Finite Group Databasebeta

← #11786744

Subgroups — #11786744

Order 1152.

631 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10594986C211yesmaximal, characteristic
2#10592820C211yesmaximal, characteristic, complemented
3C2×C3×#825C211yesmaximal, characteristic
4#10591224C211yesmaximal, characteristic, complemented
5#10591088C211yesmaximal, characteristic
6C3×#4554C211yesmaximal, characteristic, complemented
7#10589162C211yesmaximal, characteristic, complemented
8#70364—31nomaximal
9#69003—31nomaximal
10C2×C3×#235C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12#64135C2211yescharacteristic
13C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C3×C4⋊Dic3C2211yescharacteristic
15C2×C3×C4×Dic3C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17Dic3×#67—22no—
18#63898—21no—
19D3×C4⋊Dic3—21no—
20#63843—21no—
21#63841—21no—
22#63645—21no—
23#63644—21no—
24C3×#825C414yescomplemented
25#63579—22no—
26#63566—21no—
27#63563—21no—
28#5827—31no—
29#5504—31no—
30C2×C3×#104—31no—
31#5232—31no—
32C2×#825D311yescharacteristic
33#5133—31no—
34#5121—31no—
35#5121—31no—
36#5121—61no—
37#4977—31no—
38#4936—31no—
39C3×#343—31no—
40#4555—31no—
41#4554D311yescharacteristic, complemented
42#4523—31no—
43C22×C32×C4C2311yescharacteristic
44C22×C3⋊Dic3—21no—
45C2×C3×#67—21no—
46C22×C3×Dic3—21no—
47C22×C3×Dic3D412yes—
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—
59#3757—21no—
60C4⋊(C3⋊Dic3)—21no—
61C4⋊(C3⋊Dic3)—21no—
62C4⋊(C3⋊Dic3)D411yescharacteristic
63C4⋊(C3⋊Dic3)D411yescharacteristic
64C4×C3⋊Dic3—21no—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3—21no—
67C3×C4×Dic3—21no—
68C3×C4×Dic3C2×C412yescomplemented
69C3×#208—21no—
70C3×#208—21no—
71C2×C3×C3⋊C8—21no—
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×M16—31no—
80C2×C3×M16C2×D311yescharacteristic
81C2×C3×M16C2×D311yescharacteristic
82C2×C3×C4⋊C4—31no—
83C2×C3×C42—31no—
84Q8×Dic3—61no—
85D4×Dic3—61no—
86D4×Dic3—62no—
87C4×#67—62no—
88#926—31no—
89#926—31no—
90#926—61no—
91#926—61no—
92C2×C4⋊Dic3—31no—
93C2×C4⋊Dic3—31no—
94C2×C4⋊Dic3—61no—
95C2×C4⋊Dic3—61no—
96C2×C4⋊Dic3C2×D311yescharacteristic
97C2×C4×Dic3C2×D311yescharacteristic
98C2×#208C2×D311yescharacteristic
99#893—61no—
100#892—61no—
101D3×C4⋊C4—61no—
102#887—61no—
103C3×#104—34no—
104#835—61no—
105#832—61no—
106C3⋊#95—62no—
107C3⋊#95—62no—
108#825Dic314yescomplemented
109#821—61no—
110#818—61no—
111#818—61no—
112#818—62no—
113#817—61no—
114#817—61no—
115#817—62no—
116#816—61no—
117C23×C32C2×D411yescharacteristic
118C22×C3×D3—21no—
119C2×C32×C4C22×C411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4C2×Q811yescharacteristic
123C2×C32×C4#4111yescharacteristic
124C2×C32×C4#4111yescharacteristic
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—41no—
128C2×C3⋊Dic3—42no—
129C3×#67—22no—
130C3×#67—22no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—41no—
135C2×C3×Dic3—41no—
136C2×C3×Dic3—42no—
137C3×D12—21no—
138C3×D12C4⋊C412yes—
139C3×C4×D3—22no—
140C3×C4×D3—22no—
141C3×Dic6—21no—
142C3×Dic6C4⋊C412yes—
143D3×Dic3—82no—
144C32×C8—22no—
145C3×C3⋊C8—41no—
146#520—91no—
147#428—91no—
148C2×#104—31no—
149#417—91no—
150#382—91no—
151#343—31no—
152#339—91no—
153C3×#41—32no—
154C3×#41—32no—
155C3×#41—61no—
156C3×#41—61no—
157C2×C3×Q8—31no—
158C2×C3×D4—31no—
159C2×C3×D4—61no—
160C22×C3×C4—21no—
161C22×C3×C4—32no—
162C22×C3×C4—61no—
163C22×C3×C4—61no—
164C22×C3×C4C22×D311yescharacteristic
165C22×C3×C4C22×D311yescharacteristic
166C2×#67—21no—
167C22×Dic3—21no—
168C22×Dic3—62no—
169C22×Dic3—61no—
