Finite Group Databasebeta

← #29833338

Subgroups — #29833338

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×#16840203C211yesmaximal, characteristic
2#21454590C211yesmaximal, characteristic
3C2×Q16×Dic61C211yesmaximal, characteristic, complemented
4#21447687C212yesmaximal, complemented
5#21447686C212yesmaximal, complemented
6#21447675C212yesmaximal, complemented
7#21447668C212yesmaximal, complemented
8#21447608C211yesmaximal, characteristic, complemented
9#21447532C211yesmaximal, characteristic
10M32×Dic61C211yesmaximal, characteristic, complemented
11C61×#2141C211yesmaximal, characteristic, complemented
12C2×#14519799C2211yescharacteristic
13#16840230—21no—
14C2×Q8×Dic61—21no—
15#16840203C418yescomplemented
16C2×#14519723C2212yes—
17C2×#14519720C2212yes—
18C2×#14519717C2211yescharacteristic
19Q16×Dic61—21no—
20Q16×Dic61—21no—
21Q16×Dic61C2212yes—
22Q16×Dic61C2212yes—
23#16839856C2211yescharacteristic
24#16839856C2211yescharacteristic
25#16839849—21no—
26#16839849—21no—
27#16839844C2211yescharacteristic
28#16839844C2211yescharacteristic
29C2×#14519711C2211yescharacteristic
30#16839756—21no—
31C2×#14519701—21no—
32C2×C8×Dic61C2211yescharacteristic
33#16839589C2212yes—
34#16839587C2212yes—
35#16839567C2212yes—
36#16839564C2212yes—
37#16839537C2212yes—
38C16×Dic61C2212yes—
39C22×Q16×C61C2211yescharacteristic
40C2×M32×C61C2211yescharacteristic
41C61×#427C2211yescharacteristic
42C61×#358C2212yes—
43C61×#358C2212yes—
44C2×#11758222—21no—
45#14519799—22no—
46#14519799C2×C414yes—
47C2×Dic244C2311yescharacteristic
48C2×#11758190—21no—
49C2×D488C2311yescharacteristic
50C22×#92418D411yescharacteristic
51Q8×Dic61—22no—
52Q8×Dic61—22no—
53Q8×Dic61—41no—
54Q8×Dic61—41no—
55#14519742—21no—
56#14519742—21no—
57#14519739—21no—
58#14519739—21no—
59#14519737—41no—
60#14519737—41no—
61#14519734—41no—
62C2×C4⋊Dic61—21no—
63C2×C4⋊Dic61—41no—
64C2×C4×Dic61—41no—
65C2×C4×Dic61D411yescharacteristic
66C42×D61—41no—
67#14519723C2×C418yescomplemented
68#14519720C2×C418yescomplemented
69#14519717C2×C414yescomplemented
70C2×#11758185C2312yes—
71#14519711—21no—
72#14519711C2312yes—
73#14519701—22no—
74#14519701—22no—
75#14519700—21no—
76#14519700—21no—
77#14519697—21no—
78#14519697—21no—
79C8×Dic61D411yescharacteristic
80C8×Dic61D411yescharacteristic
81C8×Dic61C2311yescharacteristic
82C8×Dic61C2311yescharacteristic
83C22×Q8×C61—21no—
84C2×Q16×C61—21no—
85C2×Q16×C61—22no—
86C2×Q16×C61C2×C414yescomplemented
87C2×Q16×C61C2312yes—
88C22×C8×C61C2311yescharacteristic
89#117×C61—21no—
90M32×C61—22no—
91C2×C16×C61C2312yes—
92#107×C61C2311yescharacteristic
93#107×C61C2311yescharacteristic
94#106×C61—21no—
95C2×Q8×C61—24no—
96C2×Q8×C61—22no—
97C2×Q8×C61—41no—
98C2×Q8×C61—42no—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C2×C22⋊D61—41no—
102C22×Dic61—41no—
103C22×Dic61D811yescharacteristic
104C22×Dic61D811yescharacteristic
105#11758222—24no—
106#11758222—42no—
