Finite Group Databasebeta

← #29835018

Subgroups — #29835018

Order 15616.

552 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#21456390C211yesmaximal, characteristic
2C2×Q16×Dic61C211yesmaximal, characteristic, complemented
3#21448014C211yesmaximal, characteristic, complemented
4#21448000C211yesmaximal, characteristic
5#21447985C212yesmaximal, complemented
6#21447883C212yesmaximal, complemented
7#21447736C211yesmaximal, characteristic
8Q32×Dic61C212yesmaximal, complemented
9#21447668C212yesmaximal, complemented
10#21447615C211yesmaximal, characteristic, complemented
11C61×#2188C211yesmaximal, characteristic, complemented
12#16840298—21no—
13C2×Q8×Dic61—21no—
14C2×#14519720C2212yes—
15#16840163C2211yescharacteristic
16#16840147C2211yescharacteristic
17#16840118—21no—
18#16839998C2211yescharacteristic
19#16839993—21no—
20Q16×Dic61—21no—
21Q16×Dic61—21no—
22Q16×Dic61C2211yescharacteristic
23Q16×Dic61C2212yes—
24Q16×Dic61C2212yes—
25#16839847—21no—
26#16839844C2211yescharacteristic
27C2×#14519711C2211yescharacteristic
28C2×#14519701—21no—
29C2×C8×Dic61C2211yescharacteristic
30#16839589C2212yes—
31#16839588C2212yes—
32#16839580C2212yes—
33#16839566C2212yes—
34#16839566C2212yes—
35#16839564C2212yes—
36C22⋊#11758185C2211yescharacteristic
37C16×Dic61C2212yes—
38C22×Q16×C61C2211yescharacteristic
39C2×Q32×C61C2212yes—
40C61×#467C2211yescharacteristic
41C61×#358C2212yes—
42C61×C22⋊C16C2211yescharacteristic
43C2×#11758201—21no—
44C2×D488C2311yescharacteristic
45C22×#92418D411yescharacteristic
46#14519772—41no—
47#14519768—21no—
48#14519766—21no—
49#14519761—41no—
50#14519751—21no—
51Q8×Dic61—22no—
52Q8×Dic61—22no—
53Q8×Dic61—21no—
54Q8×Dic61—41no—
55Q8×Dic61—41no—
56#14519739—21no—
57#14519737—41no—
58#14519735—41no—
59C2×C4⋊Dic61—21no—
60C2×C4⋊Dic61—41no—
61C2×C4×Dic61—41no—
62C2×C4×Dic61D411yescharacteristic
63#14519720—24no—
64C2×#11758185C2312yes—
65#14519711—21no—
66#14519711C2311yescharacteristic
67#14519711C2312yes—
68#14519708—21no—
69#14519701—21no—
70#14519701—22no—
71#14519701—22no—
72#14519699—21no—
73#14519697—21no—
74C8×Dic61D411yescharacteristic
75C8×Dic61D411yescharacteristic
76C8×Dic61C2311yescharacteristic
77C8×Dic61C2311yescharacteristic
78C22×Q8×C61—21no—
79C2×Q16×C61—24no—
80C2×Q16×C61—21no—
81C2×Q16×C61C2311yescharacteristic
82C2×Q16×C61C2312yes—
83C22×C8×C61C2311yescharacteristic
84C22⋊Q8×C61—21no—
85Q32×C61—24no—
86C2×C16×C61C2312yes—
87#107×C61C2311yescharacteristic
88#103×C61—21no—
89C2×Q8×C61—24no—
90C2×Q8×C61—22no—
91C2×Q8×C61—21no—
92C2×Q8×C61—42no—
93C2×Q8×C61—41no—
94C22×C4×C61—41no—
95C22×C4×C61C2×D411yescharacteristic
96C2×C22⋊D61—41no—
97C22×Dic61—41no—
98C22×Dic61D811yescharacteristic
99C22×Dic61D811yescharacteristic
100C2×D244—21no—
101C2×C4×D61—41no—
102Q16×C61—24no—
103Q16×C61—22no—
104Q16×C61—22no—
105Q16×C61#4114yes—
106C2×C8×C61—22no—
107C2×C8×C61C2×D411yescharacteristic
108C2×C8×C61C2411yescharacteristic
109C4⋊C4×C61—21no—
110C4⋊C4×C61—41no—
111C22⋊C4×C61—41no—
112#11758201—42no—
113#11758198—41no—
