Finite Group Databasebeta

← #29836367

Subgroups — #29836367

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#21456406C211yesmaximal, characteristic, complemented
2#21456392C211yesmaximal, characteristic
3#21447982C212yesmaximal, complemented
4#21447963C211yesmaximal, characteristic, complemented
5#21447933C212yesmaximal
6#21447926C212yesmaximal, complemented
7#21447884C212yesmaximal, complemented
8#21447869C211yesmaximal, characteristic, complemented
9#21447865C211yesmaximal, characteristic, complemented
10#21447573C211yesmaximal, characteristic, complemented
11C61×#2227C211yesmaximal, characteristic, complemented
12#16840317—21no—
13#16840299—21no—
14C2×#14519723C2212yes—
15C2×#14519722C2212yes—
16C2×#14519716C2212yes—
17#16840163C2211yescharacteristic
18#16840147C2211yescharacteristic
19#16840142C2211yescharacteristic
20#16840118—21no—
21#16840114—21no—
22#16840004—21no—
23#16840004C2212yes—
24#16840000C2211yescharacteristic
25#16839999C2211yescharacteristic
26#16839994—21no—
27#16839856—21no—
28#16839856C2212yes—
29C4×#11758201—21no—
30C4×D488C2211yescharacteristic
31C4×C8×D61C2211yescharacteristic
32#16839589C2212yes—
33C4⋊#11758185C2211yescharacteristic
34#16839580C2212yes—
35#16839567C2212yes—
36#16839565C2212yes—
37#16839562C2212yes—
38C61×#500C2211yescharacteristic
39C4×Q16×C61C2211yescharacteristic
40C61×#366C2212yes—
41C61×C4⋊C16C2211yescharacteristic
42C61×#358C2212yes—
43C2×Dic244—21no—
44C2×Dic244C2312yes—
45C2×#11758201—21no—
46C2×D488C2311yescharacteristic
47C2×C8×D61D411yescharacteristic
48C2×C8×D61C2311yescharacteristic
49#14519777—21no—
50#14519772—41no—
51#14519768—21no—
52#14519768—41no—
53#14519766—21no—
54#14519762—41no—
55#14519757—22no—
56#14519753—21no—
57#14519753—41no—
58#14519752—21no—
59Q8×Dic61—41no—
60#14519742—22no—
61C4×D244—21no—
62C4×D244—41no—
63#14519735—41no—
64#14519734—41no—
65C42×D61D411yescharacteristic
66#14519723—42no—
67#14519722—42no—
68#14519716—42no—
69C2×#11758185C2312yes—
70#14519711C2311yescharacteristic
71#14519708—21no—
72#14519701—21no—
73#14519700—22no—
74#14519700—22no—
75#14519699—21no—
76#14519697—21no—
77C4×#92418D411yescharacteristic
78C8×Dic61D411yescharacteristic
79C8×Dic61C2311yescharacteristic
80C2×Q16×C61C2311yescharacteristic
81C4⋊Q8×C61—21no—
82C4×Q8×C61—21no—
83C2×C16×C61C2312yes—
84#107×C61—21no—
85#107×C61C2311yescharacteristic
86#107×C61C2312yes—
87#103×C61—21no—
88C4×C8×C61C2311yescharacteristic
89C2×Q8×C61—21no—
90C2×Q8×C61—41no—
91C2×D244—21no—
92C2×D244—41no—
93C2×C4×D61—41no—
94C2×C4×D61D811yescharacteristic
95C2×C4×D61D811yescharacteristic
96C2×C4×D61C2×D411yescharacteristic
97C2×Dic122—22no—
98Q16×C61—21no—
99Q16×C61—21no—
100C2×C8×C61C2×D411yescharacteristic
101C2×C8×C61C2411yescharacteristic
102C4⋊C4×C61—21no—
103C4⋊C4×C61—22no—
104C4⋊C4×C61—41no—
105C4⋊C4×C61—41no—
106C42×C61—41no—
107C42×C61C2×D411yescharacteristic
108#11758201—42no—
109#11758198—42no—
110#11758198—41no—
111#11758198—41no—
