Finite Group Databasebeta

← #29580276

Subgroups — #29580276

Order 15616.

890 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21451626C211yesmaximal, characteristic
2C2×#16839673C211yesmaximal, characteristic
3#21445754C211yesmaximal, characteristic, complemented
4C2×#16839499C211yesmaximal, characteristic, complemented
5C2×#16839485C211yesmaximal, characteristic
6#21443699C211yesmaximal, characteristic, complemented
7C2×C61×#451C211yesmaximal, characteristic, complemented
8C22×#11758198—21no—
9C22×#11758198C2211yescharacteristic
10C2×#14519701—22no—
11C2×C22⋊#92418C2211yescharacteristic
12#16839697—21no—
13#16839684—22no—
14#16839673—22no—
15#16839673—22no—
16#16839673—21no—
17#16839634—21no—
18#16839622C2211yescharacteristic
19#16839617—21no—
20C2×#14519684—21no—
21C2×#14519684C2211yescharacteristic
22#16839499—22no—
23#16839485—22no—
24C23×Q8×C61C2211yescharacteristic
25C22×Q16×C61—21no—
26C2×C22⋊Q8×C61C2211yescharacteristic
27C61×#451—22no—
28C61×#451—22no—
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—44no—
48#14519701—44no—
49C22⋊#92418—21no—
50C22⋊#92418—21no—
51#14519684—22no—
52#14519684—24no—
53#14519684—21no—
54#14519684—21no—
55#14519684D412yes—
56C22×Q8×C61—21no—
57C22×Q8×C61—21no—
58C22×Q8×C61—21no—
59C22×Q8×C61—21no—
60C22×Q8×C61—21no—
61C22×Q8×C61—44no—
62C22×Q8×C61D411yescharacteristic
63C22×Q8×C61D412yes—
64C23×C4×C61—21no—
65C23×C4×C61C2311yescharacteristic
66C2×Q16×C61—24no—
67C2×Q16×C61—44no—
68C22×C8×C61—21no—
69C22⋊Q8×C61—21no—
70C22⋊Q8×C61—21no—
71C22⋊Q8×C61—21no—
72C22⋊Q8×C61—21no—
73C2×C4⋊C4×C61—21no—
74C2×C4⋊C4×C61D411yescharacteristic
75C2×C22⋊C4×C61—21no—
76#103×C61—24no—
77C22⋊C8×C61—21no—
78C22⋊C8×C61D412yes—
79C24×C61C2×D411yescharacteristic
80C23×D61—21no—
81C23×D61—21no—
82C23×D61—21no—
83C23×D61—21no—
84C23×D61—21no—
85C23×D61—21no—
86C23×D61—42no—
87C23×D61—42no—
88C23×D61Q1612yescomplemented
89C2×Q8×C61—24no—
90C2×Q8×C61—24no—
91C2×Q8×C61—22no—
92C2×Q8×C61—21no—
93C2×Q8×C61—22no—
94C2×Q8×C61—21no—
95C2×Q8×C61—44no—
96C2×Q8×C61—42no—
97C2×Q8×C61—42no—
98C2×Q8×C61—44no—
99C2×Q8×C61—42no—
100C2×Q8×C61—44no—
101C2×Q8×C61—44no—
102C2×Q8×C61—41no—
103C2×Q8×C61—41no—
104C22×C4×C61—24no—
105C22×C4×C61—22no—
106C22×C4×C61—21no—
107C22×C4×C61—21no—
108C22×C4×C61—42no—
109C22×C4×C61—42no—
110C22×C4×C61—41no—
111C22×C4×C61—41no—
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61C2×D411yescharacteristic
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61#4111yescharacteristic
118C22×Dic61—22no—
119C22×Dic61—21no—
120C22×Dic61—21no—
121C22×Dic61—22no—
122C22×Dic61—21no—
123C22×Dic61—42no—
124C22×Dic61—42no—
125C22×Dic61—42no—
126C22×Dic61—41no—
127C22×Dic61—41no—
128Q16×C61—48no—
129C2×C8×C61—22no—
130C2×C8×C61—42no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—41no—
134C4⋊C4×C61—41no—
135C22⋊C4×C61—41no—
136C22⋊C4×C61—41no—
137C22⋊Dic61—41no—
138C22⋊Dic61—41no—
139#11758198—48no—
140#11758198—41no—
141#11758198—41no—
142#11758198—41no—
143#11758198—41no—
144#11758198—82no—
145#11758198—82no—
146C4⋊Dic61—42no—
147#11758196—82no—
148C2×#92418—42no—
149C2×#92418—41no—
150C23×C61—22no—
151C23×C61—22no—
152C23×C61#12011yescharacteristic
153C23×C61#12011yescharacteristic
154C23×C61#12111yescharacteristic
155C23×C61C2×Q1611yescharacteristic
156C23×C61#13511yescharacteristic
157C23×C61#13711yescharacteristic
158C23×C61#13711yescharacteristic
159C22×D61—22no—
160C22×D61—22no—
161C22×D61—22no—
162C22×D61—44no—
163C22×D61—42no—
164C22×D61—44no—
165C22×D61—42no—
166C22×D61—41no—
167C22×D61—41no—
168C22×D61—41no—
169C22×D61—42no—
170C22×D61—42no—
171C22×D61—41no—
172C22×D61—82no—
173C22×D61—82no—
