Finite Group Databasebeta

← #29834665

Subgroups — #29834665

Order 15616.

598 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456331C211yesmaximal, characteristic, complemented
2C2×D8×Dic61C211yesmaximal, characteristic
3#21447978C212yesmaximal, complemented
4#21447974C212yesmaximal, complemented
5#21447964C212yesmaximal, complemented
6#21447954C212yesmaximal, complemented
7#21447608C211yesmaximal, characteristic, complemented
8#21447607C211yesmaximal, characteristic, complemented
9#21447574C211yesmaximal, characteristic, complemented
10#21447535C211yesmaximal, characteristic
11C61×#2140C211yesmaximal, characteristic
12#16840315—21no—
13C2×D4×Dic61—21no—
14C2×#14519717C2211yescharacteristic
15#16840166C2211yescharacteristic, complemented
16#16840166C2211yescharacteristic, complemented
17#16840163C2211yescharacteristic, complemented
18#16840163C2211yescharacteristic, complemented
19#16840162—21no—
20#16840159—21no—
21D8×Dic61—21no—
22D8×Dic61—21no—
23D8×Dic61C2212yes—
24D8×Dic61C2212yes—
25#16839800C2211yescharacteristic
26#16839798—21no—
27C2×#14519711C2211yescharacteristic
28C2×#14519698—21no—
29C2×C8×Dic61C2211yescharacteristic
30#16839590C2212yescomplemented
31#16839589C2212yescomplemented
32#16839588C2212yescomplemented
33#16839587C2212yescomplemented
34#16839580C2212yescomplemented
35#16839580C2212yescomplemented
36#16839563C2212yes—
37#16839563C2212yes—
38C22×D8×C61C2211yescharacteristic
39C2×M32×C61C2211yescharacteristic
40C61×#427C2211yescharacteristic
41C61×#357C2212yes—
42C61×#357C2212yes—
43C23×Dic61—41no—
44C22×#92418D411yescharacteristic
45#14519776—21no—
46#14519776—21no—
47#14519768—21no—
48#14519768—21no—
49#14519754—41no—
50#14519754—41no—
51C4×Dic122—41no—
52C4×Dic122—41no—
53D4×Dic61—22no—
54D4×Dic61—22no—
55D4×Dic61—41no—
56D4×Dic61—41no—
57#14519736—21no—
58C2×C4⋊Dic61—21no—
59C2×C22⋊Dic61—41no—
60C2×C4×Dic61D411yescharacteristic
61#14519717—22no—
62C2×#11758185C2312yes—
63#14519711—21no—
64#14519711C2312yes—
65#14519711C2311yescharacteristic
66#14519711C2311yescharacteristic
67#14519710—21no—
68#14519710—21no—
69#14519709—21no—
70#14519708—21no—
71#14519708—21no—
72#14519698—22no—
73#14519698—22no—
74C8×Dic61D411yescharacteristic
75C8×Dic61D411yescharacteristic
76C8×Dic61C2311yescharacteristic
77C8×Dic61C2311yescharacteristic
78C22×D4×C61—21no—
79C2×D8×C61—21no—
80C2×D8×C61—22no—
81C2×D8×C61Q814yescomplemented
82C2×D8×C61C2312yes—
83C22×C8×C61C2311yescharacteristic
84#117×C61—21no—
85M32×C61—22no—
86C2×C16×C61C2312yes—
87#107×C61C2311yescharacteristic
88#107×C61C2311yescharacteristic
89#106×C61—21no—
90C24×C61—41no—
91C2×D4×C61—24no—
92C2×D4×C61—22no—
93C2×D4×C61—41no—
94C2×D4×C61—42no—
95C22×C4×C61C2×D411yescharacteristic
96C22×Dic61—41no—
97C22×Dic61—42no—
98C22×Dic61—42no—
99C22×Dic61—84no—
100C22×Dic61D811yescharacteristic
101C22×Dic61D811yescharacteristic
102C2×Dic122—41no—
103C2×Dic122—41no—
104D8×C61—24no—
105D8×C61—22no—
106D8×C61C2×Q814yes—
107C2×C8×C61C2×D411yescharacteristic
108C2×C8×C61C2×Q812yes—
109C2×C8×C61#4112yes—
110C2×C8×C61C2411yescharacteristic
111C4⋊C4×C61—21no—
112C4⋊C4×C61—21no—
113C22⋊C4×C61—41no—
114C42×C61—41no—
115C22⋊Dic61—41no—
116C22⋊Dic61—42no—
117C22⋊Dic61—42no—
118C4⋊Dic61—22no—
119C4⋊Dic61—21no—
