Finite Group Databasebeta

← #30612381

Subgroups — #30612381

Order 15616.

688 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462628C211yesmaximal, characteristic, complemented
2#21462623C211yesmaximal, characteristic, complemented
3#21461433C212yesmaximal, complemented
4#21461429C212yesmaximal, complemented
5#21461425C212yesmaximal, complemented
6#21461421C212yesmaximal, complemented
7#21461366C211yesmaximal, characteristic, complemented
8#21461112C211yesmaximal, characteristic, complemented
9#21461110C211yesmaximal, characteristic, complemented
10C2×#92420⋊C8C211yesmaximal, characteristic, complemented
11#21454587C211yesmaximal, characteristic
12#16840813C2212yes—
13#16840811C2212yes—
14#16840797C2212yes—
15#16840793—22no—
16#16840790C2212yes—
17#16840789—22no—
18#16840780—21no—
19#16840779—21no—
20#16840778C2211yescharacteristic
21#16840778C2211yescharacteristic
22C4×#11758217—21no—
23C4×D61⋊C8—21no—
24C2×C4×#92420C2211yescharacteristic
25C2×C4×#92420C2211yescharacteristic
26#16840625C2212yes—
27#16840623C2212yes—
28#16840612C2212yes—
29Q8⋊#92420C2212yes—
30#16840610C2212yes—
31D4⋊#92420C2212yes—
32#92420⋊C8C2212yes—
33#92420⋊C8C2212yes—
34C2×#14519797C411yescharacteristic
35#41×Dic61C2211yescharacteristic
36#16840230C2211yescharacteristic
37#16839854C411yescharacteristic
38#16839854C2211yescharacteristic
39#16839853C411yescharacteristic
40#16839853C2211yescharacteristic
41#16839847C411yescharacteristic
42#16839847C2211yescharacteristic
43#16839842C411yescharacteristic
44#16839842C2211yescharacteristic
45#16839798C411yescharacteristic
46#16839759C411yescharacteristic
47C2×C8×Dic61C2211yescharacteristic
48C61×#416C411yescharacteristic
49C2×#11758217—22no—
50C2×D61⋊C8—22no—
51C22×#92420—21no—
52C22×#92420D412yes—
53#14519844—22no—
54#14519844—22no—
55C4⋊#92420C2312yes—
56C4⋊#92420C2312yes—
57#14519834—22no—
58C22⋊#92420—21no—
59C22⋊#92420—21no—
60C22⋊#92420—22no—
61#14519832—22no—
62C4×#92420—22no—
63C4×#92420—21no—
64C4×#92420—21no—
65C4×#92420—21no—
66C4×#92420D411yescharacteristic
67C4×#92420D411yescharacteristic
68C4×#92420C2312yes—
69C4×#92420C2312yes—
70C2×#11758222C2×C411yescharacteristic
71#14519797—24no—
72C2×#11758202C2×C411yescharacteristic
73C2×#11758200C2×C411yescharacteristic
74C2×#11758201C2×C411yescharacteristic
75C2×#11758199C2×C411yescharacteristic
76C22×#92418C2×C411yescharacteristic
77Q8×Dic61C2×C411yescharacteristic
78Q8×Dic61C2311yescharacteristic
79#14519742C2×C411yescharacteristic
80#14519742C2311yescharacteristic
81D4×Dic61—21no—
82D4×Dic61—21no—
83D4×Dic61C2×C411yescharacteristic
84D4×Dic61C2311yescharacteristic
85#14519739C2×C411yescharacteristic
86#14519739C2311yescharacteristic
87#14519737—21no—
88#14519737—21no—
89#14519736—21no—
90#14519736C2×C411yescharacteristic
91#14519734—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61C2311yescharacteristic
94C42×D61—21no—
95#14519710C2×C411yescharacteristic
96#14519710C2×C411yescharacteristic
97#14519708C2×C411yescharacteristic
