Finite Group Databasebeta

← #29572002

Subgroups — #29572002

Order 15616.

696 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459684C211yesmaximal, characteristic
2#21444469C211yesmaximal, characteristic
3#21444382C211yesmaximal, characteristic, complemented
4#21444378C211yesmaximal, characteristic
5#21444221C211yesmaximal, characteristic, complemented
6#21444214C211yesmaximal, characteristic, complemented
7C61×#1805C211yesmaximal, characteristic, complemented
8#16840497—21no—
9#16840490—21no—
10#16840485C412yescomplemented
11#16840484—22no—
12#16840472C412yescomplemented
13#16840453—22no—
14#16840359—22no—
15#16840336—21no—
16#16840319C2211yescharacteristic
17#16840312C2211yescharacteristic
18#16840296—22no—
19#16840243—21no—
20#16839815C2211yescharacteristic
21#16839815C2211yescharacteristic
22#16839509—21no—
23#16839505—21no—
24#16839498—22no—
25#16839495—22no—
26Q8⋊#92418—22no—
27#16839492—22no—
28C2×C4×Q8×C61C2211yescharacteristic
29C2×C4⋊C8×C61C2211yescharacteristic
30C2×#103×C61—21no—
31C61×#382C2211yescharacteristic
32C61×#340—21no—
33C2×#11758222—21no—
34#14519777C2×C411yescharacteristic
35#14519777C2×C411yescharacteristic
36#14519777D411yescharacteristic
37#14519775—22no—
38#14519774D411yescharacteristic
39#14519772—24no—
40#14519771—24no—
41#14519769—21no—
42#14519769—22no—
43#14519767—24no—
44#14519766—22no—
45#14519765—22no—
46#14519765—21no—
47#14519762—24no—
48#14519760—24no—
49#14519756—21no—
50#14519756D412yes—
51#14519755—24no—
52#14519753—22no—
53#14519753—22no—
54#14519751—21no—
55#14519751D412yes—
56C4×Dic122—21no—
57C4×Dic122—22no—
58C4×Dic122D412yes—
59#14519742—22no—
60C4×D244—22no—
61C4×D244—21no—
62C4×D244C2×C412yes—
63#14519739—22no—
64C4×C22⋊D61—24no—
65#14519736—21no—
66#14519735—22no—
67#14519734—24no—
68C4⋊C4×D61—22no—
69C4⋊#92418—21no—
70C4⋊#92418—21no—
71C22⋊#92418—21no—
72C22⋊#92418—21no—
73C4×#92418—21no—
74C4×#92418—21no—
75C22×Q8×C61D411yescharacteristic
76C22×C8×C61—21no—
77C4×Q8×C61—21no—
78C4×Q8×C61—21no—
79C4×Q8×C61C2×C412yes—
80C4×Q8×C61D412yes—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—21no—
83C2×C4⋊C4×C61D411yescharacteristic
84C2×C42×C61—21no—
85C2×C42×C61C2311yescharacteristic
86C4⋊C8×C61—21no—
87C4⋊C8×C61—21no—
88#103×C61—42no—
89#95×C61—21no—
90#95×C61—21no—
91C2×Q8×C61—21no—
92C2×Q8×C61—21no—
93C2×Q8×C61—21no—
94C2×Q8×C61—21no—
95C2×Q8×C61—21no—
96C2×Q8×C61C22⋊C412yes—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C2×C22⋊D61—22no—
104#11758222—41no—
105#11758222—41no—
106#11758222—41no—
107#11758222—41no—
108C2×D244—21no—
109C2×D244—21no—
110C2×D244C22⋊C412yes—
111C2×C4×D61—22no—
112C2×C4×D61—24no—
113C2×Dic122—21no—
114C2×Dic122—21no—
115C2×Dic122C22⋊C412yes—
116C2×C8×C61—41no—
117C2×C8×C61—41no—
118C2×C8×C61—41no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—41no—
123C4⋊C4×C61—41no—
124C4⋊C4×C61C22⋊C412yes—
125C42×C61—22no—
126C42×C61—22no—
127C42×C61C22×C411yescharacteristic
128C42×C61C2×D411yescharacteristic
129C42×C61C2×D411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C22⋊Dic61—22no—
132#11758198—24no—
133#11758198—24no—
134#11758198—24no—
135#11758198—22no—
136#11758198—22no—
137C4⋊Dic61—21no—
138C4⋊Dic61—21no—
139C4⋊Dic61—21no—
140C4⋊Dic61—21no—
141C4⋊Dic61C22⋊C412yes—
142C4⋊Dic61C22⋊C412yes—
143#11758196—24no—
