Finite Group Databasebeta

← #30619280

Subgroups — #30619280

Order 15616.

578 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462611C212yesmaximal
2#21462006C212yesmaximal
3#21461887C212yesmaximal
4#21461886C212yesmaximal
5#21461885C212yesmaximal
6#21461884C212yesmaximal
7#21461743C211yesmaximal, characteristic
8#21461537C211yesmaximal, characteristic
9#21456332C211yesmaximal, characteristic
10#16840814—22no—
11#16840814—22no—
12#16840812—22no—
13#16840812—22no—
14#16840769C2212yes—
15#16840769C2212yes—
16#16840766C2212yes—
17#16840727C2212yes—
18#16840727C2212yes—
19#16840721C2211yescharacteristic
20#16840698C2211yescharacteristic
21#16840638C2212yes—
22#16840638C2211yescharacteristic
23#16840638C2211yescharacteristic
24#16840632C2212yes—
25#16840632C2212yes—
26#16840632C2212yes—
27#16840632C2212yes—
28(C2×#6923)⋊C8C2212yes—
29(C2×#6923)⋊C8C2212yes—
30#16840318C2212yes—
31#16840166C411yescharacteristic, complemented
32#16840166C411yescharacteristic, complemented
33#16840164C411yescharacteristic, complemented
34#16840161C412yescomplemented
35#16840159C412yescomplemented
36#16839799C2211yescharacteristic
37#16839799C2211yescharacteristic
38#16839797C2212yes—
39C4×C8×D61C2211yescharacteristic
40C61×#499C411yescharacteristic, complemented
41C2×#11758213C2312yes—
42C2×#11758213C2312yes—
43C2×D61⋊C8C2312yes—
44C2×C4×#6923C2312yes—
45#14519844—22no—
46#14519844—22no—
47C4⋊#92420—22no—
48C4⋊#92420—22no—
49C4⋊#92420—22no—
50C4⋊#92420—22no—
51#14519832—22no—
52#14519832—22no—
53#14519832D411yescharacteristic
54#14519832D412yes—
55#14519832D411yescharacteristic
56C4×#92420—22no—
57C4×#92420—22no—
58(C2×#6923)⋊C4—21no—
59(C2×#6923)⋊C4—21no—
60C2×C8×D61C2311yescharacteristic
61C2×C8×D61C2311yescharacteristic
62#14519776C2×C412yes—
63#14519776C2×C412yes—
64#14519769C2×C412yes—
65#14519755—22no—
66#14519755—22no—
67#14519742—22no—
68#14519742—22no—
69#14519735C2312yes—
70#14519735C2312yes—
71C42×D61C2311yescharacteristic
72#14519711C2×C411yescharacteristic
73#14519711C2×C411yescharacteristic
74#14519710C2×C412yes—
75#14519710C2×C412yes—
76#14519709C2×C412yes—
77#14519708C2×C412yes—
78#14519708C2×C412yes—
79C4×#92418C2×C411yescharacteristic
80C8×Dic61C2×C411yescharacteristic
81C8×Dic61C2×C411yescharacteristic
82#125×C61C2×C412yes—
83#107×C61C2×C411yescharacteristic
84#107×C61C2×C411yescharacteristic
85#106×C61C2×C412yes—
86C4×C8×C61C2×C411yescharacteristic
87C22×#6923—22no—
88C2×C4×D61#4111yescharacteristic
89C2×C4×D61#4111yescharacteristic
90C2×C4×D61C2411yescharacteristic
91C2×Dic122—22no—
92C2×Dic122—22no—
93C2×#92420—22no—
94C2×#92420—22no—
95C2×#92420C2×D412yes—
96C2×#92420C2×D412yes—
97C4×#6923—22no—
98C4×#6923—22no—
99#11758213—22no—
100#11758213—22no—
101#11758213—22no—
102#11758213—22no—
103D61⋊C8—22no—
104D61⋊C8—22no—