114#11758198—41no—
115#11758198—41no—
116#11758198—41no—
117C4⋊Dic61—21no—
118C4⋊Dic61—22no—
119C4⋊Dic61—41no—
120C4⋊Dic61—42no—
121C4⋊Dic61—42no—
122C4⋊Dic61—41no—
123C4⋊Dic61—41no—
124#11758196—41no—
125C4×Dic61—42no—
126C4×Dic61—42no—
127C4×Dic61—41no—
128C4×Dic61D811yescharacteristic
129C4×Dic61D811yescharacteristic
130C4×Dic61C2×D411yescharacteristic
131C4×Dic61C2×D411yescharacteristic
132C2×#92418—41no—
133C2×#92418C2×D411yescharacteristic
134C2×#92418C2×D411yescharacteristic
135D488—22no—
136C16×C61—22no—
137#11758185—22no—
138C23×C61C2×D811yescharacteristic
139C22×D61—41no—
140Q8×C61—24no—
141Q8×C61—44no—
142Q8×C61—42no—
143Q8×C61—41no—
144Q8×C61—41no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—44no—
148C2×C4×C61—42no—
149C2×C4×C61—41no—
150C2×C4×C61—41no—
151C2×C4×C61C2×D811yescharacteristic
152C2×C4×C61C22×D411yescharacteristic
153C22⋊D61—82no—
154C2×Dic61—22no—
155C2×Dic61—22no—
156C2×Dic61—42no—
157C2×Dic61—41no—
158C2×Dic61—82no—
159C2×Dic61C2×D811yescharacteristic
160C2×Dic61C2×D811yescharacteristic
161C2×Dic61C2×D811yescharacteristic
162C2×Dic61C2×D811yescharacteristic
163D244—41no—
164D244—41no—
165C4×D61—82no—
166C8×C61—21no—
167C8×C61C2×#4112yes—
168#92418—21no—
169#92418—21no—
170#33668—611nomaximal, Sylow
171C22×C61—21no—
172C22×C61—21no—
173C22×C61#50812yes—
174C22×C61C22×D811yescharacteristic
175C2×D61—81no—
176C2×D61—81no—
177C2×D61—81no—
178C4×C61—21no—
179C4×C61—44no—
180C4×C61—81no—
181C4×C61—81no—
182C4×C61#52511yescharacteristic
183C4×C61#57511yescharacteristic
184Dic61—21no—
185Dic61—21no—
186Dic61—21no—
187Dic61—82no—
188Dic61—81no—
189Dic61#51012yes—
190#3168—611no—
191C2×C4×Q16—611no—
192#2235—611no—
193#2232—611no—
194#2229—612no—
195#2211—612no—
196#2188D6111yescharacteristic, complemented
197#2185—611no—
198C4×Q32—612no—
199C4×SD32—612no—
200#2162—611no—
201C2×C61—22no—
202C2×C61#310011yescharacteristic
203C2×C61C2×#50811yescharacteristic
204C2×C61C2×#51011yescharacteristic
205D61—161no—
206C22×Q16C2×D6111yescharacteristic
207#529—1221no—
208C2×C4×Q8—1221no—
209C2×Q32C2×D6112yes—
210C2×SD32—612no—
211#499—611no—
212#495—611no—
213#488—1221no—
214#467C2×D6111yescharacteristic
215#466—611no—
216#465—1221no—
217C4×Q16—611no—
218C4×Q16—612no—
219C4×Q16—612no—
220C4×Q16—1221no—
221C4×Q16—1221no—
222C4×SD16—1221no—
223C4×D8—611no—
224C2×#107—611no—
225C2×#103—1221no—
226C2×C4×C8—611no—
227#367—612no—
228#366—612no—
229C4⋊C16—612no—
230#358—612no—
231#358—612no—
232#358C2×D6112yes—
233#357—612no—
234C22⋊C16—611no—
235C22⋊C16C2×D6111yescharacteristic
236C4×C16—612no—
237C61#3366811yescharacteristic, complemented, Sylow
238C22×Q8—21no—
239C2×Q16—24no—
240C2×Q16—21no—
241C2×Q16C22×D6111yescharacteristic
242C2×Q16C22×D6112yes—
243C2×SD16—1221no—
244C2×D8—611no—
245C22×C8—611no—
246C22×C8C22×D6111yescharacteristic
247#126—2441no—
248#125—1221no—
249#124—1221no—
250#123—2441no—
251C22⋊Q8—21no—
252#121—1221no—
253C4×Q8—1222no—
254C4×Q8—1222no—
255C4×Q8—1221no—
256C4×Q8—2441no—
257C4×Q8—2441no—
258C4×D4—1221no—
