Finite Group Databasebeta

← #29650607

Subgroups — #29650607

Order 15616.

594 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456425C211yesmaximal, characteristic
2#21454788C211yesmaximal, characteristic
3#21454749C211yesmaximal, characteristic, complemented
4#21444727C212yesmaximal, complemented
5#21444720C212yesmaximal, complemented
6#21444625C212yesmaximal, complemented
7#21444618C212yesmaximal, complemented
8#21444291C211yesmaximal, characteristic, complemented
9#21444282C211yesmaximal, characteristic
10M16×#92418C211yesmaximal, characteristic, complemented
11C61×#1569C211yesmaximal, characteristic, complemented
12#16840319C2211yescharacteristic
13#16840296C2211yescharacteristic
14#16840171C411yescharacteristic
15#16840156C411yescharacteristic
16#16840131C411yescharacteristic
17#16840122C411yescharacteristic
18#16840020C412yes—
19#16840008C412yes—
20#16839889—21no—
21#16839889C2212yes—
22#16839889C2211yescharacteristic
23#16839886—21no—
24#16839886C2212yes—
25#16839886C2211yescharacteristic
26C4×Dic244C2212yes—
27C4×#11758190C2212yes—
28#16839839C2211yescharacteristic
29#16839830—21no—
30#16839827C2211yescharacteristic
31#16839825—21no—
32#16839824—21no—
33#16839815C2211yescharacteristic
34#16839790—21no—
35#16839720—21no—
36#16839719—21no—
37C2×C4×#92418C2211yescharacteristic
38#16839492C2212yes—
39#16839492C2212yes—
40#16839492C2212yes—
41#16839489C2212yes—
42#16839483C2212yes—
43C8×#92418C2212yes—
44C61×#432C2211yescharacteristic
45C61×#423C2211yescharacteristic
46C4×M16×C61C2211yescharacteristic
47C61×#335C2212yes—
48C61×#334C2212yes—
49C2×Dic244C2×C412yes—
50C2×#11758190C2×C412yes—
51C2×#11758189—21no—
52C22×#92418D411yescharacteristic
53C22×#92418D411yescharacteristic
54C2×C8×D61—21no—
55#14519777C2×C411yescharacteristic
56#14519777C2×C411yescharacteristic
57#14519771—21no—
58#14519769—21no—
59#14519769C2×C411yescharacteristic
60#14519765C2×C411yescharacteristic
61#14519760—21no—
62#14519756—21no—
63#14519756C2×C412yes—
64#14519756C2×C411yescharacteristic
65#14519751C2×C411yescharacteristic
66C4×Dic122—21no—
67C4×Dic122C2311yescharacteristic
68C4×Dic122C2312yes—
69C4×D244C2311yescharacteristic
70C4×C22⋊D61—21no—
71#14519736—21no—
72#14519734—21no—
73#14519711C2×C412yes—
74#14519709C2×C412yes—
75C4⋊#92418—21no—
76#14519706—22no—
77#14519706—22no—
78#14519706—21no—
79#14519706—21no—
80#14519702C2×C412yes—
81#14519702C2×C412yes—
82#14519702C2×C412yes—
83#14519699C2×C412yes—
84C22⋊#92418—21no—
85#14519689—21no—
86#14519689—21no—
87#14519687—21no—
88#14519687—22no—
89C4×#92418D411yescharacteristic
90C4×#92418D411yescharacteristic
91C4×#92418C2312yes—
92C4×#92418C2311yescharacteristic
93C4×#92418C2311yescharacteristic
94C8×Dic61—21no—
95C8×Dic61—22no—
96C2×M16×C61—21no—
97C2×M16×C61C2×C412yes—
98C2×C42×C61C2311yescharacteristic
99C4⋊C8×C61—21no—
100C4⋊C8×C61C2311yescharacteristic
101C4⋊C8×C61C2312yes—
102C4⋊C8×C61C2311yescharacteristic
103C22⋊C8×C61—21no—
104#97×C61—21no—
105#97×C61C2×C411yescharacteristic
106#97×C61C2×C411yescharacteristic
107C4×C8×C61C2312yes—
108C22×C4×C61C22×C411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61#4111yescharacteristic
111C2×C22⋊D61—21no—
112C2×D244C22×C411yescharacteristic
113C2×C4×D61—21no—
114C2×Dic122—21no—
115C2×Dic122C22×C411yescharacteristic
116C2×Dic122C22×C412yes—
117M16×C61—24no—
118M16×C61—42no—
119C2×C8×C61—21no—
120C2×C8×C61—21no—
121C2×C8×C61—22no—
122C2×C8×C61C22×C412yes—
123C2×C8×C61C22×C412yes—
124C42×C61C22×C411yescharacteristic
