Finite Group Databasebeta

← #30620745

Subgroups — #30620745

Order 15616.

780 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840787C212yesmaximal
2#21461921C212yesmaximal, complemented
3#21461876C212yesmaximal, complemented
4#21461875C212yesmaximal, complemented
5#21461713C212yesmaximal
6#21461676C211yesmaximal, characteristic
7#21461619C212yesmaximal, complemented
8#21461546C211yesmaximal, characteristic
9#21448224C211yesmaximal, characteristic
10#16840787—28no—
11C2×#14519844C2212yes—
12C2×#14519844C2212yes—
13C2×C4⋊#92420C2212yes—
14C2×#14519834C2211yescharacteristic
15C2×#14519834C2212yes—
16C2×#14519834C2211yescharacteristic
17C2×#14519832C2212yes—
18C2×#14519832C2212yes—
19C2×C4×#92420C2212yes—
20#16840628C2212yes—
21#16840628C2212yes—
22#16840628C2212yes—
23#16840628C2211yescharacteristic
24#16840628C2212yes—
25#16840628C2211yescharacteristic
26#16840628C2212yes—
27C2×#14519735C2212yes—
28C2×C42×D61C2211yescharacteristic
29#16839632C411yescharacteristic
30#16839627C412yes—
31#16839619C412yes—
32#16839619C411yescharacteristic
33#16839619C411yescharacteristic
34C4⋊C4×Dic61C2211yescharacteristic
35C4⋊C4×Dic61C2212yes—
36C4⋊C4×Dic61C2211yescharacteristic
37C61×#384C411yescharacteristic
38C22×#92420C2312yes—
39C22×#92420C2312yes—
40C22×#92420C2312yes—
41C22×#92420C2312yes—
42#14519844—24no—
43#14519844—24no—
44C4⋊#92420—24no—
45#14519834—24no—
46#14519834—22no—
47#14519834D414yes—
48#14519832—24no—
49#14519832—24no—
50C4×#92420—24no—
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54#14519735—24no—
55#14519735—24no—
56C2×C4⋊Dic61C2×C412yes—
57C2×C4⋊Dic61C2×C412yes—
58C2×C4⋊Dic61C2×C411yescharacteristic
59C2×C4⋊Dic61C2×C411yescharacteristic
60C2×#11758198C2×C412yes—
61C2×#11758198C2×C412yes—
62C2×C4×Dic61C2×C411yescharacteristic
63C2×C4×Dic61C2312yes—
64C2×C4×Dic61C2311yescharacteristic
65C2×C4×Dic61C2312yes—
66C2×C4×Dic61C2311yescharacteristic
67C42×D61—24no—
68#14519684C2×C412yes—
69#14519684C2×C412yes—
70#14519684C2×C412yes—
71#14519684C2×C412yes—
72C2×C4⋊C4×C61C2×C412yes—
73C2×C4⋊C4×C61C2×C411yescharacteristic
74C2×C4⋊C4×C61C2×C411yescharacteristic
75C2×C4⋊C4×C61C2×C412yes—
76C2×C42×C61C2×C411yescharacteristic
77C23×D61C22×C411yescharacteristic
78C22×C4×C61C22×C412yes—
79C22×C4×C61C22×C412yes—
80C22×C4×C61C22×C411yescharacteristic
81C22×C4×C61C22×C411yescharacteristic
82C22×C4×C61C22×C411yescharacteristic
83C22×Dic61C22×C411yescharacteristic
84C22×Dic61C22×C412yes—
85C22×Dic61C22×C412yes—
86C22×Dic61C22×C411yescharacteristic
87C22×Dic61C2411yescharacteristic
88C2×C4×D61—22no—
89C2×C4×D61—22no—
90C2×C4×D61—22no—
91C2×C4×D61—22no—
92C2×C4×D61—22no—
93C2×C4×D61—41no—
94C2×C4×D61—41no—
95C2×C4×D61—41no—
96C2×C4×D61C2×D412yes—
97C2×C4×D61#4112yes—
98C2×#92420—22no—
99C2×#92420—22no—
100C2×#92420—22no—
101C2×#92420—22no—
102C2×#92420—22no—
103C2×#92420—22no—
104C2×#92420—22no—
105C2×#92420—22no—
106C2×#92420—22no—
107C2×#92420—22no—
108C2×#92420—22no—
109C2×#92420C2×D414yes—
110C4⋊C4×C61—24no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—41no—
113C4⋊C4×C61—42no—
114C42×C61—22no—
115#11758198—24no—