107C2×D244—21no—
108C2×C4×D61—41no—
109C2×C4×D61—41no—
110C2×C4×D61—41no—
111C2×Dic122—21no—
112Q16×C61—24no—
113Q16×C61—22no—
114Q16×C61C22×C414yes—
115C2×C8×C61C22×C412yes—
116C2×C8×C61C2×D411yescharacteristic
117C2×C8×C61#4112yes—
118C2×C8×C61C2411yescharacteristic
119C4⋊C4×C61—21no—
120C4⋊C4×C61—21no—
121C22⋊C4×C61—41no—
122C42×C61—41no—
123#11758198—41no—
124#11758198—41no—
125C4⋊Dic61—22no—
126C4⋊Dic61—41no—
127C4⋊Dic61—42no—
128C4⋊Dic61—42no—
129#11758196—41no—
130#11758196—41no—
131C4×Dic61—41no—
132C4×Dic61—41no—
133C4×Dic61—42no—
134C4×Dic61—42no—
135C4×Dic61D811yescharacteristic
136C4×Dic61D811yescharacteristic
137C4×Dic61C2×D411yescharacteristic
138C4×Dic61C2×D411yescharacteristic
139C2×#92418—22no—
140C2×#92418C2×D411yescharacteristic
141C2×#92418C2×D411yescharacteristic
142Dic244—21no—
143Dic244C22×C412yes—
144D488—21no—
145D488C22×C412yes—
146#11758190—24no—
147C16×C61—22no—
148#11758185C22×C414yes—
149C23×C61C2×D811yescharacteristic
150C22×D61—41no—
151Q8×C61—24no—
152Q8×C61—42no—
153Q8×C61—44no—
154C2×C4×C61—42no—
155C2×C4×C61—44no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61C4×D412yes—
159C2×C4×C61C2×D811yescharacteristic
160C2×C4×C61#13512yes—
161C2×C4×C61C22×D411yescharacteristic
162C22⋊D61—44no—
163C2×Dic61—22no—
164C2×Dic61—41no—
165C2×Dic61—41no—
166C2×Dic61—42no—
167C2×Dic61—82no—
168C2×Dic61C2×D811yescharacteristic
169C2×Dic61C2×D811yescharacteristic
170C2×Dic61C2×D811yescharacteristic
171C2×Dic61C2×D811yescharacteristic
172D244—22no—
173D244—41no—
174C4×D61—44no—
175C4×D61—82no—
176C4×D61—82no—
177Dic122—22no—
178Dic122—41no—
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—44no—
193C4×C61—81no—
194C4×C61—81no—
195C4×C61C4×D812yes—
196C4×C61C2×C4×D411yescharacteristic
197C4×C61C2×#13511yescharacteristic
198Dic61—21no—
199Dic61—21no—
200Dic61—21no—
201Dic61—42no—
202Dic61—82no—
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#2159—611no—
212#2141D6111yescharacteristic, complemented
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×Q16C2×D6111yescharacteristic
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×#107—611no—
241#416—1221no—
242C2×#103—1221no—
243C2×C4×C8—611no—
244#367—612no—
245#366—612no—
246#358—612no—
247#358C2×D6112yes—
248#358C2×D6112yes—
249#357—612no—
250#346—612no—
251C4×C16—612no—
252C61C4×#51011yescharacteristic, complemented, Sylow
253C2×#41—1221no—
254C22×Q8—21no—
255#135—614no—
256#135—1222no—
257C2×Q16—21no—
258C2×Q16—22no—
259C2×Q16—611no—
260C2×Q16C4×D6114yescomplemented
261C2×Q16C22×D6112yes—
262C2×SD16—1221no—
263C2×D8—611no—
264C22×C8—611no—
265C22×C8C22×D6111yescharacteristic
266C4×Q8—1221no—
267C4×Q8—1221no—
268C4×Q8—1222no—
269C4×Q8—1222no—
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—1221no—
279C2×C4⋊C4—2441no—
280C2×C42—611no—
281C2×C42—2441no—
282C2×C42—2441no—
283Q32—618no—
284SD32—618no—
285M32—22no—
286M32—614no—
287C2×C16—612no—
288C2×C16C22×D6112yes—