144#11758196—24no—
145#11758196—24no—
146#11758196—22no—
147#11758196—22no—
148C4×Dic61—24no—
149C4×Dic61—22no—
150C2×#92418—41no—
151C2×#92418—41no—
152C23×C61#12011yescharacteristic
153C22×D61—22no—
154Q8×C61—22no—
155Q8×C61—22no—
156Q8×C61—41no—
157Q8×C61—41no—
158C2×C4×C61—22no—
159C2×C4×C61—22no—
160C2×C4×C61—41no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—81no—
165C2×C4×C61—81no—
166C2×C4×C61—81no—
167C2×C4×C61—81no—
168C2×C4×C61C2×C22⋊C411yescharacteristic
169C2×C4×C61C2×C22⋊C411yescharacteristic
170C2×C4×C61C2×C22⋊C411yescharacteristic
171C2×C4×C61#12011yescharacteristic
172C2×C4×C61#12011yescharacteristic
173C2×C4×C61#12011yescharacteristic
174C2×C4×C61#13511yescharacteristic
175C2×C4×C61#13511yescharacteristic
176C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C61#13711yescharacteristic
178C22⋊D61—42no—
179C22⋊D61—42no—
180C2×Dic61—22no—
181C2×Dic61—22no—
182C2×Dic61—22no—
183D244—42no—
184D244—41no—
185D244—41no—
186D244—81no—
187C4×D61—42no—
188C4×D61—42no—
189C4×D61—84no—
190Dic122—42no—
191Dic122—41no—
192Dic122—41no—
193Dic122—42no—
194C8×C61—81no—
195#92418—81no—
196#92418—81no—
197#8431—611nomaximal, Sylow
198C22×C61—21no—
199C22×C61—21no—
200C22×C61#37911yescharacteristic
201C22×C61#44911yescharacteristic
202C22×C61#45211yescharacteristic
203C2×D61—42no—
204C2×D61—42no—
205C2×D61—42no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—21no—
209C4×C61—21no—
210C4×C61—42no—
211C4×C61—42no—
212C4×C61—161no—
213C4×C61#41611yescharacteristic
214C4×C61#41711yescharacteristic
215C4×C61#44911yescharacteristic
216C4×C61#45211yescharacteristic
217Dic61—42no—
218Dic61—42no—
219Dic61—42no—
220#3507—611no—
221#1805D6111yescharacteristic, complemented
222#1540—611no—
223#1518—611no—
224#1517—611no—
225#1481—611no—
226#1479—611no—
227C2×C61—21no—
228C2×C61—21no—
229C2×C61#178811yescharacteristic
230C2×C61#179411yescharacteristic
231C2×C61#179611yescharacteristic
232D61—82no—
233#559—1221no—
234#557—1221no—
235#556—612no—
236#556—1222no—
237#554—612no—
238#552—1222no—
239#539—1222no—
240#536—1221no—
241#533—611no—
242#532—611no—
243#529—1222no—
244#520—1221no—
245C2×C4×Q8C2×D6111yescharacteristic
246#431—611no—
247#431—611no—
248C2×C4⋊C8C2×D6111yescharacteristic
249C2×#103—21no—
250#382C2×D6111yescharacteristic
251#340—21no—
252#332—1221no—
253#331—1221no—
254#328—1222no—
255#327—1222no—
256Q8⋊C8—1222no—
257Q8⋊C8—1222no—
258C61#843111yescharacteristic, complemented, Sylow
259C2×#41—1221no—
260C22×Q8C22⋊D6111yescharacteristic
261C22×C8—21no—
262C4⋊Q8—611no—
263C4⋊Q8—611no—
264C4⋊Q8—611no—
265C4⋊Q8—1222no—
266#127—611no—
267#126—1224no—
268#126—1224no—
269#125—1222no—
270#125—1224no—
271#125—1221no—
272#124—1222no—
273#124—1222no—
274#124—1221no—
275#123—1224no—
276#123—1224no—
277C22⋊Q8—612no—
278C22⋊Q8—1224no—
279C22⋊Q8—1222no—
280C22⋊Q8—1222no—
281C22⋊Q8—1221no—
282#121—612no—
283#121—1221no—
284C4×Q8—21no—
285C4×Q8—21no—
286C4×Q8—612no—
287C4×Q8—1222no—
288C4×Q8—1222no—
289C4×Q8—1221no—
290C4×Q8C4×D6112yes—
291C4×Q8D24412yes—
292C4×D4—612no—
293C4×D4—1222no—
294C4×D4—1222no—
295C4×D4—1224no—
296C4×D4—1221no—
297#117—1224no—
298#117—1221no—
299#117—1222no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—1222no—
303C2×C4⋊C4C22⋊D6111yescharacteristic
304C2×C42—21no—
305C2×C42C22×D6111yescharacteristic
