Finite Group Databasebeta

← #30610342

Subgroups — #30610342

Order 15616.

1170 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462719C212yesmaximal, complemented
2#21461090C212yesmaximal, complemented
3#21461088C214yesmaximal, complemented
4#21461080C212yesmaximal, complemented
5#21461078C211yesmaximal, characteristic, complemented
6#21461060C212yesmaximal, complemented
7#21461058C211yesmaximal, characteristic, complemented
8C2×#16840422C211yesmaximal, characteristic
9#16840814—24no—
10#16840812—22no—
11#16840811—22no—
12#16840783C2212yes—
13#16840782C2214yes—
14#16840779C2212yes—
15#16840778C2214yes—
16#16840606C2211yescharacteristic
17#16840606C2211yescharacteristic
18#16840606C2212yes—
19#16840606C2214yes—
20#16840606C2214yes—
21#16840605C2212yes—
22#16840605C2212yes—
23C22×#11758222C411yescharacteristic
24C22×Q8×D61C411yescharacteristic
25#16840422—22no—
26#16840422—22no—
27#16840422—24no—
28C2×#14519766C2211yescharacteristic
29C2×#14519763C2212yes—
30C2×#14519755C412yes—
31C2×#14519752C411yescharacteristic
32C2×#14519750C412yes—
33C2×#14519742C2211yescharacteristic
34C2×#14519739C2212yes—
35C2×C4×C22⋊D61C2211yescharacteristic
36C2×C22⋊Q8×C61C411yescharacteristic
37#14519844—22no—
38#14519844—24no—
39C4⋊#92420—24no—
40C4⋊#92420—22no—
41C22⋊#92420C2312yes—
42C22⋊#92420C2314yes—
43C22⋊#92420C2312yes—
44#14519832—24no—
45#14519832—22no—
46C4×#92420—24no—
47C4×#92420—22no—
48C2×#11758222—21no—
49C2×#11758222—22no—
50C2×#11758222—21no—
51C2×#11758222—44no—
52C22×Dic122C2×C411yescharacteristic
53C22×Dic122C2×C412yes—
54C2×Q8×D61—21no—
55C2×Q8×D61—22no—
56C2×Q8×D61—21no—
57C2×Q8×D61—44no—
58C22×D244C2×C412yes—
59C22×D244C2×C411yescharacteristic
60C22×C22⋊D61C2×C411yescharacteristic
61C22×C22⋊D61C2×C412yes—
62C22×C4×D61C2×C411yescharacteristic
63C22×C4×D61C2×C412yes—
64C22×C4×D61C2×C412yes—
65#14519766—22no—
66#14519766—22no—
67#14519763—24no—
68#14519763—24no—
69#14519755—24no—
70#14519755—24no—
71#14519752—22no—
72#14519752—22no—
73#14519750—24no—
74#14519750—24no—
75#14519742D412yes—
76#14519742D412yes—
77#14519742D414yes—
78#14519739—24no—
79#14519739—24no—
80C4×C22⋊D61—22no—
81C4×C22⋊D61—22no—
82C2×#11758196C2×C411yescharacteristic
83C2×#11758196C2×C412yes—
84C2×C22⋊Dic61C2311yescharacteristic
85C2×#11758198C2×C412yes—
86C2×#11758198C2×C412yes—
87C2×#11758198C2311yescharacteristic
88C2×#11758198C2312yes—
89C2×C4×Dic61C2311yescharacteristic
90C2×C4×Dic61C2312yes—
91C22×Q8×C61C2×C411yescharacteristic
92C23×C4×C61C2×C411yescharacteristic
93C22⋊Q8×C61—22no—
94C22⋊Q8×C61—22no—
95C2×C4⋊C4×C61C2×C412yes—
96C2×C4⋊C4×C61C2×C411yescharacteristic
97C2×C22⋊C4×C61C2×C412yes—
98C24×C61C22×C411yescharacteristic
99C23×D61C22×C412yes—
100C23×D61C22×C411yescharacteristic
101C2×Q8×C61—21no—
102C2×Q8×C61—21no—
103C2×Q8×C61—42no—
104C22×C4×C61—21no—
105C22×C4×C61—21no—
106C22×C4×C61—42no—
107C22×C4×C61C22×C411yescharacteristic
