Finite Group Databasebeta

← #29576055

Subgroups — #29576055

Order 15616.

684 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 113 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21451627C211yesmaximal, characteristic
2#21448432C211yesmaximal, characteristic
3#21444392C211yesmaximal, characteristic, complemented
4#21444390C211yesmaximal, characteristic, complemented
5C2×#16839485C211yesmaximal, characteristic
6#21443686C211yesmaximal, characteristic, complemented
7C61×#1289C211yesmaximal, characteristic, complemented
8C2×#14519756C2211yescharacteristic
9C22×#11758198C2211yescharacteristic
10C2×C22⋊C4×D61—21no—
11C2×C22⋊#92418C2211yescharacteristic
12#16839699C412yescomplemented
13#16839684—22no—
14#16839675C412yescomplemented
15#16839637—21no—
16#16839636—22no—
17#16839633—22no—
18#16839622C2211yescharacteristic
19#16839620—21no—
20#16839617—21no—
21#16839500—22no—
22#16839498—22no—
23#16839485—22no—
24C2×C22⋊C8×C61C2211yescharacteristic
25C61×#380C2211yescharacteristic
26C61×#380C2211yescharacteristic
27C24×D61D411yescharacteristic
28C22×Dic122C2×C411yescharacteristic, complemented
29C23×Dic61D411yescharacteristic
30C22×C4×D61—22no—
31C22×#92418—21no—
32#14519756—21no—
33#14519756—21no—
34#14519756—21no—
35#14519756—21no—
36C2×C4⋊Dic61—21no—
37C2×C4⋊Dic61—22no—
38C2×C4⋊Dic61C2×C411yescharacteristic, complemented
39C2×#11758196—21no—
40C2×#11758196—22no—
41C2×C22⋊Dic61—21no—
42C2×C22⋊Dic61—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198—21no—
46C2×#11758198—21no—
47C2×#11758198—22no—
48C2×#11758198—22no—
49C2×#11758198—22no—
50C2×#11758198C2×C412yes—
51C2×#11758198D412yes—
52C22⋊C4×D61—42no—
53C22⋊C4×D61—42no—
54C22⋊#92418—21no—
55C22⋊#92418—21no—
56#14519684—22no—
57#14519684—22no—
58#14519684D412yes—
59C23×C4×C61C2311yescharacteristic
60C22×C8×C61—21no—
61C2×C4⋊C4×C61—21no—
62C2×C4⋊C4×C61—21no—
63C2×C22⋊C4×C61—21no—
64C2×C22⋊C4×C61—21no—
65C22⋊C8×C61—21no—
66C22⋊C8×C61—21no—
67#95×C61C2×C412yescomplemented
68#95×C61D412yes—
69C24×C61C2×D411yescharacteristic
70C23×D61—21no—
71C23×D61—21no—
72C23×D61—21no—
73C23×D61—21no—
74C23×D61—81no—
75C23×D61—81no—
76C23×D61C22⋊C412yes—
77C23×D61SD1612yes—
78C23×D61Q1612yes—
79C22×C4×C61—22no—
80C22×C4×C61—22no—
81C22×C4×C61—81no—
82C22×C4×C61C22×C411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61#4111yescharacteristic
87C22×C4×C61#4111yescharacteristic
88C22×Dic61—21no—
89C22×Dic61—21no—
90C22×Dic61—21no—
91C22×Dic61—21no—
92C22×Dic61—22no—
93C22×Dic61—81no—
94C22×Dic61C22⋊C412yes—
95C2×C4×D61—44no—
96C2×C4×D61—82no—
97C2×C4×D61—82no—
98C2×Dic122—21no—
99C2×Dic122—21no—
100C2×Dic122—81no—
101C2×C8×C61—41no—
102C2×C8×C61—81no—
103C4⋊C4×C61—81no—
104C4⋊C4×C61—81no—
105C22⋊C4×C61—41no—
106C22⋊C4×C61—41no—
107C22⋊C4×C61—41no—
108C22⋊C4×C61—41no—
109C22⋊Dic61—41no—
110C22⋊Dic61—41no—
111C22⋊Dic61—41no—
112C22⋊Dic61—41no—
113#11758198—22no—
114#11758198—22no—
115#11758198—22no—
116#11758198—22no—
117#11758198—44no—
118#11758198—82no—
119#11758198—82no—
120C4⋊Dic61—21no—
121C4⋊Dic61—21no—
122C4⋊Dic61—82no—
123C4⋊Dic61—81no—
124#11758196—41no—
125#11758196—41no—
126#11758196—82no—
127C2×#92418—41no—
128C2×#92418—81no—
129C23×C61—41no—
130C23×C61—41no—
131C23×C61C2×C22⋊C411yescharacteristic
132C23×C61#12011yescharacteristic
133C23×C61#12311yescharacteristic
134C23×C61C2×SD1611yescharacteristic
135C23×C61C2×Q1611yescharacteristic
136C23×C61AGL(1,ℤ/8)11yescharacteristic
137C23×C61#13711yescharacteristic
