Finite Group Databasebeta

← #29577225

Subgroups — #29577225

Order 15616.

774 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 774 conjugacy classes of subgroups — too many to draw. Only its 115 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#21448660C211yesmaximal, characteristic, complemented
2#21446656C211yesmaximal, characteristic, complemented
3#21445763C211yesmaximal, characteristic, complemented
4#21444335C211yesmaximal, characteristic, complemented
5#21443724C211yesmaximal, characteristic, complemented
6#21443704C211yesmaximal, characteristic, complemented
7C61×#3006C211yesmaximal, characteristic
8C2×Q8×Dic61—21no—
9C2×#14519704—22no—
10C2×#14519692—22no—
11#16839726C2211yescharacteristic
12#16839665—22no—
13#16839665—21no—
14#16839622—21no—
15#16839622C2211yescharacteristic
16C4×C22⋊Dic61—21no—
17C4×C22⋊Dic61C2211yescharacteristic
18C4×C22⋊Dic61C2211yescharacteristic
19#16839496—21no—
20#16839496—21no—
21C22×#41×C61C2211yescharacteristic
22C2×AGL(1,ℤ/8)×C61C412yescomplemented
23C2×#121×C61C2211yescharacteristic
24C61×#449—22no—
25C61×#448—21no—
26C61×#447—21no—
27C61×#417C412yescomplemented
28C61×#407C2211yescharacteristic
29C2×#11758194—21no—
30Q8×Dic61—44no—
31C2×C4⋊Dic61—22no—
32C2×C4⋊Dic61—21no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61—22no—
35C2×C22⋊Dic61—21no—
36C2×C22⋊Dic61—22no—
37C2×C22⋊Dic61D412yes—
38C2×C4×Dic61—22no—
39C2×C4×Dic61—21no—
40C2×C4×Dic61—21no—
41#14519704—44no—
42#14519692—44no—
43C22⋊#92418—21no—
44C22⋊#92418—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684D411yescharacteristic
48#14519684D411yescharacteristic
49C2×#41×C61—21no—
50C2×#41×C61—21no—
51C2×#41×C61—44no—
52C2×#41×C61C2×C412yes—
53C2×#41×C61D412yes—
54C22×Q8×C61D411yescharacteristic
55C22×D4×C61—21no—
56C22×D4×C61—21no—
57C22×D4×C61C2×C411yescharacteristic
58C22×D4×C61D411yescharacteristic
59C23×C4×C61—21no—
60C23×C4×C61C2311yescharacteristic
61AGL(1,ℤ/8)×C61—44no—
62C2×SD16×C61—22no—
63C2×D8×C61—22no—
64C2×M16×C61C2×C412yes—
65#121×C61—21no—
66#121×C61—21no—
67#121×C61—21no—
68#121×C61—21no—
69C2×C4⋊C4×C61C2×C411yescharacteristic
70C2×C22⋊C4×C61—21no—
71#103×C61—22no—
72#102×C61—22no—
73C22⋊C8×C61—21no—
74C22⋊C8×C61—21no—
75C24×C61—21no—
76C24×C61—21no—
77C24×C61C2×D411yescharacteristic
78#41×C61—24no—
79#41×C61—24no—
80#41×C61—42no—
81#41×C61—42no—
82#41×C61—44no—
83#41×C61—44no—
84C2×Q8×C61—24no—
85C2×Q8×C61—21no—
86C2×Q8×C61C22⋊C412yes—
87C2×D4×C61—22no—
88C2×D4×C61—22no—
89C2×D4×C61—21no—
90C2×D4×C61—21no—
91C2×D4×C61—21no—
92C2×D4×C61—42no—
93C2×D4×C61—44no—
94C2×D4×C61—42no—
95C2×D4×C61—42no—
96C2×D4×C61—42no—
97C2×D4×C61—41no—
98C2×D4×C61—41no—
99C2×D4×C61C22⋊C412yes—
100C22×C4×C61—21no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—21no—
104C22×C4×C61—42no—
105C22×C4×C61—44no—
106C22×C4×C61C22×C411yescharacteristic
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61#4111yescharacteristic
111C22×C4×C61#4111yescharacteristic
112C22×Dic61—21no—
113C22×Dic61—22no—
114C22×Dic61—21no—
115C22×Dic61—41no—
116SD16×C61—44no—
117D8×C61—44no—
118M16×C61—42no—
119C2×C8×C61—22no—
120C4⋊C4×C61—21no—
121C4⋊C4×C61—21no—
122C22⋊C4×C61—41no—
123C22⋊C4×C61—41no—
124C22⋊Dic61—22no—
125C22⋊Dic61—42no—
126C22⋊Dic61—44no—
127C22⋊Dic61—41no—
128C22⋊Dic61—42no—
129C22⋊Dic61—82no—
130C4⋊Dic61—44no—
131C4⋊Dic61—42no—
132C4×Dic61—44no—
