Finite Group Databasebeta

← #11878924

Subgroups — #11878924

Order 1152.

822 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#869C211yesmaximal, characteristic
2#10593873C211yesmaximal, characteristic
3#10592548C211yesmaximal, characteristic, complemented
4C3×#4849C211yesmaximal, characteristic, complemented
5C3×#4833C211yesmaximal, characteristic
6#10590952C212yesmaximal
7#10590949C212yesmaximal
8#10590896C212yesmaximal, complemented
9#10590890C212yesmaximal, complemented
10#10590388C211yesmaximal, characteristic, complemented
11#10590385C211yesmaximal, characteristic, complemented
12#80480—31nomaximal
13#79313—31nomaximal
14C2×D3×Dic6C2212yes—
15C2×#3746C2212yes—
16#64078C2211yescharacteristic
17C4×#634C2211yescharacteristic
18C2×C3×Dic12C2212yes—
19C3×C42×D3C2211yescharacteristic
20C3×#869C2211yescharacteristic
21C3×C4×Dic6C2211yescharacteristic
22#63918—21no—
23#63918—21no—
24#63918C2211yescharacteristic
25C4×#627—21no—
26D3×C4⋊Dic3—21no—
27D3×C4⋊Dic3—21no—
28#63867—21no—
29#63866C2211yescharacteristic
30#63866C2212yes—
31#63857C2211yescharacteristic
32#63856C2211yescharacteristic
33#63855—21no—
34#63851—21no—
35C2×#3719C2212yes—
36C2×#3719C2212yes—
37C32×C4⋊C8C2211yescharacteristic
38#63643C2212yes—
39C3×(C2×D3)⋊C8C2212yes—
40C3×#816C2212yes—
41C3×C8×Dic3C2212yes—
42C3×C4⋊(C3⋊C8)C2211yescharacteristic
43#63567C2212yes—
44#63566C2212yes—
45D3×C4⋊Q8—31no—
46#5578—31no—
47#5470—31no—
48C2×#869—31no—
49C3×#462—31no—
50C3×#431—31no—
51#5186—32no—
52#5183—32no—
53#5090—31no—
54#5031—31no—
55#4849—31no—
56#4849—31no—
57#4849—31no—
58#4849—31no—
59#4849—61no—
60#4849D311yescharacteristic, complemented
61#4833D311yescharacteristic
62#4809—32no—
63#4792—32no—
64#4739—32no—
65#4739—32no—
66#4699—32no—
67#4681—32no—
68C2×#634C2312yes—
69C2×#634C2311yescharacteristic
70C2×C3×C4×D3D411yescharacteristic
71C2×C3×C4×D3D411yescharacteristic
72C2×C3×C4×D3C2311yescharacteristic
73C2×C3×Dic6C2311yescharacteristic
74C2×C3×Dic6C2312yes—
75C2×#627—22no—
76C2×#627—21no—
77C2×D3×Dic3—22no—
78D3×Dic6—42no—
79D3×Dic6—42no—
80C2×C32×C8C2312yes—
81C32×C42C2311yescharacteristic
82C4⋊(C3⋊Dic3)—21no—
83C4⋊(C3⋊Dic3)—21no—
84C4⋊(C3⋊Dic3)C2311yescharacteristic
85#3750—21no—
86C4×C3⋊Dic3—21no—
87#3746—42no—
88C3×C4⋊Dic3—21no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3C2311yescharacteristic
91C3×Dic3⋊C4—21no—
92C3×C4×Dic3—21no—
93C3×C4×Dic3D411yescharacteristic
94C3×C4×Dic3D411yescharacteristic
95C3×C4×Dic3C2311yescharacteristic
96C2×C3×C3⋊C8C2312yes—
