Finite Group Databasebeta

← #29596953

Subgroups — #29596953

Order 15616.

582 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839995C211yesmaximal, characteristic
2C2×#16839799C211yesmaximal, characteristic, complemented
3C2×#16839563C211yesmaximal, characteristic, complemented
4C2×#16839541C211yesmaximal, characteristic, complemented
5#21443911C211yesmaximal, characteristic, complemented
6#21443910C211yesmaximal, characteristic, complemented
7C2×C61×#357C211yesmaximal, characteristic, complemented
8C22×C8×D61C2211yescharacteristic
9C2×#14519748—21no—
10C2×#14519735—21no—
11C22×#11758185C2211yescharacteristic
12#16839995—22no—
13#16839995C414yescomplemented
14#16839799—22no—
15#16839799—22no—
16C2×#14519711C2211yescharacteristic
17C2×#14519711C2211yescharacteristic
18C2×#14519708—21no—
19C2×#14519698—21no—
20#16839563—22no—
21#16839541—22no—
22C22×D8×C61C2211yescharacteristic
23C22×C16×C61C2211yescharacteristic
24C2×#107×C61C2211yescharacteristic
25C61×#357—22no—
26C22×C22⋊D61—41no—
27C22×C4×D61D411yescharacteristic
28C22×#92418D411yescharacteristic
29C2×C8×D61—21no—
30C2×C8×D61—21no—
31C2×C8×D61—21no—
32C2×C8×D61—21no—
33C2×C8×D61C2×C412yes—
34C2×C8×D61D412yes—
35#14519748—24no—
36#14519748—42no—
37#14519735—24no—
38#14519735—42no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61—21no—
41C2×C22⋊Dic61—41no—
42C2×#11758198—41no—
43C2×C4×Dic61—41no—
44C2×#11758185—21no—
45C2×#11758185—21no—
46C2×#11758185—21no—
47#14519711—21no—
48#14519711—21no—
49#14519711C2×C412yes—
50#14519711D412yescomplemented
51#14519708—24no—
52#14519698—24no—
53C22×D4×C61—21no—
54C2×D8×C61—21no—
55C2×D8×C61—21no—
56C2×D8×C61—21no—
57C2×D8×C61—21no—
58C2×D8×C61—21no—
59C2×D8×C61C2×C412yes—
60C22×C8×C61C2311yescharacteristic
61C2×C4⋊C4×C61—21no—
62C2×C16×C61—21no—
63C2×C16×C61—21no—
64C2×C16×C61—21no—
65#107×C61—21no—
66#107×C61D412yescomplemented
67C24×C61—41no—
68C23×D61D811yescharacteristic
69C2×D4×C61—22no—
70C2×D4×C61—41no—
71C2×D4×C61—41no—
72C2×D4×C61—41no—
73C2×D4×C61—41no—
74C2×D4×C61—41no—
75C22×C4×C61—41no—
76C22×C4×C61C2×D411yescharacteristic
77C2×C22⋊D61—44no—
78C2×C22⋊D61—82no—
79C2×C22⋊D61—82no—
80C22×Dic61—41no—
81C22×Dic61—41no—
82C22×Dic61D811yescharacteristic
83C2×C4×D61—21no—
84C2×C4×D61—21no—
85C2×C4×D61—21no—
86C2×C4×D61—21no—
87C2×C4×D61C22⋊C412yes—
88C2×C4×D61SD1612yes—
89D8×C61—22no—
90D8×C61—22no—
91D8×C61—41no—
92D8×C61—41no—
93C2×C8×C61C22×C411yescharacteristic
94C2×C8×C61C2×D411yescharacteristic
95C2×C8×C61C2×D411yescharacteristic
96C2×C8×C61C2×D411yescharacteristic
97C2×C8×C61#4111yescharacteristic
98C2×C8×C61#4111yescharacteristic
99C4⋊C4×C61—22no—
100C4⋊C4×C61—41no—
101C22⋊Dic61—44no—
102#11758198—44no—
103C4⋊Dic61—22no—
104C4⋊Dic61—22no—
105C4⋊Dic61—41no—
106C4⋊Dic61—41no—
107C4×Dic61—44no—
108C2×#92418—21no—
109C2×#92418—21no—
110C2×#92418C22⋊C412yes—
111C8×D61—22no—
112C8×D61—22no—
113C8×D61—22no—
114C8×D61—22no—
115C16×C61—41no—
116#11758185—41no—
117C23×C61—42no—
118C23×C61—81no—
119C23×C61—81no—
120C23×C61—81no—
121C23×C61—81no—
122C23×C61—81no—
123C23×C61—81no—
124C23×C61C2×D811yescharacteristic
125C22×D61—21no—
126C22×D61—21no—
127C22×D61—21no—
128C22×D61—21no—
129C22×D61—21no—
130C22×D61—21no—
131C22×D61#10212yes—
132D4×C61—42no—
133D4×C61—42no—
