Finite Group Databasebeta

← #29833341

Subgroups — #29833341

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840205C211yesmaximal, characteristic
2#21454590C211yesmaximal, characteristic
3C2×#16839856C211yesmaximal, characteristic, complemented
4#21447689C212yesmaximal, complemented
5#21447682C212yesmaximal, complemented
6#21447677C212yesmaximal, complemented
7#21447670C212yesmaximal, complemented
8#21447538C211yesmaximal, characteristic, complemented
9#21447529C211yesmaximal, characteristic
10M32×Dic61C211yesmaximal, characteristic, complemented
11C61×#2159C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13C22×Dic244C2211yescharacteristic
14#16840230—21no—
15C2×#14519742—21no—
16#16840205C418yescomplemented
17C2×Dic488C2212yes—
18C2×#14519722C2212yes—
19C2×#14519717C2211yescharacteristic
20#16839856—21no—
21#16839856—21no—
22#16839856C2212yes—
23#16839856C2211yescharacteristic
24#16839856C2212yes—
25#16839856C2211yescharacteristic
26#16839849—21no—
27#16839849—21no—
28#16839844C2211yescharacteristic
29#16839844C2211yescharacteristic
30#16839756—21no—
31C2×#14519700—21no—
32C2×C8×Dic61C2211yescharacteristic
33#16839565C2212yes—
34#16839565C2212yes—
35#16839565C2212yes—
36#16839562C2212yes—
37#16839537C2212yes—
38C16×Dic61C2212yes—
39C2×M32×C61C2211yescharacteristic
40C61×#427C2211yescharacteristic
41C2×#107×C61C2211yescharacteristic
42C61×#367C2212yes—
43C61×#366C2212yes—
44C2×#11758222—21no—
45C22×Dic122—21no—
46#14519799—22no—
47#14519799C2×C414yes—
48C2×Dic244—21no—
49C2×Dic244—22no—
50C2×Dic244C2×C414yescomplemented
51C2×Dic244C2312yes—
52C2×Dic244C2311yescharacteristic
53C2×#11758190—21no—
54C2×D488C2311yescharacteristic
55C22×#92418D411yescharacteristic
56#14519742—22no—
57#14519742—22no—
58#14519742—21no—
59#14519742—21no—
60#14519742—41no—
61#14519742—41no—
62#14519739—21no—
63#14519739—21no—
64#14519737—41no—
65#14519737—41no—
66#14519734—41no—
67C2×#11758196—41no—
68C2×C4×Dic61—41no—
69C2×C4×Dic61D411yescharacteristic
70C42×D61—41no—
71Dic488C2×C418yescomplemented
72#14519722C2×C418yescomplemented
73#14519717—22no—
74C2×#11758185C2312yes—
75#14519700—22no—
76#14519700—22no—
77#14519700—21no—
78#14519700—21no—
79#14519697—21no—
80#14519697—21no—
81C8×Dic61D411yescharacteristic
82C8×Dic61D411yescharacteristic
83C8×Dic61C2311yescharacteristic
84C8×Dic61C2311yescharacteristic
85C22×C8×C61C2311yescharacteristic
86#117×C61—21no—
87C2×C4⋊C4×C61—21no—
88M32×C61C2×C414yescomplemented
89C2×C16×C61C2312yes—
90#107×C61—21no—
91#107×C61C2311yescharacteristic
92#107×C61C2312yes—
93#107×C61C2311yescharacteristic
94#106×C61—21no—
95C22×C4×C61—41no—
96C22×C4×C61C2×D411yescharacteristic
97C2×C22⋊D61—41no—
98C22×Dic61—41no—
99C22×Dic61D811yescharacteristic
100C22×Dic61D811yescharacteristic
101#11758222—24no—
102#11758222—42no—
103C2×D244—21no—
104C2×C4×D61—41no—
105C2×C4×D61—41no—
106C2×C4×D61—41no—
107C2×Dic122—24no—
108C2×Dic122—22no—
109C2×Dic122—21no—
110C2×Dic122—41no—
111C2×Dic122—42no—
112C2×C8×C61C22×C412yes—
113C2×C8×C61C2×D411yescharacteristic
114C2×C8×C61#4112yes—
115C2×C8×C61C2411yescharacteristic
116C4⋊C4×C61—21no—
117C4⋊C4×C61—21no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—41no—
120C22⋊C4×C61—41no—
121C42×C61—41no—
122#11758198—41no—
123#11758198—41no—
124#11758196—42no—
125#11758196—42no—
