Finite Group Databasebeta

← #30598214

Subgroups — #30598214

Order 15616.

1276 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1276 conjugacy classes of subgroups — too many to draw. Only its 146 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#21461642C211yesmaximal, characteristic
2C2×#16840615C214yesmaximal, complemented
3#21460993C211yesmaximal, characteristic
4C22×#14519757C211yesmaximal, characteristic
5C22×C22⋊#6923—21no—
6C2×#14519834—21no—
7C2×#14519834C2212yes—
8C2×C22⋊#92420—21no—
9C2×(C2×#6923)⋊C4—21no—
10C2×(C2×#6923)⋊C4C2212yes—
11#16840615—28no—
12C23×Dic122C411yescharacteristic, complemented
13C23×C4×D61C2211yescharacteristic
14C2×#14519757—28no—
15C2×#14519757—21no—
16C2×#14519757—21no—
17C2×#14519757C412yescomplemented
18C2×#14519757C2212yes—
19C22×C4⋊Dic61—21no—
20C22×#11758198—21no—
21C22×C4⋊C4×C61C411yescharacteristic, complemented
22C23×#6923—22no—
23C22×#92420—22no—
24C2×C22⋊#6923—24no—
25C2×C22⋊#6923—42no—
26C2×C22⋊#6923—42no—
27#14519834—24no—
28#14519834—42no—
29C22⋊#92420—42no—
30(C2×#6923)⋊C4—24no—
31(C2×#6923)⋊C4—42no—
32C24×D61D411yescharacteristic
33C22×Dic122—21no—
34C22×Dic122—24no—
35C22×Dic122—42no—
36C22×Dic122—42no—
37C22×Dic122C2×C412yes—
38C23×Dic61—21no—
39C23×Dic61D411yescharacteristic
40C22×C4×D61—24no—
41C22×C4×D61—42no—
42C22×C4×D61—42no—
43C22×C4×D61C2×C411yescharacteristic
44C22×C4×D61D411yescharacteristic
45C22×C4×D61D411yescharacteristic
46C22×C4×D61C2311yescharacteristic
47#14519757—28no—
48#14519757—28no—
49#14519757—24no—
50#14519757—24no—
51#14519757—24no—
52#14519757D418yes—
53C2×C4⋊Dic61—28no—
54C2×C4⋊Dic61—22no—
55C2×C4⋊Dic61—22no—
56C2×#11758198—28no—
57C2×#11758198—22no—
58C2×#11758198—22no—
59C23×C4×C61—21no—
60C23×C4×C61C2×C411yescharacteristic
61C2×C4⋊C4×C61—21no—
62C2×C4⋊C4×C61—24no—
63C2×C4⋊C4×C61C2×C412yes—
64C24×C61C22⋊C411yescharacteristic
65C23×D61—22no—
66C23×D61—22no—
67C23×D61—22no—
68C23×D61—22no—
69C23×D61—41no—
70C23×D61—41no—
71C23×D61C22⋊C411yescharacteristic
72C23×D61SD1612yes—
73C23×D61Q1612yes—
74C23×D61C2×D411yescharacteristic
75C22×#6923—24no—
76C22×#6923—42no—
77C22×#6923—42no—
78C22×#6923—42no—
79C22×#6923—42no—
80C22×#6923—42no—
81C22×#6923—42no—
82C22×C4×C61—22no—
83C22×C4×C61—22no—
84C22×C4×C61—22no—
85C22×C4×C61—24no—
86C22×C4×C61—22no—
87C22×C4×C61—42no—
88C22×C4×C61—42no—
89C22×C4×C61C22⋊C411yescharacteristic
90C22×C4×C61C22×C411yescharacteristic
91C22×Dic61—24no—
92C22×Dic61—22no—
93C22×Dic61—22no—
94C22×Dic61—22no—
95C22×Dic61—41no—
96C22×Dic61—42no—
97C22×Dic61—42no—
98C22×Dic61C22⋊C411yescharacteristic
99C22×Dic61C2×D411yescharacteristic
100C2×C4×D61—22no—
101C2×C4×D61—22no—
102C2×C4×D61—22no—
103C2×C4×D61—22no—
104C2×C4×D61—22no—
105C2×C4×D61—41no—
106C2×C4×D61—41no—
107C2×C4×D61—41no—
