Finite Group Databasebeta

← #11789916

Subgroups — #11789916

Order 1152.

697 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10592638C211yesmaximal, characteristic
2C3×#4928C211yesmaximal, characteristic, complemented
3C2×C3×#803C211yesmaximal, characteristic, complemented
4C2×#63564C211yesmaximal, characteristic
5C2×#63564C211yesmaximal, characteristic
6#10589164C211yesmaximal, characteristic, complemented
7#10589164C211yesmaximal, characteristic, complemented
8#69818—31nomaximal
9#69617—31nomaximal
10C22×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×C42C2211yescharacteristic
12C2×#3752—21no—
13C2×#3752—21no—
14C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C4⋊Dic3C2211yescharacteristic
16C2×C3×C4⋊Dic3C2211yescharacteristic
17C2×C3×Dic3⋊C4—21no—
18C22×C3×C3⋊C8C2211yescharacteristic
19C22×C3×C3⋊C8C2211yescharacteristic
20C3×C3⋊#95—21no—
21C3×#803—22no—
22#63564—22no—
23#63564C414yescomplemented
24C2×C3×#97—31no—
25C3×#389—31no—
26C2×#821—31no—
27C2×#821—31no—
28C2×#809—31no—
29C2×#809—31no—
30#4939—31no—
31#4939—31no—
32#4939—61no—
33#4928D311yescharacteristic, complemented
34C2×#803D311yescharacteristic, complemented
35#4551—31no—
36#4551—31no—
37#4529—31no—
38#4529—31no—
39C23×C3⋊D3—21no—
40C22×C32×C4D411yescharacteristic
41C22×C32×C4D411yescharacteristic
42C22×C32×C4C2311yescharacteristic
43C22×C3⋊Dic3—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)—21no—
49C2×C4⋊(C3⋊D3)C2×C412yes—
50C22×C3×Dic3—21no—
51C22×C3×Dic3—21no—
52C32×C42—21no—
53C32×C42—21no—
54#3752—42no—
55#3752—42no—
56C4⋊(C3⋊Dic3)—21no—
57C4⋊(C3⋊Dic3)—21no—
58C3×C4⋊Dic3—21no—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3—21no—
61C3×C4⋊Dic3C2×C412yes—
62C3×Dic3⋊C4—42no—
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—21no—
66C2×C3×C3⋊C8—21no—
67C2×C3×C3⋊C8C2×C412yes—
68C2×C3×C3⋊C8D412yes—
69#1936—91noSylow
70C22×D12—31no—
71C22×D12—31no—
72C22×D12—61no—
73C22×C3×C8—31no—
74C22×C3×C8—31no—
75C2×C3×C4⋊C4—31no—
76C2×C3×C4⋊C4—31no—
77C2×C3×C4⋊C4—61no—
78C2×C3×C42—21no—
79C2×C3×C42C2×D311yescharacteristic
80C2×C3×C42C2×D311yescharacteristic
81C2×(C2×D3)⋊C4—62no—
82C2×(C2×D3)⋊C4—62no—
83C2×(C2×D3)⋊C4—61no—
84C2×(C2×D3)⋊C4—61no—
85C2×(C2×D3)⋊C4—61no—
86C2×(C2×D3)⋊C4—61no—
87C2×C4⋊Dic3—31no—
88C2×C4⋊Dic3—31no—
89C2×C4⋊Dic3—61no—
90C2×C4⋊Dic3C2×D311yescharacteristic
91C2×C4⋊Dic3C2×D311yescharacteristic
92C2×Dic3⋊C4—21no—
93C22×C3⋊C8C2×D311yescharacteristic
94C22×C3⋊C8C2×D311yescharacteristic
95C3×#97—62no—
96C3×#95—61no—
97C3⋊#95—21no—
98#821—34no—
99#821—62no—
100#809—34no—
101#809—62no—
102#803—22no—
103C23×C32C2×D411yescharacteristic
104C22×C3⋊D3—22no—
105C22×C3⋊D3—41no—
106C22×C3⋊D3—41no—
107C22×C3⋊D3—41no—
108C22×C3⋊D3—41no—
109C22×C3⋊D3—41no—
110C22×C3⋊D3—41no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4C22×C411yescharacteristic
118C2×C32×C4C2×D411yescharacteristic
119C2×C32×C4#4111yescharacteristic
120C2×C32×C4#4111yescharacteristic
121C2×C32×C4#4111yescharacteristic
122C2×C32×C4#4111yescharacteristic
