Finite Group Databasebeta

← #29833473

Subgroups — #29833473

Order 15616.

646 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454736C211yesmaximal, characteristic, complemented
2#21454703C211yesmaximal, characteristic, complemented
3#21447718C214yesmaximal, complemented
4#21447711C214yesmaximal, complemented
5C2×#16839585C211yesmaximal, characteristic, complemented
6#21447568C211yesmaximal, characteristic, complemented
7C2×#16839572C211yesmaximal, characteristic, complemented
8#21447423C211yesmaximal, characteristic, complemented
9C61×#3414C211yesmaximal, characteristic
10#16840529—21no—
11#16840245—21no—
12C2×#14519717C2211yescharacteristic
13#16839884C2214yes—
14#16839879—22no—
15#16839874C2212yescomplemented
16#16839869C2212yescomplemented
17SD16×Dic61—22no—
18C2×#14519713C2211yescharacteristic
19C2×#14519709C2211yescharacteristic
20#16839779—21no—
21#16839763—21no—
22#16839717C2211yescharacteristic
23#16839585C2214yescomplemented
24#16839578C2212yescomplemented
25#16839575C2212yescomplemented
26#16839572C2214yes—
27#16839539C2214yescomplemented
28C2×#135×C61C2211yescharacteristic
29C2×#135×C61C2211yescharacteristic
30C61×#510C412yescomplemented
31C61×#510C2212yes—
32C61×#509C412yescomplemented
33C61×#509C2212yes—
34C61×#508C414yescomplemented
35C2×M32×C61C2211yescharacteristic
36#14519784—24no—
37#14519784—42no—
38C2×#11758194—21no—
39C2×#11758194—42no—
40C2×#11758194D411yescharacteristic
41Q8×Dic61—22no—
42D4×Dic61—22no—
43D4×Dic61—42no—
44#14519736—41no—
45#14519736D411yescharacteristic
46C2×C4⋊Dic61—21no—
47C2×C4⋊Dic61—41no—
48#14519717—21no—
49#14519717C2312yes—
50C2×#11758185—21no—
51#14519713—21no—
52#14519713C2312yes—
53#14519709—21no—
54#14519709C2312yes—
55#14519704—24no—
56#14519701—22no—
57#14519698—22no—
58#14519687C2312yes—
59C8×Dic61D412yes—
60C2×#41×C61—21no—
61C2×#41×C61—21no—
62#135×C61—21no—
63#135×C61—21no—
64#135×C61—21no—
65#135×C61—21no—
66#135×C61C2×C412yes—
67#135×C61C2×C412yes—
68#135×C61C2×C412yescomplemented
69#135×C61C2312yes—
70C2×Q16×C61C2×C411yescharacteristic, complemented
71C2×Q16×C61C2311yescharacteristic
72C2×SD16×C61—21no—
73C2×SD16×C61—21no—
74C2×D8×C61C2×C411yescharacteristic, complemented
75C2×D8×C61C2311yescharacteristic
76C22×C8×C61C2311yescharacteristic
77Q32×C61C2×C414yescomplemented
78SD32×C61C2×C414yescomplemented
79SD32×C61C2×C414yescomplemented
80D16×C61C2×C414yescomplemented
81M32×C61C2×C412yescomplemented
82M32×C61C2312yes—
83C2×C16×C61C2×C412yescomplemented
84#41×C61—22no—
85#41×C61—22no—
86#41×C61—22no—
87#41×C61—22no—
88#41×C61—41no—
89#41×C61—41no—
90#41×C61—41no—
91#41×C61—41no—
92C2×Q8×C61—21no—
93C2×Q8×C61—21no—
94C2×D4×C61—21no—
95C2×D4×C61—21no—
96C2×D4×C61—41no—
97C2×D4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C22×Dic61—42no—
102C22×Dic61—41no—
103Q16×C61—21no—
104Q16×C61—21no—
105Q16×C61C22×C412yes—
106Q16×C61C22×C412yes—
107SD16×C61—22no—
108SD16×C61—22no—
109SD16×C61—22no—
110SD16×C61—22no—
111D8×C61—21no—
112D8×C61—21no—
113D8×C61C22×C412yes—
114D8×C61C22×C412yes—
115C2×C8×C61C22×C411yescharacteristic
116C2×C8×C61C22×C411yescharacteristic
117C2×C8×C61C2×D411yescharacteristic
118C2×C8×C61#4111yescharacteristic
119C2×C8×C61#4111yescharacteristic
120C2×C8×C61C2411yescharacteristic
121C22⋊Dic61—21no—
122C22⋊Dic61—42no—
123C4⋊Dic61—21no—
124C4⋊Dic61—22no—
125C4⋊Dic61—42no—
126C4⋊Dic61—41no—
127C4×Dic61—42no—
128C4×Dic61C2×D412yes—
129#11758194—21no—
130#11758194—21no—
131#11758194—22no—
132#11758194—44no—
133#11758194—41no—
134C2×#92418—44no—
135C2×#92418—41no—
136C2×#92418C2×D412yes—
137C16×C61C22×C414yes—
138#11758185—22no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61AGL(1,ℤ/8)11yescharacteristic
142Q8×C61—22no—
143Q8×C61—22no—
144Q8×C61—41no—
145Q8×C61—41no—
146D4×C61—22no—
147D4×C61—22no—
148D4×C61—42no—
149D4×C61—42no—
150D4×C61—42no—
151D4×C61—42no—
152D4×C61—41no—
153D4×C61—41no—
154C2×C4×C61—41no—
155C2×C4×C61—41no—
156C2×C4×C61—42no—
157C2×C4×C61—42no—
158C2×C4×C61—42no—
159C2×C4×C61—42no—
160C2×C4×C61C4×D411yescharacteristic
161C2×C4×C61C4×D411yescharacteristic
162C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C61C22×D411yescharacteristic
166C2×Dic61—22no—
167C2×Dic61—42no—
168C2×Dic61—162no—
169C2×Dic61—161no—
170C8×C61C4×D411yescharacteristic
171C8×C61C4×D411yescharacteristic
172C8×C61C23×C411yescharacteristic
173C8×C61C2×#4111yescharacteristic
174#92418—22no—
175#92418—42no—
176#33469—611nomaximal, Sylow
177C22×C61—41no—
178C22×C61—42no—
179C22×C61—42no—
180C22×C61—81no—
181C22×C61—81no—
182C22×C61—161no—
183C22×C61#44211yescharacteristic
184C22×C61#44211yescharacteristic