108C22×C4×C61C22×C412yes—
109C22×C4×C61C22×C411yescharacteristic
110C22×C4×C61C22×C412yes—
111C2×C22⋊D61—22no—
112C2×C22⋊D61—22no—
113C2×C22⋊D61—22no—
114C2×C22⋊D61—22no—
115C2×C22⋊D61—41no—
116C2×C22⋊D61—44no—
117C22×Dic61C22×C412yes—
118C22×Dic61C22×C411yescharacteristic
119C22×Dic61C22×C411yescharacteristic
120C22×Dic61C2411yescharacteristic
121Q8×D61—44no—
122Q8×D61—48no—
123Q8×D61—44no—
124#11758222—44no—
125#11758222—48no—
126#11758222—44no—
127C2×D244—24no—
128C2×D244—21no—
129C2×D244—21no—
130C2×D244—24no—
131C2×D244—42no—
132C2×D244—42no—
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×C4×D61—22no—
136C2×C4×D61—22no—
137C2×C4×D61—22no—
138C2×C4×D61—22no—
139C2×C4×D61—44no—
140C2×C4×D61—44no—
141C2×C4×D61—42no—
142C2×Dic122—24no—
143C2×Dic122—22no—
144C2×Dic122—22no—
145C2×Dic122—44no—
146C2×Dic122C22⋊C411yescharacteristic
147C2×Dic122C22⋊C412yes—
148C2×Dic122C22⋊C411yescharacteristic
149C2×#92420—22no—
150C2×#92420—24no—
151C2×#92420—22no—
152C4⋊C4×C61—24no—
153C4⋊C4×C61C22⋊C412yes—
154C4⋊C4×C61C22⋊C412yes—
155C22⋊C4×C61—24no—
156C22⋊Dic61—22no—
157#11758198—24no—
158#11758198—24no—
159#11758198—22no—
160#11758198—24no—
161#11758196—22no—
162#11758196C22⋊C414yes—
163#11758196C22⋊C414yes—
164C4×Dic61C2×D412yes—
165C4×Dic61C2×D414yes—
166C4×Dic61C2×D414yes—
167C4×Dic61C2×D412yes—
168C23×C61—21no—
169C23×C61—21no—
170C23×C61—21no—
171C23×C61—41no—
172C23×C61—41no—
173C23×C61C23×C411yescharacteristic
174C22×D61—21no—
175C22×D61—21no—
176C22×D61—22no—
177C22×D61—22no—
178C22×D61—22no—
179C22×D61—41no—
180C22×D61—41no—
181C22×D61—42no—
182C22×D61—42no—
183C22×D61#13112yes—
184Q8×C61—42no—
185Q8×C61—42no—
186C2×C4×C61—22no—
187C2×C4×C61—22no—
188C2×C4×C61—22no—
189C2×C4×C61—21no—
190C2×C4×C61—22no—
191C2×C4×C61—42no—
192C2×C4×C61—42no—
193C2×C4×C61—42no—
194C2×C4×C61—41no—
195C2×C4×C61C2×C22⋊C414yes—
196C2×C4×C61C2×C22⋊C411yescharacteristic
197C2×C4×C61C2×C22⋊C411yescharacteristic
198C22⋊D61—44no—
199C22⋊D61—44no—
200C22⋊D61—44no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—24no—
204C2×Dic61—21no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61C2×C22⋊C412yes—
208C2×Dic61C2×C22⋊C412yes—
209C2×Dic61C2×C22⋊C412yes—
210C2×Dic61C22×D412yes—
211D244—48no—
212D244—42no—
213D244—42no—
214C4×D61—44no—
215C4×D61—44no—
216C4×D61—42no—
217C4×D61—42no—
218C4×D61—44no—
219C4×D61—44no—
220Dic122—22no—
221Dic122—24no—
222Dic122—44no—
223Dic122—44no—
224Dic122#10014yes—
225#92420—42no—
226#92420—44no—
227#92420—42no—
228#17425—611nomaximal, Sylow
229C22×C61—21no—
230C22×C61—21no—
231C22×C61—41no—
232C22×C61—41no—
233C22×C61—41no—
234C22×C61—81no—
235C22×C61—81no—
236C22×C61C22×C22⋊C411yescharacteristic
237C22×C61C2×#13111yescharacteristic
238C22×C61#56811yescharacteristic
239C2×D61—22no—
240C2×D61—42no—
241C2×D61—41no—
242C2×D61—41no—
243C2×D61—42no—
244C2×D61—42no—
245C2×D61—42no—
