Finite Group Databasebeta

← #29836364

Subgroups — #29836364

Order 15616.

490 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456405C211yesmaximal, characteristic, complemented
2#21447980C212yesmaximal, complemented
3#21447971C211yesmaximal, characteristic, complemented
4#21447966C211yesmaximal, characteristic, complemented
5#21447930C212yesmaximal
6#21447925C212yesmaximal, complemented
7#21447878C212yesmaximal, complemented
8#21447869C211yesmaximal, characteristic, complemented
9#21447864C211yesmaximal, characteristic, complemented
10#21447575C211yesmaximal, characteristic, complemented
11C61×#3168C211yesmaximal, characteristic
12#16840319—21no—
13C2×#14519723C2212yes—
14C2×#14519721C2212yes—
15C2×#14519717C2212yes—
16#16840167C2211yescharacteristic
17#16840142C2211yescharacteristic
18#16840113—21no—
19#16840003—21no—
20#16840003C2212yes—
21C4×Dic244—21no—
22C4×Dic244C2212yes—
23#16839589C2212yes—
24#16839587C2212yes—
25C4⋊#11758185C2211yescharacteristic
26C4⋊#11758185C2211yescharacteristic
27C4⋊#11758185C2212yes—
28#16839566C2212yes—
29#16839565C2212yes—
30#16839565C2212yes—
31#16839563C2212yes—
32C61×#529—21no—
33C61×#499C2211yescharacteristic
34C61×#495C2211yescharacteristic
35C61×#488—21no—
36C61×#467C2211yescharacteristic
37C61×#466C2211yescharacteristic
38C61×#465—21no—
39C4×Q16×C61C2211yescharacteristic
40C4×SD16×C61—21no—
41C4×D8×C61C2211yescharacteristic
42C2×C4×C8×C61C2211yescharacteristic
43C2×Dic244—21no—
44C2×Dic244C2312yes—
45#14519777—21no—
46#14519777—21no—
47#14519769—41no—
48#14519756—22no—
49#14519756—41no—
50C4×Dic122—22no—
51C4×Dic122—41no—
52#14519736—41no—
53#14519723—42no—
54#14519721—42no—
55#14519717—42no—
56C2×#11758185C2312yes—
57C2×#11758185C2312yes—
58#14519711—21no—
59#14519711C2312yes—
60#14519702—22no—
61#14519702—22no—
62C2×Q16×C61C2311yescharacteristic
63C2×SD16×C61—21no—
64C2×D8×C61C2311yescharacteristic
65C22×C8×C61D411yescharacteristic
66C22×C8×C61C2311yescharacteristic
67#126×C61—41no—
68#125×C61—21no—
69#124×C61—21no—
70#123×C61—41no—
71C22⋊Q8×C61—21no—
72#121×C61—21no—
73C4×Q8×C61—21no—
74C4×D4×C61—21no—
75C4×D4×C61—41no—
76#117×C61—41no—
77C2×C42×C61D411yescharacteristic
78#107×C61C2311yescharacteristic
79#107×C61C2311yescharacteristic
80#106×C61—21no—
81#103×C61—21no—
82#103×C61—21no—
83#102×C61—21no—
84#102×C61—21no—
85C4×C8×C61D411yescharacteristic
86C4×C8×C61D411yescharacteristic
87C4×C8×C61C2311yescharacteristic
88C4×C8×C61C2311yescharacteristic
89C2×Q8×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61—41no—
92C22×C4×C61—41no—
93C22×C4×C61D811yescharacteristic
94C22×C4×C61D811yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C2×Dic122—22no—
97C2×Dic122—41no—
98Q16×C61—21no—
99Q16×C61—21no—
100SD16×C61—42no—
101D8×C61—21no—
102D8×C61—21no—
103C2×C8×C61—21no—
104C2×C8×C61—21no—
105C2×C8×C61—41no—
106C2×C8×C61C2×D411yescharacteristic
107C2×C8×C61C2×D411yescharacteristic
108C2×C8×C61C2×D411yescharacteristic
109C2×C8×C61C2411yescharacteristic
110C4⋊C4×C61—21no—
111C4⋊C4×C61—21no—
112C4⋊C4×C61—41no—
113C4⋊C4×C61—41no—
114C4⋊C4×C61—41no—
