Finite Group Databasebeta

← #11870814

Subgroups — #11870814

Order 1152.

822 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5728C211yesmaximal, characteristic
2C4×D3×Dic6C211yesmaximal, characteristic
3#10592789C211yesmaximal, characteristic, complemented
4C3×#5090C211yesmaximal, characteristic, complemented
5#10591653C212yesmaximal
6Dic3×#216C212yesmaximal, complemented
7#10591551C212yesmaximal
8#10591537C212yesmaximal, complemented
9#10590310C211yesmaximal, characteristic, complemented
10#10590284C211yesmaximal, characteristic, complemented
11#10590248C211yesmaximal, characteristic
12#83368—31nomaximal
13#80456—31nomaximal
14C2×D3×Dic6C2211yescharacteristic
15C32×C4⋊Q8C2211yescharacteristic
16C2×#3756C2212yes—
17C4×#634C2211yescharacteristic
18C3×#946C2212yes—
19C3×Q8×Dic3C2212yes—
20C2×C3×#216C2212yes—
21C3×#897—21no—
22C3×#892—21no—
23C3×#890—21no—
24C3×#887—21no—
25C3×C42×D3C2211yescharacteristic
26C3×#869C2211yescharacteristic
27C3×C4×Dic6C2211yescharacteristic
28C4×#627—21no—
29D3×C4⋊Dic3C2211yescharacteristic
30D3×Dic3⋊C4—21no—
31C4×D3×Dic3—21no—
32#63844C2211yescharacteristic
33Dic3×Dic6C2211yescharacteristic
34#63837—21no—
35#63662C2212yes—
36C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
37C3×#835C2212yes—
38C3×C4⋊(C3⋊C8)C2211yescharacteristic
39#63576C2212yes—
40#63575C2212yes—
41#63567C2212yes—
42#63565C2212yes—
43(C2×D3)⋊(C3⋊C8)C2212yes—
44Dic3×C3⋊C8C2212yes—
45C3×#533—31no—
46#5728D311yescharacteristic
47C4×D3×Q8—31no—
48C2×C4×Dic6—31no—
49C3×#462—31no—
50#5186—32no—
51Dic3×Q16—32no—
52#5091—31no—
53#5091—31no—
54#5090—31no—
55#5090—31no—
56#5090—61no—
57#5090D311yescharacteristic, complemented
58#5050—32no—
59#5044—32no—
60C4×#216—32no—
61C4×#216—32no—
62#5021—31no—
63#4925—31no—
64#4850—31no—
65#4833—31no—
66#4811—32no—
67#4790—32no—
68C2×C32×Q8C2312yes—
69C2×#634C2311yescharacteristic
70C2×C3×C4×D3D411yescharacteristic
71C2×C3×C4×D3D411yescharacteristic
72C2×C3×C4×D3C2311yescharacteristic
73C2×C3×Dic6—21no—
74C2×C3×Dic6C2311yescharacteristic
75C2×#627—21no—
76C2×D3×Dic3—21no—
77C2×D3×Dic3—21no—
78D3×Dic6—22no—
79D3×Dic6—22no—
80C32×C4⋊C4—21no—
81C32×C4⋊C4—21no—
82C32×C42C2311yescharacteristic
83#3756—24no—
84C4⋊(C3⋊Dic3)C2311yescharacteristic
85#3750—21no—
86C4×C3⋊Dic3—21no—
87C2×C3⋊(C3⋊C8)C2312yes—
88C3×#216—24no—
89C3×(C2×D3)⋊C4—22no—
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3—22no—
92C3×C4⋊Dic3C2312yes—
93C3×C4⋊Dic3C2311yescharacteristic
94C3×Dic3⋊C4—21no—
95C3×Dic3⋊C4—22no—
96C3×C4×Dic3—21no—
97C3×C4×Dic3—22no—
98C3×C4×Dic3D411yescharacteristic
99C3×C4×Dic3D411yescharacteristic
100C3×C4×Dic3C2311yescharacteristic
101C2×C3×C3⋊C8C2312yes—
102#3724—21no—
103Dic3⋊Dic3—21no—
104Dic3⋊Dic3—21no—
105Dic32—21no—
106Dic32—21no—
107#3104—91noSylow
108C2×D3×Q8—31no—
109C22×Dic6—31no—
110C2×C3×Q16—32no—
111C3×C4⋊Q8—21no—
112C3×C4⋊Q8—31no—
113C3×C4⋊Q8C2×D311yescharacteristic
114C3×C4⋊Q8C2×D311yescharacteristic
115C3×#125—61no—
116C3×C22⋊Q8—32no—
117C3×C22⋊Q8—61no—
118C3×C4×Q8—32no—
119C3×C4×Q8—31no—
120C3×C4×Q8—61no—
121C3×#117—61no—
122C2×C3×C42—31no—