125C42×C61C2×D411yescharacteristic
126C42×C61#4111yescharacteristic
127C42×C61C2411yescharacteristic
128C22⋊Dic61—22no—
129C22⋊Dic61—21no—
130#11758198—21no—
131#11758198—21no—
132#11758198—21no—
133#11758198—21no—
134C4⋊Dic61—21no—
135C4⋊Dic61—21no—
136C4⋊Dic61—21no—
137C4⋊Dic61C22×C412yes—
138C4⋊Dic61C22×C411yescharacteristic
139C4⋊Dic61C22×C411yescharacteristic
140#11758196—21no—
141#11758196—21no—
142#11758196—21no—
143#11758196—22no—
144#11758196—22no—
145#11758196—21no—
146C4×Dic61—21no—
147C4×Dic61—22no—
148C2×#92418—22no—
149C2×#92418—22no—
150C2×#92418—41no—
151C2×#92418C2×D411yescharacteristic
152C2×#92418C2×D411yescharacteristic
153C2×#92418C2×D411yescharacteristic
154C2×#92418C2×D411yescharacteristic
155Dic244—24no—
156#11758190—24no—
157#11758189—42no—
158C8×D61—42no—
159C23×C61C4×D411yescharacteristic
160C22×D61—21no—
161C2×C4×C61—22no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—41no—
165C2×C4×C61C4×D411yescharacteristic
166C2×C4×C61C4×D411yescharacteristic
167C2×C4×C61C4×D411yescharacteristic
168C2×C4×C61C23×C411yescharacteristic
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C2×#4111yescharacteristic
171C22⋊D61—42no—
172C2×Dic61—21no—
173C2×Dic61—22no—
174C2×Dic61—22no—
175C2×Dic61—21no—
176C2×Dic61—21no—
177D244—22no—
178C4×D61—41no—
179C4×D61—41no—
180Dic122—22no—
181Dic122—22no—
182Dic122—22no—
183Dic122—42no—
184C8×C61—22no—
185C8×C61—41no—
186C8×C61—41no—
187C8×C61—42no—
188C8×C61#13114yes—
189#92418—21no—
190#92418—21no—
191#92418—21no—
192#92418—42no—
193#92418#13712yes—
194#17965—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61—21no—
198C22×C61C2×C4×D411yescharacteristic
199C2×D61—41no—
200C2×D61—41no—
201C2×D61—41no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—21no—
206C4×C61C2×#13112yes—
207C4×C61C2×#13711yescharacteristic
208C4×C61#57511yescharacteristic
209Dic61—42no—
210Dic61—41no—
211Dic61—41no—
212Dic61—42no—
213Dic61—41no—
214#3172—611no—
215#2966—611no—
216#2960—611no—
217#1593—612no—
218#1592—612no—
219C8×Q16—612no—
220C8×SD16—612no—
221#1569D6111yescharacteristic, complemented
222#1497—611no—
223#1495—611no—
224C8×M16—611no—
225C2×C61—41no—
226C2×C61#293111yescharacteristic
227C2×C61C4×#13711yescharacteristic
228C2×C61#295511yescharacteristic
229D61—81no—
230#533—611no—
231#529—611no—
232#501—611no—
233#497—611no—
234#491—611no—
235#489—611no—
236#470—612no—
237#468—612no—
238#446—612no—
239#446—611no—
240#446—1221no—
241C8×Q8—612no—
242C8×Q8—611no—
243C8×Q8—1221no—
244C4×Q16—612no—
245C4×SD16—612no—
246#436—611no—
247#435—1221no—
248C8×D4—611no—
249C8×D4—1221no—
250#433—1221no—
251#432C2×D6111yescharacteristic
252#431—611no—
253#424—1221no—
254#423C2×D6111yescharacteristic
255#405—1221no—
256#405—1221no—
257C4×M16C2×D6111yescharacteristic
258C2×C4×C8—611no—
259#335C2×D6112yes—
260#334C2×D6112yes—
261Q8⋊C8—612no—
262Q8⋊C8—612no—
263Q8⋊C8—612no—
264D4⋊C8—612no—
265#322—612no—
266C82—612no—
267C61#1796511yescharacteristic, complemented, Sylow
268C2×Q16—612no—
269C2×SD16—612no—
270C2×M16—21no—
271C2×M16—1221no—
272C2×M16C4×D6112yes—
273C22×C8—611no—
274C22×C8—611no—
275C22×C8—1221no—
276C4⋊Q8—611no—
277C4⋊Q8—611no—
278#126—1221no—
279#125—611no—
280#125—1221no—
281#124—611no—
282#123—1221no—
283C22⋊Q8—611no—
284C22⋊Q8—612no—
285C22⋊Q8—1221no—
286#121—611no—
287C4×Q8—611no—
288C4×Q8—612no—
289C4×Q8—1221no—
290C4×D4—611no—
291C4×D4—1221no—
292#117—1221no—
293#117—1221no—
294C2×C42C22×D6111yescharacteristic
295#107—612no—
