Finite Group Databasebeta

← #30600428

Subgroups — #30600428

Order 15616.

902 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 902 conjugacy classes of subgroups — too many to draw. Only its 106 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
1C2×#16840705C211yesmaximal, characteristic
2#21461642C211yesmaximal, characteristic
3C2×#16840615C212yesmaximal, complemented
4#21460692C212yesmaximal, complemented
5#21454107C211yesmaximal, characteristic
6C22×C22⋊#6923—21no—
7C2×#14519834—21no—
8C2×#14519834C2212yes—
9C2×C22⋊#92420—21no—
10#16840705—24no—
11C2×(C2×#6923)⋊C4—21no—
12C2×(C2×#6923)⋊C4C2212yes—
13#16840615—24no—
14C23×C4×D61C2211yescharacteristic
15C2×#14519757C411yescharacteristic, complemented
16C2×#14519757C2211yescharacteristic
17C2×#14519756C411yescharacteristic, complemented
18#16839637—21no—
19#16839636—21no—
20#16839636—21no—
21#16839619C411yescharacteristic, complemented
22#16839619C2211yescharacteristic
23#16839617—21no—
24C61×#380C411yescharacteristic, complemented
25C23×#6923—22no—
26C22×#92420—22no—
27C2×C22⋊#6923—24no—
28C2×C22⋊#6923—42no—
29C2×C22⋊#6923—42no—
30#14519834—22no—
31#14519834—22no—
32#14519834—42no—
33C22⋊#92420—42no—
34(C2×#6923)⋊C4—24no—
35(C2×#6923)⋊C4—22no—
36(C2×#6923)⋊C4D414yes—
37C24×D61D411yescharacteristic
38C22×Dic122C2×C411yescharacteristic
39C23×Dic61D411yescharacteristic
40C22×C4×D61—22no—
41C22×C4×D61—22no—
42C22×C4×D61—42no—
43C22×C4×D61—42no—
44C22×C4×D61C2×C411yescharacteristic
45C22×C4×D61D411yescharacteristic
46C22×C4×D61D411yescharacteristic
47C22×C4×D61C2311yescharacteristic
48#14519757—22no—
49#14519757—22no—
50#14519757—42no—
51#14519756—42no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2×C411yescharacteristic
55C2×#11758196—21no—
56C2×C22⋊Dic61—21no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61#14519684—21no—
62#14519684—21no—
63C23×C4×C61C2×C411yescharacteristic
64C2×C4⋊C4×C61C2×C411yescharacteristic
65C2×C4⋊C4×C61C2×C411yescharacteristic
66C2×C22⋊C4×C61—21no—
67#95×C61—21no—
68C24×C61C22⋊C411yescharacteristic
69C23×D61—22no—
70C23×D61—22no—
71C23×D61—21no—
72C23×D61—21no—
73C23×D61—22no—
74C23×D61—22no—
75C23×D61—41no—
76C23×D61—41no—
77C23×D61C22⋊C411yescharacteristic
78C23×D61C2×D411yescharacteristic
79C22×#6923—24no—
80C22×#6923—42no—
81C22×#6923—42no—
82C22×#6923—42no—
83C22×#6923—42no—
84C22×#6923—42no—
85C22×#6923—42no—
86C22×C4×C61—22no—
87C22×C4×C61—22no—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—41no—
91C22×C4×C61C22⋊C411yescharacteristic
92C22×C4×C61C22×C411yescharacteristic
93C22×Dic61—22no—
94C22×Dic61—22no—
95C22×Dic61—21no—
96C22×Dic61—21no—
97C22×Dic61—41no—
98C22×Dic61C22⋊C411yescharacteristic
99C22×Dic61C2×D411yescharacteristic
100C2×C4×D61—22no—
101C2×C4×D61—22no—
