Finite Group Databasebeta

← #29712791

Subgroups — #29712791

Order 15616.

1276 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840333C211yesmaximal, characteristic
2C2×#16840103C212yesmaximal, complemented
3#21447022C212yesmaximal, complemented
4#21447015C212yesmaximal, complemented
5#21446126C212yesmaximal, complemented
6#21446016C212yesmaximal, complemented
7#21445753C212yesmaximal, complemented
8#21445731C211yesmaximal, characteristic
9C61×#2311C211yesmaximal, characteristic
10C23×Dic122C2211yescharacteristic
11#16840333—28no—
12C2×#14519775—21no—
13C2×#14519764—21no—
14C2×#14519758—21no—
15C2×#14519758—21no—
16C2×#14519756C2212yes—
17C2×#14519756C2212yes—
18C2×#14519736C2211yescharacteristic
19#16840103—28no—
20C2×#14519710C2212yes—
21C2×#14519708C2212yes—
22C2×#14519700C2212yes—
23C2×#14519700C2212yes—
24C2×#14519700C2212yes—
25C2×#14519700C2212yes—
26C2×C22⋊#92418C2212yes—
27C2×C22⋊#92418C2212yes—
28C2×C22⋊#92418C2211yescharacteristic
29C2×C22⋊#92418C2211yescharacteristic
30#16839524C2212yes—
31#16839524C2212yes—
32C22×#41×C61C2211yescharacteristic
33C2×#117×C61C2211yescharacteristic
34C22×C4⋊C4×C61C2211yescharacteristic
35C61×#391—21no—
36C61×#390—21no—
37C61×#386—21no—
38C61×#386—21no—
39C61×#380C2212yes—
40C61×#380C2212yes—
41C22×Dic122—21no—
42C22×Dic122—24no—
43C22×Dic122—24no—
44C22×Dic122—41no—
45C22×Dic122—41no—
46C22×Dic122C2312yes—
47C23×Dic61—21no—
48C22×#92418C2312yes—
49C22×#92418C2312yes—
50#14519775—28no—
51#14519764—28no—
52#14519758—28no—
53#14519758—28no—
54#14519756—24no—
55#14519756—24no—
56#14519756—24no—
57#14519756D418yes—
58#14519736—24no—
59C2×C4⋊Dic61C2312yes—
60C2×#11758196—22no—
61C2×#11758196—22no—
62C2×C22⋊Dic61—21no—
63C2×C22⋊Dic61—22no—
64C2×C22⋊Dic61—22no—
65C2×C4×Dic61—21no—
66#14519710—24no—
67#14519708—24no—
68#14519700—24no—
69#14519700—24no—
70#14519700—42no—
71#14519700—42no—
72C22⋊#92418—24no—
73C22⋊#92418—24no—
74C22⋊#92418—41no—
75C22⋊#92418—41no—
76C2×#41×C61—24no—
77C2×#41×C61—24no—
78C2×#41×C61—41no—
79C2×#41×C61—41no—
80C22×Q8×C61D411yescharacteristic
81C22×D4×C61D411yescharacteristic
82C22×D4×C61D411yescharacteristic
83C22×D4×C61D411yescharacteristic
84C23×C4×C61—21no—
85C23×C4×C61C2311yescharacteristic
86C23×C4×C61C2311yescharacteristic
87C23×C4×C61C2311yescharacteristic
88#117×C61—24no—
89C2×C4⋊C4×C61—21no—
90C2×C4⋊C4×C61—41no—
91C2×C4⋊C4×C61—41no—
92C2×C4⋊C4×C61C2312yes—
93C2×C4⋊C4×C61C2312yes—
94C2×C22⋊C4×C61—21no—
95C2×C22⋊C4×C61—22no—
96C2×C22⋊C4×C61—22no—
97C2×C42×C61—21no—
98#95×C61—22no—
99#95×C61—22no—
100C24×C61C2×D411yescharacteristic
101C24×C61C2×D411yescharacteristic
102C24×C61C2×D411yescharacteristic
103#41×C61—48no—
104#41×C61—48no—
105C2×Q8×C61—22no—
106C2×Q8×C61—22no—
107C2×Q8×C61—41no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—22no—