296#106—612no—
297C4⋊C8—21no—
298C4⋊C8—1222no—
299C4⋊C8—1222no—
300C4⋊C8—1221no—
301C4⋊C8—1221no—
302C4⋊C8—1221no—
303C4⋊C8C22×D6112yes—
304C4⋊C8C22×D6111yescharacteristic
305C4⋊C8C22×D6111yescharacteristic
306#103—612no—
307#103—612no—
308#103—612no—
309#102—612no—
310C22⋊C8—21no—
311C22⋊C8—1221no—
312C22⋊C8—1221no—
313C22⋊C8—1221no—
314#97—21no—
315#97—1221no—
316#97—1222no—
317#97C4×D6111yescharacteristic
318#97C4×D6111yescharacteristic
319C4×C8—611no—
320C4×C8—611no—
321C4×C8—611no—
322C4×C8—611no—
323C4×C8—612no—
324C4×C8—1221no—
325C4×C8—1222no—
326C4×C8C22×D6112yes—
327C2×Q8—611no—
328C2×Q8—612no—
329C2×Q8—1221no—
330C2×D4—611no—
331C2×D4—1221no—
332C22×C4—1221no—
333C22×C4C2×C4×D6111yescharacteristic
334C22×C4D4×D6111yescharacteristic
335C22×C4#1175822611yescharacteristic
336Q16—1224no—
337SD16—1224no—
338M16—24no—
339M16—42no—
340M16—2442no—
341C2×C8—21no—
342C2×C8—22no—
343C2×C8—21no—
344C2×C8—611no—
345C2×C8—611no—
346C2×C8—611no—
347C2×C8—611no—
348C2×C8—1222no—
349C2×C8—1222no—
350C2×C8—2442no—
351C2×C8—2441no—
352C2×C8C2×C4×D6112yes—
353C2×C8C2×C4×D6112yes—
354C4⋊C4—612no—
355C4⋊C4—611no—
356C4⋊C4—611no—
357C4⋊C4—1222no—
358C4⋊C4—1221no—
359C4⋊C4—1221no—
360C4⋊C4—1221no—
361C4⋊C4—1221no—
362C4⋊C4—1222no—
363C4⋊C4—1221no—
364C4⋊C4—1221no—
365C4⋊C4—1221no—
366C22⋊C4—1222no—
367C22⋊C4—1221no—
368C22⋊C4—1221no—
369C22⋊C4—1221no—
370C22⋊C4—1221no—
371C22⋊C4—1221no—
372C42—1221no—
373C42—1222no—
374C42C2×C4×D6111yescharacteristic
375C42D4×D6111yescharacteristic
376C42#1175822611yescharacteristic
377C42C23×D6111yescharacteristic
378C23—1221no—
379C23#1451973911yescharacteristic
380Q8—1222no—
381Q8—1222no—
382Q8—1222no—
383Q8—2442no—
384D4—1222no—
385D4—2442no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—41no—
390C2×C4—1221no—
391C2×C4—1221no—
392C2×C4—1222no—
393C2×C4—1222no—
394C2×C4—1221no—
395C2×C4—2441no—
396C2×C4—2441no—
397C2×C4#1451973911yescharacteristic
398C2×C4#1451973911yescharacteristic
399C2×C4#1451973911yescharacteristic
400C2×C4C22×C4×D6111yescharacteristic
401C2×C4C2×D4×D6111yescharacteristic
402C2×C4C2×#1175822611yescharacteristic
403C8—22no—
404C8—41no—
405C8—42no—
406C8—41no—
407C8—612no—
408C8—1221no—
409C8—1221no—
410C8—1221no—
411C8—2442no—
412C8#1451978514yes—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—2441no—
417C22—2441no—
418C22—2441no—
419C22C2×#1451973911yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—2441no—
425C4—2441no—
426C4—2442no—
427C4—2441no—
428C4—2442no—
429C4C2×#1451978512yes—
430C4#137×D6111yescharacteristic
431C4#1684055811yescharacteristic
432C2—41no—
433C2—4881no—
434C2#2145453011yescharacteristic
435C2#2145463711yescharacteristic
436C2#2145472111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1569C2
socleC22×C61C2×C4×D4
prime radical#14519751C2×C4
cosocleC42×C61C24
hypercenter#1569D61
hypocenterC61#17965
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1569

Complements

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

NH ≅ G/Ntype
3. #21454749C2semidirect
4. #21444727C2semidirect
5. #21444720C2semidirect
6. #21444625C2semidirect
7. #21444618C2semidirect
8. #21444291C2semidirect
10. M16×#92418C2semidirect
11. C61×#1569C2semidirect
221. #1569D61semidirect
267. C61#17965semidirect

Maximal subgroups

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

subgroupindex
#214564252
#214547882
#214547492
#214447272
#214447202
#214446252
#214446182
#214442912
#214442822
M16×#924182
C61×#15692
#1796561