Finite Group Databasebeta

← #29695943

Subgroups — #29695943

Order 15616.

1338 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840567C211yesmaximal, characteristic
2#21454735C212yesmaximal, complemented
3#21454296C212yesmaximal, complemented
4#21446727C212yesmaximal, complemented
5#21446300C214yesmaximal, complemented
6#21445790C212yesmaximal, complemented
7#21445619C211yesmaximal, characteristic, complemented
8C61×#1813C211yesmaximal, characteristic, complemented
9C2×#14519823C2212yes—
10#16840567—24no—
11#16840567—24no—
12C2×#14519805C414yescomplemented
13C2×#14519805C2212yes—
14C2×#14519799C412yescomplemented
15C2×#14519799C2211yescharacteristic
16C22×D488C412yescomplemented
17C22×D488C2211yescharacteristic
18#16840528C2214yescomplemented
19#16839883—24no—
20#16839878—24no—
21#16839778C2214yescomplemented
22#16839768C2214yescomplemented
23#16839746C2211yescharacteristic
24#16839745—24no—
25C2×#14519693C2214yescomplemented
26#100×D61—24no—
27#16839736C2211yescharacteristic
28#16839717C2212yescomplemented
29#16839533C2212yescomplemented
30C2×#131×C61C2211yescharacteristic
31C2×#108×C61C2212yescomplemented
32C61×#413C2212yes—
33C61×#338C2212yescomplemented
34#14519823—28no—
35#14519823—44no—
36C2×#11758226—22no—
37C2×#11758222C2×C414yes—
38C2×#11758222C2312yes—
39C22×D244C2×C414yes—
40C22×D244C2312yes—
41C2×D4×D61—22no—
42C2×D4×D61—24no—
43#14519805—28no—
44#14519805—216no—
45#14519799—24no—
46#14519799—24no—
47#14519799D414yes—
48#14519799D414yes—
49C2×Dic244C2×C411yescharacteristic
50C2×#11758190C2×C414yes—
51C2×#11758190C2×C412yes—
52C2×D488—22no—
53C2×D488—22no—
54C2×D488—24no—
55C2×D488—24no—
56C2×D488C2×C412yes—
57C2×D488C2×C414yes—
58C2×D488C2×C412yes—
59C2×D488C2×C411yescharacteristic
60#14519785—28no—
61#14519783—44no—
62C2×#11758194C2312yes—
63M16×D61—28no—
64#14519736C2312yes—
65#14519703—28no—
66#14519699—44no—
67#14519696D412yes—
68#14519694D412yes—
69#14519693—28no—
70#14519692—22no—
71#14519687—22no—
72C8×Dic61—22no—
73C2×#41×C61C2311yescharacteristic
74#131×C61—22no—
75#131×C61—22no—
76C2×M16×C61C2×C412yes—
77C2×M16×C61C2314yes—
78C2×M16×C61C2311yescharacteristic
79C22×C8×C61C2×C412yes—
80C22×C8×C61C2311yescharacteristic
81#108×C61—22no—
82#108×C61—22no—
83#101×C61—22no—
84#100×C61—22no—
85C23×D61—24no—
86C23×D61—22no—
87#41×C61—22no—
88#41×C61—22no—
89C2×Q8×C61C2×D411yescharacteristic
90C2×D4×C61C2×D411yescharacteristic
91C2×D4×C61#4112yes—
92C22×C4×C61C22×C412yes—
93C22×C4×C61C2411yescharacteristic
94C2×C22⋊D61—22no—
95C2×C22⋊D61—24no—
96#11758226—48no—
97D4×D61—48no—
98D4×D61—416no—
99#11758222—24no—
100#11758222—28no—
101#11758222—44no—
102#11758222C2×D418yes—
103C2×D244—24no—
104C2×D244—24no—
105C2×D244—28no—
106C2×D244—24no—
107C2×D244—22no—
108C2×D244—44no—
109C2×D244C22×C414yes—
110C2×D244C22×C412yes—
111C2×C4×D61—22no—
112C2×C4×D61—24no—
113C2×Dic122C22×C412yes—
114M16×C61—22no—
115M16×C61—24no—
116M16×C61—24no—
117M16×C61—24no—
118C2×C8×C61—22no—
119C2×C8×C61—22no—
120C2×C8×C61—24no—
121C2×C8×C61C22×C412yes—
122C2×C8×C61C22×C411yescharacteristic
123C2×C8×C61C22×C411yescharacteristic
124C2×C8×C61C2×D412yes—
125C2×C8×C61C2×D412yes—
126C22⋊Dic61—22no—
127C4⋊Dic61—22no—
128C4×Dic61—22no—
129#11758194—22no—
130#11758194C2×D414yes—