102C2×C4×D61—21no—
103C2×C4×D61—21no—
104C2×C4×D61—22no—
105C2×C4×D61—22no—
106C2×C4×D61—41no—
107C2×C4×D61—41no—
108C2×C4×D61—41no—
109C2×C4×D61—41no—
110C2×C4×D61—41no—
111C2×C4×D61—41no—
112C2×C4×D61—48no—
113C2×C4×D61—48no—
114C2×C4×D61C2×D412yes—
115C2×C4×D61#4112yes—
116C2×Dic122—22no—
117C2×Dic122—41no—
118C2×Dic122—41no—
119C22⋊#6923—48no—
120C2×#92420—42no—
121C2×#92420—42no—
122C2×#92420—42no—
123C4⋊C4×C61—22no—
124C4⋊C4×C61—41no—
125C22⋊C4×C61—42no—
126C22⋊Dic61—42no—
127#11758198—42no—
128#11758198—42no—
129#11758198—41no—
130#11758198—41no—
131#11758198—42no—
132C4⋊Dic61—42no—
133C4⋊Dic61—42no—
134C4⋊Dic61—41no—
135#11758196—42no—
136C23×C61—21no—
137C23×C61—21no—
138C23×C61—21no—
139C23×C61—21no—
140C23×C61—21no—
141C23×C61—21no—
142C23×C61#9912yes—
143C23×C61C2×C22⋊C411yescharacteristic
144C22×D61—21no—
145C22×D61—21no—
146C22×D61—21no—
147C22×D61—21no—
148C22×D61—24no—
149C22×D61—24no—
150C22×D61—21no—
151C22×D61—21no—
152C22×D61—21no—
153C22×D61—21no—
154C22×D61—21no—
155C22×D61—21no—
156C22×D61—44no—
157C22×D61—44no—
158C22×D61—44no—
159C22×D61—44no—
160C22×D61—42no—
161C22×D61—42no—
162C22×D61—42no—
163C22×D61—42no—
164C22×D61#9912yes—
165C22×D61#12011yescharacteristic
166C22×D61#12311yescharacteristic
167C22×D61#13512yes—
168C22×D61#13712yes—
169C2×#6923—44no—
170C2×#6923—44no—
171C2×#6923—44no—
172C2×#6923—82no—
173C2×#6923—82no—
174C2×#6923—82no—
175C2×#6923—82no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—44no—
182C2×C4×C61—41no—
183C2×C4×C61—41no—
184C2×C4×C61—41no—
185C2×C4×C61—41no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61C4×D412yes—
189C2×Dic61—21no—
190C2×Dic61—21no—
191C2×Dic61—21no—
192C2×Dic61—21no—
193C2×Dic61—21no—
194C2×Dic61—44no—
195C2×Dic61—41no—
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61#12112yes—
202C4×D61—42no—
203C4×D61—42no—
204C4×D61—44no—
205C4×D61—44no—
206C4×D61—42no—
207C4×D61—42no—
208Dic122—82no—
209#92420—82no—
210#9154—611nomaximal, Sylow
211C22×C61—22no—
212C22×C61—22no—
213C22×C61—22no—
214C22×C61—41no—
215C22×C61—41no—
216C22×C61—41no—
217C22×C61—41no—
218C22×C61#38611yescharacteristic
219C22×C61C2×#9911yescharacteristic
220C22×C61C2×#9911yescharacteristic
221C22×C61#41611yescharacteristic
222C22×C61#41911yescharacteristic
223C22×C61#42111yescharacteristic
224C22×C61#42111yescharacteristic
225C2×D61—21no—
226C2×D61—21no—
227C2×D61—21no—
228C2×D61—21no—
229C2×D61—22no—
230C2×D61—22no—
231C2×D61—22no—
232C2×D61—22no—
233C2×D61—22no—
234C2×D61—22no—
235C2×D61—22no—
236C2×D61—48no—
237C2×D61—48no—
238C2×D61—41no—
239C2×D61—41no—
240C2×D61—41no—
241C2×D61—41no—
242C2×D61#45212yes—
243C2×D61#48512yes—
244#6923—84no—
245C4×C61—22no—
246C4×C61—41no—
247C4×C61—81no—
248C4×C61—81no—
249Dic61—22no—
250Dic61—41no—
