Finite Group Databasebeta

← #29663546

Subgroups — #29663546

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#21456940C211yesmaximal, characteristic
3#21454565C211yesmaximal, characteristic
4#21445391C212yesmaximal, complemented
5#21445363C212yesmaximal, complemented
6#21445267C212yesmaximal, complemented
7#21445239C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444476C211yesmaximal, characteristic, complemented
10C2×#16839508C211yesmaximal, characteristic, complemented
11C61×#3139C211yesmaximal, characteristic, complemented
12C2×#14519785—21no—
13#16840312C2211yescharacteristic
14#16840165—21no—
15#16840110—21no—
16#16840075—22no—
17#16840049C2212yes—
18#16840047—21no—
19#16840039C2212yes—
20#16840039C2212yes—
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#16839511C2211yescharacteristic
34#16839511C2211yescharacteristic
35#16839508C2212yes—
36#16839508C2212yes—
37#16839508C2211yescharacteristic
38#16839508C2211yescharacteristic
39C61×#533C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C61×#491—22no—
42C61×#487C2212yes—
43C61×#487C2212yes—
44C61×#419—21no—
45C61×#419—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#14519710—22no—
62#14519710—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#103×C61—22no—
98#103×C61—22no—
99#103×C61—22no—
100#103×C61—22no—
101#97×C61D412yes—
102C4×C8×C61C2312yes—
103C2×Q8×C61—22no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61—41no—
107C2×Q8×C61—42no—
108C2×Q8×C61—42no—
109C22×C4×C61—41no—
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61#4111yescharacteristic
113C2×C22⋊D61—22no—
114#11758222—42no—
115#11758222—42no—
116C2×D244—21no—
117C2×D244—21no—
118C2×D244C2×D412yes—
119C2×C4×D61—22no—
120C2×Dic122—21no—
121M16×C61—22no—
122M16×C61—22no—
123C2×C8×C61C2×D412yes—
124C2×C8×C61C2×D412yes—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—41no—
132C4⋊C4×C61—41no—
133C4⋊C4×C61—41no—
134C4⋊C4×C61—41no—
135C22⋊C4×C61—41no—
136C42×C61—41no—
137C42×C61C2×D411yescharacteristic
138C42×C61C2×D411yescharacteristic
139C42×C61#4111yescharacteristic
140C42×C61C2411yescharacteristic
141#11758201—82no—
142#11758198—22no—
143#11758198—22no—
144C4⋊Dic61—21no—
145C4⋊Dic61C2×D412yes—
146C2×#92418—21no—
147C2×#92418—21no—
148C2×#92418—21no—
149C2×#92418—21no—
150C2×#92418—22no—
151C2×#92418—42no—
152C2×#92418—81no—
153#11758189—44no—
154C8×D61—44no—
155C23×C61#12111yescharacteristic
156C22×D61—22no—
157Q8×C61—42no—
158Q8×C61—42no—
159Q8×C61—41no—
160Q8×C61—41no—
161Q8×C61—81no—
162Q8×C61—82no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—41no—
166C2×C4×C61—41no—
167C2×C4×C61—41no—
168C2×C4×C61—42no—
169C2×C4×C61—82no—
170C2×C4×C61#12111yescharacteristic
171C2×C4×C61#12111yescharacteristic
172C2×C4×C61#12111yescharacteristic
173C2×C4×C61#13512yes—
174C2×C4×C61AGL(1,ℤ/8)12yes—
175C2×C4×C61C22×D411yescharacteristic
176C2×C4×C61C22×D411yescharacteristic
177C2×C4×C61C2×#4111yescharacteristic
178C22⋊D61—44no—
179C2×Dic61—22no—
180D244—22no—
181D244—22no—
182D244—42no—
183D244—81no—
184C4×D61—42no—
185C4×D61—42no—
186Dic122—42no—
187C8×C61—22no—
188C8×C61—22no—
189#92418—41no—
190#92418—41no—
191#92418—41no—
192#92418—42no—
193#92418—41no—
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—82no—
206C4×C61—81no—
207C4×C61—81no—
208C4×C61#46312yes—
209C4×C61C2×#13511yescharacteristic
210C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
211C4×C61#57711yescharacteristic
212C4×C61#57811yescharacteristic
213Dic61—42no—
214#3246—611no—
215#3226—611no—
216#3139D6111yescharacteristic, complemented
217#2935—611no—
218#1703—612no—
219#1699—612no—
220#1683—612no—
221#1679—612no—
222#1565—611no—
223#1542—611no—
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⋊Q8C2×D6111yescharacteristic
234#500—1221no—
235#491—22no—
236#487C2×D6112yes—
237#487C2×D6112yes—