123#946C2×D312yes—
124Q8×Dic3—31no—
125Q8×Dic3C2×D312yes—
126C2×#216—32no—
127C2×#216—32no—
128C2×#216—62no—
129C2×#216C2×D312yes—
130C2×C4⋊Dic3—31no—
131C2×Dic3⋊C4—61no—
132C2×C4×Dic3—61no—
133#897—21no—
134#892—21no—
135D3×C4⋊C4—31no—
136D3×C4⋊C4—61no—
137#890—21no—
138#887—21no—
139#887—31no—
140#887—61no—
141C42×D3—61no—
142C42×D3C2×D311yescharacteristic
143#869C2×D311yescharacteristic
144C4×Dic6—31no—
145C4×Dic6—31no—
146C4×Dic6—31no—
147C4×Dic6—31no—
148C4×Dic6—61no—
149C4×Dic6—61no—
150C4×Dic6—61no—
151C4×Dic6—61no—
152C4×Dic6C2×D311yescharacteristic
153C3×C4⋊C8—31no—
154C3×#103—32no—
155#835—32no—
156#835—32no—
157#835—32no—
158#835—62no—
159#835C2×D312yes—
160#830—32no—
161(C2×D3)⋊C8—32no—
162#818—32no—
163#816—32no—
164C8×Dic3—32no—
165#810—32no—
166#810—32no—
167#807—32no—
168#807—32no—
169#807—32no—
170C4⋊(C3⋊C8)—31no—
171C4⋊(C3⋊C8)—31no—
172C4⋊(C3⋊C8)—61no—
173C4⋊(C3⋊C8)C2×D311yescharacteristic
174C4×C3⋊C8—32no—
175C22×C3×D3C2×D411yescharacteristic
176C32×Q8—22no—
177C32×Q8—22no—
178C2×C32×C4—22no—
179C2×C32×C4C2×D411yescharacteristic
180C2×C32×C4C2×D411yescharacteristic
181C2×C32×C4C2411yescharacteristic
182C2×C3⋊Dic3—21no—
183C2×C3⋊Dic3—21no—
184#634—21no—
185#634#4112yes—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—22no—
189C2×C3×Dic3C2×D411yescharacteristic
190C2×C3×Dic3C2×D411yescharacteristic
191C2×C3×Dic3C2×D411yescharacteristic
192C3×C4×D3—21no—
193C3×C4×D3—21no—
194C3×C4×D3—41no—
195C3×C4×D3Q1614yescomplemented
196C3×Dic6—21no—
197C3×Dic6—42no—
198C3×Dic6#4112yes—
199#627—24no—
200D3×Dic3—24no—
201D3×Dic3—42no—
202C3⋊(C3⋊C8)—22no—
203C3×C3⋊C8—22no—
204#533—31no—
205C2×C4×Q8—91no—
206#484—92no—
207#477—91no—
208#477—91no—
209#462—31no—
210#462—91no—
211#451—92no—
212C4×Q16—92no—
213C4×Q16—92no—
214#431—91no—
215C2×C3×Q8—22no—
216C2×C3×Q8—31no—
217C2×C3×Q8—61no—
218C2×C3×Q8C22×D312yes—
219C2×C3×Q8C22×D312yes—
220C22×C3×C4—31no—
221C22×C3×C4—31no—
222C22×C3×C4—31no—
223C22×Dic3—61no—
224C22×Dic3—61no—
225D3×Q8—62no—
226D3×Q8—62no—
227C2×C4×D3—61no—
228C2×C4×D3—61no—
229C2×C4×D3#6711yescharacteristic
230C2×C4×D3#6711yescharacteristic
231C2×C4×D3C22×D311yescharacteristic
232C2×Dic6—21no—
233C2×Dic6—31no—
234C2×Dic6—31no—
235C2×Dic6—61no—
236C2×Dic6—62no—
237C2×Dic6—61no—
238C2×Dic6—61no—
239C2×Dic6—62no—
240C2×Dic6C22×D311yescharacteristic
241C3×Q16—64no—
242C2×C3×C8—32no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—32no—
248C3×C4⋊C4—31no—
249C3×C4⋊C4—41no—
250C3×C4⋊C4—41no—
251C3×C4⋊C4—62no—
252C3×C4⋊C4—62no—
253C3×C4⋊C4—61no—
254C3×C4⋊C4—61no—
255C3×C22⋊C4—62no—
256C3×C42—21no—
257C3×C42—31no—
258C3×C42—31no—
259C3×C42—31no—
260C3×C42—62no—
261C3×C42—61no—
262C3×C42C22×D311yescharacteristic
263C3×C42C22×D311yescharacteristic
264#216—24no—
265#216—64no—
266#216—64no—
267#216—124no—
268(C2×D3)⋊C4—22no—
269C4⋊Dic3—21no—
270C4⋊Dic3—22no—
271C4⋊Dic3—31no—
272C4⋊Dic3—31no—
273C4⋊Dic3—61no—
274C4⋊Dic3—61no—
275C4⋊Dic3—61no—
276C4⋊Dic3C22×D311yescharacteristic
277C4⋊Dic3C22×D312yes—
278Dic3⋊C4—21no—
279Dic3⋊C4—22no—
280Dic3⋊C4—61no—
281Dic3⋊C4—61no—
282Dic3⋊C4—61no—