98#14519708C2×C411yescharacteristic
99#14519702C2×C411yescharacteristic
100#14519702C2×C411yescharacteristic
101#14519699C2×C411yescharacteristic
102#14519699C2×C411yescharacteristic
103C8×Dic61C2×C411yescharacteristic
104C8×Dic61C2×C411yescharacteristic
105C8×Dic61C2311yescharacteristic
106C8×Dic61C2311yescharacteristic
107C2×#41×C61C2×C411yescharacteristic
108C22×C8×C61C2×C411yescharacteristic
109#117×C61C2×C411yescharacteristic
110#103×C61C2×C411yescharacteristic
111#103×C61C2×C411yescharacteristic
112#102×C61C2×C411yescharacteristic
113#102×C61C2×C411yescharacteristic
114#41×C61—22no—
115#41×C61—22no—
116C2×Q8×C61C22×C411yescharacteristic
117C2×D4×C61—21no—
118C2×D4×C61C22×C411yescharacteristic
119C22×C4×C61—21no—
120C22×C4×C61C22×C411yescharacteristic
121C2×C22⋊D61—21no—
122C22×Dic61—21no—
123C22×Dic61—21no—
124C22×Dic61C2×D411yescharacteristic
125C22×Dic61#4111yescharacteristic
126#11758222—22no—
127#11758222M1614yescomplemented
128C2×D244C22×C411yescharacteristic
129C2×C4×D61—21no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×Dic122C22×C411yescharacteristic
133#11758217—44no—
134C2×#92420—24no—
135C2×#92420—22no—
136C2×#92420—22no—
137C2×#92420—22no—
138C2×#92420—42no—
139C2×#92420C2×D412yes—
140C2×#92420C2×D412yes—
141D61⋊C8—44no—
142C2×C8×C61—22no—
143C2×C8×C61C22×C411yescharacteristic
144C2×C8×C61C22×C411yescharacteristic
145C4⋊C4×C61C22×C411yescharacteristic
146C4⋊C4×C61C22×C411yescharacteristic
147C22⋊C4×C61—21no—
148C42×C61—21no—
149C22⋊Dic61—21no—
150C22⋊Dic61—21no—
151C22⋊Dic61—21no—
152#11758202—22no—
153#11758201—22no—
154#11758200—22no—
155#11758199—22no—
156#11758198—21no—
157#11758198—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61C22×C411yescharacteristic
161C4⋊Dic61C22×C411yescharacteristic
162#11758196—21no—
163#11758196—21no—
164C4×Dic61—21no—
165C4×Dic61—21no—
166C4×Dic61—21no—
167C4×Dic61—21no—
168C4×Dic61—21no—
169C4×Dic61C22×C411yescharacteristic
170C4×Dic61C2×D411yescharacteristic
171C4×Dic61#4111yescharacteristic
172C4×Dic61C2411yescharacteristic
173C2×#92418—22no—
174C2×#92418C22×C411yescharacteristic
175C2×#92418C22×C411yescharacteristic
176C23×C61—21no—
177C23×C61C4×D411yescharacteristic
178C22×D61—21no—
179Q8×C61—21no—
180Q8×C61—21no—
181D4×C61—21no—
182D4×C61—21no—
183D4×C61—42no—
184C2×C4×C61—21no—
185C2×C4×C61—21no—
186C2×C4×C61—21no—
187C2×C4×C61—42no—
188C2×C4×C61C4×D411yescharacteristic
189C2×C4×C61C2×M1612yes—
190C2×C4×C61#13112yes—
191C2×C4×C61C23×C411yescharacteristic
192C22⋊D61—24no—
193C2×Dic61—21no—
194C2×Dic61—22no—
195C2×Dic61—22no—
196C2×Dic61—21no—
197C2×Dic61—21no—
198C2×Dic61—42no—
199C2×Dic61—42no—
200C2×Dic61C4×D411yescharacteristic
201C2×Dic61C4×D411yescharacteristic
202C2×Dic61C22×D411yescharacteristic
203C2×Dic61C2×#4111yescharacteristic
204D244—21no—
205D244C2×M1612yes—
206C4×D61—24no—
207C4×D61—42no—
208C4×D61—42no—
209Dic122—21no—
210Dic122C2×M1612yes—
211#92420—22no—
212#92420—42no—
213#92420—42no—
214#92420—42no—
215#92420#13514yescomplemented
