Finite Group Databasebeta

← #29779581

Subgroups — #29779581

Order 15616.

910 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448309C211yesmaximal, characteristic
2#21447228C211yesmaximal, characteristic, complemented
3#21447223C211yesmaximal, characteristic, complemented
4#21446957C211yesmaximal, characteristic, complemented
5#21446952C211yesmaximal, characteristic, complemented
6#21446515C212yesmaximal, complemented
7#21446508C212yesmaximal, complemented
8#21445911C212yesmaximal, complemented
9#21445757C212yesmaximal
10#21445581C211yesmaximal, characteristic, complemented
11C61×#2857C211yesmaximal, characteristic, complemented
12C22×#11758222C2211yescharacteristic
13C2×#14519736C2211yescharacteristic
14C2×#14519700C2212yes—
15C2×#14519700C2212yes—
16C2×#14519697C2212yes—
17C2×#14519697C2212yes—
18C2×C22⋊#92418C2212yes—
19C2×#14519688C2212yes—
20#16839668C2211yescharacteristic
21#16839668C2211yescharacteristic
22#16839664C2211yescharacteristic
23#16839664C2211yescharacteristic
24#16839641—21no—
25#16839641—21no—
26C4×#11758198—21no—
27C4×#11758198—21no—
28#16839524C2212yes—
29#16839524C2212yes—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839517C2211yescharacteristic
33#16839517C2211yescharacteristic
34C2×C4⋊Q8×C61C2211yescharacteristic
35C2×C22⋊Q8×C61—21no—
36C22×C42×C61C2211yescharacteristic
37C61×#391—21no—
38C61×#389—21no—
39C61×#389C2212yes—
40C61×#384C2211yescharacteristic
41C61×#380—21no—
42C61×#380C2212yes—
43C2×#11758222—22no—
44C2×#11758222—24no—
45C2×#11758222—22no—
46C2×#11758222—42no—
47C22×Dic122C2311yescharacteristic
48C22×D244C2311yescharacteristic
49C22×C22⋊D61—21no—
50C22×C4×D61—21no—
51C22×#92418C2312yes—
52C22×#92418C2312yes—
53#14519736—24no—
54C2×C4⋊Dic61C2311yescharacteristic
55C2×C4⋊Dic61C2311yescharacteristic
56C2×C22⋊Dic61—21no—
57C2×#11758198—21no—
58C2×#11758198—21no—
59C2×#11758198—21no—
60C2×#11758198—21no—
61C2×C4×Dic61—21no—
62#14519700—24no—
63#14519700—42no—
64#14519697—24no—
65#14519697—42no—
66C22⋊#92418—24no—
67#14519688—42no—
68#14519684—21no—
69#14519684—21no—
70#14519684—21no—
71#14519684—21no—
72C22×Q8×C61—21no—
73C23×C4×C61D411yescharacteristic
74C23×C4×C61D411yescharacteristic
75C23×C4×C61C2311yescharacteristic
76C4⋊Q8×C61—42no—
77C22⋊Q8×C61—44no—
78C2×C4⋊C4×C61—21no—
79C2×C4⋊C4×C61—21no—
80C2×C4⋊C4×C61—21no—
81C2×C4⋊C4×C61—22no—
82C2×C4⋊C4×C61C2312yes—
83C2×C4⋊C4×C61C2312yes—
84C2×C22⋊C4×C61—22no—
85C2×C42×C61—42no—
86C2×C42×C61D411yescharacteristic
87C2×C42×C61D411yescharacteristic
88C2×C42×C61C2311yescharacteristic
89C2×C42×C61C2311yescharacteristic
90#95×C61—22no—
91#95×C61—22no—
92C24×C61C2×D411yescharacteristic
93C23×D61—21no—
94C2×Q8×C61—42no—
95C2×Q8×C61—42no—
96C2×Q8×C61—42no—
97C22×C4×C61—21no—
98C22×C4×C61—21no—
99C22×C4×C61—22no—
100C22×C4×C61—22no—
101C22×C4×C61—22no—
102C22×C4×C61—22no—
103C22×C4×C61—22no—
104C22×C4×C61—41no—