185C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
186C4×C61—42no—
187C4×C61—42no—
188C4×C61#44211yescharacteristic
189C4×C61#44211yescharacteristic
190C4×C61C2×C4×D411yescharacteristic
191C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
192Dic61—42no—
193Dic61—82no—
194#3414D6111yescharacteristic
195#2957—611no—
196#2952—611no—
197AGL(1,ℤ/16)—614no—
198#2180—614no—
199C2×#365—611no—
200#2148—611no—
201C2×#360—611no—
202#2110—611no—
203C2×C61—21no—
204C2×C61—21no—
205C2×C61—21no—
206C2×C61—82no—
207C2×C61—82no—
208C2×C61C2×#44211yescharacteristic
209C2×#135C2×D6111yescharacteristic
210C2×#135C2×D6111yescharacteristic
211#568—1221no—
212#520—1221no—
213#510Dic6112yescomplemented
214#510C2×D6112yes—
215#509Dic6112yescomplemented
216#509C2×D6112yes—
217#508Dic6114yescomplemented
218C2×M32—611no—
219C2×M32C2×D6111yescharacteristic
220#444—614no—
221#443—1222no—
222#442—612no—
223#441—612no—
224C4×SD16—1222no—
225C2×#108—611no—
226C2×#106—611no—
227#421—1221no—
228#417—1221no—
229#405—611no—
230#365—614no—
231#362—612no—
232#361—612no—
233#360—614no—
234#347—614no—
235C61#3346911yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C2×#41—21no—
238#135—21no—
239#135—21no—
240#135—21no—
241#135—21no—
242#135C2×Dic6112yes—
243#135C2×Dic6112yes—
244#135C2×Dic6112yescomplemented
245#135C22×D6112yes—
246C2×Q16C2×Dic6111yescharacteristic, complemented
247C2×Q16C22×D6111yescharacteristic
248C2×SD16—21no—
249C2×SD16—21no—
250C2×D8C2×Dic6111yescharacteristic, complemented
251C2×D8C22×D6111yescharacteristic
252#131—1224no—
253#131—2442no—
254C2×M16—611no—
255C2×M16—1221no—
256C2×M16—2442no—
257C22×C8C22×D6111yescharacteristic
258C4×Q8—1222no—
259C4×D4—1222no—
260C4×D4—2442no—
261#117—611no—
262#117—2441no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—2441no—
265Q32C2×Dic6114yescomplemented
266SD32C2×Dic6114yescomplemented
267SD32C2×Dic6114yescomplemented
268D16C2×Dic6114yescomplemented
269M32—612no—
270M32—1221no—
271M32C2×Dic6112yescomplemented
272M32C22×D6112yes—
273C2×C16—1221no—
274C2×C16C2×Dic6112yescomplemented
275#108—612no—
276#108—1221no—
277#106—612no—
278#106—1221no—
279#104—1224no—
280#103—1222no—
281#102—1222no—
282#97—612no—
283C4×C8—612no—
284#41—22no—
285#41—22no—
286#41—22no—
287#41—22no—
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—41no—
297C2×D4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—2442no—
301C22×C4—2441no—
302C22×C4D4×D6111yescharacteristic
303Q16—21no—
304Q16—21no—
305Q16C22×Dic6112yes—
306Q16C22×Dic6112yes—
307SD16—22no—
308SD16—22no—
309SD16—22no—
310SD16—22no—
311D8—21no—
312D8—21no—
313D8C22×Dic6112yes—
314D8C22×Dic6112yes—
315M16—1221no—
316M16—1221no—
317M16—1222no—
318M16—2444no—
319M16—2441no—
320C2×C8—612no—
321C2×C8—2444no—
322C2×C8—2441no—
323C2×C8D4×D6111yescharacteristic
324C2×C8#1175822411yescharacteristic
325C2×C8#1175822411yescharacteristic
326C2×C8C22×Dic6111yescharacteristic
327C2×C8C22×Dic6111yescharacteristic
328C2×C8C23×D6111yescharacteristic
329C4⋊C4—1221no—
330C4⋊C4—1222no—
331C4⋊C4—2442no—
332C4⋊C4—2441no—
333C22⋊C4—1221no—
334C22⋊C4—2442no—
335C42—612no—
336C42—2442no—
337C16—1222no—
338C16C22×Dic6114yes—
339C23—41no—
340C23—41no—
341C23#1451980111yescharacteristic
342Q8—22no—
343Q8—22no—
344Q8—41no—
345Q8—41no—
346D4—22no—
347D4—22no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—42no—
352D4—41no—
353D4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—1222no—
361C2×C4—9761no—
362C2×C4—9762no—
363C2×C4—2442no—
364C2×C4D4×Dic6111yescharacteristic
365C2×C4D4×Dic6111yescharacteristic
366C2×C4#1451980111yescharacteristic
367C2×C4#1451980111yescharacteristic
368C2×C4#1451980111yescharacteristic
369C2×C4C2×D4×D6111yescharacteristic
370C8—1222no—
371C8—2442no—
372C8D4×Dic6111yescharacteristic
373C8D4×Dic6111yescharacteristic
374C8C23×Dic6111yescharacteristic
375C8C2×#1175822411yescharacteristic
376C22—41no—
377C22—42no—
378C22—42no—
379C22—81no—
380C22—81no—
381C22—161no—
382C22#1683987411yescharacteristic
383C22#1683987411yescharacteristic
384C22C2×#1451980111yescharacteristic
385C4—42no—
386C4—42no—
387C4—4882no—
388C4—2442no—
389C4#1683987411yescharacteristic
390C4#1683987411yescharacteristic
391C4C2×D4×Dic6111yescharacteristic
392C4C2×#1451980111yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—82no—
397C2—82no—
398C2C2×#1683987411yescharacteristic
subgroupG/H
centerC2C2×#16839874
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
Fitting, prime radicalC61×#3414C2
socleC2×C61C2×#442
cosocleC2×C8×C61C24
hypercenter#3414D61
hypocenterC61#33469
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ #3414

