Finite Group Databasebeta

← #11783918

Subgroups — #11783918

Order 1152.

725 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10594954C211yesmaximal, characteristic
2C2×#63652C211yesmaximal, characteristic
3C3×Dic3×M16C211yesmaximal, characteristic, complemented
4#10591222C211yesmaximal, characteristic, complemented
5C2×#63570C211yesmaximal, characteristic
6C3×#4555C211yesmaximal, characteristic, complemented
7#10589160C211yesmaximal, characteristic, complemented
8#69866—31nomaximal
9#68729—31nomaximal
10C2×#3828C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C2×C4×C3⋊Dic3C2211yescharacteristic
13C3×#926C2211yescharacteristic
14C2×C3×C4×Dic3C2211yescharacteristic
15C2×C3×C4×Dic3C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17#63952—22no—
18#63894C412yescomplemented
19C4×#624—21no—
20#63874—21no—
21#63844C412yescomplemented
22#63652C414yescomplemented
23C3×#815—21no—
24C3×C8×Dic3—21no—
25#63579—22no—
26#63570C414yescomplemented
27#5522—31no—
28C4×#235—31no—
29C3×C4×M16—31no—
30C2×#825—31no—
31C2×#825—31no—
32C2×#825—31no—
33C2×#825—61no—
34Dic3×M16D311yescharacteristic, complemented
35#4932—31no—
36#4932—31no—
37C2×#805—31no—
38C3×#339—31no—
39#4555—31no—
40#4555D311yescharacteristic, complemented
41#4532—31no—
42C22×C32×C4C2311yescharacteristic
43C2×#638—21no—
44C22×C3⋊Dic3—21no—
45#3828—21no—
46#3828—21no—
47#3828C2×C412yes—
48#3828C2×C412yes—
49C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
50C2×C4×C3⋊D3—21no—
51C2×#634C2×C411yescharacteristic
52C22×C3×Dic3—21no—
53C22×C3×Dic3D412yes—
54C2×#624—22no—
55C32×M16—21no—
56C32×M16C2×C412yescomplemented
57C2×C32×C8—21no—
58C4×C3⋊Dic3—21no—
59C4×C3⋊Dic3C2×C412yescomplemented
60C3×#217—21no—
61C3×C4⋊Dic3C2×C412yes—
62C3×C4×Dic3—21no—
63C3×C4×Dic3—21no—
64C3×C4×Dic3C2×C412yes—
65C3×C4×Dic3C2×C412yescomplemented
66C3×C4×Dic3D411yescharacteristic
67C3×C4×Dic3D411yescharacteristic
68C3×#208—21no—
69C3×#208C2×C412yescomplemented
70C2×C3×C3⋊C8—21no—
71#3724—22no—
72#3722—22no—
73Dic32—22no—
74#1759—91noSylow
75C2×#235—31no—
76C2×#235—31no—
77C2×#235—61no—
78C2×C3×M16—21no—
79C2×C3×M16—31no—
80C2×C3×M16C2×D311yescharacteristic
81C2×C3×M16C2×D311yescharacteristic
82C3×#117—31no—
83C2×C3×C42—31no—
84C2×C3×C42—31no—
85C4×#67—62no—
86#926C2×D311yescharacteristic
87C2×C4×Dic3—31no—
88C2×C4×Dic3—31no—
89C2×C4×Dic3—61no—
90C2×C4×Dic3C2×D311yescharacteristic
91C2×C4×Dic3C2×D311yescharacteristic
92C2×#208C2×D311yescharacteristic
93#893—32no—
94#887—32no—
95#67⋊C4—62no—
96C4×D12—32no—
97#872—61no—
98#872—61no—
99C42×D3—61no—
100C42×D3—61no—
101C4×Dic6—32no—
102C3×#97—61no—
103C3×C4×C8—61no—
104C3⋊#95—62no—
105C3⋊#95—62no—
106#825—34no—
107#825—34no—
108#825—34no—
109#825—64no—
110#815—21no—
111C8×Dic3—21no—
112#805—34no—
113C23×C32C2×D411yescharacteristic
