Finite Group Databasebeta

← #29834996

Subgroups — #29834996

Order 15616.

596 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456396C211yesmaximal, characteristic, complemented
2C4×#14519800C211yesmaximal, characteristic, complemented
3#21448009C212yesmaximal, complemented
4#21447971C211yesmaximal, characteristic, complemented
5#21447967C211yesmaximal, characteristic, complemented
6#21447851C211yesmaximal, characteristic, complemented
7#21447741C212yesmaximal, complemented
8SD32×Dic61C212yesmaximal, complemented
9#21447663C212yesmaximal, complemented
10#21447617C211yesmaximal, characteristic, complemented
11C61×#2207C211yesmaximal, characteristic, complemented
12C2×#14519800C2211yescharacteristic
13#16840310—21no—
14C4×#11758226—21no—
15C2×#14519718C2212yes—
16#16840167C2211yescharacteristic
17#16840108—21no—
18#16839995—21no—
19#16839995C2212yes—
20Q16×Dic61C2211yescharacteristic
21C4×#11758201—21no—
22#16839847—21no—
23C4×D488C2211yescharacteristic
24D8×Dic61—21no—
25D8×Dic61C2212yes—
26#16839844C2211yescharacteristic
27#16839799C2211yescharacteristic
28#16839757—21no—
29C4×C8×D61C2211yescharacteristic
30#16839590C2212yes—
31#16839587C2212yes—
32C4⋊#11758185C2211yescharacteristic
33#16839566C2212yes—
34#16839564C2212yes—
35#16839563C2212yes—
36#16839541C2212yes—
37C16×Dic61C2212yes—
38C2×SD32×C61C2212yes—
39C61×#493C2211yescharacteristic
40C4×Q16×C61C2211yescharacteristic
41C61×C4⋊C16C2211yescharacteristic
42C61×#357C2212yes—
43C2×#11758226—21no—
44C2×#11758224—41no—
45#14519800—22no—
46#14519800—22no—
47C2×#11758201—21no—
48C2×D488C2311yescharacteristic
49C2×C8×D61D411yescharacteristic
50C2×C8×D61C2311yescharacteristic
51#14519777—21no—
52#14519764—41no—
53#14519748—22no—
54#14519748—41no—
55Q8×Dic61—21no—
56C4×D244—21no—
57C4×D244—41no—
58D4×Dic61—22no—
59D4×Dic61—41no—
60#14519739—21no—
61#14519739—41no—
62#14519735—21no—
63#14519735—41no—
64C42×D61—41no—
65C42×D61D411yescharacteristic
66#14519718—24no—
67C2×#11758185C2312yes—
68#14519711—21no—
69#14519711C2312yes—
70#14519711C2311yescharacteristic
71#14519708—21no—
72#14519701—21no—
73#14519699—21no—
74#14519698—22no—
75#14519698—22no—
76#14519697—21no—
77C4×#92418D411yescharacteristic
78C8×Dic61D411yescharacteristic
79C8×Dic61C2311yescharacteristic
80C2×Q16×C61C2311yescharacteristic
81C2×D8×C61—21no—
82C2×D8×C61C2312yes—
83#127×C61—21no—
84C4×Q8×C61—21no—
85SD32×C61—24no—
86C2×C16×C61C2312yes—
87#107×C61C2311yescharacteristic
88#103×C61—21no—
89C4×C8×C61C2311yescharacteristic
90C2×Q8×C61—21no—
91C2×D4×C61—22no—
92C2×D4×C61—41no—
93C2×D4×C61—41no—
94C2×C22⋊D61—42no—
95C22×Dic61—42no—
96#11758226—24no—
97#11758226—42no—
98#11758224—84no—
99C2×D244—21no—
100C2×D244—41no—
101C2×C4×D61—41no—
102C2×C4×D61—41no—
103C2×C4×D61—41no—
104C2×C4×D61D811yescharacteristic
105C2×C4×D61D811yescharacteristic
106C2×C4×D61C2×D411yescharacteristic
107C2×Dic122—41no—
108Q16×C61—21no—
109Q16×C61#4112yescomplemented
110D8×C61—22no—
111D8×C61—22no—
112D8×C61—42no—
113C2×C8×C61C2×D411yescharacteristic
114C2×C8×C61C2411yescharacteristic
115C4⋊C4×C61—21no—
116C4⋊C4×C61—41no—
117C42×C61—41no—
118C42×C61C2×D411yescharacteristic
119C22⋊Dic61—42no—
120C22⋊Dic61—42no—
121#11758201—24no—
122#11758198—41no—