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#431—611no—
251C2×C4⋊C8—611no—
252C2×C4⋊C8—611no—
253#419—21no—
254#419—21no—
255#408—1221no—
256C4×M16C2×D6111yescharacteristic
257#335—612no—
258#334—612no—
259#333—611no—
260#333—611no—
261#332—612no—
262#332—612no—
263#332—611no—
264#332—611no—
265C61#1923511yescharacteristic, complemented, Sylow
266C2×#41—1221no—
267C22×Q8—21no—
268C2×SD16—1222no—
269#131—2444no—
270C2×M16—1222no—
271C2×M16C22⋊D6111yescharacteristic
272C2×M16C22⋊D6111yescharacteristic
273C22×C8—1221no—
274C22×C8—1221no—
275C22×C8—1222no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—21no—
279C4⋊Q8—611no—
280C4⋊Q8C22×D6112yes—
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—21no—
296C2×C4⋊C4—21no—
297C2×C42C22×D6111yescharacteristic
298#107—1222no—
299#107—1222no—
300#106—1222no—
301C4⋊C8—611no—
302C4⋊C8—611no—
303C4⋊C8—612no—
304C4⋊C8—611no—
305C4⋊C8—611no—
306C4⋊C8—1221no—
307C4⋊C8—1221no—
308#103—22no—
309#103—22no—
310#103—22no—
311#103—22no—
312#103—1222no—
313#102—1222no—
314#102—1222no—
315#102—1222no—
316C22⋊C8—1221no—
317C22⋊C8—1222no—
318C22⋊C8—1222no—
319#97C22⋊D6112yes—
320C4×C8—1221no—
321C4×C8C22×D6112yes—
322#41—2442no—
323#41—2442no—
324C2×Q8—21no—
325C2×Q8—22no—
326C2×Q8—21no—
327C2×Q8—41no—
328C2×Q8—42no—
329C2×Q8—42no—
330C2×Q8—1221no—
331C2×D4—612no—
332C2×D4—1222no—
333C2×D4—1221no—
334C2×D4—1221no—
335C22×C4—41no—
336C22×C4—1222no—
337C22×C4D4×D6111yescharacteristic
338C22×C4#1175822411yescharacteristic
339C22×C4C2×C22⋊D6111yescharacteristic
340SD16—4882no—
341M16—22no—
342M16—22no—
343M16—2444no—
344C2×C8—1221no—
345C2×C8—1221no—
346C2×C8—1222no—
347C2×C8—1221no—
348C2×C8—1221no—
349C2×C8—4881no—
350C2×C8—2444no—
351C2×C8—2442no—
352C2×C8C2×C22⋊D6112yes—
353C2×C8C2×C22⋊D6112yes—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—612no—
365C4⋊C4—1221no—
366C22⋊C4—41no—
367C22⋊C4—1222no—
368C22⋊C4—1222no—
369C42—41no—
370C42D4×D6111yescharacteristic
371C42#1175822411yescharacteristic
372C42C2×C22⋊D6111yescharacteristic
373C42C23×D6111yescharacteristic
374C23—1222no—
375C23#1451974811yescharacteristic
376Q8—42no—
377Q8—42no—
378Q8—41no—
379Q8—41no—
380Q8—81no—
381Q8—82no—
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—42no—
394C2×C4—82no—
395C2×C4—1222no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4#1451974811yescharacteristic
399C2×C4#1451974811yescharacteristic
400C2×C4#1451974811yescharacteristic
401C2×C4#1451979712yes—
402C2×C4#1451980212yes—
403C2×C4C2×D4×D6111yescharacteristic
404C2×C4C22×C22⋊D6111yescharacteristic
405C2×C4C2×#1175822411yescharacteristic
406C8—22no—
407C8—22no—
408C8—2441no—
409C8—2441no—
410C8—2441no—
411C8—2442no—
412C8—2441no—
413C22—21no—
414C22—21no—
415C22—2442no—
416C22—2442no—
417C22—2442no—
418C22#1683997412yes—
419C22C2×#1451974811yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—82no—
424C4—81no—
425C4—81no—
426C4—2442no—
427C4#1683997412yes—
428C4C2×#1451979711yescharacteristic
429C4C2×#1451980211yescharacteristic
430C4#1684057111yescharacteristic
431C4#1684057811yescharacteristic
432C2—22no—
433C2—4882no—
434C2C2×#1683997411yescharacteristic
435C2#2145550411yescharacteristic
436C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3139C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenter#3139D61
hypocenterC61#19235
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
4. #21445391C2semidirect
5. #21445363C2semidirect
6. #21445267C2semidirect
7. #21445239C2semidirect
8. #21444571C2semidirect
9. #21444476C2semidirect
10. C2×#16839508C2semidirect
11. C61×#3139C2semidirect
216. #3139D61semidirect
265. C61#19235semidirect

Maximal subgroups

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

subgroupindex
#214571312
#214569402
#214545652
#214453912
#214453632
#214452672
#214452392
#214445712
#214444762
C2×#168395082
C61×#31392
#1923561