126#11758196—41no—
127#11758196—41no—
128C4×Dic61—41no—
129C4×Dic61—41no—
130C4×Dic61—42no—
131C4×Dic61—42no—
132C4×Dic61D811yescharacteristic
133C4×Dic61D811yescharacteristic
134C4×Dic61C2×D411yescharacteristic
135C4×Dic61C2×D411yescharacteristic
136C2×#92418—22no—
137C2×#92418C2×D411yescharacteristic
138C2×#92418C2×D411yescharacteristic
139Dic244—24no—
140Dic244—22no—
141Dic244—21no—
142Dic244C22×C414yes—
143Dic244C22×C412yes—
144D488—21no—
145D488C22×C412yes—
146#11758190—24no—
147C16×C61C22×C414yes—
148#11758185—22no—
149C23×C61C2×D811yescharacteristic
150C22×D61—41no—
151C2×C4×C61—42no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61—82no—
155C2×C4×C61C4×D412yes—
156C2×C4×C61C2×D811yescharacteristic
157C2×C4×C61#13512yes—
158C2×C4×C61C22×D411yescharacteristic
159C22⋊D61—44no—
160C2×Dic61—22no—
161C2×Dic61—41no—
162C2×Dic61—42no—
163C2×Dic61—41no—
164C2×Dic61—44no—
165C2×Dic61C2×D811yescharacteristic
166C2×Dic61C2×D811yescharacteristic
167C2×Dic61C2×D811yescharacteristic
168C2×Dic61C2×D811yescharacteristic
169D244—22no—
170D244—41no—
171C4×D61—44no—
172C4×D61—82no—
173C4×D61—82no—
174Dic122—22no—
175Dic122—24no—
176Dic122—42no—
177Dic122—41no—
178Dic122—44no—
179C8×C61C4×D412yes—
180C8×C61C23×C411yescharacteristic
181C8×C61C2×#4111yescharacteristic
182#92418—21no—
183#92418—21no—
184C4×#510—611nomaximal, Sylow
185C22×C61—21no—
186C22×C61—21no—
187C22×C61C4×D812yes—
188C22×C61C22×D811yescharacteristic
189C2×D61—42no—
190C2×D61—81no—
191C2×D61—81no—
192C4×C61—81no—
193C4×C61—81no—
194C4×C61—82no—
195C4×C61C4×D812yes—
196C4×C61C2×C4×D411yescharacteristic
197C4×C61C2×#13511yescharacteristic
198Dic61—21no—
199Dic61—21no—
200Dic61—21no—
201Dic61—44no—
202Dic61—42no—
203Dic61#51012yescomplemented
204C2×#510—611no—
205C4×#135—611no—
206C2×C4×Q16—611no—
207#2176—612no—
208#2175—612no—
209C4×Q32—612no—
210C4×SD32—612no—
211#2159D6111yescharacteristic, complemented
212#2141—611no—
213#2139—611no—
214C4×M32—611no—
215C2×C61—22no—
216C2×C61C2×C4×D811yescharacteristic
217C2×C61C2×#51011yescharacteristic
218C2×C61#341411yescharacteristic
219D61—82no—
220C2×#135—611no—
221C22×Q16—611no—
222C4×#41—1221no—
223C2×C4×Q8—1221no—
224#510—618no—
225C2×Q32—612no—
226C2×SD32—612no—
227C2×M32—611no—
228C2×M32C2×D6111yescharacteristic
229C4×Q16—612no—
230C4×Q16—612no—
231C4×Q16—611no—
232C4×Q16—611no—
233C4×Q16—1221no—
234C4×Q16—1221no—
235C4×SD16—1221no—
236C4×SD16—1221no—
237C4×D8—611no—
238C4×D8—611no—
239#427C2×D6111yescharacteristic
240C2×#107C2×D6111yescharacteristic
241#416—1221no—
242C2×#103—1221no—
243C2×C4×C8—611no—
244#367C2×D6112yes—
245#366C2×D6112yes—
246#358—612no—
247#358—612no—
248#358—612no—
249#357—612no—
250#346—612no—
251C4×C16—612no—
252C61C4×#51011yescharacteristic, complemented, Sylow
253C2×#41—1221no—
254C22×Q8—1221no—
255#135—614no—
256#135—1222no—
257C2×Q16—614no—
258C2×Q16—612no—
259C2×Q16—611no—
260C2×Q16—1221no—
261C2×Q16—1222no—
262C2×SD16—1221no—
263C2×D8—611no—
264C22×C8—611no—
265C22×C8C22×D6111yescharacteristic
266C4×Q8—1222no—
267C4×Q8—1222no—
268C4×Q8—1221no—
269C4×Q8—1221no—
270C4×Q8—2441no—
271C4×Q8—2441no—
272C4×D4—1221no—
273C4×D4—1221no—
274C4×D4—2441no—
275C4×D4—2441no—
276#117—21no—
277#117—2441no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—2441no—
280C2×C42—611no—
281C2×C42—2441no—