112#11758198—41no—
113#11758198—41no—
114#11758198—41no—
115C4⋊Dic61—21no—
116C4⋊Dic61—41no—
117C4⋊Dic61—41no—
118C4⋊Dic61—41no—
119C4⋊Dic61—42no—
120#11758196—41no—
121#11758196—41no—
122#11758196—42no—
123C4×Dic61—42no—
124C4×Dic61D811yescharacteristic
125C4×Dic61D811yescharacteristic
126C4×Dic61C2×D411yescharacteristic
127C2×#92418C2×D411yescharacteristic
128C2×#92418C2×D411yescharacteristic
129Dic244—22no—
130Dic244—22no—
131Dic244—42no—
132D488—21no—
133D488—21no—
134C8×D61—21no—
135C8×D61—21no—
136C8×D61—41no—
137C16×C61—22no—
138#11758185—22no—
139C22×D61—41no—
140C22×D61C2×D811yescharacteristic
141Q8×C61—41no—
142Q8×C61—41no—
143Q8×C61—82no—
144C2×C4×C61—42no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61C2×D811yescharacteristic
148C2×C4×C61C2×D811yescharacteristic
149C2×C4×C61C22×D411yescharacteristic
150C2×Dic61—42no—
151C2×Dic61—41no—
152C2×Dic61C2×D811yescharacteristic
153C2×Dic61C2×D811yescharacteristic
154C2×Dic61C2×D811yescharacteristic
155D244—41no—
156D244—41no—
157D244—82no—
158C4×D61—21no—
159C4×D61—21no—
160C4×D61—22no—
161C4×D61—41no—
162C4×D61—82no—
163Dic122—42no—
164Dic122—42no—
165C8×C61—21no—
166C8×C61H(5,𝔽2)11yescharacteristic
167C8×C612−1+411yescharacteristic
168#92418—21no—
169#92418—21no—
170#33818—611nomaximal, Sylow
171C22×C61C22×D811yescharacteristic
172C2×D61—21no—
173C2×D61—21no—
174C2×D61—21no—
175C2×D61—81no—
176C2×D61—81no—
177C2×D61—81no—
178C4×C61—21no—
179C4×C61—81no—
180C4×C61—82no—
181C4×C61—81no—
182C4×C61#51012yes—
183C4×C61#53711yescharacteristic
184C4×C61#57511yescharacteristic
185Dic61—21no—
186Dic61—21no—
187Dic61—21no—
188Dic61—81no—
189Dic61—82no—
190#3170—611no—
191#3168—611no—
192#2229—612no—
193#2227D6111yescharacteristic, complemented
194#2226—611no—
195#2220—612no—
196#2219—612no—
197#2211—612no—
198#2209—611no—
199#2208—611no—
200#2150—611no—
201C2×C61#324511yescharacteristic
202C2×C61C2×#51011yescharacteristic
203C2×C61#341411yescharacteristic
204D61—41no—
205D61—161no—
206#533—1221no—
207#529—1221no—
208C2×Q32—612no—
209C2×SD32—612no—
210C2×M32—612no—
211#500C2×D6111yescharacteristic
212#499—611no—
213#495—611no—
214#494—611no—
215#488—1221no—
216#487—1221no—
217#467—611no—
218#467—612no—
219#467—1221no—
220#466—611no—
221#465—1221no—
222C4×Q16—612no—
223C4×Q16—1221no—
224C4×Q16C2×D6111yescharacteristic
225C4×SD16—1221no—
226C4×D8—611no—
227C2×C4×C8—611no—
228#367—612no—
229#366C2×D6112yes—
230C4⋊C16—612no—
231C4⋊C16—611no—
232C4⋊C16C2×D6111yescharacteristic
233#358—612no—
234#358—612no—
235#358C2×D6112yes—
236#357—612no—
237C61#3381811yescharacteristic, complemented, Sylow
238C2×Q16—612no—
239C2×Q16—1221no—
240C2×Q16C22×D6111yescharacteristic
241C2×SD16—1221no—
242C2×D8—611no—
243C22×C8—611no—
244C22×C8—611no—
245C4⋊Q8—21no—
246C4⋊Q8—1221no—
247#126—2441no—
248#125—1221no—
249#125—2441no—
250#124—1221no—
251#123—2441no—
252C22⋊Q8—1222no—
253C22⋊Q8—1221no—
254C22⋊Q8—2441no—
255#121—1221no—
256C4×Q8—21no—