131C2×#92418—22no—
132Dic244C22⋊C412yes—
133Dic244C22⋊C412yes—
134D488—28no—
135D488—22no—
136D488—22no—
137D488—24no—
138D488C22⋊C412yes—
139D488C22⋊C412yes—
140#11758190—28no—
141#11758190C22⋊C418yes—
142#11758189—48no—
143C8×D61—48no—
144C23×C61C4×D412yes—
145C23×C61C2×#4111yescharacteristic
146C22×D61—22no—
147C22×D61—24no—
148C22×D61—48no—
149C22×D61—44no—
150C22×D61—164no—
151C22×D61—162no—
152Q8×C61—22no—
153D4×C61—24no—
154D4×C61—22no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61C4×D412yes—
158C2×C4×C61C23×C411yescharacteristic
159C2×C4×C61C22×D412yes—
160C2×C4×C61C22×D411yescharacteristic
161C2×C4×C61H(5,𝔽2)12yes—
162C22⋊D61—28no—
163C22⋊D61—48no—
164C2×Dic61—22no—
165C2×Dic61—42no—
166D244—24no—
167D244—22no—
168D244—24no—
169D244—48no—
170D244—44no—
171D244C2×C22⋊C414yes—
172C4×D61—28no—
173C4×D61—48no—
174Dic122—22no—
175Dic122C2×C22⋊C414yes—
176C8×C61—22no—
177C8×C61—44no—
178C8×C61C2×C22⋊C412yes—
179C8×C61C2×C22⋊C411yescharacteristic
180C8×C61C2×C22⋊C411yescharacteristic
181#92418—24no—
182#22458—611nomaximal, Sylow
183C22×C61—22no—
184C22×C61—41no—
185C22×C61—42no—
186C22×C61C2×C4×D411yescharacteristic
187C22×C61#54612yes—
188C2×D61—24no—
189C2×D61—44no—
190C2×D61—82no—
191C2×D61—84no—
192C2×D61—168no—
193C2×D61—164no—
194C4×C61—22no—
195C4×C61C22×C22⋊C411yescharacteristic
196C4×C61#51811yescharacteristic
197C4×C61D4212yes—
198Dic61—24no—
199Dic61—82no—
200C2×#576—611no—
201#2957—612no—
202#2888—612no—
203#1978—612no—
204#1903—614no—
205#1813D6111yescharacteristic, complemented
206#1795—612no—
207#1757—611no—
208C2×C61—22no—
209C2×C61—21no—
210C2×C61—42no—
211C2×C61D4×C22⋊C411yescharacteristic
212D61—44no—
213D61—84no—
214C2×H(5,𝔽2)—612no—
215#576—1224no—
216#576—1224no—
217C2×AGL(1,ℤ/8)—614no—
218C2×AGL(1,ℤ/8)—612no—
219C2×#135—612no—
220C2×#135—611no—
221C22×D8—612no—
222C22×D8—611no—
223#568—614no—
224C2×#131C2×D6111yescharacteristic
225#444—1224no—
226#443—1224no—
227C2×#108C2×D6112yescomplemented
228#421—614no—
229#418—614no—
230#413—611no—
231#413—1224no—
232#413C2×D6112yes—
233C2×#100—614no—
234C2×#100—1224no—
235#410—611no—
236#405—612no—
237#344—612no—
238#338C2×D6112yescomplemented
239C61#2245811yescharacteristic, complemented, Sylow
240H(5,𝔽2)—1228no—
241H(5,𝔽2)—2444no—
242C2×#41—614no—
243C2×#41—612no—
244C2×#41—1222no—
245C2×#41C22×D6111yescharacteristic
246C22×D4—614no—
247C22×D4—612no—
248C22×D4—1222no—
249C22×D4—1224no—
250AGL(1,ℤ/8)—12216no—
251AGL(1,ℤ/8)—1228no—
252#135—614no—
253#135—614no—
254#135—1224no—
255#135—1224no—
256C2×Q16—611no—
257C2×SD16—612no—
258C2×SD16—614no—
259C2×D8—611no—
260C2×D8—612no—
261C2×D8—614no—
262C2×D8—612no—
263C2×D8—1224no—
264C2×D8—1224no—
265C2×D8—1222no—
266C2×D8—1222no—
267#131—22no—
268#131—22no—
269#131—1228no—
270#131—2444no—
271C2×M16—612no—
272C2×M16—1228no—
273C2×M16C4×D6112yes—
274C2×M16C22×D6114yes—
275C2×M16C22×D6111yescharacteristic
276C22×C8C4×D6112yes—
277C22×C8C22×D6111yescharacteristic
278#117—612no—
279#108—22no—
280#108—22no—
281#104—1228no—
282#102—2444no—
283#101—22no—
284#101—612no—
285#100—22no—
286#100—612no—
287#100—1228no—
288#99—1222no—
289#97—1222no—
290C4×C8—1222no—
291C24—1222no—
292C24—1224no—
293#41—22no—
294#41—22no—
295#41—618no—
296#41—1228no—
