Finite Group Databasebeta

← #11860376

Subgroups — #11860376

Order 1152.

1452 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
1C2×#64124C211yesmaximal, characteristic
2C4×Q8×C3⋊D3C211yesmaximal, characteristic, complemented
3C4×#3830C211yesmaximal, characteristic
4C32×#440C211yesmaximal, characteristic, complemented
5SD16×C3⋊Dic3C211yesmaximal, characteristic, complemented
6#10592729C211yesmaximal, characteristic, complemented
7#10592717C211yesmaximal, characteristic, complemented
8#10592580C211yesmaximal, characteristic, complemented
9#10592572C211yesmaximal, characteristic, complemented
10#10592501C211yesmaximal, characteristic, complemented
11#10592489C211yesmaximal, characteristic, complemented
12#10592470C211yesmaximal, characteristic
13#10592457C211yesmaximal, characteristic, complemented
14#10592415C211yesmaximal, characteristic, complemented
15#10592406C211yesmaximal, characteristic, complemented
16#79774—34nomaximal
17C2×Q8×C3⋊D3C2211yescharacteristic
18C2×#3830C2211yescharacteristic
19C2×C32×SD16C2211yescharacteristic
20C32×C4×Q8C2211yescharacteristic
21C32×C4×D4C2211yescharacteristic
22Q8×C3⋊Dic3C2211yescharacteristic
23C2×#3756C2211yescharacteristic
24D4×C3⋊Dic3C2211yescharacteristic
25C2×#3754C2211yescharacteristic
26C4×#638—21no—
27C2×C4×C3⋊Dic3—21no—
28#64124C418yescomplemented
29C2×#3746C2211yescharacteristic
30C2×#3743C2211yescharacteristic
31C2×#3743C2211yescharacteristic
32#64101C2211yescharacteristic
33C4⋊C4×C3⋊D3—21no—
34C4⋊C4×C3⋊D3C2211yescharacteristic
35#64096—21no—
36#64096C2211yescharacteristic
37#64096C2211yescharacteristic
38#64090—21no—
39#64086—21no—
40C42×C3⋊D3—21no—
41C42×C3⋊D3C2211yescharacteristic
42C4×#634—21no—
43C4×#634C2211yescharacteristic
44C4×#634C2211yescharacteristic
45C32×#107C2211yescharacteristic
46C32×#103C2211yescharacteristic
47C32×#102C2211yescharacteristic
48C32×#97C2211yescharacteristic
49#63662C2211yescharacteristic
50#63659C2211yescharacteristic
51#63644C2211yescharacteristic
52#63643C2211yescharacteristic
53#63643C2211yescharacteristic
54C8×C3⋊Dic3C2211yescharacteristic
55C8×C3⋊Dic3C2211yescharacteristic
56#63637C2211yescharacteristic
57#63637C2211yescharacteristic
58#63635C2211yescharacteristic
59#63634C2211yescharacteristic
60#63630C2211yescharacteristic
61C2×#915—34no—
62C4×D3×Q8—34no—
63C4×#237—34no—
64C3×#440D314yescomplemented
65Dic3×SD16—34no—
66#5030—34no—
67#5018—34no—
68#4881—34no—
69#4873—34no—
70#4802—34no—
71#4790—34no—
72#4771—34no—
73#4758—34no—
74#4716—34no—
75#4707—34no—
