Finite Group Databasebeta

← #29663524

Subgroups — #29663524

Order 15616.

640 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457131C211yesmaximal, characteristic
2#21456947C211yesmaximal, characteristic
3#21454565C211yesmaximal, characteristic
4#21445339C212yesmaximal, complemented
5#21445315C212yesmaximal, complemented
6#21445215C212yesmaximal, complemented
7#21445191C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444476C211yesmaximal, characteristic, complemented
10C2×#16839511C211yesmaximal, characteristic, complemented
11C61×#3165C211yesmaximal, characteristic, complemented
12C2×#14519785—21no—
13#16840312C2211yescharacteristic
14#16840158—21no—
15#16840110—21no—
16#16840080—22no—
17#16840049C2212yes—
18#16840049C2212yes—
19#16840049C2212yes—
20#16840047—21no—
21#16839993—22no—
22#16839964C2212yes—
23#16839962—21no—
24#16839841C2212yes—
25#16839835—22no—
26C8×D244C2212yes—
27#16839815C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29C2×C4⋊#92418C2211yescharacteristic
30#16839730—21no—
31#16839515C2212yes—
32#16839513C2212yes—
33#16839511C2212yes—
34#16839511C2211yescharacteristic
35#16839511C2212yes—
36#16839511C2211yescharacteristic
37#16839508C2211yescharacteristic
38#16839508C2211yescharacteristic
39C61×#533C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C61×#501—22no—
42C61×#500C2212yes—
43C61×#498C2212yes—
44C61×#428—21no—
45C61×#428—21no—
46C4×M16×C61C2211yescharacteristic
47C2×#11758222—21no—
48C2×#11758201—22no—
49#14519785—44no—
50C2×#11758189—22no—
51C22×#92418—21no—
52C22×#92418—21no—
53C2×C8×D61—22no—
54#14519777D411yescharacteristic
55#14519774D411yescharacteristic
56#14519765—21no—
57#14519751—21no—
58#14519751D412yes—
59C4×D244—21no—
60C4×D244C2312yes—
61#14519708—22no—
62#14519708—22no—
63#14519708—22no—
64C4⋊#92418—21no—
65C4⋊#92418—21no—
66C4⋊#92418C2312yes—
67C4⋊#92418C2311yescharacteristic
68C4⋊#92418C2311yescharacteristic
69C4⋊#92418C2311yescharacteristic
70C4⋊#92418C2311yescharacteristic
71#14519701—22no—
72#14519697—22no—
73#14519697—22no—
74#14519697—22no—
75C22⋊#92418—21no—
76#14519689—22no—
77#14519689—22no—
78C4×#92418—21no—
79C22×Q8×C61—21no—
80C2×M16×C61D411yescharacteristic
81C2×M16×C61D411yescharacteristic
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61—21no—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61C2312yes—
86C4⋊Q8×C61C2311yescharacteristic
87C4⋊Q8×C61C2311yescharacteristic
88#125×C61—21no—
89C22⋊Q8×C61—21no—
90C22⋊Q8×C61—21no—
91C4×Q8×C61—21no—
92C4×Q8×C61—21no—
93#117×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C42×C61C2311yescharacteristic
97#107×C61—22no—
98#107×C61—22no—
99#106×C61—22no—
100#106×C61—22no—
101#97×C61D412yes—
102C4×C8×C61C2312yes—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—22no—
106C2×Q8×C61—41no—
107C2×Q8×C61—82no—
108C22×C4×C61—41no—
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C2×C22⋊D61—22no—
113#11758222—42no—
114#11758222—42no—
115C2×D244—21no—
116C2×D244—21no—
117C2×D244C2×D412yes—
118C2×C4×D61—22no—
119C2×Dic122—21no—
120M16×C61—22no—
121M16×C61SD1614yescomplemented
122C2×C8×C61C2×D412yes—
123C2×C8×C61C2×D412yes—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—41no—
131C4⋊C4×C61—41no—
132C4⋊C4×C61—41no—
133C4⋊C4×C61—41no—
134C22⋊C4×C61—41no—