97C3×Dic12—42no—
98#3724—21no—
99Dic3⋊Dic3—21no—
100Dic3⋊Dic3—21no—
101Dic3⋊Dic3—21no—
102Dic3⋊Dic3—21no—
103Dic3⋊Dic3—22no—
104Dic3⋊Dic3—22no—
105(C2×D3)⋊Dic3—21no—
106(C2×D3)⋊Dic3—21no—
107#3719—42no—
108#3719—42no—
109#3209—91noSylow
110C2×D3×Q8—32no—
111C22×Dic6—32no—
112C2×C3×Q16—32no—
113C3×C4⋊Q8—31no—
114C3×C4×Q8—31no—
115C2×C3×C42—31no—
116#946—31no—
117#944—31no—
118#944—32no—
119C2×#216—32no—
120C2×#216—32no—
121#926—61no—
122C2×C4⋊Dic3—61no—
123C2×C4⋊Dic3—61no—
124#924—31no—
125#924—31no—
126#924—61no—
127C2×Dic12—32no—
128C2×Dic12—32no—
129C2×Dic12—32no—
130C2×Dic12—32no—
131C2×Dic12—62no—
132C2×Dic12C2×D312yes—
133#897—31no—
134#896—61no—
135D3×C4⋊C4—61no—
136D3×C4⋊C4—61no—
137#889—61no—
138#888—31no—
139#888—61no—
140#888—61no—
141#872—61no—
142C42×D3C2×D311yescharacteristic
143#870—61no—
144#869—31no—
145#869—32no—
146#869—31no—
147#869—31no—
148#869—31no—
149#869—61no—
150#869—61no—
151#869—61no—
152#869C2×D311yescharacteristic
153C4×Dic6—31no—
154C4×Dic6—31no—
155C4×Dic6—61no—
156C4×Dic6—61no—
157C4×Dic6—61no—
158C4×Dic6C2×D311yescharacteristic
159C3×C4⋊C8—21no—
160C3×C4⋊C8—31no—
161C3×C4⋊C8C2×D311yescharacteristic
162C3×C4⋊C8C2×D311yescharacteristic
163C3×#103—32no—
164C3×C22⋊C8—32no—
165C3×C4×C8—32no—
166#835—32no—
167(C2×D3)⋊C8C2×D312yes—
168#816—32no—
169#816—32no—
170#816—32no—
171#816—32no—
172#816—62no—
173#816C2×D312yes—
174C8×Dic3C2×D312yes—
175#810—32no—
176C4⋊(C3⋊C8)C2×D311yescharacteristic
177C22×C3×D3C2×D411yescharacteristic
178C2×C32×C4C2×D411yescharacteristic
179C2×C32×C4C2×D411yescharacteristic
180C2×C32×C4C2411yescharacteristic
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3—21no—
183C2×C3⋊Dic3—22no—
184#634—21no—
185#634—21no—
186#634—22no—
187#634—22no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—22no—
191C2×C3×Dic3C2×D411yescharacteristic
192C2×C3×Dic3C2×D411yescharacteristic
193C2×C3×Dic3C2×D411yescharacteristic
194C3×C4×D3—21no—
195C3×C4×D3—21no—
196C3×C4×D3—22no—
197C3×C4×D3Q1614yescomplemented
198C3×Dic6—21no—
199C3×Dic6—22no—
200C3×Dic6—21no—
201C3×Dic6—22no—
202#627—44no—
203#627—42no—
204D3×Dic3—44no—
205C32×C8—22no—
206C3×C3⋊C8—22no—
207#533—91no—
208C2×C4⋊Q8—91no—
209#494—92no—
210#487—92no—
211#462—31no—
212#462—91no—
213#462—91no—
214#462—91no—
215#451—92no—
216#451—92no—
217#431—31no—
218C2×C3×Q8—32no—
219C2×C3×Q8—31no—
220C22×C3×C4—31no—
221C22×C3×C4—31no—
222C22×C3×C4—31no—
223C22×Dic3—62no—
224D3×Q8—122no—
225D3×Q8—122no—
226C2×C4×D3—62no—
227C2×C4×D3D1211yescharacteristic