257C4×Q8—1222no—
258C4×Q8—2441no—
259C4×D4—1221no—
260C4×D4—2441no—
261#117—2441no—
262#117—2441no—
263C2×C42—611no—
264Q32—2442no—
265SD32—2442no—
266M32—2442no—
267C2×C16—612no—
268C2×C16C22×D6112yes—
269#107—21no—
270#107—611no—
271#107C22×D6112yes—
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×C8C22×D6111yescharacteristic
284C2×Q8—21no—
285C2×Q8—41no—
286C2×Q8—1222no—
287C2×D4—1221no—
288C2×D4—2441no—
289C22×C4—611no—
290C22×C4—611no—
291C22×C4—611no—
292C22×C4—2441no—
293Q16—21no—
294Q16—21no—
295Q16—1222no—
296Q16—1222no—
297Q16—2442no—
298SD16—2442no—
299D8—1221no—
300D8—1221no—
301C2×C8—611no—
302C2×C8—611no—
303C2×C8—1221no—
304C2×C8—1221no—
305C2×C8—2441no—
306C2×C8D4×D6111yescharacteristic
307C2×C8C23×D6111yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—1221no—
313C4⋊C4—2441no—
314C4⋊C4—2441no—
315C4⋊C4—2442no—
316C4⋊C4—2441no—
317C4⋊C4—2442no—
318C4⋊C4—2441no—
319C4⋊C4—2441no—
320C22⋊C4—2441no—
321C22⋊C4—2441no—
322C22⋊C4—2441no—
323C22⋊C4—2442no—
324C22⋊C4—2441no—
325C22⋊C4—2441no—
326C42—41no—
327C42—611no—
328C42—611no—
329C42—611no—
330C42—2442no—
331C42D4×D6111yescharacteristic
332C16—22no—
333C16—1222no—
334C23—611no—
335C23—2441no—
336Q8—41no—
337Q8—41no—
338Q8—82no—
339Q8—2442no—
340Q8—2442no—
341D4—4882no—
342D4—2441no—
343D4—2441no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—611no—
348C2×C4—611no—
349C2×C4—611no—
350C2×C4—1222no—
351C2×C4—1221no—
352C2×C4—1221no—
353C2×C4—4882no—
354C2×C4—2441no—
355C2×C4—2442no—
356C2×C4—2441no—
357C2×C4D8×D6111yescharacteristic
358C2×C4D8×D6111yescharacteristic
359C2×C4C2×D4×D6111yescharacteristic
360C8—21no—
361C8—1221no—
362C8—1221no—
363C8#1451982311yescharacteristic
364C8#1451982511yescharacteristic
365C22—1221no—
366C22—1221no—
367C22—1221no—
368C22—4881no—
369C22—4881no—
370C22—4881no—
371C22C2×D8×D6111yescharacteristic
372C4—21no—
373C4—81no—
374C4—82no—
375C4—81no—
376C4—1221no—
377C4—1221no—
378C4—1221no—
379C4—4881no—
380C4—4882no—
381C4#1684020412yes—
382C4#1684034211yescharacteristic
383C4#1684056111yescharacteristic
384C2—9761no—
385C2—2441no—
386C2#2145712811yescharacteristic
387C2C2×#1684020411yescharacteristic
388C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2227C2
socleC22×C61C22×D8
prime radical#16839999C22
hypercenter#2227D61
hypocenterC61#33818
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2227

Complements

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

NH ≅ G/Ntype
1. #21456406C2semidirect
3. #21447982C2semidirect
4. #21447963C2semidirect
6. #21447926C2semidirect
7. #21447884C2semidirect
8. #21447869C2semidirect
9. #21447865C2semidirect
10. #21447573C2semidirect
11. C61×#2227C2semidirect
193. #2227D61semidirect
237. C61#33818semidirect

Maximal subgroups

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

subgroupindex
#214564062
#214563922
#214479822
#214479632
#214479332
#214479262
#214478842
#214478692
#214478652
#214475732
C61×#22272
#3381861