Finite Group Databasebeta

← #29580309

Subgroups — #29580309

Order 15616.

778 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840098C211yesmaximal, characteristic, complemented
2#21451624C211yesmaximal, characteristic
3C2×#16839698C211yesmaximal, characteristic
4#21445910C211yesmaximal, characteristic, complemented
5C2×#16839500C211yesmaximal, characteristic, complemented
6#21443699C211yesmaximal, characteristic, complemented
7C2×C61×#323C211yesmaximal, characteristic
8C2×#14519756C2211yescharacteristic
9C22×C4⋊Dic61C2211yescharacteristic
10C22×#11758196—21no—
11C22×C22⋊Dic61—21no—
12#16840098—22no—
13#16840098—22no—
14C2×#14519711—21no—
15C2×#14519702—21no—
16#16839698—21no—
17#16839698—22no—
18#16839698—22no—
19#16839685—22no—
20#16839674—21no—
21#16839635—21no—
22#16839622C2211yescharacteristic
23#16839618—21no—
24C2×#14519684C2211yescharacteristic
25#16839525—22no—
26#16839500—22no—
27C22×C22⋊C4×C61C2211yescharacteristic
28C2×C22⋊C8×C61C2211yescharacteristic
29C2×C22⋊C8×C61C2211yescharacteristic
30C61×#323—22no—
31C22×Dic122D411yescharacteristic
32C23×Dic61—21no—
33C23×Dic61—21no—
34#14519756—21no—
35#14519756—21no—
36#14519756—21no—
37#14519756—21no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61—21no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61D412yes—
44C2×C4⋊Dic61D411yescharacteristic
45C2×C4⋊Dic61D412yes—
46C2×#11758196—24no—
47C2×#11758196—21no—
48C2×#11758196—22no—
49C2×#11758196—44no—
50C2×C22⋊Dic61—24no—
51C2×C22⋊Dic61—21no—
52C2×C22⋊Dic61—21no—
53C2×C22⋊Dic61—22no—
54C2×C22⋊Dic61—22no—
55C2×C22⋊Dic61—44no—
56#14519711—24no—
57#14519702—24no—
58#14519684—22no—
59#14519684—21no—
60#14519684—21no—
61#14519684D412yes—
62C25×C61D411yescharacteristic
63C23×C4×C61D411yescharacteristic
64C23×C4×C61C2311yescharacteristic
65C22×C8×C61—21no—
66C22×C8×C61—21no—
67C2×C22⋊C4×C61—21no—
68C2×C22⋊C4×C61—21no—
69C2×C22⋊C4×C61—21no—
70C2×C22⋊C4×C61—21no—
71C2×C22⋊C4×C61—21no—
72C2×C22⋊C4×C61—21no—
73C22⋊C8×C61—21no—
74C22⋊C8×C61—21no—
75C22⋊C8×C61—21no—
76C22⋊C8×C61D412yes—
77C24×C61—21no—
78C24×C61—21no—
79C24×C61—21no—
80C24×C61—21no—
81C24×C61—21no—
82C24×C61—21no—
83C24×C61—42no—
84C24×C61—42no—
85C24×C61Q1612yes—
86C24×C61C2×D411yescharacteristic
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61—42no—
91C22×C4×C61—42no—
92C22×C4×C61C2×D411yescharacteristic
93C22×C4×C61C2×D411yescharacteristic
94C22×C4×C61C2×D411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61#4111yescharacteristic
98C22×Dic61—24no—
99C22×Dic61—22no—
100C22×Dic61—22no—
101C22×Dic61—21no—
102C22×Dic61—22no—
103C22×Dic61—21no—
104C22×Dic61—42no—
105C22×Dic61—41no—
106C22×Dic61—41no—
107C22×Dic61—42no—
108C22×Dic61—42no—
109C22×Dic61—41no—
110C22×Dic61—41no—
111C2×Dic122—21no—
112C2×Dic122—21no—
113C2×Dic122—42no—
114C2×C8×C61—22no—
115C2×C8×C61—42no—
116C2×C8×C61—42no—
117C2×C8×C61—41no—
118C22⋊C4×C61—41no—
119C22⋊C4×C61—41no—
120C22⋊C4×C61—41no—
121C22⋊C4×C61—41no—
122C22⋊Dic61—48no—
123C22⋊Dic61—41no—
124C22⋊Dic61—41no—
125C22⋊Dic61—41no—
126C22⋊Dic61—41no—
127C22⋊Dic61—82no—
128C22⋊Dic61—82no—
129C4⋊Dic61—22no—
130C4⋊Dic61—21no—
131C4⋊Dic61—22no—
132C4⋊Dic61—21no—
133C4⋊Dic61—41no—
134C4⋊Dic61—41no—
135C4⋊Dic61—42no—
136#11758196—48no—
137#11758196—41no—
138#11758196—41no—
139#11758196—82no—
140C23×C61—22no—
141C23×C61—22no—
142C23×C61—22no—
143C23×C61—22no—
144C23×C61—22no—
145C23×C61—44no—
146C23×C61—42no—
147C23×C61—44no—
148C23×C61—42no—
149C23×C61—41no—
150C23×C61—41no—
151C23×C61—41no—
152C23×C61—42no—