138C22×D61—22no—
139C22×D61—22no—
140C22×D61—22no—
141C22×D61—22no—
142C22×D61—22no—
143C22×D61—22no—
144C22×D61—42no—
145C22×D61—42no—
146C22×D61—42no—
147C22×D61—42no—
148C22×D61—82no—
149C22×D61—82no—
150C22×D61—81no—
151C22×D61—81no—
152C22×D61—82no—
153C22×D61—82no—
154C22×D61#10314yes—
155C2×C4×C61—41no—
156C2×C4×C61—41no—
157C2×C4×C61—41no—
158C2×C4×C61—41no—
159C2×C4×C61—41no—
160C2×C4×C61—41no—
161C2×C4×C61—42no—
162C2×C4×C61—42no—
163C2×C4×C61—42no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61—42no—
167C2×C4×C61C4×D411yescharacteristic
168C2×C4×C61C4×D411yescharacteristic
169C2×C4×C61#12111yescharacteristic
170C2×C4×C61#12111yescharacteristic
171C2×Dic61—22no—
172C2×Dic61—42no—
173C2×Dic61—42no—
174C2×Dic61—42no—
175C2×Dic61—42no—
176C2×Dic61—42no—
177C2×Dic61—42no—
178C2×Dic61—41no—
179C2×Dic61—41no—
180C2×Dic61—42no—
181C2×Dic61—81no—
182C4×D61—88no—
183Dic122—81no—
184Dic122—81no—
185C8×C61—81no—
186#92418—81no—
187#9199—611nomaximal, Sylow
188C22×C61—21no—
189C22×C61—21no—
190C22×C61—21no—
191C22×C61—21no—
192C22×C61—21no—
193C22×C61—21no—
194C22×C61—41no—
195C22×C61—41no—
196C22×C61—41no—
197C22×C61—41no—
198C22×C61#38611yescharacteristic
199C22×C61C2×#10311yescharacteristic
200C22×C61#41711yescharacteristic
201C22×C61#45011yescharacteristic
202C22×C61#45111yescharacteristic
203C22×C61#48311yescharacteristic
204C22×C61#48411yescharacteristic
205C2×D61—24no—
206C2×D61—44no—
207C2×D61—44no—
208C2×D61—42no—
209C2×D61—42no—
210C2×D61—44no—
211C2×D61—82no—
212C2×D61—82no—
213C2×D61—82no—
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—82no—
219C4×C61—82no—
220Dic61—42no—
221Dic61—82no—
222Dic61—81no—
223Dic61—81no—
224#2626—611no—
225#2315—611no—
226#1520—611no—
227#1520—611no—
228C2×#323—611no—
229#1289—611no—
230#1289D6111yescharacteristic, complemented
231C2×C61—21no—
232C2×C61—21no—
233C2×C61—21no—
234C2×C61—21no—
235C2×C61—21no—
236C2×C61—21no—
237C2×C61#187211yescharacteristic
238C2×C61#191111yescharacteristic
239C2×C61#191311yescharacteristic
240D61—44no—
241C2×C22⋊Q8—611no—
242C22×C22⋊C4—611no—
243C22×C22⋊C4—1221no—
244C2×C22⋊C8—611no—
245C2×C22⋊C8C2×D6111yescharacteristic
246#399—612no—
247#396—1222no—
248#393—612no—
249#385—1222no—
250#385—1221no—
251#385—1222no—
252#380—611no—
253#380—1221no—
254#380C2×D6111yescharacteristic
255#380C2×D6111yescharacteristic
256#379—1221no—
257#328—1222no—
258#328—1222no—
259#323—1222no—
260C61#919911yescharacteristic, complemented, Sylow
261C25—611no—
262C22×Q8—611no—
263C23×C4—611no—
264C23×C4—1222no—
265C23×C4C22×D6111yescharacteristic
266C22×C8—21no—
267C22×C8—1221no—
268C22⋊Q8—1221no—
269C22⋊Q8—1221no—
270C22⋊Q8—1221no—
271C22⋊Q8—1221no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—611no—
275C2×C4⋊C4—1222no—
276C2×C4⋊C4—1222no—
277C2×C4⋊C4—1221no—
278C2×C4⋊C4—1221no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—612no—
282C2×C22⋊C4—612no—
283C2×C22⋊C4—1221no—
284C2×C22⋊C4—1221no—
285C2×C22⋊C4—1221no—
286C2×C22⋊C4—1221no—
287C2×C22⋊C4—1222no—
288C2×C22⋊C4—1222no—
289C2×C22⋊C4—1222no—
290C2×C22⋊C4—1221no—
291C2×C22⋊C4—1221no—
292C2×C22⋊C4—2442no—
293C2×C22⋊C4—2442no—
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—1221no—
297C22⋊C8—1221no—
298#95—612no—
299#95—1222no—
300#95—1222no—
301#95C4×D6112yescomplemented
302#95D24412yes—
303C24—612no—
304C24—612no—
305C24—612no—
306C24—1221no—
307C24—1221no—
308C24—1221no—
309C24—1221no—
310C24—4881no—
311C24—4881no—
312C24D4×D6111yescharacteristic
