Finite Group Databasebeta

← #29834671

Subgroups — #29834671

Order 15616.

534 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#21456330C211yesmaximal, characteristic
2C4×Q16×D61C211yesmaximal, characteristic
3#21447986C212yesmaximal, complemented
4#21447985C212yesmaximal, complemented
5#21447972C211yesmaximal, characteristic, complemented
6#21447971C211yesmaximal, characteristic, complemented
7#21447958C211yesmaximal, characteristic, complemented
8#21447606C212yesmaximal
9#21447573C211yesmaximal, characteristic
10#21447537C212yesmaximal
11C61×#2225C211yesmaximal, characteristic, complemented
12C2×Q16×D61C2211yescharacteristic
13#16840317—21no—
14C4×Q8×D61—21no—
15C2×#14519716C2212yes—
16#16840167C2211yescharacteristic
17#16840163—21no—
18#16840163C2212yes—
19#16840158—21no—
20C4×Dic244C2211yescharacteristic
21Q16×Dic61C2211yescharacteristic
22#16839856C2211yescharacteristic
23C4×#11758202—21no—
24#16839854—21no—
25#16839799—21no—
26#16839799C2212yes—
27#107×D61C2211yescharacteristic
28#103×D61—21no—
29C4×C8×D61C2211yescharacteristic
30#16839589C2212yes—
31#16839588C2212yes—
32#16839587C2212yes—
33C4⋊#11758185C2211yescharacteristic
34#16839580C2212yes—
35#16839567C2212yes—
36#16839566C2212yes—
37#16839565C2212yes—
38C61×#500C2211yescharacteristic
39C4×Q16×C61C2211yescharacteristic
40C61×#367C2212yes—
41C61×C4⋊C16C2211yescharacteristic
42C61×#358C2212yes—
43C2×Q8×D61—21no—
44C2×Dic244C2311yescharacteristic
45C2×#11758202—21no—
46Q16×D61—22no—
47Q16×D61Q814yescomplemented
48C2×C8×D61D411yescharacteristic
49C2×C8×D61C2311yescharacteristic
50#14519777—21no—
51#14519768—22no—
52#14519757—41no—
53#14519753—41no—
54Q8×Dic61—21no—
55Q8×Dic61—41no—
56C4×Dic122—21no—
57C4×Dic122—41no—
58#14519742—21no—
59#14519742—41no—
60#14519742—41no—
61#14519735—22no—
62C4⋊C4×D61—21no—
63C4⋊C4×D61—41no—
64C42×D61—41no—
65C42×D61D411yescharacteristic
66#14519716—24no—
67C2×#11758185C2312yes—
68#14519711—21no—
69#14519711C2311yescharacteristic
70#14519711C2312yes—
71#14519710—21no—
72#14519708—22no—
73#14519708—22no—
74#14519702—21no—
75#14519701—21no—
76#14519700—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×C61C2312yes—
86#107×C61C2311yescharacteristic
87#103×C61—21no—
88C4×C8×C61C2311yescharacteristic
89C2×Q8×C61—21no—
90C2×Q8×C61—41no—
91Q8×D61—24no—
92Q8×D61—42no—
93C2×C4×D61—41no—
94C2×C4×D61—41no—
95C2×C4×D61D811yescharacteristic
96C2×C4×D61D811yescharacteristic
97C2×C4×D61C2×D411yescharacteristic
98C2×Dic122—21no—
99C2×Dic122—41no—
100C2×Dic122—41no—
101Q16×C61—21no—
102Q16×C61C2×Q812yes—
103C2×C8×C61C2×D411yescharacteristic
104C2×C8×C61C2411yescharacteristic
105C4⋊C4×C61—21no—
106C4⋊C4×C61—22no—
107C4⋊C4×C61—41no—
108C4⋊C4×C61—41no—
109C42×C61—41no—
110C42×C61C2×D411yescharacteristic
111#11758202—24no—
112#11758198—42no—
113C4⋊Dic61—21no—
114C4⋊Dic61—22no—
115C4⋊Dic61—41no—
116C4⋊Dic61—41no—
117C4⋊Dic61—42no—
118#11758196—41no—
119#11758196—41no—