120C4⋊Dic61—21no—
121C4⋊Dic61—41no—
122C4⋊Dic61—41no—
123C4⋊Dic61—41no—
124#11758196—41no—
125#11758196—41no—
126#11758196—41no—
127#11758196—41no—
128C4×Dic61—41no—
129C4×Dic61D811yescharacteristic
130C4×Dic61D811yescharacteristic
131C4×Dic61C2×D411yescharacteristic
132C4×Dic61C2×D411yescharacteristic
133C2×#92418—22no—
134C2×#92418C2×D411yescharacteristic
135C2×#92418C2×D411yescharacteristic
136C16×C61—22no—
137#11758185—22no—
138C23×C61—42no—
139C23×C61—44no—
140C23×C61—82no—
141C23×C61—82no—
142C23×C61C2×D811yescharacteristic
143D4×C61—24no—
144D4×C61—42no—
145D4×C61—44no—
146C2×C4×C61—41no—
147C2×C4×C61—41no—
148C2×C4×C61C22⋊Q812yes—
149C2×C4×C61C2×D811yescharacteristic
150C2×C4×C61#13712yes—
151C2×C4×C61C22×D411yescharacteristic
152C2×Dic61—22no—
153C2×Dic61—41no—
154C2×Dic61—41no—
155C2×Dic61—41no—
156C2×Dic61—42no—
157C2×Dic61—84no—
158C2×Dic61—82no—
159C2×Dic61—84no—
160C2×Dic61C2×D811yescharacteristic
161C2×Dic61C2×D811yescharacteristic
162C2×Dic61C2×D811yescharacteristic
163C2×Dic61C2×D811yescharacteristic
164Dic122—82no—
165Dic122—82no—
166C8×C61C22⋊Q812yes—
167C8×C61C22×Q811yescharacteristic
168C8×C61C2×#4111yescharacteristic
169#92418—21no—
170#92418—21no—
171#33627—611nomaximal, Sylow
172C22×C61—21no—
173C22×C61—21no—
174C22×C61—44no—
175C22×C61—88no—
176C22×C61—82no—
177C22×C61—82no—
178C22×C61#47412yes—
179C22×C61C22×D811yescharacteristic
180C4×C61—81no—
181C4×C61—81no—
182C4×C61#47412yes—
183C4×C61C2×C22⋊Q811yescharacteristic
184C4×C61C2×#13711yescharacteristic
185Dic61—21no—
186Dic61—21no—
187Dic61—21no—
188Dic61—81no—
189Dic61—82no—
190Dic61—81no—
191Dic61#50912yes—
192#3160—611no—
193C2×C4×D8—611no—
194#2228—612no—
195#2228—612no—
196#2226—612no—
197#2224—612no—
198#2159—611no—
199#2159—611no—
200#2150—611no—
201#2140—611no—
202#2140D6111yescharacteristic
203C2×C61—22no—
204C2×C61—84no—
205C2×C61C2×#47411yescharacteristic
206C2×C61C2×#50911yescharacteristic
207C2×C61#341411yescharacteristic
208C22×D8C2×D6111yescharacteristic
209#533—1221no—
210C2×C4×D4—1221no—
211C2×M32—611no—
212C2×M32C2×D6111yescharacteristic
213#500—611no—
214#500—611no—
215#499—611no—
216#499—611no—
217#499—1221no—
218#498—1221no—
219C4×D8—612no—
220C4×D8—612no—
221C4×D8—1221no—
222C4×D8—1221no—
223#427—611no—
224#427—1221no—
225#427C2×D6111yescharacteristic
226C2×#107—611no—
227C2×#102—1221no—
228C2×C4×C8—611no—
229#367—612no—
230#367—612no—
231#366—612no—
232#366—612no—
233C4⋊C16—612no—
234C4⋊C16—612no—
235#357—612no—
236#357—612no—
237#357C2×D6112yes—
238#357C2×D6112yes—
239C61#3362711yescharacteristic, complemented, Sylow
240C22×D4—21no—
241C23×C4—2441no—
242C2×D8—21no—
243C2×D8—22no—
244C2×D8Dic12214yescomplemented
245C2×D8C22×D6112yes—
246C22×C8—611no—
247C22×C8C22×D6111yescharacteristic
248C4⋊Q8—1221no—
249C4⋊Q8—1221no—
250#125—1221no—
251#125—1221no—
252C22⋊Q8—2441no—
253C22⋊Q8—2441no—
254C4×Q8—2441no—
255C4×Q8—2441no—
256C4×D4—1222no—
257C4×D4—1222no—
258C4×D4—2441no—
259C4×D4—2441no—
260#117—21no—
261#117—1221no—
262C2×C4⋊C4—1221no—
263C2×C22⋊C4—2441no—
264C2×C42—611no—
265M32—22no—
266M32—1222no—
267C2×C16—612no—
268C2×C16C22×D6112yes—
269#107—611no—
270#107—611no—
271#107—612no—
272#107—1221no—
273#107—1221no—
274#107—1221no—