115C4⋊C4×C61—41no—
116C22⋊C4×C61—41no—
117C22⋊C4×C61—41no—
118C22⋊C4×C61—41no—
119C22⋊C4×C61—41no—
120C22⋊C4×C61—41no—
121C42×C61—41no—
122C42×C61D811yescharacteristic
123C42×C61D811yescharacteristic
124C42×C61C2×D411yescharacteristic
125C42×C61C2×D411yescharacteristic
126C22⋊Dic61—42no—
127C4⋊Dic61—22no—
128C4⋊Dic61—41no—
129C4⋊Dic61—41no—
130C4⋊Dic61—41no—
131#11758196—42no—
132#11758196—42no—
133C4×Dic61—42no—
134Dic244—22no—
135Dic244—22no—
136Dic244—42no—
137#11758185—22no—
138#11758185—22no—
139C23×C61—41no—
140C23×C61C2×D811yescharacteristic
141Q8×C61—41no—
142Q8×C61—41no—
143D4×C61—41no—
144D4×C61—41no—
145D4×C61—82no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—22no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61—41no—
152C2×C4×C61—41no—
153C2×C4×C61—82no—
154C2×C4×C61C2×D811yescharacteristic
155C2×C4×C61C2×D811yescharacteristic
156C2×C4×C61C2×D811yescharacteristic
157C2×C4×C61C2×D811yescharacteristic
158C2×C4×C61C2×D811yescharacteristic
159C2×C4×C61C22×D411yescharacteristic
160C2×Dic61—42no—
161C2×Dic61—42no—
162Dic122—42no—
163Dic122—42no—
164Dic122—82no—
165C8×C61—21no—
166C8×C61—21no—
167C8×C61—21no—
168C8×C61H(5,𝔽2)11yescharacteristic
169C8×C612−1+411yescharacteristic
170#33818—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—81no—
175C22×C61—81no—
176C22×C61—81no—
177C22×C61C22×D811yescharacteristic
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—21no—
182C4×C61—81no—
183C4×C61—81no—
184C4×C61—81no—
185C4×C61#51012yes—
186C4×C61#53711yescharacteristic
187C4×C61#57511yescharacteristic
188Dic61—82no—
189Dic61—82no—
190#3170—611no—
191#3168D6111yescharacteristic
192#2229—612no—
193#2227—611no—
194#2226—611no—
195#2220—612no—
196#2219—612no—
197#2211—612no—
198#2209—611no—
199#2208—611no—
200#2150—611no—
201C2×C61—41no—
202C2×C61—161no—
203C2×C61#324511yescharacteristic
204C2×C61C2×#51011yescharacteristic
205C2×C61#341411yescharacteristic
206#533—1221no—
207#529—21no—
208C2×Q32—612no—
209C2×SD32—612no—
210C2×M32—612no—
211#500—611no—
212#499C2×D6111yescharacteristic
213#495C2×D6111yescharacteristic
214#494—611no—
215#488—21no—
216#487—1221no—
217#467—612no—
218#467—1221no—
219#467C2×D6111yescharacteristic
220#466C2×D6111yescharacteristic
221#465—21no—
222C4×Q16—612no—
223C4×Q16—1221no—
224C4×Q16C2×D6111yescharacteristic
225C4×SD16—21no—
226C4×D8C2×D6111yescharacteristic
227C2×C4×C8C2×D6111yescharacteristic
228#367—612no—
229#366—612no—
230C4⋊C16—611no—
231C4⋊C16—611no—
232C4⋊C16—612no—
233#358—612no—
234#358—612no—
235#358—612no—
236#357—612no—
237C61#3381811yescharacteristic, complemented, Sylow
238C2×Q16—612no—
239C2×Q16—1221no—
240C2×Q16C22×D6111yescharacteristic
241C2×SD16—21no—
242C2×D8C22×D6111yescharacteristic
243C22×C8C22⋊D6111yescharacteristic
244C22×C8C22×D6111yescharacteristic
245C4⋊Q8—1221no—
246C4⋊Q8—1221no—
247#126—41no—
248#125—21no—
249#125—2441no—
250#124—21no—
251#123—41no—
252C22⋊Q8—21no—
253C22⋊Q8—1222no—
254C22⋊Q8—2441no—
255#121—21no—
256C4×Q8—21no—
257C4×Q8—1222no—
258C4×Q8—2441no—
259C4×D4—21no—
260C4×D4—41no—
261#117—41no—
