Finite Group Databasebeta

← #29580290

Subgroups — #29580290

Order 15616.

734 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21451626C211yesmaximal, characteristic
2C2×#16839688C211yesmaximal, characteristic
3#21445905C211yesmaximal, characteristic, complemented
4C2×#16839499C211yesmaximal, characteristic, complemented
5C2×#16839485C211yesmaximal, characteristic
6#21443699C211yesmaximal, characteristic, complemented
7C2×C61×#483C211yesmaximal, characteristic, complemented
8C22×#11758198—21no—
9C22×#11758198C2211yescharacteristic
10C2×C22⋊#92418C2211yescharacteristic
11#16839697—21no—
12#16839688—22no—
13#16839688—22no—
14#16839684—22no—
15#16839673—21no—
16#16839634—21no—
17#16839622C2211yescharacteristic
18#16839617—21no—
19C2×#14519684—21no—
20C2×#14519684C2211yescharacteristic
21#16839524—22no—
22#16839499—22no—
23#16839485—22no—
24C2×C22⋊Q8×C61C2211yescharacteristic
25C22×C4⋊C4×C61C2211yescharacteristic
26C61×#483—22no—
27C61×#483—22no—
28C2×#106×C61—21no—
29C2×#103×C61—21no—
30C2×C22⋊C8×C61C2211yescharacteristic
31C24×D61D411yescharacteristic
32C23×Dic61—21no—
33C23×Dic61D411yescharacteristic
34C22×#92418—21no—
35C2×C4⋊Dic61D412yes—
36C2×#11758196—22no—
37C2×C22⋊Dic61—21no—
38C2×#11758198—24no—
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198—21no—
42C2×#11758198—21no—
43C2×#11758198—21no—
44C2×#11758198—21no—
45C2×#11758198—22no—
46C2×#11758198—22no—
47C2×#11758198—42no—
48C2×#11758198—42no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51#14519684—22no—
52#14519684—24no—
53#14519684—21no—
54#14519684—21no—
55#14519684D412yes—
56C22×Q8×C61D411yescharacteristic
57C23×C4×C61—21no—
58C23×C4×C61C2311yescharacteristic
59C22×C8×C61—21no—
60C22⋊Q8×C61—21no—
61C22⋊Q8×C61—21no—
62C22⋊Q8×C61—21no—
63C22⋊Q8×C61—21no—
64C2×C4⋊C4×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61D412yes—
71C2×C4⋊C4×C61D411yescharacteristic
72C2×C22⋊C4×C61—21no—
73#106×C61—24no—
74#103×C61—24no—
75C22⋊C8×C61—21no—
76C22⋊C8×C61D412yes—
77C24×C61C2×D411yescharacteristic
78C23×D61—21no—
79C23×D61—21no—
80C23×D61—21no—
81C23×D61—21no—
82C23×D61—21no—
83C23×D61—21no—
84C23×D61—42no—
85C23×D61—42no—
86C23×D61SD1612yes—
87C2×Q8×C61—21no—
88C2×Q8×C61—21no—
89C2×Q8×C61—42no—
90C22×C4×C61—21no—
91C22×C4×C61—21no—
92C22×C4×C61—22no—
93C22×C4×C61—42no—
94C22×C4×C61—41no—
95C22×C4×C61—41no—
96C22×C4×C61—41no—
97C22×C4×C61—41no—
98C22×C4×C61—41no—
99C22×C4×C61—41no—
100C22×C4×C61C2×D411yescharacteristic
101C22×C4×C61C2×D411yescharacteristic
102C22×C4×C61C2×D411yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61#4111yescharacteristic
106C22×Dic61—22no—
107C22×Dic61—21no—
108C22×Dic61—21no—
109C22×Dic61—22no—
110C22×Dic61—21no—
111C22×Dic61—42no—
112C22×Dic61—42no—
113C22×Dic61—42no—
114C22×Dic61—41no—
115C22×Dic61—41no—
116C2×C8×C61—22no—
117C2×C8×C61—42no—
118C4⋊C4×C61—22no—
119C4⋊C4×C61—21no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—21no—
122C4⋊C4×C61—41no—
123C4⋊C4×C61—41no—
124C4⋊C4×C61—41no—
125C4⋊C4×C61—41no—
126C22⋊C4×C61—41no—
127C22⋊C4×C61—41no—
128C22⋊Dic61—41no—
129C22⋊Dic61—41no—
130#11758198—48no—
131#11758198—41no—
132#11758198—41no—
133#11758198—41no—
134#11758198—41no—
135#11758198—82no—
136#11758198—82no—
137C4⋊Dic61—42no—
138#11758196—82no—
139C2×#92418—42no—
140C2×#92418—41no—
141C23×C61—22no—
142C23×C61—22no—
143C23×C61#12011yescharacteristic
144C23×C61#12011yescharacteristic
145C23×C61#12111yescharacteristic
146C23×C61C2×SD1611yescharacteristic
147C23×C61#13511yescharacteristic
148C23×C61AGL(1,ℤ/8)11yescharacteristic
