Finite Group Databasebeta

← #29577226

Subgroups — #29577226

Order 15616.

774 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 97 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#21455069C211yesmaximal, characteristic
2#21448656C211yesmaximal, characteristic, complemented
3#21445764C211yesmaximal, characteristic, complemented
4#21444335C211yesmaximal, characteristic, complemented
5#21443725C211yesmaximal, characteristic, complemented
6#21443705C211yesmaximal, characteristic, complemented
7C61×#1964C211yesmaximal, characteristic, complemented
8C22×#11758222C2211yescharacteristic
9C2×#14519803C412yescomplemented
10C2×#14519742—21no—
11#16839905—22no—
12#16839900—21no—
13#16839895—21no—
14C2×#14519705—22no—
15#16839761C412yescomplemented
16#16839726C2211yescharacteristic
17#16839726C2211yescharacteristic
18#16839663—21no—
19#16839662—22no—
20#16839621—21no—
21C4×C22⋊Dic61C2211yescharacteristic
22#16839609—21no—
23#16839496—21no—
24#16839496—21no—
25C2×#121×C61C2211yescharacteristic
26C2×#99×C61—22no—
27C61×#380C2211yescharacteristic
28C4×C22⋊C4×C61C2211yescharacteristic
29C2×#11758222—21no—
30C2×#11758222—21no—
31C2×#11758222—44no—
32C2×#11758222C2×C412yes—
33C2×#11758222D412yes—
34C22×Dic122D411yescharacteristic
35C22×D244D411yescharacteristic
36C22×C22⋊D61—21no—
37C22×C4×D61—21no—
38#14519803—44no—
39C2×#11758200—22no—
40C2×#11758199—22no—
41C2×#11758194—21no—
42C2×#11758194C2×C412yes—
43#14519742—44no—
44C2×#11758196—22no—
45C2×C22⋊Dic61—21no—
46C2×#11758198—22no—
47C2×#11758198—22no—
48C2×C4×Dic61—22no—
49C2×C4×Dic61—21no—
50#14519705—44no—
51#14519700—22no—
52#14519697—22no—
53C22⋊#92418—21no—
54C22⋊#92418—21no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57#14519684D411yescharacteristic
58#14519684D411yescharacteristic
59C22×D4×C61—21no—
60C22×D4×C61C2×C411yescharacteristic
61C23×C4×C61C2311yescharacteristic
62#121×C61—21no—
63#121×C61—21no—
64#121×C61—21no—
65#121×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61C2×C411yescharacteristic
68C2×C22⋊C4×C61—21no—
69C2×C22⋊C4×C61—21no—
70C2×C22⋊C4×C61D412yes—
71C2×C42×C61—21no—
72#99×C61—44no—
73#95×C61—21no—
74#95×C61—21no—
75C24×C61—21no—
76C24×C61C2×D411yescharacteristic
77C23×D61—21no—
78C2×D4×C61—21no—
79C2×D4×C61—21no—
80C2×D4×C61—42no—
81C2×D4×C61—42no—
82C2×D4×C61—42no—
83C2×D4×C61—41no—
84C2×D4×C61—41no—
85C22×C4×C61—21no—
86C22×C4×C61—22no—
87C22×C4×C61—42no—
88C22×C4×C61—41no—
89C22×C4×C61C22×C411yescharacteristic
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61#4111yescharacteristic
94C22×C4×C61#4111yescharacteristic
95C2×C22⋊D61—22no—
96C2×C22⋊D61—22no—
97C2×C22⋊D61—44no—
98C22×Dic61—21no—
99C22×Dic61—21no—
100C22×Dic61—21no—
101#11758222—24no—
102#11758222—24no—
103#11758222—42no—
104#11758222—42no—
105#11758222—44no—
106#11758222—44no—
107C2×D244—21no—
108C2×D244—42no—
109C2×D244C22⋊C412yes—
110C2×C4×D61—22no—
111C2×C4×D61—22no—
112C2×C4×D61—44no—
113C2×Dic122—21no—
114C2×Dic122—24no—
115C2×Dic122C22⋊C412yes—
116C4⋊C4×C61—21no—
117C4⋊C4×C61—21no—
118C4⋊C4×C61—42no—
119C22⋊C4×C61—22no—
120C22⋊C4×C61—41no—
121C22⋊C4×C61—41no—
122C22⋊C4×C61—41no—
123C22⋊C4×C61—42no—
124C42×C61—42no—
125C22⋊Dic61—42no—
126#11758200—44no—
127#11758199—44no—
128#11758198—44no—
129#11758198—82no—
130#11758196—44no—
131C4×Dic61—44no—
132C4×Dic61—42no—
133#11758194—42no—
134#11758194—42no—
135C2×#92418—22no—