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61—41no—
115C2×D4×C61—41no—
116C2×D4×C61—41no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—22no—
124C22×C4×C61—22no—
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—22no—
128C22×C4×C61—22no—
129C22×C4×C61—42no—
130C22×C4×C61—82no—
131C22×C4×C61C2×D411yescharacteristic
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D414yes—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61#4114yes—
136C22×C4×C61C2411yescharacteristic
137C22×Dic61—24no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—42no—
141C22×Dic61—42no—
142C2×Dic122—22no—
143C2×Dic122—22no—
144C2×Dic122—22no—
145C2×Dic122—22no—
146C2×Dic122—24no—
147C2×Dic122—22no—
148C2×Dic122—48no—
149C2×Dic122—48no—
150C2×Dic122—42no—
151C2×Dic122—42no—
152C2×Dic122C2×D414yes—
153C4⋊C4×C61—22no—
154C4⋊C4×C61—22no—
155C4⋊C4×C61—22no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—42no—
158C22⋊C4×C61—42no—
159C22⋊C4×C61—82no—
160C22⋊C4×C61—82no—
161C42×C61—42no—
162C22⋊Dic61—24no—
163C22⋊Dic61—28no—
164C22⋊Dic61—28no—
165C4⋊Dic61—22no—
166C4⋊Dic61C2×D414yes—
167#11758196—28no—
168#11758196—28no—
169C4×Dic61—24no—
170C2×#92418—22no—
171C2×#92418—22no—
172C2×#92418—22no—
173C2×#92418—22no—
174C2×#92418—22no—
175C2×#92418—22no—
176C23×C61—21no—
177C23×C61—21no—
178C23×C61—21no—
179C23×C61—21no—
180C23×C61—22no—
181C23×C61—22no—
182C23×C61—22no—
183C23×C61—22no—
184C23×C61—21no—
185C23×C61—21no—
186C23×C61—21no—
187C23×C61—22no—
188C23×C61—22no—
189C23×C61#12012yes—
190C23×C61#12312yes—
191C23×C61C22×D411yescharacteristic
192Q8×C61—44no—
193D4×C61—44no—
194D4×C61—44no—
195D4×C61—44no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—21no—
199C2×C4×C61—21no—
200C2×C4×C61—24no—
201C2×C4×C61—24no—
202C2×C4×C61—24no—
203C2×C4×C61—24no—
204C2×C4×C61—21no—
205C2×C4×C61—21no—
206C2×C4×C61—21no—
207C2×C4×C61—42no—
208C2×C4×C61—42no—
209C2×C4×C61—42no—
210C2×C4×C61—42no—
211C2×C4×C61—42no—
212C2×C4×C61—42no—
213C2×C4×C61—42no—
214C2×C4×C61—42no—
215C2×C4×C61—84no—
216C2×C4×C61#12012yes—
217C2×C4×C61#12312yes—
218C2×C4×C61C22×D412yes—
219C2×C4×C61C2×#4112yes—
220C2×C4×C61H(5,𝔽2)11yescharacteristic
221C2×C4×C61H(5,𝔽2)11yescharacteristic
222C2×Dic61—24no—
223C2×Dic61—24no—
224C2×Dic61—48no—
225C2×Dic61—42no—
226C2×Dic61—42no—
227C2×Dic61—42no—
228C2×Dic61—42no—
229Dic122—24no—
230Dic122—24no—
231Dic122—48no—
232Dic122—42no—
233Dic122—42no—
234#92418—42no—
235#92418—42no—
236#24016—611nomaximal, Sylow
237C22×C61—24no—
238C22×C61—22no—
239C22×C61—22no—
240C22×C61—24no—
241C22×C61—42no—
242C22×C61—41no—
243C22×C61—41no—
244C22×C61—41no—
245C22×C61—41no—
246C22×C61—42no—
247C22×C61—42no—
248C22×C61—41no—
249C22×C61—41no—
250C22×C61—41no—
251C22×C61—42no—
252C22×C61C2×#12011yescharacteristic
253C22×C61C2×#12311yescharacteristic
