Finite Group Databasebeta

← #11794548

Subgroups — #11794548

Order 1152.

697 conjugacy classes of subgroups.

The subgroup lattice has 560 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×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
4C2×#63575C211yesmaximal, characteristic, complemented
5C2×#63564C211yesmaximal, characteristic
6C2×#63564C211yesmaximal, characteristic
7#10589164C211yesmaximal, characteristic, complemented
8#70484—31nomaximal
9#69630—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×C4⋊(C3⋊C8)—22no—
22#63575—22no—
23#63564—22no—
24#63564—22no—
25C2×C3×C4⋊C8—31no—
26C3×#389—31no—
27C2×#821—31no—
28C2×#821—31no—
29C2×#818—31no—
30C2×#809—31no—
31C2×#809—31no—
32C2×#807—31no—
33#4939—31no—
34#4939—31no—
35#4939—61no—
36#4928D311yescharacteristic, complemented
37C2×C4⋊(C3⋊C8)D311yescharacteristic, complemented
38#4551—31no—
39#4529—31no—
40C23×C3⋊D3—21no—
41C22×C32×C4D411yescharacteristic
42C22×C32×C4D411yescharacteristic
43C22×C32×C4C2311yescharacteristic
44C22×C3⋊Dic3—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)—21no—
50C2×C4⋊(C3⋊D3)Q812yes—
51C22×C3×Dic3—21no—
52C22×C3×Dic3—21no—
53C32×C42—21no—
54C32×C42—21no—
55#3752—42no—
56#3752—42no—
57C4⋊(C3⋊Dic3)—21no—
58C4⋊(C3⋊Dic3)—21no—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3—21no—
61C3×C4⋊Dic3—21no—
62C3×C4⋊Dic3D412yes—
63C3×Dic3⋊C4—42no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—21no—
66C2×C3×C3⋊C8—21no—
67C2×C3×C3⋊C8—21no—
68C2×C3×C3⋊C8—21no—
69C2×C3×C3⋊C8D412yes—
70#2067—91noSylow
71C22×D12—31no—
72C22×D12—31no—
73C22×D12—61no—
74C22×C3×C8—31no—
75C22×C3×C8—31no—
76C2×C3×C4⋊C4—31no—
77C2×C3×C4⋊C4—31no—
78C2×C3×C4⋊C4—61no—
79C2×C3×C42—21no—
80C2×C3×C42C2×D311yescharacteristic
81C2×C3×C42C2×D311yescharacteristic
82C2×(C2×D3)⋊C4—62no—
83C2×(C2×D3)⋊C4—62no—
84C2×(C2×D3)⋊C4—61no—
85C2×(C2×D3)⋊C4—61no—
86C2×(C2×D3)⋊C4—61no—
87C2×(C2×D3)⋊C4—61no—
88C2×C4⋊Dic3—31no—
89C2×C4⋊Dic3—31no—
90C2×C4⋊Dic3—61no—
91C2×C4⋊Dic3C2×D311yescharacteristic
92C2×C4⋊Dic3C2×D311yescharacteristic
93C2×Dic3⋊C4—21no—
94C22×C3⋊C8C2×D311yescharacteristic
95C22×C3⋊C8C2×D311yescharacteristic
96C3×C4⋊C8—62no—
97C3×#95—61no—
98C3⋊#95—21no—
99#821—62no—
100#821—62no—
101#818—62no—
102#809—62no—
103#809—62no—
104#807—62no—
105C4⋊(C3⋊C8)—22no—
106C23×C32C2×D411yescharacteristic
107C22×C3⋊D3—22no—
108C22×C3⋊D3—41no—
109C22×C3⋊D3—41no—
110C22×C3⋊D3—41no—
111C22×C3⋊D3—41no—
112C22×C3⋊D3—41no—
113C22×C3⋊D3—41no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4D812yes—
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4C2×Q811yescharacteristic
123C2×C32×C4#4111yescharacteristic
124C2×C32×C4#4111yescharacteristic
125C2×C32×C4#4111yescharacteristic
126C2×C3⋊Dic3—41no—
127C2×C3⋊Dic3—41no—
128C2×C3⋊Dic3—41no—
129C4⋊(C3⋊D3)—22no—
130C4⋊(C3⋊D3)—22no—
131C4⋊(C3⋊D3)—41no—
132C4⋊(C3⋊D3)—41no—
133C2×C3×Dic3—22no—
134C2×C3×Dic3—41no—
135C2×C3×Dic3—41no—
136C2×C3×Dic3—41no—
137C2×C3×Dic3—41no—
138C2×C3×Dic3—41no—
