Finite Group Databasebeta

← #30620326

Subgroups — #30620326

Order 15616.

1132 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#21462327C212yesmaximal, complemented
2#21462008C212yesmaximal, complemented
3#21461843C214yesmaximal
4#21461716C212yesmaximal, complemented
5#21461680C211yesmaximal, characteristic
6C4×#14519836C211yesmaximal, characteristic
7#21461518C212yesmaximal, complemented
8C2×#16840453C211yesmaximal, characteristic
9C2×#14519856C2214yes—
10C2×#14519836C2212yes—
11C2×#14519836C2212yes—
12#16840713—22no—
13#16840711—24no—
14#16840705C2214yes—
15#16840704C2212yes—
16C4⋊C4×#6923—22no—
17C4×C22⋊#6923C2212yes—
18#16840640C2211yescharacteristic
19#16840640C2214yes—
20#16840640C2212yes—
21#16840640C2211yescharacteristic
22#16840640C2214yes—
23#16840453—24no—
24#16840453—22no—
25#16840453—22no—
26C2×#14519771C412yescomplemented
27C2×#14519769C411yescharacteristic, complemented
28C2×#14519767C412yescomplemented
29C2×#14519762C2212yes—
30C2×#14519753C411yescharacteristic, complemented
31C2×#14519753C2211yescharacteristic
32C2×C4×D244C2211yescharacteristic
33C2×#14519735C411yescharacteristic, complemented
34C2×#14519734C2212yes—
35C2×C4⋊C4×D61C2211yescharacteristic
36C2×C4×Q8×C61C411yescharacteristic, complemented
37#14519856—28no—
38#14519856—44no—
39C2×C4⋊#6923—22no—
40C2×C4⋊#6923—24no—
41C2×C22⋊#6923C2314yes—
42C2×C22⋊#6923C2312yes—
43C2×C22⋊#6923C2312yes—
44C2×C4×#6923—24no—
45C2×C4×#6923—22no—
46#14519836—24no—
47#14519836—42no—
48(C2×#6923)⋊C4—22no—
49(C2×#6923)⋊C4—24no—
50(C2×#6923)⋊C4—24no—
51(C2×#6923)⋊C4—22no—
52C22×D244C2311yescharacteristic
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61C2312yes—
55C22×C4×D61C2311yescharacteristic
56#14519771—24no—
57#14519771—24no—
58#14519769—22no—
59#14519769—22no—
60#14519767—24no—
61#14519767—24no—
62#14519762—24no—
63#14519762—24no—
64#14519753—22no—
65#14519753—22no—
66#14519753—22no—
67#14519753—22no—
68C4×D244—22no—
69C4×D244—22no—
70#14519735—22no—
71#14519735—21no—
72#14519735—21no—
73#14519734—24no—
74#14519734—24no—
75C2×C4⋊Dic61C2×C411yescharacteristic
76C2×C4⋊Dic61C2×C411yescharacteristic
77C2×C4⋊Dic61C2×C411yescharacteristic
78C2×#11758196C2×C412yes—
79C2×#11758196C2×C412yes—
80C2×#11758196C2×C412yes—
81C2×#11758196C2×C412yes—
82C2×#11758196C2×C412yes—
83C4⋊C4×D61D414yes—
84C4⋊C4×D61D412yes—
85C4⋊C4×D61D412yes—
86C2×#11758198C2×C412yes—
87C2×#11758198C2×C411yescharacteristic
88C2×#11758198C2×C412yes—
89C2×#11758198C2311yescharacteristic
90C2×#11758198C2312yes—
91C2×C4×Dic61C2×C412yes—
92C22×Q8×C61C2×C411yescharacteristic
93C4×Q8×C61—22no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C2×C4⋊C4×C61C2×C412yes—
97C2×C4⋊C4×C61C2×C411yescharacteristic
98C2×C42×C61C2×C412yes—
99C2×C42×C61C2×C411yescharacteristic
100C23×D61C22×C411yescharacteristic
101C23×D61C2411yescharacteristic
102C22×#6923—22no—
103C22×#6923—22no—
104C22×#6923—24no—
105C2×Q8×C61—21no—
106C2×Q8×C61—21no—
