Finite Group Databasebeta

← #30601810

Subgroups — #30601810

Order 15616.

856 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 856 conjugacy classes of subgroups — too many to draw. Only its 120 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#21461676C211yesmaximal, characteristic, complemented
2C2×#16840614C212yesmaximal, complemented
3C2×#16840613C212yesmaximal, complemented
4C4×(C2×#6923)⋊C4C211yesmaximal, characteristic, complemented
5#21454163C211yesmaximal, characteristic
6C22×C4×#6923—21no—
7C2×#14519844—21no—
8C2×#14519834C2212yes—
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—
27C2×C4×#6923—24no—
28C2×C4×#6923—42no—
29C2×C4×#6923—42no—
30#14519844—42no—
31#14519834—22no—
32#14519834—22no—
33(C2×#6923)⋊C4—24no—
34(C2×#6923)⋊C4—22no—
35(C2×#6923)⋊C4D414yescomplemented
36C22×D244—21no—
37C22×D244C2×C411yescharacteristic
38C22×C22⋊D61—21no—
39C22×C22⋊D61—21no—
40C22×C22⋊D61—21no—
41C22×C4×D61D411yescharacteristic
42C22×C4×D61D411yescharacteristic
43C22×C4×D61C2311yescharacteristic
44#14519773—42no—
45#14519752—22no—
46#14519752—22no—
47#14519750—44no—
48#14519750—42no—
49#14519750—42no—
50#14519748—22no—
51#14519748—22no—
52C2×C4⋊Dic61C2×C411yescharacteristic
53C2×#11758196—21no—
54C2×#11758196—21no—
55C2×C22⋊Dic61—21no—
56C2×#11758198—21no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×C4×Dic61C2×C411yescharacteristic
60C2×C4×Dic61D411yescharacteristic
61C2×C4×Dic61D411yescharacteristic
62C42×D61—42no—
63C22×D4×C61C2×C411yescharacteristic
64C2×C4⋊C4×C61C2×C411yescharacteristic
65C2×C22⋊C4×C61—21no—
66C2×C22⋊C4×C61—21no—
67C2×C42×C61C2×C411yescharacteristic
68C24×C61—21no—
69C23×D61—21no—
70C23×D61C2×D411yescharacteristic
71C22×#6923—24no—
72C22×#6923—24no—
73C22×#6923—24no—
74C22×#6923—42no—
75C22×#6923—42no—
76C22×#6923—42no—
77C22×#6923—42no—
78C2×D4×C61—22no—
79C2×D4×C61—41no—
80C2×D4×C61—41no—
81C22×C4×C61—21no—
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22⋊C411yescharacteristic
84C22×C4×C61C22×C411yescharacteristic
85C2×C22⋊D61—42no—
86C2×C22⋊D61—42no—
87C2×C22⋊D61—42no—
88C2×C22⋊D61—41no—
89C2×C22⋊D61—41no—
90C2×C22⋊D61—42no—
91C2×C22⋊D61—42no—
92C2×C22⋊D61—42no—
93C2×C22⋊D61—42no—
94C2×C22⋊D61—42no—
95C22×Dic61—21no—
96C22×Dic61C22⋊C411yescharacteristic
97C22×Dic61C22⋊C411yescharacteristic
98C22×Dic61C2×D411yescharacteristic
99C2×D244—22no—
100C2×D244—41no—
101C2×D244—41no—
102C2×D244—42no—
103C2×D244—41no—
104C2×D244—41no—
105C2×D244—42no—
106C2×C4×D61—22no—
107C2×C4×D61—22no—
108C2×C4×D61—22no—
109C2×C4×D61—22no—
110C2×C4×D61—41no—
111C2×C4×D61—41no—
112C2×C4×D61—41no—
113C2×C4×D61D812yes—
114C2×C4×D61SD1612yes—
115C2×C4×D61C2×D412yes—
116C2×C4×D61#4112yes—
117C2×#92420—42no—
118C2×#92420—42no—
119C2×#92420—42no—
120C4×#6923—48no—
121C4⋊C4×C61—22no—
122C22⋊C4×C61—42no—
123C22⋊C4×C61—42no—
124C42×C61—41no—
125C22⋊Dic61—41no—
126C22⋊Dic61—41no—
127#11758198—42no—
128#11758198—42no—
129#11758198—41no—
130#11758198—41no—
131C4⋊Dic61—22no—
132#11758196—42no—
133#11758196—42no—
134C4×Dic61—22no—
135C4×Dic61—22no—
136C4×Dic61—41no—
137C23×C61—41no—
138C23×C61—41no—
139C23×C61—41no—
140C23×C61—41no—
141C23×C61—41no—
142C23×C61—41no—
143C23×C61—41no—
144C23×C61C2×C22⋊C411yescharacteristic
145C22×D61—21no—
146C22×D61—21no—
147C22×D61—21no—
148C22×D61—21no—
149C22×D61—41no—
150C22×D61—41no—
151C22×D61—41no—
152C22×D61—41no—
153C22×D61—41no—
154C22×D61—41no—
155C22×D61—41no—
156C22×D61#12112yes—
157C22×D61C2×D811yescharacteristic
158C22×D61C2×SD1611yescharacteristic