216C8×C61—21no—
217C8×C61—21no—
218#92418—21no—
219#92418—21no—
220#17863—611nomaximal, Sylow
221C22×C61—21no—
222C22×C61—21no—
223C22×C61—41no—
224C22×C61—41no—
225C22×C61—41no—
226C22×C61#43512yes—
227C22×C61C2×C4×D411yescharacteristic
228C2×D61—22no—
229C2×D61—41no—
230C2×D61—41no—
231C4×C61—41no—
232C4×C61—41no—
233C4×C61—41no—
234C4×C61#43512yes—
235C4×C61C22×M1611yescharacteristic
236C4×C61C2×#13111yescharacteristic
237Dic61—22no—
238Dic61—21no—
239Dic61—21no—
240Dic61—41no—
241Dic61—41no—
242Dic61#44412yes—
243Dic61C2×#13512yes—
244#2965—611no—
245C8×#41—611no—
246C4×#135—611no—
247#1584—612no—
248#1583—612no—
249#1582—612no—
250#1581—612no—
251#1564—611no—
252#1476—611no—
253#1476—611no—
254C2×#322—611no—
255C2×C61—22no—
256C2×C61—81no—
257C2×C61C2×#43511yescharacteristic
258C2×C61C4×#13511yescharacteristic
259C2×C61C2×#44411yescharacteristic
260D61—42no—
261C2×#135—611no—
262C4×#41—611no—
263C4×#41—611no—
264#446—612no—
265C8×Q8—612no—
266C4×Q16—611no—
267C4×Q16—611no—
268C4×SD16—611no—
269C4×SD16—611no—
270C4×SD16—611no—
271C4×SD16—611no—
272C4×D8—611no—
273C4×D8—611no—
274#436—612no—
275#435—1222no—
276C8×D4—612no—
277C8×D4—1222no—
278#431—611no—
279#431—611no—
280#431—1221no—
281#431—1221no—
282#427—611no—
283#416—611no—
284#416#692311yescharacteristic
285C4×M16—1221no—
286C2×C4×C8—611no—
287C2×C4×C8—611no—
288C2×C4×C8—611no—
289C2×C4×C8—1221no—
290#335—612no—
291#334—612no—
292Q8⋊C8—612no—
293Q8⋊C8—612no—
294D4⋊C8—612no—
295D4⋊C8—612no—
296#322—612no—
297#322—612no—
298C61#1786311yescharacteristic, complemented, Sylow
299C2×#41—611no—
300C2×#41C2×#692311yescharacteristic
301#135—1224no—
302C2×Q16—611no—
303C2×SD16—611no—
304C2×SD16—611no—
305C2×D8—611no—
306C2×M16—1222no—
307C22×C8—612no—
308C22×C8—611no—
309C22×C8—1222no—
310C22×C8—1221no—
311C22×C8C2×#692311yescharacteristic
312C4×Q8—611no—
313C4×Q8—611no—
314C4×Q8—611no—
315C4×Q8—611no—
316C4×D4—611no—
317C4×D4—611no—
318C4×D4—611no—
319C4×D4—611no—
320C4×D4—1221no—
321C4×D4—1221no—
322C4×D4—1221no—
323C4×D4—1221no—
324#117—611no—
325#117—1221no—
326#117—1221no—
327#117C2×#692311yescharacteristic
328C2×C42—611no—
329C2×C42—1221no—
330C2×C42—1221no—
331#107—611no—
332#107—611no—
333#106—611no—
334#106—611no—
335C4⋊C8—612no—
336C4⋊C8—612no—
337C4⋊C8—1222no—
338C4⋊C8—1222no—
339#103—611no—
340#103—611no—
341#103C2×#692311yescharacteristic
342#103C2×#692311yescharacteristic
343#102—611no—
344#102—611no—
345#102C2×#692311yescharacteristic
346#102C2×#692311yescharacteristic
347C22⋊C8—1222no—
348C22⋊C8—1222no—
349C22⋊C8—1221no—
350C22⋊C8—1221no—
351#97—1222no—
352C4×C8—611no—
353C4×C8—611no—
354C4×C8—612no—
355C4×C8—611no—
356C4×C8—611no—
357C4×C8—611no—
358C4×C8—611no—
359C4×C8—612no—
360C4×C8—1221no—
361C4×C8—1221no—
362C4×C8—1221no—
363C4×C8—1222no—
364#41—22no—
365#41—22no—
366#41—614no—
367#41—1222no—
368C2×Q8—611no—
369C2×Q8C22×#692311yescharacteristic
370C2×D4—21no—
371C2×D4—611no—
372C2×D4—1221no—
373C2×D4C22×#692311yescharacteristic
374C22×C4—21no—