153C23×C61—42no—
154C23×C61—41no—
155C23×C61—82no—
156C23×C61—82no—
157C23×C61—81no—
158C23×C61—81no—
159C23×C61#12011yescharacteristic
160C23×C61#12011yescharacteristic
161C23×C61#12111yescharacteristic
162C23×C61C2×Q1611yescharacteristic
163C23×C61#13511yescharacteristic
164C23×C61#13711yescharacteristic
165C23×C61#13711yescharacteristic
166C2×C4×C61—22no—
167C2×C4×C61—22no—
168C2×C4×C61—41no—
169C2×C4×C61—41no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—42no—
173C2×C4×C61—41no—
174C2×C4×C61—42no—
175C2×C4×C61—81no—
176C2×C4×C61#12011yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61#12711yescharacteristic
180C2×Dic61—24no—
181C2×Dic61—44no—
182C2×Dic61—44no—
183C2×Dic61—42no—
184C2×Dic61—42no—
185C2×Dic61—42no—
186C2×Dic61—42no—
187C2×Dic61—41no—
188C2×Dic61—41no—
189C2×Dic61—42no—
190C2×Dic61—42no—
191C2×Dic61—42no—
192C2×Dic61—42no—
193C2×Dic61—42no—
194C2×Dic61—42no—
195C2×Dic61—82no—
196C2×Dic61—82no—
197Dic122—42no—
198Dic122—81no—
199C8×C61—42no—
200C8×C61—81no—
201#9928—611nomaximal, Sylow
202C22×C61—22no—
203C22×C61—22no—
204C22×C61—22no—
205C22×C61—22no—
206C22×C61—21no—
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—21no—
210C22×C61—44no—
211C22×C61—44no—
212C22×C61—42no—
213C22×C61—82no—
214C22×C61—82no—
215C22×C61—82no—
216C22×C61—82no—
217C22×C61—81no—
218C22×C61—81no—
219C22×C61#39211yescharacteristic
220C22×C61#45111yescharacteristic
221C22×C61#45111yescharacteristic
222C22×C61#45211yescharacteristic
223C22×C61#45211yescharacteristic
224C22×C61#45312yes—
225C22×C61#45812yes—
226C22×C61#46711yescharacteristic
227C22×C61#47011yescharacteristic
228C4×C61—21no—
229C4×C61—21no—
230C4×C61—21no—
231C4×C61—21no—
232C4×C61—81no—
233Dic61—44no—
234Dic61—82no—
235Dic61—81no—
236Dic61—82no—
237Dic61—81no—
238C2×#484—611no—
239#2626—611no—
240C2×#399—611no—
241#1824—611no—
242C2×#328—611no—
243C2×#323D6111yescharacteristic
244#1294—611no—
245C2×C61—21no—
246C2×C61—21no—
247C2×C61—21no—
248C2×C61—21no—
249C2×C61—82no—
250C2×C61#161412yes—
251C2×C61#161812yes—
252C2×C61#200111yescharacteristic
253C2×C61#200211yescharacteristic
254C2×C61#202211yescharacteristic
255C2×C22⋊Q8—611no—
256C22×C4⋊C4—611no—
257C22×C4⋊C4—1221no—
258C22×C22⋊C4—1221no—
259C22×C22⋊C4C2×D6111yescharacteristic
260#484—1222no—
261#484—1222no—
262C2×#107—1221no—
263C2×#103—1221no—
264C2×C22⋊C8C2×D6111yescharacteristic
265C2×C22⋊C8C2×D6111yescharacteristic
266#399—1222no—
267#399—1221no—
268#399—1222no—
269#396—1222no—
270#393—1221no—
271#385—1221no—
272#380—611no—
273#379—1221no—
274C2×#95—611no—
275#340—1222no—
276#328—1222no—
277#323—22no—
278C61#992811yescharacteristic, complemented, Sylow
279C25D24411yescharacteristic
280C22×Q8—611no—
281C23×C4—1221no—
282C23×C4—1221no—
283C23×C4D24411yescharacteristic
284C23×C4C22×D6111yescharacteristic
285C22×C8—21no—
286C22×C8—21no—
287C22⋊Q8—1221no—
288C22⋊Q8—1221no—
289C22⋊Q8—1221no—
290C22⋊Q8—1221no—
291C2×C4⋊C4—612no—
292C2×C4⋊C4—611no—
293C2×C4⋊C4—612no—
294C2×C4⋊C4—1221no—
295C2×C4⋊C4—1221no—
296C2×C4⋊C4—1221no—
297C2×C4⋊C4—1221no—
298C2×C4⋊C4—1222no—
299C2×C4⋊C4—1224no—
300C2×C4⋊C4—1221no—
301C2×C4⋊C4—1221no—
302C2×C4⋊C4—2444no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—1222no—
310C2×C22⋊C4—1222no—
311C2×C22⋊C4—1224no—
312C2×C22⋊C4—1221no—
313C2×C22⋊C4—1221no—
314C2×C22⋊C4—2444no—
315#107—1224no—
316#103—1224no—
317C22⋊C8—21no—
318C22⋊C8—21no—
319C22⋊C8—21no—
320C22⋊C8C22⋊D6112yes—
321#95—612no—
322#95—1222no—
323#95—1221no—
324#95—1221no—
325C24—21no—