174C22×D61—81no—
175C22×D61—81no—
176Q8×C61—24no—
177Q8×C61—24no—
178Q8×C61—44no—
179Q8×C61—42no—
180Q8×C61—42no—
181Q8×C61—44no—
182Q8×C61—42no—
183Q8×C61—81no—
184C2×C4×C61—22no—
185C2×C4×C61—22no—
186C2×C4×C61—24no—
187C2×C4×C61—41no—
188C2×C4×C61—41no—
189C2×C4×C61—41no—
190C2×C4×C61—41no—
191C2×C4×C61—42no—
192C2×C4×C61—44no—
193C2×C4×C61—42no—
194C2×C4×C61—44no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61#12011yescharacteristic
200C2×C4×C61#12111yescharacteristic
201C2×C4×C61#12111yescharacteristic
202C2×C4×C61#12711yescharacteristic
203C2×Dic61—42no—
204C2×Dic61—42no—
205C2×Dic61—42no—
206C2×Dic61—41no—
207C2×Dic61—42no—
208C2×Dic61—42no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61—42no—
212C2×Dic61—82no—
213C2×Dic61—82no—
214C2×Dic61—81no—
215C8×C61—42no—
216#92418—81no—
217#9924—611nomaximal, Sylow
218C22×C61—22no—
219C22×C61—22no—
220C22×C61—22no—
221C22×C61—22no—
222C22×C61—21no—
223C22×C61—21no—
224C22×C61—21no—
225C22×C61—21no—
226C22×C61#39211yescharacteristic
227C22×C61#45111yescharacteristic
228C22×C61#45111yescharacteristic
229C22×C61#45211yescharacteristic
230C22×C61#45211yescharacteristic
231C22×C61#45512yes—
232C22×C61B(4,𝔽2)12yes—
233C22×C61#46711yescharacteristic
234C22×C61#47011yescharacteristic
235C2×D61—44no—
236C2×D61—44no—
237C2×D61—42no—
238C2×D61—82no—
239C2×D61—82no—
240C2×D61—82no—
241C2×D61—81no—
242C2×D61—81no—
243C2×D61—82no—
244C4×C61—21no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—44no—
249C4×C61—81no—
250C4×C61—81no—
251Dic61—82no—
252Dic61—82no—
253Dic61—81no—
254C2×#451D6111yescharacteristic, complemented
255#2626—611no—
256C2×#393—611no—
257#1787—611no—
258C2×#328—611no—
259C2×#323—611no—
260#1294—611no—
261C2×C61—21no—
262C2×C61—21no—
263C2×C61—21no—
264C2×C61—21no—
265C2×C61#160212yes—
266C2×C61#160612yes—
267C2×C61#200111yescharacteristic
268C2×C61#200411yescharacteristic
269C2×C61#202111yescharacteristic
270D61—82no—
271C23×Q8C2×D6111yescharacteristic
272C22×Q16—21no—
273C2×C22⋊Q8C2×D6111yescharacteristic
274C22×C22⋊C4—611no—
275C22×C22⋊C4—1221no—
276#451—22no—
277#451—22no—
278C2×#103—21no—
279C2×#103—1222no—
280C2×C22⋊C8—611no—
281C2×C22⋊C8C2×D6111yescharacteristic
282#399—1221no—
283#396—1222no—
284#393—1222no—
285#393—1221no—
286#393—1222no—
287#385—1221no—
288#380—611no—
289#379—1221no—
290C2×#95—611no—
291C2×#95—1221no—
292#328—1222no—
293#323—1222no—
294C61#992411yescharacteristic, complemented, Sylow
295C25—611no—
296C22×Q8—21no—
297C22×Q8—21no—
298C22×Q8—21no—
299C22×Q8—21no—
300C22×Q8—21no—
301C22×Q8—44no—
302C22×Q8C22⋊D6111yescharacteristic
303C22×Q8C22⋊D6112yes—
304C23×C4—21no—
305C23×C4—611no—
306C23×C4—1221no—
307C23×C4C22×D6111yescharacteristic
308C2×Q16—24no—
309C2×Q16—44no—
310C22×C8—21no—
311C22×C8—1221no—
312C22⋊Q8—21no—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315C22⋊Q8—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—612no—
318C2×C4⋊C4—1222no—
319C2×C4⋊C4C22⋊D6111yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—1222no—
322C2×C22⋊C4—1222no—
323C2×C22⋊C4—1224no—
324C2×C22⋊C4—1221no—
325C2×C22⋊C4—1221no—
326C2×C22⋊C4—1221no—
327C2×C22⋊C4—1221no—
328C2×C22⋊C4—1221no—
329C2×C22⋊C4—1221no—
330C2×C22⋊C4—1221no—
331C2×C22⋊C4—2444no—
332#103—24no—
333#103—2444no—
334C22⋊C8—21no—
335C22⋊C8—1221no—
336C22⋊C8—1221no—
337C22⋊C8C22⋊D6112yes—
338#95—612no—
339#95—1224no—
340#95—1222no—
341#95—1221no—
342#95—1221no—
343C24—612no—
344C24—1221no—
345C24—1221no—
346C24—1221no—
347C24—1221no—
348C24—1221no—
349C24—1221no—
350C24—2442no—
351C24—2442no—
352C24D4×D6111yescharacteristic
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—22no—