105C22×C4×C61—41no—
106C22×C4×C61—41no—
107C22×C4×C61—41no—
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61C2×D411yescharacteristic
110C22×C4×C61C2×D411yescharacteristic
111C22×C4×C61C2×D411yescharacteristic
112C22×C4×C61C2×D411yescharacteristic
113C22×C4×C61#4114yes—
114C22×C4×C61C2411yescharacteristic
115C2×C22⋊D61—24no—
116C2×C22⋊D61—42no—
117C2×C22⋊D61—42no—
118C22×Dic61—21no—
119C22×Dic61—21no—
120C22×Dic61—21no—
121#11758222—48no—
122#11758222—48no—
123C2×D244—22no—
124C2×D244—22no—
125C2×D244—21no—
126C2×D244#4112yes—
127C2×C4×D61—24no—
128C2×C4×D61—42no—
129C2×C4×D61—42no—
130C2×Dic122—22no—
131C2×Dic122—22no—
132C2×Dic122—21no—
133C2×Dic122#4112yes—
134C4⋊C4×C61—22no—
135C4⋊C4×C61—22no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—42no—
139C4⋊C4×C61—44no—
140C4⋊C4×C61—42no—
141C4⋊C4×C61—42no—
142C22⋊C4×C61—44no—
143C42×C61—22no—
144C42×C61—22no—
145C42×C61—41no—
146C42×C61—41no—
147C22⋊Dic61—42no—
148#11758198—42no—
149#11758198—42no—
150#11758198—42no—
151#11758198—42no—
152C4⋊Dic61—22no—
153C4⋊Dic61—22no—
154C4×Dic61—42no—
155C2×#92418—22no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—22no—
159C2×#92418—22no—
160C2×#92418—22no—
161C23×C61—21no—
162C23×C61—21no—
163C23×C61—21no—
164C23×C61—21no—
165C23×C61—21no—
166C23×C61—21no—
167C23×C61#12412yes—
168C23×C61C22×D411yescharacteristic
169C22×D61—22no—
170C22×D61—41no—
171C22×D61—41no—
172C22×D61—41no—
173C22×D61—41no—
174C22×D61—41no—
175C22×D61—41no—
176Q8×C61—84no—
177C2×C4×C61—21no—
178C2×C4×C61—21no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—21no—
182C2×C4×C61—21no—
183C2×C4×C61—24no—
184C2×C4×C61—24no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—21no—
188C2×C4×C61—21no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—44no—
192C2×C4×C61—44no—
193C2×C4×C61—42no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—42no—
199C2×C4×C61#12412yes—
200C2×C4×C61AGL(1,ℤ/8)12yes—
201C2×C4×C61#13712yes—
202C2×C4×C61C2×#4111yescharacteristic
203C2×C4×C61C2×#4111yescharacteristic
204C2×C4×C61C2×#4112yes—
205C2×C4×C61H(5,𝔽2)12yes—
206C22⋊D61—48no—
207C2×Dic61—22no—
208C2×Dic61—41no—
209C2×Dic61—41no—
210C2×Dic61—41no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—41no—
214C2×Dic61—41no—
215C2×Dic61—41no—
216D244—24no—
217D244—42no—
218C4×D61—48no—
219Dic122—24no—
220Dic122—42no—
221#92418—42no—
222#92418—42no—
223#29113—611nomaximal, Sylow
224C22×C61—22no—
225C22×C61—22no—
226C22×C61—22no—
227C22×C61—41no—
228C22×C61—41no—
229C22×C61—41no—
230C22×C61—41no—
231C22×C61#52511yescharacteristic
232C22×C61C2×#12411yescharacteristic
233C22×C61#53411yescharacteristic
234C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
235C22×C61C2×#13711yescharacteristic
236C22×C61#57511yescharacteristic
237C22×C61#57511yescharacteristic
238C2×D61—42no—
239C2×D61—42no—
240C2×D61—42no—
241C2×D61—81no—
242C2×D61—81no—
