Finite Group Databasebeta

← #29597405

Subgroups — #29597405

Order 15616.

510 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455400C211yesmaximal, characteristic
2C2×#16839573C212yesmaximal, complemented
3#21445844C211yesmaximal, characteristic, complemented
4#21443925C212yesmaximal, complemented
5C22×M32×C61C211yesmaximal, characteristic, complemented
6C2×#14519799C2211yescharacteristic
7#16840273—21no—
8#16840003C411yescharacteristic, complemented
9#16840003C411yescharacteristic, complemented
10#16839998C411yescharacteristic, complemented
11#16839998C411yescharacteristic, complemented
12#16839991—21no—
13#16839991—21no—
14#16839800C2211yescharacteristic
15C2×#14519709C2212yes—
16#16839759—21no—
17#16839622—21no—
18#16839573—24no—
19#16839525—21no—
20C23×C8×C61C2211yescharacteristic
21C2×M32×C61—21no—
22C2×M32×C61—24no—
23C2×M32×C61C2212yes—
24C22×C16×C61—21no—
25C2×#11758222—21no—
26#14519799—22no—
27#14519799—22no—
28C2×Dic244C2×C411yescharacteristic
29C2×#11758190—21no—
30C2×D488C2×C411yescharacteristic
31#14519760—41no—
32#14519756—21no—
33#14519756—21no—
34#14519751—21no—
35#14519751—21no—
36#14519746—41no—
37C4×C22⋊D61—41no—
38C4×C22⋊D61—41no—
39#14519736—21no—
40C2×C4⋊Dic61—22no—
41C2×C22⋊Dic61—41no—
42#14519711C2×C411yescharacteristic
43#14519711C2×C411yescharacteristic
44#14519709—21no—
45#14519709—22no—
46#14519709—22no—
47#14519702—21no—
48#14519702—21no—
49#14519699—21no—
50#14519699—21no—
51#14519684—41no—
52C23×C4×C61D411yescharacteristic
53C22×C8×C61—22no—
54C22×C8×C61—22no—
55C22×C8×C61C2×C411yescharacteristic
56C22×C8×C61C2×C411yescharacteristic
57C22×C8×C61D411yescharacteristic
58C22×C8×C61D411yescharacteristic
59C22×C8×C61D411yescharacteristic
60C22×C8×C61C2311yescharacteristic
61M32×C61—24no—
62M32×C61—22no—
63M32×C61D414yescomplemented
64C2×C16×C61—24no—
65C2×C16×C61—22no—
66C24×C61D811yescharacteristic
67C22×C4×C61—22no—
68C22×C4×C61—22no—
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61C22⋊C411yescharacteristic
71C22×C4×C61D811yescharacteristic
72C22×C4×C61SD1611yescharacteristic
73C22×C4×C61SD1611yescharacteristic
74C22×C4×C61C2×D411yescharacteristic
75C2×C22⋊D61—41no—
76C2×C22⋊D61—41no—
77C2×C22⋊D61—41no—
78C22×Dic61—42no—
79#11758222—42no—
80#11758222—42no—
81C2×D244—21no—
82C2×C4×D61—41no—
83C2×Dic122—21no—
84C2×C8×C61—22no—
85C2×C8×C61—22no—
86C2×C8×C61—22no—
87C2×C8×C61—21no—
88C2×C8×C61—21no—
89C2×C8×C61—21no—
90C2×C8×C61—21no—
91C2×C8×C61C22⋊C411yescharacteristic
92C2×C8×C61C22⋊C411yescharacteristic
93C2×C8×C61C22×C411yescharacteristic
94C2×C8×C61C2×D412yes—
95C2×C8×C61C2×D411yescharacteristic
96C2×C8×C61#4112yes—
97C22⋊Dic61—41no—
98C22⋊Dic61—41no—
99C22⋊Dic61—41no—
100C22⋊Dic61—82no—
101#11758198—41no—
102#11758198—41no—
103C4⋊Dic61—21no—
104C4⋊Dic61—21no—
105C4⋊Dic61—42no—
106C4⋊Dic61—42no—
107#11758196—41no—
108#11758196—41no—
109C4×Dic61—41no—
110Dic244—22no—
111D488—22no—
112#11758190—41no—
113#11758190—41no—
114C16×C61—24no—
115C23×C61—22no—