254C22×C61#53411yescharacteristic
255C22×C61#57712yes—
256C22×C61#57812yes—
257C4×C61—22no—
258C4×C61—22no—
259C4×C61—41no—
260C4×C61—82no—
261C4×C61—82no—
262C4×C61#54612yes—
263C4×C61#54612yes—
264Dic61—44no—
265Dic61—44no—
266C2×#536—611no—
267C2×#485—612no—
268#2311D6111yescharacteristic
269#2036—612no—
270#2035—612no—
271#1872—612no—
272#1854—612no—
273#1787—612no—
274#1780—611no—
275C2×C61—42no—
276C2×C61—42no—
277C2×C61—81no—
278C2×C61#240511yescharacteristic
279C2×C61#301312yes—
280C2×C61#309712yes—
281C2×C61#320011yescharacteristic
282C2×C61#320211yescharacteristic
283C22×#41C2×D6111yescharacteristic
284C23×Q8—611no—
285#536—1228no—
286C2×C4⋊Q8—1221no—
287C2×#124—1221no—
288C2×#123—1221no—
289C2×#123—1221no—
290C2×C22⋊Q8—612no—
291C2×C22⋊Q8—612no—
292C2×#117—611no—
293C2×#117C2×D6111yescharacteristic
294C22×C4⋊C4C2×D6111yescharacteristic
295#485—1228no—
296C2×#107—612no—
297C2×#106—612no—
298C2×#103—612no—
299C2×#103—612no—
300C2×#103—612no—
301C2×#103—612no—
302C2×C22⋊C8—612no—
303C2×C22⋊C8—612no—
304C2×C22⋊C8—611no—
305C2×C22⋊C8—611no—
306#391—21no—
307#390—21no—
308#386—21no—
309#386—21no—
310#380C2×D6112yes—
311#380C2×D6112yes—
312#340—612no—
313#340—612no—
314C61#2401611yescharacteristic, complemented, Sylow
315C2×#41—24no—
316C2×#41—24no—
317C2×#41—41no—
318C2×#41—41no—
319C22×Q8—612no—
320C22×Q8—1224no—
321C22×Q8—1221no—
322C22×Q8—1224no—
323C22×Q8—2441no—
324C22×Q8—2441no—
325C22×Q8C22⋊D6111yescharacteristic
326C22×D4C22⋊D6111yescharacteristic
327C22×D4C22⋊D6111yescharacteristic
328C22×D4C22⋊D6111yescharacteristic
329C23×C4—21no—
330C23×C4—1221no—
331C23×C4C22×D6111yescharacteristic
332C23×C4C22×D6111yescharacteristic
333C23×C4C22×D6111yescharacteristic
334C22×C8—612no—
335C22×C8—612no—
336C4⋊Q8—1228no—
337#124—1228no—
338#123—1228no—
339#123—1228no—
340C22⋊Q8—618no—
341C22⋊Q8—1224no—
342C22⋊Q8—1224no—
343C22⋊Q8—1224no—
344#117—24no—
345#117—1224no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—41no—
348C2×C4⋊C4—41no—
349C2×C4⋊C4—612no—
350C2×C4⋊C4—1222no—
351C2×C4⋊C4—1222no—
352C2×C4⋊C4C22×D6112yes—
353C2×C4⋊C4C22×D6112yes—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—1221no—
358C2×C22⋊C4—1222no—
359C2×C22⋊C4—1222no—
360C2×C42—21no—
361C2×C42—1221no—
362#107—1224no—
363#106—1224no—
364#103—1224no—
365#103—1224no—
366#103—2442no—
367#103—2442no—
368C22⋊C8—1224no—
369C22⋊C8—1224no—
370C22⋊C8—2441no—
371C22⋊C8—2441no—
372#95—22no—
373#95—22no—
374C24C2×C22⋊D6111yescharacteristic
375C24C2×C22⋊D6111yescharacteristic
376C24C2×C22⋊D6111yescharacteristic
377#41—48no—
378#41—48no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—41no—
382C2×Q8—614no—
383C2×Q8—1224no—
384C2×Q8—1222no—
385C2×Q8—1222no—
386C2×Q8—1222no—
387C2×Q8—1222no—
388C2×Q8—1222no—
389C2×Q8—2448no—
390C2×Q8—2448no—
391C2×Q8—2442no—
392C2×Q8—2442no—