243C2×D61—81no—
244C2×D61—81no—
245C4×C61—22no—
246C4×C61—22no—
247C4×C61—22no—
248C4×C61—41no—
249C4×C61—41no—
250C4×C61—82no—
251C4×C61—82no—
252C4×C61#55112yes—
253C4×C61#55112yes—
254Dic61—42no—
255Dic61—81no—
256Dic61—81no—
257#2857D6111yescharacteristic, complemented
258#2297—611no—
259#2072—611no—
260#2071—611no—
261#2024—611no—
262#2023—611no—
263#1936—612no—
264#1935—612no—
265#1825—612no—
266#1788—612no—
267#1749—611no—
268C2×C61—42no—
269C2×C61#256111yescharacteristic
270C2×C61#311511yescharacteristic
271C2×C61#311611yescharacteristic
272C2×C61#313511yescharacteristic
273C2×C61#314111yescharacteristic
274C2×C61#319411yescharacteristic
275C2×C61#319611yescharacteristic
276D61—82no—
277C22×#41—611no—
278C2×C4⋊Q8C2×D6111yescharacteristic
279C2×C22⋊Q8—21no—
280C2×#117—611no—
281C22×C42C2×D6111yescharacteristic
282C2×#103—612no—
283C2×#103—612no—
284C2×#102—612no—
285C2×#102—612no—
286C2×C22⋊C8—612no—
287C2×#97—612no—
288#391—21no—
289#391—611no—
290#391—611no—
291#390—611no—
292#390—611no—
293#389—21no—
294#389C2×D6112yes—
295#386—1221no—
296#386—1221no—
297#384C2×D6111yescharacteristic
298#380—21no—
299#380C2×D6112yes—
300C4×C22⋊C4—1221no—
301C4×C22⋊C4—1221no—
302#340—612no—
303#340—612no—
304#340—612no—
305#340—612no—
306#336—611no—
307#336—611no—
308C61#2911311yescharacteristic, complemented, Sylow
309C2×#41—1222no—
310C2×#41—1222no—
311C2×#41—1224no—
312C2×#41—2442no—
313C22×Q8—21no—
314C22×Q8—611no—
315C22×D4—611no—
316C22×D4—1221no—
317C23×C4—1221no—
318C23×C4C22⋊D6111yescharacteristic
319C23×C4C22⋊D6111yescharacteristic
320C23×C4C22×D6111yescharacteristic
321C22×C8—612no—
322C22×C8—612no—
323C4⋊Q8—42no—
324C22⋊Q8—44no—
325#117—1224no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—611no—
331C2×C4⋊C4—611no—
332C2×C4⋊C4C22×D6112yes—
333C2×C4⋊C4C22×D6112yes—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—1221no—
336C2×C22⋊C4—1221no—
337C2×C22⋊C4—1221no—
338C2×C22⋊C4—1221no—
339C2×C22⋊C4—1221no—
340C2×C42—42no—
341C2×C42—1221no—
342C2×C42C22⋊D6111yescharacteristic
343C2×C42C22⋊D6111yescharacteristic
344C2×C42C22×D6111yescharacteristic
345C2×C42C22×D6111yescharacteristic
346#103—1224no—
347#103—2442no—
348#102—1224no—
349#102—2442no—
350C22⋊C8—1224no—
351#97—2442no—
352#95—22no—
353#95—22no—
354#95—1221no—
355#95—1221no—
356#95—1221no—
357#95—1221no—
358C24—1221no—
359C24C2×C22⋊D6111yescharacteristic
360#41—2448no—
361#41—2448no—
362C2×Q8—42no—
363C2×Q8—42no—
364C2×Q8—42no—
365C2×Q8—612no—
366C2×Q8—1221no—
367C2×Q8—1222no—
368C2×Q8—1222no—
369C2×D4—612no—
370C2×D4—1224no—
371C2×D4—1221no—
372C2×D4—1222no—
373C2×D4—1222no—
374C2×D4—2442no—
375C2×D4—2442no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—1224no—
388C22×C4—1221no—
389C22×C4—1221no—
390C22×C4—1221no—
391C22×C4—2442no—
392C22×C4—2442no—
393C22×C4#1175822614yes—
394C22×C4C2×C22⋊D6111yescharacteristic