123#11758198—41no—
124#11758198—41no—
125C4⋊Dic61—22no—
126C4⋊Dic61—21no—
127C4⋊Dic61—41no—
128C4⋊Dic61—41no—
129C4×Dic61—41no—
130C4×Dic61—41no—
131C4×Dic61D811yescharacteristic
132C4×Dic61D811yescharacteristic
133C4×Dic61C2×D411yescharacteristic
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136D488—21no—
137D488#4112yescomplemented
138C8×D61—21no—
139C8×D61—21no—
140C8×D61—41no—
141C16×C61—22no—
142#11758185—22no—
143C23×C61—42no—
144C22×D61—41no—
145C22×D61C2×D811yescharacteristic
146Q8×C61—22no—
147Q8×C61—41no—
148D4×C61—42no—
149D4×C61—42no—
150D4×C61—82no—
151D4×C61—82no—
152C2×C4×C61—41no—
153C2×C4×C61—41no—
154C2×C4×C61C2×D811yescharacteristic
155C2×C4×C61C2×D811yescharacteristic
156C2×C4×C61C22×D411yescharacteristic
157C22⋊D61—84no—
158C2×Dic61—41no—
159C2×Dic61—42no—
160C2×Dic61—84no—
161C2×Dic61C2×D811yescharacteristic
162C2×Dic61C2×D811yescharacteristic
163C2×Dic61C2×D811yescharacteristic
164D244—22no—
165D244—44no—
166D244—41no—
167C4×D61—21no—
168C4×D61—21no—
169C4×D61—22no—
170C4×D61—22no—
171C4×D61—44no—
172C4×D61—82no—
173C4×D61—82no—
174Dic122—82no—
175C8×C61—21no—
176C8×C61C2×#4111yescharacteristic
177C8×C61C2×#4111yescharacteristic
178#92418—21no—
179#92418—21no—
180#33666—611nomaximal, Sylow
181C22×C61—82no—
182C22×C61—82no—
183C22×C61—82no—
184C22×C61C22×D811yescharacteristic
185C2×D61—21no—
186C2×D61—21no—
187C2×D61—21no—
188C2×D61—42no—
189C2×D61—81no—
190C2×D61—81no—
191C4×C61—21no—
192C4×C61—42no—
193C4×C61—81no—
194C4×C61#50812yes—
195C4×C61#52511yescharacteristic
196C4×C61#57511yescharacteristic
197Dic61—21no—
198Dic61—21no—
199Dic61—81no—
200Dic61—82no—
201Dic61#50912yes—
202#3169—611no—
203C4×#135—611no—
204#2234—612no—
205#2227—611no—
206#2226—611no—
207#2207D6111yescharacteristic, complemented
208#2206—611no—
209#2186—612no—
210C4×SD32—612no—
211C4×D16—612no—
212#2162—611no—
213C2×C61—162no—
214C2×C61#310011yescharacteristic
215C2×C61C2×#50811yescharacteristic
216C2×C61C2×#50911yescharacteristic
217D61—41no—
218D61—82no—
219C2×#135—611no—
220#532—1221no—
221C4×#41—1221no—
222C2×SD32C2×D6112yes—
223C2×D16—612no—
224#500—611no—
225#493C2×D6111yescharacteristic
226#486—1221no—
227#466—612no—
228#466—1221no—
229C4×Q16—611no—
230C4×Q16C2×D6111yescharacteristic
231C4×SD16—1221no—
232C4×SD16—1221no—
233C4×D8—612no—
234C4×D8—611no—
235C4×D8—611no—
236C4×D8—1221no—
237#427—611no—
238#416—1221no—
239C2×C4×C8—611no—
240#367—612no—
241#366—612no—
242C4⋊C16—611no—
243C4⋊C16C2×D6111yescharacteristic
244#358—612no—
245#357—612no—
246#357—612no—
247#357C2×D6112yes—
248C22⋊C16—612no—
249C4×C16—612no—
250C61#3366611yescharacteristic, complemented, Sylow
251C2×#41—1221no—
252C2×#41—2441no—
253#135—1222no—
254#135—1222no—
255C2×Q16C22×D6111yescharacteristic
256C2×SD16—1221no—
257C2×D8—21no—
258C2×D8—611no—
259C2×D8C22×D6112yes—
260C22×C8—611no—
261C22×C8—611no—
262C4⋊Q8—1221no—
263#127—21no—
264#124—2441no—
265#121—1222no—
266#121—2441no—
267C4×Q8—21no—
268C4×Q8—1221no—
269C4×D4—1221no—
270C4×D4—1222no—
271C4×D4—1221no—
272C4×D4—2441no—
273C4×D4—2441no—
274C4×D4—2441no—
275#117—1221no—
276#117—2441no—
277C2×C42—611no—
278C2×C42—2441no—
279SD32—24no—
280D16—1224no—
281C2×C16—612no—
282C2×C16C22×D6112yes—
283#107—611no—
284#107—612no—