135C42×C61—41no—
136C42×C61C2×D411yescharacteristic
137C42×C61C2×D411yescharacteristic
138C42×C61#4111yescharacteristic
139C42×C61C2411yescharacteristic
140#11758201—82no—
141#11758198—22no—
142#11758198—22no—
143C4⋊Dic61—21no—
144C4⋊Dic61C2×D412yes—
145C2×#92418—21no—
146C2×#92418—21no—
147C2×#92418—21no—
148C2×#92418—21no—
149C2×#92418—22no—
150C2×#92418—42no—
151C2×#92418—81no—
152#11758189—44no—
153C8×D61—44no—
154C23×C61#12111yescharacteristic
155C22×D61—22no—
156Q8×C61—41no—
157Q8×C61—41no—
158Q8×C61—82no—
159Q8×C61—81no—
160Q8×C61—82no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61—41no—
166C2×C4×C61—42no—
167C2×C4×C61—82no—
168C2×C4×C61#12111yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61C2×SD1612yes—
172C2×C4×C61#13712yes—
173C2×C4×C61C22×D411yescharacteristic
174C2×C4×C61C22×D411yescharacteristic
175C2×C4×C61C2×#4111yescharacteristic
176C22⋊D61—44no—
177C2×Dic61—22no—
178D244—22no—
179D244—22no—
180D244—42no—
181D244—81no—
182C4×D61—42no—
183C4×D61—42no—
184Dic122—42no—
185C8×C61—22no—
186C8×C61C2×SD1614yes—
187#92418—41no—
188#92418—41no—
189#92418—41no—
190#92418—42no—
191#92418—41no—
192#19233—611nomaximal, Sylow
193C22×C61—21no—
194C22×C61—21no—
195C22×C61#46112yes—
196C22×C61C2×#12111yescharacteristic
197C2×D61—42no—
198C2×D61—42no—
199C2×D61—42no—
200C4×C61—21no—
201C4×C61—21no—
202C4×C61—21no—
203C4×C61—81no—
204C4×C61—82no—
205C4×C61—81no—
206C4×C61#46112yes—
207C4×C61C22×SD1611yescharacteristic
208C4×C61C2×#13711yescharacteristic
209C4×C61#57711yescharacteristic
210C4×C61#57811yescharacteristic
211Dic61—42no—
212#3246—611no—
213#3227—611no—
214#3165D6111yescharacteristic, complemented
215#2935—611no—
216#1695—612no—
217#1691—612no—
218#1675—612no—
219#1671—612no—
220#1565—611no—
221#1542—611no—
222C2×#333—611no—
223C2×C61—22no—
224C2×C61C2×#46111yescharacteristic
225C2×C61#306311yescharacteristic
226C2×C61#306811yescharacteristic
227D61—82no—
228C2×#131—1221no—
229#533C2×D6111yescharacteristic
230#532—611no—
231C2×C4⋊Q8C2×D6111yescharacteristic
232#501—22no—
233#500C2×D6112yes—
234#498—1221no—
235#498C2×D6112yes—
236#486—1221no—
237#481—1222no—
238#476—612no—
239#476—612no—
240#476—612no—
241#475—1221no—
242#465—1222no—
243#461—612no—
244#460—1221no—
245#436—612no—
246#435—1222no—
247C8×D4—612no—
248#431—611no—
249#428—21no—
250#428—21no—
251C2×C4⋊C8—611no—
252C2×C4⋊C8—611no—
253#408—1221no—
254C4×M16C2×D6111yescharacteristic
255#335—612no—
256#334—612no—
257#333—612no—
258#333—612no—
259#333—611no—
260#333—611no—
261#332—611no—
262#332—611no—
263C61#1923311yescharacteristic, complemented, Sylow
264C2×#41—1221no—
265C22×Q8—21no—
266C2×SD16—1222no—
267#131—2444no—
268C2×M16—1222no—
269C2×M16C22⋊D6111yescharacteristic
270C2×M16C22⋊D6111yescharacteristic
271C22×C8—1221no—
272C22×C8—1221no—
273C22×C8—1222no—
274C4⋊Q8—21no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8—611no—
278C4⋊Q8C22×D6112yes—
279C4⋊Q8C22×D6111yescharacteristic
280C4⋊Q8C22×D6111yescharacteristic
281#127—611no—
282#125—21no—
283#124—1221no—
284C22⋊Q8—21no—
285C22⋊Q8—21no—
286#121—612no—
287#121—1221no—
288C4×Q8—21no—
289C4×Q8—21no—
290C4×D4—612no—
291C4×D4—1221no—
292#117—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C42C22×D6111yescharacteristic
296#107—22no—