133C4×Dic61—42no—
134C4×Dic61—42no—
135#11758194—42no—
136C2×#92418—41no—
137C23×C61—21no—
138C23×C61—21no—
139C23×C61—22no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61—42no—
143C23×C61—42no—
144C23×C61—42no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61—81no—
149C23×C61—81no—
150C23×C61C4×D411yescharacteristic
151C23×C61#12011yescharacteristic
152C23×C61#12111yescharacteristic
153Q8×C61—24no—
154Q8×C61—22no—
155Q8×C61#9914yescomplemented
156D4×C61—22no—
157D4×C61—22no—
158D4×C61—44no—
159D4×C61—44no—
160D4×C61—42no—
161D4×C61—42no—
162D4×C61—81no—
163D4×C61—82no—
164D4×C61—81no—
165D4×C61—81no—
166C2×C4×C61—24no—
167C2×C4×C61—22no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—44no—
175C2×C4×C61—44no—
176C2×C4×C61—42no—
177C2×C4×C61—41no—
178C2×C4×C61C2×C22⋊C411yescharacteristic
179C2×C4×C61C4×D411yescharacteristic
180C2×C4×C61#12111yescharacteristic
181C2×C4×C61#12311yescharacteristic
182C2×Dic61—22no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—42no—
186C2×Dic61—41no—
187C2×Dic61—42no—
188C2×Dic61—42no—
189C2×Dic61—41no—
190C2×Dic61—42no—
191C2×Dic61—41no—
192C2×Dic61—81no—
193C2×Dic61—81no—
194C8×C61—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—42no—
206C22×C61—42no—
207C22×C61—42no—
208C22×C61—81no—
209C22×C61—81no—
210C22×C61—82no—
211C22×C61—82no—
212C22×C61—81no—
213C22×C61—81no—
214C22×C61—81no—
215C22×C61—162no—
216C22×C61#38611yescharacteristic
217C22×C61#45311yescharacteristic
218C22×C61#45411yescharacteristic
219C4×C61—21no—
220C4×C61—24no—
221C4×C61—21no—
222C4×C61—21no—
223C4×C61—81no—
224C4×C61C2×#9912yes—
225Dic61—42no—
226Dic61—82no—
227Dic61—81no—
228Dic61—82no—
229#3006D6111yescharacteristic
230#2341—611no—
231#1964—611no—
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—82no—
241C2×C61—161no—
242C2×C61#188311yescharacteristic
243C2×C61#189811yescharacteristic
244C2×C61#190611yescharacteristic
245C22×#41C2×D6111yescharacteristic
246C2×AGL(1,ℤ/8)Dic6112yescomplemented
247C2×#121C2×D6111yescharacteristic
248C2×C4×Q8—1221no—
249#449—22no—
250#448—21no—
251#447—21no—
252C2×#104—1222no—
253#417Dic6112yescomplemented
254C2×#99—1222no—
255#407—611no—
256#407C2×D6111yescharacteristic
257#390—1222no—
258#390—1221no—
259#380—611no—
260#380—1221no—
261C4×C22⋊C4—611no—
262C4×C22⋊C4—611no—
263C4×C22⋊C4—1221no—
264#327—1221no—
265#327—1221no—
266C61#936811yescharacteristic, complemented, Sylow
267C2×#41—21no—
268C2×#41—21no—
269C2×#41—44no—
270C2×#41C2×Dic6112yes—
271C2×#41C22⋊D6112yes—
272C22×Q8C22⋊D6111yescharacteristic
273C22×D4—21no—
274C22×D4—21no—
275C22×D4C2×Dic6111yescharacteristic
276C22×D4C22⋊D6111yescharacteristic
277C23×C4—21no—
278C23×C4C22×D6111yescharacteristic
279AGL(1,ℤ/8)—44no—
280C2×SD16—22no—
281C2×D8—22no—
282C2×M16—1221no—
283C2×M16C2×Dic6112yes—
284#121—21no—
285#121—21no—
286#121—21no—
287#121—21no—
288C4×Q8—2444no—
289C2×C4⋊C4—1222no—
290C2×C4⋊C4—1221no—
291C2×C4⋊C4C2×Dic6111yescharacteristic
292C2×C22⋊C4—21no—
293C2×C22⋊C4—612no—
294C2×C22⋊C4—1221no—
295C2×C22⋊C4—1222no—
296C2×C22⋊C4—1222no—
297C2×C22⋊C4—1221no—
298C2×C42—1222no—
299C2×C42—1221no—
300C2×C42—1221no—
301#104—2444no—
302#103—22no—
303#102—22no—
304#99—2444no—
305C22⋊C8—21no—
306C22⋊C8—21no—
307C22⋊C8—1221no—
308C22⋊C8—1221no—
309#95—611no—
310#95—611no—
311#95—1221no—
312#95—1221no—
313C24—21no—
314C24—21no—