76C2×C32×Q8C2311yescharacteristic
77C2×C32×D4C2311yescharacteristic
78C22×C32×C4—21no—
79C2×#638—21no—
80C22×C3⋊Dic3—21no—
81C22×C3⋊Dic3—21no—
82Q8×C3⋊D3—22no—
83Q8×C3⋊D3C2×C414yescomplemented
84#3830—22no—
85#3830C2×C414yes—
86C2×C4×C3⋊D3—21no—
87C2×C4×C3⋊D3—21no—
88C2×C4×C3⋊D3D411yescharacteristic
89C2×C4×C3⋊D3D411yescharacteristic
90C2×C4×C3⋊D3C2311yescharacteristic
91C2×#634—21no—
92C2×#634C2311yescharacteristic
93C2×#634C2311yescharacteristic
94C32×SD16C2×C414yescomplemented
95C2×C32×C8C2311yescharacteristic
96C2×C32×C8C2311yescharacteristic
97C32×C4⋊C4—21no—
98C32×C4⋊C4C2311yescharacteristic
99C32×C4⋊C4C2311yescharacteristic
100C32×C22⋊C4—21no—
101C32×C42—21no—
102C32×C42C2311yescharacteristic
103#3757—21no—
104#3756C2×C414yescomplemented
105#3754C2×C414yescomplemented
106#3752—21no—
107C4⋊(C3⋊Dic3)—21no—
108C4⋊(C3⋊Dic3)C2311yescharacteristic
109C4⋊(C3⋊Dic3)C2311yescharacteristic
110#3750—21no—
111#3750—21no—
112#3750—21no—
113#3750—21no—
114#3750—21no—
115C4×C3⋊Dic3—21no—
116C4×C3⋊Dic3—21no—
117C4×C3⋊Dic3—21no—
118C4×C3⋊Dic3—21no—
119C4×C3⋊Dic3—21no—
120C4×C3⋊Dic3—21no—
121C4×C3⋊Dic3D411yescharacteristic
122C4×C3⋊Dic3D411yescharacteristic
123C4×C3⋊Dic3C2311yescharacteristic
124C2×C3⋊(C3⋊C8)C2311yescharacteristic
125C2×C3⋊(C3⋊C8)C2311yescharacteristic
126#3746C2×C414yescomplemented
127#3743—22no—
128#3743C2×C414yescomplemented
129C4×#137—91noSylow
130C2×D3×Q8—34no—
131C2×#237—34no—
132C2×C3×SD16C2×D314yes—
133C3×C4×Q8C2×D314yes—
134C3×C4×D4C2×D314yes—
135Q8×Dic3—34no—
136C2×#216—34no—
137D4×Dic3—34no—
138C2×#214—34no—
139C4×#67—64no—
140C2×C4×Dic3—64no—
141#915—332no—
142C2×Dic12—34no—
143C2×ΓL(1,𝔽25)—34no—
144C2×ΓL(1,𝔽25)—34no—
145#892—34no—
146D3×C4⋊C4—34no—
147D3×C4⋊C4—64no—
148#887—34no—
149#887—34no—
150#887—64no—
151#67⋊C4—64no—
152#877—64no—
153C42×D3—34no—
154C42×D3—64no—
155C4×Dic6—34no—
156C4×Dic6—34no—
157C4×Dic6—64no—
158C3×#107C2×D314yes—
159C3×#103C2×D314yes—
160C3×#102C2×D314yes—
161C3×#97C2×D314yes—
162#835—34no—
163#832—34no—
164#817—34no—
165#816—34no—
166#816—34no—
167C8×Dic3—34no—
168C8×Dic3—34no—
169#810—34no—
170#810—34no—
171#808—34no—
172#807—34no—
173#803—34no—
174C23×C32—21no—
175C22×C3⋊D3C2×D411yescharacteristic
176C32×Q8—21no—
177C32×Q8C22×C412yes—
178C32×D4—21no—
179C32×D4C22×C412yes—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—42no—
184C2×C32×C4C2×D411yescharacteristic
185C2×C32×C4C2×D411yescharacteristic
186C2×C32×C4C2411yescharacteristic
187#638—24no—
188C2×C3⋊Dic3—21no—
189C2×C3⋊Dic3—21no—
190C2×C3⋊Dic3—24no—
