Finite Group Databasebeta

← #30601764

Subgroups — #30601764

Order 15616.

770 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 770 conjugacy classes of subgroups — too many to draw. Only its 104 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#21461546C211yesmaximal, characteristic, complemented
2C2×#16840614C212yesmaximal, complemented
3C2×#16840613C212yesmaximal, complemented
4#21460995C211yesmaximal, characteristic, complemented
5#21454163C211yesmaximal, characteristic
6C2×#14519834C2212yes—
7C2×#14519832—21no—
8C2×(C2×#6923)⋊C4—21no—
9C2×(C2×#6923)⋊C4—21no—
10C2×(C2×#6923)⋊C4C2212yes—
11#16840628—21no—
12#16840614—24no—
13#16840613—24no—
14C2×#14519773C411yescharacteristic, complemented
15C2×#14519752C2211yescharacteristic
16C2×#14519750—21no—
17C2×#14519750—21no—
18C2×#14519748C2211yescharacteristic
19C2×C42×D61C2211yescharacteristic
20#16839664C411yescharacteristic, complemented
21#16839662—21no—
22#16839662—21no—
23#16839631C411yescharacteristic, complemented
24C61×#390C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27#14519834—22no—
28#14519834D414yescomplemented
29#14519832—42no—
30(C2×#6923)⋊C4—24no—
31(C2×#6923)⋊C4—22no—
32(C2×#6923)⋊C4—22no—
33(C2×#6923)⋊C4—42no—
34C22×D244—21no—
35C22×D244C2×C411yescharacteristic
36C22×C22⋊D61—21no—
37C22×C22⋊D61—21no—
38C22×C22⋊D61—21no—
39C22×C4×D61D411yescharacteristic
40C22×C4×D61D411yescharacteristic
41C22×C4×D61C2311yescharacteristic
42#14519773—42no—
43#14519752—22no—
44#14519752—22no—
45#14519750—44no—
46#14519750—44no—
47#14519748—22no—
48#14519748—22no—
49C2×C4⋊Dic61C2×C411yescharacteristic
50C2×#11758196—21no—
51C2×#11758196—21no—
52C2×C22⋊Dic61—21no—
53C2×#11758198—21no—
54C2×#11758198—21no—
55C2×#11758198—21no—
56C2×C4×Dic61C2×C411yescharacteristic
57C2×C4×Dic61D411yescharacteristic
58C2×C4×Dic61D411yescharacteristic
59C42×D61—42no—
60C22×D4×C61C2×C411yescharacteristic
61C2×C4⋊C4×C61C2×C411yescharacteristic
62C2×C22⋊C4×C61—21no—
63C2×C22⋊C4×C61—21no—
64C2×C42×C61C2×C411yescharacteristic
65C24×C61—21no—
66C23×D61—21no—
67C23×D61C2×D411yescharacteristic
68C22×#6923—24no—
69C22×#6923—42no—
70C22×#6923—42no—
71C22×#6923—42no—
72C22×#6923—42no—
73C22×#6923—42no—
74C22×#6923—42no—
75C2×D4×C61—22no—
76C2×D4×C61—41no—
77C2×D4×C61—41no—
78C22×C4×C61—21no—
79C22×C4×C61C22⋊C411yescharacteristic
80C22×C4×C61C22⋊C411yescharacteristic
81C22×C4×C61C22×C411yescharacteristic
82C2×C22⋊D61—42no—
83C2×C22⋊D61—42no—
84C2×C22⋊D61—42no—
85C2×C22⋊D61—42no—
86C2×C22⋊D61—41no—
87C2×C22⋊D61—41no—
88C2×C22⋊D61—42no—
89C2×C22⋊D61—42no—
90C2×C22⋊D61—42no—
91C2×C22⋊D61—42no—
92C22×Dic61—21no—
93C22×Dic61C22⋊C411yescharacteristic
94C22×Dic61C22⋊C411yescharacteristic
95C22×Dic61C2×D411yescharacteristic
96C2×D244—22no—
97C2×D244—41no—
98C2×D244—41no—
99C2×D244—41no—
100C2×D244—41no—
101C2×D244—42no—
102C2×D244—42no—
103C2×C4×D61—22no—
104C2×C4×D61—22no—
105C2×C4×D61—22no—
106C2×C4×D61—22no—
107C2×C4×D61—41no—
108C2×C4×D61—41no—
109C2×C4×D61—41no—
110C2×C4×D61—41no—
111C2×C4×D61C2×D412yes—
112C2×C4×D61#4112yes—
113C2×#92420—24no—
114C2×#92420—42no—
115C2×#92420—42no—
116C4⋊C4×C61—22no—
117C22⋊C4×C61—42no—
118C22⋊C4×C61—42no—
119C42×C61—41no—
120C22⋊Dic61—42no—
121#11758198—42no—
122#11758198—42no—
123#11758198—42no—
124C4⋊Dic61—22no—
125#11758196—42no—
126#11758196—42no—
127C4×Dic61—22no—
128C4×Dic61—22no—
129C4×Dic61—41no—
130C23×C61—41no—
131C23×C61—41no—
132C23×C61—41no—
133C23×C61—41no—
134C23×C61—41no—
135C23×C61—41no—
136C23×C61—41no—
137C23×C61C2×C22⋊C411yescharacteristic
138C22×D61—21no—
139C22×D61—21no—
140C22×D61—21no—
