Finite Group Databasebeta

← #29663550

Subgroups — #29663550

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#21457136C211yesmaximal, characteristic
2#21456940C211yesmaximal, characteristic
3#21456199C211yesmaximal, characteristic, complemented
4#21454562C211yesmaximal, characteristic
5#21445393C212yesmaximal, complemented
6#21445365C212yesmaximal, complemented
7#21445269C212yesmaximal, complemented
8#21445241C212yesmaximal, complemented
9#21444569C211yesmaximal, characteristic, complemented
10#21444477C211yesmaximal, characteristic, complemented
11C2×C61×#332C211yesmaximal, characteristic, complemented
12C2×#14519783—21no—
13#16840319C2211yescharacteristic
14#16840312C2211yescharacteristic
15#16840165—21no—
16#16840130—22no—
17#16840112C2212yes—
18#16840112C2212yes—
19#16840110—21no—
20#16840075—22no—
21#16840053C2212yes—
22#16840048—21no—
23#16840039C2212yes—
24#16840039C2212yes—
25#16839991—22no—
26#16839968C2212yes—
27#16839963—21no—
28#16839839C2212yes—
29#16839832—22no—
30#16839827C2212yes—
31#16839815C2211yescharacteristic
32#16839771—21no—
33#16839771—21no—
34#16839728—21no—
35C4×#11758194C2211yescharacteristic
36#16839514C2212yes—
37#16839512C2212yes—
38#16839510C2211yescharacteristic
39#16839510C2211yescharacteristic
40#16839507C2211yescharacteristic
41#16839507C2211yescharacteristic
42C2×C4⋊Q8×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×C4⋊C8×C61C2211yescharacteristic
45C61×#332C2212yes—
46C61×#332C2212yes—
47C2×#11758222—21no—
48C2×#11758190—22no—
49#14519783—44no—
50C2×#11758194D411yescharacteristic
51C2×#11758194D411yescharacteristic
52C2×#11758189—22no—
53C2×C8×D61—22no—
54#14519777D411yescharacteristic
55#14519777C2311yescharacteristic
56#14519777C2311yescharacteristic
57#14519774D411yescharacteristic
58#14519769—21no—
59#14519765—21no—
60#14519756—21no—
61#14519756—21no—
62#14519751—21no—
63#14519751D412yes—
64C4×Dic122—21no—
65C4×Dic122—21no—
66C4×D244—21no—
67C4×D244C2312yes—
68#14519736—21no—
69#14519710—22no—
70#14519710—22no—
71#14519709—22no—
72C4⋊#92418C2311yescharacteristic
73C4⋊#92418C2311yescharacteristic
74#14519702—22no—
75#14519701—22no—
76#14519701—22no—
77#14519701—22no—
78#14519701—22no—
79#14519699—22no—
80#14519697—22no—
81#14519697—22no—
82C22⋊#92418—21no—
83#14519689—22no—
84#14519689—22no—
85#14519688D412yes—
86C4×#92418—21no—
87C4×#92418C2312yes—
88C22×Q8×C61—21no—
89C22×C8×C61—21no—
90C22×C8×C61—21no—
91C4⋊Q8×C61—21no—
92C4⋊Q8×C61—21no—
93C4⋊Q8×C61—21no—
94C4⋊Q8×C61C2312yes—
95C2×C4⋊C4×C61—21no—
96C2×C4⋊C4×C61—21no—
97C2×C42×C61C2311yescharacteristic
98C4⋊C8×C61—21no—
99C4⋊C8×C61—21no—
100C4⋊C8×C61C2312yes—
101C4⋊C8×C61C2311yescharacteristic
102C4⋊C8×C61C2311yescharacteristic
103C2×Q8×C61—22no—
104C2×Q8×C61—41no—
105C2×Q8×C61—42no—
106C2×Q8×C61—42no—
107C22×C4×C61—41no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61#4111yescharacteristic
111C2×C22⋊D61—22no—
112#11758222—42no—
113#11758222—42no—
114C2×D244—21no—
115C2×D244—21no—
116C2×D244C2×D412yes—
