Finite Group Databasebeta

← #29833061

Subgroups — #29833061

Order 15616.

658 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840180C211yesmaximal, characteristic
2#21454416C211yesmaximal, characteristic
3#21454415C211yesmaximal, characteristic
4#21447607C211yesmaximal, characteristic, complemented
5#21447606C212yesmaximal, complemented
6#21447604C212yesmaximal, complemented
7#21447601C211yesmaximal, characteristic, complemented
8#21447600C212yesmaximal, complemented
9C2×#16839589C211yesmaximal, characteristic, complemented
10#367×D61C212yesmaximal, complemented
11C61×#2159C211yesmaximal, characteristic, complemented
12C2×#14519783C2211yescharacteristic
13#16840243—21no—
14#16840242—21no—
15#16840180C418yescomplemented
16C2×#14519716C2212yes—
17C22×#11758185C2211yescharacteristic
18C2×C16×D61C2212yes—
19#16839805—21no—
20#16839803—22no—
21#16839802—21no—
22#16839801—22no—
23#16839800C2211yescharacteristic
24#16839799C2212yes—
25#16839799C2212yes—
26C2×#14519711C2211yescharacteristic
27#107×D61C2212yes—
28#107×D61C2212yes—
29#16839590C2212yes—
30#16839589C2211yescharacteristic
31#16839589C2212yes—
32#16839589C2212yes—
33#16839589C2211yescharacteristic
34#16839588C2212yes—
35#16839587C2211yescharacteristic
36#16839587C2211yescharacteristic
37C2×M32×C61C2211yescharacteristic
38C61×#427C2211yescharacteristic
39C2×#107×C61C2211yescharacteristic
40C61×#367C2212yes—
41C61×#366C2212yes—
42C2×#11758222D411yescharacteristic
43#14519783C2×C414yes—
44#14519783Q814yes—
45C2×#11758194D411yescharacteristic
46C2×#11758189D412yes—
47C2×C8×D61C2312yes—
48C4×Dic122—41no—
49#14519742—41no—
50C4×D244—41no—
51#14519739—41no—
52C4×C22⋊D61—42no—
53#14519737—42no—
54#14519736—21no—
55#14519735—22no—
56#14519735—22no—
57C2×C4⋊Dic61—21no—
58C4⋊C4×D61—22no—
59C4⋊C4×D61—22no—
60#14519716C2×C418yescomplemented
61C2×#11758185C2×C414yescomplemented
62C2×#11758185C2311yescharacteristic
63C2×#11758185C2311yescharacteristic
64C16×D61C2×C418yescomplemented
65#14519711—21no—
66#14519711C2311yescharacteristic
67#14519711C2312yes—
68#14519711C2311yescharacteristic
69#14519710—22no—
70#14519710—22no—
71#14519709—21no—
72#14519708—22no—
73#14519708—22no—
74C22×C8×C61C2311yescharacteristic
75#117×C61—21no—
76C2×C4⋊C4×C61—21no—
77M32×C61C2×C414yescomplemented
78C2×C16×C61C2312yes—
79#107×C61—21no—
80#107×C61C2311yescharacteristic
81#107×C61C2312yes—
82#107×C61C2311yescharacteristic
83#106×C61—21no—
84C22×C4×C61—41no—
85C22×C4×C61C2×D411yescharacteristic
86C2×C22⋊D61D812yes—
87C22×Dic61—41no—
88#11758222C4⋊C414yes—
89#11758222Q1614yes—
90C2×D244D811yescharacteristic
91C2×C4×D61—42no—
92C2×C4×D61—42no—
93C2×C4×D61C2×D412yes—
94C2×Dic122D811yescharacteristic
95C2×C8×C61C22×C412yes—
96C2×C8×C61C2×D411yescharacteristic
97C2×C8×C61C2×Q812yes—
98C2×C8×C61C2411yescharacteristic
99C4⋊C4×C61—21no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—22no—