108C2×C4×D61—41no—
109C2×C4×D61—41no—
110C2×C4×D61—41no—
111C2×C4×D61—416no—
112C2×C4×D61C2×D414yes—
113C2×Dic122—24no—
114C2×Dic122—22no—
115C2×Dic122—42no—
116C2×Dic122—416no—
117C2×Dic122—42no—
118C2×Dic122—42no—
119C2×Dic122—42no—
120C2×Dic122C22⋊C414yescomplemented
121C22⋊#6923—48no—
122C2×#92420—42no—
123C2×#92420—42no—
124C2×#92420—42no—
125C4⋊C4×C61—24no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61C22⋊C414yescomplemented
128#11758198—28no—
129#11758198—28no—
130C4⋊Dic61—28no—
131C4⋊Dic61—28no—
132C23×C61—21no—
133C23×C61—21no—
134C23×C61—21no—
135C23×C61—21no—
136C23×C61#9912yes—
137C23×C61#10312yes—
138C23×C61#10412yes—
139C23×C61C2×C22⋊C411yescharacteristic
140C22×D61—21no—
141C22×D61—21no—
142C22×D61—21no—
143C22×D61—21no—
144C22×D61—24no—
145C22×D61—24no—
146C22×D61—21no—
147C22×D61—21no—
148C22×D61—21no—
149C22×D61—21no—
150C22×D61—21no—
151C22×D61—21no—
152C22×D61—24no—
153C22×D61—24no—
154C22×D61—42no—
155C22×D61—42no—
156C22×D61—44no—
157C22×D61—44no—
158C22×D61—44no—
159C22×D61—44no—
160C22×D61#10312yes—
161C22×D61#12012yes—
162C22×D61C2×SD1611yescharacteristic
163C22×D61C2×Q1611yescharacteristic
164C22×D61#13712yes—
165C2×#6923—44no—
166C2×#6923—44no—
167C2×#6923—44no—
168C2×#6923—82no—
169C2×#6923—82no—
170C2×#6923—82no—
171C2×#6923—82no—
172C2×C4×C61—28no—
173C2×C4×C61—21no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—21no—
177C2×C4×C61—24no—
178C2×C4×C61—48no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61#9912yes—
182C2×C4×C61C2×C22⋊C412yes—
183C2×Dic61—21no—
184C2×Dic61—21no—
185C2×Dic61—21no—
186C2×Dic61—24no—
187C2×Dic61—21no—
188C2×Dic61—21no—
189C2×Dic61—44no—
190C2×Dic61—48no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61#12012yes—
194C4×D61—44no—
195C4×D61—44no—
196C4×D61—44no—
197C4×D61—44no—
198Dic122—48no—
199Dic122—44no—
200Dic122—44no—
201#92420—82no—
202#8329—611nomaximal, Sylow
203C22×C61—22no—
204C22×C61—22no—
205C22×C61—22no—
206C22×C61—22no—
207C22×C61—22no—
208C22×C61—22no—
209C22×C61#32814yescomplemented
210C22×C61#37911yescharacteristic
211C22×C61C2×#9911yescharacteristic
212C22×C61C2×#9911yescharacteristic
213C22×C61C2×#10311yescharacteristic
214C22×C61#41911yescharacteristic
215C22×C61C2×#10411yescharacteristic
216C22×C61#42111yescharacteristic
217C2×D61—21no—
218C2×D61—21no—
219C2×D61—22no—
220C2×D61—22no—
221C2×D61—21no—
222C2×D61—21no—
223C2×D61—22no—
224C2×D61—22no—
225C2×D61—22no—
226C2×D61—22no—
227C2×D61—22no—
228C2×D61—48no—
229C2×D61—48no—
230C2×D61—41no—
231C2×D61—41no—
232C2×D61—41no—
233C2×D61—41no—
234C2×D61#44812yes—
235C2×D61#45112yes—
236#6923—84no—
237C4×C61—22no—
238C4×C61—22no—
239C4×C61—44no—
240Dic61—22no—
241Dic61—44no—
242Dic61—41no—
243C22×C22⋊Q8—611no—
244#1779—611no—
245C2×#328—614no—