282C2×C42—2441no—
283Q32—618no—
284SD32—618no—
285M32—1222no—
286M32C4×D6114yescomplemented
287C2×C16—612no—
288C2×C16C22×D6112yes—
289#107—21no—
290#107C22×D6112yes—
291#107C22×D6111yescharacteristic
292#107C22×D6111yescharacteristic
293#106—21no—
294#103—1222no—
295#103—1222no—
296#103—1221no—
297#103—1221no—
298#102—1221no—
299#102—1221no—
300C4×C8—611no—
301C4×C8—611no—
302C4×C8—611no—
303C4×C8—611no—
304#41—1224no—
305#41—2442no—
306C2×Q8—1221no—
307C2×Q8—1222no—
308C2×Q8—1224no—
309C2×Q8—2441no—
310C2×Q8—2442no—
311C2×D4—1221no—
312C2×D4—2441no—
313C22×C4—41no—
314C22×C4—611no—
315C22×C4—611no—
316C22×C4—2441no—
317C22×C4—2441no—
318C22×C4—2441no—
319C22×C4—2441no—
320C22×C4D4×D6111yescharacteristic
321Q16—614no—
322Q16—612no—
323Q16—1222no—
324Q16—1221no—
325Q16—1224no—
326SD16—1224no—
327D8—612no—
328D8—1221no—
329C2×C8—611no—
330C2×C8—611no—
331C2×C8—1222no—
332C2×C8C2×C4×D6112yes—
333C2×C8D4×D6111yescharacteristic
334C2×C8#1175822612yes—
335C2×C8C23×D6111yescharacteristic
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—41no—
340C4⋊C4—2442no—
341C4⋊C4—2442no—
342C4⋊C4—2441no—
343C4⋊C4—2441no—
344C22⋊C4—41no—
345C22⋊C4—2441no—
346C22⋊C4—2441no—
347C42—41no—
348C42—611no—
349C42—611no—
350C42—611no—
351C42—611no—
352C42—2441no—
353C42—2441no—
354C42—2442no—
355C42—2442no—
356C16—1222no—
357C16C2×C4×D6114yes—
358C23—2441no—
359C23D8×D6111yescharacteristic
360Q8—1222no—
361Q8—1224no—
362Q8—2442no—
363Q8—2441no—
364Q8—2444no—
365D4—1222no—
366D4—2444no—
367D4—2441no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—82no—
372C2×C4—611no—
373C2×C4—611no—
374C2×C4—611no—
375C2×C4—611no—
376C2×C4—1222no—
377C2×C4—4882no—
378C2×C4—4882no—
379C2×C4—2441no—
380C2×C4—2441no—
381C2×C4—2442no—
382C2×C4—2444no—
383C2×C4—2444no—
384C2×C4#1451973912yes—
385C2×C4D8×D6111yescharacteristic
386C2×C4#1451980012yes—
387C2×C4C2×D4×D6111yescharacteristic
388C8—1221no—
389C8—1221no—
390C8#1451973912yes—
391C8C22×C4×D6111yescharacteristic
392C8C2×#1175822611yescharacteristic
393C22—21no—
394C22—21no—
395C22—4881no—
396C22—4881no—
397C22—2442no—
398C22#1683984412yes—
399C22C2×D8×D6111yescharacteristic
400C4—81no—
401C4—82no—
402C4—81no—
403C4—612no—
404C4—1221no—
405C4—1221no—
406C4—1221no—
407C4—2442no—
408C4—2444no—
409C4#1683984412yes—
410C4C2×#1451973911yescharacteristic
411C4C2×#1451980011yescharacteristic
412C2—22no—
413C2—4882no—
414C2C2×#1683984411yescharacteristic
415C2#510×D6111yescharacteristic
416C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2159C2
socleC22×C61C22×D8
prime radicalC2×#14519799C22
cosocleC2×C8×C61C24
hypercenter#2159D61
hypocenterC61C4×#510
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
3. C2×#16839856C2semidirect
4. #21447689C2semidirect
5. #21447682C2semidirect
6. #21447677C2semidirect
7. #21447670C2semidirect
8. #21447538C2semidirect
10. M32×Dic61C2semidirect
11. C61×#2159C2semidirect
16. #16840205C4semidirect
50. C2×Dic244C2×C4semidirect
71. Dic488C2×C4semidirect
72. #14519722C2×C4semidirect
88. M32×C61C2×C4semidirect
203. Dic61#510semidirect
211. #2159D61semidirect
252. C61C4×#510semidirect
286. M32C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402052
#214545902
C2×#168398562
#214476892
#214476822
#214476772
#214476702
#214475382
#214475292
M32×Dic612
C61×#21592
C4×#51061