141C22×D61—21no—
142C22×D61—41no—
143C22×D61—41no—
144C22×D61—41no—
145C22×D61—41no—
146C22×D61—41no—
147C22×D61—41no—
148C22×D61—41no—
149C22×D61#12411yescharacteristic
150C22×D61#12711yescharacteristic
151C22×D61AGL(1,ℤ/8)12yes—
152C22×D61AGL(1,ℤ/8)12yes—
153C2×#6923—44no—
154C2×#6923—44no—
155C2×#6923—44no—
156C2×#6923—82no—
157C2×#6923—82no—
158C2×#6923—82no—
159C2×#6923—82no—
160D4×C61—82no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—21no—
165C2×C4×C61—21no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61C4×D412yes—
173C22⋊D61—84no—
174C22⋊D61—84no—
175C22⋊D61—82no—
176C22⋊D61—82no—
177C2×Dic61—21no—
178C2×Dic61—21no—
179C2×Dic61—21no—
180C2×Dic61—21no—
181C2×Dic61—21no—
182C2×Dic61—21no—
183C2×Dic61—41no—
184C2×Dic61—41no—
185C2×Dic61—41no—
186C2×Dic61#9912yes—
187C2×Dic61#10412yes—
188C2×Dic61#12112yes—
189D244—82no—
190D244—82no—
191D244—82no—
192C4×D61—42no—
193C4×D61—42no—
194C4×D61—44no—
195C4×D61—44no—
196#92420—44no—
197#9684—611nomaximal, Sylow
198C22×C61—81no—
199C22×C61—81no—
200C22×C61—81no—
201C22×C61—81no—
202C22×C61—81no—
203C22×C61—81no—
204C22×C61—81no—
205C22×C61#39011yescharacteristic
206C22×C61C2×#9911yescharacteristic
207C22×C61#41011yescharacteristic
208C22×C61#41811yescharacteristic
209C22×C61#41811yescharacteristic
210C22×C61C2×#10411yescharacteristic
211C22×C61#42111yescharacteristic
212C2×D61—21no—
213C2×D61—21no—
214C2×D61—21no—
215C2×D61—21no—
216C2×D61—22no—
217C2×D61—22no—
218C2×D61—22no—
219C2×D61—22no—
220C2×D61—81no—
221C2×D61—81no—
222C2×D61—81no—
223C2×D61—81no—
224C2×D61—81no—
225C2×D61—81no—
226C2×D61—81no—
227C2×D61#49012yes—
228C2×D61#49612yes—
229#6923—84no—
230C4×C61—22no—
231C4×C61—41no—
232C4×C61—41no—
233C4×C61—81no—
234Dic61—22no—
235Dic61—22no—
236Dic61—22no—
237Dic61—81no—
238#2856—611no—
239#1749—611no—
240C2×#327—612no—
241C2×#327—612no—
242#1434—611no—
243C2×C61—161no—
244C2×C61#150511yescharacteristic
245C2×C61#150711yescharacteristic
246C2×C61#151011yescharacteristic
247C2×C61#151411yescharacteristic
248C2×C61#196111yescharacteristic
249C2×C61#196411yescharacteristic
250C2×C61#198011yescharacteristic
251D61—22no—
252D61—22no—
253D61—161no—
254C2×#127—611no—
255C2×#121—611no—
256C2×#121—611no—
257C2×#121—1221no—
258C2×#121—1221no—
259C22×C42—611no—
260C2×C22⋊C8—612no—
261C2×#97—1221no—
262#390—611no—
263#390—1221no—
264#390—1221no—
265#390#692311yescharacteristic, complemented
266#384—611no—
267C2×#95—612no—
268C2×#95—1221no—
269C2×#95—1221no—
270#336—1221no—
271#327—1224no—
272#327—1224no—
273C61#968411yescharacteristic, complemented, Sylow
274C22×D4—611no—
275C22×D4—1221no—
276C22×D4—1221no—
277C22×D4—1221no—
278C22×D4—1221no—
279C22×D4C2×#692311yescharacteristic
280C23×C4—611no—
281C23×C4—611no—
282C23×C4—611no—
283C23×C4—1222no—
284C22×C8—1222no—
285#127—2442no—
286#121—1222no—
287#121—1222no—
288#121—1222no—
289#121—1222no—
290#121—2444no—
291#121—2444no—
292C2×C4⋊C4—611no—
293C2×C4⋊C4—1221no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4C2×#692311yescharacteristic
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—1221no—
299C2×C22⋊C4—1221no—
300C2×C22⋊C4—1221no—
301C2×C22⋊C4—1221no—
302C2×C42—611no—
303C2×C42—611no—
304C2×C42—611no—
305C2×C42—2442no—
306C2×C42C2×#692311yescharacteristic
307C22⋊C8—614no—
308C22⋊C8—1222no—
309#97—2442no—
310#95—1222no—
311#95—1222no—
312#95—1224no—
313#95—2442no—
314C24—21no—
315C24—611no—
316C24—1221no—
317C2×D4—22no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—1222no—
321C2×D4—2441no—
322C2×D4—2441no—
323C2×D4—2441no—
324C2×D4—2441no—
325C2×D4—2442no—