297#41—1224no—
298#41—2444no—
299#41—2448no—
300C2×Q8—612no—
301C2×Q8D4×D6111yescharacteristic
302C2×D4—612no—
303C2×D4—614no—
304C2×D4—1228no—
305C2×D4—1222no—
306C2×D4—1224no—
307C2×D4—1224no—
308C2×D4—1222no—
309C2×D4—1224no—
310C2×D4—1224no—
311C2×D4—2444no—
312C2×D4—2448no—
313C2×D4—24416no—
314C2×D4D4×D6111yescharacteristic
315C2×D4#1175822612yes—
316C22×C4—1222no—
317C22×C4—1224no—
318C22×C4C2×C4×D6112yes—
319C22×C4C23×D6111yescharacteristic
320Q16—612no—
321Q16—612no—
322SD16—618no—
323SD16—1228no—
324D8—612no—
325D8—612no—
326D8—1228no—
327D8—1222no—
328D8—1222no—
329D8—1224no—
330M16—22no—
331M16—24no—
332M16—24no—
333M16—24no—
334M16—614no—
335M16—1222no—
336M16—2448no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—24no—
340C2×C8—1222no—
341C2×C8—2448no—
342C2×C8C2×C4×D6112yes—
343C2×C8C2×C4×D6111yescharacteristic
344C2×C8C2×C4×D6111yescharacteristic
345C2×C8D4×D6112yes—
346C2×C8D4×D6112yes—
347C4⋊C4—1222no—
348C22⋊C4—1222no—
349C42—1222no—
350C23—1222no—
351C23—1224no—
352C23—9762no—
353C23—9764no—
354C23—2448no—
355C23—2444no—
356C23#1451973912yes—
357C23C2×#1175822611yescharacteristic
358Q8—22no—
359Q8—614no—
360Q8—1222no—
361D4—24no—
362D4—22no—
363D4—614no—
364D4—1228no—
365D4—1224no—
366D4—1222no—
367D4—1224no—
368D4—2448no—
369D4—2448no—
370D4—2444no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—1222no—
374C2×C4—1228no—
375C2×C4—2448no—
376C2×C4—2442no—
377C2×C4#1451973912yes—
378C2×C4C22×C4×D6111yescharacteristic
379C2×C4C2×D4×D6112yes—
380C2×C4C2×D4×D6111yescharacteristic
381C2×C4#1451982312yes—
382C8—22no—
383C8—44no—
384C8—1224no—
385C8C22⋊C4×D6111yescharacteristic
386C8C22⋊C4×D6111yescharacteristic
387C8C22⋊C4×D6112yes—
388C22—22no—
389C22—41no—
390C22—42no—
391C22—1224no—
392C22—9764no—
393C22—9768no—
394C22—4884no—
395C22—4882no—
396C22—2444no—
397C22C2×#1451973911yescharacteristic
398C22#1684040312yes—
399C4—22no—
400C4—1224no—
401C4—4882no—
402C4C2×C22⋊C4×D6111yescharacteristic
403C4#1684023611yescharacteristic
404C4#1684039612yes—
405C2—22no—
406C2—21no—
407C2—42no—
408C2—4884no—
409C2—2444no—
410C2#2144863911yescharacteristic
subgroupG/H
centerC2#21448639
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1813C2
socleC2×C61D4×C22⋊C4
prime radicalC2×#16840567C2
cosocleC22×C4×C61C24
hypercenter#1813D61
hypocenterC61#22458
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1813

Complements

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

NH ≅ G/Ntype
2. #21454735C2semidirect
3. #21454296C2semidirect
4. #21446727C2semidirect
5. #21446300C2semidirect
6. #21445790C2semidirect
7. #21445619C2semidirect
8. C61×#1813C2semidirect
12. C2×#14519805C4semidirect
14. C2×#14519799C4semidirect
16. C22×D488C4semidirect
18. #16840528C22semidirect
21. #16839778C22semidirect
22. #16839768C22semidirect
25. C2×#14519693C22semidirect
28. #16839717C22semidirect
29. #16839533C22semidirect
31. C2×#108×C61C22semidirect
33. C61×#338C22semidirect
205. #1813D61semidirect
227. C2×#108C2×D61semidirect
238. #338C2×D61semidirect
239. C61#22458semidirect

Maximal subgroups

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

subgroupindex
C2×#168405672
#214547352
#214542962
#214467272
#214463002
#214457902
#214456192
C61×#18132
#2245861