136C2×#92418—41no—
137C23×C61—21no—
138C23×C61—21no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61—42no—
142C23×C61—41no—
143C23×C61—81no—
144C23×C61—81no—
145C23×C61C4×D411yescharacteristic
146C23×C61#12011yescharacteristic
147C23×C61#12111yescharacteristic
148C22×D61—22no—
149C22×D61—42no—
150C22×D61—42no—
151C22×D61—41no—
152C22×D61—41no—
153D4×C61—42no—
154D4×C61—81no—
155D4×C61—82no—
156D4×C61—81no—
157D4×C61—81no—
158C2×C4×C61—22no—
159C2×C4×C61—22no—
160C2×C4×C61—22no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—42no—
165C2×C4×C61—42no—
166C2×C4×C61—42no—
167C2×C4×C61—42no—
168C2×C4×C61—41no—
169C2×C4×C61—42no—
170C2×C4×C61—81no—
171C2×C4×C61—81no—
172C2×C4×C61C2×C22⋊C411yescharacteristic
173C2×C4×C61C4×D411yescharacteristic
174C2×C4×C61#12111yescharacteristic
175C2×C4×C61#12311yescharacteristic
176C22⋊D61—44no—
177C22⋊D61—44no—
178C2×Dic61—24no—
179C2×Dic61—22no—
180C2×Dic61—22no—
181C2×Dic61—41no—
182C2×Dic61—41no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186D244—22no—
187D244—22no—
188D244—42no—
189C4×D61—44no—
190C4×D61—44no—
191Dic122—24no—
192Dic122—22no—
193Dic122#9914yescomplemented
194#92418—42no—
195#92418—81no—
196#9368—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—21no—
200C22×C61—21no—
201C22×C61—21no—
202C22×C61—21no—
203C22×C61—21no—
204C22×C61—21no—
205C22×C61—81no—
206C22×C61—81no—
207C22×C61—81no—
208C22×C61—81no—
209C22×C61—81no—
210C22×C61—162no—
211C22×C61#38611yescharacteristic
212C22×C61#45311yescharacteristic
213C22×C61#45411yescharacteristic
214C2×D61—42no—
215C2×D61—42no—
216C2×D61—42no—
217C2×D61—82no—
218C2×D61—82no—
219C4×C61—21no—
220C4×C61—21no—
221C4×C61—21no—
222C4×C61—82no—
223C4×C61—82no—
224C4×C61—81no—
225C4×C61C2×#9912yes—
226Dic61—24no—
227Dic61—42no—
228Dic61—81no—
229#3006—611no—
230#2341—611no—
231#1964D6111yescharacteristic, complemented
232#1789—611no—
233#1507—611no—
234#1305—611no—
235#1297—611no—
236C2×C61—21no—
237C2×C61—21no—
238C2×C61—41no—
239C2×C61—41no—
240C2×C61—161no—
241C2×C61#188311yescharacteristic
242C2×C61#189811yescharacteristic
243C2×C61#190611yescharacteristic
244D61—82no—
245C22×#41—611no—
246C2×AGL(1,ℤ/8)—612no—
247C2×#121C2×D6111yescharacteristic
248C2×C4×Q8—1221no—
249#449—1222no—
250#448—1221no—
251#447—1221no—
252C2×#104—1222no—
253#417—612no—
254C2×#99—22no—
255#407—611no—
256#407—611no—
257#390—1221no—
258#390—1222no—
259#380—1221no—
260#380C2×D6111yescharacteristic
261C4×C22⋊C4—611no—
262C4×C22⋊C4—1221no—
263C4×C22⋊C4C2×D6111yescharacteristic
264#327—1221no—
265#327—1221no—
266C61#936811yescharacteristic, complemented, Sylow
267C2×#41—612no—
268C2×#41—612no—
269C2×#41—1221no—
270C2×#41—1221no—
271C2×#41—2444no—
272C22×Q8—611no—
273C22×D4—21no—
274C22×D4—611no—
275C22×D4—1221no—
276C22×D4C4×D6111yescharacteristic
277C23×C4—1221no—
278C23×C4C22×D6111yescharacteristic
279AGL(1,ℤ/8)—2444no—
280C2×SD16—1222no—
281C2×D8—1222no—
282C2×M16—612no—
283C2×M16—1221no—
284#121—21no—
285#121—21no—
286#121—21no—
287#121—21no—
288C4×Q8—2444no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—1222no—
291C2×C4⋊C4C4×D6111yescharacteristic
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—1222no—
295C2×C22⋊C4—1222no—
296C2×C22⋊C4—1221no—
297C2×C22⋊C4C22⋊D6112yes—
298C2×C42—21no—
299C2×C42—1222no—
300C2×C42—1221no—
301#104—2444no—
302#103—1222no—
303#102—1222no—