102C4⋊C4×C61—41no—
103C22⋊C4×C61—41no—
104C42×C61—41no—
105C22⋊Dic61—41no—
106#11758198—42no—
107#11758198—42no—
108C4⋊Dic61—21no—
109C4⋊Dic61—22no—
110C4⋊Dic61—21no—
111C4⋊Dic61—41no—
112#11758196—42no—
113#11758196—42no—
114C4×Dic61—41no—
115C4×Dic61—42no—
116C4×Dic61—42no—
117#11758194C4⋊C414yes—
118C2×#92418C2×D412yes—
119#11758189C4⋊C418yes—
120C8×D61C22×C414yes—
121C8×D61C2×Q814yes—
122C16×C61C22×C414yes—
123#11758185C22×C412yes—
124#11758185C22×C412yes—
125C23×C61C2×D811yescharacteristic
126C22×D61C2×D812yes—
127C2×C4×C61—41no—
128C2×C4×C61—42no—
129C2×C4×C61—41no—
130C2×C4×C61—161no—
131C2×C4×C61C2×C4⋊C412yes—
132C2×C4×C61C2×D811yescharacteristic
133C2×C4×C61C2×Q1612yes—
134C2×C4×C61C22×D411yescharacteristic
135C22⋊D61#10718yes—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—42no—
139C2×Dic61—161no—
140C2×Dic61C2×D812yes—
141D244#10714yes—
142C4×D61—162no—
143C4×D61—162no—
144C4×D61C2×C4⋊C414yes—
145C4×D61C2×Q1614yes—
146Dic122#10714yes—
147C8×C61C2×C4⋊C412yes—
148C8×C61C23×C411yescharacteristic
149C8×C61C22×Q811yescharacteristic
150#92418C2×C4⋊C414yes—
151#33411—611nomaximal, Sylow
152C22×C61—21no—
153C22×C61—21no—
154C22×C61C2×#10712yes—
155C22×C61C22×D811yescharacteristic
156C2×D61—22no—
157C2×D61—22no—
158C2×D61C2×#10714yes—
159C4×C61—81no—
160C4×C61—82no—
161C4×C61—81no—
162C4×C61C2×#10712yes—
163C4×C61C22×C4⋊C411yescharacteristic
164C4×C61C22×Q1611yescharacteristic
165Dic61—81no—
166Dic61—82no—
167Dic61—81no—
168Dic61C2×#10714yes—
169C2×#504—611no—
170#2913—611no—
171#2913—611no—
172#2159—611no—
173#2159—612no—
174#2159—612no—
175#2159D6111yescharacteristic, complemented
176#2158—611no—
177#2158—612no—
178C2×#367—611no—
179C2×#367—612no—
180C2×C61—22no—
181C2×C61C22×#10711yescharacteristic
182C2×C61#341611yescharacteristic
183C2×C61#341611yescharacteristic
184D61—24no—
185C2×#131—611no—
186#520—1221no—
187#520—1221no—
188#504—618no—
189C2×M32—612no—
190C2×M32C2×D6111yescharacteristic
191C22×C16—611no—
192C22×C16—612no—
193#428—1222no—
194#428—1222no—
195#428—1221no—
196#428—1221no—
197#427—611no—
198#427—612no—
199#427—612no—
200#427C2×D6111yescharacteristic
201C2×#107—611no—
202C2×#107—612no—
203C2×#107—612no—
204C2×#107C2×D6111yescharacteristic
205#367—612no—
206#367—611no—
207#367—612no—
208#367—612no—
209#367—611no—
210#367C2×D6112yes—
211#366—612no—
212#366—611no—
213#366—611no—
214#366C2×D6112yes—
215C61#3341111yescharacteristic, complemented, Sylow
216C2×#41—611no—
217#131—614no—
218#131—614no—
219C2×M16—611no—
220C2×M16—612no—
221C22×C8—612no—
222C22×C8C22×D6111yescharacteristic
223C4×Q8—2441no—
224C4×Q8—2441no—
225C4×D4—2441no—
226C4×D4—2442no—
227C4×D4—2441no—
228C4×D4—2442no—
229#117—21no—
230#117—1222no—
231#117—1222no—
232#117—1221no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—1222no—
235C2×C4⋊C4—1222no—
236C2×C4⋊C4—1221no—
237M32—618no—