251Dic61—81no—
252Dic61—81no—
253#2848—611no—
254C2×#381—611no—
255#1779—611no—
256C2×#328—612no—
257#1294—612no—
258C2×C61—42no—
259C2×C61#151111yescharacteristic
260C2×C61#151111yescharacteristic
261C2×C61#151611yescharacteristic
262C2×C61#151611yescharacteristic
263C2×C61#187311yescharacteristic
264C2×C61#189711yescharacteristic
265C2×C61#189811yescharacteristic
266D61—22no—
267D61—22no—
268D61—42no—
269C24×C4—611no—
270C2×C22⋊Q8—611no—
271C2×C22⋊Q8—611no—
272C2×C22⋊Q8—611no—
273C22×C22⋊C4—1221no—
274C2×C22⋊C8—612no—
275C2×C22⋊C8—1221no—
276C2×C22⋊C8—1221no—
277#385—1221no—
278#385—1221no—
279#385—1221no—
280#381—1224no—
281#380—611no—
282#380—611no—
283#380#692311yescharacteristic, complemented
284#379—1221no—
285C2×#95—612no—
286C2×#95—1221no—
287#328—1224no—
288C61#915411yescharacteristic, complemented, Sylow
289C25—611no—
290C22×Q8—611no—
291C23×C4—611no—
292C23×C4—611no—
293C23×C4—611no—
294C23×C4—611no—
295C23×C4—611no—
296C23×C4—1222no—
297C23×C4—1222no—
298C23×C4—1222no—
299C23×C4—2442no—
300C23×C4—2442no—
301C23×C4C2×#692311yescharacteristic
302C22×C8—1222no—
303C22⋊Q8—1222no—
304C22⋊Q8—1222no—
305C22⋊Q8—2442no—
306C22⋊Q8—2442no—
307C2×C4⋊C4—611no—
308C2×C4⋊C4—1221no—
309C2×C4⋊C4—1221no—
310C2×C4⋊C4—1221no—
311C2×C4⋊C4C2×#692311yescharacteristic
312C2×C4⋊C4C2×#692311yescharacteristic
313C2×C22⋊C4—21no—
314C2×C22⋊C4—1221no—
315C2×C22⋊C4—1221no—
316C2×C22⋊C4—1221no—
317C2×C22⋊C4—1221no—
318C2×C22⋊C4—1224no—
319C2×C22⋊C4—1221no—
320C2×C22⋊C4—2442no—
321C2×C22⋊C4—2442no—
322C22⋊C8—1222no—
323C22⋊C8—1222no—
324C22⋊C8—2442no—
325C22⋊C8—2442no—
326#95—21no—
327#95—614no—
328#95—1221no—
329#95—1221no—
330#95—1222no—
331#95—1224no—
332C24—611no—
333C24—611no—
334C24—1221no—
335C24—1222no—
336C24—1221no—
337C24—1222no—
338C24—1222no—
339C24—1222no—
340C24—2441no—
341C24—2441no—
342C24C22⋊#692311yescharacteristic
343C2×Q8—1222no—
344C2×Q8—2441no—
345C2×Q8—2441no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—41no—
351C22×C4—611no—
352C22×C4—612no—
353C22×C4—612no—
354C22×C4—611no—
355C22×C4—1222no—
356C22×C4—1221no—
357C22×C4—1222no—
358C22×C4—1222no—
359C22×C4—1221no—
360C22×C4—1222no—
361C22×C4—1222no—
362C22×C4—1222no—
363C22×C4—1224no—
364C22×C4—1221no—
365C22×C4—1221no—
366C22×C4—2441no—
367C22×C4—2441no—
368C22×C4—2441no—
369C22×C4—2441no—
370C22×C4—2441no—
371C22×C4—2441no—
372C22×C4—2448no—
373C22×C4—2448no—
374C22×C4—2441no—
375C22×C4—2442no—
376C22×C4—2442no—
377C22×C4—2442no—
378C22×C4—2442no—
379C22×C4—2442no—
380C22×C4—2442no—
381C22×C4C22⋊#692311yescharacteristic
382C22×C4C22×#692311yescharacteristic
383C2×C8—2442no—
384C2×C8—2442no—
385C2×C8—2442no—
386C4⋊C4—22no—
387C4⋊C4—41no—
388C4⋊C4—2441no—
389C4⋊C4—2442no—
390C4⋊C4—2442no—
391C4⋊C4—2442no—
392C22⋊C4—42no—
393C22⋊C4—2442no—