246C22⋊#95—611no—
247C2×C61—24no—
248C2×C61C2×#32811yescharacteristic
249C2×C61#150811yescharacteristic
250C2×C61#151111yescharacteristic
251C2×C61#152011yescharacteristic
252C2×C61#178111yescharacteristic
253C2×C61#178711yescharacteristic
254C2×C61#178911yescharacteristic
255D61—22no—
256D61—22no—
257D61—42no—
258C23×Q8—611no—
259C24×C4—611no—
260C2×C22⋊Q8—612no—
261C2×C22⋊Q8—612no—
262C2×C22⋊Q8—1228no—
263C2×C22⋊Q8—1221no—
264C2×C22⋊Q8—1221no—
265C22×C4⋊C4—1221no—
266C22×C4⋊C4#692311yescharacteristic, complemented
267C22×C22⋊C4—1221no—
268C22×C22⋊C4—1221no—
269C2×C22⋊C8—612no—
270C2×C22⋊C8—1221no—
271C2×C22⋊C8—1221no—
272C2×#95—612no—
273C2×#95—1221no—
274#328—1228no—
275C61#832911yescharacteristic, complemented, Sylow
276C25—611no—
277C22×Q8—612no—
278C22×Q8—1224no—
279C22×Q8—1221no—
280C22×Q8—2442no—
281C22×Q8—2442no—
282C23×C4—21no—
283C23×C4—611no—
284C23×C4—611no—
285C23×C4—611no—
286C23×C4—611no—
287C23×C4—611no—
288C23×C4—1224no—
289C23×C4—1221no—
290C23×C4—1222no—
291C23×C4—2442no—
292C23×C4—2442no—
293C23×C4C2×#692311yescharacteristic
294C22×C8—1222no—
295C22⋊Q8—618no—
296C22⋊Q8—1224no—
297C22⋊Q8—1224no—
298C22⋊Q8—1228no—
299C22⋊Q8—1228no—
300C22⋊Q8—1224no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—1222no—
304C2×C4⋊C4—1222no—
305C2×C4⋊C4—1228no—
306C2×C4⋊C4C2×#692312yes—
307C2×C22⋊C4—1222no—
308C2×C22⋊C4—1222no—
309C2×C22⋊C4—1228no—
310C2×C22⋊C4—1224no—
311C2×C22⋊C4—2442no—
312C2×C22⋊C4—2442no—
313C22⋊C8—1224no—
314C22⋊C8—2442no—
315C22⋊C8—2442no—
316#95—1224no—
317#95—2442no—
318C24—611no—
319C24—611no—
320C24—612no—
321C24—612no—
322C24—1222no—
323C24—1222no—
324C24—1222no—
325C24—1222no—
326C24—2441no—
327C24—2441no—
328C24C22⋊#692311yescharacteristic
329C2×Q8—614no—
330C2×Q8—1224no—
331C2×Q8—1222no—
332C2×Q8—24416no—
333C2×Q8—2442no—
334C2×Q8—2442no—
335C2×Q8—2442no—
336C2×Q8—2442no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—611no—
345C22×C4—614no—
346C22×C4—611no—
347C22×C4—1222no—
348C22×C4—1222no—
349C22×C4—1222no—
350C22×C4—1224no—
351C22×C4—1222no—
352C22×C4—1222no—
353C22×C4—1222no—
354C22×C4—1222no—
355C22×C4—1222no—
356C22×C4—1224no—
357C22×C4—2441no—
358C22×C4—2441no—
359C22×C4—2441no—
360C22×C4—2441no—
361C22×C4—2441no—
362C22×C4—2441no—
363C22×C4—24416no—
364C22×C4—2441no—
365C22×C4—2442no—
366C22×C4—2442no—
367C22×C4—2442no—
368C22×C4—2442no—
369C22×C4—2442no—
370C22×C4—2442no—
371C22×C4—2442no—
372C22×C4—2442no—
373C22×C4C22⋊#692311yescharacteristic
374C22×C4C22×#692311yescharacteristic
375C2×C8—2442no—
376C2×C8—2442no—
377C2×C8—2442no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—1228no—
381C4⋊C4—1228no—
382C4⋊C4C22⋊#692314yescomplemented
383C22⋊C4—1228no—
384C22⋊C4—1228no—
385C22⋊C4—2448no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—611no—