105C2×C8×C61C22×C411yescharacteristic
106C2×C8×C61C22×C411yescharacteristic
107C4⋊C4×C61—22no—
108C4⋊C4×C61—22no—
109C4⋊C4×C61C22×C412yes—
110C4⋊C4×C61C22×C412yes—
111C42×C61C22×C411yescharacteristic
112#11758198—22no—
113#11758198—22no—
114C4⋊Dic61C22×C412yes—
115C4⋊Dic61C22×C412yes—
116#11758196—22no—
117#11758196—22no—
118#11758196—22no—
119#11758196—22no—
120C4×Dic61—22no—
121C4×Dic61—22no—
122C4×Dic61C22×C411yescharacteristic
123C4×Dic61C2×D411yescharacteristic
124C4×Dic61C2×D411yescharacteristic
125C2×#92418C22×C411yescharacteristic
126C2×#92418C22×C411yescharacteristic
127C8×D61—22no—
128C8×D61—22no—
129C22×D61C2×#4111yescharacteristic
130C2×#6923—42no—
131C2×#6923—42no—
132C2×#6923—42no—
133C2×C4×C61—22no—
134C2×C4×C61—22no—
135C2×C4×C61C4×D411yescharacteristic
136C2×C4×C61C4×D411yescharacteristic
137C2×C4×C61C23×C411yescharacteristic
138C2×Dic61—22no—
139C2×Dic61—22no—
140C2×Dic61C4×D411yescharacteristic
141C2×Dic61C4×D411yescharacteristic
142C2×Dic61C22×D411yescharacteristic
143C4×D61—41no—
144C4×D61—41no—
145C4×D612−1+412yes—
146C4×D612−1+412yes—
147Dic122—82no—
148Dic122—82no—
149#92420—22no—
150#92420—22no—
151#92420—42no—
152#92420—42no—
153C8×C61—21no—
154C8×C61—21no—
155#92418—21no—
156#92418—21no—
157#22258—611nomaximal, Sylow
158C22×C61C2×C4×D411yescharacteristic
159C2×D61—21no—
160C2×D61—21no—
161C2×D61#54112yes—
162#6923—82no—
163C4×C61—21no—
164C4×C61—21no—
165C4×C61—42no—
166C4×C61—42no—
167C4×C61#51911yescharacteristic
168C4×C61#51911yescharacteristic
169Dic61—21no—
170Dic61—21no—
171Dic61—42no—
172Dic61—42no—
173Dic61#53612yes—
174#3160—611no—
175#2961—612no—
176#1995—612no—
177#1922—612no—
178#1922—612no—
179#1922—612no—
180#1922—612no—
181#1837—611no—
182#1746—611no—
183C2×C61#232711yescharacteristic
184C2×C61#295711yescharacteristic
185C2×C61#295711yescharacteristic
186D61—22no—
187#533—612no—
188#500—611no—
189#500—611no—
190#499—612no—
191#499—611no—
192#499#692311yescharacteristic, complemented
193#498—612no—
194#446—1222no—
195#446—1222no—
196#446—1222no—
197#446—1222no—
198#427—611no—
199#427—611no—
200#427—612no—
201#424—612no—
202#424—612no—
203#423—612no—
204#405—612no—
205#405—612no—
206C2×#97—611no—
207C2×C4×C8—611no—
208#376—611no—
209#341—612no—
210#341—611no—
211#341—611no—
212#339—612no—
213#339—612no—
214#339—612no—
215#339—612no—
216#336—612no—
217#336—612no—
218C61#2225811yescharacteristic, complemented, Sylow
219C2×M16—612no—
220C2×M16—612no—
221C22×C8—612no—
222C22×C8—611no—
223C22×C8—611no—
224C4⋊Q8—612no—
225C4⋊Q8—612no—
226#125—612no—
227#125C2×#692312yes—
228C22⋊Q8—1222no—
229C22⋊Q8—1222no—
230C4×Q8—1222no—
231C4×Q8—1222no—
232#117—612no—
233#117—612no—
234C2×C42—611no—
235C2×C42—612no—
236#107—611no—
237#107—611no—
238#107—612no—
239#107—612no—
240#107C2×#692311yescharacteristic