191C2×C3⋊Dic3—21no—
192C2×C3⋊Dic3—21no—
193C2×C3⋊Dic3—42no—
194C2×C3⋊Dic3C2×D411yescharacteristic
195C2×C3⋊Dic3C2×D411yescharacteristic
196C2×C3⋊Dic3C2×D411yescharacteristic
197C4×C3⋊D3—24no—
198C4×C3⋊D3—22no—
199C4×C3⋊D3—41no—
200C4×C3⋊D3—42no—
201C4×C3⋊D3C22×C412yes—
202C4×C3⋊D3#4112yes—
203#634—21no—
204#634—21no—
205#634—24no—
206#634C22×C412yes—
207#634C22×C412yes—
208C32×C8—21no—
209C32×C8C22×C412yes—
210C3⋊(C3⋊C8)—21no—
211C3⋊(C3⋊C8)C22×C412yes—
212C2×#137—91no—
213C4×#41—91no—
214C2×C4×Q8—91no—
215#441—91no—
216#441—91no—
217#440—91no—
218#440C3⋊D311yescharacteristic, complemented
219C4×Q16—91no—
220C4×Q16—91no—
221C4×SD16—91no—
222C4×SD16—91no—
223#428—91no—
224#419—91no—
225#417—91no—
226C4×M16—91no—
227C2×C3×Q8C22×D314yes—
228C2×C3×D4C22×D314yes—
229C22×C3×C4—24no—
230C2×#67—64no—
231C22×Dic3—64no—
232C22×Dic3—64no—
233D3×Q8—316no—
234D3×Q8—68no—
235#237—316no—
236#237—68no—
237C2×C4×D3—34no—
238C2×C4×D3—34no—
239C2×C4×D3—34no—
240C2×C4×D3—64no—
241C2×C4×D3—64no—
242C2×Dic6—34no—
243C2×Dic6—34no—
244C2×Dic6—64no—
245C3×SD16C4×D3116yescomplemented
246C2×C3×C8C22×D314yes—
247C2×C3×C8C22×D314yes—
248C3×C4⋊C4—24no—
249C3×C4⋊C4C22×D314yes—
250C3×C4⋊C4C22×D314yes—
251C3×C22⋊C4—24no—
252C3×C42—24no—
253C3×C42C22×D314yes—
254#217—64no—
255#216—316no—
256#214—316no—
257(C2×D3)⋊C4—64no—
258C4⋊Dic3—34no—
259C4⋊Dic3—34no—
260C4⋊Dic3—64no—
261Dic3⋊C4—64no—
262Dic3⋊C4—64no—
263Dic3⋊C4—64no—
264Dic3⋊C4—64no—
265Dic3⋊C4—64no—
266C4×Dic3—34no—
267C4×Dic3—34no—
268C4×Dic3—34no—
269C4×Dic3—64no—
270C4×Dic3—64no—
271C4×Dic3—64no—
272C4×Dic3—64no—
273C4×Dic3—64no—
274C4×Dic3—64no—
275C2×C3⋊C8—34no—
276C2×C3⋊C8—34no—
277Dic12—316no—
278ΓL(1,𝔽25)—316no—
279ΓL(1,𝔽25)—68no—
280C22×C32—22no—
281C22×C32—41no—
282C22×C32—41no—
283C22×C32C22×D411yescharacteristic
284C2×C3⋊D3—21no—
285C2×C3⋊D3—21no—
286C2×C3⋊D3C4×D412yes—
287C32×C4—22no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—41no—
291C32×C4—41no—
292C32×C4C23×C411yescharacteristic
293C32×C4C2×#4111yescharacteristic
294C3⋊Dic3—22no—
295C3⋊Dic3—22no—
296C3⋊Dic3—21no—
297C3⋊Dic3—41no—
298C3⋊Dic3—41no—
299C3⋊Dic3C4×D412yes—
300C3⋊Dic3#13712yescomplemented
301C2×#41—91no—
302C22×Q8—91no—
303#137—98no—
304C2×Q16—91no—
305C2×Q16—91no—
306C2×SD16—91no—
307C2×SD16C2×C3⋊D311yescharacteristic
308C2×M16—91no—
309C2×M16—91no—
310C4×Q8—91no—
311C4×Q8—91no—
312C4×Q8—91no—
313C4×Q8—91no—
314C4×Q8—91no—
315C4×Q8—181no—
316C4×Q8—181no—
317C4×Q8C2×C3⋊D311yescharacteristic
318C4×D4—91no—
319C4×D4—181no—
320C4×D4—181no—
321C4×D4C2×C3⋊D311yescharacteristic