116#11758198—24no—
117C4⋊Dic61—24no—
118C4⋊Dic61—24no—
119C4⋊Dic61—41no—
120C4⋊Dic61—41no—
121C4×Dic61—22no—
122C4×Dic61—22no—
123C4×Dic61—22no—
124C4×Dic61—22no—
125C4×Dic61—22no—
126C4×Dic61—22no—
127C4×Dic61#4114yes—
128C23×C61C23×C411yescharacteristic
129C22×D61—21no—
130C22×D61—21no—
131C22×D61—21no—
132C22×D61—21no—
133C22×D61—21no—
134C22×D61—21no—
135C22×D61C4×D412yes—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—22no—
143C2×C4×C61—22no—
144C2×C4×C61—22no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61#11712yes—
149C2×C4×C61C4×D412yes—
150C2×C4×C61#13112yes—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C2×Dic61—21no—
154C2×Dic61—22no—
155C2×Dic61—22no—
156C2×Dic61—22no—
157C2×Dic61—22no—
158C2×Dic61—22no—
159C2×Dic61—22no—
160C2×Dic61—21no—
161C2×Dic61—21no—
162C2×Dic61#11712yes—
163C2×Dic61C22×D411yescharacteristic
164C2×Dic61C2×#4112yes—
165C2×Dic61C2×#4111yescharacteristic
166C2×Dic612−1+412yes—
167C4×D61—24no—
168C4×D61—24no—
169C4×D61—44no—
170C4×D61—44no—
171#92420—24no—
172#92420—42no—
173#92420—42no—
174#92420—42no—
175#22602—611nomaximal, Sylow
176C22×C61C2×#11711yescharacteristic
177C22×C61C2×C4×D411yescharacteristic
178C22×C61#51911yescharacteristic
179C22×C61C2×#13111yescharacteristic
180C22×C61#56811yescharacteristic
181C22×C61#56811yescharacteristic
182C22×C61#56811yescharacteristic
183C2×D61—22no—
184C2×D61—22no—
185C2×D61—22no—
186C2×D61—41no—
187C2×D61—41no—
188C2×D61—41no—
189C2×D61—41no—
190C4×C61—22no—
191C4×C61—22no—
192C4×C61—22no—
193C4×C61—42no—
194C4×C61—42no—
195Dic61—22no—
196Dic61—41no—
197Dic61—42no—
198Dic61—42no—
199Dic61#54712yes—
200Dic61#55212yes—
201C2×#433—612no—
202#2285—611no—
203#1940—612no—
204#1917—612no—
205#1916—612no—
206#1818—612no—
207#1797—611no—
208#1771—612no—
209#1749—611no—
210C2×C61#234411yescharacteristic
211C2×C61#292111yescharacteristic
212C2×C61#292411yescharacteristic
213C2×C61#293211yescharacteristic
214C2×C61#293511yescharacteristic
215C2×C61#298511yescharacteristic
216C2×C61#298711yescharacteristic
217D61—42no—
218C2×#117—612no—
219C22×C42—611no—
220#433—1228no—
221C2×C4⋊C8—612no—
222C2×C4⋊C8—612no—
223C2×C4⋊C8—612no—
224C2×C22⋊C8—611no—
225C2×C22⋊C8—612no—
226C2×C22⋊C8—611no—
227C2×#97—612no—
228C2×#97—612no—
229C2×C4×C8—612no—
230#384—611no—
231#384#692311yescharacteristic
232#382—612no—
233#380—612no—
234#380—611no—
235#380—611no—
236C4×C4⋊C4—612no—
237C4×C4⋊C4—611no—
238C4×C4⋊C4—611no—
239#336—611no—
240#336—611no—
241#336—612no—
242#336—612no—
243#336—612no—
244#336—612no—
245#336—612no—
246C61#2260211yescharacteristic, complemented, Sylow
247C23×C4—611no—
248C23×C4—611no—
249C23×C4—611no—
250C22×C8—612no—
251C22×C8—612no—
252C22×C8—612no—
253C22×C8—612no—
254#117—1224no—
255#117—1224no—
256C2×C4⋊C4—612no—
257C2×C4⋊C4—612no—
258C2×C4⋊C4—611no—
259C2×C4⋊C4—611no—
260C2×C4⋊C4C2×#692312yes—
261C2×C4⋊C4C2×#692311yescharacteristic
262C2×C4⋊C4C2×#692311yescharacteristic
263C2×C4⋊C4C2×#692312yes—
264C2×C22⋊C4—612no—
265C2×C22⋊C4—612no—
266C2×C42—611no—
267C2×C42—612no—
268C2×C42—612no—
269C2×C42—611no—
270C2×C42—611no—
271C2×C42—1224no—