315C24D4×D6111yescharacteristic
316#41—24no—
317#41—24no—
318#41—42no—
319#41—42no—
320#41—44no—
321#41—44no—
322C2×Q8—24no—
323C2×Q8—21no—
324C2×Q8C22⋊Dic6112yes—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—42no—
331C2×D4—44no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4C22⋊Dic6112yes—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—42no—
343C22×C4—44no—
344C22×C4—1221no—
345C22×C4—1221no—
346C22×C4—1222no—
347C22×C4—2441no—
348C22×C4D4×D6111yescharacteristic
349C22×C4#1175822411yescharacteristic
350C22×C4#1175822411yescharacteristic
351C22×C4C22×Dic6111yescharacteristic
352C22×C4C2×C22⋊D6111yescharacteristic
353C22×C4C2×C22⋊D6111yescharacteristic
354SD16—44no—
355D8—44no—
356M16—42no—
357M16—2442no—
358C2×C8—22no—
359C2×C8—2441no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—2444no—
363C4⋊C4—2442no—
364C22⋊C4—41no—
365C22⋊C4—41no—
366C22⋊C4—1222no—
367C22⋊C4—4882no—
368C22⋊C4—2442no—
369C22⋊C4—2444no—
370C22⋊C4—2441no—
371C22⋊C4—2442no—
372C42—2442no—
373C42—2444no—
374C42—2442no—
375C23—21no—
376C23—21no—
377C23—22no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—81no—
387C23—81no—
388C23D4×Dic6111yescharacteristic
389C23#1451974511yescharacteristic
390C23#1451974911yescharacteristic
391Q8—24no—
392Q8—22no—
393Q8#1451969214yescomplemented
394D4—22no—
395D4—22no—
396D4—44no—
397D4—44no—
398D4—42no—
399D4—42no—
400D4—81no—
401D4—82no—
402D4—81no—
403D4—81no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—1222no—
417C2×C4—4881no—
418C2×C4—4881no—
419C2×C4—2441no—
420C2×C4—2442no—
421C2×C4—2442no—
422C2×C4—2441no—
423C2×C4—2442no—
424C2×C4—2441no—
425C2×C4—2442no—
426C2×C4—2441no—
427C2×C4—2441no—
428C2×C4C2×C22⋊Dic6111yescharacteristic
429C2×C4D4×Dic6111yescharacteristic
430C2×C4#1451974911yescharacteristic
431C2×C4#1451975811yescharacteristic
432C8—42no—
433C8—4881no—
434C22—21no—
435C22—21no—
436C22—21no—
437C22—21no—
438C22—21no—
439C22—21no—
440C22—21no—
441C22—21no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—81no—
446C22—81no—
447C22—82no—
448C22—82no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—162no—
453C22#1683964211yescharacteristic
454C22#1683992811yescharacteristic
455C22#1683993311yescharacteristic
456C4—21no—
457C4—24no—
458C4—21no—
459C4—21no—
460C4—81no—
461C4—4881no—
462C4—4882no—
463C4—4882no—
464C4—2442no—
465C4C2×#1451969212yes—
466C2—21no—
467C2—21no—
468C2—41no—
469C2—41no—
470C2—82no—
471C2—161no—
472C2#2144619711yescharacteristic
473C2#2144627311yescharacteristic
474C2#2144632211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
Fitting, prime radicalC61×#3006C2
socleC22×C61#386
cosocleC23×C4×C61C23
hypercenter#3006D61
hypocenterC61#9368
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3006

Complements

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

NH ≅ G/Ntype
1. #21448660C2semidirect
2. #21446656C2semidirect
3. #21445763C2semidirect
4. #21444335C2semidirect
5. #21443724C2semidirect
6. #21443704C2semidirect
22. C2×AGL(1,ℤ/8)×C61C4semidirect
27. C61×#417C4semidirect
155. Q8×C61#99semidirect
246. C2×AGL(1,ℤ/8)Dic61semidirect
253. #417Dic61semidirect
266. C61#9368semidirect
393. Q8#14519692semidirect

Maximal subgroups

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

subgroupindex
#214486602
#214466562
#214457632
#214443352
#214437242
#214437042
C61×#30062
#936861