356C2×Q8—21no—
357C2×Q8—22no—
358C2×Q8—21no—
359C2×Q8—44no—
360C2×Q8—42no—
361C2×Q8—42no—
362C2×Q8—44no—
363C2×Q8—42no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—41no—
367C2×Q8—41no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—1221no—
377C22×C4—1222no—
378C22×C4—1221no—
379C22×C4—1222no—
380C22×C4—1221no—
381C22×C4—2442no—
382C22×C4—2442no—
383C22×C4—2441no—
384C22×C4—2441no—
385C22×C4—2442no—
386C22×C4D4×D6111yescharacteristic
387C22×C4D4×D6111yescharacteristic
388C22×C4#1175822411yescharacteristic
389C22×C4C2×C22⋊D6111yescharacteristic
390C22×C4C2×C22⋊D6111yescharacteristic
391C22×C4C2×C22⋊D6111yescharacteristic
392Q16—48no—
393C2×C8—22no—
394C2×C8—42no—
395C2×C8—2442no—
396C2×C8—2441no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—4882no—
402C4⋊C4—2442no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—4882no—
406C22⋊C4—4882no—
407C22⋊C4—2448no—
408C22⋊C4—2441no—
409C22⋊C4—2441no—
410C22⋊C4—2441no—
411C22⋊C4—2441no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C23—22no—
415C23—22no—
416C23—1222no—
417C23—1222no—
418C23—1222no—
419C23—4881no—
420C23—4881no—
421C23—4882no—
422C23—4882no—
423C23—2442no—
424C23—2442no—
425C23—2444no—
426C23—2444no—
427C23—2442no—
428C23—2442no—
429C23—2441no—
430C23—2441no—
431C23—2441no—
432C23—2441no—
433C23#1451974511yescharacteristic
434C23#1451974511yescharacteristic
435C23#1451974811yescharacteristic
436C23Q16×D6111yescharacteristic
437C23#1451980011yescharacteristic
438C23#1451980911yescharacteristic
439C23#1451980911yescharacteristic
440Q8—24no—
441Q8—24no—
442Q8—42no—
443Q8—44no—
444Q8—42no—
445Q8—44no—
446Q8—42no—
447Q8—81no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—24no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—4882no—
464C2×C4—4882no—
465C2×C4—4881no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4—2442no—
469C2×C4—2442no—
470C2×C4—2442no—
471C2×C4—2442no—
472C2×C4—2442no—
473C2×C4—2442no—
474C2×C4—2441no—
475C2×C4#1451974611yescharacteristic
476C2×C4#1451974911yescharacteristic
477C2×C4#1451974911yescharacteristic
478C2×C4#1451977311yescharacteristic
479C8—42no—
480C8—4881no—
481C22—21no—
482C22—21no—
483C22—22no—
484C22—22no—
485C22—21no—
486C22—21no—
487C22—22no—
488C22—22no—
489C22—4882no—
490C22—4882no—
491C22—4882no—
492C22—4881no—
493C22—4881no—
494C22—4882no—
495C22—2444no—
496C22—2444no—
497C22—2442no—
498C22#1683967011yescharacteristic
499C22#1683991611yescharacteristic
500C22#1683991611yescharacteristic
501C22#1683992311yescharacteristic
502C22#1683992311yescharacteristic
503C22#1683993812yes—
504C22#1683994812yes—
505C22#1684000011yescharacteristic
506C22#1684001711yescharacteristic
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—44no—
512C4—81no—
513C4—81no—
514C4—4882no—
515C4—4881no—
516C4—4882no—
517C2—21no—
518C2—21no—
519C2—21no—
520C2—21no—
521C2—4882no—
522C2#2144477412yes—
523C2#2144479812yes—
524C2#2144682611yescharacteristic
525C2#2144684511yescharacteristic
526C2#2144694211yescharacteristic
subgroupG/H
centerC23#14519745
commutator, cosocleC23×C4×C61C23
FrattiniC23×C4C22×D61
FittingC2×C61×#451C2
socleC23×C61#120
prime radicalC24×D61D4
hypercenterC2×#451D61
hypocenterC61#9924
LCSG ▷ C23×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
3. #21445754C2semidirect
4. C2×#16839499C2semidirect
6. #21443699C2semidirect
7. C2×C61×#451C2semidirect
88. C23×D61Q16semidirect
254. C2×#451D61semidirect
294. C61#9924semidirect

Maximal subgroups

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

subgroupindex
#214516262
C2×#168396732
#214457542
C2×#168394992
C2×#168394852
#214436992
C2×C61×#4512
#992461