Finite Group Databasebeta

← #29834975

Subgroups — #29834975

Order 15616.

546 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456403C211yesmaximal, characteristic
2C4×Q16×D61C211yesmaximal, characteristic
3#21448020C212yesmaximal
4#21447972C211yesmaximal, characteristic, complemented
5#21447971C211yesmaximal, characteristic, complemented
6#21447866C211yesmaximal, characteristic, complemented
7#21447751C212yesmaximal
8Q32×Dic61C212yesmaximal, complemented
9#21447675C212yesmaximal, complemented
10#21447617C211yesmaximal, characteristic
11C61×#2209C211yesmaximal, characteristic, complemented
12C2×Q16×D61C2211yescharacteristic
13#16840317—21no—
14C4×Q8×D61—21no—
15C2×#14519723C2212yes—
16#16840167C2211yescharacteristic
17#16840112—21no—
18#16840000—21no—
19#16840000C2212yes—
20C4×Dic244C2211yescharacteristic
21Q16×Dic61—21no—
22Q16×Dic61C2212yes—
23Q16×Dic61C2211yescharacteristic
24#16839856C2211yescharacteristic
25C4×#11758202—21no—
26#16839854—21no—
27#107×D61C2211yescharacteristic
28#103×D61—21no—
29C4×C8×D61C2211yescharacteristic
30#16839588C2212yes—
31#16839587C2212yes—
32C4⋊#11758185C2211yescharacteristic
33#16839567C2212yes—
34#16839566C2212yes—
35#16839566C2212yes—
36#16839541C2212yes—
37C16×Dic61C2212yes—
38C2×Q32×C61C2212yes—
39C61×#494C2211yescharacteristic
40C4×Q16×C61C2211yescharacteristic
41C61×C4⋊C16C2211yescharacteristic
42C61×#358C2212yes—
43C2×Q8×D61—21no—
44C2×Dic244C2311yescharacteristic
45C2×#11758202—21no—
46Q16×D61—22no—
47Q16×D61—22no—
48C2×C8×D61D411yescharacteristic
49C2×C8×D61C2311yescharacteristic
50#14519777—21no—
51#14519768—41no—
52#14519757—41no—
53#14519753—22no—
54Q8×Dic61—21no—
55Q8×Dic61—22no—
56C4×Dic122—21no—
57C4×Dic122—41no—
58#14519742—21no—
59#14519742—41no—
60#14519742—41no—
61#14519735—41no—
62C4⋊C4×D61—21no—
63C4⋊C4×D61—41no—
64C42×D61—41no—
65C42×D61D411yescharacteristic
66#14519723—24no—
67C2×#11758185C2312yes—
68#14519711—21no—
69#14519711C2312yes—
70#14519711C2311yescharacteristic
71#14519710—21no—
72#14519702—21no—
73#14519701—22no—
74#14519701—22no—
75#14519701—21no—
76#14519700—21no—
77C4×#92418D411yescharacteristic
78C8×Dic61D411yescharacteristic
79C8×Dic61C2311yescharacteristic
80C2×Q16×C61—21no—
81C2×Q16×C61C2312yes—
82C2×Q16×C61C2311yescharacteristic
83C4⋊Q8×C61—21no—
84C4×Q8×C61—21no—
85Q32×C61—24no—
86C2×C16×C61C2312yes—
87#107×C61C2311yescharacteristic
88#103×C61—21no—
89C4×C8×C61C2311yescharacteristic
90C2×Q8×C61—21no—
91C2×Q8×C61—22no—
92Q8×D61—24no—
93Q8×D61—42no—
94C2×C4×D61—41no—
95C2×C4×D61—41no—
96C2×C4×D61D811yescharacteristic
97C2×C4×D61D811yescharacteristic
98C2×C4×D61C2×D411yescharacteristic
99C2×Dic122—21no—
100C2×Dic122—41no—
101C2×Dic122—41no—
102Q16×C61—21no—
103Q16×C61—22no—
104Q16×C61—22no—
105Q16×C61—42no—
106Q16×C61#4112yes—
107C2×C8×C61C2×D411yescharacteristic
108C2×C8×C61C2411yescharacteristic
109C4⋊C4×C61—21no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—41no—
113C42×C61—41no—
114C42×C61C2×D411yescharacteristic
115#11758202—24no—
116#11758198—42no—