283Dic3⋊C4—61no—
284Dic3⋊C4—61no—
285Dic3⋊C4—61no—
286Dic3⋊C4—121no—
287C4×Dic3—21no—
288C4×Dic3—22no—
289C4×Dic3—61no—
290C4×Dic3—61no—
291C4×Dic3—61no—
292C4×Dic3—61no—
293C4×Dic3—61no—
294C4×Dic3—61no—
295C4×Dic3—121no—
296C4×Dic3#6711yescharacteristic
297C4×Dic3#6711yescharacteristic
298C4×Dic3C22×D311yescharacteristic
299C2×C3⋊C8—32no—
300C2×C3⋊C8—32no—
301C2×C3⋊C8—62no—
302C2×C3⋊C8C22×D312yes—
303C22×C32C22×D411yescharacteristic
304C2×C3×D3—21no—
305C2×C3×D3—21no—
306C2×C3×D3C2×Q1612yes—
307C32×C4—21no—
308C32×C4—42no—
309C32×C4C2×Q1612yes—
310C32×C4C2×#4111yescharacteristic
311C32×C4C2×#4111yescharacteristic
312C3⋊Dic3—22no—
313C3⋊Dic3—41no—
314C3×Dic3—22no—
315C3×Dic3—21no—
316C3×Dic3—21no—
317C3×Dic3—41no—
318C3×Dic3—42no—
319C3×Dic3C2×Q1612yes—
320C22×Q8—91no—
321C2×Q16—32no—
322C2×Q16—92no—
323C4⋊Q8—31no—
324C4⋊Q8D3211yescharacteristic
325#125—61no—
326C22⋊Q8—32no—
327C22⋊Q8—61no—
328C4×Q8—32no—
329C4×Q8—31no—
330C4×Q8—61no—
331C4×Q8—91no—
332C4×Q8—91no—
333C4×Q8—91no—
334C4×Q8—181no—
335C4×Q8—181no—
336#117—61no—
337C2×C4⋊C4—91no—
338C2×C4⋊C4—181no—
339C2×C42—31no—
340C2×C42—181no—
341#107—92no—
342#107—92no—
343C4⋊C8—31no—
344C4⋊C8—91no—
345#103—32no—
346#103—92no—
347#103—92no—
348#103—92no—
349C22⋊C8—92no—
350C4×C8—92no—
351C23×C3—31no—
352C22×D3C2×#6711yescharacteristic
353C3×Q8—22no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—22no—
357C3×Q8—32no—
358C3×Q8—42no—
359C3×Q8—42no—
360C3×Q8—61no—
361C3×Q8—122no—
362C2×C3×C4—22no—
363C2×C3×C4—22no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—31no—
368C2×C3×C4—34no—
369C2×C3×C4—31no—
370C2×C3×C4—31no—
371C2×C3×C4—42no—
372C2×C3×C4—61no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—62no—
376C2×C3×C4—61no—
377C2×C3×C4—121no—
378C2×C3×C4D3×D411yescharacteristic
379C2×C3×C4D3×D411yescharacteristic
380C2×C3×C4C2×#6711yescharacteristic
381C2×C3×C4C2×#6711yescharacteristic
382C2×C3×C4C23×D311yescharacteristic
383C2×C3×C4C23×D311yescharacteristic
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—22no—
387C2×Dic3—64no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390C2×Dic3—61no—
391C2×Dic3—61no—
392C2×Dic3—121no—
393C2×Dic3—121no—
394C2×Dic3—122no—
395C2×Dic3C2×#6711yescharacteristic
396C2×Dic3C2×#6711yescharacteristic
397C2×Dic3C2×#6711yescharacteristic
398C4×D3—21no—
399C4×D3—21no—
400C4×D3—41no—
401C4×D3—64no—
402C4×D3—122no—
403C4×D3#21614yescomplemented
404Dic6—21no—
405Dic6—32no—
406Dic6—32no—
407Dic6—42no—
408Dic6—62no—
409Dic6—64no—
410Dic6—64no—
411Dic6—61no—
412Dic6—61no—
413Dic6—121no—
414Dic6#23712yes—
415C3×C8—62no—
416C3⋊C8—22no—
417C3⋊C8—62no—
418C3⋊C8—62no—
419C3⋊C8—122no—
420C2×C32#52511yescharacteristic
421C2×C32C22×Q1611yescharacteristic
422C2×C32#57511yescharacteristic
423C3×D3—22no—
424C2×Q8—31no—
425C2×Q8—61no—
426C2×Q8—91no—
427C2×Q8—181no—
428C2×Q8—182no—
429C2×Q8—182no—
430C2×Q8C2×D3212yes—
431C22×C4—31no—
432C22×C4—31no—
433C22×C4—31no—
434C22×C4—181no—
435C22×C4—181no—
436Q16—64no—
437Q16—184no—
438C2×C8—32no—
439C2×C8—92no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—31no—
