Finite Group Databasebeta

← #11787907

Subgroups — #11787907

Order 1152.

783 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5516C211yesmaximal, characteristic, complemented
2C2×#63864C211yesmaximal, characteristic
3#10592700C211yesmaximal, characteristic
4#10591096C211yesmaximal, characteristic
5C2×#63560C211yesmaximal, characteristic, complemented
6C3×#4532C211yesmaximal, characteristic, complemented
7#10589161C211yesmaximal, characteristic, complemented
8#71661—31nomaximal
9#68766—31nomaximal
10C22×C4⋊(C3⋊D3)C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C32×#117C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C3×#926C2211yescharacteristic
15C2×C3×C4×Dic3C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C3×#887—22no—
18C3×#877—22no—
19C3×C4×Dic6—22no—
20C2×#3722—22no—
21#63864—22no—
22#63864C414yescomplemented
23#63636—22no—
24#63564—22no—
25#63560—22no—
26C3×#519—31no—
27C2×#947—31no—
28#5516D311yescharacteristic, complemented
29C2×#875—31no—
30C2×#836—31no—
31C2×#824—31no—
32#5131—31no—
33#5001—31no—
34#5001—31no—
35#5001—31no—
36#5001—61no—
37C3×#339—31no—
38#4532—31no—
39#4532D311yescharacteristic, complemented
40#4523—31no—
41C23×C3⋊D3—21no—
42C22×C32×C4C2311yescharacteristic
43C2×C4⋊(C3⋊D3)—21no—
44C2×C4⋊(C3⋊D3)—21no—
45C2×C4⋊(C3⋊D3)—21no—
46C2×C4⋊(C3⋊D3)—21no—
47C2×C4⋊(C3⋊D3)—21no—
48C2×C4⋊(C3⋊D3)C2×C412yes—
49C22×C3×Dic3—21no—
50C22×C3×Dic3D412yes—
51C2×C3×Dic6—21no—
52C2×C3×Dic6—21no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6C2×C412yes—
56C2×C3×Dic6D412yes—
57C32×C4⋊C4D412yes—
58C32×C22⋊C4—21no—
59C32×C42—21no—
60#3748—21no—
61#3748—21no—
62C2×C3⋊(C3⋊C8)—21no—
63C3×#217—21no—
64C3×C4⋊Dic3D412yes—
65C3×Dic3⋊C4—24no—
66C3×C4×Dic3—24no—
67C3×C4×Dic3—21no—
68C3×C4×Dic3—21no—
69C3×C4×Dic3C2×C412yes—
70C3×#208—21no—
71C3×#208—21no—
72C2×C3×C3⋊C8—21no—
73#3722—28no—
74#1888—91noSylow
75C22×C3×Q8—31no—
76C22×D12—31no—
77C22×D12—31no—
78C22×D12—61no—
79C22×Dic6C2×D311yescharacteristic
80C2×C3×M16—31no—
81C3×C4×Q8—62no—
82C3×C4×Q8—62no—
83C3×#117—21no—
84C3×#117—31no—
85C3×#117—62no—
86C3×#117C2×D311yescharacteristic
87C3×#117C2×D311yescharacteristic
88C2×C3×C42—31no—
89#947—34no—
90#947—62no—
91C2×(C2×D3)⋊C4—62no—
92C2×(C2×D3)⋊C4—62no—
93#926C2×D311yescharacteristic
94C2×C4×Dic3C2×D311yescharacteristic
95C2×#208—31no—
96C2×#208—31no—
97C2×#208—61no—
98C2×#208C2×D311yescharacteristic
99#887—22no—
100#877—22no—
101#875—34no—
102#875—62no—
103C4×Dic6—22no—
104#836—62no—
105#824—62no—
106#821—62no—
107#809—62no—
108#809—64no—
109#809—62no—
110#809—62no—
111C23×C32C2×D411yescharacteristic
112C22×C3⋊D3—22no—
113C22×C3⋊D3—41no—
114C22×C3⋊D3—41no—
115C22×C3⋊D3—44no—
116C2×C32×C4—22no—
117C2×C32×C4C22×C411yescharacteristic
118C2×C32×C4C2×D411yescharacteristic
119C2×C32×C4C2×D411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4#4111yescharacteristic
122C2×C32×C4#4111yescharacteristic
123C4⋊(C3⋊D3)—22no—
124C4⋊(C3⋊D3)—22no—
125C4⋊(C3⋊D3)—41no—
126C4⋊(C3⋊D3)—41no—
127C2×C3×Dic3—22no—
128C2×C3×Dic3—22no—
129C2×C3×Dic3—22no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—42no—
132C2×C3×Dic3C22⋊C414yes—
133C3×Dic6—22no—
134C3×Dic6—22no—
135C3×Dic6—22no—
136C3×Dic6—22no—
137C3⋊(C3⋊C8)—41no—
138C3×C3⋊C8—41no—
139C2×#124—91no—
140#519—31no—
141#418—91no—
142#418—91no—
143C2×#101—91no—
144#339—31no—
145#339—91no—
146C23×D3—61no—
147C23×D3—61no—
148C23×D3—121no—
149C2×C3×Q8—32no—
150C2×C3×Q8—32no—
151C2×C3×Q8—61no—
152C2×C3×Q8—61no—
153C2×C3×Q8—61no—
154C2×C3×Q8—61no—
155C22×C3×C4—21no—