393C2×D4—22no—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—22no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—42no—
415C22×C4—82no—
416C22×C4—1224no—
417C22×C4—1222no—
418C22×C4—1222no—
419C22×C4—2442no—
420C22×C4—2442no—
421C22×C4#1175822214yes—
422C22×C4D4×D6114yes—
423C22×C4C2×C22⋊D6111yescharacteristic
424C22×C4C2×C22⋊D6111yescharacteristic
425C22×C4C2×C22⋊D6111yescharacteristic
426C22×C4C23×D6111yescharacteristic
427C2×C8—1222no—
428C2×C8—1222no—
429C2×C8—1222no—
430C2×C8—1222no—
431C2×C8—1222no—
432C2×C8—1222no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—42no—
438C4⋊C4—614no—
439C4⋊C4—1228no—
440C4⋊C4—1222no—
441C4⋊C4—1228no—
442C22⋊C4—42no—
443C22⋊C4—82no—
444C22⋊C4—82no—
445C22⋊C4—1224no—
446C22⋊C4—1228no—
447C22⋊C4—1228no—
448C42—42no—
449C42—1224no—
450C23—21no—
451C23—21no—
452C23—22no—
453C23—22no—
454C23—21no—
455C23—21no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—21no—
462C23—22no—
463C23#1451974512yes—
464C23#1451976012yes—
465C23C22×C22⋊D6111yescharacteristic
466Q8—44no—
467Q8—1224no—
468Q8—1224no—
469Q8—2448no—
470Q8—2442no—
471Q8—2442no—
472D4—44no—
473D4—44no—
474D4—44no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—84no—
495C2×C4—1224no—
496C2×C4—1224no—
497C2×C4—2448no—
498C2×C4—2442no—
499C2×C4—2442no—
500C2×C4—2442no—
501C2×C4—2442no—
502C2×C4#1451974512yes—
503C2×C4#1451976012yes—
504C2×C4C2×D4×D6112yes—
505C2×C4C2×#1175822212yes—
506C2×C4#1451982211yescharacteristic
507C2×C4#1451982211yescharacteristic
508C8—2442no—
509C8—2442no—
510C22—24no—
511C22—24no—
512C22—22no—
513C22—22no—
514C22—42no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—42no—
520C22—42no—
521C22—41no—
522C22—41no—
523C22—42no—
524C22—41no—
525C22C2×#1451974511yescharacteristic
526C22C2×#1451976011yescharacteristic
527C22#1684032111yescharacteristic
528C22#1684057312yes—
529C22#1684057712yes—
530C4—22no—
531C4—22no—
532C4—41no—
533C4—82no—
534C4—82no—
535C4—2444no—
536C4—2444no—
537C4#1684040212yes—
538C4#1684040212yes—
539C2—42no—
540C2—42no—
541C2—81no—
542C2#2144914611yescharacteristic
543C2#2145513112yes—
544C2#2145578812yes—
545C2#2145669111yescharacteristic
546C2#2145671011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2311C2
socleC23×C61C22×D4
prime radicalC22×#41×C61C22
hypercenter#2311D61
hypocenterC61#24016
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×#41 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C2×#16840103C2semidirect
3. #21447022C2semidirect
4. #21447015C2semidirect
5. #21446126C2semidirect
6. #21446016C2semidirect
7. #21445753C2semidirect
314. C61#24016semidirect

Maximal subgroups

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

subgroupindex
C2×#168403332
C2×#168401032
#214470222
#214470152
#214461262
#214460162
#214457532
#214457312
C61×#23112
#2401661