123C2×C3⋊Dic3—41no—
124C2×C3⋊Dic3—41no—
125C2×C3⋊Dic3—41no—
126C4⋊(C3⋊D3)—22no—
127C4⋊(C3⋊D3)—22no—
128C4⋊(C3⋊D3)—41no—
129C4⋊(C3⋊D3)—41no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—41no—
132C2×C3×Dic3—41no—
133C2×C3×Dic3—41no—
134C2×C3×Dic3—41no—
135C2×C3×Dic3—41no—
136C3×C3⋊C8—22no—
137C3×C3⋊C8—22no—
138C2×#102—91no—
139C2×#102—91no—
140C2×#97—31no—
141#390—91no—
142#389—31no—
143#340—91no—
144#340—91no—
145C23×D3—62no—
146C23×D3—61no—
147C23×D3—61no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—61no—
152C22×C3×C4—61no—
153C22×C3×C4D1211yescharacteristic
154C22×C3×C4D1211yescharacteristic
155C22×C3×C4#6711yescharacteristic
156C22×C3×C4#6711yescharacteristic
157C22×C3×C4C22×D311yescharacteristic
158C22×C3×C4C22×D311yescharacteristic
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—62no—
162C22×Dic3—61no—
163C22×Dic3—61no—
164C2×D12—32no—
165C2×D12—32no—
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—62no—
176C2×D12—61no—
177C2×D12—61no—
178C2×D12—121no—
179C2×D12—121no—
180C2×D12—121no—
181C2×D12—121no—
182C2×C3×C8—32no—
183C2×C3×C8—32no—
184C2×C3×C8—61no—
185C2×C3×C8—61no—
186C2×C3×C8—61no—
187C2×C3×C8—61no—
188C3×C4⋊C4—32no—
189C3×C4⋊C4—61no—
190C3×C4⋊C4—61no—
191C3×C4⋊C4—61no—
192C3×C4⋊C4—122no—
193C3×C42—21no—
194C3×C42—21no—
195C3×C42—21no—
196C3×C42—21no—
197C3×C42—41no—
198C3×C42—41no—
199(C2×D3)⋊C4—124no—
200(C2×D3)⋊C4—124no—
201(C2×D3)⋊C4—122no—
202(C2×D3)⋊C4—122no—
203(C2×D3)⋊C4—122no—
204(C2×D3)⋊C4—122no—
205C4⋊Dic3—21no—
206C4⋊Dic3—21no—
207C4⋊Dic3—21no—
208C4⋊Dic3—61no—
209C4⋊Dic3—61no—
210C4⋊Dic3—61no—
211C4⋊Dic3—61no—
212C4⋊Dic3—121no—
213C4⋊Dic3—121no—
214C4⋊Dic3C4×D312yes—
215Dic3⋊C4—42no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8C4×D312yes—
221C2×C3⋊C8#6712yes—
222C22×C32C4×D411yescharacteristic
223C22×C32#12111yescharacteristic
224C22×C32#12411yescharacteristic
225C22×C32AGL(1,ℤ/8)11yescharacteristic
226C22×C32AGL(1,ℤ/8)11yescharacteristic
227C22×C32AGL(1,ℤ/8)11yescharacteristic
228C22×C32#13711yescharacteristic
229C2×C3⋊D3—42no—
230C2×C3⋊D3—42no—
231C2×C3⋊D3—42no—
232C2×C3⋊D3—81no—
233C2×C3⋊D3—81no—
234C2×C3⋊D3—81no—
235C2×C3⋊D3—81no—
236C32×C4—41no—
237C32×C4—41no—
238C32×C4#11711yescharacteristic
239C32×C4C4×D411yescharacteristic
240C32×C4#12311yescharacteristic
241C32×C4#12611yescharacteristic
242C3⋊Dic3—81no—
243C3×Dic3—42no—
244C3×Dic3—81no—
245C22×D4—91no—
246C22×C8—31no—
247C22×C8—31no—
248C2×C4⋊C4—31no—
249C2×C4⋊C4—31no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4—91no—
252C2×C22⋊C4—181no—
253C2×C22⋊C4—181no—
254C2×C42D3211yescharacteristic
255#102—94no—
256#102—182no—
257#97—62no—
258#95—61no—
259C23×C3—21no—
260C23×C3C2×D1211yescharacteristic
261C23×C3C2×#6711yescharacteristic
262C22×D3—64no—
263C22×D3—62no—
264C22×D3—62no—
265C22×D3—122no—
266C22×D3—122no—
267C22×D3—122no—
268C22×D3—122no—
269C22×D3—122no—
270C22×D3—122no—
271C22×D3—121no—
272C22×D3—121no—
273C22×D3—121no—
274C22×D3—121no—
275C22×D3—121no—
276C22×D3—121no—
277C22×D3—121no—