139C3×C3⋊C8—22no—
140C3×C3⋊C8—41no—
141C2×#107—91no—
142C2×C4⋊C8—31no—
143C2×#102—91no—
144C2×#102—91no—
145#390—91no—
146#389—31no—
147#340—91no—
148C23×D3—62no—
149C23×D3—61no—
150C23×D3—61no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—61no—
155C22×C3×C4—61no—
156C22×C3×C4D1211yescharacteristic
157C22×C3×C4D1211yescharacteristic
158C22×C3×C4#6711yescharacteristic
159C22×C3×C4#6711yescharacteristic
160C22×C3×C4C22×D311yescharacteristic
161C22×C3×C4C22×D311yescharacteristic
162C22×Dic3—21no—
163C22×Dic3—21no—
164C22×Dic3—62no—
165C22×Dic3—61no—
166C22×Dic3—61no—
167C2×D12—32no—
168C2×D12—32no—
169C2×D12—62no—
170C2×D12—61no—
171C2×D12—61no—
172C2×D12—61no—
173C2×D12—61no—
174C2×D12—61no—
175C2×D12—61no—
176C2×D12—61no—
177C2×D12—61no—
178C2×D12—62no—
179C2×D12—61no—
180C2×D12—61no—
181C2×D12—121no—
182C2×D12—121no—
183C2×D12—121no—
184C2×D12—121no—
185C2×C3×C8—32no—
186C2×C3×C8—61no—
187C2×C3×C8—61no—
188C2×C3×C8—61no—
189C2×C3×C8—61no—
190C2×C3×C8—61no—
191C3×C4⋊C4—32no—
192C3×C4⋊C4—61no—
193C3×C4⋊C4—61no—
194C3×C4⋊C4—61no—
195C3×C4⋊C4—122no—
196C3×C42—21no—
197C3×C42—21no—
198C3×C42—21no—
199C3×C42—21no—
200C3×C42—41no—
201C3×C42—41no—
202(C2×D3)⋊C4—124no—
203(C2×D3)⋊C4—122no—
204(C2×D3)⋊C4—122no—
205(C2×D3)⋊C4—124no—
206(C2×D3)⋊C4—122no—
207(C2×D3)⋊C4—122no—
208C4⋊Dic3—21no—
209C4⋊Dic3—21no—
210C4⋊Dic3—21no—
211C4⋊Dic3—61no—
212C4⋊Dic3—61no—
213C4⋊Dic3—61no—
214C4⋊Dic3—61no—
215C4⋊Dic3—121no—
216C4⋊Dic3—121no—
217C4⋊Dic3D1212yes—
218Dic3⋊C4—42no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8—21no—
222C2×C3⋊C8—21no—
223C2×C3⋊C8—21no—
224C2×C3⋊C8#6712yes—
225C22×C32#12111yescharacteristic
226C22×C32#12111yescharacteristic
227C22×C32C22⋊Q811yescharacteristic
228C22×C32C2×D811yescharacteristic
229C22×C32#13511yescharacteristic
230C22×C32AGL(1,ℤ/8)11yescharacteristic
231C22×C32#13711yescharacteristic
232C2×C3⋊D3—42no—
233C2×C3⋊D3—42no—
234C2×C3⋊D3—42no—
235C2×C3⋊D3—81no—
236C2×C3⋊D3—81no—
237C2×C3⋊D3—81no—
238C2×C3⋊D3—81no—
239C32×C4—22no—
240C32×C4—41no—
241C32×C4#12111yescharacteristic
242C32×C4C22⋊Q811yescharacteristic
243C32×C4C22⋊Q811yescharacteristic
244C32×C4#12611yescharacteristic
245C3⋊Dic3—81no—
246C3×Dic3—42no—
247C3×Dic3—81no—
248C22×D4—91no—
249C22×C8—31no—
250C22×C8—31no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—61no—
254C2×C4⋊C4—91no—
255C2×C22⋊C4—181no—
256C2×C22⋊C4—181no—
257C2×C42D3211yescharacteristic
258#107—182no—
259C4⋊C8—62no—
260#102—182no—
261#102—182no—
262#95—61no—
263C23×C3—21no—
264C23×C3C2×D1211yescharacteristic
265C23×C3C2×#6711yescharacteristic
266C22×D3—64no—
267C22×D3—62no—
268C22×D3—62no—
269C22×D3—122no—
270C22×D3—122no—
271C22×D3—122no—
272C22×D3—122no—
273C22×D3—122no—
274C22×D3—122no—
275C22×D3—121no—
276C22×D3—121no—
277C22×D3—121no—
278C22×D3—121no—
279C22×D3—121no—
280C22×D3—121no—
281C22×D3—121no—
282C22×D3—121no—
283C22×D3—121no—
284C22×D3—121no—
285C22×D3—121no—
286C22×D3—121no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—22no—
