Finite Group Databasebeta

← #29663565

Subgroups — #29663565

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21457126C211yesmaximal, characteristic
2#21456215C211yesmaximal, characteristic, complemented
3#21454563C211yesmaximal, characteristic, complemented
4#21445396C212yesmaximal, complemented
5#21445354C212yesmaximal, complemented
6#21445272C212yesmaximal, complemented
7#21445230C212yesmaximal, complemented
8#21444572C211yesmaximal, characteristic, complemented
9C2×#16839511C211yesmaximal, characteristic, complemented
10#21444466C211yesmaximal, characteristic, complemented
11C61×#3227C211yesmaximal, characteristic, complemented
12C2×#14519784—21no—
13#16840312C2211yescharacteristic
14#16840124—22no—
15#16840114C2212yes—
16#16840110C2212yes—
17#16840042C2212yes—
18#16840039—21no—
19#16839996—22no—
20#16839957C2212yes—
21#16839954—21no—
22#16839840C2212yes—
23#16839834—22no—
24D4×#92418C2212yes—
25C2×C4⋊#92418C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27#16839761—21no—
28#16839761—21no—
29#16839729—21no—
30C4×#11758194C2211yescharacteristic
31#16839514C2212yes—
32#16839512C2212yes—
33#16839511C2212yes—
34#16839511C2212yes—
35#16839507C2211yescharacteristic
36#16839507C2211yescharacteristic
37#16839505C2211yescharacteristic
38#16839505C2211yescharacteristic
39C61×#532C2211yescharacteristic
40C2×C4⋊Q8×C61C2211yescharacteristic
41C61×#498—21no—
42C61×#486—21no—
43C61×#481—22no—
44C61×#476C2212yes—
45C61×#476C2212yes—
46C61×#431C2211yescharacteristic
47C2×#11758222—21no—
48C2×#11758199—22no—
49#14519784—44no—
50C2×#11758194—22no—
51C2×#11758194D411yescharacteristic
52C2×#11758194D411yescharacteristic
53C22×#92418—21no—
54C22×#92418—21no—
55C22×#92418—22no—
56#14519777C2311yescharacteristic
57#14519774C2311yescharacteristic
58#14519765—21no—
59#14519751—21no—
60#14519751—21no—
61C4×D244—21no—
62C4×D244—21no—
63#14519710—22no—
64C4⋊#92418—21no—
65C4⋊#92418—21no—
66C4⋊#92418C2312yes—
67C4⋊#92418C2311yescharacteristic
68C4⋊#92418C2311yescharacteristic
69#14519700—22no—
70#14519700—22no—
71#14519698—22no—
72#14519697—22no—
73#14519697—22no—
74#14519697—22no—
75C22⋊#92418—22no—
76C22⋊#92418—22no—
77#14519688D412yes—
78C4×#92418C2312yes—
79C2×#41×C61—21no—
80C22×Q8×C61—21no—
81C4⋊Q8×C61—21no—
82C4⋊Q8×C61—21no—
83C4⋊Q8×C61—21no—
84C4⋊Q8×C61D411yescharacteristic
85C4⋊Q8×C61C2312yes—
86#127×C61D411yescharacteristic
87#124×C61—21no—
88#121×C61—21no—
89#121×C61D412yes—
90C4×D4×C61—21no—
91C4×D4×C61C2312yes—
92C2×C4⋊C4×C61—21no—
93C2×C4⋊C4×C61—21no—
94C2×C42×C61C2311yescharacteristic
95#106×C61—22no—
96#106×C61—22no—
97C4⋊C8×C61C2311yescharacteristic
98C4⋊C8×C61C2311yescharacteristic
99#102×C61—22no—
100#102×C61—22no—
101C22⋊C8×C61—21no—
102C4×C8×C61—21no—
103#41×C61—42no—
104#41×C61—42no—
105C2×Q8×C61—21no—
106C2×Q8×C61—22no—
107C2×Q8×C61—41no—
108C2×Q8×C61—82no—
109C2×D4×C61—22no—
110C2×D4×C61—21no—
111C2×D4×C61—21no—
112C2×D4×C61C2×D412yes—
113C22×C4×C61—22no—
114C22×C4×C61—41no—
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61#4111yescharacteristic
118C2×C22⋊D61—41no—
119#11758222—42no—
120#11758222—42no—
121C2×D244—21no—
122C2×D244—21no—
123C2×D244—21no—
124C2×C4×D61—41no—
125C2×Dic122—21no—
126C2×C8×C61—21no—
127C2×C8×C61—21no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—41no—
132C4⋊C4×C61—41no—
133C4⋊C4×C61C2×D412yes—
134C22⋊C4×C61—22no—
135C22⋊C4×C61—22no—
136C42×C61C2×D411yescharacteristic
137C42×C61C2×D411yescharacteristic
138C42×C61#4111yescharacteristic
