Finite Group Databasebeta

← #30614650

Subgroups — #30614650

Order 15616.

906 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840786C212yesmaximal
2#21461619C212yesmaximal, complemented
3#21461618C212yesmaximal, complemented
4#21461611C212yesmaximal, complemented
5C4⋊C4×#92420C212yesmaximal, complemented
6#21461549C211yesmaximal, characteristic
7#21461549C211yesmaximal, characteristic
8C4×C22⋊#92420C212yesmaximal
9#21448298C211yesmaximal, characteristic
10#16840786C4116yescomplemented
11C2×#14519844C2212yes—
12C2×C4⋊#92420C2212yes—
13C2×C22⋊#92420C2212yes—
14C2×C22⋊#92420C2212yes—
15C2×#14519832C2211yescharacteristic
16C2×#14519832C2211yescharacteristic
17C2×#14519832C2212yes—
18C2×C4×#92420C2212yes—
19C2×C4×#92420C2212yes—
20C2×C4×#92420C2211yescharacteristic
21C2×C4×#92420C2211yescharacteristic
22C2×C4×#92420C2212yes—
23#16840628C2212yes—
24#16840628C2212yes—
25#16840628C2212yes—
26#16840628C2212yes—
27C2×#14519733C2211yescharacteristic
28C2×#14519733C2211yescharacteristic
29C22×C4×Dic61C2211yescharacteristic
30#16839659C412yes—
31#16839657C412yes—
32#16839637C411yescharacteristic
33#16839635C411yescharacteristic
34#16839634C411yescharacteristic
35C4⋊C4×Dic61C2212yes—
36#16839613C2212yes—
37C61×#385C411yescharacteristic
38C22×#92420C2312yes—
39C22×#92420C2312yes—
40C22×#92420C2312yes—
41C22×#92420C2312yes—
42#14519844C2×C418yescomplemented
43C4⋊#92420C2×C418yescomplemented
44C22⋊#92420C2×C418yes—
45C22⋊#92420C2×C418yes—
46#14519832—41no—
47#14519832—41no—
48#14519832C2×C418yescomplemented
49C4×#92420—42no—
50C4×#92420—42no—
51C4×#92420—41no—
52C4×#92420—41no—
53C4×#92420C2×C418yescomplemented
54C23×Dic61C2×C411yescharacteristic
55C23×Dic61C2311yescharacteristic
56#14519733—41no—
57#14519733—41no—
58#14519733C2×C418yes—
59C2×C4⋊Dic61C2×C412yes—
60C2×#11758196C2×C412yes—
61C2×#11758196C2×C412yes—
62C2×#11758196C2×C411yescharacteristic
63C2×#11758196C2×C411yescharacteristic
64C2×#11758196C2×C412yes—
65C2×C22⋊Dic61C2×C411yescharacteristic
66C2×C22⋊Dic61C2×C411yescharacteristic
67C2×C4×Dic61—41no—
68C2×C4×Dic61—41no—
69C2×C4×Dic61C2×C412yes—
70C2×C4×Dic61C2312yes—
71C2×C4×Dic61C2311yescharacteristic
72C2×C4×Dic61C2311yescharacteristic
73C2×C4×Dic61C2312yes—
74#14519684C2×C412yes—
75#14519684C2×C412yes—
76#14519684C2×C412yes—
77C23×C4×C61C2×C411yescharacteristic
78C2×C4⋊C4×C61C2×C412yes—
79C2×C22⋊C4×C61C2×C411yescharacteristic
80C2×C22⋊C4×C61C2×C411yescharacteristic
81#95×C61C2×C412yes—
82C24×C61C22×C411yescharacteristic
83C22×C4×C61—24no—
84C22×C4×C61—41no—
85C22×C4×C61C22×C411yescharacteristic
86C22×C4×C61C22×C411yescharacteristic
87C22×C4×C61C22×C412yes—
88C22×C4×C61C22×C412yes—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—22no—
92C22×Dic61—24no—
93C22×Dic61—41no—
94C22×Dic61C22×C411yescharacteristic
95C22×Dic61C22×C411yescharacteristic
96C22×Dic61C22×C411yescharacteristic
97C22×Dic61C22×C412yes—
98C22×Dic61C22×C414yes—
99C22×Dic61C22×C412yes—
100C22×Dic61C2411yescharacteristic
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×#92420C22×C414yes—
110C2×#92420C22×C414yes—
111C2×#92420C22×C414yes—
