Finite Group Databasebeta

← #29571510

Subgroups — #29571510

Order 15616.

802 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840317C211yesmaximal, characteristic
2#21445796C211yesmaximal, characteristic
3#21445796C211yesmaximal, characteristic
4C2×#16839511C211yesmaximal, characteristic, complemented
5C2×#16839509C211yesmaximal, characteristic, complemented
6C2×#16839507C211yesmaximal, characteristic, complemented
7C2×C61×#333C211yesmaximal, characteristic, complemented
8#16840317—22no—
9#16840317—22no—
10#16840317—22no—
11#16840317—22no—
12C2×#14519777C2211yescharacteristic
13C2×#14519768—21no—
14C2×#14519757—21no—
15C2×#14519753C412yes—
16C2×Q8×Dic61C412yes—
17C2×#14519742—21no—
18C2×#14519735—21no—
19C2×C42×D61C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×C4⋊#92418C2211yescharacteristic
22C2×#14519689—21no—
23C2×#14519689—21no—
24#16839516—21no—
25#16839516—21no—
26#16839511—22no—
27#16839509—22no—
28#16839507—22no—
29C2×C4⋊Q8×C61C2211yescharacteristic
30C2×C4⋊C8×C61C2211yescharacteristic
31C2×C4⋊C8×C61C2211yescharacteristic
32C61×#333—22no—
33C22×Dic122—21no—
34C22×C4×D61C2×C411yescharacteristic
35C22×C4×D61D411yescharacteristic
36C22×C4×D61D411yescharacteristic
37C22×#92418—21no—
38C22×#92418—21no—
39#14519777—21no—
40#14519777—21no—
41#14519777D412yes—
42#14519777D412yes—
43#14519768—24no—
44#14519768—24no—
45#14519757—24no—
46#14519757—24no—
47#14519753—24no—
48#14519753—24no—
49Q8×Dic61—24no—
50Q8×Dic61—24no—
51#14519742—24no—
52#14519742—24no—
53#14519735—24no—
54#14519735—24no—
55C2×C4⋊Dic61—21no—
56C2×C4⋊Dic61—22no—
57C2×C4⋊Dic61C2×C412yes—
58C2×#11758196—22no—
59C2×#11758198—22no—
60C2×C4×Dic61—22no—
61C2×C4×Dic61C2×C411yescharacteristic
62C2×C4×Dic61D411yescharacteristic
63C2×C4×Dic61D411yescharacteristic
64C42×D61—21no—
65C42×D61—21no—
66C42×D61—21no—
67C42×D61—21no—
68C4⋊#92418—21no—
69C4⋊#92418—21no—
70C4⋊#92418—21no—
71C4⋊#92418—21no—
72#14519689—42no—
73#14519689—42no—
74C22×Q8×C61C2×C412yes—
75C22×C8×C61—21no—
76C22×C8×C61—21no—
77C4⋊Q8×C61—21no—
78C4⋊Q8×C61—21no—
79C4⋊Q8×C61D412yes—
80C4⋊Q8×C61D412yes—
81C2×C4⋊C4×C61—21no—
82C2×C4⋊C4×C61—21no—
83C2×C42×C61C2311yescharacteristic
84C4⋊C8×C61—21no—
85C4⋊C8×C61—21no—
86C4⋊C8×C61—21no—
87C4⋊C8×C61—21no—
88C23×D61C22⋊C411yescharacteristic
89C2×Q8×C61—22no—
90C2×Q8×C61—42no—
91C2×Q8×C61—42no—
92C2×Q8×C61C22⋊C414yes—
93C22×C4×C61—22no—
94C22×C4×C61C22×C411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×Dic61—22no—
98C22×Dic61C22⋊C411yescharacteristic
99C22×Dic61C22⋊C411yescharacteristic
100C22×Dic61C22⋊C411yescharacteristic
101C2×C4×D61—21no—
102C2×C4×D61—21no—
103C2×C4×D61—21no—
104C2×C4×D61—21no—
105C2×C4×D61—21no—
106C2×C4×D61—21no—
107C2×C4×D61—21no—
108C2×C4×D61—21no—
109C2×C4×D61—21no—
110C2×C4×D61—21no—
111C2×C4×D61—21no—
112C2×C4×D61—21no—
113C2×C4×D61—41no—
114C2×C4×D61—41no—
115C2×C4×D61SD1612yes—
116C2×C4×D61Q1612yes—
117C2×Dic122—22no—
118C2×Dic122—22no—
119C2×Dic122—41no—
120C2×Dic122—41no—
121C2×Dic122—41no—