114C22×C3⋊D3—21no—
115C2×C32×C4C22×C411yescharacteristic
116C2×C32×C4C22×C411yescharacteristic
117C2×C32×C4C22×C411yescharacteristic
118C2×C32×C4C2×D411yescharacteristic
119C2×C32×C4#4111yescharacteristic
120C2×C32×C4#4111yescharacteristic
121#638—22no—
122#638—22no—
123C2×C3⋊Dic3—21no—
124C2×C3⋊Dic3—22no—
125C2×C3⋊Dic3—42no—
126C4⋊(C3⋊D3)—21no—
127C4⋊(C3⋊D3)C4212yes—
128C4×C3⋊D3—22no—
129C4×C3⋊D3—22no—
130#634—21no—
131#634C4212yes—
132C2×C3×Dic3—22no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—42no—
135C2×C3×Dic3—42no—
136C2×C3×Dic3C22⋊C412yes—
137C2×C3×Dic3C22⋊C412yes—
138#624—44no—
139C32×C8—22no—
140C3×C3⋊C8—22no—
141C4×#41—91no—
142C2×#104—91no—
143C2×#104—91no—
144C4×M16—31no—
145#376—91no—
146#339—31no—
147#339—91no—
148C22×C3×C4—21no—
149C22×C3×C4—32no—
150C22×C3×C4—61no—
151C22×C3×C4C22×D311yescharacteristic
152C22×C3×C4C22×D311yescharacteristic
153C2×#67—61no—
154C2×#67—61no—
155C2×#67—121no—
156C22×Dic3—21no—
157C22×Dic3—61no—
158C22×Dic3—61no—
159C22×Dic3—121no—
160C22×Dic3D1212yes—
161#235—32no—
162#235—32no—
163#235—32no—
164#235—32no—
165#235—62no—
166#235—62no—
167#235—61no—
168#235—61no—
169#235—61no—
170#235—61no—
171#235—121no—
172#235—121no—
173C2×D12—31no—
174C2×D12—31no—
175C2×D12—61no—
176C2×C4×D3—61no—
177C2×C4×D3—61no—
178C2×C4×D3—62no—
179C2×C4×D3—62no—
180C2×C4×D3—121no—
181C2×Dic6—31no—
182C2×Dic6—31no—
183C2×Dic6—61no—
184C3×M16—21no—
185C3×M16—21no—
186C3×M16—22no—
187C3×M16—32no—
188C3×M16—41no—
189C3×M16—61no—
190C3×M16C4×D312yescomplemented
191C3×M16C4×D312yescomplemented
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—41no—
195C2×C3×C8—61no—
196C3×C4⋊C4—32no—
197C3×C22⋊C4—61no—
198C3×C42—32no—
199C3×C42—31no—
200C3×C42—31no—
201C3×C42—32no—
202C3×C42—61no—
203C3×C42—61no—
204#217—21no—
205(C2×D3)⋊C4—62no—
206(C2×D3)⋊C4—62no—
207C4⋊Dic3C4×D312yes—
208Dic3⋊C4—62no—
209Dic3⋊C4—62no—
210C4×Dic3—21no—
211C4×Dic3—21no—
212C4×Dic3—32no—
213C4×Dic3—32no—
214C4×Dic3—62no—
215C4×Dic3—62no—
216C4×Dic3—61no—
217C4×Dic3—61no—
218C4×Dic3—62no—
219C4×Dic3—121no—
220C4×Dic3C4×D312yes—
221C4×Dic3C4×D312yescomplemented
222C4×Dic3#6711yescharacteristic
223C4×Dic3#6711yescharacteristic
224#208—21no—
225#208C4×D312yescomplemented
226C2×C3⋊C8—21no—
227C22×C32—22no—
228C22×C32C2×C22⋊C411yescharacteristic
229C22×C32C4×D411yescharacteristic
230C22×C32C4×D411yescharacteristic
231C2×C3⋊D3—22no—
232C2×C3⋊D3—42no—
233C32×C4C2×C4211yescharacteristic
234C32×C4#11711yescharacteristic
235C32×C4C4×D411yescharacteristic
236C32×C4C4×D411yescharacteristic
237C3⋊Dic3—22no—
238C3⋊Dic3—42no—
239C3×Dic3—22no—
240C3×Dic3—22no—
241C3×Dic3—42no—
242C3×Dic3—42no—
243C2×#41—91no—
244C2×M16—31no—
245C2×M16D3211yescharacteristic
246C4×Q8—92no—
247C4×D4—92no—
248C4×D4—182no—
249#117—31no—
250#117—181no—