170C22×Dic3—61no—
171C22×Dic3—121no—
172C22×Dic3D1212yes—
173#235—21no—
174#235—21no—
175#235Dic612yes—
176#235C2×Dic312yes—
177C2×D12D1211yescharacteristic
178C2×C4×D3—21no—
179C2×C4×D3—62no—
180C2×Dic6D1211yescharacteristic
181C3×M16—22no—
182C3×M16—21no—
183C3×M16—21no—
184C3×M16—22no—
185C3×M16—61no—
186C3×M16—61no—
187C3×M16C4×D312yescomplemented
188C3×M16C2×Dic312yescomplemented
189C2×C3×C8—22no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—61no—
193C3×C4⋊C4—61no—
194C3×C4⋊C4—61no—
195C3×C42—32no—
196C3×C42—61no—
197#217—62no—
198#217—61no—
199#217—61no—
200#217—121no—
201(C2×D3)⋊C4—62no—
202C4⋊Dic3—21no—
203C4⋊Dic3—21no—
204C4⋊Dic3—31no—
205C4⋊Dic3—31no—
206C4⋊Dic3—31no—
207C4⋊Dic3—31no—
208C4⋊Dic3—61no—
209C4⋊Dic3—61no—
210C4⋊Dic3—62no—
211C4⋊Dic3—62no—
212C4⋊Dic3—61no—
213C4⋊Dic3—61no—
214C4⋊Dic3—61no—
215C4⋊Dic3—61no—
216C4⋊Dic3—121no—
217Dic3⋊C4—62no—
218C4×Dic3—21no—
219C4×Dic3—62no—
220C4×Dic3—62no—
221C4×Dic3—61no—
222C4×Dic3—61no—
223C4×Dic3—121no—
224C4×Dic3C2×Dic312yescomplemented
225#208—21no—
226#208—21no—
227C2×C3⋊C8—21no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32C4×D411yescharacteristic
231C22×C32#12011yescharacteristic
232C22×C32C22⋊Q811yescharacteristic
233C2×C3×D3—22no—
234C2×C3×D3—41no—
235C2×C3×D3—41no—
236C32×C4C2×C4⋊C411yescharacteristic
237C32×C4C4×D411yescharacteristic
238C32×C4C22⋊Q811yescharacteristic
239C32×C4#12311yescharacteristic
240C3⋊Dic3—41no—
241C3⋊Dic3—41no—
242C3⋊Dic3—81no—
243C3×Dic3—22no—
244C3×Dic3—42no—
245C3×Dic3—42no—
246C2×#41—31no—
247C2×M16—31no—
248C2×M16D3211yescharacteristic
249C4×Q8—181no—
250C4×D4—181no—
251C4×D4—182no—
252#117—91no—
253#117—181no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—91no—
256C2×C4⋊C4—181no—
257C2×C42—31no—
258#107—181no—
259#106—181no—
260#104—34no—
261#103—181no—
262#102—181no—
263#95—182no—
264C23×C3—21no—
265C23×C3—61no—
266C23×C3C2×D1211yescharacteristic
267C23×C3D3×D411yescharacteristic
268C22×D3—21no—
269C3×Q8—32no—
270C3×Q8—61no—
271C3×D4—32no—
272C3×D4—62no—
273C3×D4—62no—
274C3×D4—61no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—61no—
282C2×C3×C4—62no—
283C2×C3×C4—62no—
284C2×C3×C4—62no—
285C2×C3×C4—62no—
286C2×C3×C4—121no—
287C2×C3×C4—121no—
288C2×C3×C4—122no—
289C2×C3×C4C2×Dic611yescharacteristic
290C2×C3×C4C2×C4×D311yescharacteristic
291C2×C3×C4C2×D1211yescharacteristic
292C2×C3×C4#23511yescharacteristic
293C2×C3×C4D3×D411yescharacteristic
294C2×C3×C4D3×D411yescharacteristic
295C2×C3×C4D3×D411yescharacteristic
296C2×C3×C4#23711yescharacteristic
297C2×C3×C4#23711yescharacteristic
298C2×C3×C4#23711yescharacteristic
299C2×C3×C4D3×Q811yescharacteristic
300C2×C3×C4C22×Dic311yescharacteristic
301#67—22no—
302#67—22no—
303C2×Dic3—21no—
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—41no—
307C2×Dic3—41no—
308C2×Dic3—42no—
309C2×Dic3—61no—
310C2×Dic3—61no—
311C2×Dic3—61no—
312C2×Dic3—61no—
313C2×Dic3—121no—
314C2×Dic3—121no—
315C2×Dic3—122no—
316C2×Dic3—122no—
317C2×Dic3—122no—
318C2×Dic3—121no—
319C2×Dic3—121no—
320C2×Dic3—242no—
321C2×Dic3—242no—
322D12—21no—
323D12C4⋊Dic312yes—
324C4×D3—22no—
325C4×D3—22no—
326C4×D3—242no—
327Dic6—21no—
328Dic6C4⋊Dic312yes—
329C3×C8—24no—
330C3×C8—22no—
331C3×C8—22no—
332C3×C8—121no—
333C3⋊C8—41no—
334C2×C32—21no—
335C2×C32—21no—
336C2×C32#38511yescharacteristic
337C2×C32#45411yescharacteristic
338C2×C32#45511yescharacteristic
339C3×D3—42no—
340#41—32no—
341#41—32no—
342#41—61no—
343#41—61no—
344C2×Q8—31no—
345C2×D4—31no—
346C2×D4—61no—
347C22×C4—32no—
