Finite Group Databasebeta

← #11786746

Subgroups — #11786746

Order 1152.

621 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10594992C211yesmaximal, characteristic
2C2×C3×#825C211yesmaximal, characteristic
3C3×#5121C211yesmaximal, characteristic, complemented
4#10591224C211yesmaximal, characteristic, complemented
5#10591087C211yesmaximal, characteristic
6#10589420C211yesmaximal, characteristic, complemented
7#10589161C211yesmaximal, characteristic, complemented
8#70681—31nomaximal
9#70364—31nomaximal
10C2×C3×#235C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C3×#926C2211yescharacteristic
15C2×C3×C4⋊Dic3C2211yescharacteristic
16C2×C3×C4×Dic3C2211yescharacteristic
17#67⋊Dic3—22no—
18Dic3×D12—21no—
19D3×C4⋊Dic3—21no—
20Dic3×Dic6—21no—
21#63841—21no—
22C3×#825C414yescomplemented
23C3×#818—21no—
24C3×#817—21no—
25#63579—22no—
26#63565—21no—
27#63562—21no—
28#5827—31no—
29#5523—31no—
30C3×#428—31no—
31C2×C3×#104—31no—
32#5232—31no—
33C2×#825D311yescharacteristic
34#5121D311yescharacteristic, complemented
35#4977—31no—
36#4966—31no—
37#4936—31no—
38#4554—31no—
39#4554—31no—
40#4554—61no—
41#4532—31no—
42#4523—31no—
43C22×C32×C4C2311yescharacteristic
44C22×C3⋊Dic3D412yes—
45C2×C3×#67—21no—
46C22×C3×Dic3—21no—
47C22×C3×Dic3—21no—
48C3×#235—21no—
49C3×#235—21no—
50C3×#235C2×C412yes—
51C3×#235Q812yes—
52C2×C3×D12D411yescharacteristic
53C2×C3×C4×D3—21no—
54C2×C3×Dic6D411yescharacteristic
55C2×D3×Dic3—22no—
56C32×M16—21no—
57C32×M16C2×C412yescomplemented
58C2×C32×C8—21no—
59C4⋊(C3⋊Dic3)—21no—
60C4⋊(C3⋊Dic3)—21no—
61#3748—21no—
62#3748—21no—
63C2×C3⋊(C3⋊C8)—21no—
64C3×#217—21no—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3D411yescharacteristic
68C3×C4⋊Dic3D411yescharacteristic
69C3×C4×Dic3—21no—
70C3×C4×Dic3—21no—
71C3×C4×Dic3C2×C412yescomplemented
72Dic3⋊Dic3—22no—
73(C2×D3)⋊Dic3—22no—
74Dic32—22no—
75#1860—91noSylow
76C2×C3×#41—31no—
77C2×#235C2×D311yescharacteristic
78C2×C3×M16—21no—
79C2×C3×M16C2×D311yescharacteristic
80C2×C3×M16C2×D311yescharacteristic
81C3×#117—31no—
82C2×C3×C4⋊C4—31no—
83C2×C3×C42—31no—
84Q8×Dic3—61no—
85D4×Dic3—61no—
86D4×Dic3—62no—
87#926—61no—
88#926C2×D311yescharacteristic
89C2×C4⋊Dic3—31no—
90C2×C4⋊Dic3—31no—
91C2×C4⋊Dic3—61no—
92C2×C4⋊Dic3—61no—
93C2×C4⋊Dic3C2×D311yescharacteristic
94C2×C4×Dic3C2×D311yescharacteristic
95C2×#208—31no—
96C2×#208—31no—
97C2×#208—61no—
98#892—61no—
99D3×C4⋊C4—61no—
100#67⋊C4—62no—
101C4×D12—61no—
102C4×Dic6—61no—
103C3×#107—61no—
104C3×#106—61no—
105C3×#104—34no—
106#835—61no—
107#832—61no—
108C3⋊#95—62no—
109C3⋊#95—62no—
110#825Dic314yescomplemented
111#818—21no—
112#817—21no—
113#810—61no—
114#809—61no—
115C23×C32C2×D411yescharacteristic
116C22×C3×D3—21no—