134D4×C61—81no—
135D4×C61—81no—
136C2×C4×C61—42no—
137C2×C4×C61—81no—
138C2×C4×C61—81no—
139C2×C4×C61C2×C22⋊C411yescharacteristic
140C2×C4×C61#12011yescharacteristic
141C2×C4×C61#12311yescharacteristic
142C2×C4×C61C2×SD1611yescharacteristic
143C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C61#13711yescharacteristic
145C22⋊D61—88no—
146C2×Dic61—21no—
147C2×Dic61—21no—
148C2×Dic61—42no—
149C2×Dic61—42no—
150C2×Dic61—81no—
151C2×Dic61—81no—
152C2×Dic61—81no—
153C2×Dic61—81no—
154C2×Dic61#10212yes—
155C4×D61—22no—
156C4×D61—22no—
157C4×D61—22no—
158C4×D61—22no—
159C8×C61C4×D411yescharacteristic
160C8×C61C4×D411yescharacteristic
161C8×C61#12111yescharacteristic
162C8×C61#12111yescharacteristic
163#92418—22no—
164#12992—611nomaximal, Sylow
165C22×C61—82no—
166C22×C61—82no—
167C22×C61—82no—
168C22×C61—161no—
169C22×C61—161no—
170C22×C61—161no—
171C22×C61—161no—
172C22×C61C2×#10211yescharacteristic
173C22×C61#44711yescharacteristic
174C22×C61#48011yescharacteristic
175C22×C61#50811yescharacteristic
176C22×C61#50811yescharacteristic
177C22×C61#50911yescharacteristic
178C22×C61#50911yescharacteristic
179C2×D61—22no—
180C2×D61—22no—
181C2×D61—22no—
182C2×D61—41no—
183C2×D61—41no—
184C2×D61—41no—
185C2×D61—41no—
186C4×C61—82no—
187C4×C61#38611yescharacteristic
188C4×C61#41711yescharacteristic
189C4×C61#44811yescharacteristic
190C4×C61#48111yescharacteristic
191Dic61—22no—
192Dic61—82no—
193Dic61—82no—
194C2×#466—611no—
195C2×#427—611no—
196C2×#357—611no—
197C2×#357D6111yescharacteristic, complemented
198C2×C22⋊C16—611no—
199#1380—611no—
200#1380—611no—
201C2×C61—162no—
202C2×C61#187411yescharacteristic
203C2×C61#213811yescharacteristic
204C2×C61#214011yescharacteristic
205C2×C61#218611yescharacteristic
206C2×C61#218611yescharacteristic
207C2×C61#223411yescharacteristic
208C2×C61#223411yescharacteristic
209D61—42no—
210C22×D8C2×D6111yescharacteristic
211C23×C8—611no—
212C2×#121—1221no—
213C2×#117—1221no—
214C22×C16—611no—
215C22×C16C2×D6111yescharacteristic
216#466—614no—
217#466—1222no—
218#427—1222no—
219#427—1222no—
220C2×#107—611no—
221C2×#107—611no—
222C2×#107C2×D6111yescharacteristic
223C2×#106—1221no—
224C2×#102—1221no—
225#357—22no—
226#357—1222no—
227C22⋊C16—1222no—
228C61#1299211yescharacteristic, complemented, Sylow
229C22×D4—21no—
230C22×D4—2441no—
231C23×C4—611no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8—21no—
235C2×D8—21no—
236C2×D8—21no—
237C2×D8C4×D6112yes—
238C22×C8—611no—
239C22×C8—612no—
240C22×C8—612no—
241C22×C8—1221no—
242C22×C8—1221no—
243C22×C8—1221no—
244C22×C8—1221no—
245C22×C8C22×D6111yescharacteristic
246#121—1224no—
247#121—2442no—
248#117—1224no—
249#117—2442no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—1221no—
252C2×C4⋊C4—1221no—
253C2×C22⋊C4—2441no—
254C2×C22⋊C4—2441no—
255C2×C42—2441no—
256C2×C16—21no—
257C2×C16—21no—
258C2×C16—21no—
259C2×C16—1221no—
260C2×C16—1221no—
261C2×C16—1221no—
262#107—21no—
263#107—612no—
264#107—612no—
265#107—1221no—
266#107—1221no—
267#107C22⋊D6112yescomplemented
268#106—1224no—
269#102—1224no—
270C24—41no—
271C24—611no—
272C2×D4—22no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—4882no—
279C2×D4—4882no—
280C2×D4—2444no—
281C22×C4—41no—
282C22×C4—611no—
283C22×C4—612no—