246C2×D61—42no—
247C2×D61—82no—
248C2×D61—82no—
249C4×C61—24no—
250C4×C61—21no—
251C4×C61—42no—
252C4×C61—41no—
253C4×C61C2×#10012yes—
254Dic61—22no—
255Dic61—22no—
256Dic61—22no—
257Dic61—41no—
258Dic61C2×#10014yes—
259C2×#548—611no—
260#2984—612no—
261#1469—614no—
262#1469—612no—
263#1466—611no—
264#1466—612no—
265#1460—611no—
266#1460—612no—
267C2×C61—21no—
268C2×C61—21no—
269C2×C61—81no—
270C2×C61#288011yescharacteristic
271C2×C61#288411yescharacteristic
272C2×C61C22×#10011yescharacteristic
273D61—42no—
274D61—82no—
275C22×#41—611no—
276C23×Q8—611no—
277#548—1222no—
278#548—1224no—
279#548—1222no—
280C2×#124—611no—
281C2×#124—612no—
282C2×C22⋊Q8—612no—
283C2×C22⋊Q8#692311yescharacteristic
284C2×#121—611no—
285C2×#121—612no—
286C2×C4×Q8—611no—
287C2×C4×D4—611no—
288C2×C4×D4—612no—
289#446—1222no—
290#446—1224no—
291C8×Q8—1222no—
292#432—614no—
293#432—612no—
294#431—614no—
295#431—612no—
296#323—612no—
297#323—612no—
298#323—611no—
299#323—611no—
300#323—612no—
301#323—614no—
302#323—614no—
303C61#1742511yescharacteristic, complemented, Sylow
304C2×#41—1221no—
305C2×#41—1222no—
306C2×#41—1221no—
307C2×#41—2444no—
308C22×Q8—611no—
309C22×Q8—612no—
310C22×Q8—1221no—
311C22×Q8—1222no—
312C22×Q8—1221no—
313C22×Q8—2444no—
314C22×Q8C2×#692311yescharacteristic
315C22×D4—611no—
316C22×D4—612no—
317C22×D4—612no—
318C22×D4—611no—
319C23×C4—612no—
320C23×C4—611no—
321C23×C4—612no—
322C23×C4C2×#692311yescharacteristic
323#124—1224no—
324#124—1224no—
325#124—1222no—
326#124—1222no—
327C22⋊Q8—22no—
328C22⋊Q8—22no—
329C22⋊Q8—1224no—
330C22⋊Q8—1224no—
331#121—1224no—
332#121—1224no—
333#121—1222no—
334#121—1222no—
335C4×Q8—612no—
336C4×Q8—612no—
337C4×Q8—614no—
338C4×D4—1224no—
339C4×D4—1224no—
340C4×D4—1222no—
341C4×D4—1222no—
342C2×C4⋊C4—612no—
343C2×C4⋊C4—611no—
344C2×C4⋊C4C2×#692312yes—
345C2×C4⋊C4C2×#692311yescharacteristic
346C2×C22⋊C4—611no—
347C2×C22⋊C4—611no—
348C2×C22⋊C4—612no—
349C2×C22⋊C4—612no—
350C2×C22⋊C4—612no—
351C2×C22⋊C4C2×#692312yes—
352C2×C42—611no—
353C2×C42—612no—
354C4⋊C8—1224no—
355C4⋊C8—1222no—
356C4⋊C8—1222no—
357C4⋊C8—1224no—
358C22⋊C8—612no—
359C22⋊C8—614no—
360C22⋊C8—612no—
361#97—1224no—
362#97—1222no—
363C4×C8—1224no—
364C4×C8—1222no—
365C24—611no—
366C24—612no—
367C24C22×#692311yescharacteristic
368#41—2444no—
369#41—2448no—
370#41—2444no—
371C2×Q8—21no—
372C2×Q8—21no—
373C2×Q8—42no—
374C2×Q8—611no—
375C2×Q8—612no—
376C2×Q8—611no—
377C2×Q8—1222no—
378C2×Q8—1222no—
379C2×Q8—1224no—
380C2×Q8—2444no—
381C2×Q8—2444no—
382C2×Q8—2448no—
383C2×Q8—2444no—
384C2×D4—1224no—
385C2×D4—1222no—
386C2×D4—1222no—
387C2×D4—1222no—
388C2×D4—1222no—
389C2×D4—1224no—
390C2×D4—1221no—
391C2×D4—1221no—
392C2×D4—2442no—
393C2×D4—2442no—
394C2×D4—2441no—
395C2×D4—2444no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—42no—
399C22×C4—611no—
400C22×C4—612no—
401C22×C4—611no—
402C22×C4—611no—
403C22×C4—1222no—
