Finite Group Databasebeta

← #29578998

Subgroups — #29578998

Order 15616.

668 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454120C211yesmaximal, characteristic
2C4×C22⋊#92418C211yesmaximal, characteristic, complemented
3C2×#16839498C212yesmaximal, complemented
4C2×#16839495C212yesmaximal
5C61×#1434C211yesmaximal, characteristic, complemented
6C2×#14519765C411yescharacteristic, complemented
7C2×#14519756C2211yescharacteristic
8C2×#14519751C2211yescharacteristic
9C2×C22⋊#92418C2212yes—
10C2×C4×#92418—21no—
11#16839668C411yescharacteristic, complemented
12#16839664C411yescharacteristic, complemented
13#16839653—21no—
14#16839653—21no—
15#16839641—21no—
16#16839641—21no—
17#16839628C411yescharacteristic, complemented
18#16839517—21no—
19#16839498—24no—
20#16839495—24no—
21C22×C42×C61C2211yescharacteristic
22C2×#95×C61—21no—
23C2×#95×C61—21no—
24C2×#95×C61C2212yes—
25C22×Dic122C2×C411yescharacteristic
26C22×D244C2×C411yescharacteristic
27C22×C22⋊D61—21no—
28C22×#92418—22no—
29#14519765—42no—
30#14519756—22no—
31#14519756—22no—
32#14519751—22no—
33#14519751—22no—
34C2×C4⋊Dic61C2×C411yescharacteristic
35C2×C4⋊Dic61C2×C411yescharacteristic
36C2×#11758196—21no—
37C2×C22⋊Dic61—21no—
38C2×#11758198—21no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C22⋊#92418—22no—
44C22⋊#92418D414yes—
45C4×#92418—42no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50C23×C4×C61—22no—
51C23×C4×C61D411yescharacteristic
52C23×C4×C61D411yescharacteristic
53C23×C4×C61C2311yescharacteristic
54C2×C42×C61—42no—
55C2×C42×C61C2×C411yescharacteristic
56C2×C42×C61C2×C411yescharacteristic
57C2×C42×C61D411yescharacteristic
58C2×C42×C61D411yescharacteristic
59#95×C61—24no—
60#95×C61—22no—
61#95×C61—22no—
62#95×C61—42no—
63C24×C61C2×D411yescharacteristic
64C23×D61—21no—
65C22×C4×C61—22no—
66C22×C4×C61—22no—
67C22×C4×C61—22no—
68C22×C4×C61—22no—
69C22×C4×C61—24no—
70C22×C4×C61—41no—
71C22×C4×C61—41no—
72C22×C4×C61—41no—
73C22×C4×C61—41no—
74C22×C4×C61—42no—
75C22×C4×C61—42no—
76C22×C4×C61—42no—
77C22×C4×C61—42no—
78C22×C4×C61—42no—
79C22×C4×C61—42no—
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22⋊C411yescharacteristic
84C22×C4×C61C22×C411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D412yes—
87C22×C4×C61#4112yes—
88C2×C22⋊D61—41no—
89C2×C22⋊D61—41no—
90C2×C22⋊D61—42no—
91C2×C22⋊D61—42no—
92C22×Dic61—21no—
93C22×Dic61—21no—
94C22×Dic61—21no—
95C2×D244—22no—
96C2×D244—41no—
97C2×D244—41no—
98C2×Dic122—22no—
99C2×Dic122—41no—
100C2×Dic122—41no—
101C42×C61—22no—
102C42×C61—22no—
103C42×C61—41no—
104C42×C61—41no—
105C22⋊Dic61—41no—
106C22⋊Dic61—41no—
107#11758198—42no—
108#11758198—42no—
109#11758198—42no—
110#11758198—42no—
111#11758198—42no—
112C4⋊Dic61—22no—
113C4⋊Dic61—22no—
114#11758196—42no—
115C2×#92418—24no—
116C2×#92418—42no—
117C2×#92418—42no—
118C23×C61—21no—
119C23×C61—21no—
120C23×C61—21no—
121C23×C61—21no—
122C23×C61C2×C22⋊C411yescharacteristic
123C23×C61#12411yescharacteristic
124C23×C61#12711yescharacteristic
125C23×C61#13512yes—
126C23×C61#13512yes—
127C22×D61—41no—
128C22×D61—41no—
129C22×D61—41no—
130C22×D61—41no—
131C22×D61—41no—
132C22×D61—41no—
