Finite Group Databasebeta

← #29663128

Subgroups — #29663128

Order 15616.

650 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456946C212yesmaximal
2#500×D61C211yesmaximal, characteristic, complemented
3#21454528C211yesmaximal, characteristic
4#21445343C211yesmaximal, characteristic, complemented
5#21445342C211yesmaximal, characteristic, complemented
6#21445339C211yesmaximal, characteristic, complemented
7#21445217C212yesmaximal, complemented
8#21445216C212yesmaximal, complemented
9#21444564C211yesmaximal, characteristic, complemented
10#21444498C212yesmaximal, complemented
11C61×#1695C211yesmaximal, characteristic, complemented
12C2×#14519785—21no—
13#16840309C2211yescharacteristic
14C4⋊Q8×D61C2212yes—
15#16840167C2211yescharacteristic
16#16840166—21no—
17#16840166—21no—
18#16840165C2212yes—
19#16840159—22no—
20#16840107—22no—
21#16840082—22no—
22#16840079—22no—
23#16840053C2212yes—
24#16840052C2212yes—
25#16840049C2212yes—
26#16839841C2211yescharacteristic
27#16839838—21no—
28D4×#92418C2211yescharacteristic
29#16839827C2211yescharacteristic
30#16839825—21no—
31#16839814C2212yes—
32#107×D61—21no—
33#107×D61—21no—
34#16839784C2211yescharacteristic
35#16839727—21no—
36C4×C8×D61C2211yescharacteristic
37#16839514C2212yes—
38#16839513C2211yescharacteristic
39#16839511C2212yes—
40#16839510C2212yes—
41#16839510C2212yes—
42C61×#500C2211yescharacteristic
43C61×#476C2212yes—
44C61×#436C2211yescharacteristic
45C61×#334C2211yescharacteristic
46C61×#333C2212yes—
47C2×#11758222—21no—
48C2×Q8×D61—22no—
49#14519785—44no—
50C2×#11758189—21no—
51C2×#11758189—21no—
52C22×#92418—21no—
53C2×C8×D61—21no—
54C2×C8×D61D411yescharacteristic
55C2×C8×D61D411yescharacteristic
56#14519777C2312yes—
57#14519776—22no—
58#14519776—22no—
59#14519776D411yescharacteristic
60#14519775—22no—
61#14519773D411yescharacteristic
62#14519763—21no—
63#14519752D411yescharacteristic
64#14519750—21no—
65#14519748D411yescharacteristic
66C4×D244C2311yescharacteristic
67C4×C22⋊D61—21no—
68C4⋊C4×D61—22no—
69C4⋊C4×D61—22no—
70C42×D61C2311yescharacteristic
71#14519711—21no—
72#14519711—21no—
73#14519710—22no—
74#14519710—22no—
75#14519709—22no—
76#14519708—22no—
77#14519708—22no—
78C4⋊#92418C2311yescharacteristic
79C4⋊#92418C2312yes—
80#14519706—21no—
81#14519699—22no—
82#14519698—22no—
83#14519697—22no—
84C22⋊#92418—21no—
85#14519689—22no—
86#14519689—21no—
87#14519689—21no—
88C4×#92418C2311yescharacteristic
89C8×Dic61—22no—
90C8×Dic61D411yescharacteristic
91C8×Dic61D411yescharacteristic
92C2×M16×C61—21no—
93C4⋊Q8×C61C2312yes—
94C4×D4×C61C2311yescharacteristic
95#107×C61—21no—
96#107×C61—21no—
97#106×C61—22no—
98C4⋊C8×C61C2311yescharacteristic
99C4⋊C8×C61C2312yes—
100#102×C61—22no—
101C22⋊C8×C61—21no—
102C4×C8×C61C2311yescharacteristic
103C2×Q8×C61—22no—
104C2×D4×C61C2×D411yescharacteristic
105C22×C4×C61—21no—
106C2×C22⋊D61—21no—
107C2×C22⋊D61—21no—
108C2×C22⋊D61—21no—
109Q8×D61—84no—
110#11758222—42no—
111#11758222—42no—
112C2×D244—21no—
113C2×D244C2×D411yescharacteristic
114C2×C4×D61—21no—
115C2×C4×D61—42no—
116C2×C4×D61C2×D411yescharacteristic
117C2×C4×D61C2×D411yescharacteristic
118C2×C4×D61#4111yescharacteristic
119C2×Dic122—21no—
120C2×Dic122—22no—
121C2×Dic122—42no—
122M16×C61—42no—
123C2×C8×C61—21no—
124C2×C8×C61—22no—
125C2×C8×C61C2×D411yescharacteristic
126C2×C8×C61C2×D411yescharacteristic
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61C2×D411yescharacteristic