156C22×C3×C4—32no—
157C22×C3×C4—61no—
158C22×C3×C4C22×D311yescharacteristic
159C22×C3×C4C22×D311yescharacteristic
160C22×Dic3—21no—
161C22×Dic3#6712yes—
162C2×D12—32no—
163C2×D12—32no—
164C2×D12—62no—
165C2×D12—62no—
166C2×D12—62no—
167C2×D12—61no—
168C2×D12—61no—
169C2×D12—61no—
170C2×D12—61no—
171C2×D12—61no—
172C2×D12—61no—
173C2×D12—61no—
174C2×D12—61no—
175C2×D12—61no—
176C2×D12—61no—
177C2×D12—121no—
178C2×D12—121no—
179C2×D12—121no—
180C2×Dic6—21no—
181C2×Dic6—21no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6C4×D312yes—
185C2×Dic6D1212yes—
186C3×M16—61no—
187C3×M16—61no—
188C2×C3×C8—61no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—32no—
191C3×C4⋊C4—64no—
192C3×C4⋊C4D1212yes—
193C3×C4⋊C4#6712yes—
194C3×C22⋊C4—21no—
195C3×C22⋊C4—21no—
196C3×C22⋊C4—41no—
197C3×C22⋊C4—61no—
198C3×C42—21no—
199C3×C42—21no—
200C3×C42—32no—
201C3×C42—41no—
202C3×C42—64no—
203C3×C42—61no—
204C3×C42—61no—
205#217—21no—
206(C2×D3)⋊C4—68no—
207(C2×D3)⋊C4—68no—
208C4⋊Dic3#6712yes—
209Dic3⋊C4—24no—
210C4×Dic3—24no—
211C4×Dic3—21no—
212C4×Dic3—21no—
213C4×Dic3C4×D312yes—
214#208—21no—
215#208—21no—
216#208—61no—
217#208—61no—
218#208—61no—
219#208—61no—
220#208—121no—
221#208—121no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—62no—
224C2×C3⋊C8—61no—
225C2×C3⋊C8—61no—
226C22×C32—22no—
227C22×C32C2×C22⋊C411yescharacteristic
228C22×C32#12011yescharacteristic
229C22×C32#12311yescharacteristic
230C2×C3⋊D3—42no—
231C2×C3⋊D3—44no—
232C2×C3⋊D3—82no—
233C2×C3⋊D3—82no—
234C32×C4—42no—
235C32×C4C4×D411yescharacteristic
236C32×C4C4×D411yescharacteristic
237C32×C4#12111yescharacteristic
238C32×C4#12111yescharacteristic
239C3×Dic3—24no—
240C3×Dic3—42no—
241C3×Dic3—42no—
242C22×Q8—31no—
243C22×D4—91no—
244C2×M16—31no—
245C2×M16—91no—
246#124—94no—
247#124—182no—
248C4×Q8—62no—
249C4×Q8—62no—
250#117—31no—
251#117—62no—
252#117D3211yescharacteristic
253C2×C22⋊C4—182no—
254C2×C42—31no—
255#102—182no—
256#102—182no—
257#101—182no—
258C23×C3—21no—
259C23×C3D3×D411yescharacteristic
260C23×C3C2×#6711yescharacteristic
261C22×D3—62no—
262C22×D3—62no—
263C22×D3—122no—
264C22×D3—122no—
265C22×D3—122no—
266C22×D3—124no—
267C22×D3—124no—
268C22×D3—121no—
269C22×D3—121no—
270C22×D3—121no—
271C22×D3—121no—
272C22×D3—244no—
273C3×Q8—62no—
274C3×Q8—62no—
275C3×Q8—62no—
276C3×Q8—62no—
277C2×C3×C4—22no—
278C2×C3×C4—22no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—34no—
286C2×C3×C4—42no—
287C2×C3×C4—62no—
288C2×C3×C4—62no—
289C2×C3×C4—62no—
290C2×C3×C4—62no—
291C2×C3×C4—122no—
292C2×C3×C4C2×C4×D311yescharacteristic
293C2×C3×C4C2×C4×D311yescharacteristic
294C2×C3×C4C2×D1211yescharacteristic
295C2×C3×C4C2×D1211yescharacteristic
296C2×C3×C4#23511yescharacteristic
297C2×C3×C4#23511yescharacteristic
298C2×C3×C4#23511yescharacteristic
299C2×C3×C4D3×D411yescharacteristic
300C2×C3×C4D3×D411yescharacteristic
301C2×C3×C4#23911yescharacteristic
302C2×C3×C4C2×#6711yescharacteristic
303C2×C3×C4C2×#6711yescharacteristic
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—42no—
309C2×Dic3(C2×D3)⋊C414yes—
310D12—64no—
311D12—62no—
312D12—62no—
313D12—62no—
314D12—62no—
315D12—122no—
316D12—122no—
317D12—122no—
318D12—121no—
319D12—121no—
320D12—121no—
321D12—121no—
322Dic6—22no—
323Dic6—22no—
324Dic6—22no—
325Dic6—22no—
326C3×C8—121no—
327C3⋊C8—41no—
328C3⋊C8—122no—
329C3⋊C8—121no—
330C3⋊C8—121no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32#38611yescharacteristic
334C2×C32#45412yes—
335C3⋊D3—82no—
336C24—181no—
337C2×Q8—32no—
338C2×Q8—32no—