394C22⋊C4—2441no—
395C22⋊C4—2441no—
396C22⋊C4—2442no—
397C22⋊C4—2442no—
398C22⋊C4—2448no—
399C22⋊C4—2442no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—611no—
407C23—611no—
408C23—612no—
409C23—612no—
410C23—612no—
411C23—1224no—
412C23—1224no—
413C23—1221no—
414C23—1221no—
415C23—1221no—
416C23—1221no—
417C23—1221no—
418C23—1221no—
419C23—1221no—
420C23—1221no—
421C23—1221no—
422C23—1221no—
423C23—2444no—
424C23—2444no—
425C23—2442no—
426C23—2442no—
427C23—2444no—
428C23—2444no—
429C23—2442no—
430C23—2442no—
431C23#1451983612yes—
432C23C2×C22⋊#692311yescharacteristic
433Q8—4882no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—44no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—612no—
447C2×C4—1221no—
448C2×C4—1221no—
449C2×C4—1221no—
450C2×C4—1221no—
451C2×C4—1221no—
452C2×C4—4882no—
453C2×C4—4882no—
454C2×C4—4882no—
455C2×C4—4882no—
456C2×C4—2442no—
457C2×C4—2442no—
458C2×C4—2444no—
459C2×C4—2444no—
460C2×C4—2442no—
461C2×C4—2442no—
462C2×C4—2444no—
463C2×C4—2444no—
464C2×C4—2444no—
465C2×C4—2444no—
466C2×C4—2441no—
467C2×C4—2441no—
468C2×C4—2441no—
469C2×C4—2441no—
470C2×C4—2441no—
471C2×C4—2441no—
472C2×C4D4×#692312yes—
473C8—4882no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—612no—
482C22—612no—
483C22—1221no—
484C22—1221no—
485C22—1221no—
486C22—1221no—
487C22—1222no—
488C22—1222no—
489C22—1222no—
490C22—1222no—
491C22—1222no—
492C22—1222no—
493C22—1222no—
494C22—2448no—
495C22—2448no—
496C22—2441no—
497C22—2441no—
498C22—2441no—
499C22—2441no—
500C22#1684071011yescharacteristic
501C22D61⋊#9911yescharacteristic
502C22C2×#1451983611yescharacteristic
503C22#1684075211yescharacteristic
504C22#1684075511yescharacteristic
505C22#1684075911yescharacteristic
506C22#1684076011yescharacteristic
507C4—22no—
508C4—41no—
509C4—81no—
510C4—81no—
511C4—1222no—
512C4—4884no—
513C4—4881no—
514C4—4881no—
515C4—2441no—
516C2—42no—
517C2—1222no—
518C2—1222no—
519C2—2442no—
520C2#2146122711yescharacteristic
521C2#2146122811yescharacteristic
522C2#2146124411yescharacteristic
523C2#2146124711yescharacteristic
524C2#2146180511yescharacteristic
525C2#2146183911yescharacteristic
526C2#2146184111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#380C4
socleC23×C61C2×C22⋊C4
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenter#380#6923
hypocenterC61#9154
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. C2×#16840615C2semidirect
4. #21460692C2semidirect
15. C2×#14519757C4semidirect
17. C2×#14519756C4semidirect
21. #16839619C4semidirect
24. C61×#380C4semidirect
283. #380#6923semidirect
288. C61#9154semidirect

Maximal subgroups

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

subgroupindex
C2×#168407052
#214616422
C2×#168406152
#214606922
#214541072
#915461