228C2×C4×D3D1211yescharacteristic
229C2×C4×D3C22×D311yescharacteristic
230C2×Dic6—32no—
231C2×Dic6—32no—
232C2×Dic6—31no—
233C2×Dic6—31no—
234C2×Dic6—61no—
235C2×Dic6—62no—
236C2×Dic6—61no—
237C2×Dic6—62no—
238C2×Dic6—61no—
239C2×Dic6—62no—
240C2×Dic6—122no—
241C2×Dic6—122no—
242C2×Dic6C22×D312yes—
243C2×Dic6C22×D311yescharacteristic
244C3×Q16—122no—
245C2×C3×C8—22no—
246C2×C3×C8—32no—
247C2×C3×C8C22×D312yes—
248C2×C3×C8C22×D312yes—
249C3×C4⋊C4—31no—
250C3×C4⋊C4—61no—
251C3×C4⋊C4—61no—
252C3×C4⋊C4—61no—
253C3×C42—21no—
254C3×C42—31no—
255C3×C42—31no—
256C3×C42—31no—
257C3×C42—61no—
258C3×C42C22×D311yescharacteristic
259C3×C42C22×D311yescharacteristic
260#217—61no—
261#217—61no—
262#216—122no—
263#216—122no—
264(C2×D3)⋊C4—61no—
265(C2×D3)⋊C4—61no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—31no—
269C4⋊Dic3—31no—
270C4⋊Dic3—61no—
271C4⋊Dic3—61no—
272C4⋊Dic3—62no—
273C4⋊Dic3—62no—
274C4⋊Dic3—61no—
275C4⋊Dic3—61no—
276C4⋊Dic3—61no—
277C4⋊Dic3—61no—
278C4⋊Dic3—61no—
279C4⋊Dic3—61no—
280C4⋊Dic3—61no—
281C4⋊Dic3—121no—
282C4⋊Dic3—121no—
283C4⋊Dic3C22×D311yescharacteristic
284Dic3⋊C4—21no—
285Dic3⋊C4—61no—
286Dic3⋊C4—61no—
287Dic3⋊C4—61no—
288Dic3⋊C4—61no—
289Dic3⋊C4—61no—
290Dic3⋊C4—61no—
291Dic3⋊C4—61no—
292Dic3⋊C4—61no—
293Dic3⋊C4—62no—
294Dic3⋊C4—62no—
295Dic3⋊C4—121no—
296C4×Dic3—21no—
297C4×Dic3—61no—
298C4×Dic3—61no—
299C4×Dic3—121no—
300C4×Dic3D1211yescharacteristic
301C4×Dic3D1211yescharacteristic
302C4×Dic3C22×D311yescharacteristic
303C2×C3⋊C8C22×D312yes—
304Dic12—42no—
305Dic12—122no—
306Dic12—122no—
307Dic12—124no—
308Dic12—122no—
309Dic12—122no—
310C22×C32C22×D411yescharacteristic
311C2×C3×D3—21no—
312C2×C3×D3—21no—
313C2×C3×D3C2×Q1612yes—
314C32×C4—21no—
315C32×C4C2×Q1612yes—
316C32×C4H(5,𝔽2)11yescharacteristic
317C32×C4H(5,𝔽2)11yescharacteristic
318C3⋊Dic3—41no—
319C3⋊Dic3—42no—
320C3⋊Dic3—41no—
321C3×Dic3—21no—
322C3×Dic3—42no—
323C3×Dic3—41no—
324C3×Dic3—41no—
325C3×Dic3C2×Q1612yes—
326C3×Dic3#13712yes—
327C22×Q8—92no—
328C2×Q16—32no—
329C2×Q16—92no—
330C2×Q16—92no—
331C2×Q16—92no—
332C4⋊Q8—31no—
333C4⋊Q8—91no—
334C4⋊Q8—92no—
335C4⋊Q8—91no—
336C4⋊Q8—91no—
337C4⋊Q8—181no—
338C4⋊Q8—181no—
339#125—181no—
340C22⋊Q8—91no—
341C22⋊Q8—91no—
342C22⋊Q8—181no—
343C4×Q8—31no—
344C4×Q8—91no—
345C4×Q8—181no—
346#117—181no—
347C2×C4⋊C4—181no—
348C2×C4⋊C4—181no—
349C2×C42—31no—
350C4⋊C8—31no—
351C4⋊C8D3211yescharacteristic
352#103—32no—
353#103—92no—
354#103—92no—
355#103—92no—
356C22⋊C8—32no—
357C4×C8—32no—
358C23×C3—31no—