117C2×C4×D61—22no—
118C2×Dic122—21no—
119C2×Dic122—21no—
120C2×Dic122—21no—
121C2×C8×C61—21no—
122C2×C8×C61—21no—
123C2×C8×C61—22no—
124C2×C8×C61—42no—
125C2×C8×C61—81no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—41no—
129C4⋊C4×C61—41no—
130C42×C61C2×D411yescharacteristic
131C42×C61C2×D411yescharacteristic
132C42×C61#4111yescharacteristic
133C42×C61C2411yescharacteristic
134C22⋊Dic61—41no—
135#11758198—22no—
136#11758198—22no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61—21no—
140C4⋊Dic61—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61C2×D412yes—
143#11758196—41no—
144#11758196—41no—
145C4×Dic61—41no—
146#11758194—22no—
147#11758194—22no—
148C2×#92418—21no—
149C2×#92418—21no—
150C2×#92418C2×D412yes—
151C2×#92418C2×D412yes—
152#11758190—82no—
153#11758189—44no—
154C8×D61—44no—
155C23×C61#12111yescharacteristic
156C22×D61—22no—
157Q8×C61—42no—
158Q8×C61—42no—
159Q8×C61—82no—
160C2×C4×C61—21no—
161C2×C4×C61—21no—
162C2×C4×C61—41no—
163C2×C4×C61—42no—
164C2×C4×C61—82no—
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—41no—
176C2×Dic61—41no—
177D244—22no—
178D244—22no—
179D244—42no—
180D244—81no—
181C4×D61—42no—
182C4×D61—42no—
183Dic122—42no—
184Dic122—41no—
185Dic122—41no—
186Dic122—81no—
187C8×C61—41no—
188C8×C61—41no—
189C8×C61—42no—
190#92418—22no—
191#92418—22no—
192#92418—41no—
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#46312yes—
207C4×C61C2×#13511yescharacteristic
208C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
209C4×C61#57711yescharacteristic
210C4×C61#57811yescharacteristic
211Dic61—42no—
212Dic61—81no—
213Dic61—81no—
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#1542—611no—
224C2×#332D6111yescharacteristic, complemented
225C2×C61—22no—
226C2×C61C2×#46311yescharacteristic
227C2×C61#306311yescharacteristic
228C2×C61#306911yescharacteristic
229D61—82no—
230C2×#131—1221no—
231#533—611no—
232#532—611no—
233C2×C4⋊Q8C2×D6111yescharacteristic
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#431—611no—
251C2×C4⋊C8C2×D6111yescharacteristic
252C2×C4⋊C8C2×D6111yescharacteristic
253#419—1221no—
254#419—1221no—
255#408—1221no—
256C4×M16—611no—
257#335—612no—
258#334—612no—
259#333—611no—
260#333—611no—
261#332—611no—
262#332—611no—
263#332C2×D6112yes—
264#332C2×D6112yes—
265C61#1923511yescharacteristic, complemented, Sylow
266C2×#41—1221no—
267C22×Q8—21no—
268C2×SD16—1222no—
269#131—2444no—
270C2×M16—611no—
271C2×M16—611no—
272C2×M16—1222no—
273C22×C8—21no—
274C22×C8—21no—
275C22×C8—1222no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8—21no—
279C4⋊Q8—611no—
280C4⋊Q8—611no—
281C4⋊Q8—611no—
282C4⋊Q8C22×D6112yes—
283#127—611no—
284#125—1221no—
285#124—1221no—
286C22⋊Q8—1221no—
287C22⋊Q8—1221no—
288#121—612no—
289#121—1221no—
290C4×Q8—1221no—
291C4×Q8—1221no—
292C4×D4—612no—
293C4×D4—1221no—
294#117—1221no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C42C22×D6111yescharacteristic
298#107—1222no—
299#107—1222no—
300#106—1222no—