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—22no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—62no—
310C2×C3×C4—121no—
311C2×C3×C4—121no—
312C2×C3×C4—121no—
313C2×C3×C4—121no—
314C2×C3×C4—121no—
315C2×C3×C4D2412yes—
316C2×C3×C4#21312yes—
317C2×C3×C4C2×D1211yescharacteristic
318C2×C3×C4#23511yescharacteristic
319C2×C3×C4#23511yescharacteristic
320C2×C3×C4D3×D411yescharacteristic
321C2×C3×C4D3×D411yescharacteristic
322C2×C3×C4#23711yescharacteristic
323C2×C3×C4#23711yescharacteristic
324C2×C3×C4D3×Q811yescharacteristic
325C2×C3×C4D3×Q811yescharacteristic
326C2×C3×C4#23911yescharacteristic
327C2×C3×C4#23911yescharacteristic
328C2×C3×C4C2×#6711yescharacteristic
329C2×Dic3—22no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—122no—
336C2×Dic3—122no—
337C2×Dic3—122no—
338C2×Dic3—121no—
339C2×Dic3—121no—
340C2×Dic3—121no—
341C2×Dic3—121no—
342C2×Dic3—121no—
343C2×Dic3—121no—
344D12—62no—
345D12—62no—
346D12—62no—
347D12—62no—
348D12—122no—
349D12—122no—
350D12—122no—
351D12—122no—
352D12—121no—
353D12—121no—
354D12—121no—
355D12—121no—
356C3×C8—62no—
357C3×C8—121no—
358C3⋊C8—22no—
359C3⋊C8—41no—
360C2×C32#39611yescharacteristic
361C2×C32#45911yescharacteristic
362C2×C32#46411yescharacteristic
363C2×C32#46611yescharacteristic
364C2×C32#46811yescharacteristic
365C2×C32#47411yescharacteristic
366C2×C32#47811yescharacteristic
367C3⋊D3—82no—
368C24—181no—
369C2×D4—92no—
370C2×D4—181no—
371C2×D4—181no—
372C2×D4—181no—
373C2×D4—181no—
374C2×D4—181no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—181no—
378C22×C4#62611yescharacteristic
379C22×C4#62611yescharacteristic
380C22×C4C2×D3211yescharacteristic
381C2×C8—32no—
382C2×C8—61no—
383C2×C8—61no—
384C2×C8—61no—
385C2×C8—61no—
386C2×C8—61no—
387C4⋊C4—32no—
388C4⋊C4—61no—
389C4⋊C4—61no—
390C4⋊C4—61no—
391C4⋊C4—122no—
392C4⋊C4—181no—
393C4⋊C4—181no—
394C22⋊C4—362no—
395C22⋊C4—362no—
396C42—21no—
397C42—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3#89511yescharacteristic
406C22×C3#89711yescharacteristic
407C22×C3C2×D2411yescharacteristic
408C22×C3#90411yescharacteristic
409C22×C3#90811yescharacteristic
410C22×C3#90911yescharacteristic
411C22×C3#93011yescharacteristic
412C22×C3#93011yescharacteristic
413C22×C3C2×#21311yescharacteristic
414C22×C3#93911yescharacteristic
415C22×C3#94211yescharacteristic
416C22×C3#94611yescharacteristic
417C22×C3#94911yescharacteristic
418C22×C3#95011yescharacteristic
419C2×D3—124no—
420C2×D3—124no—
421C2×D3—124no—
422C2×D3—122no—
423C2×D3—122no—
424C2×D3—122no—
425C2×D3—122no—
426C2×D3—122no—
427C2×D3—122no—
428C2×D3—242no—
429C2×D3—242no—
430C2×D3—242no—
431C2×D3—242no—
432C2×D3—241no—
433C2×D3—241no—
434C2×D3—241no—
435C2×D3—241no—
436C2×D3—241no—
437C2×D3—241no—
438C2×D3—241no—
439C2×D3—241no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—42no—
447C3×C4—42no—
448C3×C4—41no—
449C3×C4—41no—
450C3×C4—122no—
451C3×C4—241no—
452C3×C4#89511yescharacteristic