359C22×D3C2×D1211yescharacteristic
360C3×Q8—61no—
361C3×Q8—62no—
362C3×Q8—61no—
363C3×Q8—62no—
364C2×C3×C4—21no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—31no—
368C2×C3×C4—31no—
369C2×C3×C4—34no—
370C2×C3×C4—31no—
371C2×C3×C4—62no—
372C2×C3×C4—61no—
373C2×C3×C4—61no—
374C2×C3×C4—61no—
375C2×C3×C4—62no—
376C2×C3×C4—61no—
377C2×C3×C4C2×D1211yescharacteristic
378C2×C3×C4C2×D1211yescharacteristic
379C2×C3×C4D3×D411yescharacteristic
380C2×C3×C4D3×D411yescharacteristic
381C2×C3×C4C23×D311yescharacteristic
382C2×C3×C4C23×D311yescharacteristic
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—22no—
386C2×Dic3—61no—
387C2×Dic3—61no—
388C2×Dic3—62no—
389C2×Dic3—61no—
390C2×Dic3—61no—
391C2×Dic3—62no—
392C2×Dic3—121no—
393C2×Dic3—121no—
394C2×Dic3—122no—
395C2×Dic3—124no—
396C2×Dic3C2×D1211yescharacteristic
397C2×Dic3C2×D1211yescharacteristic
398C2×Dic3C2×D1211yescharacteristic
399C4×D3—21no—
400C4×D3—21no—
401C4×D3—22no—
402C4×D3—124no—
403C4×D3Dic1214yescomplemented
404Dic6—21no—
405Dic6—22no—
406Dic6—21no—
407Dic6—22no—
408Dic6—62no—
409Dic6—61no—
410Dic6—62no—
411Dic6—61no—
412Dic6—61no—
413Dic6—62no—
414Dic6—62no—
415Dic6—61no—
416Dic6—62no—
417Dic6—121no—
418Dic6—122no—
419Dic6—122no—
420Dic6—124no—
421Dic6—122no—
422Dic6—124no—
423Dic6—122no—
424C3×C8—22no—
425C3×C8—22no—
426C3×C8—42no—
427C3×C8—62no—
428C3⋊C8—22no—
429C2×C32#53511yescharacteristic
430C2×C32C22×Q1611yescharacteristic
431C2×C32C2×#13711yescharacteristic
432C3×D3—22no—
433C2×Q8—32no—
434C2×Q8—31no—
435C2×Q8—92no—
436C2×Q8—91no—
437C2×Q8—181no—
438C2×Q8—182no—
439C2×Q8—362no—
440C2×Q8—362no—
441C22×C4—31no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—182no—
445Q16—122no—
446Q16—362no—
447Q16—362no—
448Q16—362no—
449C2×C8—32no—
450C2×C8C2×D3212yes—
451C4⋊C4—31no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—91no—
456C4⋊C4—182no—
457C4⋊C4—182no—
458C4⋊C4—181no—
459C4⋊C4—181no—
460C4⋊C4—181no—
461C4⋊C4—181no—
462C4⋊C4—181no—
463C4⋊C4—181no—
464C4⋊C4—181no—
465C4⋊C4—181no—
466C22⋊C4—181no—
467C22⋊C4—181no—
468C42—31no—
469C42—31no—
470C42—31no—
471C42—61no—
472C42—181no—
473C42C2×D3211yescharacteristic
474C22×C3—21no—
475C22×C3—32no—
476C22×C3—61no—
477C22×C3—61no—
478C22×C3C22×D1211yescharacteristic
479C22×C3C2×D3×D411yescharacteristic
480C2×D3—21no—
481C2×D3—21no—
482C2×D3C2×Dic1212yes—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—22no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—32no—
489C3×C4—32no—
490C3×C4—41no—
491C3×C4—61no—
492C3×C4—121no—