159C22×D61AGL(1,ℤ/8)12yes—
160C2×#6923—28no—
161C2×#6923—44no—
162C2×#6923—44no—
163C2×#6923—44no—
164C2×#6923—44no—
165C2×#6923—44no—
166C2×#6923—44no—
167D4×C61—82no—
168C2×C4×C61—21no—
169C2×C4×C61—21no—
170C2×C4×C61—21no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—41no—
178C2×C4×C61#10212yes—
179C2×C4×C61C4×D412yes—
180C22⋊D61—84no—
181C22⋊D61—82no—
182C22⋊D61—82no—
183C22⋊D61—84no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190C2×Dic61—41no—
191C2×Dic61#9912yes—
192C2×Dic61#10212yes—
193C2×Dic61#10412yes—
194C2×Dic61#10412yes—
195C2×Dic61#12411yescharacteristic
196C2×Dic61#12711yescharacteristic
197D244—82no—
198D244—82no—
199D244—82no—
200C4×D61—24no—
201C4×D61—24no—
202C4×D61—42no—
203C4×D61—42no—
204C4×D61—44no—
205#92420—82no—
206#9712—611nomaximal, Sylow
207C22×C61—81no—
208C22×C61—81no—
209C22×C61—81no—
210C22×C61—81no—
211C22×C61—81no—
212C22×C61—81no—
213C22×C61—81no—
214C22×C61#39011yescharacteristic
215C22×C61C2×#9911yescharacteristic
216C22×C61#41011yescharacteristic
217C22×C61C2×#10211yescharacteristic
218C22×C61#41811yescharacteristic
219C22×C61C2×#10411yescharacteristic
220C22×C61C2×#10411yescharacteristic
221C2×D61—22no—
222C2×D61—22no—
223C2×D61—21no—
224C2×D61—21no—
225C2×D61—21no—
226C2×D61—21no—
227C2×D61—22no—
228C2×D61—22no—
229C2×D61—81no—
230C2×D61—81no—
231C2×D61—81no—
232C2×D61—81no—
233C2×D61—81no—
234C2×D61—81no—
235C2×D61—81no—
236C2×D61#45912yes—
237C2×D61#46012yes—
238#6923—48no—
239C4×C61—22no—
240C4×C61—22no—
241C4×C61—41no—
242C4×C61—81no—
243Dic61—21no—
244Dic61—21no—
245Dic61—22no—
246Dic61—81no—
247Dic61#32714yescomplemented
248#2856—611no—
249#1797—611no—
250C2×#327—612no—
251C2×#327—612no—
252C4×#95—611no—
253C2×C61—161no—
254C2×C61C2×#32711yescharacteristic
255C2×C61#150211yescharacteristic
256C2×C61#150511yescharacteristic
257C2×C61#150711yescharacteristic
258C2×C61#196211yescharacteristic
259C2×C61#196411yescharacteristic
260C2×C61#198211yescharacteristic
261D61—22no—
262D61—22no—
263D61—161no—
264C2×#127—611no—
265C2×#121—611no—
266C2×#121—611no—
267C2×#121—1221no—
268C2×#121—1221no—
269C22×C42—611no—
270C22×C42—1221no—
271C2×C4⋊C8—1221no—
272C2×C22⋊C8—612no—
273#390—611no—
274#390—1221no—
275#390—1221no—
276#390#692311yescharacteristic, complemented
277#384—611no—
278C2×#95—612no—
279C2×#95—1221no—
280#336—1221no—
281#327—1224no—
282#327—1224no—
283C61#971211yescharacteristic, complemented, Sylow
284C22×D4—611no—
285C22×D4—1221no—
286C22×D4—1221no—
287C22×D4—1221no—
288C22×D4—1221no—
289C22×D4C2×#692311yescharacteristic
290C23×C4—611no—
291C23×C4—611no—
292C23×C4—611no—
293C23×C4—1222no—
294C22×C8—1222no—
295#127—2442no—
296#121—1222no—
297#121—1222no—
298#121—1222no—
299#121—1222no—
300#121—2442no—
301#121—2442no—
302#121—2444no—
303C2×C4⋊C4—611no—
304C2×C4⋊C4—1221no—
305C2×C4⋊C4—1221no—
306C2×C4⋊C4C2×#692311yescharacteristic
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—1221no—
310C2×C22⋊C4—1221no—
311C2×C22⋊C4—1221no—
312C2×C22⋊C4—1221no—
313C2×C42—611no—
314C2×C42—611no—
315C2×C42—611no—
316C2×C42—1224no—
317C2×C42—2442no—
318C2×C42—2442no—
319C2×C42—2442no—
320C2×C42C2×#692311yescharacteristic
321C4⋊C8—2442no—
322C22⋊C8—1222no—
323C22⋊C8—1222no—
324#95—614no—
325#95—1222no—
326#95—1224no—
327C24—21no—
328C24—611no—
329C24—1221no—
330C2×D4—22no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—1222no—
334C2×D4—2441no—
335C2×D4—2441no—
336C2×D4—2442no—
337C2×D4—2441no—
338C2×D4—2441no—
339C2×D4—2442no—
340C2×D4—2442no—
341C2×D4—2442no—
342C2×D4—2442no—
