Finite Group Databasebeta

← #29779559

Subgroups — #29779559

Order 15616.

906 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454169C211yesmaximal, characteristic
2C2×#16839669C211yesmaximal, characteristic
3#21447230C212yesmaximal, complemented
4#21446958C212yesmaximal, complemented
5#21446511C212yesmaximal, complemented
6#21446506C212yesmaximal, complemented
7#21445906C211yesmaximal, characteristic
8#21445903C211yesmaximal, characteristic
9#21445754C211yesmaximal, characteristic
10#21445583C211yesmaximal, characteristic
11C61×#1787C211yesmaximal, characteristic
12C2×#14519776C2211yescharacteristic
13C2×#14519754—21no—
14C22×C4⋊Dic61C2211yescharacteristic
15C22×C4×Dic61C2211yescharacteristic
16C2×#14519701C2212yes—
17C2×#14519701C2212yes—
18C2×C22⋊#92418C2211yescharacteristic
19C2×#14519687C2212yes—
20#16839669—21no—
21#16839669—21no—
22#16839669C2212yes—
23#16839669C2212yes—
24#16839666—21no—
25#16839661C2211yescharacteristic
26#16839659—21no—
27#16839658C2211yescharacteristic
28#16839631C2211yescharacteristic
29#16839622C2211yescharacteristic
30#16839620—21no—
31C2×#14519684—21no—
32C2×#14519684—21no—
33#16839525C2212yes—
34#16839525C2212yes—
35#16839524C2212yes—
36#16839524C2212yes—
37#16839516C2212yes—
38C23×Q8×C61C2211yescharacteristic
39C2×#103×C61C2212yes—
40C2×#103×C61C2212yes—
41C2×C22⋊C8×C61C2211yescharacteristic
42C61×#380C2211yescharacteristic
43C22×Dic122—21no—
44C23×Dic61—21no—
45C23×Dic61D411yescharacteristic
46C23×Dic61D411yescharacteristic
47C22×#92418C2312yes—
48#14519776—42no—
49#14519754—44no—
50C2×C4⋊Dic61—22no—
51C2×C4⋊Dic61—22no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61—21no—
54C2×C4⋊Dic61C2312yes—
55C2×C4⋊Dic61C2311yescharacteristic
56C2×C4⋊Dic61C2311yescharacteristic
57C2×#11758196—21no—
58C2×#11758196—21no—
59C2×#11758196—21no—
60C2×#11758196—21no—
61C2×C22⋊Dic61—21no—
62C2×C4×Dic61—42no—
63C2×C4×Dic61D411yescharacteristic
64C2×C4×Dic61D411yescharacteristic
65C2×C4×Dic61C2311yescharacteristic
66C2×C4×Dic61C2311yescharacteristic
67#14519701—24no—
68#14519701—42no—
69C22⋊#92418—22no—
70#14519687—42no—
71#14519684—21no—
72#14519684—22no—
73#14519684—22no—
74#14519684—22no—
75#14519684—22no—
76#14519684—21no—
77#14519684—21no—
78C22×Q8×C61—24no—
79C22×Q8×C61—21no—
80C22×Q8×C61—22no—
81C22×Q8×C61—22no—
82C22×Q8×C61—42no—
83C22×Q8×C61C2312yes—
84C23×C4×C61—21no—
85C23×C4×C61C2311yescharacteristic
86C22×C8×C61C2312yes—
87C2×C4⋊C4×C61C2311yescharacteristic
88C2×C4⋊C4×C61C2311yescharacteristic
89C2×C22⋊C4×C61—21no—
90#103×C61—24no—
91#103×C61—42no—
92C22⋊C8×C61—22no—
93#95×C61—21no—
94C24×C61C2×D411yescharacteristic
95C2×Q8×C61—24no—
96C2×Q8×C61—22no—
97C2×Q8×C61—22no—
98C2×Q8×C61—22no—
99C2×Q8×C61—22no—
100C2×Q8×C61—22no—
101C2×Q8×C61—48no—
102C2×Q8×C61—48no—
103C2×Q8×C61—42no—
104C2×Q8×C61—42no—
105C2×Q8×C61#4114yes—
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—24no—
109C22×C4×C61—22no—
110C22×C4×C61—21no—
111C22×C4×C61—21no—
112C22×C4×C61—42no—
113C22×C4×C61—42no—
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61#4112yes—
116C22×C4×C61#4112yes—
117C22×C4×C61C2411yescharacteristic
118C22×Dic61—22no—
119C22×Dic61—22no—
120C22×Dic61—22no—
121C22×Dic61—22no—
122C22×Dic61—22no—
123C22×Dic61—21no—
124C22×Dic61—21no—
125C22×Dic61—41no—
126C22×Dic61—41no—
127C22×Dic61—42no—
128C22×Dic61—42no—
129C22×Dic61—42no—
130C22×Dic61C2×D411yescharacteristic
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61C2×D411yescharacteristic
133C22×Dic61C2×D411yescharacteristic
134C2×Dic122—42no—
135C2×Dic122—42no—
136C2×Dic122—42no—
137C2×C8×C61—22no—
138C2×C8×C61—22no—
139C2×C8×C61—22no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C22⋊C4×C61—42no—
