Finite Group Databasebeta

← #29663533

Subgroups — #29663533

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#21456949C211yesmaximal, characteristic
3#21456366C211yesmaximal, characteristic, complemented
4#21454562C211yesmaximal, characteristic
5#21445341C212yesmaximal, complemented
6#21445317C212yesmaximal, complemented
7#21445217C212yesmaximal, complemented
8#21445193C212yesmaximal, complemented
9#21444572C211yesmaximal, characteristic, complemented
10C2×#16839510C211yesmaximal, characteristic, complemented
11C61×#1542C211yesmaximal, characteristic, complemented
12C2×#14519783—21no—
13#16840312C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15#16840172—22no—
16#16840165C2212yes—
17#16840160—21no—
18#16840158C2212yes—
19#16840109—21no—
20#16840081—22no—
21#16840053C2212yes—
22#16840053C2212yes—
23#16840049C2212yes—
24#16840047—21no—
25#16839993—22no—
26#16839964C2212yes—
27#16839962—21no—
28#16839839C2212yes—
29#16839832—22no—
30#16839827C2212yes—
31#16839805—21no—
32#16839805—21no—
33C2×C4⋊#92418C2211yescharacteristic
34#16839728—21no—
35C4×#11758194C2211yescharacteristic
36#16839514C2212yes—
37#16839512C2212yes—
38#16839510C2212yes—
39#16839510C2212yes—
40C61×#533C2211yescharacteristic
41C61×#431C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C61×#333C2211yescharacteristic
44C61×#333C2211yescharacteristic
45C61×#332C2211yescharacteristic
46C61×#332C2211yescharacteristic
47C2×#11758222—21no—
48C22×Dic122—21no—
49C2×#11758201—22no—
50#14519783—44no—
51C2×#11758194D411yescharacteristic
52C2×#11758194D411yescharacteristic
53C2×#11758189—22no—
54C22×#92418—21no—
55C2×C8×D61—22no—
56#14519777—21no—
57#14519777—21no—
58#14519777—21no—
59#14519777D411yescharacteristic
60#14519777C2312yes—
61#14519774D411yescharacteristic
62#14519765—21no—
63#14519751—21no—
64#14519751D412yes—
65C4×D244—21no—
66C4×D244C2312yes—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69#14519710—22no—
70#14519710—22no—
71#14519709—22no—
72#14519709—22no—
73#14519708—22no—
74#14519708—22no—
75#14519708—22no—
76C4⋊#92418—21no—
77C4⋊#92418C2312yes—
78#14519701—22no—
79#14519699—22no—
80#14519699—22no—
81#14519697—22no—
82#14519689—22no—
83#14519689—22no—
84#14519688D412yes—
85C4×#92418C2312yes—
86C22×C8×C61—21no—
87C4⋊Q8×C61C2311yescharacteristic
88C4⋊Q8×C61C2311yescharacteristic
89#125×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61—21no—
94#117×C61—21no—
95C2×C42×C61C2311yescharacteristic
96C4⋊C8×C61—21no—
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2311yescharacteristic
99C4⋊C8×C61C2311yescharacteristic
100C4⋊C8×C61C2311yescharacteristic
101C22⋊C8×C61—21no—
102C4×C8×C61—21no—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61#4111yescharacteristic
108C2×C22⋊D61—22no—
109C22×Dic61—41no—
110#11758222—42no—
111#11758222—42no—
112C2×D244—21no—
113C2×D244—21no—
114C2×D244C2×D412yes—
115C2×C4×D61—22no—
116C2×Dic122—21no—
117C2×Dic122—22no—
118C2×Dic122—41no—
119C2×Dic122—82no—
120C2×C8×C61—21no—
121C2×C8×C61—21no—
122C2×C8×C61—21no—
123C2×C8×C61—21no—
124C2×C8×C61—42no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—41no—
130C4⋊C4×C61—41no—
131C22⋊C4×C61—41no—
132C42×C61—41no—
133C42×C61C2×D411yescharacteristic
134C42×C61C2×D411yescharacteristic
135C42×C61#4111yescharacteristic
136C42×C61C2411yescharacteristic
137#11758201—82no—
138#11758198—22no—
139#11758198—22no—
140C4⋊Dic61—21no—
141C4⋊Dic61—22no—
142C4⋊Dic61—22no—
143C4⋊Dic61—41no—
144C4⋊Dic61—41no—
145C4⋊Dic61C2×D412yes—
146#11758194—22no—
147#11758194SD1614yescomplemented
148C2×#92418—22no—
149C2×#92418—81no—
150C2×#92418C2×D412yes—
151C2×#92418C2×D412yes—
152#11758189—44no—
153C8×D61—44no—
154C23×C61#12111yescharacteristic
155C22×D61—22no—
156Q8×C61—41no—
157Q8×C61—41no—
158Q8×C61—81no—
159C2×C4×C61—21no—
160C2×C4×C61—21no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61C2×SD1612yes—
168C2×C4×C61#13712yes—
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C2×#4111yescharacteristic
172C22⋊D61—44no—
173C2×Dic61—22no—
174C2×Dic61—42no—
175C2×Dic61—82no—
176D244—22no—
177D244—22no—
178D244—42no—
179D244—81no—
180C4×D61—42no—
181C4×D61—42no—
182Dic122—42no—
183Dic122—82no—
184Dic122—82no—
185C8×C61—41no—
186C8×C61—41no—
187C8×C61—41no—
188C8×C61—41no—
189#92418—22no—
190#92418—42no—
191#92418C2×SD1614yes—
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—81no—
205C4×C61#46112yes—
206C4×C61C22×SD1611yescharacteristic
207C4×C61C2×#13711yescharacteristic
208C4×C61#57711yescharacteristic