322#117—91no—
323#117—181no—
324C2×C4⋊C4—91no—
325C2×C4⋊C4—181no—
326C2×C42—91no—
327C2×C42—181no—
328C2×C42—181no—
329#107—91no—
330#107C2×C3⋊D311yescharacteristic
331#106—91no—
332#106—91no—
333#103—91no—
334#103—91no—
335#103—91no—
336#103—91no—
337#103—91no—
338#103C2×C3⋊D311yescharacteristic
339#102—91no—
340#102C2×C3⋊D311yescharacteristic
341#97—91no—
342#97C2×C3⋊D311yescharacteristic
343C4×C8—91no—
344C4×C8—91no—
345C23×C3—24no—
346C22×D3—34no—
347C3×Q8—24no—
348C3×Q8C2×C4×D318yes—
349C3×D4—24no—
350C3×D4C2×C4×D318yes—
351C2×C3×C4—24no—
352C2×C3×C4—24no—
353C2×C3×C4—24no—
354C2×C3×C4—48no—
355C2×C3×C4D3×D414yes—
356C2×C3×C4D3×D414yes—
357C2×C3×C4C23×D314yes—
358#67—616no—
359C2×Dic3—34no—
360C2×Dic3—34no—
361C2×Dic3—34no—
362C2×Dic3—616no—
363C2×Dic3—64no—
364C2×Dic3—64no—
365C2×Dic3—64no—
366C2×Dic3—64no—
367C2×Dic3—128no—
368C4×D3—38no—
369C4×D3—38no—
370C4×D3—68no—
371C4×D3—616no—
372C4×D3—124no—
373C4×D3—128no—
374Dic6—38no—
375Dic6—38no—
376Dic6—616no—
377Dic6—64no—
378Dic6—64no—
379C3×C8—24no—
380C3×C8C2×C4×D318yes—
381C3⋊C8—38no—
382C3⋊C8—64no—
383C2×C32—42no—
384C2×C32C2×C4×D411yescharacteristic
385C2×C32C2×#13711yescharacteristic
386C2×C32#57511yescharacteristic
387C3⋊D3—22no—
388#41—94no—
389#41—182no—
390C2×Q8—94no—
391C2×Q8—91no—
392C2×Q8—91no—
393C2×Q8—181no—
394C2×Q8—182no—
395C2×Q8C22×C3⋊D311yescharacteristic
396C2×D4—181no—
397C2×D4C22×C3⋊D311yescharacteristic
398C22×C4—21no—
399C22×C4—91no—
400C22×C4—91no—
401C22×C4—91no—
402C22×C4—181no—
403C22×C4—181no—
404C22×C4—181no—
405C22×C4—181no—
406Q16—94no—
407Q16—94no—
408SD16—94no—
409SD16C4×C3⋊D314yescomplemented
410M16—94no—
411M16—182no—
412C2×C8—91no—
413C2×C8—91no—
414C2×C8C22×C3⋊D311yescharacteristic
415C2×C8C22×C3⋊D311yescharacteristic
416C4⋊C4—21no—
417C4⋊C4—91no—
418C4⋊C4—91no—
419C4⋊C4—181no—
420C4⋊C4—181no—
421C4⋊C4—181no—
422C4⋊C4—181no—
423C4⋊C4—181no—
424C4⋊C4—181no—
425C4⋊C4C22×C3⋊D311yescharacteristic
426C4⋊C4C22×C3⋊D311yescharacteristic
427C22⋊C4—21no—
428C22⋊C4—181no—
429C22⋊C4—181no—
430C42—21no—
431C42—91no—
432C42—91no—
433C42—91no—
434C42—181no—
435C42—181no—
436C42—181no—
437C42—181no—
438C42—181no—
439C42—181no—
440C42C22×C3⋊D311yescharacteristic
441C22×C3—28no—
442C22×C3—44no—
443C22×C3—44no—
444C22×C3C2×D3×D414yes—
445C2×D3—38no—
446C2×D3—64no—
447C2×D3—64no—
448C3×C4—24no—
449C3×C4—24no—
450C3×C4—28no—
451C3×C4—44no—
452C3×C4—44no—
453C3×C4C22×C4×D314yes—
454C3×C4D3×#4114yes—
455Dic3—38no—
456Dic3—38no—
457Dic3—64no—
458Dic3—68no—
459Dic3—68no—