116C23×C61—22no—
117C23×C61—41no—
118C23×C61#10211yescharacteristic
119C23×C61#10211yescharacteristic
120C23×C61C2×D811yescharacteristic
121C22×D61—41no—
122C2×C4×C61—22no—
123C2×C4×C61—21no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—21no—
127C2×C4×C61—22no—
128C2×C4×C61—22no—
129C2×C4×C61#10211yescharacteristic
130C2×C4×C61#10211yescharacteristic
131C2×C4×C61C2×C22⋊C411yescharacteristic
132C2×C4×C61#12112yes—
133C2×C4×C61C2×SD1611yescharacteristic
134C2×C4×C61#13512yes—
135C22⋊D61—82no—
136C22⋊D61—82no—
137C22⋊D61—82no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—82no—
142C2×Dic61—82no—
143C2×Dic61—82no—
144D244—41no—
145D244—41no—
146C4×D61—82no—
147Dic122—41no—
148Dic122—41no—
149C8×C61—21no—
150C8×C61—21no—
151C8×C61C4×D412yes—
152C8×C61#12011yescharacteristic
153C8×C61#12311yescharacteristic
154#13065—611nomaximal, Sylow
155C22×C61—21no—
156C22×C61—21no—
157C22×C61—21no—
158C22×C61—21no—
159C22×C61—21no—
160C22×C61—21no—
161C22×C61—21no—
162C22×C61—44no—
163C22×C61#36012yescomplemented
164C22×C61C2×#10211yescharacteristic
165C22×C61#46612yes—
166C2×D61—81no—
167C2×D61—81no—
168C2×D61—81no—
169C4×C61—21no—
170C4×C61—21no—
171C4×C61#38611yescharacteristic
172C4×C61#41611yescharacteristic
173C4×C61#46512yes—
174Dic61—81no—
175Dic61—81no—
176Dic61—81no—
177Dic61—162no—
178C22×M32D6111yescharacteristic, complemented
179#3050—611no—
180C2×#360—612no—
181#1807—611no—
182#1386—612no—
183C2×C61—22no—
184C2×C61—22no—
185C2×C61—41no—
186C2×C61#189311yescharacteristic
187C2×C61C2×#36011yescharacteristic
188C2×C61#214511yescharacteristic
189D61—161no—
190C2×#135—611no—
191C23×C8C2×D6111yescharacteristic
192#525—1221no—
193C2×M32—21no—
194C2×M32—24no—
195C2×M32C2×D6112yes—
196C22×C16—21no—
197#467—611no—
198#467—611no—
199#466—611no—
200#466—611no—
201#465—1221no—
202#465—1221no—
203#427—611no—
204C2×#106—612no—
205#416—1221no—
206#380—1221no—
207#360—1224no—
208#340—1221no—
209C61#1306511yescharacteristic, complemented, Sylow
210C2×#41—1221no—
211C23×C4D24411yescharacteristic
212#135—1222no—
213#135—1222no—
214C2×Q16—611no—
215C2×SD16—1221no—
216C2×D8—611no—
217C22×C8—22no—
218C22×C8—22no—
219C22×C8C4×D6111yescharacteristic
220C22×C8C4×D6111yescharacteristic
221C22×C8D24411yescharacteristic
222C22×C8C22⋊D6111yescharacteristic
223C22×C8C22⋊D6111yescharacteristic
224C22×C8C22×D6111yescharacteristic
225#123—2441no—
226C22⋊Q8—1221no—
227C22⋊Q8—1221no—
228#121—1221no—
229#121—1221no—
230#120—2441no—
231C4×D4—2441no—
232C4×D4—2441no—
233#117—1221no—
234C2×C4⋊C4—1222no—
235C2×C22⋊C4—2441no—
236M32—24no—
237M32—22no—
238M32C22⋊D6114yescomplemented
239C2×C16—24no—
240C2×C16—22no—
241#107—611no—
242#107—611no—
243#106—1221no—
244#106—1222no—
245#106—1222no—
246#103—1221no—
247#103—1221no—
248#102—1221no—
249#102—1221no—
250#95—2441no—
251C24D48811yescharacteristic
252#41—2442no—
253#41—2442no—
254C2×Q8—1221no—
255C2×D4—1221no—
256C2×D4—2441no—
257C2×D4—2441no—