117C4⋊Dic61—22no—
118C4⋊Dic61—21no—
119C4⋊Dic61—41no—
120C4⋊Dic61—41no—
121C4⋊Dic61—42no—
122#11758196—41no—
123#11758196—41no—
124#11758196—41no—
125#11758196—42no—
126C4×Dic61—41no—
127C4×Dic61—41no—
128C4×Dic61—41no—
129C4×Dic61—42no—
130C4×Dic61D811yescharacteristic
131C4×Dic61D811yescharacteristic
132C4×Dic61C2×D411yescharacteristic
133C2×#92418C2×D411yescharacteristic
134C2×#92418C2×D411yescharacteristic
135Dic244—21no—
136Dic244#4112yes—
137C8×D61—21no—
138C8×D61—21no—
139C8×D61—41no—
140C16×C61—22no—
141#11758185—22no—
142C22×D61C2×D811yescharacteristic
143Q8×C61—22no—
144Q8×C61—41no—
145Q8×C61—42no—
146Q8×C61—42no—
147C2×C4×C61—41no—
148C2×C4×C61—41no—
149C2×C4×C61—42no—
150C2×C4×C61C2×D811yescharacteristic
151C2×C4×C61C2×D811yescharacteristic
152C2×C4×C61C22×D411yescharacteristic
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—42no—
156C2×Dic61C2×D811yescharacteristic
157C2×Dic61C2×D811yescharacteristic
158C2×Dic61C2×D811yescharacteristic
159C4×D61—21no—
160C4×D61—21no—
161C4×D61—41no—
162C4×D61—44no—
163C4×D61—82no—
164C4×D61Q3214yescomplemented
165Dic122—22no—
166Dic122—44no—
167Dic122—41no—
168Dic122—82no—
169C8×C61—21no—
170C8×C61C2×#4111yescharacteristic
171C8×C61C2×#4111yescharacteristic
172#92418—21no—
173#92418—21no—
174#33664—611nomaximal, Sylow
175C22×C61C22×D811yescharacteristic
176C2×D61—21no—
177C2×D61—21no—
178C2×D61C2×Q3212yes—
179C4×C61—21no—
180C4×C61—42no—
181C4×C61—82no—
182C4×C61—81no—
183C4×C61C2×Q3212yes—
184C4×C61#52511yescharacteristic
185C4×C61#57511yescharacteristic
186Dic61—21no—
187Dic61—21no—
188Dic61—42no—
189Dic61—82no—
190Dic61—81no—
191Dic61C2×Q3212yes—
192#3170—611no—
193C2×C4×Q16—611no—
194#2236—612no—
195#2227—611no—
196#2227—611no—
197#2209—611no—
198#2209D6111yescharacteristic, complemented
199#2188—612no—
200C4×Q32—612no—
201C4×Q32—612no—
202#2162—611no—
203C2×C61#310011yescharacteristic
204C2×C61C22×Q3211yescharacteristic
205C2×C61#341411yescharacteristic
206D61—22no—
207C22×Q16—611no—
208#533—1221no—
209C2×C4×Q8—1221no—
210C2×Q32—612no—
211C2×Q32C2×D6112yes—
212#500—611no—
213#494C2×D6111yescharacteristic
214#487—1221no—
215#467—612no—
216#467—1221no—
217C4×Q16—612no—
218C4×Q16—611no—
219C4×Q16—611no—
220C4×Q16—611no—
221C4×Q16—1221no—
222C4×Q16—1221no—
223C4×Q16—1221no—
224C4×Q16C2×D6111yescharacteristic
225C2×#107—611no—
226C2×#103—1221no—
227C2×C4×C8—611no—
228#366—612no—
229#366—612no—
230C4⋊C16—611no—
231C4⋊C16C2×D6111yescharacteristic
232#358—612no—
233#358—612no—
234#358—612no—
235#358C2×D6112yes—
236C22⋊C16—612no—
237C4×C16—612no—
238C61#3366411yescharacteristic, complemented, Sylow
239C22×Q8—1221no—
240C2×Q16—21no—
241C2×Q16—611no—
242C2×Q16—1221no—
243C2×Q16—1222no—
244C2×Q16—1222no—
245C2×Q16C22×D6112yes—
246C2×Q16C22×D6111yescharacteristic
247C22×C8—611no—
248C22×C8—611no—
249C4⋊Q8—21no—
250C4⋊Q8—1221no—
251#125—2441no—
252C22⋊Q8—1222no—
253C22⋊Q8—2441no—
254C4×Q8—21no—
255C4×Q8—1221no—
256C4×Q8—1221no—
257C4×Q8—1222no—
258C4×Q8—1221no—
259C4×Q8—2441no—
260C4×Q8—2441no—
