Finite Group Databasebeta

← #30598291

Subgroups — #30598291

Order 15616.

1126 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 1126 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
1#21461662C212yesmaximal, complemented
2C4×#14519835C211yesmaximal, characteristic, complemented
3#21461215C212yesmaximal, complemented
4#21461074C211yesmaximal, characteristic, complemented
5C2×#16840431C211yesmaximal, characteristic
6C2×#14519842—24no—
7C2×#14519835—21no—
8#16840733C2212yes—
9C4×C22⋊#6923C2212yes—
10#16840640—21no—
11#16840613—24no—
12#16840608—21no—
13#16840605—21no—
14C22×#11758226C2211yescharacteristic
15#16840431—24no—
16#16840431—24no—
17C2×#14519773—21no—
18C2×#14519773C411yescharacteristic, complemented
19C2×#14519751—21no—
20C2×#14519751C411yescharacteristic, complemented
21C2×#14519748—21no—
22C2×#14519748C2211yescharacteristic
23C2×Q8×Dic61C411yescharacteristic, complemented
24C2×C4×C22⋊D61—21no—
25C2×C4×C22⋊D61C2211yescharacteristic
26C22×#41×C61C411yescharacteristic, complemented
27C2×#11758213—22no—
28C2×C22⋊#6923—22no—
29C2×C4×#6923—22no—
30#14519842—48no—
31#14519835—24no—
32#14519834—22no—
33C22⋊#92420—22no—
34(C2×#6923)⋊C4—22no—
35C2×#11758226—22no—
36C2×#11758226—44no—
37C2×#11758226D414yes—
38C22×D244—21no—
39C22×D244C2×C411yescharacteristic
40C22×C22⋊D61—21no—
41C22×C22⋊D61—21no—
42C22×C22⋊D61—21no—
43C22×C22⋊D61—21no—
44C22×C22⋊D61D411yescharacteristic
45C22×C4×D61—21no—
46C22×C4×D61C2311yescharacteristic
47#14519773—28no—
48#14519773—22no—
49#14519773—22no—
50#14519751—28no—
51#14519751—22no—
52#14519751—22no—
53#14519748—28no—
54#14519748—22no—
55#14519748D414yes—
56Q8×Dic61—22no—
57Q8×Dic61—22no—
58C4×C22⋊D61—28no—
59C4×C22⋊D61—22no—
60C4×C22⋊D61—22no—
61C2×C4⋊Dic61—21no—
62C2×C4⋊Dic61C2×C411yescharacteristic
63C2×#11758196D411yescharacteristic
64C2×C22⋊Dic61—21no—
65C2×C22⋊Dic61D411yescharacteristic
66C2×#11758198—21no—
67C2×#11758198D411yescharacteristic
68C2×C4×Dic61—21no—
69C2×C4×Dic61C2×C411yescharacteristic
70C2×#41×C61—22no—
71C2×#41×C61—22no—
72C2×#41×C61—44no—
73C22×Q8×C61C2×C411yescharacteristic
74C22×D4×C61—21no—
75C22×D4×C61C2×C411yescharacteristic
76C23×C4×C61—21no—
77C23×C4×C61C2×C411yescharacteristic
78C24×C61—21no—
79C24×C61C22⋊C411yescharacteristic
80C23×D61—21no—
81C23×D61C2×D411yescharacteristic
82C22×#6923—42no—
83#41×C61—48no—
84#41×C61—44no—
85#41×C61—44no—
86C2×Q8×C61—24no—
87C2×Q8×C61—21no—
88C2×Q8×C61C22⋊C412yes—
89C2×D4×C61—24no—
90C2×D4×C61—21no—
91C2×D4×C61—44no—
92C2×D4×C61—42no—
93C2×D4×C61C22⋊C412yes—
94C22×C4×C61—21no—
95C22×C4×C61—24no—
96C22×C4×C61—21no—
97C22×C4×C61—21no—
98C22×C4×C61—44no—
99C22×C4×C61—42no—
100C22×C4×C61C22⋊C411yescharacteristic