404C22×C4—1222no—
405C22×C4—1222no—
406C22×C4—1222no—
407C22×C4—1221no—
408C22×C4—1221no—
409C22×C4—2442no—
410C22×C4—2444no—
411C22×C4—2444no—
412C22×C4C22×#692312yes—
413C22×C4C22×#692311yescharacteristic
414C22×C4C22×#692312yes—
415C22×C4C22×#692311yescharacteristic
416C2×C8—1222no—
417C2×C8—1224no—
418C2×C8—1222no—
419C4⋊C4—24no—
420C4⋊C4—614no—
421C4⋊C4—614no—
422C4⋊C4—1222no—
423C4⋊C4C22⋊#692312yes—
424C4⋊C4C22⋊#692312yes—
425C22⋊C4—24no—
426C22⋊C4—1222no—
427C22⋊C4—1224no—
428C22⋊C4—1224no—
429C22⋊C4—1222no—
430C22⋊C4—1224no—
431C42—614no—
432C42—612no—
433C42—612no—
434C42—614no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—41no—
439C23—41no—
440C23—612no—
441C23—1221no—
442C23—1221no—
443C23—1222no—
444C23—1222no—
445C23—1222no—
446C23—2442no—
447C23—2442no—
448C23—2441no—
449C23—2441no—
450C23C23×#692311yescharacteristic
451Q8—42no—
452Q8—42no—
453Q8—614no—
454Q8—1224no—
455Q8—1222no—
456Q8—2444no—
457Q8—2444no—
458D4—2448no—
459D4—2442no—
460D4—2444no—
461D4—2444no—
462D4—2444no—
463D4—2442no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—21no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—612no—
474C2×C4—612no—
475C2×C4—612no—
476C2×C4—612no—
477C2×C4—1224no—
478C2×C4—1221no—
479C2×C4—1221no—
480C2×C4—1221no—
481C2×C4—2442no—
482C2×C4—2444no—
483C2×C4—2444no—
484C2×C4—2444no—
485C2×C4—2444no—
486C2×C4—2441no—
487C2×C4—2442no—
488C2×C4—2441no—
489C2×C4C2×C22⋊#692314yes—
490C2×C4C2×C22⋊#692311yescharacteristic
491C2×C4C2×C22⋊#692311yescharacteristic
492C8—2442no—
493C8—2444no—
494C8—2442no—
495C22—21no—
496C22—21no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—81no—
501C22—81no—
502C22—1222no—
503C22—4882no—
504C22—4882no—
505C22—2442no—
506C22—2442no—
507C22—2442no—
508C22—2442no—
509C22—2442no—
510C22—2441no—
511C22—2441no—
512C22C22×C22⋊#692311yescharacteristic
513C22D61⋊#13111yescharacteristic
514C22#1684084111yescharacteristic
515C4—24no—
516C4—21no—
517C4—41no—
518C4—42no—
519C4—614no—
520C4—1222no—
521C4—1222no—
522C4—1222no—
523C4—2441no—
524C4D61⋊#10012yes—
525C2—21no—
526C2—21no—
527C2—81no—
528C2—4882no—
529C2—2442no—
530C2#2146240911yescharacteristic
531C2#2146242011yescharacteristic
532C2C2×D61⋊#10011yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC23×C61C23×C4
FrattiniC23C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#16840422C2
cosocleC22×Dic61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61#17425
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #21462719C2semidirect
2. #21461090C2semidirect
3. #21461088C2semidirect
4. #21461080C2semidirect
5. #21461078C2semidirect
6. #21461060C2semidirect
7. #21461058C2semidirect
303. C61#17425semidirect

Maximal subgroups

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

subgroupindex
#214627192
#214610902
#214610882
#214610802
#214610782
#214610602
#214610582
C2×#168404222
#1742561