375C22×C4—611no—
376C22×C4—611no—
377C22×C4—1221no—
378C22×C4—1221no—
379C22×C4—1221no—
380C22×C4—1221no—
381C22×C4—1221no—
382C22×C4C22×#692311yescharacteristic
383Q16—1222no—
384SD16—1222no—
385SD16—1222no—
386D8—1222no—
387M16—2444no—
388C2×C8—22no—
389C2×C8—612no—
390C2×C8—612no—
391C2×C8—611no—
392C2×C8—611no—
393C2×C8—1224no—
394C2×C8—1222no—
395C2×C8—1222no—
396C2×C8—1222no—
397C2×C8—1222no—
398C2×C8—2444no—
399C2×C8—2442no—
400C2×C8C22×#692311yescharacteristic
401C2×C8C22×#692311yescharacteristic
402C4⋊C4—611no—
403C4⋊C4—611no—
404C4⋊C4—1221no—
405C4⋊C4—1221no—
406C4⋊C4—1221no—
407C4⋊C4—1221no—
408C4⋊C4C22×#692311yescharacteristic
409C4⋊C4C22×#692311yescharacteristic
410C22⋊C4—21no—
411C22⋊C4—1221no—
412C22⋊C4—1221no—
413C22⋊C4—1221no—
414C22⋊C4—1221no—
415C22⋊C4—1221no—
416C42—21no—
417C42—611no—
418C42—611no—
419C42—611no—
420C42—611no—
421C42—1221no—
422C42—1221no—
423C42—1221no—
424C42—1221no—
425C42—1221no—
426C23—21no—
427C23—1221no—
428C23D4×#692311yescharacteristic
429Q8—21no—
430Q8—21no—
431Q8—612no—
432Q8—1221no—
433D4—21no—
434D4—21no—
435D4—42no—
436D4—612no—
437D4—1224no—
438D4—1221no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—42no—
443C2×C4—611no—
444C2×C4—611no—
445C2×C4—611no—
446C2×C4—611no—
447C2×C4—1222no—
448C2×C4—1222no—
449C2×C4—1221no—
450C2×C4—1224no—
451C2×C4—1221no—
452C2×C4—1221no—
453C2×C4—2442no—
454C2×C4—2442no—
455C2×C4—2442no—
456C2×C4—2442no—
457C2×C4D4×#692311yescharacteristic
458C2×C4D61⋊M1612yes—
459C2×C4#1451986412yes—
460C2×C4C23×#692311yescharacteristic
461C8—21no—
462C8—21no—
463C8—614no—
464C8—1222no—
465C8—1221no—
466C8—1221no—
467C8—2442no—
468C8—2442no—
469C8—2442no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—1222no—
476C22—2441no—
477C22—2441no—
478C22#1684079412yes—
479C22C2×D4×#692311yescharacteristic
480C4—41no—
481C4—41no—
482C4—41no—
483C4—612no—
484C4—612no—
485C4—1222no—
486C4—1221no—
487C4—1221no—
488C4—2441no—
489C4—2441no—
490C4#1684079412yes—
491C4C2×D61⋊M1611yescharacteristic
492C4C2×#1451986411yescharacteristic
493C2—22no—
494C2—81no—
495C2—2442no—
496C2C2×#1684079411yescharacteristic
497C2#135×#692311yescharacteristic
498C2D61⋊#44411yescharacteristic
subgroupG/H
centerC2×C4D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#416C4
socleC22×C61C2×C4×D4
prime radicalC2×#14519797C4
cosocleC4×Dic61C24
hypercenter#416#6923
hypocenterC61#17863
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #21462628C2semidirect
2. #21462623C2semidirect
3. #21461433C2semidirect
4. #21461429C2semidirect
5. #21461425C2semidirect
6. #21461421C2semidirect
7. #21461366C2semidirect
8. #21461112C2semidirect
9. #21461110C2semidirect
10. C2×#92420⋊C8C2semidirect
127. #11758222M16semidirect
215. #92420#135semidirect
298. C61#17863semidirect

Maximal subgroups

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

subgroupindex
#214626282
#214626232
#214614332
#214614292
#214614252
#214614212
#214613662
#214611122
#214611102
C2×#92420⋊C82
#214545872
#1786361