272C2×C42C2×#692311yescharacteristic
273C4⋊C8—1224no—
274C4⋊C8—1224no—
275C4⋊C8—1224no—
276C22⋊C8—614no—
277C22⋊C8—1222no—
278C22⋊C8—1224no—
279#97—1224no—
280#97—1224no—
281C4×C8—1224no—
282#95—612no—
283#95—612no—
284#95—612no—
285#95—612no—
286C24—611no—
287C22×C4—611no—
288C22×C4—611no—
289C22×C4—612no—
290C22×C4—612no—
291C22×C4—611no—
292C22×C4—612no—
293C22×C4—612no—
294C22×C4—1222no—
295C22×C4—1222no—
296C22×C4—1222no—
297C22×C4—1222no—
298C22×C4—1222no—
299C22×C4—2441no—
300C22×C4—2441no—
301C22×C4—2441no—
302C22×C4C22×#692312yes—
303C22×C4C22×#692312yes—
304C22×C4C22×#692311yescharacteristic
305C22×C4C22×#692311yescharacteristic
306C22×C4C22×#692311yescharacteristic
307C2×C8—614no—
308C2×C8—1222no—
309C2×C8—1222no—
310C2×C8—1222no—
311C2×C8—1222no—
312C2×C8—1222no—
313C2×C8—1222no—
314C2×C8—1222no—
315C2×C8—1222no—
316C2×C8—1222no—
317C2×C8—1222no—
318C2×C8—1222no—
319C4⋊C4—24no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—1224no—
324C4⋊C4—1224no—
325C4⋊C4—2441no—
326C4⋊C4—2441no—
327C22⋊C4—1224no—
328C22⋊C4—1224no—
329C42—22no—
330C42—614no—
331C42—1222no—
332C42—1222no—
333C42—1222no—
334C42—1222no—
335C42—1222no—
336C42—1222no—
337C23—612no—
338C23—1221no—
339C23—1221no—
340C23—1221no—
341C23—1221no—
342C23—1221no—
343C23—1221no—
344C23C23×#692311yescharacteristic
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—612no—
358C2×C4—611no—
359C2×C4—611no—
360C2×C4—612no—
361C2×C4—612no—
362C2×C4—1224no—
363C2×C4—1224no—
364C2×C4—1221no—
365C2×C4—1221no—
366C2×C4—1221no—
367C2×C4—1222no—
368C2×C4—1222no—
369C2×C4—1222no—
370C2×C4—1222no—
371C2×C4—1221no—
372C2×C4—1222no—
373C2×C4—1222no—
374C2×C4—1221no—
375C2×C4—2444no—
376C2×C4—2444no—
377C2×C4#1451985612yes—
378C2×C4D4×#692312yes—
379C2×C4#1451986512yes—
380C8—1224no—
381C8—2442no—
382C8—2442no—
383C8—2442no—
384C22—1222no—
385C22—1222no—
386C22—1222no—
387C22—2441no—
388C22—2441no—
389C22—2441no—
390C22—2441no—
391C22C2×#1451985611yescharacteristic
392C22C2×D4×#692311yescharacteristic
393C22#1684083011yescharacteristic
394C22C2×#1451986511yescharacteristic
395C22#1684084011yescharacteristic
396C22#1684084011yescharacteristic
397C22#1684084211yescharacteristic
398C4—22no—
399C4—22no—
400C4—22no—
401C4—42no—
402C4—42no—
403C4—612no—
404C4—612no—
405C4—1222no—
406C4—2442no—
407C4—2442no—
408C4—2441no—
409C2—2442no—
410C2#2146232711yescharacteristic
411C2#2146250111yescharacteristic
412C2#2146251011yescharacteristic
413C2#2146254811yescharacteristic
414C2#2146256511yescharacteristic
415C2#2146272711yescharacteristic
416C2#2146273311yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC61×#384C4
prime radicalC23×D61C22×C4
cosocleC22×Dic61C24
hypercenter#384#6923
hypocenterC61#22602
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
2. #21461921C2semidirect
3. #21461876C2semidirect
4. #21461875C2semidirect
7. #21461619C2semidirect
246. C61#22602semidirect

Maximal subgroups

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

subgroupindex
C2×#168407872
#214619212
#214618762
#214618752
#214617132
#214616762
#214616192
#214615462
#214482242
#2260261