313C2×Q8—1221no—
314C2×Q8—1221no—
315C2×Q8—4881no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—81no—
319C22×C4—612no—
320C22×C4—1221no—
321C22×C4—1221no—
322C22×C4—1221no—
323C22×C4—1221no—
324C22×C4—1222no—
325C22×C4—4882no—
326C22×C4—4882no—
327C22×C4—4881no—
328C22×C4—2444no—
329C22×C4C2×C4×D6111yescharacteristic
330C22×C4C2×D24411yescharacteristic
331C22×C4#1175822211yescharacteristic
332C22×C4D4×D6111yescharacteristic
333C22×C4D4×D6111yescharacteristic
334C22×C4#1175822611yescharacteristic
335C2×C8—41no—
336C2×C8—81no—
337C2×C8—4881no—
338C2×C8—2441no—
339C4⋊C4—81no—
340C4⋊C4—81no—
341C4⋊C4—1221no—
342C4⋊C4—1221no—
343C4⋊C4—4882no—
344C4⋊C4—4881no—
345C4⋊C4—4882no—
346C4⋊C4—2441no—
347C4⋊C4—2441no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—1222no—
353C22⋊C4—1222no—
354C22⋊C4—1222no—
355C22⋊C4—1222no—
356C22⋊C4—4882no—
357C22⋊C4—4882no—
358C22⋊C4—2444no—
359C22⋊C4—2441no—
360C22⋊C4—2441no—
361C22⋊C4—2441no—
362C22⋊C4—2441no—
363C23—41no—
364C23—41no—
365C23—614no—
366C23—1222no—
367C23—1222no—
368C23—1222no—
369C23—1222no—
370C23—1222no—
371C23—1222no—
372C23—4881no—
373C23—4881no—
374C23—4882no—
375C23—4882no—
376C23—4882no—
377C23—4882no—
378C23—2442no—
379C23—2442no—
380C23—2442no—
381C23—2442no—
382C23C22⋊C4×D6111yescharacteristic
383C23#1451974711yescharacteristic
384C23#1451976211yescharacteristic
385C23SD16×D6111yescharacteristic
386C23Q16×D6111yescharacteristic
387C23#1451980111yescharacteristic
388C23#1451980611yescharacteristic
389Q8—4881no—
390Q8—4881no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—1222no—
404C2×C4—4888no—
405C2×C4—4881no—
406C2×C4—2442no—
407C2×C4—2442no—
408C2×C4—2442no—
409C2×C4—2442no—
410C2×C4—2441no—
411C2×C4—2441no—
412C2×C4—2442no—
413C2×C4—2442no—
414C2×C4—2442no—
415C2×C4#1451973911yescharacteristic
416C2×C4C4×D24411yescharacteristic
417C2×C4#1451975011yescharacteristic
418C2×C4#1451975211yescharacteristic
419C8—81no—
420C8—4881no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—21no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—1224no—
432C22—4882no—
433C22—4882no—
434C22—4882no—
435C22—2444no—
436C22—2444no—
437C22—2444no—
438C22—2442no—
439C22—2442no—
440C22#1683964011yescharacteristic
441C22#103×D6111yescharacteristic
442C22#1683976011yescharacteristic
443C22#1683991411yescharacteristic
444C22#1683991911yescharacteristic
445C22#1684009411yescharacteristic
446C22#1684009911yescharacteristic
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—82no—
452C4—82no—
453C4—4882no—
454C4—4881no—
455C4—4881no—
456C4—2442no—
457C2—21no—
458C2—21no—
459C2—21no—
460C2—21no—
461C2—21no—
462C2—21no—
463C2—2444no—
464C2#2144612511yescharacteristic
465C2#2144635011yescharacteristic
466C2#2144636011yescharacteristic
subgroupG/H
centerC22#16839640
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1289C2
socleC22×C61#386
prime radicalC24×D61D4
cosocleC23×C4×C61C23
hypercenter#1289D61
hypocenterC61#9199
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
3. #21444392C2semidirect
4. #21444390C2semidirect
6. #21443686C2semidirect
7. C61×#1289C2semidirect
12. #16839699C4semidirect
14. #16839675C4semidirect
28. C22×Dic122C2×C4semidirect
38. C2×C4⋊Dic61C2×C4semidirect
67. #95×C61C2×C4semidirect
230. #1289D61semidirect
260. C61#9199semidirect
301. #95C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214516272
#214484322
#214443922
#214443902
C2×#168394852
#214436862
C61×#12892
#919961