Finite Group Databasebeta

← #29663568

Subgroups — #29663568

Order 15616.

674 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457126C211yesmaximal, characteristic
2#21456744C211yesmaximal, characteristic, complemented
3#21454565C211yesmaximal, characteristic, complemented
4#21445377C212yesmaximal, complemented
5#21445363C212yesmaximal, complemented
6#21445253C212yesmaximal, complemented
7#21445239C212yesmaximal, complemented
8#21444571C211yesmaximal, characteristic, complemented
9#21444468C211yesmaximal, characteristic, complemented
10C2×#16839508C211yesmaximal, characteristic, complemented
11C61×#3141C211yesmaximal, characteristic, complemented
12C2×#14519785—21no—
13#16840312C2211yescharacteristic
14#16840136—21no—
15#16840112—21no—
16#16840042—21no—
17#16840039C2212yes—
18#16839996—22no—
19#16839964C2212yes—
20#16839964C2212yes—
21#16839957—21no—
22#16839954C2212yes—
23#16839901—22no—
24#16839841C2212yes—
25#16839835—22no—
26C8×D244C2212yes—
27#16839815C2211yescharacteristic
28C2×C4⋊#92418C2211yescharacteristic
29C2×C4⋊#92418C2211yescharacteristic
30#16839730—21no—
31#16839515C2212yes—
32#16839513C2212yes—
33#16839508C2212yes—
34#16839508C2212yes—
35#16839508C2211yescharacteristic
36#16839508C2211yescharacteristic
37#16839506C2211yescharacteristic
38#16839506C2211yescharacteristic
39C61×#532C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C61×#489—22no—
42C61×#487C2212yes—
43C61×#486C2212yes—
44C61×#417—21no—
45C61×#417—21no—
46C4×M16×C61C2211yescharacteristic
47C2×#11758222—21no—
48C2×#11758201—22no—
49C2×#11758201—22no—
50C2×#11758199—22no—
51#14519785—44no—
52C2×#11758189—22no—
53C22×#92418—21no—
54C22×#92418—21no—
55C2×C8×D61—22no—
56#14519777D411yescharacteristic
57#14519774D411yescharacteristic
58#14519765—21no—
59#14519751—21no—
60#14519751D412yes—
61C4×D244—21no—
62C4×D244C2312yes—
63#14519710—22no—
64C4⋊#92418—21no—
65C4⋊#92418—21no—
66C4⋊#92418C2312yes—
67C4⋊#92418C2311yescharacteristic
68C4⋊#92418C2311yescharacteristic
69C4⋊#92418C2311yescharacteristic
70C4⋊#92418C2311yescharacteristic
71#14519701—22no—
72#14519701—22no—
73#14519698—22no—
74#14519697—22no—
75C22⋊#92418—21no—
76#14519689—22no—
77#14519689—22no—
78C4×#92418—21no—
79C2×#41×C61—21no—
80C22×Q8×C61—21no—
81C2×M16×C61D411yescharacteristic
82C2×M16×C61D411yescharacteristic
83C4⋊Q8×C61—21no—
84C4⋊Q8×C61—21no—
85C4⋊Q8×C61—21no—
86C4⋊Q8×C61C2312yes—
87C4⋊Q8×C61C2311yescharacteristic
88#127×C61C2311yescharacteristic
89#124×C61—21no—
90#121×C61—21no—
91#121×C61—21no—
92C4×D4×C61—21no—
93C4×D4×C61—21no—
94C2×C4⋊C4×C61—21no—
95C2×C4⋊C4×C61—21no—
96C2×C42×C61C2311yescharacteristic
97#103×C61—22no—
98#103×C61—22no—
99#102×C61—22no—
100#102×C61—22no—
101#97×C61D412yes—
102C4×C8×C61C2312yes—
103#41×C61—42no—
104#41×C61—42no—
105C2×Q8×C61—21no—
106C2×Q8×C61—22no—
107C2×Q8×C61—41no—
108C2×Q8×C61—81no—
109C2×Q8×C61—81no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61—21no—
113C2×D4×C61—41no—
114C22×C4×C61—41no—
115C22×C4×C61—41no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C2×C22⋊D61—22no—
120#11758222—42no—
121#11758222—42no—
122C2×D244—21no—
123C2×D244—21no—
124C2×D244C2×D412yes—
125C2×C4×D61—22no—
126C2×Dic122—21no—
127M16×C61—22no—
128M16×C61—22no—
129C2×C8×C61C2×D412yes—
130C2×C8×C61C2×D412yes—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—21no—
135C4⋊C4×C61—41no—
136C4⋊C4×C61—41no—
137C22⋊C4×C61—41no—
138C22⋊C4×C61—41no—
139C42×C61C2×D411yescharacteristic
140C42×C61C2×D411yescharacteristic