107C2×Q8×C61—42no—
108C22×C4×C61C22×C411yescharacteristic
109C22×C4×C61C22×C411yescharacteristic
110C22×C4×C61C22×C412yes—
111C22×C4×C61C22×C411yescharacteristic
112C22×C4×C61C22×C412yes—
113C22×Dic61C22×C412yes—
114C22×Dic61C22×C411yescharacteristic
115C22×Dic61C22×C412yes—
116C22×Dic61C22×C411yescharacteristic
117C2×D244—22no—
118C2×D244—22no—
119C2×D244—21no—
120C2×D244—21no—
121C2×C4×D61—24no—
122C2×C4×D61—22no—
123C2×C4×D61—22no—
124C2×C4×D61—21no—
125C2×C4×D61—21no—
126C2×C4×D61—42no—
127C2×C4×D61—42no—
128C2×C4×D61C2×D414yes—
129C2×C4×D61C2×D414yes—
130C2×C4×D61C2×D412yes—
131C2×C4×D61C2×D412yes—
132C22⋊#6923—22no—
133C22⋊#6923—22no—
134C22⋊#6923—24no—
135C22⋊#6923—24no—
136C22⋊#6923—22no—
137C22⋊#6923#4114yes—
138C4⋊#6923—48no—
139C4⋊#6923—44no—
140C4×#6923—48no—
141C4×#6923—44no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61—22no—
144C4⋊C4×C61C22⋊C412yes—
145C4⋊C4×C61C22⋊C411yescharacteristic
146C4⋊C4×C61C22⋊C411yescharacteristic
147C42×C61—22no—
148C42×C61—22no—
149C42×C61—21no—
150C42×C61—21no—
151#11758198—22no—
152#11758198—24no—
153#11758198—22no—
154#11758198—22no—
155#11758198—22no—
156#11758198—24no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—22no—
160C4⋊Dic61C22⋊C412yes—
161C4⋊Dic61C22⋊C412yes—
162#11758196—24no—
163#11758196—24no—
164#11758196—24no—
165#11758196—24no—
166#11758196C22⋊C414yes—
167#11758196C22⋊C414yes—
168C4×Dic61—22no—
169C4×Dic61—22no—
170C23×C61C23×C411yescharacteristic
171C22×D61—21no—
172C22×D61—21no—
173C22×D61—21no—
174C22×D61—41no—
175C22×D61—41no—
176C22×D61—41no—
177C22×D61—41no—
178C22×D61C22×D412yes—
179C22×D61C2×#4112yes—
180C22×D61H(5,𝔽2)11yescharacteristic
181C22×D612−1+411yescharacteristic
182C2×#6923—24no—
183C2×#6923—42no—
184C2×#6923—42no—
185C2×#6923—42no—
186C2×#6923—44no—
187C2×#6923—42no—
188C2×#6923—44no—
189C2×#6923—44no—
190C2×#6923—42no—
191Q8×C61—42no—
192Q8×C61—42no—
193C2×C4×C61—22no—
194C2×C4×C61—22no—
195C2×C4×C61—22no—
196C2×C4×C61—21no—
197C2×C4×C61—21no—
198C2×C4×C61—22no—
199C2×C4×C61—21no—
200C2×C4×C61—21no—
201C2×C4×C61—41no—
202C2×C4×C61—42no—
203C2×C4×C61C2×C22⋊C412yes—
204C2×C4×C61C2×C22⋊C412yes—
205C2×C4×C61C2×C22⋊C411yescharacteristic
206C2×C4×C61C2×C22⋊C411yescharacteristic
207C2×Dic61—22no—
208C2×Dic61—22no—
209C2×Dic61—21no—
210C2×Dic61—22no—
211C2×Dic61—41no—
212C2×Dic61—41no—
213C2×Dic61—42no—
214C2×Dic61C2×C22⋊C414yes—
215C2×Dic61C2×C22⋊C412yes—
216D244—42no—
217D244—42no—
218C4×D61—24no—
219C4×D61—24no—
220C4×D61—24no—
221C4×D61—24no—
222C4×D61—24no—
223C4×D61—42no—
224C4×D61—42no—
225C4×D61—42no—
226#22478—611nomaximal, Sylow
227C22×C61—21no—
228C22×C61—21no—
229C22×C61C22×C22⋊C411yescharacteristic
230C22×C61C4×#4111yescharacteristic
231C22×C61#52011yescharacteristic
232C2×D61—22no—
233C2×D61—22no—
234C2×D61—42no—
235C2×D61—41no—
236C2×D61—41no—
237C2×D61—41no—
238C2×D61—41no—
239C2×D61—41no—
240C2×D61—81no—
241C2×D61—81no—
242C2×D61#54612yes—
243C2×D61#54712yes—