326C2×D4—2442no—
327C2×D4—2442no—
328C2×D4—2442no—
329C2×D4—2442no—
330C2×D4—2442no—
331C2×D4—2442no—
332C2×D4—2442no—
333C2×D4—2441no—
334C2×D4—2441no—
335C2×D4—2442no—
336C2×D4—2442no—
337C22×C4—21no—
338C22×C4—611no—
339C22×C4—611no—
340C22×C4—611no—
341C22×C4—612no—
342C22×C4—612no—
343C22×C4—1222no—
344C22×C4—1222no—
345C22×C4—1222no—
346C22×C4—1222no—
347C22×C4—1224no—
348C22×C4—1221no—
349C22×C4—2441no—
350C22×C4—2441no—
351C22×C4—2441no—
352C22×C4—2441no—
353C22×C4—2442no—
354C22×C4—2442no—
355C22×C4—2442no—
356C22×C4—2442no—
357C22×C4—2442no—
358C22×C4—2442no—
359C22×C4C22⋊#692311yescharacteristic
360C22×C4C22⋊#692311yescharacteristic
361C22×C4C22×#692311yescharacteristic
362C2×C8—1224no—
363C2×C8—2442no—
364C2×C8—2442no—
365C4⋊C4—22no—
366C4⋊C4—1222no—
367C4⋊C4—2442no—
368C4⋊C4—2442no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4—2442no—
372C22⋊C4—2442no—
373C22⋊C4—2442no—
374C22⋊C4—2442no—
375C42—41no—
376C42—1222no—
377C42—1222no—
378C42—2441no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—611no—
387C23—611no—
388C23—612no—
389C23—612no—
390C23—1221no—
391C23—1221no—
392C23—1221no—
393C23—1221no—
394C23—2441no—
395C23—2441no—
396C23—2441no—
397C23—2441no—
398C23—2441no—
399C23—2441no—
400C23—2441no—
401C23C2×C22⋊#692311yescharacteristic
402D4—82no—
403D4—4882no—
404D4—4882no—
405D4—4882no—
406D4—4884no—
407D4—4884no—
408D4—4882no—
409D4—4882no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—612no—
422C2×C4—612no—
423C2×C4—612no—
424C2×C4—1221no—
425C2×C4—1221no—
426C2×C4—1221no—
427C2×C4—1221no—
428C2×C4—1221no—
429C2×C4—1221no—
430C2×C4—4882no—
431C2×C4—4882no—
432C2×C4—4882no—
433C2×C4—4882no—
434C2×C4—2442no—
435C2×C4—2442no—
436C2×C4—2444no—
437C2×C4—2444no—
438C2×C4—2444no—
439C2×C4—2444no—
440C2×C4—2444no—
441C2×C4—2441no—
442C2×C4—2441no—
443C2×C4—2441no—
444C2×C4D4×#692312yes—
445C8—2444no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—612no—
454C22—612no—
455C22—1221no—
456C22—1221no—
457C22—1222no—
458C22—1222no—
459C22—1221no—
460C22—1221no—
461C22—1222no—
462C22—1222no—
463C22—4881no—
464C22—4881no—
465C22—4881no—
466C22—4881no—
467C22—4881no—
468C22—4881no—
469C22—4881no—
470C22#1684071411yescharacteristic
471C22D61⋊#9911yescharacteristic
472C22#1684074111yescharacteristic
473C22#1684075411yescharacteristic
474C22#1684075411yescharacteristic
475C22D61⋊#10411yescharacteristic
476C22#1684075911yescharacteristic
477C4—22no—
478C4—41no—
479C4—41no—
480C4—81no—
481C4—1222no—
482C4—1222no—
483C4—1222no—
484C4—4884no—
485C4—4881no—
486C2—161no—
487C2—1222no—
488C2—1222no—
489C2—9761no—
490C2#2146120911yescharacteristic
491C2#2146121411yescharacteristic
492C2#2146122511yescharacteristic
493C2#2146123811yescharacteristic
494C2#2146195111yescharacteristic
495C2#2146196011yescharacteristic
496C2#2146198411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#390C4
socleC23×C61C2×C22⋊C4
prime radical#21454163C2
cosocleC22×C4×D61C23
hypercenter#390#6923
hypocenterC61#9684
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #390

Complements

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

NH ≅ G/Ntype
1. #21461546C2semidirect
2. C2×#16840614C2semidirect
3. C2×#16840613C2semidirect
4. #21460995C2semidirect
14. C2×#14519773C4semidirect
20. #16839664C4semidirect
23. #16839631C4semidirect
24. C61×#390C4semidirect
28. #14519834D4semidirect
265. #390#6923semidirect
273. C61#9684semidirect

Maximal subgroups

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

subgroupindex
#214615462
C2×#168406142
C2×#168406132
#214609952
#214541632
#968461