241#107C2×#692311yescharacteristic
242#106—612no—
243#106—612no—
244#106—612no—
245#106C2×#692312yes—
246C4⋊C8—1222no—
247C4⋊C8—1222no—
248C4⋊C8—1222no—
249C4⋊C8—1222no—
250C4⋊C8—1222no—
251C4⋊C8—1222no—
252#97—611no—
253#97—612no—
254#97—611no—
255#97—1222no—
256#97—1222no—
257C4×C8—611no—
258C4×C8—611no—
259C4×C8—611no—
260C4×C8—1222no—
261C4×C8—1222no—
262C4×C8C2×#692311yescharacteristic
263#95—1221no—
264#95—1221no—
265C2×Q8—1222no—
266C2×Q8—1222no—
267C22×C4—611no—
268C22×C4—611no—
269C22×C4—611no—
270C22×C4—1222no—
271M16—1222no—
272M16—1222no—
273M16—1222no—
274M16—1222no—
275C2×C8—612no—
276C2×C8—612no—
277C2×C8—611no—
278C2×C8—611no—
279C2×C8—1222no—
280C2×C8—1222no—
281C2×C8—1222no—
282C2×C8—1222no—
283C2×C8—1222no—
284C2×C8—1222no—
285C2×C8C22×#692311yescharacteristic
286C2×C8C22×#692311yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—612no—
290C4⋊C4—612no—
291C4⋊C4—1222no—
292C4⋊C4—1222no—
293C4⋊C4—1222no—
294C4⋊C4—1222no—
295C4⋊C4C22×#692312yes—
296C4⋊C4C22×#692312yes—
297C22⋊C4—1222no—
298C22⋊C4—1222no—
299C42—611no—
300C42—611no—
301C42—611no—
302C42—1222no—
303C42—1222no—
304C42—1222no—
305C42—1222no—
306C42C22×#692311yescharacteristic
307C23—611no—
308Q8—4882no—
309Q8—4882no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—611no—
313C2×C4—612no—
314C2×C4—612no—
315C2×C4—611no—
316C2×C4—611no—
317C2×C4—1222no—
318C2×C4—1222no—
319C2×C4—2441no—
320C2×C4—2441no—
321C2×C4—2442no—
322C2×C4—2442no—
323C2×C4—2442no—
324C2×C4D4×#692311yescharacteristic
325C2×C4D4×#692311yescharacteristic
326C2×C4C23×#692311yescharacteristic
327C8—21no—
328C8—21no—
329C8—1222no—
330C8—1222no—
331C8—1221no—
332C8—1221no—
333C8—2442no—
334C8—2442no—
335C22—612no—
336C22—1221no—
337C22—1221no—
338C22C2×D4×#692311yescharacteristic
339C4—21no—
340C4—21no—
341C4—42no—
342C4—42no—
343C4—612no—
344C4—1221no—
345C4—1221no—
346C4—4882no—
347C4—2442no—
348C4—2442no—
349C4#1684083011yescharacteristic
350C4#1684083011yescharacteristic
351C2—1222no—
352C2#2146231011yescharacteristic
353C2#2146259611yescharacteristic
354C2#2146259811yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#499C4
socleC22×C61C2×C4×D4
prime radicalC22×D61C2×#41
cosocleC2×C4×D61C24
hypercenter#499#6923
hypocenterC61#22258
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #499

Complements

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

NH ≅ G/Ntype
31. #16840166C4semidirect
32. #16840166C4semidirect
33. #16840164C4semidirect
34. #16840161C4semidirect
35. #16840159C4semidirect
40. C61×#499C4semidirect
192. #499#6923semidirect
218. C61#22258semidirect

Maximal subgroups

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

subgroupindex
#214626112
#214620062
#214618872
#214618862
#214618852
#214618842
#214617432
#214615372
#214563322
#2225861