Complements

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

NH ≅ G/Ntype
1. #21454736C2semidirect
2. #21454703C2semidirect
3. #21447718C2semidirect
4. #21447711C2semidirect
5. C2×#16839585C2semidirect
6. #21447568C2semidirect
7. C2×#16839572C2semidirect
8. #21447423C2semidirect
15. #16839874C22semidirect
16. #16839869C22semidirect
23. #16839585C22semidirect
24. #16839578C22semidirect
25. #16839575C22semidirect
27. #16839539C22semidirect
30. C61×#510C4semidirect
32. C61×#509C4semidirect
34. C61×#508C4semidirect
68. #135×C61C2×C4semidirect
70. C2×Q16×C61C2×C4semidirect
74. C2×D8×C61C2×C4semidirect
77. Q32×C61C2×C4semidirect
78. SD32×C61C2×C4semidirect
79. SD32×C61C2×C4semidirect
80. D16×C61C2×C4semidirect
81. M32×C61C2×C4semidirect
83. C2×C16×C61C2×C4semidirect
213. #510Dic61semidirect
215. #509Dic61semidirect
217. #508Dic61semidirect
235. C61#33469semidirect
244. #135C2×Dic61semidirect
246. C2×Q16C2×Dic61semidirect
250. C2×D8C2×Dic61semidirect
265. Q32C2×Dic61semidirect
266. SD32C2×Dic61semidirect
267. SD32C2×Dic61semidirect
268. D16C2×Dic61semidirect
271. M32C2×Dic61semidirect
274. C2×C16C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214547362
#214547032
#214477182
#214477112
C2×#168395852
#214475682
C2×#168395722
#214474232
C61×#34142
#3346961