143C22⋊Dic61—42no—
144C4⋊Dic61—22no—
145C4⋊Dic61—22no—
146C4⋊Dic61—22no—
147C4⋊Dic61—22no—
148C4⋊Dic61—42no—
149C4⋊Dic61—42no—
150#11758196—42no—
151#11758196—42no—
152#11758196—42no—
153#11758196—42no—
154C4×Dic61—22no—
155C4×Dic61—22no—
156C4×Dic61—41no—
157C4×Dic61—41no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—22no—
161C23×C61—21no—
162C23×C61—21no—
163C23×C61—21no—
164C23×C61—21no—
165C23×C61#12412yes—
166C23×C61AGL(1,ℤ/8)12yes—
167C23×C61#13712yes—
168C23×C61C22×D411yescharacteristic
169Q8×C61—24no—
170Q8×C61—24no—
171Q8×C61—48no—
172Q8×C61—42no—
173Q8×C61—42no—
174C2×C4×C61—21no—
175C2×C4×C61—21no—
176C2×C4×C61—24no—
177C2×C4×C61—24no—
178C2×C4×C61—24no—
179C2×C4×C61—48no—
180C2×C4×C61—42no—
181C2×C4×C61—42no—
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×C61#12412yes—
189C2×C4×C61C2×#4112yes—
190C2×C4×C61C2×#4111yescharacteristic
191C2×C4×C61C2×#4111yescharacteristic
192C2×C4×C61H(5,𝔽2)12yes—
193C2×Dic61—21no—
194C2×Dic61—21no—
195C2×Dic61—21no—
196C2×Dic61—21no—
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—44no—
204C2×Dic61—44no—
205C2×Dic61—41no—
206C2×Dic61—42no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—42no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—42no—
214C2×Dic61—84no—
215C2×Dic61#13712yes—
216C2×Dic61#13712yes—
217Dic122—84no—
218C8×C61—42no—
219#92418—42no—
220#29111—611nomaximal, Sylow
221C22×C61—22no—
222C22×C61—22no—
223C22×C61—22no—
224C22×C61—22no—
225C22×C61—22no—
226C22×C61—22no—
227C22×C61#48914yes—
228C22×C61#52511yescharacteristic
229C22×C61C2×#12411yescharacteristic
230C22×C61#53411yescharacteristic
231C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
232C22×C61C2×#13711yescharacteristic
233C22×C61C2×#13711yescharacteristic
234C22×C61C2×#13711yescharacteristic
235C4×C61—22no—
236C4×C61—44no—
237C4×C61—81no—
238C4×C61—81no—
239C4×C61#55112yes—
240C4×C61#55112yes—
241Dic61—22no—
242Dic61—22no—
243Dic61—41no—
244Dic61—41no—
245Dic61—81no—
246Dic61—82no—
247Dic61—81no—
248#2857—611no—
249C2×#391—611no—
250#2072—612no—
251#2024—612no—
252#1935—612no—
253#1935—612no—
254#1823—611no—
255#1823—611no—
256#1787—611no—
257#1787D6111yescharacteristic
258#1749—611no—
259C2×C61—24no—
260C2×C61#256111yescharacteristic
261C2×C61#311211yescharacteristic
262C2×C61#311311yescharacteristic
263C2×C61C2×#48911yescharacteristic
264C2×C61#313911yescharacteristic
265C2×C61#319311yescharacteristic
266C2×C61#319511yescharacteristic
267C23×Q8C2×D6111yescharacteristic
268C2×C4⋊Q8—611no—
269C2×C22⋊Q8—1221no—
270C22×C4⋊C4—611no—
271C22×C42—611no—
272C2×#103—612no—
273C2×#103—612no—
274C2×#103C2×D6112yes—
275C2×#103C2×D6112yes—
276C2×C22⋊C8—611no—
277C2×C22⋊C8C2×D6111yescharacteristic
278C2×#97—612no—
279#391—612no—
280#391—612no—
281#391—1221no—
282#391—1221no—
283#391—1221no—
284#389—611no—
285#389—611no—
286#389—1221no—
287#384—611no—
288#380—611no—
289#380—1221no—
290#380C2×D6111yescharacteristic
291C2×#95—1221no—
292C2×#95—1221no—
293#340—612no—
294#340—612no—
295#340—612no—
296#340—612no—
297#336—612no—
298C61#2911111yescharacteristic, complemented, Sylow
299C22×Q8—24no—
300C22×Q8—21no—
301C22×Q8—22no—
302C22×Q8—22no—
303C22×Q8—42no—
304C22×Q8—1221no—
305C22×Q8C22×D6112yes—
306C23×C4—21no—
307C23×C4—611no—
308C23×C4—611no—
309C23×C4—1221no—
310C23×C4C22×D6111yescharacteristic
311C22×C8—612no—
312C22×C8C22×D6112yes—
313C4⋊Q8—2442no—
314C22⋊Q8—2444no—
315C2×C4⋊C4—612no—
316C2×C4⋊C4—611no—
317C2×C4⋊C4—611no—
318C2×C4⋊C4—1222no—
319C2×C4⋊C4—1222no—
320C2×C4⋊C4—1221no—
321C2×C4⋊C4—1221no—
322C2×C4⋊C4—1221no—
323C2×C4⋊C4—1221no—
324C2×C4⋊C4—1221no—
325C2×C4⋊C4—1221no—
326C2×C4⋊C4C22×D6111yescharacteristic
