Finite Group Databasebeta

← #29835591

Subgroups — #29835591

Order 15616.

510 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456389C211yesmaximal, characteristic
2#21448102C211yesmaximal, characteristic, complemented
3#21448088C211yesmaximal, characteristic, complemented
4#21448064C211yesmaximal, characteristic, complemented
5#21448058C211yesmaximal, characteristic, complemented
6#21447934C212yesmaximal
7#21447925C212yesmaximal, complemented
8C4×#14519723C212yesmaximal, complemented
9C4×#14519720C212yesmaximal, complemented
10C4×#14519717C211yesmaximal, characteristic, complemented
11C61×#3170C211yesmaximal, characteristic, complemented
12#16840296—21no—
13C2×#14519723C2212yes—
14C2×#14519720C2212yes—
15C2×#14519717C2212yes—
16#16840164C2211yescharacteristic
17#16840146C2211yescharacteristic
18#16840117—21no—
19#16840003C2211yescharacteristic
20#16839998C2211yescharacteristic
21#16839991—21no—
22C4×Dic244C2211yescharacteristic
23C4×#11758190—21no—
24C4×D488C2211yescharacteristic
25#16839589C2212yes—
26#16839587C2212yes—
27#16839566C2212yes—
28#16839566C2212yes—
29#16839565C2212yes—
30#16839562C2212yes—
31#16839538C2211yescharacteristic
32#16839538C2211yescharacteristic
33C4×#11758185C2212yes—
34C61×#533—21no—
35C61×#500C2211yescharacteristic
36C61×#494C2211yescharacteristic
37C61×#487—21no—
38C61×#467—21no—
39C61×#467C2212yes—
40C4×Q16×C61—21no—
41C4×Q16×C61C2212yes—
42C2×C4×C8×C61C2211yescharacteristic
43C2×Dic244C2311yescharacteristic
44C2×#11758190—21no—
45C2×D488C2311yescharacteristic
46#14519771—41no—
47#14519769—21no—
48#14519765—21no—
49#14519760—41no—
50#14519756—21no—
51#14519751—21no—
52C4×Dic122—21no—
53C4×D244—21no—
54C4×C22⋊D61—41no—
55#14519734—41no—
56#14519723—24no—
57#14519720—24no—
58#14519717—24no—
59C2×#11758185C2312yes—
60C2×#11758185C2312yes—
61#14519711C2311yescharacteristic
62#14519711C2311yescharacteristic
63#14519709—21no—
64#14519702—21no—
65#14519702—21no—
66#14519699—21no—
67#14519699—21no—
68C2×Q16×C61—21no—
69C2×Q16×C61C2312yes—
70C22×C8×C61D411yescharacteristic
71C22×C8×C61C2311yescharacteristic
72C4⋊Q8×C61—21no—
73C4⋊Q8×C61—21no—
74#125×C61—41no—
75C22⋊Q8×C61—22no—
76C22⋊Q8×C61—41no—
77C4×Q8×C61—22no—
78C4×Q8×C61—41no—
79#117×C61—41no—
80C2×C42×C61D411yescharacteristic
81#107×C61—21no—
82#107×C61C2312yes—
83#103×C61—22no—
84#103×C61—22no—
85C4×C8×C61D411yescharacteristic
86C4×C8×C61D411yescharacteristic
87C4×C8×C61C2311yescharacteristic
88C4×C8×C61C2311yescharacteristic
89C2×Q8×C61—22no—
90C2×Q8×C61—41no—
91C22×C4×C61D811yescharacteristic
92C22×C4×C61D811yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C2×C22⋊D61—41no—
95C2×D244—21no—
96C2×C4×D61—41no—
97C2×Dic122—21no—
98Q16×C61—22no—
99Q16×C61—22no—
100Q16×C61—42no—
101C2×C8×C61—22no—
102C2×C8×C61—22no—
103C2×C8×C61C2×D411yescharacteristic
104C2×C8×C61C2×D411yescharacteristic
105C2×C8×C61C2×D411yescharacteristic
106C2×C8×C61C2411yescharacteristic
107C4⋊C4×C61—22no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61—42no—
111C4⋊C4×C61—42no—
112C4⋊C4×C61—41no—
113C22⋊C4×C61—42no—
114C42×C61—42no—
115C42×C61D811yescharacteristic