139C42×C61C2411yescharacteristic
140#11758199—82no—
141#11758198—41no—
142#11758198—41no—
143C4⋊Dic61—21no—
144C4⋊Dic61—21no—
145#11758194—22no—
146#11758194—22no—
147#11758194—44no—
148C2×#92418—21no—
149C2×#92418—21no—
150C2×#92418—22no—
151C2×#92418—44no—
152C2×#92418—42no—
153C2×#92418—81no—
154C2×#92418C2×D412yes—
155C2×#92418C2×D412yes—
156C23×C61—22no—
157C23×C61#12111yescharacteristic
158C22×D61—41no—
159Q8×C61—42no—
160Q8×C61—82no—
161Q8×C61—82no—
162D4×C61—22no—
163D4×C61—22no—
164D4×C61—44no—
165D4×C61—42no—
166D4×C61—81no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—22no—
170C2×C4×C61—41no—
171C2×C4×C61—42no—
172C2×C4×C61—42no—
173C2×C4×C61—42no—
174C2×C4×C61—82no—
175C2×C4×C61#12111yescharacteristic
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61#13512yes—
179C2×C4×C61AGL(1,ℤ/8)12yes—
180C2×C4×C61C22×D411yescharacteristic
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C2×#4111yescharacteristic
183C22⋊D61—82no—
184C2×Dic61—41no—
185D244—41no—
186D244—41no—
187D244—41no—
188D244—41no—
189D244—81no—
190C4×D61—82no—
191Dic122—41no—
192Dic122—41no—
193C8×C61—41no—
194C8×C61—41no—
195#92418—22no—
196#92418—22no—
197#92418—41no—
198#92418—41no—
199#92418—42no—
200#19236—611nomaximal, Sylow
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—42no—
204C22×C61—42no—
205C22×C61—42no—
206C22×C61#46312yes—
207C22×C61C2×#12111yescharacteristic
208C2×D61—81no—
209C2×D61—81no—
210C2×D61—81no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61—21no—
214C4×C61—42no—
215C4×C61—82no—
216C4×C61#46312yes—
217C4×C61C2×#13511yescharacteristic
218C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
219C4×C61#57711yescharacteristic
220C4×C61#57811yescharacteristic
221Dic61—81no—
222#3245—611no—
223#3227D6111yescharacteristic, complemented
224#3141—611no—
225#2935—611no—
226#1703—612no—
227#1697—612no—
228#1683—612no—
229#1677—612no—
230#1565—611no—
231C2×#333—611no—
232#1540—611no—
233C2×C61—22no—
234C2×C61—82no—
235C2×C61C2×#46311yescharacteristic
236C2×C61#306311yescharacteristic
237C2×C61#306811yescharacteristic
238D61—161no—
239C2×#131—1221no—
240#532—611no—
241#532C2×D6111yescharacteristic
242C2×C4⋊Q8C2×D6111yescharacteristic
243#498—21no—
244#489—1222no—
245#487—612no—
246#486—21no—
247#486—612no—
248#481—22no—
249#476C2×D6112yes—
250#476C2×D6112yes—
251#474—612no—
252#474—1221no—
253#466—1222no—
254#459—612no—
255#459—1221no—
256#436—612no—
257#435—1222no—
258C8×D4—612no—
259#431C2×D6111yescharacteristic
260C2×C4⋊C8—611no—
261C2×C4⋊C8—611no—
262#417—1221no—
263#417—1221no—
264#408—1221no—
265C4×M16—611no—
266#335—612no—
267#334—612no—
268#333—612no—
269#333—612no—
270#332—611no—
271#332—611no—
272#331—611no—
273#331—611no—
274C61#1923611yescharacteristic, complemented, Sylow
275C2×#41—21no—
276C2×#41—1221no—
277C22×Q8—21no—
278C2×D8—1222no—
279#131—2444no—
280C2×M16—611no—
281C2×M16—611no—
282C2×M16—1222no—
283C22×C8—1222no—
284C22×C8—1221no—
285C22×C8—1221no—
286C4⋊Q8—21no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—611no—
290C4⋊Q8C22⋊D6111yescharacteristic
291C4⋊Q8C22×D6112yes—
292#127—611no—
293#127C22⋊D6111yescharacteristic
294#124—21no—
295#124—1221no—
296#121—21no—
297#121—1221no—
298#121—1221no—
299#121C22⋊D6112yes—
300C4×D4—21no—
301C4×D4—1221no—
302C4×D4—1221no—
303C4×D4C22×D6112yes—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C42C22×D6111yescharacteristic
307#107—1222no—
308#106—22no—
309#106—22no—
310C4⋊C8—611no—