391C23—611no—
392C23—612no—
393C23—612no—
394C23—612no—
395C23—1224no—
396C23—1224no—
397C23—1221no—
398C23—1221no—
399C23—1221no—
400C23—1221no—
401C23—1221no—
402C23—1221no—
403C23—1221no—
404C23—1221no—
405C23—1221no—
406C23—1224no—
407C23—1224no—
408C23—1221no—
409C23—2442no—
410C23—2442no—
411C23—2444no—
412C23—2444no—
413C23—2444no—
414C23—2444no—
415C23#1451983612yes—
416C23#1451984012yes—
417C23#1451984112yes—
418C23C2×C22⋊#692311yescharacteristic
419Q8—2444no—
420Q8—2448no—
421Q8—2444no—
422C2×C4—28no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—48no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—612no—
432C2×C4—1221no—
433C2×C4—1221no—
434C2×C4—1221no—
435C2×C4—1221no—
436C2×C4—1224no—
437C2×C4—1221no—
438C2×C4—4882no—
439C2×C4—4882no—
440C2×C4—4882no—
441C2×C4—4882no—
442C2×C4—2444no—
443C2×C4—2444no—
444C2×C4—2444no—
445C2×C4—2444no—
446C2×C4—2444no—
447C2×C4—2444no—
448C2×C4—2444no—
449C2×C4—2444no—
450C2×C4—2448no—
451C2×C4—2442no—
452C2×C4—2442no—
453C2×C4#1451983612yes—
454C2×C4C2×C22⋊#692312yes—
455C8—4882no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—612no—
463C22—612no—
464C22—1221no—
465C22—1221no—
466C22—1222no—
467C22—1222no—
468C22—1221no—
469C22—1221no—
470C22—1222no—
471C22—1222no—
472C22—1222no—
473C22—1222no—
474C22—1222no—
475C22—2448no—
476C22—2448no—
477C22—2441no—
478C22—2441no—
479C22—2441no—
480C22—2441no—
481C22#1684061514yescomplemented
482C22#1684070311yescharacteristic
483C22C2×#1451983611yescharacteristic
484C22C2×#1451983611yescharacteristic
485C22C2×#1451984011yescharacteristic
486C22#1684075511yescharacteristic
487C22C2×#1451984111yescharacteristic
488C22#1684076011yescharacteristic
489C4—22no—
490C4—22no—
491C4—44no—
492C4—1222no—
493C4—4884no—
494C4—2441no—
495C4—2444no—
496C2—24no—
497C2—1222no—
498C2—1222no—
499C2—2442no—
500C2C2×#1684061511yescharacteristic
501C2#2146121811yescharacteristic
502C2#2146122711yescharacteristic
503C2#2146125811yescharacteristic
504C2#2146164711yescharacteristic
505C2#2146165811yescharacteristic
506C2#2146166011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×C4⋊C4×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×D61D4
cosocleC22×C4×D61C23
hypercenterC22×C4⋊C4#6923
hypocenterC61#8329
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16840615C2semidirect
12. C23×Dic122C4semidirect
17. C2×#14519757C4semidirect
21. C22×C4⋊C4×C61C4semidirect
120. C2×Dic122C22⋊C4semidirect
127. C4⋊C4×C61C22⋊C4semidirect
209. C22×C61#328semidirect
266. C22×C4⋊C4#6923semidirect
275. C61#8329semidirect
382. C4⋊C4C22⋊#6923semidirect
481. C22#16840615semidirect

Maximal subgroups

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

subgroupindex
#214616422
C2×#168406152
#214609932
C22×#145197572
#832961