116C42×C61D811yescharacteristic
117C42×C61C2×D411yescharacteristic
118C42×C61C2×D411yescharacteristic
119C22⋊Dic61—41no—
120#11758198—41no—
121#11758198—41no—
122#11758198—41no—
123#11758198—41no—
124C4⋊Dic61—21no—
125C4⋊Dic61—21no—
126#11758196—41no—
127#11758196—41no—
128#11758196—41no—
129#11758196—41no—
130C4×Dic61—41no—
131Dic244—22no—
132D488—22no—
133#11758190—42no—
134#11758185—22no—
135#11758185#4114yes—
136C23×C61C2×D811yescharacteristic
137C22×D61—41no—
138Q8×C61—42no—
139Q8×C61—42no—
140Q8×C61—82no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—22no—
144C2×C4×C61—41no—
145C2×C4×C61—42no—
146C2×C4×C61—42no—
147C2×C4×C61C2×D811yescharacteristic
148C2×C4×C61C2×D811yescharacteristic
149C2×C4×C61C2×D811yescharacteristic
150C2×C4×C61C2×D811yescharacteristic
151C2×C4×C61C2×D811yescharacteristic
152C2×C4×C61C22×D411yescharacteristic
153C22⋊D61—82no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—41no—
157D244—41no—
158D244—41no—
159C4×D61—82no—
160Dic122—41no—
161Dic122—41no—
162C8×C61—21no—
163C8×C61—21no—
164C8×C61#13512yes—
165C8×C61C2×#4112yes—
166#33738—611nomaximal, Sylow
167C22×C61—21no—
168C22×C61—21no—
169C22×C61—21no—
170C22×C61C22×D811yescharacteristic
171C2×D61—81no—
172C2×D61—81no—
173C2×D61—81no—
174C4×C61—21no—
175C4×C61—21no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—82no—
179C4×C61—82no—
180C4×C61#51012yes—
181C4×C61#53211yescharacteristic
182C4×C61C2×#13511yescharacteristic
183Dic61—81no—
184Dic61—81no—
185Dic61—81no—
186#3170D6111yescharacteristic, complemented
187#3168—611no—
188#2255—611no—
189#2251—611no—
190#2245—611no—
191#2243—611no—
192#2220—612no—
193#2219—612no—
194C4×Q32—612no—
195C4×SD32—612no—
196C4×M32—611no—
197C2×C61—41no—
198C2×C61#316911yescharacteristic
199C2×C61C2×#51011yescharacteristic
200C2×C61#341411yescharacteristic
201D61—161no—
202#533—21no—
203#529—1221no—
204C2×Q32—612no—
205C2×SD32—612no—
206C2×M32—612no—
207#500C2×D6111yescharacteristic
208#499—611no—
209#495—611no—
210#494C2×D6111yescharacteristic
211#488—1221no—
212#487—21no—
213#467—21no—
214#467—611no—
215#467C2×D6112yes—
216#466—611no—
217#465—1221no—
218C4×Q16—21no—
219C4×Q16—611no—
220C4×Q16C2×D6112yes—
221C4×SD16—1221no—
222C4×D8—611no—
223C2×C4×C8C2×D6111yescharacteristic
224#367—612no—
225#366—612no—
226#358—612no—
227#358—612no—
228#358—612no—
229#357—612no—
230#346—611no—
231#346—611no—
232C4×C16—612no—
233C61#3373811yescharacteristic, complemented, Sylow
234C2×Q16—21no—
235C2×Q16—611no—
236C2×Q16C22×D6112yes—
237C2×SD16—1221no—
238C2×D8—611no—
239C22×C8C22⋊D6111yescharacteristic
240C22×C8C22×D6111yescharacteristic
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243#126—2441no—
244#125—41no—
245#125—1221no—
246#124—1221no—
247#123—2441no—
248C22⋊Q8—22no—
249C22⋊Q8—41no—
250C22⋊Q8—1221no—
251#121—1221no—
252C4×Q8—22no—
253C4×Q8—41no—
254C4×Q8—1221no—
255C4×D4—1221no—
256C4×D4—2441no—
257#117—41no—
258#117—2441no—
259C2×C42C22⋊D6111yescharacteristic
260Q32—1224no—
261SD32—1224no—
262M32—1224no—
263C2×C16—612no—
264C2×C16—612no—
265#107—21no—
266#107—611no—
267#107—611no—