289#107—612no—
290#107—1221no—
291#107C22×D6111yescharacteristic
292#107C22×D6111yescharacteristic
293#106—21no—
294#103—1221no—
295#103—1221no—
296#103—1222no—
297#103—1222no—
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—22no—
307C2×Q8—24no—
308C2×Q8—41no—
309C2×Q8—42no—
310C2×Q8—1221no—
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—22no—
322Q16—24no—
323Q16—612no—
324Q16—1221no—
325Q16C2×C4×D6114yes—
326SD16—1224no—
327D8—612no—
328D8—1221no—
329C2×C8—611no—
330C2×C8—611no—
331C2×C8—1222no—
332C2×C8C2×C4×D6112yes—
333C2×C8D4×D6111yescharacteristic
334C2×C8#1175822412yes—
335C2×C8C23×D6111yescharacteristic
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—1222no—
339C4⋊C4—2442no—
340C4⋊C4—2442no—
341C4⋊C4—2441no—
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—22no—
357C16—614no—
358C23—2441no—
359C23D8×D6111yescharacteristic
360Q8—24no—
361Q8—42no—
362Q8—44no—
363Q8—1222no—
364Q8—2441no—
365D4—1222no—
366D4—2444no—
367D4—2441no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—611no—
373C2×C4—611no—
374C2×C4—611no—
375C2×C4—611no—
376C2×C4—1222no—
377C2×C4—4882no—
378C2×C4—4882no—
379C2×C4—4882no—
380C2×C4—2441no—
381C2×C4—2442no—
382C2×C4—2441no—
383C2×C4—2444no—
384C2×C4#1451973712yes—
385C2×C4D8×D6111yescharacteristic
386C2×C4#1451979812yes—
387C2×C4C2×D4×D6111yescharacteristic
388C8—1221no—
389C8—1221no—
390C8#1451973712yes—
391C8C22×C4×D6111yescharacteristic
392C8C2×#1175822411yescharacteristic
393C22—21no—
394C22—21no—
395C22—4881no—
396C22—4881no—
397C22—2442no—
398C22#1683984212yes—
399C22C2×D8×D6111yescharacteristic
400C4—44no—
401C4—81no—
402C4—81no—
403C4—612no—
404C4—1221no—
405C4—1221no—
406C4—1221no—
407C4—4882no—
408C4—2442no—
409C4#1683984212yes—
410C4C2×#1451973711yescharacteristic
411C4C2×#1451979811yescharacteristic
412C2—22no—
413C2—4882no—
414C2C2×#1683984211yescharacteristic
415C2#510×D6111yescharacteristic
416C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutatorC8×C61C23×C4
FrattiniC2×C8C23×D61
FittingC61×#2141C2
socleC22×C61C22×D8
prime radicalC2×#14519799C22
cosocleC2×C8×C61C24
hypercenter#2141D61
hypocenterC61C4×#510
LCSG ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2141

Complements

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

NH ≅ G/Ntype
3. C2×Q16×Dic61C2semidirect
4. #21447687C2semidirect
5. #21447686C2semidirect
6. #21447675C2semidirect
7. #21447668C2semidirect
8. #21447608C2semidirect
10. M32×Dic61C2semidirect
11. C61×#2141C2semidirect
15. #16840203C4semidirect
67. #14519723C2×C4semidirect
68. #14519720C2×C4semidirect
69. #14519717C2×C4semidirect
86. C2×Q16×C61C2×C4semidirect
203. Dic61#510semidirect
212. #2141D61semidirect
252. C61C4×#510semidirect
260. C2×Q16C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168402032
#214545902
C2×Q16×Dic612
#214476872
#214476862
#214476752
#214476682
#214476082
#214475322
M32×Dic612
C61×#21412
C4×#51061