130C22⋊C4×C61—21no—
131C42×C61C2411yescharacteristic
132C22⋊Dic61—21no—
133#11758198—21no—
134#11758198—21no—
135C4⋊Dic61—22no—
136C4⋊Dic61—22no—
137C4⋊Dic61C2×D411yescharacteristic
138#11758196—21no—
139#11758196—42no—
140#11758196—42no—
141C4×Dic61C2×D411yescharacteristic
142C4×Dic61C2×D411yescharacteristic
143C4×Dic61#4111yescharacteristic
144C2×#92418—22no—
145C2×#92418—21no—
146C2×#92418—42no—
147C2×#92418C2×D411yescharacteristic
148C2×#92418C2×D411yescharacteristic
149#11758189—42no—
150#11758189—42no—
151C8×D61—22no—
152C8×D61—42no—
153C8×D61SD1614yes—
154C23×C61—21no—
155C22×D61—21no—
156C22×D61#12111yescharacteristic
157Q8×C61—82no—
158D4×C61—22no—
159C2×C4×C61—21no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—42no—
163C2×C4×C61C22×D411yescharacteristic
164C2×C4×C61C22×D411yescharacteristic
165C2×C4×C61C2×#4111yescharacteristic
166C22⋊D61—42no—
167C22⋊D61—42no—
168C22⋊D61—42no—
169C2×Dic61—21no—
170C2×Dic61—42no—
171C2×Dic61#12111yescharacteristic
172C2×Dic61#12111yescharacteristic
173C2×Dic61#12111yescharacteristic
174D244—22no—
175D244—42no—
176C4×D61—22no—
177C4×D61—22no—
178C4×D61—41no—
179C4×D61—41no—
180C4×D61—84no—
181C4×D61C2×SD1612yes—
182C4×D61#13712yes—
183Dic122—42no—
184Dic122—84no—
185Dic122—82no—
186C8×C61—21no—
187C8×C61—41no—
188C8×C61—42no—
189C8×C61C2×SD1612yes—
190#92418—21no—
191#92418—41no—
192#92418—42no—
193#92418C2×SD1612yes—
194#19193—611nomaximal, Sylow
195C22×C61—41no—
196C22×C61—41no—
197C22×C61—41no—
198C22×C61C2×#12111yescharacteristic
199C2×D61—21no—
200C2×D61—21no—
201C2×D61—41no—
202C2×D61—41no—
203C2×D61—41no—
204C2×D61#46112yes—
205C4×C61—21no—
206C4×C61—41no—
207C4×C61—82no—
208C4×C61C22×SD1611yescharacteristic
209C4×C61C2×#13711yescharacteristic
210C4×C61#57812yes—
211Dic61—21no—
212Dic61—41no—
213Dic61—82no—
214Dic61#46112yes—
215Dic61#47212yes—
216#3227—612no—
217C2×#500—611no—
218#2931—611no—
219#1695—611no—
220#1695—611no—
221#1695—611no—
222#1695D6111yescharacteristic, complemented
223#1675—612no—
224#1675—612no—
225#1564—611no—
226#1548—612no—
227C2×C61—81no—
228C2×C61C2×#46111yescharacteristic
229C2×C61#305911yescharacteristic
230C2×C61C2×#47211yescharacteristic
231D61—22no—
232D61—81no—
233C2×#131—1221no—
234#532—611no—
235C2×C4⋊Q8—612no—
236#500—612no—
237#500—611no—
238#500—1221no—
239#500—1221no—
240#500C2×D6111yescharacteristic
241#498—1222no—
242#486—1222no—
243#481—1222no—
244#481—1222no—
245#476—612no—
246#476—612no—
247#476—612no—
248#476C2×D6112yes—
249#436—611no—
250#436—1221no—
251#436C2×D6111yescharacteristic
252C8×D4—611no—
253C8×D4—611no—
254C8×D4—1221no—
255#431—612no—
256C2×#107—1221no—
257C2×#107—1221no—
258#423—611no—
259#408—1221no—
260C2×C4×C8—611no—
261#335—612no—
262#334—611no—
263#334C2×D6111yescharacteristic
264#333—612no—
265#333—612no—
266#333—612no—
267#333C2×D6112yes—
268C61#1919311yescharacteristic, complemented, Sylow
269C2×#41—1221no—
270C22×Q8—1222no—
271#131—2444no—
272C2×M16—21no—
273C2×M16—1221no—
274C2×M16—1221no—
275C22×C8—611no—
276C22×C8—611no—
277C22×C8—1221no—
278C22×C8—1221no—
279C4⋊Q8—611no—
280C4⋊Q8—612no—
281C4⋊Q8—1222no—
282C4⋊Q8—1222no—
283C4⋊Q8—1222no—
284C4⋊Q8C22×D6112yes—
285#127—611no—
286#124—1221no—
287#121—611no—
288#121—611no—
289#121—1221no—
290C4×D4—611no—
291C4×D4—1221no—
292C4×D4C22×D6111yescharacteristic
293C2×C4⋊C4—1222no—
294C2×C4⋊C4—1222no—
295C2×C42—611no—