262#117—2441no—
263C2×C42C22⋊D6111yescharacteristic
264Q32—2442no—
265SD32—2442no—
266M32—2442no—
267C2×C16—612no—
268C2×C16—612no—
269#107—612no—
270#107—1221no—
271#107C22×D6111yescharacteristic
272#107C22×D6111yescharacteristic
273#106—21no—
274#103—21no—
275#103—21no—
276#103—1222no—
277#103—1222no—
278#102—21no—
279#102—21no—
280C4×C8C22⋊D6111yescharacteristic
281C4×C8C22⋊D6111yescharacteristic
282C4×C8C22×D6111yescharacteristic
283C4×C8C22×D6111yescharacteristic
284C2×Q8—21no—
285C2×Q8—1222no—
286C2×Q8—2441no—
287C2×D4—21no—
288C2×D4—41no—
289C22×C4—41no—
290C22×C4#1175819911yescharacteristic
291C22×C4#1175819911yescharacteristic
292C22×C4C2×C22⋊D6111yescharacteristic
293Q16—21no—
294Q16—21no—
295Q16—1222no—
296Q16—1222no—
297Q16—2442no—
298SD16—42no—
299D8—21no—
300D8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—41no—
304C2×C8C2×C22⋊D6111yescharacteristic
305C2×C8C2×C22⋊D6111yescharacteristic
306C2×C8C2×C22⋊D6111yescharacteristic
307C2×C8C23×D6111yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—1222no—
315C4⋊C4—2442no—
316C4⋊C4—2442no—
317C4⋊C4—2441no—
318C4⋊C4—2441no—
319C4⋊C4—2441no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—2442no—
326C42—41no—
327C42—2442no—
328C42#1175819911yescharacteristic
329C42#1175819911yescharacteristic
330C42C2×C22⋊D6111yescharacteristic
331C42C2×C22⋊D6111yescharacteristic
332C16—1222no—
333C16—1222no—
334C23—41no—
335C23C2×#1175819911yescharacteristic
336Q8—41no—
337Q8—41no—
338Q8—4882no—
339Q8—2442no—
340Q8—2442no—
341D4—41no—
342D4—41no—
343D4—82no—
344C2×C4—22no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—82no—
352C2×C4—2442no—
353C2×C4—2442no—
354C2×C4C2×#1175819911yescharacteristic
355C2×C4C2×#1175819911yescharacteristic
356C2×C4C2×#1175819911yescharacteristic
357C2×C4C2×#1175819911yescharacteristic
358C2×C4C2×#1175819911yescharacteristic
359C2×C4C22×C22⋊D6111yescharacteristic
360C8—21no—
361C8—21no—
362C8—21no—
363C8#1451982311yescharacteristic
364C8#1451982411yescharacteristic
365C22—21no—
366C22—21no—
367C22—21no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22C22×#1175819911yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—4882no—
380C4—4882no—
381C4#1684020212yes—
382C4#1684034011yescharacteristic
383C4#1684056011yescharacteristic
384C2—41no—
385C2—161no—
386C2#2145712611yescharacteristic
387C2C2×#1684020211yescharacteristic
388C2#2145895311yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3168C2
socleC22×C61C22×D8
prime radicalC61×#466C22
hypercenter#3168D61
hypocenterC61#33818
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
1. #21456405C2semidirect
2. #21447980C2semidirect
3. #21447971C2semidirect
4. #21447966C2semidirect
6. #21447925C2semidirect
7. #21447878C2semidirect
8. #21447869C2semidirect
9. #21447864C2semidirect
10. #21447575C2semidirect
237. C61#33818semidirect

Maximal subgroups

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

subgroupindex
#214564052
#214479802
#214479712
#214479662
#214479302
#214479252
#214478782
#214478692
#214478642
#214475752
C61×#31682
#3381861