133C22×D61—41no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—21no—
137C2×C4×C61—21no—
138C2×C4×C61—21no—
139C2×C4×C61—21no—
140C2×C4×C61—21no—
141C2×C4×C61—21no—
142C2×C4×C61—21no—
143C2×C4×C61—21no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—21no—
147C2×C4×C61—21no—
148C2×C4×C61—42no—
149C2×C4×C61—42no—
150C2×C4×C61—44no—
151C2×C4×C61—44no—
152C2×C4×C61—44no—
153C2×C4×C61—44no—
154C2×C4×C61—44no—
155C2×C4×C61—82no—
156C2×C4×C61—82no—
157C2×C4×C61—82no—
158C2×C4×C61—82no—
159C2×C4×C61#9912yes—
160C2×C4×C61#10412yes—
161C2×C4×C61C4×D412yes—
162C2×C4×C61#12112yes—
163C22⋊D61—82no—
164C22⋊D61—82no—
165C2×Dic61—41no—
166C2×Dic61—41no—
167C2×Dic61—41no—
168C2×Dic61—41no—
169C2×Dic61—41no—
170C2×Dic61—41no—
171C2×Dic61—41no—
172C2×Dic61—41no—
173C2×Dic61—41no—
174D244—82no—
175Dic122—82no—
176#92418—44no—
177#9681—611nomaximal, Sylow
178C22×C61—21no—
179C22×C61—21no—
180C22×C61—21no—
181C22×C61—21no—
182C22×C61—22no—
183C22×C61—22no—
184C22×C61—22no—
185C22×C61—22no—
186C22×C61#39011yescharacteristic
187C22×C61C2×#9911yescharacteristic
188C22×C61#41011yescharacteristic
189C22×C61#41611yescharacteristic
190C22×C61#41611yescharacteristic
191C22×C61C2×#10411yescharacteristic
192C22×C61#42111yescharacteristic
193C22×C61#48812yes—
194C22×C61#49512yes—
195C2×D61—81no—
196C2×D61—81no—
197C2×D61—81no—
198C2×D61—81no—
199C2×D61—81no—
200C2×D61—81no—
201C2×D61—81no—
202C4×C61—22no—
203C4×C61—22no—
204C4×C61—22no—
205C4×C61—22no—
206C4×C61—41no—
207C4×C61—41no—
208C4×C61—84no—
209Dic61—81no—
210Dic61—81no—
211Dic61—81no—
212#2850—611no—
213C4×C22⋊C8—611no—
214C2×#328—612no—
215C2×#327—612no—
216#1434D6111yescharacteristic, complemented
217C2×C61—22no—
218C2×C61—22no—
219C2×C61#150011yescharacteristic
220C2×C61#150111yescharacteristic
221C2×C61#151611yescharacteristic
222C2×C61#152111yescharacteristic
223C2×C61#195811yescharacteristic
224C2×C61#196411yescharacteristic
225C2×C61#198011yescharacteristic
226D61—161no—
227C2×#124—611no—
228C2×C22⋊Q8—611no—
229C2×#121—611no—
230C22×C42C2×D6111yescharacteristic
231C2×C22⋊C8—612no—
232C2×C4×C8—1221no—
233#391—611no—
234#390—611no—
235#388—1221no—
236#388—1221no—
237#386—1221no—
238#386—1221no—
239#382—611no—
240C2×#95—21no—
241C2×#95—21no—
242C2×#95C2×D6112yes—
243#336—1221no—
244#328—1224no—
245#327—1224no—
246C61#968111yescharacteristic, complemented, Sylow
247C22×Q8—611no—
248C22×D4—611no—
249C22×D4—1221no—
250C23×C4—22no—
251C23×C4C22⋊D6111yescharacteristic
252C23×C4C22⋊D6111yescharacteristic
253C23×C4C22×D6111yescharacteristic
254C22×C8—1222no—
255#124—2442no—
256C22⋊Q8—1222no—
257C22⋊Q8—1222no—
258#121—1222no—
259#121—1222no—
260C2×C4⋊C4—611no—
261C2×C4⋊C4—611no—
262C2×C4⋊C4—1221no—
263C2×C22⋊C4—1221no—
264C2×C22⋊C4—1221no—
265C2×C22⋊C4—1221no—
266C2×C22⋊C4—1221no—
267C2×C22⋊C4—1221no—
268C2×C22⋊C4—1221no—
269C2×C42—42no—
270C2×C42C4×D6111yescharacteristic
271C2×C42C4×D6111yescharacteristic
272C2×C42D24411yescharacteristic
273C2×C42D24411yescharacteristic
274C22⋊C8—614no—
275C22⋊C8—1222no—
276C4×C8—2442no—
277#95—24no—
278#95—22no—
279#95—22no—
280#95—42no—
281#95—1221no—
282#95—1221no—
283#95—1221no—