460Dic3—124no—
461Dic3—124no—
462C32C4×#13711yescharacteristic, complemented, Sylow
463C23—21no—
464C23—91no—
465Q8—21no—
466Q8—92no—
467Q8—92no—
468Q8—184no—
469Q8—181no—
470Q8—181no—
471Q8C2×C4×C3⋊D312yes—
472D4—21no—
473D4—184no—
474D4C2×C4×C3⋊D312yes—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—42no—
479C2×C4—91no—
480C2×C4—91no—
481C2×C4—92no—
482C2×C4—92no—
483C2×C4—91no—
484C2×C4—182no—
485C2×C4—184no—
486C2×C4—184no—
487C2×C4—181no—
488C2×C4—181no—
489C2×C4—181no—
490C2×C4—181no—
491C2×C4—361no—
492C2×C4—362no—
493C2×C4—362no—
494C2×C4D4×C3⋊D311yescharacteristic
495C2×C4D4×C3⋊D311yescharacteristic
496C2×C4C23×C3⋊D311yescharacteristic
497C8—21no—
498C8—92no—
499C8—181no—
500C8C2×C4×C3⋊D312yes—
501C2×C3—48no—
502C2×C3C4×D3×D414yes—
503C2×C3#577314yes—
504C2×C3D3×#13714yes—
505D3—68no—
506C22—22no—
507C22—41no—
508C22—41no—
509C22—92no—
510C22—181no—
511C22—181no—
512C22C2×D4×C3⋊D311yescharacteristic
513C4—21no—
514C4—21no—
515C4—22no—
516C4—41no—
517C4—41no—
518C4—92no—
519C4—92no—
520C4—181no—
521C4—182no—
522C4—182no—
523C4—361no—
524C4—361no—
525C4C22×C4×C3⋊D311yescharacteristic
526C4#41×C3⋊D311yescharacteristic
527C3#7977414yescomplemented
528C2—42no—
529C2—182no—
530C2C4×D4×C3⋊D311yescharacteristic
531C2#1059624411yescharacteristic
532C2C3⋊D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×C3⋊D3
commutatorC32×C4C23×C4
FrattiniC2×C4C23×C3⋊D3
FittingC32×#440C2
socleC22×C32C22×D4
prime radicalC2×#3830C22
cosocleC2×C32×C4C24
hypercenter#440C3⋊D3
hypocenterC32C4×#137
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
2. C4×Q8×C3⋊D3C2semidirect
4. C32×#440C2semidirect
5. SD16×C3⋊Dic3C2semidirect
6. #10592729C2semidirect
7. #10592717C2semidirect
8. #10592580C2semidirect
9. #10592572C2semidirect
10. #10592501C2semidirect
11. #10592489C2semidirect
13. #10592457C2semidirect
14. #10592415C2semidirect
15. #10592406C2semidirect
28. #64124C4semidirect
64. C3×#440D3semidirect
83. Q8×C3⋊D3C2×C4semidirect
94. C32×SD16C2×C4semidirect
104. #3756C2×C4semidirect
105. #3754C2×C4semidirect
126. #3746C2×C4semidirect
128. #3743C2×C4semidirect
218. #440C3⋊D3semidirect
245. C3×SD16C4×D3semidirect
300. C3⋊Dic3#137semidirect
409. SD16C4×C3⋊D3semidirect
462. C32C4×#137semidirect
527. C3#79774semidirect

Maximal subgroups

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

subgroupindex
C2×#641242
C4×Q8×C3⋊D32
C4×#38302
C32×#4402
SD16×C3⋊Dic32
#105927292
#105927172
#105925802
#105925722
#105925012
#105924892
#105924702
#105924572
#105924152
#105924062
#797743