141C42×C61#4111yescharacteristic
142C42×C61C2411yescharacteristic
143#11758201—82no—
144#11758201—82no—
145#11758199—82no—
146#11758198—22no—
147#11758198—22no—
148C4⋊Dic61—21no—
149C4⋊Dic61C2×D412yes—
150C2×#92418—21no—
151C2×#92418—21no—
152C2×#92418—21no—
153C2×#92418—21no—
154C2×#92418—22no—
155C2×#92418—42no—
156C2×#92418—81no—
157#11758189—44no—
158C8×D61—44no—
159C23×C61—41no—
160C23×C61#12111yescharacteristic
161C22×D61—22no—
162Q8×C61—41no—
163Q8×C61—41no—
164Q8×C61—42no—
165Q8×C61—42no—
166Q8×C61—82no—
167D4×C61—41no—
168D4×C61—41no—
169D4×C61—41no—
170D4×C61—41no—
171D4×C61—81no—
172D4×C61—82no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—42no—
178C2×C4×C61—82no—
179C2×C4×C61—82no—
180C2×C4×C61#12111yescharacteristic
181C2×C4×C61#12111yescharacteristic
182C2×C4×C61#12111yescharacteristic
183C2×C4×C61#13512yes—
184C2×C4×C61AGL(1,ℤ/8)12yes—
185C2×C4×C61C22×D411yescharacteristic
186C2×C4×C61C22×D411yescharacteristic
187C2×C4×C61C2×#4111yescharacteristic
188C22⋊D61—44no—
189C2×Dic61—22no—
190D244—22no—
191D244—22no—
192D244—42no—
193D244—81no—
194C4×D61—42no—
195C4×D61—42no—
196Dic122—42no—
197C8×C61—22no—
198C8×C61—22no—
199#92418—41no—
200#92418—41no—
201#92418—42no—
202#92418—41no—
203#92418—41no—
204#19237—611nomaximal, Sylow
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—81no—
208C22×C61—81no—
209C22×C61—81no—
210C22×C61#46312yes—
211C22×C61C2×#12111yescharacteristic
212C2×D61—42no—
213C2×D61—42no—
214C2×D61—42no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—81no—
219C4×C61—82no—
220C4×C61#46312yes—
221C4×C61C2×#13511yescharacteristic
222C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
223C4×C61#57611yescharacteristic
224C4×C61#57711yescharacteristic
225Dic61—42no—
226#3245—611no—
227#3206—611no—
228#3141D6111yescharacteristic, complemented
229#2935—611no—
230#1701—612no—
231#1699—612no—
232#1681—612no—
233#1679—612no—
234#1565—611no—
235#1540—611no—
236C2×#332—611no—
237C2×C61—22no—
238C2×C61—161no—
239C2×C61C2×#46311yescharacteristic
240C2×C61#306211yescharacteristic
241C2×C61#306911yescharacteristic
242D61—82no—
243C2×#131—1221no—
244#532—611no—
245#532C2×D6111yescharacteristic
246C2×C4⋊Q8C2×D6111yescharacteristic
247#492—1221no—
248#489—22no—
249#487—1221no—
250#487C2×D6112yes—
251#486C2×D6112yes—
252#474—612no—
253#474—1221no—
254#466—1222no—
255#461—612no—
256#461—612no—
257#459—612no—
258#459—1221no—
259#448—1222no—
260#436—612no—
261#435—1222no—
262C8×D4—612no—
263#431—611no—
264C2×C4⋊C8—611no—
265C2×C4⋊C8—611no—
266#417—21no—
267#417—21no—
268#408—1221no—
269C4×M16C2×D6111yescharacteristic
270#335—612no—
271#334—612no—
272#332—612no—
273#332—612no—
274#332—611no—
275#332—611no—
276#331—611no—
277#331—611no—
278C61#1923711yescharacteristic, complemented, Sylow
279C2×#41—21no—
280C2×#41—1221no—
281C22×Q8—21no—
282C2×SD16—1222no—
283C2×SD16—1222no—
284C2×D8—1222no—
285#131—2444no—
286C2×M16—1222no—
287C2×M16C22⋊D6111yescharacteristic
288C2×M16C22⋊D6111yescharacteristic
289C22×C8—1221no—
290C22×C8—1221no—
291C22×C8—1222no—
292C4⋊Q8—21no—
293C4⋊Q8—21no—
294C4⋊Q8—21no—
295C4⋊Q8—611no—
296C4⋊Q8C22×D6112yes—
297C4⋊Q8C22×D6111yescharacteristic
298#127—611no—
299#127C22×D6111yescharacteristic
300#124—21no—
301#124—1221no—
302#121—21no—
303#121—21no—
304#121—612no—
305#121—1221no—
306C4×D4—21no—
307C4×D4—21no—
308C4×D4—612no—
309C4×D4—1221no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C42C22×D6111yescharacteristic
313#107—1222no—