311C4⋊C8—611no—
312C4⋊C8—612no—
313C4⋊C8—1221no—
314C4⋊C8—1221no—
315C4⋊C8C22×D6111yescharacteristic
316C4⋊C8C22×D6111yescharacteristic
317#103—1222no—
318#103—1222no—
319#102—22no—
320#102—22no—
321#102—1222no—
322#102—1222no—
323#102—1222no—
324#102—1222no—
325C22⋊C8—21no—
326C22⋊C8—1222no—
327C22⋊C8—1222no—
328#97—612no—
329C4×C8—21no—
330C4×C8—612no—
331#41—42no—
332#41—42no—
333#41—2442no—
334#41—2442no—
335C2×Q8—21no—
336C2×Q8—22no—
337C2×Q8—41no—
338C2×Q8—82no—
339C2×Q8—1221no—
340C2×D4—22no—
341C2×D4—21no—
342C2×D4—21no—
343C2×D4—1221no—
344C2×D4—1221no—
345C2×D4—1221no—
346C2×D4—2441no—
347C2×D4C2×C22⋊D6112yes—
348C22×C4—22no—
349C22×C4—41no—
350C22×C4—2441no—
351C22×C4D4×D6111yescharacteristic
352C22×C4#1175822411yescharacteristic
353C22×C4C2×C22⋊D6111yescharacteristic
354D8—4882no—
355M16—1222no—
356M16—1222no—
357M16—2444no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—612no—
361C2×C8—612no—
362C2×C8—1221no—
363C2×C8—1221no—
364C2×C8—1222no—
365C2×C8—4881no—
366C2×C8—2444no—
367C2×C8—2442no—
368C4⋊C4—21no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—1221no—
374C4⋊C4—1221no—
375C4⋊C4C2×C22⋊D6112yes—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—2441no—
379C22⋊C4—2441no—
380C42D4×D6111yescharacteristic
381C42#1175822411yescharacteristic
382C42C2×C22⋊D6111yescharacteristic
383C42C23×D6111yescharacteristic
384C23—22no—
385C23—2441no—
386C23#1451974911yescharacteristic
387Q8—42no—
388Q8—82no—
389Q8—82no—
390Q8—2441no—
391Q8—2441no—
392D4—22no—
393D4—22no—
394D4—44no—
395D4—42no—
396D4—81no—
397D4—4881no—
398D4—4882no—
399D4—2441no—
400D4—2441no—
401D4—2441no—
402D4—2441no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—82no—
411C2×C4—4882no—
412C2×C4—2441no—
413C2×C4#1451974911yescharacteristic
414C2×C4#1451974911yescharacteristic
415C2×C4#1451974911yescharacteristic
416C2×C4#1451979812yes—
417C2×C4#1451980412yes—
418C2×C4C2×D4×D6111yescharacteristic
419C2×C4C22×C22⋊D6111yescharacteristic
420C2×C4C2×#1175822411yescharacteristic
421C8—41no—
422C8—41no—
423C8—1222no—
424C8—1222no—
425C8—2441no—
426C8—2441no—
427C8—2442no—
428C22—21no—
429C22—21no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—4881no—
434C22—4881no—
435C22—4881no—
436C22#1683997912yes—
437C22C2×#1451974911yescharacteristic
438C4—21no—
439C4—21no—
440C4—21no—
441C4—42no—
442C4—82no—
443C4—4881no—
444C4#1683997912yes—
445C4C2×#1451979811yescharacteristic
446C4C2×#1451980411yescharacteristic
447C4#1684057011yescharacteristic
448C4#1684057711yescharacteristic
449C2—22no—
450C2—82no—
451C2—9761no—
452C2C2×#1683997911yescharacteristic
453C2#2145550611yescharacteristic
454C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3227C2
socleC22×C61C2×#121
prime radical#21457126C2
hypercenter#3227D61
hypocenterC61#19236
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3227

Complements

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

NH ≅ G/Ntype
2. #21456215C2semidirect
3. #21454563C2semidirect
4. #21445396C2semidirect
5. #21445354C2semidirect
6. #21445272C2semidirect
7. #21445230C2semidirect
8. #21444572C2semidirect
9. C2×#16839511C2semidirect
10. #21444466C2semidirect
11. C61×#3227C2semidirect
223. #3227D61semidirect
274. C61#19236semidirect

Maximal subgroups

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

subgroupindex
#214571262
#214562152
#214545632
#214453962
#214453542
#214452722
#214452302
#214445722
C2×#168395112
#214444662
C61×#32272
#1923661