209C4×C61#57811yescharacteristic
210Dic61—42no—
211Dic61—82no—
212#3246—611no—
213#3227—611no—
214#3165—611no—
215#2935—611no—
216#1695—612no—
217#1691—612no—
218#1675—612no—
219#1671—612no—
220#1565—611no—
221#1542D6111yescharacteristic, complemented
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⋊Q8—611no—
232#501—1222no—
233#500—612no—
234#498—612no—
235#498—1221no—
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#431C2×D6111yescharacteristic
249#428—1221no—
250#428—1221no—
251C2×C4⋊C8—611no—
252C2×C4⋊C8C2×D6111yescharacteristic
253#408—1221no—
254C4×M16—611no—
255#335—612no—
256#334—612no—
257#333—612no—
258#333—612no—
259#333C2×D6111yescharacteristic
260#333C2×D6111yescharacteristic
261#332C2×D6111yescharacteristic
262#332C2×D6111yescharacteristic
263C61#1923311yescharacteristic, complemented, Sylow
264C2×#41—1221no—
265C22×Q8—1221no—
266C2×SD16—1222no—
267#131—2444no—
268C2×M16—611no—
269C2×M16—611no—
270C2×M16—1222no—
271C22×C8—21no—
272C22×C8—1222no—
273C22×C8—1221no—
274C4⋊Q8—611no—
275C4⋊Q8—612no—
276C4⋊Q8—1221no—
277C4⋊Q8—1221no—
278C4⋊Q8—1221no—
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—1221no—
294C2×C4⋊C4—1221no—
295C2×C42C22×D6111yescharacteristic
296#107—1222no—
297#107—1222no—
298#106—1222no—
299#106—1222no—
300#106—1222no—
301#106—1222no—
302#106—1222no—
303C4⋊C8—21no—
304C4⋊C8—612no—
305C4⋊C8—1221no—
306C4⋊C8C22×D6111yescharacteristic
307C4⋊C8C22×D6111yescharacteristic
308C4⋊C8C22×D6111yescharacteristic
309C4⋊C8C22×D6111yescharacteristic
310#103—1222no—
311#102—1222no—
312#102—1222no—
313#102—1222no—
314C22⋊C8—21no—
315C22⋊C8—1222no—
316C22⋊C8—1222no—
317#97—612no—
318C4×C8—21no—
319C4×C8—612no—
320#41—2442no—
321#41—2442no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—1221no—
325C2×Q8—1222no—
326C2×Q8—4882no—
327C2×Q8—2441no—
328C2×D4—612no—
329C2×D4—1222no—
330C2×D4—1221no—
331C2×D4—1221no—
332C22×C4—1222no—
333C22×C4—2441no—
334C22×C4#1175822211yescharacteristic
335C22×C4D4×D6111yescharacteristic
336C22×C4D4×D6111yescharacteristic
337SD16—4882no—
338M16—614no—
339M16—1222no—
340M16—2444no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—42no—
346C2×C8—612no—
347C2×C8—612no—
348C2×C8—1222no—
349C2×C8—4881no—
350C2×C8—2444no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—612no—
358C4⋊C4—1221no—
359C4⋊C4—1222no—
360C4⋊C4—1222no—
361C4⋊C4—2441no—
362C4⋊C4—2441no—
363C22⋊C4—41no—
364C22⋊C4—1222no—
365C22⋊C4—1222no—
366C42—41no—
367C42#1175822211yescharacteristic
368C42D4×D6111yescharacteristic
369C42D4×D6111yescharacteristic
370C42C23×D6111yescharacteristic
371C23—1222no—
372C23#1451975011yescharacteristic
373Q8—41no—
374Q8—41no—
375Q8—81no—
376Q8—4882no—
377Q8—4882no—
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—1222no—
390C2×C4—4882no—
391C2×C4—2442no—
392C2×C4—2442no—
393C2×C4—2442no—
394C2×C4#1451975011yescharacteristic
395C2×C4#1451975011yescharacteristic
396C2×C4#1451975011yescharacteristic
397C2×C4SD16×D6112yes—
398C2×C4#1451980912yes—
399C2×C4C2×D4×D6111yescharacteristic
400C2×C4C2×D4×D6111yescharacteristic
401C2×C4C2×#1175822211yescharacteristic
402C8—41no—
403C8—41no—
404C8—41no—
405C8—41no—
406C8—614no—
407C8—1222no—
408C8—2442no—
409C22—21no—
410C22—21no—
411C22—2442no—
412C22—2442no—
413C22—2442no—
414C22#1683996612yes—
415C22C2×#1451975011yescharacteristic
416C4—21no—
417C4—21no—
418C4—21no—
419C4—81no—
420C4—81no—
421C4—4882no—
422C4—2442no—
423C4#1683996612yes—
424C4C2×SD16×D6111yescharacteristic
425C4C2×#1451980911yescharacteristic
426C4#1684056911yescharacteristic
427C4#1684057611yescharacteristic
428C2—22no—
429C2—4882no—
430C2C2×#1683996611yescharacteristic
431C2#2145551311yescharacteristic
432C2#2145555011yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1542C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#1542D61
hypocenterC61#19233
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1542

Complements

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

NH ≅ G/Ntype
3. #21456366C2semidirect
5. #21445341C2semidirect
6. #21445317C2semidirect
7. #21445217C2semidirect
8. #21445193C2semidirect
9. #21444572C2semidirect
10. C2×#16839510C2semidirect
11. C61×#1542C2semidirect
147. #11758194SD16semidirect
221. #1542D61semidirect
263. C61#19233semidirect

Maximal subgroups

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

subgroupindex
#214571312
#214569492
#214563662
#214545622
#214453412
#214453172
#214452172
#214451932
#214445722
C2×#168395102
C61×#15422
#1923361