304#99—44no—
305C22⋊C8—1221no—
306C22⋊C8—1221no—
307C22⋊C8—1221no—
308C22⋊C8—1221no—
309#95—21no—
310#95—21no—
311#95—611no—
312#95—611no—
313C24—21no—
314C24—1221no—
315C24C2×C22⋊D6111yescharacteristic
316#41—1224no—
317#41—1224no—
318#41—2442no—
319#41—2442no—
320#41—2444no—
321#41—2444no—
322C2×Q8—612no—
323C2×Q8—1224no—
324C2×Q8—1221no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—612no—
333C2×D4—1222no—
334C2×D4—1222no—
335C2×D4—1221no—
336C2×D4—2442no—
337C2×D4—2444no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—1221no—
343C22×C4—1221no—
344C22×C4—1222no—
345C22×C4—1222no—
346C22×C4—1221no—
347C22×C4—2444no—
348C22×C4C2×C4×D6111yescharacteristic
349C22×C4#1175822211yescharacteristic
350C22×C4D4×D6111yescharacteristic
351C22×C4D4×D6111yescharacteristic
352C22×C4D4×D6111yescharacteristic
353C22×C4#1175822411yescharacteristic
354SD16—2444no—
355D8—2444no—
356M16—2442no—
357M16—2442no—
358C2×C8—1222no—
359C2×C8—2441no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—42no—
363C4⋊C4—2444no—
364C22⋊C4—22no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—42no—
369C22⋊C4—4882no—
370C22⋊C4—2444no—
371C22⋊C4—2442no—
372C42—42no—
373C42—2442no—
374C42—2444no—
375C23—21no—
376C23—21no—
377C23—41no—
378C23—41no—
379C23—42no—
380C23—41no—
381C23—81no—
382C23—81no—
383C23—1222no—
384C23—2442no—
385C23—2442no—
386C23—2441no—
387C23—2441no—
388C23C4×C22⋊D6111yescharacteristic
389C23#1451974511yescharacteristic
390C23#1451974811yescharacteristic
391Q8—614no—
392Q8—1224no—
393Q8—1222no—
394D4—42no—
395D4—81no—
396D4—82no—
397D4—81no—
398D4—81no—
399D4—1222no—
400D4—1222no—
401D4—2442no—
402D4—2444no—
403D4—2444no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—1224no—
419C2×C4—1222no—
420C2×C4—1222no—
421C2×C4—2444no—
422C2×C4—2444no—
423C2×C4—2441no—
424C2×C4—2441no—
425C2×C4—2441no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4C22⋊C4×D6111yescharacteristic
429C2×C4#1451973711yescharacteristic
430C2×C4#1451975011yescharacteristic
431C2×C4#1451975911yescharacteristic
432C8—4881no—
433C8—2442no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—162no—
448C22—4882no—
449C22—4882no—
450C22—2442no—
451C22—2442no—
452C22—2442no—
453C22#1683963811yescharacteristic
454C22#1683992911yescharacteristic
455C22#1683993411yescharacteristic
456C4—21no—
457C4—21no—
458C4—21no—
459C4—82no—
460C4—82no—
461C4—81no—
462C4—1224no—
463C4—4881no—
464C4—2442no—
465C4#99×D6112yes—
466C2—21no—
467C2—21no—
468C2—41no—
469C2—41no—
470C2—161no—
471C2—4882no—
472C2#2144619811yescharacteristic
473C2#2144626911yescharacteristic
474C2#2144631711yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1964C2
socleC22×C61#386
prime radical#21455069C2
cosocleC23×C4×C61C23
hypercenter#1964D61
hypocenterC61#9368
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1964

Complements

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

NH ≅ G/Ntype
2. #21448656C2semidirect
3. #21445764C2semidirect
4. #21444335C2semidirect
5. #21443725C2semidirect
6. #21443705C2semidirect
7. C61×#1964C2semidirect
9. C2×#14519803C4semidirect
15. #16839761C4semidirect
193. Dic122#99semidirect
231. #1964D61semidirect
266. C61#9368semidirect

Maximal subgroups

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

subgroupindex
#214550692
#214486562
#214457642
#214443352
#214437252
#214437052
C61×#19642
#936861