Finite Group Databasebeta

← #29663555

Subgroups — #29663555

Order 15616.

640 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#21457131C211yesmaximal, characteristic
2#21456942C211yesmaximal, characteristic
3#21456201C211yesmaximal, characteristic, complemented
4#21454562C211yesmaximal, characteristic
5#21445397C212yesmaximal, complemented
6#21445369C212yesmaximal, complemented
7#21445273C212yesmaximal, complemented
8#21445245C212yesmaximal, complemented
9#21444572C211yesmaximal, characteristic, complemented
10C2×#16839509C211yesmaximal, characteristic, complemented
11C61×#1542C211yesmaximal, characteristic, complemented
12C2×#14519783—21no—
13#16840312C2211yescharacteristic
14C2×#14519777C2211yescharacteristic
15#16840167—21no—
16#16840132—22no—
17#16840114C2212yes—
18#16840114C2212yes—
19#16840109—21no—
20#16840076—22no—
21#16840049C2212yes—
22#16840047—21no—
23#16840043C2212yes—
24#16840043C2212yes—
25#16839993—22no—
26#16839964C2212yes—
27#16839962—21no—
28#16839839C2212yes—
29#16839832—22no—
30#16839827C2212yes—
31C2×C4⋊#92418C2211yescharacteristic
32#16839770—21no—
33#16839770—21no—
34#16839728—21no—
35C4×#11758194C2211yescharacteristic
36#16839514C2212yes—
37#16839512C2212yes—
38#16839509C2212yes—
39#16839509C2212yes—
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#14519711—22no—
70#14519711—22no—
71#14519708—22no—
72C4⋊#92418—21no—
73C4⋊#92418C2312yes—
74#14519701—22no—
75#14519700—22no—
76#14519700—22no—
77#14519700—22no—
78#14519700—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—42no—
119C2×Dic122—42no—
120C2×Dic122—41no—
121C2×C8×C61—21no—
122C2×C8×C61—21no—
123C2×C8×C61—21no—
124C2×C8×C61—21no—
125C2×C8×C61—42no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—41no—
131C4⋊C4×C61—41no—
132C22⋊C4×C61—41no—
133C42×C61—41no—
134C42×C61C2×D411yescharacteristic
135C42×C61C2×D411yescharacteristic
136C42×C61#4111yescharacteristic
137C42×C61C2411yescharacteristic
138#11758201—82no—
139#11758198—22no—
140#11758198—22no—
141C4⋊Dic61—21no—
142C4⋊Dic61—22no—
143C4⋊Dic61—22no—
144C4⋊Dic61—41no—
145C4⋊Dic61—41no—
146C4⋊Dic61C2×D412yes—
147#11758194—22no—
148#11758194—22no—
149C2×#92418—22no—
150C2×#92418—81no—
151C2×#92418C2×D412yes—
152C2×#92418C2×D412yes—
153#11758189—44no—
154C8×D61—44no—
155C23×C61#12111yescharacteristic
156C22×D61—22no—
157Q8×C61—41no—
158Q8×C61—41no—
159Q8×C61—81no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—41no—
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61#13512yes—
169C2×C4×C61AGL(1,ℤ/8)12yes—
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C22×D411yescharacteristic
172C2×C4×C61C2×#4111yescharacteristic
173C22⋊D61—44no—
174C2×Dic61—22no—
175C2×Dic61—42no—
176C2×Dic61—82no—
177D244—22no—
178D244—22no—
179D244—42no—
180D244—81no—
181C4×D61—42no—
182C4×D61—42no—
183Dic122—42no—
184Dic122—42no—
185Dic122—42no—
186Dic122—82no—
187C8×C61—41no—
188C8×C61—41no—
189C8×C61—41no—
190C8×C61—41no—
191#92418—22no—
192#92418—22no—
193#92418—42no—
194#19235—611nomaximal, Sylow
195C22×C61—21no—
196C22×C61—21no—
197C22×C61#46312yes—
198C22×C61C2×#12111yescharacteristic
199C2×D61—42no—
200C2×D61—42no—
201C2×D61—42no—
202C4×C61—21no—
203C4×C61—21no—
204C4×C61—21no—
205C4×C61—81no—
206C4×C61—81no—
207C4×C61#46312yes—
208C4×C61C2×#13511yescharacteristic
209C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
210C4×C61#57711yescharacteristic
211C4×C61#57811yescharacteristic
212Dic61—42no—
213Dic61—82no—
214#3246—611no—
215#3226—611no—
216#3139—611no—
217#2935—611no—
218#1703—612no—
219#1699—612no—
220#1683—612no—
221#1679—612no—
222#1565—611no—
223#1542D6111yescharacteristic, complemented
224C2×#332—611no—
225C2×C61—22no—
226C2×C61C2×#46311yescharacteristic
227C2×C61#306311yescharacteristic
228C2×C61#306911yescharacteristic
229D61—82no—
230C2×#131—1221no—
231#533C2×D6111yescharacteristic
232#532—611no—
233C2×C4⋊Q8—611no—
234#500—1221no—
235#491—1222no—
236#487—612no—
237#487—612no—
238#486—1221no—
239#480—1222no—
240#476—612no—
241#475—1221no—
242#474—612no—
243#474—612no—
244#465—1222no—
245#461—612no—
246#460—1221no—
247#436—612no—
248#435—1222no—
249C8×D4—612no—
250#431C2×D6111yescharacteristic
251C2×C4⋊C8—611no—