275#107C22×D6111yescharacteristic
276#107C22×D6111yescharacteristic
277#106—21no—
278#106—1221no—
279#106—1221no—
280#106—1221no—
281#102—1222no—
282#102—1222no—
283C4×C8—611no—
284C4×C8—611no—
285C4×C8—611no—
286C4×C8—611no—
287C24—41no—
288C2×Q8—2441no—
289C2×Q8—2441no—
290C2×D4—22no—
291C2×D4—24no—
292C2×D4—41no—
293C2×D4—42no—
294C22×C4—611no—
295C22×C4—611no—
296C22×C4—4884no—
297C22×C4—2442no—
298C22×C4—2442no—
299C22×C4—2441no—
300C22×C4D4×D6111yescharacteristic
301D8—22no—
302D8—24no—
303D8C2×Dic12214yes—
304C2×C8—611no—
305C2×C8—611no—
306C2×C8—1222no—
307C2×C8C2×Dic12212yes—
308C2×C8D4×D6111yescharacteristic
309C2×C8#1175822412yes—
310C2×C8C23×D6111yescharacteristic
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—1222no—
314C4⋊C4—1221no—
315C4⋊C4—1221no—
316C4⋊C4—2441no—
317C4⋊C4—2441no—
318C4⋊C4—2441no—
319C4⋊C4—2441no—
320C4⋊C4—2441no—
321C4⋊C4—2441no—
322C4⋊C4—2441no—
323C22⋊C4—41no—
324C22⋊C4—2442no—
325C22⋊C4—2442no—
326C22⋊C4—2441no—
327C42—41no—
328C42—611no—
329C42—611no—
330C42—611no—
331C42—611no—
332C42—2441no—
333C16—22no—
334C16—1222no—
335C23—42no—
336C23—44no—
337C23—82no—
338C23—82no—
339C23D8×D6111yescharacteristic
340Q8—4882no—
341Q8—4882no—
342D4—24no—
343D4—44no—
344D4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—611no—
348C2×C4—611no—
349C2×C4—611no—
350C2×C4—611no—
351C2×C4—1222no—
352C2×C4—4884no—
353C2×C4—4884no—
354C2×C4—4882no—
355C2×C4—2441no—
356C2×C4—2441no—
357C2×C4—2441no—
358C2×C4—2442no—
359C2×C4#1451975412yes—
360C2×C4D8×D6111yescharacteristic
361C2×C4#1451980612yes—
362C2×C4C2×D4×D6111yescharacteristic
363C8—1221no—
364C8—1221no—
365C8#1451975412yes—
366C8C22×Dic12211yescharacteristic
367C8C2×#1175822411yescharacteristic
368C22—21no—
369C22—21no—
370C22—44no—
371C22—88no—
372C22—82no—
373C22—82no—
374C22#1684004012yes—
375C22C2×D8×D6111yescharacteristic
376C4—81no—
377C4—81no—
378C4—612no—
379C4—1221no—
380C4—1221no—
381C4—1221no—
382C4—4881no—
383C4—4882no—
384C4—4881no—
385C4#1684004012yes—
386C4C2×#1451975411yescharacteristic
387C4C2×#1451980611yescharacteristic
388C2—22no—
389C2—84no—
390C2C2×#1684004011yescharacteristic
391C2#509×D6111yescharacteristic
392C2#2145895111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2140C2
socleC22×C61C22×D8
prime radicalC22×D8×C61C22
hypercenter#2140D61
hypocenterC61#33627
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2140

Complements

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

NH ≅ G/Ntype
1. #21456331C2semidirect
3. #21447978C2semidirect
4. #21447974C2semidirect
5. #21447964C2semidirect
6. #21447954C2semidirect
7. #21447608C2semidirect
8. #21447607C2semidirect
9. #21447574C2semidirect
15. #16840166C22semidirect
16. #16840166C22semidirect
17. #16840163C22semidirect
18. #16840163C22semidirect
30. #16839590C22semidirect
31. #16839589C22semidirect
32. #16839588C22semidirect
33. #16839587C22semidirect
34. #16839580C22semidirect
35. #16839580C22semidirect
81. C2×D8×C61Q8semidirect
239. C61#33627semidirect
244. C2×D8Dic122semidirect

Maximal subgroups

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

subgroupindex
#214563312
C2×D8×Dic612
#214479782
#214479742
#214479642
#214479542
#214476082
#214476072
#214475742
#214475352
C61×#21402
#3362761