284#95—1221no—
285C24—1221no—
286C24C2×C22⋊D6111yescharacteristic
287C2×Q8—1222no—
288C2×Q8—2441no—
289C2×Q8—2441no—
290C2×D4—1222no—
291C2×D4—2441no—
292C2×D4—2441no—
293C2×D4—2441no—
294C2×D4—2441no—
295C2×D4—2442no—
296C2×D4—2442no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—1221no—
313C22×C4—1221no—
314C22×C4—1221no—
315C22×C4#1175819811yescharacteristic
316C22×C4#1175819811yescharacteristic
317C22×C4#1175819811yescharacteristic
318C22×C4#1175819811yescharacteristic
319C22×C4C2×C4×D6111yescharacteristic
320C22×C4C2×D24411yescharacteristic
321C22×C4D4×D6112yes—
322C22×C4#1175822612yes—
323C2×C8—1224no—
324C2×C8—2442no—
325C2×C8—2442no—
326C4⋊C4—1222no—
327C4⋊C4—1222no—
328C4⋊C4—2442no—
329C22⋊C4—2442no—
330C22⋊C4—2442no—
331C22⋊C4—2441no—
332C22⋊C4—2441no—
333C22⋊C4—2442no—
334C22⋊C4—2442no—
335C22⋊C4—2442no—
336C42—22no—
337C42—22no—
338C42—41no—
339C42—41no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—2441no—
345C23—2441no—
346C23—2441no—
347C23—2441no—
348C23—2441no—
349C23—2441no—
350C23—2441no—
351C23C2×#1175819811yescharacteristic
352C23#1451976611yescharacteristic
353C23#1451977311yescharacteristic
354C23#1451979712yes—
355C23#1451979712yes—
356Q8—4882no—
357D4—4882no—
358D4—4882no—
359D4—4882no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—2441no—
386C2×C4—2441no—
387C2×C4—2441no—
388C2×C4—2441no—
389C2×C4—2441no—
390C2×C4—2441no—
391C2×C4—2441no—
392C2×C4—2441no—
393C2×C4—2441no—
394C2×C4#1451969112yes—
395C2×C4#1451970512yes—
396C2×C4#1451973912yes—
397C2×C4#1451975212yes—
398C8—2444no—
399C22—22no—
400C22—22no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—22no—
405C22—21no—
406C22—21no—
407C22—4881no—
408C22—4881no—
409C22—4881no—
410C22—4881no—
411C22—4881no—
412C22—4881no—
413C22—4881no—
414C22#1683966311yescharacteristic
415C22C2×#1451969111yescharacteristic
416C22#1683973511yescharacteristic
417C22#1683975611yescharacteristic
418C22#1683975611yescharacteristic
419C22C2×#1451970511yescharacteristic
420C22#1683978011yescharacteristic
421C22#1684011912yes—
422C22#1684014412yes—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—41no—
428C4—41no—
429C4—84no—
430C4—4881no—
431C4—4881no—
432C4—4881no—
433C2—22no—
434C2—22no—
435C2—9761no—
436C2#2144430311yescharacteristic
437C2#2144430811yescharacteristic
438C2#2144437311yescharacteristic
439C2#2144439411yescharacteristic
440C2#2144662811yescharacteristic
441C2#2144665411yescharacteristic
442C2#2144673711yescharacteristic
subgroupG/H
centerC23C2×#11758198
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1434C2
socleC23×C61C2×C22⋊C4
prime radicalC2×#14519751C22
cosocleC23×C4×C61C23
hypercenter#1434D61
hypocenterC61#9681
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊#92418C2semidirect
3. C2×#16839498C2semidirect
5. C61×#1434C2semidirect
6. C2×#14519765C4semidirect
11. #16839668C4semidirect
12. #16839664C4semidirect
17. #16839628C4semidirect
216. #1434D61semidirect
246. C61#9681semidirect

Maximal subgroups

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

subgroupindex
#214541202
C4×C22⋊#924182
C2×#168394982
C2×#168394952
C61×#14342
#968161