327C2×C4⋊C4C22×D6111yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—1221no—
330C2×C42—611no—
331C2×C42—611no—
332C2×C42—611no—
333C2×C42—611no—
334C2×C42—2442no—
335#103—24no—
336#103—42no—
337#103—1224no—
338#103—2442no—
339C22⋊C8—22no—
340C22⋊C8—1222no—
341#97—2442no—
342#95—21no—
343#95—1222no—
344#95—1222no—
345#95—1222no—
346#95—1222no—
347#95—1221no—
348#95—1221no—
349#95—1221no—
350C24D4×D6111yescharacteristic
351C2×Q8—24no—
352C2×Q8—22no—
353C2×Q8—22no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—2442no—
362C2×Q8—2442no—
363C2×Q8—2442no—
364C2×Q8#1175822414yes—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—611no—
374C22×C4—611no—
375C22×C4—611no—
376C22×C4—611no—
377C22×C4—1222no—
378C22×C4—1222no—
379C22×C4—1222no—
380C22×C4—1222no—
381C22×C4—1222no—
382C22×C4—1221no—
383C22×C4—1221no—
384C22×C4—2441no—
385C22×C4—2441no—
386C22×C4—2442no—
387C22×C4—2442no—
388C22×C4—2442no—
389C22×C4#1175822212yes—
390C22×C4D4×D6111yescharacteristic
391C22×C4#1175822412yes—
392C22×C4C23×D6111yescharacteristic
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—1222no—
397C2×C8—1222no—
398C2×C8—1222no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—1222no—
402C4⋊C4—1222no—
403C4⋊C4—1222no—
404C4⋊C4—1222no—
405C4⋊C4—2442no—
406C4⋊C4—2442no—
407C4⋊C4—2442no—
408C4⋊C4—2442no—
409C4⋊C4—2442no—
410C4⋊C4—2442no—
411C22⋊C4—42no—
412C22⋊C4—2442no—
413C42—1222no—
414C42—1222no—
415C42—2441no—
416C42—2441no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23#1451976312yes—
422C23#1451980412yes—
423C23#1451980912yes—
424C23C2×D4×D6111yescharacteristic
425Q8—24no—
426Q8—24no—
427Q8—48no—
428Q8—42no—
429Q8—42no—
430Q8—4884no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—48no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—612no—
446C2×C4—612no—
447C2×C4—1221no—
448C2×C4—1221no—
449C2×C4—1221no—
450C2×C4—1221no—
451C2×C4—1221no—
452C2×C4—1221no—
453C2×C4—1221no—
454C2×C4—1221no—
455C2×C4—1221no—
456C2×C4—1221no—
457C2×C4—4884no—
458C2×C4—2444no—
459C2×C4—2444no—
460C2×C4—2441no—
461C2×C4—2442no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2442no—
465C2×C4—2441no—
466C2×C4—2441no—
467C2×C4—2442no—
468C2×C4—2441no—
469C2×C4#1451976312yes—
470C2×C4C2×#1175822412yes—
471C2×C4C2×#1175822411yescharacteristic
472C2×C4C2×#1175822211yescharacteristic
473C2×C4#1451982212yes—
474C8—42no—
475C8—2442no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22#1684012314yes—
483C22#1684026811yescharacteristic
484C22C2×#1451976311yescharacteristic
485C22#1684032211yescharacteristic
486C22C2×#1451980411yescharacteristic
487C22#137×D6111yescharacteristic
488C22#137×D6111yescharacteristic
489C22C2×#1451980911yescharacteristic
490C4—22no—
491C4—44no—
492C4—81no—
493C4—81no—
494C4—1222no—
495C4—1222no—
496C4—4882no—
497C4—4881no—
498C4—4881no—
499C4—2441no—
500C4—2441no—
501C4#1684043812yes—
502C4#1684044312yes—
503C2—24no—
504C2#2145097411yescharacteristic
505C2#2145593611yescharacteristic
506C2#2145594411yescharacteristic
507C2C2×#1684012311yescharacteristic
508C2#2145619711yescharacteristic
509C2#2145662311yescharacteristic
510C2#2145664111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1787C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1787D61
hypocenterC61#29111
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
3. #21447230C2semidirect
4. #21446958C2semidirect
5. #21446511C2semidirect
6. #21446506C2semidirect
298. C61#29111semidirect

Maximal subgroups

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

subgroupindex
#214541692
C2×#168396692
#214472302
#214469582
#214465112
#214465062
#214459062
#214459032
#214457542
#214455832
C61×#17872
#2911161