Finite Group Databasebeta

← #29577992

Subgroups — #29577992

Order 15616.

812 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 812 conjugacy classes of subgroups — too many to draw. Only its 107 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#21449134C211yesmaximal, characteristic
2#21445747C211yesmaximal, characteristic, complemented
3#21444145C212yesmaximal, complemented
4#21443728C212yesmaximal, complemented
5C61×#2879C211yesmaximal, characteristic, complemented
6C2×#14519745C411yescharacteristic
7C2×#14519745C411yescharacteristic
8C22×#11758198C2211yescharacteristic
9C2×#14519696—21no—
10C2×#14519694—21no—
11#16839726C2212yes—
12#16839673—21no—
13#16839670—21no—
14#16839642C411yescharacteristic
15#16839642C411yescharacteristic
16#16839641—21no—
17#16839641—21no—
18#16839625C2211yescharacteristic
19#16839485—24no—
20C22×#41×C61C2211yescharacteristic
21C22×M16×C61—21no—
22C61×#408—24no—
23C61×#407—21no—
24C61×#407C2212yes—
25C24×D61C2×C411yescharacteristic
26C22×C22⋊D61—21no—
27C22×C22⋊D61—21no—
28C23×Dic61C2×C411yescharacteristic
29C2×#11758194—22no—
30#14519745—42no—
31#14519745—42no—
32C2×C22⋊Dic61—21no—
33C2×C22⋊Dic61—21no—
34C2×C22⋊Dic61C2×C411yescharacteristic
35C2×C22⋊Dic61C2×C411yescharacteristic
36C2×#11758198—21no—
37C2×#11758198—21no—
38C2×#11758198—22no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—22no—
42#14519696—42no—
43#14519694—42no—
44C22⋊#92418—22no—
45C22⋊#92418—22no—
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50C2×#41×C61—22no—
51C2×#41×C61—22no—
52C2×#41×C61—42no—
53C2×#41×C61—42no—
54C22×Q8×C61D411yescharacteristic
55C22×D4×C61C2×C411yescharacteristic
56C22×D4×C61C2×C411yescharacteristic
57C22×D4×C61D411yescharacteristic
58C23×C4×C61D411yescharacteristic
59C23×C4×C61D411yescharacteristic
60C23×C4×C61C2311yescharacteristic
61C2×M16×C61—22no—
62C2×M16×C61—22no—
63C2×M16×C61—42no—
64C2×M16×C61—42no—
65C22×C8×C61—22no—
66C22⋊C8×C61—24no—
67C22⋊C8×C61—22no—
68C22⋊C8×C61D414yescomplemented
69C24×C61C22⋊C411yescharacteristic
70C24×C61C22⋊C411yescharacteristic
71C24×C61C22×C411yescharacteristic
72C23×D61—21no—
73C23×D61—22no—
74C23×D61—42no—
75C23×D61—41no—
76C23×D61—41no—
77C23×D61—81no—
78#41×C61—44no—
79#41×C61—44no—
80#41×C61—48no—
81C2×Q8×C61—22no—
82C2×Q8×C61—22no—
83C2×Q8×C61—41no—
84C2×D4×C61—22no—
85C2×D4×C61—22no—
86C2×D4×C61—22no—
87C2×D4×C61—22no—
88C2×D4×C61—22no—
89C2×D4×C61—22no—
90C2×D4×C61—41no—
91C2×D4×C61—41no—
92C2×D4×C61—41no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61—21no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—41no—
100C22×C4×C61—41no—
101C22×C4×C61—41no—
102C22×C4×C61—41no—
103C22×C4×C61C22⋊C411yescharacteristic
104C22×C4×C61C22⋊C411yescharacteristic
105C22×C4×C61C2×D411yescharacteristic
106C22×C4×C61C2×D411yescharacteristic
107C22×C4×C61C2×D412yes—
108C22×C4×C61#4112yes—
109C2×C22⋊D61—42no—
110C2×C22⋊D61—42no—
111C2×C22⋊D61—42no—
112C2×C22⋊D61—42no—
113C2×C22⋊D61—81no—
114C2×C22⋊D61—81no—
115C22×Dic61—21no—
116C22×Dic61—21no—
117C22×Dic61—21no—
118C22×Dic61—22no—
119C22×Dic61—81no—
120M16×C61—44no—
121M16×C61—42no—
122M16×C61—42no—
123C2×C8×C61—24no—
124C2×C8×C61—44no—
125C22⋊Dic61—41no—
126C22⋊Dic61—41no—
127C22⋊Dic61—81no—
128C22⋊Dic61—81no—
129#11758198—42no—
130#11758198—42no—
131#11758198—41no—
132#11758198—42no—