251C2×C42—31no—
252C2×C42—31no—
253C2×C42—91no—
254C2×C42—181no—
255#104—94no—
256#104—94no—
257#97—61no—
258C4×C8—61no—
259#95—182no—
260C23×C3—21no—
261C23×C3C2×D1211yescharacteristic
262C23×C3D3×D411yescharacteristic
263C22×D3—61no—
264C22×D3—61no—
265C22×D3—121no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—32no—
273C2×C3×C4—32no—
274C2×C3×C4—62no—
275C2×C3×C4—62no—
276C2×C3×C4—122no—
277C2×C3×C4—122no—
278C2×C3×C4C2×C4×D311yescharacteristic
279C2×C3×C4C2×C4×D311yescharacteristic
280C2×C3×C4C2×C4×D311yescharacteristic
281C2×C3×C4C2×C4×D311yescharacteristic
282C2×C3×C4C2×C4×D311yescharacteristic
283C2×C3×C4C2×C4×D311yescharacteristic
284C2×C3×C4#23511yescharacteristic
285C2×C3×C4#23511yescharacteristic
286C2×C3×C4D3×D411yescharacteristic
287C2×C3×C4#23711yescharacteristic
288C2×C3×C4#23911yescharacteristic
289C2×C3×C4C2×#6711yescharacteristic
290#67—62no—
291#67—62no—
292#67—62no—
293#67—62no—
294#67—122no—
295#67—122no—
296C2×Dic3—22no—
297C2×Dic3—22no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—61no—
301C2×Dic3—61no—
302C2×Dic3—62no—
303C2×Dic3—62no—
304C2×Dic3—122no—
305C2×Dic3—121no—
306C2×Dic3—122no—
307C2×Dic3—122no—
308C2×Dic3—124no—
309C2×Dic3(C2×D3)⋊C412yes—
310C2×Dic3(C2×D3)⋊C412yes—
311D12—32no—
312D12—32no—
313D12—62no—
314D12—61no—
315D12—61no—
316D12—121no—
317C4×D3—62no—
318C4×D3—62no—
319C4×D3—62no—
320C4×D3—62no—
321C4×D3—122no—
322C4×D3—122no—
323C4×D3—124no—
324C4×D3—124no—
325Dic6—32no—
326Dic6—32no—
327Dic6—62no—
328Dic6—61no—
329Dic6—61no—
330Dic6—121no—
331C3×C8—22no—
332C3×C8—22no—
333C3×C8—42no—
334C3×C8—62no—
335C3⋊C8—22no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32C4×C22⋊C411yescharacteristic
339C2×C32#42112yes—
340C3⋊D3—42no—
341#41—92no—
342#41—92no—
343#41—181no—
344#41—181no—
345C2×Q8—91no—
346C2×D4—91no—
347C2×D4—181no—
348C22×C4—32no—
349C22×C4—61no—
350C22×C4—182no—
351C22×C4—181no—
352C22×C4—181no—
353C22×C4C2×D3211yescharacteristic
354M16—21no—
355M16—32no—
356M16—61no—
357M16#62412yescomplemented
358C2×C8—21no—
359C2×C8—61no—
360C4⋊C4—32no—
361C4⋊C4—182no—
362C22⋊C4—61no—
363C22⋊C4—182no—
364C42—32no—
365C42—31no—
366C42—31no—
367C42—32no—
368C42—61no—
369C42—61no—
370C42—92no—
371C42—182no—
372C42—181no—
373C22×C3—22no—
374C22×C3—22no—
375C22×C3—21no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—42no—
379C22×C3C4×D1211yescharacteristic
380C22×C3C4×D1211yescharacteristic
381C22×C3D3×C22⋊C411yescharacteristic
382C22×C3#67⋊C411yescharacteristic
383C22×C3#89311yescharacteristic
384C22×C3C2×(C2×D3)⋊C411yescharacteristic
385C2×D3—62no—
386C2×D3—62no—
387C2×D3—122no—
388C2×D3—122no—
389C2×D3—122no—
390C2×D3—242no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—62no—
396C3×C4—62no—
397C3×C4—122no—
398C3×C4—122no—
399C3×C4C42×D311yescharacteristic