348C22×C4—61no—
349C22×C4—61no—
350C22×C4—182no—
351C22×C4—181no—
352C22×C4C2×D3211yescharacteristic
353M16—21no—
354M16—61no—
355M16—61no—
356M16D3×Dic312yescomplemented
357C2×C8—21no—
358C2×C8—61no—
359C4⋊C4—61no—
360C4⋊C4—61no—
361C4⋊C4—91no—
362C4⋊C4—91no—
363C4⋊C4—182no—
364C4⋊C4—181no—
365C4⋊C4—181no—
366C22⋊C4—182no—
367C22⋊C4—181no—
368C42—32no—
369C42—61no—
370C42—182no—
371C42—181no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—41no—
380C22×C3—41no—
381C22×C3—62no—
382C22×C3—121no—
383C22×C3—121no—
384C22×C3C4×D1211yescharacteristic
385C22×C3#87811yescharacteristic
386C22×C3#88211yescharacteristic
387C22×C3#88211yescharacteristic
388C22×C3#89711yescharacteristic
389C22×C3D4×Dic311yescharacteristic
390C2×D3—22no—
391C2×D3—41no—
392C2×D3—41no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—62no—
398C3×C4—122no—
399C3×C4—122no—
400C3×C4#87811yescharacteristic
401C3×C4#67⋊C411yescharacteristic
402C3×C4#88311yescharacteristic
403C3×C4#88611yescharacteristic
404C3×C4D3×C4⋊C411yescharacteristic
405C3×C4#89611yescharacteristic
406C3×C4C2×C4⋊Dic311yescharacteristic
407C3×C4D4×Dic311yescharacteristic
408Dic3—22no—
409Dic3—42no—
410Dic3—42no—
411Dic3—121no—
412Dic3—121no—
413Dic3—121no—
414Dic3—121no—
415Dic3—241no—
416Dic3—241no—
417Dic3—242no—
418Dic3—241no—
419Dic3—241no—
420C32#186011yescharacteristic, complemented, Sylow
421C23—61no—
422C23D3×D1211yescharacteristic
423Q8—32no—
424Q8—61no—
425D4—32no—
426D4—62no—
427D4—62no—
428D4—61no—
429C2×C4—61no—
430C2×C4—62no—
431C2×C4—62no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—122no—
437C2×C4—181no—
438C2×C4—181no—
439C2×C4—362no—
440C2×C4—361no—
441C2×C4—722no—
442C2×C4D3×Dic611yescharacteristic
443C2×C4#379511yescharacteristic
444C2×C4D3×D1211yescharacteristic
445C2×C4C2×D3×Dic311yescharacteristic
446C2×C4#380511yescharacteristic
447C2×C4#381111yescharacteristic
448C8—22no—
449C8—121no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—41no—
458C2×C3—41no—
459C2×C3—122no—
460C2×C3#466811yescharacteristic
461C2×C3#474611yescharacteristic
462C2×C3#475311yescharacteristic
463C2×C3#482511yescharacteristic
464C2×C3#482711yescharacteristic
465C2×C3#495011yescharacteristic
466D3—42no—
467C22—21no—
468C22—21no—
469C22—62no—
470C22—121no—
471C22—121no—
472C22Dic3×D1211yescharacteristic
473C22#6389311yescharacteristic
474C22#6392311yescharacteristic
475C4—62no—
476C4—122no—
477C4—122no—
478C4—361no—
479C4—361no—
480C4—721no—
481C4D3×C4⋊Dic311yescharacteristic
482C4#6389111yescharacteristic
483C4#6389611yescharacteristic
484C4#67⋊Dic311yescharacteristic
485C3—21no—
486C3#6900311yescharacteristic, complemented
487C3#7036411yescharacteristic, complemented
488C2—21no—
489C2—21no—
490C2—122no—
491C2#1059101611yescharacteristic
492C2#1059102211yescharacteristic
493C2#1059137911yescharacteristic
subgroupG/H
centerC22#63923
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
2. #10592820C2semidirect
4. #10591224C2semidirect
6. C3×#4554C2semidirect
7. #10589162C2semidirect
24. C3×#825C4semidirect
41. #4554D3semidirect
57. C32×M16C2×C4semidirect
68. C3×C4×Dic3C2×C4semidirect
108. #825Dic3semidirect
187. C3×M16C4×D3semidirect
188. C3×M16C2×Dic3semidirect
224. C4×Dic3C2×Dic3semidirect
356. M16D3×Dic3semidirect
420. C32#1860semidirect
486. C3#69003semidirect
487. C3#70364semidirect

Maximal subgroups

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

subgroupindex
#105949862
#105928202
C2×C3×#8252
#105912242
#105910882
C3×#45542
#105891622
#703643
#690033