112C2×#92420C22×C414yes—
113C4⋊C4×C61—24no—
114C22⋊C4×C61—41no—
115C22⋊C4×C61C4214yes—
116C22⋊Dic61—41no—
117C22⋊Dic61C4214yes—
118C4⋊Dic61—24no—
119#11758196—24no—
120#11758196—24no—
121#11758196—41no—
122#11758196—41no—
123#11758196C4218yes—
124C4×Dic61—22no—
125C4×Dic61—22no—
126C4×Dic61—21no—
127C4×Dic61—21no—
128C4×Dic61—21no—
129C4×Dic61—21no—
130C4×Dic61—42no—
131C4×Dic61C22×C414yes—
132C4×Dic61C22×C414yes—
133C23×C61—21no—
134C23×C61—21no—
135C23×C61—22no—
136C23×C61—22no—
137C23×C61C2×C4212yes—
138C23×C61C23×C411yescharacteristic
139C2×C4×C61—22no—
140C2×C4×C61—22no—
141C2×C4×C61—22no—
142C2×C4×C61—22no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—22no—
146C2×C4×C61—22no—
147C2×C4×C61—44no—
148C2×C4×C61C2×C4214yes—
149C2×C4×C61#13114yes—
150C2×Dic61—22no—
151C2×Dic61—21no—
152C2×Dic61—21no—
153C2×Dic61—22no—
154C2×Dic61—22no—
155C2×Dic61—22no—
156C2×Dic61—24no—
157C2×Dic61—24no—
158C2×Dic61—21no—
159C2×Dic61—21no—
160C2×Dic61—22no—
161C2×Dic61—22no—
162C2×Dic61—44no—
163C2×Dic61C2×C4214yes—
164C2×Dic61C2×C4212yes—
165C2×Dic61#13114yes—
166C2×Dic61C23×C412yes—
167C2×Dic61H(5,𝔽2)11yescharacteristic
168C2×Dic61H(5,𝔽2)11yescharacteristic
169C2×Dic612−1+412yes—
170#92420—24no—
171#92420—24no—
172#92420—24no—
173#92420—24no—
174#20540—611nomaximal, Sylow
175C22×C61—24no—
176C22×C61—22no—
177C22×C61—41no—
178C22×C61—41no—
179C22×C61—42no—
180C22×C61C22×C4211yescharacteristic
181C22×C61#51811yescharacteristic
182C22×C61#51911yescharacteristic
183C22×C61C2×#13112yes—
184C22×C61C2×#13112yes—
185C4×C61—24no—
186C4×C61—24no—
187C4×C61—41no—
188C4×C61—41no—
189Dic61—24no—
190Dic61—24no—
191Dic61—22no—
192Dic61—41no—
193Dic61—41no—
194Dic61#52012yes—
195Dic61#52012yes—
196C2×#433—612no—
197#2296—611no—
198#1771—612no—
199#1771—612no—
200C8×C4⋊C4—612no—
201C8×C4⋊C4—612no—
202#1750—611no—
203#1750—611no—
204C4×C22⋊C8—612no—
205C2×C61—42no—
206C2×C61#227411yescharacteristic
207C2×C61C4×#13112yes—
208C2×C61#297711yescharacteristic
209C2×C61#297711yescharacteristic
210C2×C61#298612yes—
211C2×#117—611no—
212C2×#117—611no—
213C22×C42—611no—
214#433—6116no—
215C2×C4⋊C8—612no—
216C2×C4⋊C8—612no—
217C2×C22⋊C8—612no—
218C2×C22⋊C8—612no—
219C2×#97—612no—
220C2×#97—611no—
221C2×#97—611no—
222C2×C4×C8—612no—
223C2×C4×C8—612no—
224C2×C4×C8—612no—
225C2×C4×C8—611no—
226C2×C4×C8—611no—
227#389—612no—
228#389—612no—
229#385—611no—
230#385—611no—
231#385—611no—
232#385#692311yescharacteristic
233C4×C4⋊C4—612no—
234C4×C4⋊C4—612no—
235#336—612no—
236#336—612no—
237#336—612no—
238#336—612no—
239C61#2054011yescharacteristic, complemented, Sylow
240C23×C4—611no—
241C23×C4—611no—
242C23×C4C2×#692311yescharacteristic
243C22×C8—612no—
244C22×C8—612no—
245C22×C8—612no—
246C22×C8—612no—
247#117—618no—
248#117—2441no—
249#117—2441no—
250C2×C4⋊C4—612no—
251C2×C4⋊C4—612no—
252C2×C4⋊C4—612no—
253C2×C4⋊C4—612no—
254C2×C4⋊C4—611no—
255C2×C4⋊C4—611no—
256C2×C4⋊C4C2×#692312yes—
257C2×C22⋊C4—611no—
258C2×C22⋊C4—611no—
259C2×C22⋊C4C2×#692311yescharacteristic
260C2×C22⋊C4C2×#692311yescharacteristic
261C2×C42—612no—
262C2×C42—611no—
263C2×C42—611no—
264C2×C42—612no—