306C4⋊C8—21no—
307C4⋊C8—21no—
308C4⋊C8—1221no—
309C4⋊C8—1221no—
310#103—42no—
311C22⋊C8—1221no—
312C22⋊C8—1221no—
313C4×C8—1221no—
314C4×C8—1221no—
315#95—21no—
316#95—21no—
317#41—2441no—
318#41—2441no—
319#41—2441no—
320#41—2441no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—612no—
327C2×Q8—1221no—
328C2×Q8—1221no—
329C2×Q8#1175819812yes—
330C2×D4—612no—
331C2×D4—1222no—
332C2×D4—1221no—
333C2×D4—1221no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—41no—
337C22×C4—1224no—
338C22×C4—1222no—
339C22×C4D4×D6111yescharacteristic
340C22×C4D4×D6111yescharacteristic
341C22×C4C2×C22⋊D6111yescharacteristic
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—2441no—
346C2×C8—2441no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—22no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—612no—
353C4⋊C4—612no—
354C4⋊C4—1224no—
355C4⋊C4—1224no—
356C4⋊C4—1224no—
357C4⋊C4—1221no—
358C4⋊C4—1221no—
359C4⋊C4—1221no—
360C4⋊C4—1222no—
361C4⋊C4—1222no—
362C4⋊C4—1221no—
363C4⋊C4#1175819812yes—
364C22⋊C4—1224no—
365C22⋊C4—1224no—
366C22⋊C4—1224no—
367C22⋊C4—1222no—
368C22⋊C4—1222no—
369C22⋊C4—1222no—
370C42—22no—
371C42—22no—
372C42—1224no—
373C42—1222no—
374C42C2×C4×D6111yescharacteristic
375C42C2×D24411yescharacteristic
376C42D4×D6111yescharacteristic
377C42D4×D6111yescharacteristic
378C23—1222no—
379C23#1451974511yescharacteristic
380Q8—22no—
381Q8—22no—
382Q8—41no—
383Q8—41no—
384Q8—2442no—
385Q8—2442no—
386Q8—2441no—
387Q8—2441no—
388D4—4881no—
389D4—2442no—
390D4—2442no—
391D4—2442no—
392D4—2441no—
393D4—2441no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—1222no—
405C2×C4—1222no—
406C2×C4—1222no—
407C2×C4—4884no—
408C2×C4—2442no—
409C2×C4—2442no—
410C2×C4C22⋊C4×D6111yescharacteristic
411C2×C4C22⋊C4×D6111yescharacteristic
412C2×C4C2×#1175819811yescharacteristic
413C2×C4#1451974511yescharacteristic
414C2×C4#1451974711yescharacteristic
415C2×C4#1451974711yescharacteristic
416C2×C4#1451979711yescharacteristic
417C2×C4#1451979711yescharacteristic
418C2×C4#1451980311yescharacteristic
419C2×C4#1451980811yescharacteristic
420C8—81no—
421C8—4881no—
422C8—4881no—
423C22—21no—
424C22—21no—
425C22—2442no—
426C22—2442no—
427C22—2442no—
428C22#1683961711yescharacteristic
429C22#1683990511yescharacteristic
430C22#1683992211yescharacteristic
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—42no—
436C4—42no—
437C4—161no—
438C4—2442no—
439C4—2442no—
440C4—2442no—
441C4#1683975611yescharacteristic
442C4#1683976111yescharacteristic
443C4#1683990511yescharacteristic
444C4#1683992211yescharacteristic
445C2—21no—
446C2—21no—
447C2—4882no—
448C2#2144575711yescharacteristic
449C2#2144578411yescharacteristic
450C2#2144579111yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1805C2
socleC22×C61#379
prime radical#16840312C22
cosocleC2×C42×C61C23
hypercenter#1805D61
hypocenterC61#8431
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
3. #21444382C2semidirect
5. #21444221C2semidirect
6. #21444214C2semidirect
7. C61×#1805C2semidirect
10. #16840485C4semidirect
12. #16840472C4semidirect
221. #1805D61semidirect
258. C61#8431semidirect

Maximal subgroups

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

subgroupindex
#214596842
#214444692
#214443822
#214443782
#214442212
#214442142
C61×#18052
#843161