261C4×Q8—2441no—
262#117—2441no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—2441no—
265C2×C42—611no—
266C2×C42—2441no—
267Q32—24no—
268Q32—1224no—
269C2×C16—612no—
270C2×C16C22×D6112yes—
271#107—611no—
272#107—612no—
273#107—1221no—
274#107—1221no—
275#107C22×D6111yescharacteristic
276#103—21no—
277#103—1221no—
278#103—1221no—
279#103—1222no—
280#103—1222no—
281#103—1221no—
282C4×C8—611no—
283C4×C8—611no—
284C4×C8—611no—
285C4×C8C22×D6111yescharacteristic
286C2×Q8—21no—
287C2×Q8—22no—
288C2×Q8—1224no—
289C2×Q8—1221no—
290C2×Q8—2441no—
291C2×Q8—2442no—
292C2×Q8—2441no—
293C22×C4—611no—
294C22×C4—611no—
295C22×C4—611no—
296C22×C4—2441no—
297C22×C4—2441no—
298Q16—21no—
299Q16—22no—
300Q16—22no—
301Q16—42no—
302Q16—612no—
303Q16—1224no—
304Q16—1221no—
305Q16#1175822412yes—
306C2×C8—611no—
307C2×C8—611no—
308C2×C8—1221no—
309C2×C8—1221no—
310C2×C8—2441no—
311C2×C8D4×D6111yescharacteristic
312C2×C8C23×D6111yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—1221no—
318C4⋊C4—1222no—
319C4⋊C4—2441no—
320C4⋊C4—2441no—
321C4⋊C4—2441no—
322C4⋊C4—2442no—
323C4⋊C4—2442no—
324C4⋊C4—2441no—
325C4⋊C4—2441no—
326C22⋊C4—2442no—
327C42—41no—
328C42—611no—
329C42—611no—
330C42—611no—
331C42—2441no—
332C42—2441no—
333C42—2442no—
334C42—2441no—
335C42D4×D6111yescharacteristic
336C16—22no—
337C16—1222no—
338C23—611no—
339Q8—22no—
340Q8—41no—
341Q8—42no—
342Q8—42no—
343Q8—1222no—
344Q8—4882no—
345Q8—2444no—
346Q8—2441no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—611no—
351C2×C4—614no—
352C2×C4—611no—
353C2×C4—611no—
354C2×C4—1221no—
355C2×C4—1221no—
356C2×C4—4882no—
357C2×C4—2441no—
358C2×C4—2444no—
359C2×C4—2441no—
360C2×C4—2442no—
361C2×C4—2441no—
362C2×C4D8×D6111yescharacteristic
363C2×C4D8×D6111yescharacteristic
364C2×C4C2×D4×D6111yescharacteristic
365C8—21no—
366C8—1221no—
367C8—1221no—
368C8#41×D6111yescharacteristic
369C8C2×#1175822411yescharacteristic
370C22—612no—
371C22—1221no—
372C22—1221no—
373C22C2×D8×D6111yescharacteristic
374C4—21no—
375C4—42no—
376C4—81no—
377C4—82no—
378C4—612no—
379C4—1221no—
380C4—1221no—
381C4—4881no—
382C4—4882no—
383C4—2442no—
384C4Q32×D6112yes—
385C4#1684027211yescharacteristic
386C4#1684055811yescharacteristic
387C2—1222no—
388C2#2145581811yescharacteristic
389C2C2×Q32×D6111yescharacteristic
390C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2209C2
socleC22×C61C22×D8
prime radicalC22×D61C2×D8
hypercenter#2209D61
hypocenterC61#33664
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2209

Complements

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

NH ≅ G/Ntype
4. #21447972C2semidirect
5. #21447971C2semidirect
6. #21447866C2semidirect
8. Q32×Dic61C2semidirect
9. #21447675C2semidirect
11. C61×#2209C2semidirect
164. C4×D61Q32semidirect
198. #2209D61semidirect
238. C61#33664semidirect

Maximal subgroups

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

subgroupindex
#214564032
C4×Q16×D612
#214480202
#214479722
#214479712
#214478662
#214477512
Q32×Dic612
#214476752
#214476172
C61×#22092
#3366461