101C22×C4×C61C22×C411yescharacteristic
102C2×C22⋊D61—24no—
103C2×C22⋊D61—22no—
104C2×C22⋊D61—24no—
105C2×C22⋊D61—24no—
106C2×C22⋊D61—24no—
107C2×C22⋊D61—22no—
108C2×C22⋊D61—41no—
109C2×C22⋊D61—42no—
110C2×C22⋊D61—42no—
111C2×C22⋊D61—82no—
112C2×C22⋊D61—82no—
113C2×C22⋊D61—82no—
114C22×Dic61—21no—
115C22×Dic61C22⋊C411yescharacteristic
116C22×Dic61C22⋊C411yescharacteristic
117C22×Dic61C2×D411yescharacteristic
118#11758226—28no—
119#11758226—44no—
120#11758226—48no—
121C2×D244—24no—
122C2×D244—21no—
123C2×D244—42no—
124C2×D244—44no—
125C2×D244C22⋊C412yes—
126C2×C4×D61—24no—
127C2×C4×D61—42no—
128C2×C4×D61—44no—
129C2×C4×D61C2×D412yes—
130C2×C4×D61C2×D412yes—
131C22⋊#6923—24no—
132C22⋊#6923—42no—
133C2×#92420—42no—
134C4×#6923—44no—
135#11758213—44no—
136C22⋊Dic61—24no—
137C22⋊Dic61—22no—
138#11758198—24no—
139#11758198—22no—
140C4⋊Dic61—24no—
141C4⋊Dic61—21no—
142C4⋊Dic61C22⋊C412yes—
143#11758196—22no—
144C4×Dic61—24no—
145C4×Dic61—21no—
146C4×Dic61—21no—
147C23×C61—21no—
148C23×C61—22no—
149C23×C61—21no—
150C23×C61—21no—
151C23×C61—42no—
152C23×C61—42no—
153C23×C61—41no—
154C23×C61—41no—
155C23×C61—41no—
156C23×C61—41no—
157C23×C61C2×C22⋊C411yescharacteristic
158C22×D61—21no—
159C22×D61—21no—
160C22×D61—22no—
161C22×D61—41no—
162C22×D61—41no—
163C22×D61—42no—
164C22×D61—42no—
165C22×D61—41no—
166C22×D61—41no—
167C22×D61#12012yes—
168C2×#6923—44no—
169C2×#6923—82no—
170C2×#6923—82no—
171Q8×C61—24no—
172Q8×C61—22no—
173Q8×C61#9914yescomplemented
174D4×C61—42no—
175D4×C61—48no—
176D4×C61—42no—
177C2×C4×C61—24no—
178C2×C4×C61—22no—
179C2×C4×C61—22no—
180C2×C4×C61—21no—
181C2×C4×C61—48no—
182C2×C4×C61—42no—
183C2×C4×C61—42no—
184C2×C4×C61—41no—
185C2×C4×C61C2×C22⋊C411yescharacteristic
186C2×C4×C61C2×C22⋊C411yescharacteristic
187C22⋊D61—44no—
188C22⋊D61—48no—
189C22⋊D61—84no—
190C22⋊D61—84no—
191C22⋊D61—84no—
192C2×Dic61—21no—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—22no—
196C2×Dic61—41no—
197C2×Dic61—42no—
198C2×Dic61—41no—
199C2×Dic61#10412yescomplemented
200C2×Dic61#12012yes—
201D244—24no—
202D244—48no—
203D244—42no—
204C4×D61—24no—
205C4×D61—42no—
206C4×D61—48no—
207#92420—82no—
208#8361—611nomaximal, Sylow
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—41no—
212C22×C61—42no—
213C22×C61—41no—
214C22×C61—42no—
215C22×C61—41no—
216C22×C61—42no—
217C22×C61—81no—
218C22×C61—81no—
219C22×C61—82no—
220C22×C61—82no—
221C22×C61#37911yescharacteristic
222C22×C61C2×#10411yescharacteristic
223C22×C61#42111yescharacteristic
224C2×D61—22no—
225C2×D61—22no—
226C2×D61—22no—
227C2×D61—42no—
228C2×D61—42no—
229C2×D61—42no—
230C2×D61—81no—
231C2×D61—81no—
232C2×D61—82no—
233C2×D61—82no—
234#6923—84no—
235C4×C61—24no—
236C4×C61—21no—