296#107—21no—
297#107—21no—
298#107—1222no—
299#107—1222no—
300#107—1221no—
301#107—1221no—
302#106—22no—
303#106—1222no—
304#106—1222no—
305#106—1222no—
306C4⋊C8—611no—
307C4⋊C8—612no—
308C4⋊C8—1221no—
309C4⋊C8C22×D6111yescharacteristic
310C4⋊C8C22×D6112yes—
311#102—22no—
312#102—1222no—
313#102—1222no—
314#102—1222no—
315C22⋊C8—21no—
316C22⋊C8—1221no—
317C22⋊C8—1222no—
318C22⋊C8—1221no—
319C22⋊C8—1221no—
320C4×C8—611no—
321C4×C8—611no—
322C4×C8—611no—
323C4×C8—1222no—
324C4×C8C22×D6111yescharacteristic
325#41—2442no—
326#41—2442no—
327C2×Q8—22no—
328C2×Q8—1221no—
329C2×Q8—1222no—
330C2×Q8—4884no—
331C2×Q8—2442no—
332C2×D4—611no—
333C2×D4—1221no—
334C2×D4—1221no—
335C2×D4—1221no—
336C2×D4—1221no—
337C2×D4D4×D6111yescharacteristic
338C22×C4—21no—
339C22×C4—611no—
340C22×C4—611no—
341C22×C4—611no—
342C22×C4—1221no—
343C22×C4—2442no—
344M16—42no—
345M16—2442no—
346M16—2442no—
347C2×C8—22no—
348C2×C8—21no—
349C2×C8—611no—
350C2×C8—614no—
351C2×C8—611no—
352C2×C8—1222no—
353C2×C8—1222no—
354C2×C8—1221no—
355C2×C8—2442no—
356C2×C8—2442no—
357C2×C8D4×D6111yescharacteristic
358C2×C8D4×D6111yescharacteristic
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—611no—
362C4⋊C4—1221no—
363C4⋊C4—1222no—
364C4⋊C4—1222no—
365C4⋊C4—2442no—
366C4⋊C4—2442no—
367C4⋊C4D4×D6111yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—1221no—
370C22⋊C4—1221no—
371C22⋊C4—1221no—
372C42—611no—
373C42—611no—
374C42—611no—
375C42C23×D6111yescharacteristic
376C23—21no—
377C23—611no—
378C23—1221no—
379Q8—82no—
380Q8—4882no—
381Q8—4884no—
382Q8—2442no—
383D4—22no—
384D4—1222no—
385D4—2442no—
386D4—2442no—
387D4—2442no—
388D4—2442no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—611no—
394C2×C4—611no—
395C2×C4—612no—
396C2×C4—612no—
397C2×C4—611no—
398C2×C4—1222no—
399C2×C4—1222no—
400C2×C4—1221no—
401C2×C4—4884no—
402C2×C4—2442no—
403C2×C4—2441no—
404C2×C4—2441no—
405C2×C4C2×D4×D6111yescharacteristic
406C2×C4C2×D4×D6111yescharacteristic
407C2×C4#41×D6111yescharacteristic
408C8—21no—
409C8—41no—
410C8—42no—
411C8—612no—
412C8—1221no—
413C8—2441no—
414C8—2442no—
415C8SD16×D6112yes—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—612no—
420C22—1221no—
421C22—1221no—
422C22—2441no—
423C22—2441no—
424C22—2441no—
425C22#121×D6111yescharacteristic
426C4—21no—
427C4—41no—
428C4—82no—
429C4—612no—
430C4—612no—
431C4—1221no—
432C4—4882no—
433C4—2441no—
434C4C2×SD16×D6111yescharacteristic
435C4#137×D6111yescharacteristic
436C4#1684057812yes—
437C2—81no—
438C2—1222no—
439C2—4881no—
440C2#461×D6111yescharacteristic
441C2#2145547211yescharacteristic
442C2#472×D6111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1695C2
socleC22×C61C2×#121
prime radical#16840309C22
hypercenter#1695D61
hypocenterC61#19193
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1695

Complements

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

NH ≅ G/Ntype
2. #500×D61C2semidirect
4. #21445343C2semidirect
5. #21445342C2semidirect
6. #21445339C2semidirect
7. #21445217C2semidirect
8. #21445216C2semidirect
9. #21444564C2semidirect
10. #21444498C2semidirect
11. C61×#1695C2semidirect
222. #1695D61semidirect
268. C61#19193semidirect

Maximal subgroups

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

subgroupindex
#214569462
#500×D612
#214545282
#214453432
#214453422
#214453392
#214452172
#214452162
#214445642
#214444982
C61×#16952
#1919361