117C2×C32×C4C22×C411yescharacteristic
118C2×C32×C4C2×D411yescharacteristic
119C2×C32×C4C2×D411yescharacteristic
120C2×C32×C4C2×Q811yescharacteristic
121C2×C32×C4#4111yescharacteristic
122C2×C32×C4#4111yescharacteristic
123C2×C3⋊Dic3—22no—
124C2×C3⋊Dic3—42no—
125C3×#67—22no—
126C3×#67—22no—
127C2×C3×Dic3—21no—
128C2×C3×Dic3—22no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—41no—
132C2×C3×Dic3—41no—
133C2×C3×Dic3—42no—
134C2×C3×Dic3—41no—
135C3×D12—21no—
136C3×D12C4⋊C412yes—
137C3×C4×D3—22no—
138C3×C4×D3—22no—
139C3×Dic6—21no—
140C3×Dic6C4⋊C412yes—
141D3×Dic3—82no—
142C32×C8—22no—
143C3⋊(C3⋊C8)—41no—
144#520—91no—
145#428—31no—
146C2×#104—31no—
147#417—91no—
148#382—91no—
149#343—91no—
150#339—91no—
151C3×#41—32no—
152C3×#41—32no—
153C3×#41—61no—
154C3×#41—61no—
155C2×C3×Q8—31no—
156C2×C3×D4—31no—
157C2×C3×D4—61no—
158C22×C3×C4—21no—
159C22×C3×C4—61no—
160C22×C3×C4—61no—
161C22×C3×C4—61no—
162C22×C3×C4C22×D311yescharacteristic
163C22×C3×C4C22×D311yescharacteristic
164C2×#67—21no—
165C22×Dic3—21no—
166C22×Dic3—21no—
167C22×Dic3—32no—
168C22×Dic3—32no—
169C22×Dic3—62no—
170C22×Dic3—62no—
171#235—21no—
172#235—21no—
173#235Dic612yes—
174#235C2×Dic312yes—
175C2×D12D1211yescharacteristic
176C2×C4×D3—21no—
177C2×C4×D3—62no—
178C2×Dic6D1211yescharacteristic
179C3×M16—21no—
180C3×M16—21no—
181C3×M16—22no—
182C3×M16—41no—
183C3×M16C4×D312yescomplemented
184C3×M16C2×Dic312yescomplemented
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—41no—
188C3×C4⋊C4—31no—
189C3×C4⋊C4—31no—
190C3×C4⋊C4—61no—
191C3×C4⋊C4—61no—
192C3×C22⋊C4—61no—
193C3×C42—32no—
194C3×C42—61no—
195C3×C42—61no—
196#217—21no—
197#217—62no—
198(C2×D3)⋊C4—62no—
199C4⋊Dic3—21no—
200C4⋊Dic3—21no—
201C4⋊Dic3—62no—
202C4⋊Dic3—61no—
203C4⋊Dic3—61no—
204C4⋊Dic3—61no—
205C4⋊Dic3—61no—
206C4⋊Dic3—121no—
207C4⋊Dic3—121no—
208C4⋊Dic3#6711yescharacteristic
209C4⋊Dic3#6711yescharacteristic
210Dic3⋊C4—62no—
211C4×Dic3—21no—
212C4×Dic3—21no—
213C4×Dic3—62no—
214C4×Dic3—62no—
215C4×Dic3C2×Dic312yescomplemented
216#208—62no—
217#208—61no—
218#208—61no—
219#208—61no—
220#208—61no—
221#208—121no—
222C2×C3⋊C8—61no—
223C2×C3⋊C8—61no—
224C2×C3⋊C8—121no—
225C22×C32—21no—
226C22×C32—21no—
227C22×C32C4×D411yescharacteristic
228C22×C32#12011yescharacteristic
229C22×C32C22⋊Q811yescharacteristic
230C2×C3×D3—22no—
231C2×C3×D3—41no—
232C2×C3×D3—41no—
233C32×C4C2×C4⋊C411yescharacteristic
234C32×C4C4×D411yescharacteristic
235C32×C4C22⋊Q811yescharacteristic
236C32×C4#12311yescharacteristic
237C3⋊Dic3—42no—
238C3×Dic3—22no—
239C3×Dic3—42no—
240C3×Dic3—41no—
241C3×Dic3—41no—
242C3×Dic3—81no—
243C2×#41—31no—
244C2×M16—91no—
245C2×M16D3211yescharacteristic
246C4×Q8—181no—
247C4×D4—181no—
248C4×D4—182no—
249#117—31no—