314C4⋊C8—611no—
315C4⋊C8—611no—
316C4⋊C8—611no—
317C4⋊C8—611no—
318C4⋊C8—612no—
319C4⋊C8—1221no—
320C4⋊C8—1221no—
321#103—22no—
322#103—22no—
323#103—1222no—
324#103—1222no—
325#102—22no—
326#102—22no—
327#102—1222no—
328#102—1222no—
329C22⋊C8—1221no—
330C22⋊C8—1222no—
331C22⋊C8—1222no—
332#97C22⋊D6112yes—
333C4×C8—1221no—
334C4×C8C22×D6112yes—
335#41—42no—
336#41—42no—
337#41—2442no—
338#41—2442no—
339C2×Q8—22no—
340C2×Q8—21no—
341C2×Q8—41no—
342C2×Q8—81no—
343C2×Q8—81no—
344C2×Q8—1221no—
345C2×D4—21no—
346C2×D4—21no—
347C2×D4—21no—
348C2×D4—41no—
349C2×D4—612no—
350C2×D4—1222no—
351C2×D4—1221no—
352C2×D4—1221no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—1222no—
356C22×C4D4×D6111yescharacteristic
357C22×C4#1175822411yescharacteristic
358C22×C4C2×C22⋊D6111yescharacteristic
359SD16—4882no—
360SD16—4882no—
361D8—4882no—
362M16—22no—
363M16—22no—
364M16—2444no—
365C2×C8—1221no—
366C2×C8—1221no—
367C2×C8—1222no—
368C2×C8—1221no—
369C2×C8—1221no—
370C2×C8—4881no—
371C2×C8—2444no—
372C2×C8—2442no—
373C2×C8C2×C22⋊D6112yes—
374C2×C8C2×C22⋊D6112yes—
375C4⋊C4—21no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—612no—
382C4⋊C4—1221no—
383C22⋊C4—41no—
384C22⋊C4—41no—
385C22⋊C4—1222no—
386C22⋊C4—1222no—
387C42D4×D6111yescharacteristic
388C42#1175822411yescharacteristic
389C42C2×C22⋊D6111yescharacteristic
390C42C23×D6111yescharacteristic
391C23—41no—
392C23—1222no—
393C23#1451974811yescharacteristic
394Q8—41no—
395Q8—41no—
396Q8—42no—
397Q8—42no—
398Q8—82no—
399Q8—2442no—
400D4—41no—
401D4—41no—
402D4—41no—
403D4—41no—
404D4—81no—
405D4—82no—
406D4—1222no—
407D4—1222no—
408D4—4881no—
409D4—2444no—
410D4—2442no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—1222no—
419C2×C4—2442no—
420C2×C4—2442no—
421C2×C4#1451974811yescharacteristic
422C2×C4#1451974811yescharacteristic
423C2×C4#1451974811yescharacteristic
424C2×C4#1451979712yes—
425C2×C4#1451980212yes—
426C2×C4C2×D4×D6111yescharacteristic
427C2×C4C22×C22⋊D6111yescharacteristic
428C2×C4C2×#1175822411yescharacteristic
429C8—22no—
430C8—22no—
431C8—2441no—
432C8—2441no—
433C8—2442no—
434C8—2441no—
435C8—2441no—
436C22—21no—
437C22—21no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—2442no—
442C22—2442no—
443C22—2442no—
444C22#1683997412yes—
445C22C2×#1451974811yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—82no—
450C4—81no—
451C4—2442no—
452C4#1683997412yes—
453C4C2×#1451979711yescharacteristic
454C4C2×#1451980211yescharacteristic
455C4#1684056611yescharacteristic
456C4#1684057211yescharacteristic
457C2—22no—
458C2—161no—
459C2—4882no—
460C2C2×#1683997411yescharacteristic
461C2#2145549311yescharacteristic
462C2#2145555211yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3141C2
socleC22×C61C2×#121
prime radical#21457126C2
hypercenter#3141D61
hypocenterC61#19237
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3141

Complements

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

NH ≅ G/Ntype
2. #21456744C2semidirect
3. #21454565C2semidirect
4. #21445377C2semidirect
5. #21445363C2semidirect
6. #21445253C2semidirect
7. #21445239C2semidirect
8. #21444571C2semidirect
9. #21444468C2semidirect
10. C2×#16839508C2semidirect
11. C61×#3141C2semidirect
228. #3141D61semidirect
278. C61#19237semidirect

Maximal subgroups

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

subgroupindex
#214571262
#214567442
#214545652
#214453772
#214453632
#214452532
#214452392
#214445712
#214444682
C2×#168395082
C61×#31412
#1923761