278C22×D3—121no—
279C22×D3—121no—
280C22×D3—121no—
281C22×D3—121no—
282C22×D3—121no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—62no—
308C2×C3×C4—121no—
309C2×C3×C4—121no—
310C2×C3×C4—121no—
311C2×C3×C4—121no—
312C2×C3×C4—121no—
313C2×C3×C4C2×C4×D311yescharacteristic
314C2×C3×C4C2×C4×D311yescharacteristic
315C2×C3×C4#23511yescharacteristic
316C2×C3×C4#23511yescharacteristic
317C2×C3×C4#23511yescharacteristic
318C2×C3×C4#23511yescharacteristic
319C2×C3×C4#23511yescharacteristic
320C2×C3×C4D3×D411yescharacteristic
321C2×C3×C4#23711yescharacteristic
322C2×C3×C4#23911yescharacteristic
323C2×C3×C4#23911yescharacteristic
324C2×C3×C4C2×#6711yescharacteristic
325C2×Dic3—22no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—122no—
332C2×Dic3—122no—
333C2×Dic3—122no—
334C2×Dic3—121no—
335C2×Dic3—121no—
336C2×Dic3—121no—
337C2×Dic3—121no—
338C2×Dic3—121no—
339C2×Dic3—121no—
340D12—62no—
341D12—62no—
342D12—62no—
343D12—62no—
344D12—122no—
345D12—122no—
346D12—122no—
347D12—122no—
348D12—121no—
349D12—121no—
350D12—121no—
351D12—121no—
352C3×C8—62no—
353C3×C8—62no—
354C3⋊C8—22no—
355C3⋊C8—22no—
356C2×C32#38811yescharacteristic
357C2×C32#44011yescharacteristic
358C2×C32#44211yescharacteristic
359C2×C32#46811yescharacteristic
360C2×C32#46911yescharacteristic
361C2×C32#48911yescharacteristic
362C2×C32#49011yescharacteristic
363C3⋊D3—82no—
364C24—181no—
365C2×D4—92no—
366C2×D4—181no—
367C2×D4—181no—
368C2×D4—181no—
369C2×D4—181no—
370C2×D4—181no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—181no—
374C22×C4#62611yescharacteristic
375C22×C4#62611yescharacteristic
376C22×C4C2×D3211yescharacteristic
377C2×C8—32no—
378C2×C8—32no—
379C2×C8—61no—
380C2×C8—61no—
381C2×C8—61no—
382C2×C8—61no—
383C4⋊C4—32no—
384C4⋊C4—61no—
385C4⋊C4—61no—
386C4⋊C4—61no—
387C4⋊C4—122no—
388C4⋊C4—181no—
389C4⋊C4—181no—
390C22⋊C4—362no—
391C22⋊C4—362no—
392C42—21no—
393C42—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3C4×D1211yescharacteristic
402C22×C3#87511yescharacteristic
403C22×C3#90811yescharacteristic
404C22×C3#90811yescharacteristic
405C22×C3#90811yescharacteristic
406C22×C3#90911yescharacteristic
407C22×C3C4×#6711yescharacteristic
408C22×C3#93011yescharacteristic
409C22×C3#93211yescharacteristic
410C22×C3#93911yescharacteristic
411C22×C3#94211yescharacteristic
412C22×C3#94711yescharacteristic
413C22×C3#94911yescharacteristic
414C22×C3#94911yescharacteristic
415C2×D3—124no—
416C2×D3—124no—
417C2×D3—124no—
418C2×D3—122no—
419C2×D3—122no—
420C2×D3—122no—
421C2×D3—122no—
422C2×D3—122no—
423C2×D3—122no—
424C2×D3—242no—
425C2×D3—242no—
426C2×D3—242no—
427C2×D3—242no—
428C2×D3—241no—
429C2×D3—241no—
430C2×D3—241no—
431C2×D3—241no—
432C2×D3—241no—
433C2×D3—241no—
434C2×D3—241no—
435C2×D3—241no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—42no—
441C3×C4—42no—
442C3×C4—41no—
443C3×C4—41no—
444C3×C4—41no—
445C3×C4—41no—
446C3×C4—122no—
447C3×C4—241no—
448C3×C4#87211yescharacteristic
449C3×C4#87611yescharacteristic
450C3×C4#89211yescharacteristic