133#11758198—41no—
134#11758194—42no—
135#11758194—42no—
136C2×#92418—42no—
137C23×C61—21no—
138C23×C61—21no—
139C23×C61—21no—
140C23×C61—21no—
141C23×C61—21no—
142C23×C61—21no—
143C23×C61—21no—
144C23×C61—41no—
145C23×C61—41no—
146C23×C61—41no—
147C23×C61—41no—
148C23×C61—41no—
149C23×C61—41no—
150C23×C61#9912yes—
151C23×C61C2×C22⋊C411yescharacteristic
152C23×C61C4×D412yes—
153C22×D61—22no—
154C22×D61—24no—
155C22×D61—41no—
156C22×D61—41no—
157C22×D61—41no—
158C22×D61—42no—
159C22×D61—41no—
160C22×D61—41no—
161C22×D61—41no—
162C22×D61—84no—
163C22×D61—84no—
164C22×D61—82no—
165C22×D61—82no—
166Q8×C61—44no—
167D4×C61—44no—
168D4×C61—44no—
169D4×C61—44no—
170C2×C4×C61—22no—
171C2×C4×C61—22no—
172C2×C4×C61—24no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61#9912yes—
187C2×C4×C61#12011yescharacteristic
188C2×C4×C61#12112yes—
189C2×C4×C61#12311yescharacteristic
190C22⋊D61—84no—
191C22⋊D61—84no—
192C2×Dic61—41no—
193C2×Dic61—41no—
194C2×Dic61—41no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—82no—
202C8×C61—44no—
203#92418—82no—
204#9511—611nomaximal, Sylow
205C22×C61—21no—
206C22×C61—21no—
207C22×C61—22no—
208C22×C61—22no—
209C22×C61—22no—
210C22×C61—22no—
211C22×C61—42no—
212C22×C61—42no—
213C22×C61—42no—
214C22×C61—42no—
215C22×C61—41no—
216C22×C61—41no—
217C22×C61—41no—
218C22×C61—41no—
219C22×C61—41no—
220C22×C61—81no—
221C22×C61—81no—
222C22×C61#38611yescharacteristic
223C22×C61C2×#9911yescharacteristic
224C22×C61C2×#9911yescharacteristic
225C2×D61—42no—
226C2×D61—42no—
227C2×D61—42no—
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—84no—
231C2×D61—81no—
232C2×D61—81no—
233C2×D61—84no—
234C4×C61—22no—
235C4×C61—21no—
236C4×C61—21no—
237C4×C61—22no—
238C4×C61—41no—
239Dic61—81no—
240Dic61—81no—
241Dic61—81no—
242#2879D6111yescharacteristic, complemented
243#2404—611no—
244#1785—611no—
245#1462—612no—
246#1306—612no—
247C2×C61—21no—
248C2×C61—21no—
249C2×C61—42no—
250C2×C61—42no—
251C2×C61—81no—
252C2×C61#189111yescharacteristic
253C2×C61#189811yescharacteristic
254C2×C61#191411yescharacteristic
255D61—82no—
256C22×#41C2×D6111yescharacteristic
257C22×M16—21no—
258C2×#120—611no—
259C2×#120—611no—
260C22×C22⋊C4—611no—
261C2×#101—1221no—
262C2×#100—1221no—
263#408—24no—
264#407—21no—
265#407—612no—
266#407C2×D6112yes—
267#393—1221no—
268#392—1221no—
269#386—611no—
270#386—611no—
271#386—1221no—
272#386—1221no—
273#381—611no—
274#323—1224no—
275C61#951111yescharacteristic, complemented, Sylow
276C25—611no—
277C2×#41—22no—
278C2×#41—22no—
279C2×#41—42no—
280C2×#41—42no—
281C22×Q8D24411yescharacteristic
282C22×D4—1221no—
283C22×D4—1221no—
284C22×D4C4×D6111yescharacteristic
285C22×D4C4×D6111yescharacteristic
286C22×D4D24411yescharacteristic
287C23×C4—611no—
288C23×C4C22⋊D6111yescharacteristic
289C23×C4C22⋊D6111yescharacteristic
290C23×C4C22×D6111yescharacteristic
291C2×M16—22no—
292C2×M16—22no—
293C2×M16—42no—
294C2×M16—42no—
295C2×M16—1222no—
296C22×C8—22no—
297#120—2442no—
298#120—2442no—
299C2×C22⋊C4—611no—
300C2×C22⋊C4—611no—
301C2×C22⋊C4—1221no—
302C2×C22⋊C4—1221no—
303C2×C22⋊C4—1221no—
304C2×C22⋊C4—1221no—
305C2×C22⋊C4—1221no—
306C2×C22⋊C4—1222no—
307C2×C22⋊C4—1222no—
308C2×C22⋊C4—1221no—
309#101—2442no—
310#100—2442no—
311C22⋊C8—24no—
312C22⋊C8—22no—
313C22⋊C8—1222no—
314C22⋊C8—1222no—