244#6923—44no—
245#6923—84no—
246#6923—82no—
247C4×C61—22no—
248C4×C61—22no—
249C4×C61—21no—
250C4×C61—41no—
251C4×C61—42no—
252C4×C61—41no—
253C4×C61#41012yes—
254Dic61—24no—
255Dic61—22no—
256Dic61—41no—
257Dic61—42no—
258C2×#552—611no—
259#2344—612no—
260#1996—612no—
261#1899—614no—
262#1820—612no—
263#1799—611no—
264C4×#99—611no—
265#1740—612no—
266C2×C61—21no—
267C2×C61—21no—
268C2×C61#233911yescharacteristic
269C2×C61C2×#41011yescharacteristic
270C2×C61#288411yescharacteristic
271D61—22no—
272D61—22no—
273D61—81no—
274#552—1222no—
275#552—1224no—
276#552—1222no—
277C2×#126—612no—
278C2×#125—612no—
279C2×#125—611no—
280C2×#123—612no—
281C2×C22⋊Q8—611no—
282C2×C22⋊Q8—611no—
283C2×C4×Q8#692311yescharacteristic, complemented
284C2×C4×D4—611no—
285C2×#117—611no—
286C2×#117—612no—
287C2×#117—614no—
288C22×C4⋊C4—611no—
289C2×#99—612no—
290C2×#99—612no—
291#389—1222no—
292#387—1224no—
293#381—614no—
294#380—612no—
295C4×C4⋊C4—1222no—
296C4×C22⋊C4—612no—
297#342—612no—
298#342—611no—
299#342—614no—
300#342—614no—
301#342—611no—
302C61#2247811yescharacteristic, complemented, Sylow
303C22×Q8C2×#692311yescharacteristic
304C22×D4—611no—
305C23×C4—612no—
306C23×C4—611no—
307C23×C4—611no—
308#126—1224no—
309#126—1224no—
310#125—1222no—
311#125—1222no—
312#125—1224no—
313#125—1224no—
314#123—1224no—
315#123—1224no—
316C22⋊Q8—1222no—
317C22⋊Q8—1222no—
318C22⋊Q8—1222no—
319C22⋊Q8—1222no—
320C4×Q8—22no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×D4—1222no—
324C4×D4—1222no—
325#117—1224no—
326#117—1224no—
327#117—1222no—
328#117—1221no—
329#117—1221no—
330#117—1228no—
331#117—2444no—
332C2×C4⋊C4—611no—
333C2×C4⋊C4—612no—
334C2×C4⋊C4—612no—
335C2×C4⋊C4—614no—
336C2×C4⋊C4—612no—
337C2×C4⋊C4—612no—
338C2×C4⋊C4—611no—
339C2×C4⋊C4—612no—
340C2×C4⋊C4—611no—
341C2×C4⋊C4—612no—
342C2×C4⋊C4—612no—
343C2×C4⋊C4—1224no—
344C2×C4⋊C4—1222no—
345C2×C4⋊C4C2×#692312yes—
346C2×C4⋊C4C2×#692311yescharacteristic
347C2×C22⋊C4—612no—
348C2×C22⋊C4—611no—
349C2×C22⋊C4—612no—
350C2×C22⋊C4—612no—
351C2×C22⋊C4—611no—
352C2×C22⋊C4—612no—
353C2×C22⋊C4—614no—
354C2×C22⋊C4—612no—
355C2×C42—612no—
356C2×C42—1224no—
357C2×C42—1222no—
358C2×C42C2×#692312yes—
359C2×C42C2×#692311yescharacteristic
360#99—1224no—
361#99—2442no—
362#95—1222no—
363#95—1224no—
364#95—1222no—
365#95—1224no—
366C24—611no—
367C24—611no—
368C2×Q8—21no—
369C2×Q8—21no—
370C2×Q8—42no—
371C2×D4—1222no—
372C2×D4—1221no—
373C2×D4—1222no—
374C2×D4—1221no—
375C22×C4—612no—
376C22×C4—611no—
377C22×C4—614no—
378C22×C4—614no—
379C22×C4—612no—
380C22×C4—612no—
381C22×C4—611no—
382C22×C4—612no—
383C22×C4—1224no—
384C22×C4—1222no—
385C22×C4—1222no—
386C22×C4—1222no—
387C22×C4—1221no—
388C22×C4—1221no—
389C22×C4—1222no—
390C22×C4—1224no—
391C22×C4—2442no—
392C22×C4—2442no—
393C22×C4C22×#692311yescharacteristic
394C22×C4C22×#692312yes—
395C22×C4C22×#692311yescharacteristic
396C22×C4C22×#692311yescharacteristic