493C3×C4—122no—
494C3×C4—121no—
495C3×C4C2×Dic1212yes—
496C3×C4D3×Q1612yes—
497C3×C4#100411yescharacteristic
498C3×C4#100911yescharacteristic
499C3×C4#100911yescharacteristic
500C3×C4#100911yescharacteristic
501Dic3—21no—
502Dic3—41no—
503Dic3—42no—
504Dic3—41no—
505Dic3—122no—
506Dic3—121no—
507Dic3—121no—
508Dic3—122no—
509Dic3—121no—
510Dic3—122no—
511Dic3—121no—
512Dic3—242no—
513Dic3—241no—
514Dic3C2×Dic1212yes—
515Dic3#90912yes—
516C32#320911yescharacteristic, complemented, Sylow
517C23—31no—
518Q8—61no—
519Q8—62no—
520Q8—61no—
521Q8—62no—
522Q8—181no—
523Q8—182no—
524Q8—181no—
525Q8—182no—
526Q8—362no—
527Q8—364no—
528C2×C4—31no—
529C2×C4—31no—
530C2×C4—34no—
531C2×C4—31no—
532C2×C4—62no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—62no—
538C2×C4—181no—
539C2×C4—181no—
540C2×C4—182no—
541C2×C4—364no—
542C2×C4D3×D1211yescharacteristic
543C2×C4D3×D1211yescharacteristic
544C2×C4C22×D3211yescharacteristic
545C8—22no—
546C8—62no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—62no—
551C2×C3#552511yescharacteristic
552C2×C3#560711yescharacteristic
553C2×C3C22×Dic1211yescharacteristic
554C2×C3C2×#90911yescharacteristic
555C2×C3C2×D3×Q1611yescharacteristic
556C2×C3D3×#13711yescharacteristic
557D3—22no—
558C22—32no—
559C22—61no—
560C22—61no—
561C22C2×D3×D1211yescharacteristic
562C4—21no—
563C4—32no—
564C4—32no—
565C4—61no—
566C4—122no—
567C4—121no—
568C4—121no—
569C4—362no—
570C4—361no—
571C4—361no—
572C4D3×Dic1212yes—
573C4#6431411yescharacteristic
574C4#6432011yescharacteristic
575C3—21no—
576C3#7931311yescharacteristic, complemented
577C3#8048011yescharacteristic, complemented
578C2—62no—
579C2#1059462811yescharacteristic
580C2C2×D3×Dic1211yescharacteristic
581C2D3×#90911yescharacteristic
subgroupG/H
centerC22C2×D3×D12
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊C8C22
socleC22×C32C22×D4
prime radicalC22×C3×D3C2×D4
hypercenterC4⋊C8D32
hypocenterC32#3209
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
3. #10592548C2semidirect
4. C3×#4849C2semidirect
8. #10590896C2semidirect
9. #10590890C2semidirect
10. #10590388C2semidirect
11. #10590385C2semidirect
60. #4849D3semidirect
197. C3×C4×D3Q16semidirect
403. C4×D3Dic12semidirect
516. C32#3209semidirect
576. C3#79313semidirect
577. C3#80480semidirect

Maximal subgroups

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

subgroupindex
D3×#8692
#105938732
#105925482
C3×#48492
C3×#48332
#105909522
#105909492
#105908962
#105908902
#105903882
#105903852
#804803
#793133