174#3167D6111yescharacteristic
175#1726—611no—
176#1568—611no—
177#1531—611no—
178#1530—611no—
179#1506—611no—
180#1505—611no—
181C2×C61—22no—
182C2×C61—161no—
183C2×C61#196211yescharacteristic
184C2×C61#197611yescharacteristic
185C2×C61#197711yescharacteristic
186#533C2×D6111yescharacteristic
187#532C2×D6111yescharacteristic
188#498Dic6111yescharacteristic, complemented
189#492Dic6111yescharacteristic, complemented
190#487Dic6111yescharacteristic, complemented
191#486Dic6111yescharacteristic, complemented
192#465—21no—
193#465—21no—
194C4×SD16—21no—
195C4×SD16—21no—
196#432—611no—
197#432—611no—
198C2×#97—611no—
199C2×#97—1221no—
200C2×C4×C8—1221no—
201C2×C4×C8C2×D6111yescharacteristic
202#376—611no—
203#335—1221no—
204#334—1221no—
205#330—1222no—
206#329—1222no—
207#328—1222no—
208#327—1222no—
209C61#976111yescharacteristic, complemented, Sylow
210C2×#41—21no—
211C2×SD16—21no—
212C2×SD16—21no—
213C22×C8—1221no—
214C22×C8C22⋊D6111yescharacteristic
215C22×C8C22⋊D6111yescharacteristic
216C4⋊Q8C2×Dic6111yescharacteristic
217C4⋊Q8C2×Dic6111yescharacteristic
218C4⋊Q8C2×Dic6111yescharacteristic
219#127C2×Dic6111yescharacteristic
220#125—21no—
221#124—21no—
222C22⋊Q8—21no—
223C22⋊Q8—21no—
224#121—21no—
225#121—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×D4—21no—
229C4×D4—21no—
230#117—21no—
231C2×C42—1221no—
232C2×C42C22×D6111yescharacteristic
233#106—21no—
234#106—21no—
235C4⋊C8—1221no—
236C4⋊C8—1221no—
237#103—21no—
238#103—21no—
239#102—21no—
240#102—21no—
241C22⋊C8—1221no—
242C22⋊C8—1221no—
243#97—612no—
244#97—1224no—
245#97—1221no—
246#97—1221no—
247#97—1221no—
248C4×C8—1224no—
249C4×C8C2×Dic6111yescharacteristic
250C4×C8C2×Dic6111yescharacteristic
251C4×C8C22⋊D6111yescharacteristic
252C4×C8C22⋊D6111yescharacteristic
253#95—612no—
254#95—1221no—
255#41—82no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—41no—
263C22×C4—41no—
264C22×C4—1222no—
265C22×C4D4×D6111yescharacteristic
266C22×C4#1175822411yescharacteristic
267C22×C4C2×C22⋊D6111yescharacteristic
268SD16—42no—
269SD16—42no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—1222no—
273C2×C8—1222no—
274C2×C8—1222no—
275C2×C8—2441no—
276C2×C8—2441no—
277C2×C8C22⋊Dic6111yescharacteristic
278C2×C8C22⋊Dic6111yescharacteristic
279C2×C8C22⋊Dic6111yescharacteristic
280C2×C8C22⋊Dic6111yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C42—41no—
293C42—1222no—
294C42—1222no—
295C42D4×D6111yescharacteristic
296C42#1175822411yescharacteristic
297C42C22×Dic6111yescharacteristic
298C42C2×C22⋊D6111yescharacteristic
299C23—41no—
300C23#1451974811yescharacteristic
301Q8—41no—
302Q8—41no—
303Q8—81no—
304Q8—81no—
305D4—41no—
306D4—41no—
307D4—81no—
308D4—81no—
309D4—82no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—82no—
318C2×C4—1224no—
319C2×C4—2442no—
320C2×C4—2442no—
321C2×C4C2×C22⋊Dic6111yescharacteristic
322C2×C4D4×Dic6111yescharacteristic
323C2×C4D4×Dic6111yescharacteristic
324C2×C4#1451974811yescharacteristic
325C2×C4#1451976411yescharacteristic
326C2×C4#1451977311yescharacteristic
327C8—21no—
328C8—21no—
329C8—1224no—
330C8—4881no—
331C8—4881no—
332C8#1451970412yes—
333C8#1451970412yes—
334C22—21no—
335C22—21no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22#1683966511yescharacteristic
340C22#1684002212yes—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—81no—
347C4—81no—
348C4—81no—
349C4—2444no—
350C4C2×#1451970411yescharacteristic
351C4C2×#1451970411yescharacteristic
352C2—22no—
353C2—161no—
354C2#2144664811yescharacteristic
355C2#2144671811yescharacteristic
356C2#2144672411yescharacteristic
subgroupG/H
centerC22#16839665
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#3167C2
socleC22×C61#390
prime radicalC61×#532C22
cosocleC2×C42×C61C23
hypercenter#3167D61
hypocenterC61#9761
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #21445505C2semidirect
2. #21444582C2semidirect
3. #21444434C2semidirect
5. #21444331C2semidirect
22. C61×#498C4semidirect
23. C61×#492C4semidirect
24. C61×#487C4semidirect
25. C61×#486C4semidirect
188. #498Dic61semidirect
189. #492Dic61semidirect
190. #487Dic61semidirect
191. #486Dic61semidirect
209. C61#9761semidirect

Maximal subgroups

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

subgroupindex
#214455052
#214445822
#214444342
#214444302
#214443312
#214443272
C61×#31672
#976161