397C22×C4C22×#692312yes—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—614no—
401C4⋊C4—614no—
402C4⋊C4—612no—
403C4⋊C4—612no—
404C4⋊C4—1224no—
405C4⋊C4—1224no—
406C4⋊C4—1224no—
407C4⋊C4—1224no—
408C4⋊C4—1222no—
409C4⋊C4—1221no—
410C4⋊C4—1221no—
411C4⋊C4—2448no—
412C4⋊C4—2444no—
413C4⋊C4C22⋊#692312yes—
414C4⋊C4C22⋊#692311yescharacteristic
415C4⋊C4C22⋊#692311yescharacteristic
416C22⋊C4—614no—
417C22⋊C4—1222no—
418C22⋊C4—1224no—
419C22⋊C4—1222no—
420C22⋊C4—1224no—
421C22⋊C4—1222no—
422C22⋊C4—1222no—
423C22⋊C4—1222no—
424C22⋊C4—1222no—
425C22⋊C4—1224no—
426C22⋊C4—1224no—
427C22⋊C4—1222no—
428C42—22no—
429C42—22no—
430C42—21no—
431C42—21no—
432C42—1222no—
433C42—1222no—
434C42—2448no—
435C42—2444no—
436C23—612no—
437C23—612no—
438C23—611no—
439C23—611no—
440C23—1221no—
441C23—1221no—
442C23—1221no—
443C23—2441no—
444C23—2441no—
445C23—2441no—
446C23—2441no—
447C23C23×#692311yescharacteristic
448Q8—42no—
449Q8—42no—
450D4—2442no—
451D4—2442no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—614no—
463C2×C4—612no—
464C2×C4—1224no—
465C2×C4—1224no—
466C2×C4—1224no—
467C2×C4—1224no—
468C2×C4—1222no—
469C2×C4—1222no—
470C2×C4—1224no—
471C2×C4—1224no—
472C2×C4—1221no—
473C2×C4—1222no—
474C2×C4—2442no—
475C2×C4—2442no—
476C2×C4—2441no—
477C2×C4—2444no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4—2444no—
481C2×C4—2442no—
482C2×C4—2444no—
483C2×C4—2442no—
484C2×C4—2442no—
485C2×C4—2442no—
486C2×C4—2442no—
487C2×C4—2441no—
488C2×C4C2×C22⋊#692312yes—
489C2×C4C2×C22⋊#692312yes—
490C2×C4C2×C22⋊#692311yescharacteristic
491C2×C4C2×C22⋊#692311yescharacteristic
492C22—21no—
493C22—21no—
494C22—612no—
495C22—612no—
496C22—1222no—
497C22—1222no—
498C22—4881no—
499C22—4881no—
500C22—2442no—
501C22—2441no—
502C22—2441no—
503C22—2441no—
504C22—2441no—
505C22—2441no—
506C22C22×C22⋊#692311yescharacteristic
507C22#41×#692311yescharacteristic
508C22#1684083111yescharacteristic
509C4—22no—
510C4—22no—
511C4—21no—
512C4—41no—
513C4—42no—
514C4—41no—
515C4—1224no—
516C4—1222no—
517C4—4884no—
518C4—4882no—
519C4—2444no—
520C4—2441no—
521C4—2442no—
522C4#1684074112yes—
523C2—21no—
524C2—21no—
525C2—1222no—
526C2—1222no—
527C2—4881no—
528C2#2146232211yescharacteristic
529C2C2×#1684074111yescharacteristic
530C2#2146242011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC22×D244C23
cosocleC23×D61C24
hypercenterC2×C4×Q8#6923
hypocenterC61#22478
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. #21462327C2semidirect
2. #21462008C2semidirect
4. #21461716C2semidirect
7. #21461518C2semidirect
26. C2×#14519771C4semidirect
27. C2×#14519769C4semidirect
28. C2×#14519767C4semidirect
30. C2×#14519753C4semidirect
33. C2×#14519735C4semidirect
36. C2×C4×Q8×C61C4semidirect
283. C2×C4×Q8#6923semidirect
302. C61#22478semidirect

Maximal subgroups

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

subgroupindex
#214623272
#214620082
#214618432
#214617162
#214616802
C4×#145198362
#214615182
C2×#168404532
#2247861