284C22×C4—612no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4—1221no—
288C22×C4—1221no—
289C22×C4—2441no—
290C22×C4—2441no—
291C22×C4D4×D6111yescharacteristic
292D8—22no—
293D8—22no—
294D8—41no—
295D8—41no—
296C2×C8—612no—
297C2×C8—1221no—
298C2×C8—1221no—
299C2×C8—1222no—
300C2×C8—1222no—
301C2×C8—1222no—
302C2×C8—1222no—
303C2×C8C2×C4×D6111yescharacteristic
304C2×C8#1175822211yescharacteristic
305C2×C8D4×D6111yescharacteristic
306C2×C8D4×D6111yescharacteristic
307C2×C8#1175822411yescharacteristic
308C2×C8C2×C22⋊D6111yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—1222no—
312C4⋊C4—1222no—
313C4⋊C4—2441no—
314C4⋊C4—2441no—
315C22⋊C4—2444no—
316C22⋊C4—2444no—
317C42—2444no—
318C16—41no—
319C16—2441no—
320C23—42no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23—612no—
328C23—1221no—
329C23—1221no—
330C23—1221no—
331C23—1221no—
332C23—1221no—
333C23—1221no—
334C23D8×D6111yescharacteristic
335D4—42no—
336D4—42no—
337D4—81no—
338D4—81no—
339D4—4888no—
340C2×C4—42no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—612no—
344C2×C4—1221no—
345C2×C4—1221no—
346C2×C4—1222no—
347C2×C4—1222no—
348C2×C4—1222no—
349C2×C4—1222no—
350C2×C4—4881no—
351C2×C4—4881no—
352C2×C4—4881no—
353C2×C4—4881no—
354C2×C4—2442no—
355C2×C4—2442no—
356C2×C4C22⋊C4×D6111yescharacteristic
357C2×C4#1451974511yescharacteristic
358C2×C4#1451975911yescharacteristic
359C2×C4SD16×D6111yescharacteristic
360C2×C4#1451980111yescharacteristic
361C2×C4#1451980611yescharacteristic
362C8—1222no—
363C8#1451973711yescharacteristic
364C8C4×C22⋊D6111yescharacteristic
365C8#1451974811yescharacteristic
366C8#1451975011yescharacteristic
367C22—82no—
368C22—82no—
369C22—82no—
370C22—161no—
371C22—161no—
372C22—161no—
373C22—161no—
374C22—1222no—
375C22—1222no—
376C22—1222no—
377C22—2441no—
378C22—2441no—
379C22—2441no—
380C22—2441no—
381C22#102×D6111yescharacteristic
382C22#1683989411yescharacteristic
383C22#1684007411yescharacteristic
384C22#1684019111yescharacteristic
385C22#1684019311yescharacteristic
386C22#1684019611yescharacteristic
387C22#1684019911yescharacteristic
388C4—82no—
389C4—1222no—
390C4—4882no—
391C4—4882no—
392C4#1683963811yescharacteristic
393C4#1683976011yescharacteristic
394C4#1683989911yescharacteristic
395C4#1684007911yescharacteristic
396C2—162no—
397C2—2442no—
398C2#2144613311yescharacteristic
399C2#2144752311yescharacteristic
400C2#2144753311yescharacteristic
401C2#2144773911yescharacteristic
402C2#2144774111yescharacteristic
403C2#2144800811yescharacteristic
404C2#2144800911yescharacteristic
subgroupG/H
centerC23D8×D61
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC2×C61×#357C2
socleC23×C61C2×D8
prime radicalC2×#16839995C2
cosocleC22×C8×C61C23
hypercenterC2×#357D61
hypocenterC61#12992
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#16839799C2semidirect
3. C2×#16839563C2semidirect
4. C2×#16839541C2semidirect
5. #21443911C2semidirect
6. #21443910C2semidirect
7. C2×C61×#357C2semidirect
13. #16839995C4semidirect
50. #14519711D4semidirect
66. #107×C61D4semidirect
197. C2×#357D61semidirect
228. C61#12992semidirect
267. #107C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399952
C2×#168397992
C2×#168395632
C2×#168395412
#214439112
#214439102
C2×C61×#3572
#1299261