122C2×Dic122—41no—
123C2×C8×C61—41no—
124C2×C8×C61—41no—
125C2×C8×C61—41no—
126C2×C8×C61—41no—
127C2×C8×C61—41no—
128C2×C8×C61—41no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—22no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C42×C61C2×D411yescharacteristic
134C42×C61C2×D411yescharacteristic
135C42×C61C2×D411yescharacteristic
136C42×C61C2×D411yescharacteristic
137#11758198—28no—
138C4⋊Dic61—28no—
139C4⋊Dic61—22no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61C22⋊C414yes—
143#11758196—28no—
144C4×Dic61—28no—
145C4×Dic61—21no—
146C4×Dic61—21no—
147C4×Dic61—21no—
148C4×Dic61—21no—
149C4×Dic61—21no—
150C4×Dic61—21no—
151C2×#92418—41no—
152C2×#92418—41no—
153C2×#92418—41no—
154C2×#92418—41no—
155C2×#92418—41no—
156C2×#92418—41no—
157C23×C61C2×C22⋊C411yescharacteristic
158C22×D61—21no—
159C22×D61—21no—
160C22×D61—21no—
161C22×D61—21no—
162C22×D61—21no—
163C22×D61—21no—
164C22×D61#10312yes—
165Q8×C61—44no—
166Q8×C61—44no—
167C2×C4×C61—21no—
168C2×C4×C61—21no—
169C2×C4×C61—24no—
170C2×C4×C61—42no—
171C2×C4×C61—42no—
172C2×C4×C61C2×C22⋊C411yescharacteristic
173C2×C4×C61C2×C22⋊C411yescharacteristic
174C2×C4×C61#12011yescharacteristic
175C2×C4×C61#12011yescharacteristic
176C2×C4×C61#12011yescharacteristic
177C2×C4×C61#12011yescharacteristic
178C2×C4×C61C2×SD1611yescharacteristic
179C2×C4×C61C2×Q1611yescharacteristic
180C2×C4×C61#13511yescharacteristic
181C2×C4×C61#13511yescharacteristic
182C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
183C2×C4×C61#13711yescharacteristic
184C2×C4×C61#13711yescharacteristic
185C2×C4×C61#13711yescharacteristic
186C2×Dic61—21no—
187C2×Dic61—21no—
188C2×Dic61—21no—
189C2×Dic61—21no—
190C2×Dic61—24no—
191C2×Dic61—21no—
192C2×Dic61—21no—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—42no—
196C2×Dic61—42no—
197C2×Dic61#10312yes—
198C4×D61—22no—
199C4×D61—22no—
200C4×D61—22no—
201C4×D61—22no—
202C4×D61—42no—
203C4×D61—42no—
204C4×D61—41no—
205C4×D61—41no—
206C4×D61—41no—
207C4×D61—41no—
208Dic122—42no—
209Dic122—42no—
210Dic122—42no—
211Dic122—42no—
212C8×C61—81no—
213C8×C61—81no—
214#92418—81no—
215#92418—81no—
216#8343—611nomaximal, Sylow
217C22×C61#37911yescharacteristic
218C22×C61#41311yescharacteristic
219C22×C61#41311yescharacteristic
220C22×C61C2×#10311yescharacteristic
221C22×C61#41611yescharacteristic
222C22×C61#41711yescharacteristic
223C22×C61#41911yescharacteristic
224C2×D61—22no—
225C2×D61—22no—
226C2×D61—22no—
227C2×D61—41no—
228C2×D61—41no—
229C2×D61—41no—
230C2×D61—41no—
231C4×C61—21no—
232C4×C61—21no—
233C4×C61—44no—
234C4×C61#44811yescharacteristic
235C4×C61#44811yescharacteristic
236C4×C61#44911yescharacteristic
237C4×C61#44911yescharacteristic
238C4×C61#45111yescharacteristic
239C4×C61#45111yescharacteristic
240C4×C61#45211yescharacteristic
241C4×C61#45211yescharacteristic
242Dic61—22no—
243Dic61—41no—
244Dic61—44no—
245Dic61—41no—
246C2×#533—611no—
247#1797—611no—
248#1797—611no—
249C2×#333—611no—
250C2×#333D6111yescharacteristic, complemented