326C24—21no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—42no—
332C24—42no—
333C24Dic24412yes—
334C24C2×D24411yescharacteristic
335C2×Q8—1221no—
336C2×Q8—1221no—
337C2×Q8—2442no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—1224no—
344C22×C4—1222no—
345C22×C4—1222no—
346C22×C4—1222no—
347C22×C4—1221no—
348C22×C4—1221no—
349C22×C4—2442no—
350C22×C4—2442no—
351C22×C4—2441no—
352C22×C4—2441no—
353C22×C4—2442no—
354C22×C4—2441no—
355C22×C4—2441no—
356C22×C4#1175822211yescharacteristic
357C22×C4D4×D6111yescharacteristic
358C22×C4D4×D6111yescharacteristic
359C22×C4D4×D6111yescharacteristic
360C22×C4D4×D6111yescharacteristic
361C22×C4C2×C22⋊D6111yescharacteristic
362C2×C8—22no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—41no—
366C4⋊C4—1222no—
367C4⋊C4—1221no—
368C4⋊C4—1222no—
369C4⋊C4—1221no—
370C4⋊C4—4882no—
371C4⋊C4—2442no—
372C4⋊C4—2448no—
373C4⋊C4—2441no—
374C4⋊C4—2441no—
375C4⋊C4—2441no—
376C4⋊C4—2441no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—4882no—
382C22⋊C4—4882no—
383C22⋊C4—2448no—
384C22⋊C4—2441no—
385C22⋊C4—2441no—
386C22⋊C4—2441no—
387C22⋊C4—2441no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—44no—
394C23—42no—
395C23—44no—
396C23—42no—
397C23—42no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—42no—
403C23—82no—
404C23—82no—
405C23—81no—
406C23—81no—
407C23#1451974711yescharacteristic
408C23#1451974711yescharacteristic
409C23#1451975111yescharacteristic
410C23C2×Dic24411yescharacteristic
411C23#1451979911yescharacteristic
412C23#1451981011yescharacteristic
413C23#1451981011yescharacteristic
414Q8—4881no—
415Q8—2442no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—81no—
426C2×C4—1224no—
427C2×C4—4882no—
428C2×C4—4882no—
429C2×C4—2442no—
430C2×C4—2442no—
431C2×C4—2442no—
432C2×C4—2442no—
433C2×C4—2442no—
434C2×C4—2441no—
435C2×C4—2441no—
436C2×C4—2442no—
437C2×C4—2442no—
438C2×C4—2442no—
439C2×C4—2444no—
440C2×C4—2442no—
441C2×C4—2444no—
442C2×C4—2442no—
443C2×C4#1451974511yescharacteristic
444C2×C4#1451975011yescharacteristic
445C2×C4#1451975011yescharacteristic
446C2×C4#1451977311yescharacteristic
447C8—42no—
448C8—81no—
449C22—22no—
450C22—22no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—22no—
456C22—22no—
457C22—44no—
458C22—42no—
459C22—44no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—81no—
465C22—81no—
466C22#1683967111yescharacteristic
467C22#1683992011yescharacteristic
468C22#1683992011yescharacteristic
469C22#1683992611yescharacteristic
470C22#1683992611yescharacteristic
471C22#1683993212yes—
472C22#1683995312yes—
473C22#1684000311yescharacteristic
474C22#1684002011yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—81no—
480C4—4882no—
481C4—4881no—
482C4—4882no—
483C4—4881no—
484C4—2444no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—82no—
490C2#2144484612yes—
491C2#2144486912yes—
492C2#2144682911yescharacteristic
493C2#2144683611yescharacteristic
494C2#2144694911yescharacteristic
subgroupG/H
centerC23#14519747
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#323C2
socleC23×C61#120
prime radicalC25×C61D4
hypercenterC2×#323D61
hypocenterC61#9928
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
1. C2×#16840098C2semidirect
4. #21445910C2semidirect
5. C2×#16839500C2semidirect
6. #21443699C2semidirect
278. C61#9928semidirect

Maximal subgroups

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

subgroupindex
C2×#168400982
#214516242
C2×#168396982
#214459102
C2×#168395002
#214436992
C2×C61×#3232
#992861