258C2×D4—2441no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—2442no—
262C22×C4—2441no—
263C22×C4#1175819011yescharacteristic
264C22×C4#1175819011yescharacteristic
265C22×C4D48811yescharacteristic
266C22×C4#1175819811yescharacteristic
267C22×C4#1175819811yescharacteristic
268C22×C4C2×D24411yescharacteristic
269Q16—1222no—
270SD16—2441no—
271SD16—2441no—
272D8—1222no—
273C2×C8—22no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8#1175819811yescharacteristic
281C2×C8#1175819811yescharacteristic
282C2×C8C2×C4×D6111yescharacteristic
283C2×C8#1175822212yes—
284C2×C8C2×C22⋊D6112yes—
285C2×C8C2×C22⋊D6111yescharacteristic
286C4⋊C4—1221no—
287C4⋊C4—1221no—
288C4⋊C4—2442no—
289C4⋊C4—2442no—
290C4⋊C4—2441no—
291C4⋊C4—2441no—
292C22⋊C4—4882no—
293C22⋊C4—2441no—
294C22⋊C4—2441no—
295C22⋊C4—2441no—
296C22⋊C4—2441no—
297C22⋊C4—2441no—
298C42—2441no—
299C16—24no—
300C23—22no—
301C23—22no—
302C23—41no—
303C23—2441no—
304C23#1451969911yescharacteristic
305C23#1451969911yescharacteristic
306C23C2×D48811yescharacteristic
307Q8—2441no—
308Q8—2441no—
309D4—4882no—
310D4—4882no—
311D4—4882no—
312D4—2441no—
313D4—2441no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—4882no—
322C2×C4—4882no—
323C2×C4—4882no—
324C2×C4—4882no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4#1451969911yescharacteristic
329C2×C4#1451969911yescharacteristic
330C2×C4C2×#1175819811yescharacteristic
331C2×C4#1451975112yes—
332C2×C4C2×#1175819011yescharacteristic
333C2×C4#1451979912yes—
334C8—21no—
335C8—21no—
336C8C4×C22⋊D6112yes—
337C8#1451974611yescharacteristic
338C8#1451976011yescharacteristic
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22—4881no—
348C22—4881no—
349C22—4881no—
350C22#1683957312yescomplemented
351C22C2×#1451969911yescharacteristic
352C22#1683999812yes—
353C4—21no—
354C4—21no—
355C4—9762no—
356C4—4881no—
357C4—4881no—
358C4—4881no—
359C4#1683964111yescharacteristic
360C4#1683975911yescharacteristic
361C4#1683999112yes—
362C2—22no—
363C2—22no—
364C2—41no—
365C2—9761no—
366C2#2144624711yescharacteristic
367C2C2×#1683957311yescharacteristic
368C2#2144755611yescharacteristic
subgroupG/H
centerC22C2×#14519699
commutatorC2×C8×C61C22×C4
FrattiniC22×C8C22×D61
FittingC22×M32×C61C2
socleC22×C61C2×#102
prime radical#21455400C2
cosocleC22×C8×C61C23
hypercenterC22×M32D61
hypocenterC61#13065
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C22×M32

Complements

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

NH ≅ G/Ntype
2. C2×#16839573C2semidirect
3. #21445844C2semidirect
4. #21443925C2semidirect
5. C22×M32×C61C2semidirect
8. #16840003C4semidirect
9. #16840003C4semidirect
10. #16839998C4semidirect
11. #16839998C4semidirect
63. M32×C61D4semidirect
163. C22×C61#360semidirect
178. C22×M32D61semidirect
209. C61#13065semidirect
238. M32C22⋊D61semidirect
350. C22#16839573semidirect

Maximal subgroups

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

subgroupindex
#214554002
C2×#168395732
#214458442
#214439252
C22×M32×C612
#1306561