259C4×D4—2441no—
260#117—2441no—
261C2×C4⋊C4—1221no—
262C2×C4⋊C4—2441no—
263C2×C42—611no—
264C2×C42—2441no—
265Q32—24no—
266SD32—1224no—
267C2×C16—612no—
268C2×C16C22×D6112yes—
269#107—612no—
270#107—611no—
271#107—1221no—
272#107C22×D6111yescharacteristic
273#106—1221no—
274#103—21no—
275#103—1222no—
276#103—1222no—
277#103—1221no—
278#102—1221no—
279#102—1221no—
280C4×C8—611no—
281C4×C8—611no—
282C4×C8—611no—
283C4×C8—611no—
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—21no—
287C2×Q8—42no—
288C2×Q8—41no—
289C2×D4—1221no—
290C2×D4—2441no—
291C22×C4—41no—
292C22×C4—611no—
293C22×C4—611no—
294C22×C4—2441no—
295C22×C4—2441no—
296C22×C4D4×D6111yescharacteristic
297Q16—24no—
298Q16—22no—
299Q16—22no—
300Q16#1175822414yes—
301SD16—2442no—
302D8—1222no—
303C2×C8—22no—
304C2×C8—611no—
305C2×C8—611no—
306C2×C8—2441no—
307C2×C8D4×D6111yescharacteristic
308C2×C8C23×D6111yescharacteristic
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—1221no—
312C4⋊C4—1222no—
313C4⋊C4—2442no—
314C4⋊C4—2442no—
315C4⋊C4—2441no—
316C4⋊C4—2441no—
317C4⋊C4—2441no—
318C4⋊C4—2441no—
319C22⋊C4—41no—
320C22⋊C4—2441no—
321C22⋊C4—2441no—
322C22⋊C4—2441no—
323C22⋊C4—2441no—
324C42—611no—
325C42—611no—
326C42—611no—
327C42—611no—
328C42—2442no—
329C42—2442no—
330C42—2441no—
331C16—22no—
332C16—1222no—
333C23—2441no—
334C23D8×D6111yescharacteristic
335Q8—24no—
336Q8—44no—
337Q8—42no—
338Q8—41no—
339Q8—41no—
340D4—4882no—
341D4—2441no—
342D4—2441no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—611no—
350C2×C4—611no—
351C2×C4—611no—
352C2×C4—611no—
353C2×C4—1222no—
354C2×C4—1222no—
355C2×C4—4882no—
356C2×C4—4882no—
357C2×C4—2442no—
358C2×C4—2441no—
359C2×C4D8×D6111yescharacteristic
360C2×C4C2×D4×D6111yescharacteristic
361C8—21no—
362C8—1221no—
363C8—1221no—
364C8C2×#1175822412yes—
365C22—21no—
366C22—21no—
367C22—4881no—
368C22—4881no—
369C22—4881no—
370C22#1684019512yes—
371C22C2×D8×D6111yescharacteristic
372C4—21no—
373C4—44no—
374C4—81no—
375C4—81no—
376C4—612no—
377C4—1221no—
378C4—1221no—
379C4—1221no—
380C4—4882no—
381C4—4881no—
382C4#1684026811yescharacteristic
383C4#1684055811yescharacteristic
384C2—22no—
385C2—9761no—
386C2#2145581411yescharacteristic
387C2C2×#1684019511yescharacteristic
388C2#510×D6111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2188C2
socleC22×C61C22×D8
prime radical#16839998C22
hypercenter#2188D61
hypocenterC61#33668
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2188

Complements

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

NH ≅ G/Ntype
2. C2×Q16×Dic61C2semidirect
3. #21448014C2semidirect
5. #21447985C2semidirect
6. #21447883C2semidirect
8. Q32×Dic61C2semidirect
9. #21447668C2semidirect
10. #21447615C2semidirect
11. C61×#2188C2semidirect
196. #2188D61semidirect
237. C61#33668semidirect

Maximal subgroups

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

subgroupindex
#214563902
C2×Q16×Dic612
#214480142
#214480002
#214479852
#214478832
#214477362
Q32×Dic612
#214476682
#214476152
C61×#21882
#3366861