315C22⋊C8C22⋊D6114yescomplemented
316#95—1221no—
317#95—1221no—
318#95—1221no—
319#95—1221no—
320C24—1222no—
321C24—1221no—
322C24—4881no—
323C24—2441no—
324C24—2441no—
325C24—2442no—
326C24#1175819811yescharacteristic
327C24#1175819811yescharacteristic
328C24C2×C4×D6111yescharacteristic
329#41—44no—
330#41—44no—
331#41—48no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—41no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—22no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—4881no—
345C2×D4—4881no—
346C2×D4—2442no—
347C2×D4—2442no—
348C2×D4—2442no—
349C2×D4—2442no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—1222no—
361C22×C4—1221no—
362C22×C4—1221no—
363C22×C4—1221no—
364C22×C4—4881no—
365C22×C4#1175819811yescharacteristic
366C22×C4#1175819811yescharacteristic
367C22×C4C2×D24411yescharacteristic
368C22×C4#1175822212yes—
369C22×C4C2×C22⋊D6111yescharacteristic
370C22×C4C2×C22⋊D6112yes—
371M16—44no—
372M16—42no—
373M16—42no—
374M16—2442no—
375M16—2442no—
376C2×C8—24no—
377C2×C8—44no—
378C2×C8—2442no—
379C22⋊C4—4881no—
380C22⋊C4—4881no—
381C22⋊C4—2442no—
382C22⋊C4—2442no—
383C22⋊C4—2441no—
384C22⋊C4—2442no—
385C22⋊C4—2441no—
386C22⋊C4—2441no—
387C22⋊C4—2441no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—1224no—
402C23—1222no—
403C23—4882no—
404C23—4882no—
405C23—4884no—
406C23—4884no—
407C23—2441no—
408C23—2441no—
409C23—2441no—
410C23—2442no—
411C23—2441no—
412C23—2441no—
413C23—2441no—
414C23#1451969112yes—
415C23C2×#1175819811yescharacteristic
416C23C4×C22⋊D6112yes—
417Q8—44no—
418D4—44no—
419D4—44no—
420D4—44no—
421D4—4884no—
422D4—4884no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—4882no—
440C2×C4—2441no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4—2441no—
445C2×C4—2441no—
446C2×C4—2441no—
447C2×C4—2441no—
448C2×C4—2441no—
449C2×C4#1451969112yes—
450C2×C4#1451974611yescharacteristic
451C2×C4#1451975112yes—
452C2×C4#1451976011yescharacteristic
453C8—44no—
454C8—4882no—
455C22—21no—
456C22—21no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—81no—
471C22—81no—
472C22—4881no—
473C22—4881no—
474C22—4884no—
475C22—4884no—
476C22—4881no—
477C22—4881no—
478C22—2442no—
479C22—2442no—
480C22—2442no—
481C22#1683964111yescharacteristic
482C22C2×#1451969111yescharacteristic
483C22C2×#1451969111yescharacteristic
484C4—21no—
485C4—22no—
486C4—21no—
487C4—22no—
488C4—41no—
489C4—4881no—
490C4—4881no—
491C4—4881no—
492C2—21no—
493C2—21no—
494C2—42no—
495C2—42no—
496C2—81no—
497C2—4882no—
498C2#2144623711yescharacteristic
499C2#2144627211yescharacteristic
500C2#2144636611yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC24×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2879C2
socleC22×C61#386
prime radical#21449134C2
cosocleC23×C4×C61C23
hypercenter#2879D61
hypocenterC61#9511
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2879

Complements

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

NH ≅ G/Ntype
2. #21445747C2semidirect
3. #21444145C2semidirect
4. #21443728C2semidirect
5. C61×#2879C2semidirect
68. C22⋊C8×C61D4semidirect
242. #2879D61semidirect
275. C61#9511semidirect
315. C22⋊C8C22⋊D61semidirect

Maximal subgroups

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

subgroupindex
#214491342
#214457472
#214441452
#214437282
C61×#28792
#951161