251C2×#332—611no—
252C2×#332—611no—
253C2×C61#154211yescharacteristic
254C2×C61#155011yescharacteristic
255C2×C61#155411yescharacteristic
256C2×C61#155911yescharacteristic
257C2×C61#178511yescharacteristic
258C2×C61#178711yescharacteristic
259C2×C61#178811yescharacteristic
260D61—42no—
261#533—1222no—
262#533—1222no—
263#533—1222no—
264#533—1222no—
265C2×C4⋊Q8—611no—
266C2×C4⋊Q8C2×D6111yescharacteristic
267C2×#125—1221no—
268C2×C22⋊Q8—612no—
269C2×C22⋊Q8—1221no—
270C2×C4×Q8—612no—
271C2×C4×Q8—1221no—
272C2×#117—1221no—
273C22×C42—611no—
274C2×C4⋊C8—611no—
275C2×C4⋊C8—611no—
276C2×C4⋊C8C2×D6111yescharacteristic
277C2×C4⋊C8C2×D6111yescharacteristic
278C2×C22⋊C8—1221no—
279C2×C22⋊C8—1221no—
280#336—1221no—
281#336—1221no—
282#333—22no—
283#333—1222no—
284#332—1222no—
285#332—1222no—
286C61#834311yescharacteristic, complemented, Sylow
287C22×Q8—1221no—
288C22×Q8C4×D6112yes—
289C23×C4—611no—
290C23×C4—611no—
291C23×C4—611no—
292C22×C8—21no—
293C22×C8—21no—
294C22×C8—1221no—
295C22×C8—1221no—
296C4⋊Q8—21no—
297C4⋊Q8—21no—
298C4⋊Q8—612no—
299C4⋊Q8—612no—
300C4⋊Q8—1221no—
301C4⋊Q8—1221no—
302C4⋊Q8D24412yes—
303C4⋊Q8C22⋊D6112yes—
304#125—1224no—
305#125—1224no—
306C22⋊Q8—1224no—
307C22⋊Q8—1224no—
308C22⋊Q8—1224no—
309C22⋊Q8—1224no—
310C4×Q8—1224no—
311C4×Q8—1224no—
312C4×Q8—1224no—
313C4×Q8—1224no—
314#117—1224no—
315#117—1224no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—612no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1222no—
321C2×C4⋊C4—1221no—
322C2×C22⋊C4—1222no—
323C2×C42—611no—
324C2×C42—611no—
325C2×C42—611no—
326C2×C42—1222no—
327C2×C42—1221no—
328C2×C42—1221no—
329C2×C42—1221no—
330C2×C42—1221no—
331C2×C42C22×D6111yescharacteristic
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—21no—
336C4⋊C8—1221no—
337C4⋊C8—1221no—
338C4⋊C8—1221no—
339C4⋊C8—1221no—
340C22⋊C8—2442no—
341C22⋊C8—2442no—
342C24—611no—
343C2×Q8—22no—
344C2×Q8—42no—
345C2×Q8—42no—
346C2×Q8—1222no—
347C2×Q8—1222no—
348C2×Q8—2441no—
349C2×Q8—2441no—
350C2×Q8—2441no—
351C2×Q8—2441no—
352C2×Q8#1175819814yes—
353C22×C4—22no—
354C22×C4—611no—
355C22×C4—612no—
356C22×C4—612no—
357C22×C4—611no—
358C22×C4—611no—
359C22×C4—1221no—
360C22×C4—1221no—
361C22×C4—1221no—
362C22×C4—1221no—
363C22×C4—1222no—
364C22×C4—1221no—
365C22×C4—1221no—
366C22×C4—1221no—
367C22×C4—1221no—
368C22×C4—1221no—
369C22×C4—1221no—
370C22×C4—1221no—
371C22×C4—1221no—
372C22×C4—2441no—
373C22×C4—2441no—
374C22×C4C2×C4×D6111yescharacteristic
375C22×C4D4×D6111yescharacteristic
376C22×C4D4×D6111yescharacteristic
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—2441no—
384C2×C8—2441no—
385C2×C8—2441no—
386C2×C8—2441no—
387C2×C8—2441no—
388C2×C8—2441no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—614no—
394C4⋊C4—1228no—
395C4⋊C4—1228no—
396C4⋊C4—1222no—
397C4⋊C4—1222no—
398C4⋊C4—1222no—
399C22⋊C4—1228no—
400C42—1228no—
401C42—1221no—
402C42—1221no—
403C42—1221no—
404C42—1221no—
405C42—1221no—
406C42—1221no—
407C42C2×D24411yescharacteristic