252C2×C4⋊C8C2×D6111yescharacteristic
253#419—1221no—
254#419—1221no—
255#408—1221no—
256C4×M16—611no—
257#335—612no—
258#334—612no—
259#333C2×D6111yescharacteristic
260#333C2×D6111yescharacteristic
261#332—612no—
262#332—612no—
263#332C2×D6111yescharacteristic
264#332C2×D6111yescharacteristic
265C61#1923511yescharacteristic, complemented, Sylow
266C2×#41—1221no—
267C22×Q8—1221no—
268C2×SD16—1222no—
269#131—2444no—
270C2×M16—611no—
271C2×M16—611no—
272C2×M16—1222no—
273C22×C8—21no—
274C22×C8—1222no—
275C22×C8—1221no—
276C4⋊Q8—611no—
277C4⋊Q8—612no—
278C4⋊Q8—1221no—
279C4⋊Q8—1221no—
280C4⋊Q8—1221no—
281C4⋊Q8C22×D6111yescharacteristic
282C4⋊Q8C22×D6111yescharacteristic
283#127—611no—
284#125—21no—
285#124—1221no—
286C22⋊Q8—21no—
287C22⋊Q8—21no—
288#121—612no—
289#121—1221no—
290C4×Q8—21no—
291C4×Q8—21no—
292C4×D4—612no—
293C4×D4—1221no—
294#117—21no—
295C2×C4⋊C4—1221no—
296C2×C4⋊C4—1221no—
297C2×C42C22×D6111yescharacteristic
298#107—1222no—
299#107—1222no—
300#106—1222no—
301C4⋊C8—21no—
302C4⋊C8—612no—
303C4⋊C8—1221no—
304C4⋊C8C22×D6111yescharacteristic
305C4⋊C8C22×D6111yescharacteristic
306C4⋊C8C22×D6111yescharacteristic
307C4⋊C8C22×D6111yescharacteristic
308#103—1222no—
309#103—1222no—
310#103—1222no—
311#103—1222no—
312#103—1222no—
313#102—1222no—
314#102—1222no—
315#102—1222no—
316C22⋊C8—21no—
317C22⋊C8—1222no—
318C22⋊C8—1222no—
319#97—612no—
320C4×C8—21no—
321C4×C8—612no—
322#41—2442no—
323#41—2442no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—1221no—
327C2×Q8—1222no—
328C2×Q8—2442no—
329C2×Q8—2442no—
330C2×Q8—2441no—
331C2×D4—612no—
332C2×D4—1222no—
333C2×D4—1221no—
334C2×D4—1221no—
335C22×C4—1222no—
336C22×C4—2441no—
337C22×C4#1175822211yescharacteristic
338C22×C4D4×D6111yescharacteristic
339C22×C4D4×D6111yescharacteristic
340SD16—4882no—
341M16—1222no—
342M16—1222no—
343M16—2444no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—42no—
349C2×C8—612no—
350C2×C8—612no—
351C2×C8—1222no—
352C2×C8—4881no—
353C2×C8—2444no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—612no—
361C4⋊C4—1221no—
362C4⋊C4—1222no—
363C4⋊C4—1222no—
364C4⋊C4—2441no—
365C4⋊C4—2441no—
366C22⋊C4—41no—
367C22⋊C4—1222no—
368C22⋊C4—1222no—
369C42—41no—
370C42#1175822211yescharacteristic
371C42D4×D6111yescharacteristic
372C42D4×D6111yescharacteristic
373C42C23×D6111yescharacteristic
374C23—1222no—
375C23#1451975011yescharacteristic
376Q8—41no—
377Q8—41no—
378Q8—81no—
379Q8—4882no—
380Q8—2442no—
381Q8—2442no—
382Q8—2442no—
383D4—1222no—
384D4—1222no—
385D4—4881no—
386D4—2444no—
387D4—2442no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—1222no—
394C2×C4—4882no—
395C2×C4—2442no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4#1451975011yescharacteristic
399C2×C4#1451975011yescharacteristic
400C2×C4#1451975011yescharacteristic
401C2×C4#1451980012yes—
402C2×C4#1451980412yes—
403C2×C4C2×D4×D6111yescharacteristic
404C2×C4C2×D4×D6111yescharacteristic
405C2×C4C2×#1175822211yescharacteristic
406C8—41no—
407C8—41no—
408C8—41no—
409C8—41no—
410C8—1222no—
411C8—1222no—
412C8—2442no—
413C22—21no—
414C22—21no—
415C22—2442no—
416C22—2442no—
417C22—2442no—
418C22#1683998012yes—
419C22C2×#1451975011yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—81no—
424C4—81no—
425C4—4882no—
426C4—2442no—
427C4#1683998012yes—
428C4C2×#1451980011yescharacteristic
429C4C2×#1451980411yescharacteristic
430C4#1684056911yescharacteristic
431C4#1684057611yescharacteristic
432C2—22no—
433C2—4882no—
434C2C2×#1683998011yescharacteristic
435C2#2145551311yescharacteristic
436C2#2145555911yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1542C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#1542D61
hypocenterC61#19235
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. #21456201C2semidirect
5. #21445397C2semidirect
6. #21445369C2semidirect
7. #21445273C2semidirect
8. #21445245C2semidirect
9. #21444572C2semidirect
10. C2×#16839509C2semidirect
11. C61×#1542C2semidirect
223. #1542D61semidirect
265. C61#19235semidirect

Maximal subgroups

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

subgroupindex
#214571312
#214569422
#214562012
#214545622
#214453972
#214453692
#214452732
#214452452
#214445722
C2×#168395092
C61×#15422
#1923561