285#107—1221no—
286#107C22×D6111yescharacteristic
287#106—1221no—
288#103—21no—
289#103—1221no—
290#102—1221no—
291#102—1222no—
292#102—1222no—
293#102—1221no—
294C4×C8—611no—
295C4×C8—611no—
296C4×C8—611no—
297C4×C8C22×D6111yescharacteristic
298#41—1224no—
299#41—4884no—
300#41—2442no—
301C2×Q8—21no—
302C2×Q8—2441no—
303C2×D4—22no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—1221no—
307C2×D4—2441no—
308C2×D4—2442no—
309C22×C4—611no—
310C22×C4—611no—
311C22×C4—611no—
312C22×C4—2441no—
313C22×C4—2441no—
314C22×C4—2442no—
315C22×C4—2441no—
316Q16—21no—
317Q16#1175822412yescomplemented
318SD16—1224no—
319D8—22no—
320D8—22no—
321D8—42no—
322D8—612no—
323D8—1221no—
324C2×C8—611no—
325C2×C8—611no—
326C2×C8—1221no—
327C2×C8—1221no—
328C2×C8—2441no—
329C2×C8D4×D6111yescharacteristic
330C2×C8C23×D6111yescharacteristic
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—1222no—
334C4⋊C4—1221no—
335C4⋊C4—2441no—
336C4⋊C4—2441no—
337C22⋊C4—2441no—
338C22⋊C4—2441no—
339C22⋊C4—2442no—
340C22⋊C4—2442no—
341C22⋊C4—2441no—
342C42—41no—
343C42—611no—
344C42—611no—
345C42—611no—
346C42—2441no—
347C42—2441no—
348C42D4×D6111yescharacteristic
349C16—22no—
350C16—1222no—
351C23—42no—
352C23—611no—
353C23—2441no—
354Q8—22no—
355Q8—41no—
356Q8—4882no—
357D4—42no—
358D4—42no—
359D4—82no—
360D4—82no—
361D4—1222no—
362D4—4884no—
363D4—2444no—
364D4—2441no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—611no—
368C2×C4—611no—
369C2×C4—611no—
370C2×C4—1222no—
371C2×C4—1222no—
372C2×C4—1221no—
373C2×C4—1221no—
374C2×C4—4882no—
375C2×C4—4884no—
376C2×C4—4882no—
377C2×C4—2444no—
378C2×C4—2441no—
379C2×C4—2442no—
380C2×C4D8×D6111yescharacteristic
381C2×C4D8×D6111yescharacteristic
382C2×C4C2×D4×D6111yescharacteristic
383C8—21no—
384C8—1221no—
385C8—1221no—
386C8#41×D6111yescharacteristic
387C8C2×#1175822411yescharacteristic
388C22—82no—
389C22—82no—
390C22—82no—
391C22—1221no—
392C22—1221no—
393C22—1221no—
394C22—4881no—
395C22—4881no—
396C22—2442no—
397C22C2×D8×D6111yescharacteristic
398C4—21no—
399C4—42no—
400C4—81no—
401C4—612no—
402C4—1221no—
403C4—1221no—
404C4—4881no—
405C4—4882no—
406C4#1684019312yes—
407C4#1684027211yescharacteristic
408C4#1684055811yescharacteristic
409C2—162no—
410C2—4882no—
411C2—2441no—
412C2#2145581811yescharacteristic
413C2C2×#1684019311yescharacteristic
414C2#509×D6111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2207C2
socleC22×C61C22×D8
hypercenter#2207D61
hypocenterC61#33666
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2207

Complements

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

NH ≅ G/Ntype
1. #21456396C2semidirect
2. C4×#14519800C2semidirect
3. #21448009C2semidirect
4. #21447971C2semidirect
5. #21447967C2semidirect
6. #21447851C2semidirect
7. #21447741C2semidirect
8. SD32×Dic61C2semidirect
9. #21447663C2semidirect
10. #21447617C2semidirect
11. C61×#2207C2semidirect
109. Q16×C61#41semidirect
137. D488#41semidirect
207. #2207D61semidirect
250. C61#33666semidirect
317. Q16#11758224semidirect

Maximal subgroups

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

subgroupindex
#214563962
C4×#145198002
#214480092
#214479712
#214479672
#214478512
#214477412
SD32×Dic612
#214476632
#214476172
C61×#22072
#3366661