408C42D4×D6111yescharacteristic
409C42D4×D6111yescharacteristic
410C42C2×C22⋊D6111yescharacteristic
411C23—612no—
412C23—1221no—
413C23—1221no—
414C23—1221no—
415C23—1221no—
416C23—1221no—
417C23—1221no—
418C23C22⋊C4×D6111yescharacteristic
419Q8—44no—
420Q8—44no—
421Q8—2442no—
422Q8—2442no—
423Q8—2442no—
424Q8—2442no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—612no—
431C2×C4—1221no—
432C2×C4—1221no—
433C2×C4—1222no—
434C2×C4—1222no—
435C2×C4—1222no—
436C2×C4—1222no—
437C2×C4—1221no—
438C2×C4—1221no—
439C2×C4—1221no—
440C2×C4—1221no—
441C2×C4—1221no—
442C2×C4—1221no—
443C2×C4—1224no—
444C2×C4—2442no—
445C2×C4—2442no—
446C2×C4—2442no—
447C2×C4—2442no—
448C2×C4—2441no—
449C2×C4—2441no—
450C2×C4—2441no—
451C2×C4—2441no—
452C2×C4C22⋊C4×D6111yescharacteristic
453C2×C4C2×#1175819811yescharacteristic
454C2×C4#1451974511yescharacteristic
455C2×C4#1451974511yescharacteristic
456C2×C4#1451974711yescharacteristic
457C2×C4#1451974711yescharacteristic
458C2×C4SD16×D6111yescharacteristic
459C2×C4Q16×D6111yescharacteristic
460C2×C4#1451979811yescharacteristic
461C2×C4#1451980011yescharacteristic
462C2×C4#1451980411yescharacteristic
463C2×C4#1451980611yescharacteristic
464C2×C4#1451980911yescharacteristic
465C2×C4#1451980911yescharacteristic
466C8—81no—
467C8—81no—
468C8—4881no—
469C8—4881no—
470C22—1222no—
471C22—1222no—
472C22—1222no—
473C22—2441no—
474C22—2441no—
475C22—2441no—
476C22—2441no—
477C22#1683961711yescharacteristic
478C22#1683974311yescharacteristic
479C22#1683974311yescharacteristic
480C22#103×D6111yescharacteristic
481C22#1683975711yescharacteristic
482C22#1683976211yescharacteristic
483C22#1683976911yescharacteristic
484C4—21no—
485C4—21no—
486C4—44no—
487C4—1222no—
488C4—2441no—
489C4—2441no—
490C4—2444no—
491C4#1683989911yescharacteristic
492C4#1683990211yescharacteristic
493C4#1683990611yescharacteristic
494C4#1683991011yescharacteristic
495C4#1683991611yescharacteristic
496C4#1683991911yescharacteristic
497C4#1683992311yescharacteristic
498C4#1683992711yescharacteristic
499C2—2442no—
500C2#2144447311yescharacteristic
501C2#2144450611yescharacteristic
502C2#2144452811yescharacteristic
503C2#2144454611yescharacteristic
504C2#2144574411yescharacteristic
505C2#2144575211yescharacteristic
506C2#2144575811yescharacteristic
subgroupG/H
centerC23C22⋊C4×D61
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC2×C61×#333C2
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C42×C61C23
hypercenterC2×#333D61
hypocenterC61#8343
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#333

Complements

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

NH ≅ G/Ntype
4. C2×#16839511C2semidirect
5. C2×#16839509C2semidirect
6. C2×#16839507C2semidirect
7. C2×C61×#333C2semidirect
250. C2×#333D61semidirect
286. C61#8343semidirect

Maximal subgroups

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

subgroupindex
C2×#168403172
#214457962
C2×#168395112
C2×#168395092
C2×#168395072
C2×C61×#3332
#834361