250#117—181no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—91no—
253C2×C4⋊C4—181no—
254C2×C42—31no—
255#107—61no—
256#106—61no—
257#104—34no—
258#103—181no—
259#102—181no—
260#95—182no—
261C23×C3—21no—
262C23×C3—61no—
263C23×C3D3×D411yescharacteristic
264C23×C3D3×D411yescharacteristic
265C22×D3—21no—
266C3×Q8—32no—
267C3×Q8—61no—
268C3×D4—32no—
269C3×D4—62no—
270C3×D4—62no—
271C3×D4—61no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—61no—
279C2×C3×C4—62no—
280C2×C3×C4—62no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—62no—
284C2×C3×C4—121no—
285C2×C3×C4—121no—
286C2×C3×C4—122no—
287C2×C3×C4—121no—
288C2×C3×C4C2×Dic611yescharacteristic
289C2×C3×C4C2×C4×D311yescharacteristic
290C2×C3×C4C2×D1211yescharacteristic
291C2×C3×C4#23511yescharacteristic
292C2×C3×C4D3×D411yescharacteristic
293C2×C3×C4D3×D411yescharacteristic
294C2×C3×C4#23711yescharacteristic
295C2×C3×C4#23711yescharacteristic
296C2×C3×C4D3×Q811yescharacteristic
297C2×C3×C4#23911yescharacteristic
298C2×C3×C4C22×Dic311yescharacteristic
299C2×C3×C4C2×#6711yescharacteristic
300#67—22no—
301#67—22no—
302C2×Dic3—21no—
303C2×Dic3—22no—
304C2×Dic3—21no—
305C2×Dic3—21no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—42no—
309C2×Dic3—41no—
310C2×Dic3—62no—
311C2×Dic3—62no—
312C2×Dic3—122no—
313C2×Dic3—122no—
314C2×Dic3—122no—
315C2×Dic3—242no—
316C2×Dic3—242no—
317D12—21no—
318D12C4⋊Dic312yes—
319C4×D3—22no—
320C4×D3—22no—
321C4×D3—242no—
322Dic6—21no—
323Dic6C4⋊Dic312yes—
324C3×C8—22no—
325C3×C8—22no—
326C3×C8—42no—
327C3⋊C8—122no—
328C3⋊C8—121no—
329C3⋊C8—121no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32#38511yescharacteristic
333C2×C32#45411yescharacteristic
334C2×C32#45511yescharacteristic
335C3×D3—42no—
336#41—32no—
337#41—32no—
338#41—61no—
339#41—61no—
340C2×Q8—31no—
341C2×D4—31no—
342C2×D4—61no—
343C22×C4—61no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—92no—
347C22×C4—182no—
348C22×C4C2×D3211yescharacteristic
349M16—21no—
350M16—181no—
351M16—181no—
352M16D3×Dic312yescomplemented
353C2×C8—21no—
354C2×C8—181no—
355C4⋊C4—31no—
356C4⋊C4—31no—
357C4⋊C4—61no—
358C4⋊C4—61no—
359C4⋊C4—182no—
360C4⋊C4—181no—
361C4⋊C4—181no—
362C22⋊C4—61no—
363C22⋊C4—182no—
364C42—32no—
365C42—61no—
366C42—61no—
367C42—182no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—62no—
378C22×C3—121no—
379C22×C3—121no—
380C22×C3#87811yescharacteristic
381C22×C3#88211yescharacteristic
382C22×C3#89311yescharacteristic
383C22×C3#89611yescharacteristic
384C22×C3D4×Dic311yescharacteristic
385C22×C3#93711yescharacteristic
386C2×D3—22no—
387C2×D3—41no—
388C2×D3—41no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—62no—
394C3×C4—122no—
395C3×C4—121no—
396C3×C4—121no—