443C4⋊C4—32no—
444C4⋊C4—62no—
445C4⋊C4—62no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—91no—
449C4⋊C4—181no—
450C4⋊C4—181no—
451C4⋊C4—181no—
452C4⋊C4—181no—
453C22⋊C4—62no—
454C42—31no—
455C42—31no—
456C42—31no—
457C42—62no—
458C42—61no—
459C42—181no—
460C42—181no—
461C42—181no—
462C42C2×D3211yescharacteristic
463C22×C3—21no—
464C22×C3—32no—
465C22×C3—61no—
466C22×C3—61no—
467C22×C3C2×D3×D411yescharacteristic
468C22×C3C22×#6711yescharacteristic
469C2×D3—21no—
470C2×D3—21no—
471C2×D3C2×#21612yes—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—22no—
477C3×C4—32no—
478C3×C4—41no—
479C3×C4—42no—
480C3×C4—42no—
481C3×C4—61no—
482C3×C4—62no—
483C3×C4—61no—
484C3×C4—82no—
485C3×C4—122no—
486C3×C4—121no—
487C3×C4D3×Q1612yes—
488C3×C4C2×#21612yes—
489C3×C4C2×#23711yescharacteristic
490C3×C4D3×#4111yescharacteristic
491C3×C4D3×#4111yescharacteristic
492C3×C4D3×#4111yescharacteristic
493Dic3—21no—
494Dic3—21no—
495Dic3—22no—
496Dic3—41no—
497Dic3—42no—
498Dic3—62no—
499Dic3—62no—
500Dic3—122no—
501Dic3—121no—
502Dic3—121no—
503Dic3—241no—
504Dic3C2×#21612yes—
505C32#310411yescharacteristic, complemented, Sylow
506C23—31no—
507Q8—22no—
508Q8—22no—
509Q8—32no—
510Q8—61no—
511Q8—92no—
512Q8—122no—
513Q8—184no—
514Q8—181no—
515C2×C4—22no—
516C2×C4—31no—
517C2×C4—34no—
518C2×C4—31no—
519C2×C4—31no—
520C2×C4—61no—
521C2×C4—61no—
522C2×C4—61no—
523C2×C4—61no—
524C2×C4—62no—
525C2×C4—121no—
526C2×C4—184no—
527C2×C4—181no—
528C2×C4—181no—
529C2×C4—362no—
530C2×C4D3×#6711yescharacteristic
531C2×C4D3×#6711yescharacteristic
532C2×C4C22×D3211yescharacteristic
533C8—62no—
534C8—182no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—62no—
539C2×C3#560611yescharacteristic
540C2×C3C2×D3×Q1611yescharacteristic
541C2×C3#577711yescharacteristic
542C2×C3C22×#21611yescharacteristic
543C2×C3#582211yescharacteristic
544C2×C3#582811yescharacteristic
545D3—22no—
546C22—32no—
547C22—61no—
548C22—61no—
549C22C2×D3×#6711yescharacteristic
550C4—21no—
551C4—32no—
552C4—42no—
553C4—61no—
554C4—62no—
555C4—61no—
556C4—121no—
557C4—122no—
558C4—182no—
559C4—361no—
560C4D3×#21612yes—
561C4D3×#23711yescharacteristic
562C4#6431211yescharacteristic
563C3—21no—
564C3#8045611yescharacteristic, complemented
565C3#8336811yescharacteristic, complemented
566C2—62no—
567C2#1059530111yescharacteristic
568C2#1059531711yescharacteristic
569C2C2×D3×#21611yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
prime radicalC22×C3×D3C2×D4
hypercenterC4⋊Q8D32
hypocenterC32#3104
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #10592789C2semidirect
4. C3×#5090C2semidirect
6. Dic3×#216C2semidirect
8. #10591537C2semidirect
9. #10590310C2semidirect
10. #10590284C2semidirect
57. #5090D3semidirect
195. C3×C4×D3Q16semidirect
403. C4×D3#216semidirect
505. C32#3104semidirect
564. C3#80456semidirect
565. C3#83368semidirect

Maximal subgroups

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

subgroupindex
C3×#57282
C4×D3×Dic62
#105927892
C3×#50902
#105916532
Dic3×#2162
#105915512
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