395C22×C4C2×C22⋊D6111yescharacteristic
396C22×C4C2×C22⋊D6111yescharacteristic
397C22×C4C2×C22⋊D6111yescharacteristic
398C22×C4C2×C22⋊D6111yescharacteristic
399C22×C4C23×D6111yescharacteristic
400C2×C8—1222no—
401C2×C8—1222no—
402C2×C8—1222no—
403C2×C8—1222no—
404C2×C8—1222no—
405C2×C8—1222no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—44no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—1222no—
415C4⋊C4—1222no—
416C22⋊C4—44no—
417C22⋊C4—2442no—
418C22⋊C4—2442no—
419C22⋊C4—2442no—
420C22⋊C4—2442no—
421C22⋊C4—2442no—
422C42—22no—
423C42—22no—
424C42—41no—
425C42—41no—
426C42—2442no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—1222no—
434C23—2441no—
435C23—2441no—
436C23—2441no—
437C23—2441no—
438C23—2441no—
439C23—2441no—
440C23#1451976612yes—
441C23C22×C22⋊D6111yescharacteristic
442Q8—84no—
443Q8—1224no—
444Q8—2442no—
445D4—1224no—
446D4—2448no—
447D4—2442no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—1222no—
471C2×C4—2448no—
472C2×C4—2441no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2441no—
476C2×C4—2441no—
477C2×C4—2441no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4#1451976612yes—
481C2×C4#1451980312yes—
482C2×C4#1451980812yes—
483C2×C4#41×D6111yescharacteristic
484C2×C4#41×D6111yescharacteristic
485C2×C4C2×#1175822612yes—
486C2×C4#1451982212yes—
487C8—2442no—
488C8—2442no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—4881no—
497C22—4881no—
498C22—4881no—
499C22—4881no—
500C22—2442no—
501C22—2442no—
502C22—2442no—
503C22#1684026911yescharacteristic
504C22C2×#1451976611yescharacteristic
505C22#1684032111yescharacteristic
506C22C2×#1451980311yescharacteristic
507C22C2×#1451980811yescharacteristic
508C22#1684056011yescharacteristic
509C22#1684056011yescharacteristic
510C4—22no—
511C4—22no—
512C4—22no—
513C4—41no—
514C4—41no—
515C4—82no—
516C4—82no—
517C4—4881no—
518C4—4881no—
519C4—2442no—
520C4#1684043912yes—
521C4#1684043912yes—
522C2—42no—
523C2—4882no—
524C2#2145097811yescharacteristic
525C2#2145596411yescharacteristic
526C2#2145597511yescharacteristic
527C2#2145616711yescharacteristic
528C2#2145621511yescharacteristic
529C2#2145663311yescharacteristic
530C2#2145664911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2857C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenter#2857D61
hypocenterC61#29113
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
2. #21447228C2semidirect
3. #21447223C2semidirect
4. #21446957C2semidirect
5. #21446952C2semidirect
6. #21446515C2semidirect
7. #21446508C2semidirect
8. #21445911C2semidirect
10. #21445581C2semidirect
11. C61×#2857C2semidirect
257. #2857D61semidirect
308. C61#29113semidirect

Maximal subgroups

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

subgroupindex
#214483092
#214472282
#214472232
#214469572
#214469522
#214465152
#214465082
#214459112
#214457572
#214455812
C61×#28572
#2911361