397C3×C4—241no—
398C3×C4#87811yescharacteristic
399C3×C4#88611yescharacteristic
400C3×C4D3×C4⋊C411yescharacteristic
401C3×C4#89411yescharacteristic
402C3×C4C2×C4⋊Dic311yescharacteristic
403C3×C4C4×#6711yescharacteristic
404C3×C4D4×Dic311yescharacteristic
405C3×C4#94611yescharacteristic
406Dic3—22no—
407Dic3—42no—
408Dic3—41no—
409Dic3—41no—
410Dic3—81no—
411Dic3—122no—
412Dic3—122no—
413Dic3—242no—
414C32#186011yescharacteristic, complemented, Sylow
415C23—61no—
416C23#380211yescharacteristic
417Q8—32no—
418Q8—61no—
419D4—32no—
420D4—62no—
421D4—62no—
422D4—61no—
423C2×C4—61no—
424C2×C4—62no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—122no—
432C2×C4—121no—
433C2×C4—182no—
434C2×C4—362no—
435C2×C4—722no—
436C2×C4D3×Dic611yescharacteristic
437C2×C4#379611yescharacteristic
438C2×C4D3×D1211yescharacteristic
439C2×C4C2×D3×Dic311yescharacteristic
440C2×C4#380411yescharacteristic
441C2×C4D3×#6711yescharacteristic
442C8—22no—
443C8—361no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—41no—
452C2×C3—41no—
453C2×C3—122no—
454C2×C3#482511yescharacteristic
455C2×C3#482711yescharacteristic
456C2×C3#495011yescharacteristic
457C2×C3#498911yescharacteristic
458C2×C3#520111yescharacteristic
459C2×C3#520211yescharacteristic
460D3—42no—
461C22—21no—
462C22—21no—
463C22—62no—
464C22—121no—
465C22—121no—
466C22#6389811yescharacteristic
467C22#6389911yescharacteristic
468C22#6392211yescharacteristic
469C4—62no—
470C4—122no—
471C4—121no—
472C4—121no—
473C4—241no—
474C4—362no—
475C4#6388711yescharacteristic
476C4D3×C4⋊Dic311yescharacteristic
477C4#6389011yescharacteristic
478C4Dic3×#6711yescharacteristic
479C3—21no—
480C3#7036411yescharacteristic, complemented
481C3#7068111yescharacteristic, complemented
482C2—21no—
483C2—21no—
484C2—122no—
485C2#1059101111yescharacteristic
486C2#1059101911yescharacteristic
487C2#1059137711yescharacteristic
subgroupG/H
centerC22#63922
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32#120
prime radicalC2×C3×#235C22
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1860
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
3. C3×#5121C2semidirect
4. #10591224C2semidirect
6. #10589420C2semidirect
7. #10589161C2semidirect
22. C3×#825C4semidirect
34. #5121D3semidirect
57. C32×M16C2×C4semidirect
71. C3×C4×Dic3C2×C4semidirect
110. #825Dic3semidirect
183. C3×M16C4×D3semidirect
184. C3×M16C2×Dic3semidirect
215. C4×Dic3C2×Dic3semidirect
352. M16D3×Dic3semidirect
414. C32#1860semidirect
480. C3#70364semidirect
481. C3#70681semidirect

Maximal subgroups

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

subgroupindex
#105949922
C2×C3×#8252
C3×#51212
#105912242
#105910872
#105894202
#105891612
#706813
#703643