268#107C22×D6112yes—
269#106—1221no—
270#103—22no—
271#103—22no—
272#103—1221no—
273#103—1221no—
274#102—1221no—
275#102—1221no—
276C4×C8C22⋊D6111yescharacteristic
277C4×C8C22⋊D6111yescharacteristic
278C4×C8C22×D6111yescharacteristic
279C4×C8C22×D6111yescharacteristic
280C2×Q8—22no—
281C2×Q8—41no—
282C2×Q8—1221no—
283C2×D4—1221no—
284C2×D4—2441no—
285C22×C4—2441no—
286C22×C4#1175819911yescharacteristic
287C22×C4#1175819911yescharacteristic
288C22×C4C2×C22⋊D6111yescharacteristic
289Q16—22no—
290Q16—22no—
291Q16—42no—
292Q16—1222no—
293SD16—2442no—
294D8—1222no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8C2×C22⋊D6111yescharacteristic
298C2×C8C2×C22⋊D6111yescharacteristic
299C2×C8C2×C22⋊D6111yescharacteristic
300C2×C8C23×D6111yescharacteristic
301C4⋊C4—22no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—1221no—
308C4⋊C4—1221no—
309C4⋊C4—2441no—
310C4⋊C4—2441no—
311C4⋊C4—2441no—
312C4⋊C4—2441no—
313C22⋊C4—42no—
314C22⋊C4—2441no—
315C22⋊C4—2441no—
316C22⋊C4—2441no—
317C22⋊C4—2441no—
318C22⋊C4—2441no—
319C42—42no—
320C42—2441no—
321C42#1175819911yescharacteristic
322C42#1175819911yescharacteristic
323C42C2×C22⋊D6111yescharacteristic
324C42C2×C22⋊D6111yescharacteristic
325C16—614no—
326C16—1222no—
327C23—2441no—
328C23C2×#1175819911yescharacteristic
329Q8—42no—
330Q8—42no—
331Q8—82no—
332Q8—2441no—
333Q8—2441no—
334D4—4882no—
335D4—2441no—
336D4—2441no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—4882no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4—2441no—
347C2×C4C2×#1175819911yescharacteristic
348C2×C4C2×#1175819911yescharacteristic
349C2×C4C2×#1175819911yescharacteristic
350C2×C4C2×#1175819911yescharacteristic
351C2×C4C2×#1175819911yescharacteristic
352C2×C4C22×C22⋊D6111yescharacteristic
353C8—21no—
354C8—21no—
355C8#1451979712yes—
356C8#41×D6112yes—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—4881no—
361C22—4881no—
362C22—4881no—
363C22C22×#1175819911yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—82no—
369C4—82no—
370C4—4881no—
371C4—4881no—
372C4—4881no—
373C4#1684020312yes—
374C4#1684030811yescharacteristic
375C4C2×#1451979711yescharacteristic
376C2—41no—
377C2—9761no—
378C2#2145639411yescharacteristic
379C2C2×#1684020311yescharacteristic
380C2#2145895311yescharacteristic
subgroupG/H
centerC2×C4C2×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#3170C2
socleC22×C61C22×D8
prime radical#16839998C22
hypercenter#3170D61
hypocenterC61#33738
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3170

Complements

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

NH ≅ G/Ntype
2. #21448102C2semidirect
3. #21448088C2semidirect
4. #21448064C2semidirect
5. #21448058C2semidirect
7. #21447925C2semidirect
8. C4×#14519723C2semidirect
9. C4×#14519720C2semidirect
10. C4×#14519717C2semidirect
11. C61×#3170C2semidirect
186. #3170D61semidirect
233. C61#33738semidirect

Maximal subgroups

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

subgroupindex
#214563892
#214481022
#214480882
#214480642
#214480582
#214479342
#214479252
C4×#145197232
C4×#145197202
C4×#145197172
C61×#31702
#3373861