265C2×C42—612no—
266C2×C42—2441no—
267C2×C42—2441no—
268C4⋊C8—618no—
269C4⋊C8—618no—
270C22⋊C8—618no—
271C22⋊C8—618no—
272#97—618no—
273#97—2441no—
274#97—2441no—
275C4×C8—618no—
276C4×C8—2442no—
277C4×C8—2442no—
278C4×C8—2441no—
279C4×C8—2441no—
280#95—612no—
281#95—612no—
282#95—612no—
283#95C2×#692312yes—
284C24C22×#692311yescharacteristic
285C22×C4—24no—
286C22×C4—41no—
287C22×C4—611no—
288C22×C4—614no—
289C22×C4—612no—
290C22×C4—612no—
291C22×C4—611no—
292C22×C4—611no—
293C22×C4—611no—
294C22×C4—1221no—
295C22×C4—1221no—
296C22×C4—1222no—
297C22×C4—1224no—
298C22×C4—2441no—
299C22×C4C22×#692311yescharacteristic
300C22×C4C22×#692311yescharacteristic
301C22×C4C22×#692312yes—
302C22×C4C22×#692312yes—
303C2×C8—614no—
304C2×C8—614no—
305C2×C8—614no—
306C2×C8—614no—
307C2×C8—1222no—
308C2×C8—1222no—
309C2×C8—1222no—
310C2×C8—1222no—
311C2×C8—1222no—
312C2×C8—1222no—
313C2×C8—1222no—
314C2×C8—1222no—
315C4⋊C4—24no—
316C4⋊C4—618no—
317C4⋊C4—1224no—
318C4⋊C4—1224no—
319C4⋊C4—1224no—
320C4⋊C4—2441no—
321C4⋊C4—2441no—
322C22⋊C4—41no—
323C22⋊C4—614no—
324C22⋊C4—2441no—
325C22⋊C4C4×#692314yes—
326C42—614no—
327C42—614no—
328C42—1222no—
329C42—1222no—
330C42—1221no—
331C42—1221no—
332C42—1221no—
333C42—1221no—
334C42—2442no—
335C23—21no—
336C23—21no—
337C23—22no—
338C23—22no—
339C23C2×C4×#692312yes—
340C23C23×#692311yescharacteristic
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—44no—
350C2×C4—612no—
351C2×C4—611no—
352C2×C4—611no—
353C2×C4—612no—
354C2×C4—614no—
355C2×C4—612no—
356C2×C4—614no—
357C2×C4—1224no—
358C2×C4—1224no—
359C2×C4—1222no—
360C2×C4—1221no—
361C2×C4—1221no—
362C2×C4—1222no—
363C2×C4—1222no—
364C2×C4—1222no—
365C2×C4—1221no—
366C2×C4—1221no—
367C2×C4—1222no—
368C2×C4—1222no—
369C2×C4—2444no—
370C2×C4C2×C4×#692314yes—
371C2×C4#1451986414yes—
372C8—1224no—
373C8—1224no—
374C8—1224no—
375C8—1224no—
376C22—24no—
377C22—22no—
378C22—41no—
379C22—41no—
380C22—42no—
381C22C22×C4×#692311yescharacteristic
382C22#1684082911yescharacteristic
383C22#1684083011yescharacteristic
384C22C2×#1451986412yes—
385C22D61⋊#13112yes—
386C4—24no—
387C4—24no—
388C4—41no—
389C4—41no—
390C4—612no—
391C4—612no—
392C4—1224no—
393C4—1224no—
394C4—1222no—
395C4—2441no—
396C4—2441no—
397C2—42no—
398C2#2146225711yescharacteristic
399C2C4×#1451986412yes—
400C2#2146269111yescharacteristic
401C2#2146269311yescharacteristic
402C2#2146272812yes—
subgroupG/H
center, FrattiniC23C23×#6923
commutatorC22×C61C22×C42
FittingC61×#385C4
socleC23×C61C23×C4
prime radicalC24×C61C22×C4
cosocleC22×Dic61C24
hypercenter#385#6923
hypocenterC61#20540
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #21461619C2semidirect
3. #21461618C2semidirect
4. #21461611C2semidirect
5. C4⋊C4×#92420C2semidirect
10. #16840786C4semidirect
42. #14519844C2×C4semidirect
43. C4⋊#92420C2×C4semidirect
48. #14519832C2×C4semidirect
53. C4×#92420C2×C4semidirect
239. C61#20540semidirect

Maximal subgroups

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

subgroupindex
C2×#168407862
#214616192
#214616182
#214616112
C4⋊C4×#924202
#214615492
C4×C22⋊#924202
#214482982
#2054061