120#11758196—41no—
121#11758196—42no—
122C4×Dic61—41no—
123C4×Dic61—41no—
124C4×Dic61—41no—
125C4×Dic61—42no—
126C4×Dic61D811yescharacteristic
127C4×Dic61D811yescharacteristic
128C4×Dic61C2×D411yescharacteristic
129C2×#92418C2×D411yescharacteristic
130C2×#92418C2×D411yescharacteristic
131Dic244—21no—
132Dic244C2×Q812yes—
133C8×D61—22no—
134C8×D61C2×Q812yes—
135C8×D61#4112yes—
136C16×C61—22no—
137#11758185—22no—
138C22×D61C2×D811yescharacteristic
139Q8×C61—22no—
140Q8×C61—41no—
141Q8×C61—82no—
142C2×C4×C61—41no—
143C2×C4×C61—41no—
144C2×C4×C61—42no—
145C2×C4×C61C2×D811yescharacteristic
146C2×C4×C61C2×D811yescharacteristic
147C2×C4×C61C22×D411yescharacteristic
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61—42no—
151C2×Dic61C2×D811yescharacteristic
152C2×Dic61C2×D811yescharacteristic
153C2×Dic61C2×D811yescharacteristic
154C4×D61—22no—
155C4×D61—41no—
156C4×D61—44no—
157C4×D61—82no—
158C4×D61C22⋊Q812yes—
159C4×D61#13712yes—
160Dic122—22no—
161Dic122—41no—
162Dic122—44no—
163Dic122—82no—
164C8×C61—21no—
165C8×C61C22×Q811yescharacteristic
166C8×C61C2×#4111yescharacteristic
167#92418—21no—
168#92418C22⋊Q812yes—
169#33628—611nomaximal, Sylow
170C22×C61C22×D811yescharacteristic
171C2×D61—21no—
172C2×D61—21no—
173C2×D61#47412yes—
174C4×C61—21no—
175C4×C61—42no—
176C4×C61—82no—
177C4×C61—81no—
178C4×C61#51012yes—
179C4×C61C2×C22⋊Q811yescharacteristic
180C4×C61C2×#13711yescharacteristic
181Dic61—21no—
182Dic61—21no—
183Dic61—42no—
184Dic61—82no—
185Dic61—81no—
186Dic61#47412yes—
187#3160—611no—
188C2×C4×Q16—611no—
189#2229—612no—
190#2229—612no—
191#2227—611no—
192#2227—611no—
193#2225—611no—
194#2225D6111yescharacteristic, complemented
195#2159—612no—
196#2150—611no—
197#2141—612no—
198C2×C61C2×#47411yescharacteristic
199C2×C61C2×#51011yescharacteristic
200C2×C61#341411yescharacteristic
201D61—22no—
202C22×Q16—611no—
203#533—1221no—
204C2×C4×Q8—1221no—
205C2×M32—612no—
206#500—611no—
207#500C2×D6111yescharacteristic
208#499—612no—
209#499—1221no—
210#498—1221no—
211C4×Q16—611no—
212C4×Q16—611no—
213C4×Q16—611no—
214C4×Q16—1221no—
215C4×Q16—1221no—
216C4×Q16C2×D6111yescharacteristic
217#427—612no—
218#427—1221no—
219C2×#107—611no—
220C2×#103—1221no—
221C2×C4×C8—611no—
222#367—612no—
223#367C2×D6112yes—
224#366—612no—
225#366—612no—
226C4⋊C16—612no—
227C4⋊C16—611no—
228C4⋊C16C2×D6111yescharacteristic
229#358—612no—
230#358—612no—
231#358—612no—
232#358C2×D6112yes—
233C61#3362811yescharacteristic, complemented, Sylow
234C22×Q8—1221no—
235C2×Q16—614no—
236C2×Q16—611no—
237C2×Q16—1222no—
238C2×Q16—1221no—
239C2×Q16C22×D6111yescharacteristic
240C22×C8—611no—
241C22×C8—611no—
242C4⋊Q8—21no—
243C4⋊Q8—1221no—
244#125—1222no—
245C22⋊Q8—2441no—
246C22⋊Q8—2441no—
247C4×Q8—21no—
248C4×Q8—1221no—
249C4×Q8—1221no—
250C4×Q8—1221no—
251C4×Q8—2441no—
252C4×Q8—2441no—
253C4×Q8—2441no—
254C4×Q8—2441no—
255#117—1222no—
256C2×C4⋊C4—1221no—
257C2×C4⋊C4—2441no—