400C3×C4C42×D311yescharacteristic
401C3×C4#87211yescharacteristic
402C3×C4#67⋊C411yescharacteristic
403C3×C4#89211yescharacteristic
404C3×C4#89311yescharacteristic
405C3×C4C4×#6711yescharacteristic
406C3×C4C4×#6711yescharacteristic
407Dic3—22no—
408Dic3—22no—
409Dic3—42no—
410Dic3—42no—
411Dic3—62no—
412Dic3—62no—
413Dic3—122no—
414Dic3—122no—
415Dic3—122no—
416Dic3—124no—
417C32#175911yescharacteristic, complemented, Sylow
418C23—181no—
419C23D3×D1211yescharacteristic
420Q8—92no—
421Q8—181no—
422D4—92no—
423D4—182no—
424D4—182no—
425D4—181no—
426C2×C4—32no—
427C2×C4—32no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—122no—
431C2×C4—122no—
432C2×C4—181no—
433C2×C4—182no—
434C2×C4—182no—
435C2×C4—182no—
436C2×C4—364no—
437C2×C4—362no—
438C2×C4#379911yescharacteristic
439C2×C4C4×D3211yescharacteristic
440C2×C4C4×D3211yescharacteristic
441C2×C4#380411yescharacteristic
442C2×C4C2×#62411yescharacteristic
443C2×C4D3×#6711yescharacteristic
444C8—22no—
445C8—62no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—22no—
452C2×C3—41no—
453C2×C3—41no—
454C2×C3#466611yescharacteristic
455C2×C3#482112yes—
456C2×C3C4×(C2×D3)⋊C411yescharacteristic
457C2×C3#513912yes—
458D3—122no—
459D3—122no—
460D3—242no—
461C22—22no—
462C22—182no—
463C22—362no—
464C22#6388011yescharacteristic
465C22#6389411yescharacteristic
466C22D3×(C2×D3)⋊C411yescharacteristic
467C4—62no—
468C4—62no—
469C4—122no—
470C4—122no—
471C4—182no—
472C4—362no—
473C4#6383611yescharacteristic
474C4C4×#62411yescharacteristic
475C4#6390411yescharacteristic
476C4#6395211yescharacteristic
477C3—21no—
478C3#6872911yescharacteristic, complemented
479C3#6986611yescharacteristic, complemented
480C2—21no—
481C2—21no—
482C2—362no—
483C2#1059100512yes—
484C2#1059123311yescharacteristic
subgroupG/H
centerC2×C4D3×#67
commutatorC32×C4C2×C42
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32C2×C22⋊C4
prime radicalC2×#3828C22
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1759
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
3. C3×Dic3×M16C2semidirect
4. #10591222C2semidirect
6. C3×#4555C2semidirect
7. #10589160C2semidirect
18. #63894C4semidirect
21. #63844C4semidirect
22. #63652C4semidirect
26. #63570C4semidirect
34. Dic3×M16D3semidirect
40. #4555D3semidirect
56. C32×M16C2×C4semidirect
59. C4×C3⋊Dic3C2×C4semidirect
65. C3×C4×Dic3C2×C4semidirect
69. C3×#208C2×C4semidirect
190. C3×M16C4×D3semidirect
191. C3×M16C4×D3semidirect
221. C4×Dic3C4×D3semidirect
225. #208C4×D3semidirect
357. M16#624semidirect
417. C32#1759semidirect
478. C3#68729semidirect
479. C3#69866semidirect

Maximal subgroups

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

subgroupindex
#105949542
C2×#636522
C3×Dic3×M162
#105912222
C2×#635702
C3×#45552
#105891602
#698663
#687293