297#107—22no—
298#106—22no—
299#106—22no—
300#106—1222no—
301#106—1222no—
302#106—1222no—
303C4⋊C8—611no—
304C4⋊C8—611no—
305C4⋊C8—611no—
306C4⋊C8—611no—
307C4⋊C8—612no—
308C4⋊C8—1221no—
309C4⋊C8—1221no—
310#103—1222no—
311#102—1222no—
312#102—1222no—
313#102—1222no—
314C22⋊C8—1221no—
315C22⋊C8—1222no—
316C22⋊C8—1222no—
317#97C22⋊D6112yes—
318C4×C8—1221no—
319C4×C8C22×D6112yes—
320#41—2442no—
321#41—2442no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×Q8—82no—
327C2×Q8—1221no—
328C2×D4—612no—
329C2×D4—1222no—
330C2×D4—1221no—
331C2×D4—1221no—
332C22×C4—41no—
333C22×C4—1222no—
334C22×C4D4×D6111yescharacteristic
335C22×C4#1175822411yescharacteristic
336C22×C4C2×C22⋊D6111yescharacteristic
337SD16—4882no—
338M16—22no—
339M16—2444no—
340M16#1175820114yescomplemented
341C2×C8—1221no—
342C2×C8—1221no—
343C2×C8—1222no—
344C2×C8—1221no—
345C2×C8—1221no—
346C2×C8—4881no—
347C2×C8—2442no—
348C2×C8—2444no—
349C2×C8C2×C22⋊D6112yes—
350C2×C8C2×C22⋊D6112yes—
351C4⋊C4—22no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—612no—
362C4⋊C4—1221no—
363C22⋊C4—41no—
364C22⋊C4—1222no—
365C22⋊C4—1222no—
366C42—41no—
367C42D4×D6111yescharacteristic
368C42#1175822411yescharacteristic
369C42C2×C22⋊D6111yescharacteristic
370C42C23×D6111yescharacteristic
371C23—1222no—
372C23#1451974811yescharacteristic
373Q8—41no—
374Q8—41no—
375Q8—81no—
376Q8—82no—
377Q8—82no—
378Q8—2442no—
379D4—1222no—
380D4—1222no—
381D4—4881no—
382D4—2444no—
383D4—2442no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—82no—
391C2×C4—1222no—
392C2×C4—2442no—
393C2×C4—2442no—
394C2×C4#1451974811yescharacteristic
395C2×C4#1451974811yescharacteristic
396C2×C4#1451974811yescharacteristic
397C2×C4C2×#1175820112yes—
398C2×C4#1451980812yes—
399C2×C4C2×D4×D6111yescharacteristic
400C2×C4C22×C22⋊D6111yescharacteristic
401C2×C4C2×#1175822411yescharacteristic
402C8—22no—
403C8—2441no—
404C8—2441no—
405C8—2441no—
406C8—2442no—
407C8—2441no—
408C8C2×#1175820114yes—
409C22—21no—
410C22—21no—
411C22—2442no—
412C22—2442no—
413C22—2442no—
414C22#1683996412yes—
415C22C2×#1451974811yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—81no—
420C4—82no—
421C4—81no—
422C4—2442no—
423C4#1683996412yes—
424C4C22×#1175820111yescharacteristic
425C4C2×#1451980811yescharacteristic
426C4#1684057111yescharacteristic
427C4#1684057811yescharacteristic
428C2—22no—
429C2—4882no—
430C2C2×#1683996411yescharacteristic
431C2#2145550411yescharacteristic
432C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3165C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#3165D61
hypocenterC61#19233
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3165

Complements

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

NH ≅ G/Ntype
4. #21445339C2semidirect
5. #21445315C2semidirect
6. #21445215C2semidirect
7. #21445191C2semidirect
8. #21444571C2semidirect
9. #21444476C2semidirect
10. C2×#16839511C2semidirect
11. C61×#3165C2semidirect
121. M16×C61SD16semidirect
214. #3165D61semidirect
263. C61#19233semidirect
340. M16#11758201semidirect

Maximal subgroups

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

subgroupindex
#214571312
#214569472
#214545652
#214453392
#214453152
#214452152
#214451912
#214445712
#214444762
C2×#168395112
C61×#31652
#1923361