237C4×C61—41no—
238C4×C61C2×#9912yes—
239Dic61—22no—
240Dic61—22no—
241Dic61—42no—
242Dic61—41no—
243C2×#550—611no—
244#1789—612no—
245C4×#99—611no—
246#1507—612no—
247#1465—611no—
248C2×C61—21no—
249C2×C61—21no—
250C2×C61—81no—
251C2×C61—82no—
252C2×C61#178111yescharacteristic
253C2×C61#178911yescharacteristic
254C2×C61#179511yescharacteristic
255D61—42no—
256D61—82no—
257C22×#41—611no—
258C22×#41#692311yescharacteristic, complemented
259#550—1224no—
260#550—1224no—
261C2×#127—611no—
262C2×#127—1221no—
263C2×#121—611no—
264C2×#121—611no—
265C2×#121—1221no—
266C2×#121—1221no—
267C2×C4×Q8—611no—
268C2×C4×D4—611no—
269C2×C4×D4—1221no—
270C2×#104—1224no—
271C2×#99—1221no—
272#407—612no—
273C4×C22⋊C4—612no—
274#342—1221no—
275#327—1224no—
276#324—1221no—
277#323—1221no—
278C61#836111yescharacteristic, complemented, Sylow
279C2×#41—22no—
280C2×#41—22no—
281C2×#41—44no—
282C2×#41—614no—
283C2×#41—1222no—
284C2×#41—2444no—
285C22×Q8C2×#692311yescharacteristic
286C22×D4—21no—
287C22×D4—611no—
288C22×D4—611no—
289C22×D4—1221no—
290C22×D4—1221no—
291C22×D4—1221no—
292C22×D4—1221no—
293C22×D4—1221no—
294C22×D4C2×#692311yescharacteristic
295C23×C4—21no—
296C23×C4—611no—
297C23×C4—1221no—
298C23×C4C2×#692311yescharacteristic
299C2×M16—1222no—
300#127—1222no—
301#127—1228no—
302#127—1222no—
303#121—614no—
304#121—1222no—
305#121—1222no—
306#121—1228no—
307#121—1222no—
308#121—1228no—
309C4×Q8—1222no—
310C4×Q8—1222no—
311C4×D4—1222no—
312C4×D4—1222no—
313C4×D4—1228no—
314C2×C4⋊C4—611no—
315C2×C4⋊C4—611no—
316C2×C4⋊C4—1221no—
317C2×C22⋊C4—611no—
318C2×C22⋊C4—611no—
319C2×C22⋊C4—1221no—
320C2×C22⋊C4—1221no—
321C2×C22⋊C4—1222no—
322C2×C42—611no—
323C2×C42—1221no—
324C2×C42—1222no—
325#104—2448no—
326#99—1224no—
327C22⋊C8—1222no—
328C22⋊C8—1222no—
329#95—1222no—
330C24—21no—
331C24—611no—
332C24—1221no—
333C24C22⋊#692311yescharacteristic
334#41—48no—
335#41—44no—
336#41—44no—
337#41—1228no—
338#41—2444no—
339#41—2448no—
340C2×Q8—24no—
341C2×Q8—21no—
342C2×Q8C22⋊#692312yes—
343C2×D4—24no—
344C2×D4—21no—
345C2×D4—44no—
346C2×D4—42no—
347C2×D4—612no—
348C2×D4—1224no—
349C2×D4—1222no—
350C2×D4—1224no—
351C2×D4—1224no—
352C2×D4—1224no—
353C2×D4—1222no—
354C2×D4—1224no—
355C2×D4—1221no—
356C2×D4—4882no—
357C2×D4—4882no—
358C2×D4—4882no—
359C2×D4—2442no—
360C2×D4—2444no—
361C2×D4—2442no—
362C2×D4—2441no—
363C2×D4—2442no—
364C2×D4C22⋊#692312yes—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—44no—
370C22×C4—42no—
371C22×C4—611no—
372C22×C4—611no—
373C22×C4—612no—
374C22×C4—612no—
375C22×C4—611no—
376C22×C4—1224no—
377C22×C4—1221no—
378C22×C4—2442no—
379C22×C4—2444no—
380C22×C4—2442no—
381C22×C4C22⋊#692311yescharacteristic
382C22×C4C22×#692311yescharacteristic