301C4⋊C8—21no—
302C4⋊C8—21no—
303C4⋊C8—611no—
304C4⋊C8—611no—
305C4⋊C8C22×D6111yescharacteristic
306C4⋊C8C22×D6111yescharacteristic
307C4⋊C8C22×D6112yes—
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—1222no—
317C22⋊C8—1222no—
318C22⋊C8—1221no—
319#97—612no—
320C4×C8—612no—
321C4×C8—1221no—
322#41—2442no—
323#41—2442no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—1221no—
329C2×Q8—1221no—
330C2×Q8—1221no—
331C2×D4—612no—
332C2×D4—1222no—
333C2×D4—1221no—
334C2×D4—1221no—
335C22×C4—41no—
336C22×C4—1222no—
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—22no—
347C2×C8—42no—
348C2×C8—81no—
349C2×C8—612no—
350C2×C8—612no—
351C2×C8—1221no—
352C2×C8—1221no—
353C2×C8—2444no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—612no—
359C4⋊C4—1221no—
360C4⋊C4—1221no—
361C4⋊C4—1221no—
362C4⋊C4—1221no—
363C4⋊C4—1221no—
364C4⋊C4—2441no—
365C4⋊C4—2441no—
366C22⋊C4—1222no—
367C22⋊C4—1222no—
368C22⋊C4—2441no—
369C42—2441no—
370C42#1175822211yescharacteristic
371C42D4×D6111yescharacteristic
372C42D4×D6111yescharacteristic
373C42C23×D6111yescharacteristic
374C23—1222no—
375C23#1451975011yescharacteristic
376Q8—42no—
377Q8—42no—
378Q8—82no—
379Q8—4881no—
380Q8—2441no—
381Q8—2441no—
382Q8—2442no—
383D4—1222no—
384D4—1222no—
385D4—4881no—
386D4—2444no—
387D4—2442no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—82no—
393C2×C4—1222no—
394C2×C4—2441no—
395C2×C4—2441no—
396C2×C4—2442no—
397C2×C4—2442no—
398C2×C4#1451975011yescharacteristic
399C2×C4#1451975011yescharacteristic
400C2×C4#1451975011yescharacteristic
401C2×C4#1451979812yes—
402C2×C4#1451980112yes—
403C2×C4C2×D4×D6111yescharacteristic
404C2×C4C2×D4×D6111yescharacteristic
405C2×C4C2×#1175822211yescharacteristic
406C8—41no—
407C8—41no—
408C8—42no—
409C8—1222no—
410C8—1222no—
411C8—2441no—
412C8—2441no—
413C22—21no—
414C22—21no—
415C22—2442no—
416C22—2442no—
417C22—2442no—
418C22#1683997612yes—
419C22C2×#1451975011yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—82no—
424C4—4881no—
425C4—4881no—
426C4—2442no—
427C4#1683997612yes—
428C4C2×#1451979811yescharacteristic
429C4C2×#1451980111yescharacteristic
430C4#1684056811yescharacteristic
431C4#1684057511yescharacteristic
432C2—22no—
433C2—4882no—
434C2C2×#1683997611yescharacteristic
435C2#2145550811yescharacteristic
436C2#2145555511yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#332C2
socleC22×C61C2×#121
prime radical#16840312C22
hypercenterC2×#332D61
hypocenterC61#19235
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
3. #21456199C2semidirect
5. #21445393C2semidirect
6. #21445365C2semidirect
7. #21445269C2semidirect
8. #21445241C2semidirect
9. #21444569C2semidirect
10. #21444477C2semidirect
11. C2×C61×#332C2semidirect
224. C2×#332D61semidirect
265. C61#19235semidirect

Maximal subgroups

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

subgroupindex
#214571362
#214569402
#214561992
#214545622
#214453932
#214453652
#214452692
#214452412
#214445692
#214444772
C2×C61×#3322
#1923561