258C2×C42—611no—
259C2×C42—2441no—
260M32—1224no—
261C2×C16—612no—
262C2×C16C22×D6112yes—
263#107—21no—
264#107—612no—
265#107—611no—
266#107—1221no—
267#107—1221no—
268#107C22×D6112yes—
269#107C22×D6111yescharacteristic
270#106—1222no—
271#106—1222no—
272#103—21no—
273#103—1221no—
274#103—1221no—
275#103—1221no—
276C4×C8—611no—
277C4×C8—611no—
278C4×C8—611no—
279C4×C8C22×D6111yescharacteristic
280C2×Q8—21no—
281C2×Q8—41no—
282C2×Q8—1224no—
283C2×Q8—1221no—
284C2×Q8—2441no—
285C2×Q8—2442no—
286C2×Q8—2441no—
287C22×C4—611no—
288C22×C4—611no—
289C22×C4—611no—
290C22×C4—2441no—
291C22×C4—2441no—
292Q16—21no—
293Q16—612no—
294Q16—1224no—
295Q16—1221no—
296Q16Q8×D6112yes—
297C2×C8—611no—
298C2×C8—612no—
299C2×C8—612no—
300C2×C8—611no—
301C2×C8—1222no—
302C2×C8D4×D6111yescharacteristic
303C2×C8C23×D6111yescharacteristic
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—1221no—
309C4⋊C4—1222no—
310C4⋊C4—2441no—
311C4⋊C4—2441no—
312C4⋊C4—2441no—
313C4⋊C4—2442no—
314C4⋊C4—2442no—
315C4⋊C4—2441no—
316C4⋊C4—2441no—
317C22⋊C4—2442no—
318C42—41no—
319C42—611no—
320C42—611no—
321C42—611no—
322C42—2441no—
323C42—2441no—
324C42—2442no—
325C42—2441no—
326C42D4×D6111yescharacteristic
327C16—22no—
328C16—1222no—
329C23—611no—
330Q8—22no—
331Q8—41no—
332Q8—82no—
333Q8—1222no—
334Q8—4882no—
335Q8—2444no—
336Q8—2441no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—611no—
341C2×C4—612no—
342C2×C4—612no—
343C2×C4—611no—
344C2×C4—611no—
345C2×C4—1222no—
346C2×C4—4882no—
347C2×C4—2441no—
348C2×C4—2444no—
349C2×C4—2441no—
350C2×C4—2442no—
351C2×C4—2441no—
352C2×C4D8×D6111yescharacteristic
353C2×C4D8×D6111yescharacteristic
354C2×C4C2×D4×D6111yescharacteristic
355C8—21no—
356C8—612no—
357C8—1221no—
358C8C2×Q8×D6111yescharacteristic
359C8#41×D6111yescharacteristic
360C22—612no—
361C22—1221no—
362C22—1221no—
363C22C2×D8×D6111yescharacteristic
364C4—21no—
365C4—42no—
366C4—82no—
367C4—81no—
368C4—612no—
369C4—1221no—
370C4—1221no—
371C4—4881no—
372C4—4882no—
373C4—2442no—
374C4#1684020112yes—
375C4C22⋊Q8×D6111yescharacteristic
376C4#137×D6111yescharacteristic
377C2—1222no—
378C2#474×D6111yescharacteristic
379C2C2×#1684020111yescharacteristic
380C2#2145895411yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2225C2
socleC22×C61C22×D8
prime radicalC22×D61C2×D8
hypercenter#2225D61
hypocenterC61#33628
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2225

Complements

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

NH ≅ G/Ntype
3. #21447986C2semidirect
4. #21447985C2semidirect
5. #21447972C2semidirect
6. #21447971C2semidirect
7. #21447958C2semidirect
11. C61×#2225C2semidirect
47. Q16×D61Q8semidirect
194. #2225D61semidirect
233. C61#33628semidirect

Maximal subgroups

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

subgroupindex
#214563302
C4×Q16×D612
#214479862
#214479852
#214479722
#214479712
#214479582
#214476062
#214475732
#214475372
C61×#22252
#3362861