451C3×C4#89311yescharacteristic
452C3×C4#89411yescharacteristic
453C3×C4#89811yescharacteristic
454C3×C4C4×#6711yescharacteristic
455C3×C4#92911yescharacteristic
456Dic3—42no—
457Dic3—81no—
458Dic3—242no—
459Dic3—241no—
460Dic3—241no—
461C32#193611yescharacteristic, complemented, Sylow
462C23—182no—
463C23—361no—
464C23—361no—
465C23—361no—
466C23—361no—
467C23—361no—
468C23—361no—
469C23C2×#62611yescharacteristic
470D4—182no—
471D4—182no—
472D4—361no—
473D4—361no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—62no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—121no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—361no—
487C2×C4—361no—
488C2×C4—361no—
489C2×C4#379811yescharacteristic
490C2×C4#379911yescharacteristic
491C2×C4#379911yescharacteristic
492C2×C4C4×D3211yescharacteristic
493C2×C4#380411yescharacteristic
494C2×C4D3×#6711yescharacteristic
495C8—62no—
496C8—62no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3#471011yescharacteristic
505C2×C3#471111yescharacteristic
506C2×C3#471311yescharacteristic
507C2×C3#471411yescharacteristic
508C2×C3#494011yescharacteristic
509C2×C3#499011yescharacteristic
510C2×C3#501411yescharacteristic
511C2×C3#502611yescharacteristic
512C2×C3#503911yescharacteristic
513C2×C3#504911yescharacteristic
514C2×C3#506711yescharacteristic
515C2×C3#508911yescharacteristic
516C2×C3#513411yescharacteristic
517C2×C3#513511yescharacteristic
518D3—244no—
519D3—242no—
520D3—242no—
521C22—362no—
522C22—362no—
523C22—362no—
524C22—721no—
525C22—721no—
526C22—721no—
527C22—721no—
528C22#6382511yescharacteristic
529C22#6382511yescharacteristic
530C22#6383011yescharacteristic
531C22#6383311yescharacteristic
532C22#6386411yescharacteristic
533C22C4×#62611yescharacteristic
534C22#6397811yescharacteristic
535C4—41no—
536C4—41no—
537C4—122no—
538C4—241no—
539C4—721no—
540C4#6383611yescharacteristic
541C4#6385011yescharacteristic
542C4#6386811yescharacteristic
543C4#6390411yescharacteristic
544C3—21no—
545C3#6961711yescharacteristic, complemented
546C3#6981811yescharacteristic, complemented
547C2—722no—
548C2#1059028211yescharacteristic
549C2#1059029411yescharacteristic
550C2#1059034611yescharacteristic
551C2#1059036211yescharacteristic
552C2#1059116511yescharacteristic
553C2#1059117811yescharacteristic
554C2#1059128911yescharacteristic
subgroupG/H
centerC23C2×#626
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×C42C22
socleC23×C32C2×D4
prime radicalC22×C4⋊(C3⋊D3)C22
cosocleC22×C32×C4C23
hypercenterC2×C42D32
hypocenterC32#1936
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C3×#4928C2semidirect
3. C2×C3×#803C2semidirect
6. #10589164C2semidirect
7. #10589164C2semidirect
23. #63564C4semidirect
33. #4928D3semidirect
34. C2×#803D3semidirect
461. C32#1936semidirect
545. C3#69617semidirect
546. C3#69818semidirect

Maximal subgroups

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

subgroupindex
#105926382
C3×#49282
C2×C3×#8032
C2×#635642
#105891642
#698183
#696173