383M16—2444no—
384C2×C8—2442no—
385C4⋊C4—612no—
386C4⋊C4—1222no—
387C4⋊C4—1224no—
388C4⋊C4—1221no—
389C22⋊C4—1222no—
390C22⋊C4—1224no—
391C22⋊C4—1222no—
392C22⋊C4—1224no—
393C22⋊C4—1224no—
394C22⋊C4—2442no—
395C42—1224no—
396C42—1221no—
397C42—1221no—
398C42—2444no—
399C23—21no—
400C23—22no—
401C23—21no—
402C23—21no—
403C23—42no—
404C23—42no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—612no—
410C23—1221no—
411C23—1221no—
412C23—1222no—
413C23—2441no—
414C23—2441no—
415C23—2442no—
416C23—2442no—
417C23—2441no—
418C23—2441no—
419C23C2×C22⋊#692311yescharacteristic
420Q8—24no—
421Q8—22no—
422Q8#1451983514yescomplemented
423D4—42no—
424D4—48no—
425D4—42no—
426D4—1224no—
427D4—4884no—
428D4—4884no—
429D4—4884no—
430D4—2448no—
431D4—2442no—
432D4—2444no—
433D4—2448no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—48no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—612no—
443C2×C4—612no—
444C2×C4—1221no—
445C2×C4—1221no—
446C2×C4—1224no—
447C2×C4—1222no—
448C2×C4—1221no—
449C2×C4—4882no—
450C2×C4—4882no—
451C2×C4—2442no—
452C2×C4—2448no—
453C2×C4—2441no—
454C2×C4—2444no—
455C2×C4—2442no—
456C2×C4—2441no—
457C2×C4C2×C22⋊#692311yescharacteristic
458C2×C4C2×C22⋊#692311yescharacteristic
459C8—4882no—
460C22—21no—
461C22—21no—
462C22—41no—
463C22—42no—
464C22—41no—
465C22—42no—
466C22—41no—
467C22—42no—
468C22—81no—
469C22—81no—
470C22—82no—
471C22—82no—
472C22—1222no—
473C22—1222no—
474C22—1222no—
475C22—4882no—
476C22—4882no—
477C22—4881no—
478C22—4881no—
479C22—2442no—
480C22—2442no—
481C22—2442no—
482C22#1684070311yescharacteristic
483C22D61⋊#10411yescharacteristic
484C22#1684075911yescharacteristic
485C4—24no—
486C4—21no—
487C4—41no—
488C4—1222no—
489C4—1222no—
490C4—4884no—
491C4—2442no—
492C4—2441no—
493C4C2×#1451983512yes—
494C2—21no—
495C2—21no—
496C2—82no—
497C2—81no—
498C2—4882no—
499C2—2442no—
500C2#2146164511yescharacteristic
501C2#2146166111yescharacteristic
502C2#2146167011yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×#41×C61C4
socleC22×C61#379
prime radicalC2×#16840431C2
cosocleC22×C4×D61C23
hypercenterC22×#41#6923
hypocenterC61#8361
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. #21461662C2semidirect
2. C4×#14519835C2semidirect
3. #21461215C2semidirect
4. #21461074C2semidirect
18. C2×#14519773C4semidirect
20. C2×#14519751C4semidirect
23. C2×Q8×Dic61C4semidirect
26. C22×#41×C61C4semidirect
173. Q8×C61#99semidirect
199. C2×Dic61#104semidirect
258. C22×#41#6923semidirect
278. C61#8361semidirect
422. Q8#14519835semidirect

Maximal subgroups

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

subgroupindex
#214616622
C4×#145198352
#214612152
#214610742
C2×#168404312
#836161