339C2×Q8—61no—
340C2×Q8—61no—
341C2×Q8—61no—
342C2×Q8—61no—
343C2×D4—92no—
344C2×D4—181no—
345C2×D4—181no—
346C2×D4—181no—
347C2×D4—181no—
348C2×D4—181no—
349C22×C4—32no—
350C22×C4—61no—
351C22×C4C2×D3211yescharacteristic
352M16—61no—
353M16—61no—
354M16—181no—
355M16—181no—
356C2×C8—61no—
357C2×C8—181no—
358C4⋊C4—32no—
359C4⋊C4—64no—
360C4⋊C4#62612yes—
361C22⋊C4—21no—
362C22⋊C4—61no—
363C22⋊C4—188no—
364C42—21no—
365C42—32no—
366C42—64no—
367C42—61no—
368C42—61no—
369C22×C3—22no—
370C22×C3—22no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—42no—
375C22×C3D3×C22⋊C411yescharacteristic
376C22×C3#88211yescharacteristic
377C22×C3#89411yescharacteristic
378C22×C3C2×(C2×D3)⋊C411yescharacteristic
379C22×C3#92911yescharacteristic
380C22×C3#95311yescharacteristic
381C2×D3—124no—
382C2×D3—124no—
383C2×D3—124no—
384C2×D3—122no—
385C2×D3—122no—
386C2×D3—244no—
387C2×D3—242no—
388C2×D3—242no—
389C2×D3—244no—
390C2×D3—244no—
391C2×D3—242no—
392C2×D3—242no—
393C3×C4—21no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—21no—
397C3×C4—42no—
398C3×C4—42no—
399C3×C4—64no—
400C3×C4—82no—
401C3×C4—122no—
402C3×C4—122no—
403C3×C4C4×D1211yescharacteristic
404C3×C4#88411yescharacteristic
405C3×C4#89311yescharacteristic
406C3×C4#89511yescharacteristic
407C3×C4C4×#6711yescharacteristic
408C3×C4C4×#6711yescharacteristic
409C3×C4#93011yescharacteristic
410C3×C4#93011yescharacteristic
411Dic3—24no—
412Dic3—42no—
413Dic3—42no—
414C32#188811yescharacteristic, complemented, Sylow
415C23—182no—
416C23—364no—
417C23—361no—
418C23—361no—
419C23D3×#6711yescharacteristic
420Q8—62no—
421Q8—62no—
422Q8—62no—
423Q8—62no—
424D4—182no—
425D4—182no—
426D4—361no—
427D4—361no—
428C2×C4—22no—
429C2×C4—34no—
430C2×C4—62no—
431C2×C4—62no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—122no—
435C2×C4#379811yescharacteristic
436C2×C4#379911yescharacteristic
437C2×C4C4×D3211yescharacteristic
438C2×C4D3×D1211yescharacteristic
439C2×C4C2×#62611yescharacteristic
440C2×C4D3×#6711yescharacteristic
441C8—121no—
442C8—361no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—22no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3#466511yescharacteristic
452C2×C3#482712yes—
453C2×C3#521311yescharacteristic
454C2×C3#524512yes—
455D3—244no—
456D3—242no—
457D3—242no—
458C22—22no—
459C22—364no—
460C22—362no—
461C22—722no—
462C22—722no—
463C22#6385011yescharacteristic
464C22D3×(C2×D3)⋊C411yescharacteristic
465C22#6397711yescharacteristic
466C4—42no—
467C4—64no—
468C4—122no—
469C4—122no—
470C4C4×#62611yescharacteristic
471C4#6390411yescharacteristic
472C4#6390611yescharacteristic
473C4#6390911yescharacteristic
474C3—21no—
475C3#6876611yescharacteristic, complemented
476C3#7166111yescharacteristic, complemented
477C2—21no—
478C2—21no—
479C2—722no—
480C2#1059138211yescharacteristic
481C2#1059168512yes—
subgroupG/H
centerC22#63977
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC32×#117C22
socleC22×C32#120
prime radicalC22×C4⋊(C3⋊D3)C22
cosocleC22×C32×C4C23
hypercenter#117D32
hypocenterC32#1888
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C3×#5516C2semidirect
5. C2×#63560C2semidirect
6. C3×#4532C2semidirect
7. #10589161C2semidirect
22. #63864C4semidirect
28. #5516D3semidirect
39. #4532D3semidirect
414. C32#1888semidirect
475. C3#68766semidirect
476. C3#71661semidirect

Maximal subgroups

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

subgroupindex
C3×#55162
C2×#638642
#105927002
#105910962
C2×#635602
C3×#45322
#105891612
#716613
#687663