453C3×C4#89611yescharacteristic
454C3×C4#89611yescharacteristic
455C3×C4#89711yescharacteristic
456C3×C4#89811yescharacteristic
457C3×C4#89811yescharacteristic
458C3×C4#93911yescharacteristic
459C3×C4#94611yescharacteristic
460Dic3—42no—
461Dic3—81no—
462Dic3—242no—
463Dic3—241no—
464Dic3—241no—
465C32#206711yescharacteristic, complemented, Sylow
466C23—182no—
467C23—361no—
468C23—361no—
469C23—361no—
470C23—361no—
471C23—361no—
472C23—361no—
473C23C2×#62611yescharacteristic
474D4—182no—
475D4—182no—
476D4—361no—
477D4—361no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—62no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—121no—
487C2×C4—121no—
488C2×C4—121no—
489C2×C4—361no—
490C2×C4—361no—
491C2×C4—361no—
492C2×C4#371412yes—
493C2×C4#379611yescharacteristic
494C2×C4#379711yescharacteristic
495C2×C4#379911yescharacteristic
496C2×C4D3×D1211yescharacteristic
497C2×C4#380411yescharacteristic
498C2×C4D3×#6711yescharacteristic
499C8—62no—
500C8—121no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3#484311yescharacteristic
509C2×C3#484511yescharacteristic
510C2×C3#484611yescharacteristic
511C2×C3#484711yescharacteristic
512C2×C3#499411yescharacteristic
513C2×C3#499411yescharacteristic
514C2×C3#501511yescharacteristic
515C2×C3#502811yescharacteristic
516C2×C3#503811yescharacteristic
517C2×C3#504911yescharacteristic
518C2×C3#506611yescharacteristic
519C2×C3#508711yescharacteristic
520C2×C3#511511yescharacteristic
521C2×C3#513411yescharacteristic
522D3—244no—
523D3—242no—
524D3—242no—
525C22—362no—
526C22—362no—
527C22—362no—
528C22—721no—
529C22—721no—
530C22—721no—
531C22—721no—
532C22C2×#371411yescharacteristic
533C22#6382511yescharacteristic
534C22#6382911yescharacteristic
535C22#6383311yescharacteristic
536C22#6387111yescharacteristic
537C22#6390911yescharacteristic
538C22#6397811yescharacteristic
539C4—22no—
540C4—41no—
541C4—122no—
542C4—241no—
543C4—721no—
544C4#6384811yescharacteristic
545C4#6388611yescharacteristic
546C4#6388811yescharacteristic
547C4#6389511yescharacteristic
548C3—21no—
549C3#6963011yescharacteristic, complemented
550C3#7048411yescharacteristic, complemented
551C2—722no—
552C2#1059030811yescharacteristic
553C2#1059031311yescharacteristic
554C2#1059034911yescharacteristic
555C2#1059036811yescharacteristic
556C2#1059116411yescharacteristic
557C2#1059117811yescharacteristic
558C2#1059131111yescharacteristic
subgroupG/H
centerC23C2×#626
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×C42C22
socleC23×C32C2×D4
prime radicalC22×C4⋊(C3⋊D3)C22
hypercenterC2×C42D32
hypocenterC32#2067
LCSG ▷ C22×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×C4⋊(C3⋊C8)C2semidirect
4. C2×#63575C2semidirect
7. #10589164C2semidirect
36. #4928D3semidirect
37. C2×C4⋊(C3⋊C8)D3semidirect
465. C32#2067semidirect
549. C3#69630semidirect
550. C3#70484semidirect

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×C4⋊(C3⋊C8)2
C2×#635752
C2×#635642
#105891642
#704843
#696303