238M32C4×D6114yescomplemented
239C2×C16—614no—
240C2×C16—611no—
241C2×C16—611no—
242C2×C16—618no—
243C2×C16C22×D6112yes—
244#107—21no—
245#107—611no—
246#107—612no—
247#107—611no—
248#107—1222no—
249#107—1222no—
250#107—1221no—
251#107C22×D6111yescharacteristic
252#107C22×D6111yescharacteristic
253#107C22×D6112yes—
254#106—21no—
255#106—1222no—
256#106—1222no—
257#106—1221no—
258#41—614no—
259#41—614no—
260C2×Q8—611no—
261C2×D4—612no—
262C2×D4—611no—
263C22×C4—41no—
264C22×C4—612no—
265C22×C4—2442no—
266C22×C4—2442no—
267C22×C4—2441no—
268C22×C4D4×D6111yescharacteristic
269M16—614no—
270M16—618no—
271C2×C8—612no—
272C2×C8—614no—
273C2×C8—614no—
274C2×C8C2×C4×D6112yes—
275C2×C8D4×D6111yescharacteristic
276C2×C8Q8×D6112yes—
277C2×C8C23×D6111yescharacteristic
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—41no—
282C4⋊C4—1221no—
283C4⋊C4—1222no—
284C4⋊C4—1221no—
285C4⋊C4—2442no—
286C4⋊C4—2442no—
287C4⋊C4—2441no—
288C22⋊C4—41no—
289C22⋊C4—2442no—
290C22⋊C4—2442no—
291C22⋊C4—2441no—
292C42—41no—
293C42—2442no—
294C42—2442no—
295C42—2441no—
296C16—612no—
297C16—612no—
298C16C2×C4×D6114yes—
299C23—612no—
300C23D8×D6111yescharacteristic
301Q8—614no—
302D4—618no—
303D4—614no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—161no—
308C2×C4—612no—
309C2×C4—614no—
310C2×C4—614no—
311C2×C4—9762no—
312C2×C4—9762no—
313C2×C4—9761no—
314C2×C4—2441no—
315C2×C4—2442no—
316C2×C4—2441no—
317C2×C4C4⋊C4×D6112yes—
318C2×C4D8×D6111yescharacteristic
319C2×C4Q16×D6112yes—
320C2×C4C2×D4×D6111yescharacteristic
321C8—614no—
322C8C4⋊C4×D6112yes—
323C8C22×C4×D6111yescharacteristic
324C8C2×Q8×D6111yescharacteristic
325C22—21no—
326C22—21no—
327C22—614no—
328C22—1222no—
329C22—1222no—
330C22#107×D6112yes—
331C22C2×D8×D6111yescharacteristic
332C4—81no—
333C4—81no—
334C4—82no—
335C4—614no—
336C4—4882no—
337C4—4881no—
338C4—4881no—
339C4#107×D6112yes—
340C4C2×C4⋊C4×D6111yescharacteristic
341C4C2×Q16×D6111yescharacteristic
342C2—22no—
343C2—1224no—
344C2C2×#107×D6111yescharacteristic
345C2#2145896411yescharacteristic
346C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2159C2
socleC22×C61C22×D8
prime radicalC2×#11758222D4
cosocleC2×C8×C61C24
hypercenter#2159D61
hypocenterC61#33411
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
4. #21447607C2semidirect
5. #21447606C2semidirect
6. #21447604C2semidirect
7. #21447601C2semidirect
8. #21447600C2semidirect
9. C2×#16839589C2semidirect
10. #367×D61C2semidirect
11. C61×#2159C2semidirect
15. #16840180C4semidirect
60. #14519716C2×C4semidirect
61. C2×#11758185C2×C4semidirect
64. C16×D61C2×C4semidirect
77. M32×C61C2×C4semidirect
175. #2159D61semidirect
215. C61#33411semidirect
238. M32C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168401802
#214544162
#214544152
#214476072
#214476062
#214476042
#214476012
#214476002
C2×#168395892
#367×D612
C61×#21592
#3341161