149C23×C61#13711yescharacteristic
150C22×D61—22no—
151C22×D61—22no—
152C22×D61—22no—
153C22×D61—44no—
154C22×D61—44no—
155C22×D61—42no—
156C22×D61—42no—
157C22×D61—41no—
158C22×D61—41no—
159C22×D61—41no—
160C22×D61—42no—
161C22×D61—42no—
162C22×D61—41no—
163C22×D61—82no—
164C22×D61—82no—
165C22×D61—81no—
166C22×D61—81no—
167Q8×C61—42no—
168Q8×C61—81no—
169C2×C4×C61—22no—
170C2×C4×C61—22no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61—41no—
174C2×C4×C61—41no—
175C2×C4×C61—41no—
176C2×C4×C61—41no—
177C2×C4×C61—42no—
178C2×C4×C61—42no—
179C2×C4×C61—42no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
182C2×C4×C61—81no—
183C2×C4×C61—81no—
184C2×C4×C61—81no—
185C2×C4×C61—81no—
186C2×C4×C61#12011yescharacteristic
187C2×C4×C61#12111yescharacteristic
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#12711yescharacteristic
190C2×Dic61—42no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61—41no—
194C2×Dic61—42no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61—42no—
198C2×Dic61—42no—
199C2×Dic61—82no—
200C2×Dic61—82no—
201C2×Dic61—81no—
202C8×C61—42no—
203#92418—81no—
204#9926—611nomaximal, Sylow
205C22×C61—22no—
206C22×C61—22no—
207C22×C61—22no—
208C22×C61—22no—
209C22×C61—21no—
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—21no—
213C22×C61#39211yescharacteristic
214C22×C61#44811yescharacteristic
215C22×C61#44911yescharacteristic
216C22×C61#45011yescharacteristic
217C22×C61#45211yescharacteristic
218C22×C61#45512yes—
219C22×C61#45812yes—
220C22×C61#46511yescharacteristic
221C22×C61#46811yescharacteristic
222C2×D61—44no—
223C2×D61—44no—
224C2×D61—42no—
225C2×D61—82no—
226C2×D61—82no—
227C2×D61—82no—
228C2×D61—81no—
229C2×D61—81no—
230C2×D61—82no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—21no—
234C4×C61—21no—
235C4×C61—82no—
236C4×C61—81no—
237C4×C61—81no—
238Dic61—82no—
239Dic61—82no—
240Dic61—81no—
241C2×#483D6111yescharacteristic, complemented
242#2626—611no—
243C2×#397—611no—
244#1823—611no—
245C2×#328—611no—
246C2×#323—611no—
247#1294—611no—
248C2×C61—21no—
249C2×C61—21no—
250C2×C61—21no—
251C2×C61—21no—
252C2×C61#160812yes—
253C2×C61#161212yes—
254C2×C61#200011yescharacteristic
255C2×C61#200411yescharacteristic
256C2×C61#202211yescharacteristic
257D61—82no—
258C2×C22⋊Q8C2×D6111yescharacteristic
259C22×C4⋊C4C2×D6111yescharacteristic
260C22×C22⋊C4—611no—
261C22×C22⋊C4—1221no—
262#483—22no—
263#483—22no—
264C2×#106—21no—
265C2×#103—21no—
266C2×C22⋊C8—611no—
267C2×C22⋊C8C2×D6111yescharacteristic
268#399—1221no—
269#397—1222no—
270#397—1222no—
271#396—1222no—
272#393—1221no—
273#385—1221no—
274#380—611no—
275#379—1221no—
276C2×#95—611no—
277C2×#95—1221no—
278#340—1222no—
279#328—1222no—
280#323—1222no—
281C61#992611yescharacteristic, complemented, Sylow
282C25—611no—
283C22×Q8C22⋊D6111yescharacteristic
284C23×C4—21no—
285C23×C4—611no—
286C23×C4—1221no—
287C23×C4C22×D6111yescharacteristic
288C22×C8—21no—
289C22×C8—1221no—
290C22⋊Q8—21no—
291C22⋊Q8—21no—
292C22⋊Q8—21no—
293C22⋊Q8—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—612no—
301C2×C4⋊C4—1222no—
302C2×C4⋊C4C22⋊D6112yes—
303C2×C4⋊C4C22⋊D6111yescharacteristic
304C2×C22⋊C4—21no—
305C2×C22⋊C4—1222no—
306C2×C22⋊C4—1222no—
307C2×C22⋊C4—1224no—
308C2×C22⋊C4—1221no—
309C2×C22⋊C4—1221no—
310C2×C22⋊C4—1221no—
311C2×C22⋊C4—1221no—
312C2×C22⋊C4—1221no—
313C2×C22⋊C4—1221no—
314C2×C22⋊C4—1221no—
315C2×C22⋊C4—2442no—
316C2×C22⋊C4—2442no—
317#106—24no—
318#103—24no—
319C22⋊C8—21no—
320C22⋊C8—1221no—
321C22⋊C8—1221no—
322C22⋊C8C22⋊D6112yes—