343C2×D4—2442no—
344C2×D4—2442no—
345C2×D4—2442no—
346C2×D4—2442no—
347C2×D4—2441no—
348C2×D4—2441no—
349C2×D4—2442no—
350C22×C4—21no—
351C22×C4—611no—
352C22×C4—611no—
353C22×C4—611no—
354C22×C4—612no—
355C22×C4—612no—
356C22×C4—612no—
357C22×C4—612no—
358C22×C4—1222no—
359C22×C4—1222no—
360C22×C4—1222no—
361C22×C4—1222no—
362C22×C4—1224no—
363C22×C4—1224no—
364C22×C4—1224no—
365C22×C4—1221no—
366C22×C4—2441no—
367C22×C4—2441no—
368C22×C4—2441no—
369C22×C4—2442no—
370C22×C4—2442no—
371C22×C4—2442no—
372C22×C4—2442no—
373C22×C4C22⋊#692311yescharacteristic
374C22×C4C22⋊#692311yescharacteristic
375C22×C4C22×#692311yescharacteristic
376C2×C8—2442no—
377C2×C8—2442no—
378C2×C8—2442no—
379C4⋊C4—22no—
380C4⋊C4—1222no—
381C4⋊C4—2442no—
382C4⋊C4—2442no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—2442no—
386C22⋊C4—2441no—
387C22⋊C4—2441no—
388C22⋊C4—2441no—
389C22⋊C4—2441no—
390C22⋊C4—2442no—
391C42—41no—
392C42—1222no—
393C42—1222no—
394C42—2441no—
395C42—2448no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—612no—
404C23—611no—
405C23—611no—
406C23—612no—
407C23—1221no—
408C23—1221no—
409C23—1221no—
410C23—1221no—
411C23—2441no—
412C23—2441no—
413C23—2441no—
414C23—2441no—
415C23—2441no—
416C23—2441no—
417C23—2441no—
418C23C2×C22⋊#692311yescharacteristic
419D4—82no—
420D4—4882no—
421D4—4882no—
422D4—4882no—
423D4—4884no—
424D4—4884no—
425D4—4882no—
426D4—4882no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—612no—
438C2×C4—612no—
439C2×C4—612no—
440C2×C4—611no—
441C2×C4—611no—
442C2×C4—612no—
443C2×C4—1224no—
444C2×C4—1224no—
445C2×C4—1221no—
446C2×C4—1221no—
447C2×C4—1221no—
448C2×C4—1221no—
449C2×C4—1228no—
450C2×C4—2442no—
451C2×C4—2442no—
452C2×C4—2444no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4—2444no—
457C2×C4—2444no—
458C2×C4—2444no—
459C2×C4—2444no—
460C2×C4—2444no—
461C2×C4—2444no—
462C2×C4#1451983912yes—
463C2×C4D4×#692312yes—
464C8—4882no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—612no—
473C22—612no—
474C22—1221no—
475C22—1221no—
476C22—1222no—
477C22—1222no—
478C22—1221no—
479C22—1221no—
480C22—1222no—
481C22—1222no—
482C22—4881no—
483C22—4881no—
484C22—4881no—
485C22—4881no—
486C22—4881no—
487C22—4881no—
488C22—4881no—
489C22#1684071411yescharacteristic
490C22D61⋊#9911yescharacteristic
491C22#1684074111yescharacteristic
492C22C2×#1451983911yescharacteristic
493C22#1684075411yescharacteristic
494C22D61⋊#10411yescharacteristic
495C22D61⋊#10411yescharacteristic
496C4—22no—
497C4—22no—
498C4—41no—
499C4—81no—
500C4—614no—
501C4—1221no—
502C4—1221no—
503C4—1222no—
504C4—4881no—
505C4—2448no—
506C2—161no—
507C2—1222no—
508C2—1222no—
509C2—9761no—
510C2D61⋊#32711yescharacteristic
511C2#2146119811yescharacteristic
512C2#2146120911yescharacteristic
513C2#2146121411yescharacteristic
514C2#2146195411yescharacteristic
515C2#2146196011yescharacteristic
516C2#2146198611yescharacteristic
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#9712
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. #21461676C2semidirect
2. C2×#16840614C2semidirect
3. C2×#16840613C2semidirect
4. C4×(C2×#6923)⋊C4C2semidirect
14. C2×#14519773C4semidirect
20. #16839664C4semidirect
23. #16839631C4semidirect
24. C61×#390C4semidirect
35. (C2×#6923)⋊C4D4semidirect
247. Dic61#327semidirect
276. #390#6923semidirect
283. C61#9712semidirect

Maximal subgroups

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

subgroupindex
#214616762
C2×#168406142
C2×#168406132
C4×(C2×#6923)⋊C42
#214541632
#971261