323#95—612no—
324#95—1224no—
325#95—1222no—
326#95—1221no—
327#95—1221no—
328C24—612no—
329C24—1221no—
330C24—1221no—
331C24—1221no—
332C24—1221no—
333C24—1221no—
334C24—1221no—
335C24—2442no—
336C24—2442no—
337C24D4×D6111yescharacteristic
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—42no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—42no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—1221no—
352C22×C4—1222no—
353C22×C4—1221no—
354C22×C4—1222no—
355C22×C4—1221no—
356C22×C4—2442no—
357C22×C4—2442no—
358C22×C4—2441no—
359C22×C4—2441no—
360C22×C4—2442no—
361C22×C4D4×D6111yescharacteristic
362C22×C4D4×D6111yescharacteristic
363C22×C4#1175822411yescharacteristic
364C22×C4C2×C22⋊D6111yescharacteristic
365C22×C4C2×C22⋊D6111yescharacteristic
366C22×C4C2×C22⋊D6111yescharacteristic
367C2×C8—22no—
368C2×C8—42no—
369C2×C8—2442no—
370C2×C8—2441no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—22no—
374C4⋊C4—21no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—4882no—
380C4⋊C4—2442no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—4882no—
384C22⋊C4—4882no—
385C22⋊C4—2448no—
386C22⋊C4—2441no—
387C22⋊C4—2441no—
388C22⋊C4—2441no—
389C22⋊C4—2441no—
390C22⋊C4—2441no—
391C22⋊C4—2441no—
392C23—22no—
393C23—22no—
394C23—1222no—
395C23—1222no—
396C23—1222no—
397C23—4881no—
398C23—4881no—
399C23—4882no—
400C23—4882no—
401C23—2442no—
402C23—2442no—
403C23—2444no—
404C23—2444no—
405C23—2442no—
406C23—2442no—
407C23—2441no—
408C23—2441no—
409C23—2441no—
410C23—2441no—
411C23#1451974511yescharacteristic
412C23#1451974511yescharacteristic
413C23#1451974811yescharacteristic
414C23SD16×D6111yescharacteristic
415C23#1451979811yescharacteristic
416C23#1451980411yescharacteristic
417C23#1451980611yescharacteristic
418Q8—42no—
419Q8—81no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—4882no—
438C2×C4—4882no—
439C2×C4—4881no—
440C2×C4—2442no—
441C2×C4—2442no—
442C2×C4—2442no—
443C2×C4—2442no—
444C2×C4—2442no—
445C2×C4—2442no—
446C2×C4—2442no—
447C2×C4—2442no—
448C2×C4—2441no—
449C2×C4#1451974611yescharacteristic
450C2×C4#1451974911yescharacteristic
451C2×C4#1451974911yescharacteristic
452C2×C4#1451977311yescharacteristic
453C8—42no—
454C8—4881no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—4882no—
464C22—4882no—
465C22—4882no—
466C22—4881no—
467C22—4881no—
468C22—4882no—
469C22—2444no—
470C22—2442no—
471C22—2444no—
472C22#1683967011yescharacteristic
473C22#1683989911yescharacteristic
474C22#1683990611yescharacteristic
475C22#1683991111yescharacteristic
476C22#1683992111yescharacteristic
477C22#1683993812yes—
478C22#1683995112yes—
479C22#1683998811yescharacteristic
480C22#1684000511yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—81no—
486C4—82no—
487C4—81no—
488C4—4882no—
489C4—4881no—
490C4—4882no—
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—4882no—
496C2#2144480812yes—
497C2#2144483212yes—
498C2#2144681911yescharacteristic
499C2#2144684511yescharacteristic
500C2#2144694711yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#483C2
socleC23×C61#120
prime radicalC24×D61D4
hypercenterC2×#483D61
hypocenterC61#9926
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
3. #21445905C2semidirect
4. C2×#16839499C2semidirect
6. #21443699C2semidirect
7. C2×C61×#483C2semidirect
241. C2×#483D61semidirect
281. C61#9926semidirect

Maximal subgroups

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

subgroupindex
#214516262
C2×#168396882
#214459052
C2×#168394992
C2×#168394852
#214436992
C2×C61×#4832
#992661