Finite Group Databasebeta

← #11802685

Subgroups — #11802685

Order 1152.

1661 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×D3×Q8C211yesmaximal, characteristic
2C2×#63855C211yesmaximal, characteristic
3C2×#63841C211yesmaximal, characteristic
4#10592928C211yesmaximal, characteristic, complemented
5C3×#5229C211yesmaximal, characteristic, complemented
6#10591294C214yesmaximal
7#10591261C214yesmaximal, complemented
8#10591221C211yesmaximal, characteristic, complemented
9#10591220C211yesmaximal, characteristic, complemented
10#84062—31nomaximal
11#76610—31nomaximal
12C22×C32×Q8C2211yescharacteristic
13C2×C3×D3×Q8—24no—
14C2×C3×D3×Q8—24no—
15C2×C3×D3×Q8C418yescomplemented
16C22×C3×C4×D3C2212yes—
17C22×C3×C4×D3C2211yescharacteristic
18C22×C3×Dic6C2212yes—
19C22×C3×Dic6C2211yescharacteristic
20C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C4×C3⋊Dic3C2211yescharacteristic
22C2×C3×C4⋊Dic3C2211yescharacteristic
23C2×C3×C4×Dic3C2211yescharacteristic
24C2×Dic3⋊Dic3C2212yes—
25C2×(C2×D3)⋊Dic3C2212yes—
26C2×(C2×D3)⋊Dic3C2212yes—
27C2×Dic32C2212yes—
28#63855—24no—
29#63841—24no—
30#63658C2214yes—
31C3×C3⋊#95C2214yes—
32#63579C2214yes—
33#63579C2214yes—
34C23×C3×Q8—31no—
35C22×D3×Q8D311yescharacteristic
36C2×#926—31no—
37C2×#926—31no—
38C2×#892—31no—
39C2×#872—31no—
40C3×#391—31no—
41#5229—31no—
42#5229—31no—
43#5229—31no—
44#5229—31no—
45#5229—34no—
46#5229—61no—
47#5229D311yescharacteristic, complemented
48#5211—34no—
49#4973—31no—
50#4929—31no—
51#4664—34no—
52#4646—34no—
53C23×C3×D3C2311yescharacteristic
54C2×C32×Q8—22no—
55C2×C32×Q8—22no—
56C2×C32×Q8C2×C414yes—
57C22×C32×C4C2312yes—
58C22×C32×C4C2311yescharacteristic
59C22×C3⋊Dic3C2312yes—
60C22×C3⋊Dic3C2312yes—
61C22×C3×Dic3C2312yes—
62C22×C3×Dic3C2312yes—
63C22×C3×Dic3C2312yes—
64C22×C3×Dic3C2311yescharacteristic
65C3×D3×Q8—232no—
66C2×C3×C4×D3—24no—
67C2×C3×C4×D3—24no—
68C2×C3×C4×D3C2×C418yes—
69C2×C3×C4×D3C2×C414yes—
70C2×C3×C4×D3D414yes—
71C2×C3×C4×D3D414yes—
72C2×C3×Dic6—22no—
73C2×C3×Dic6—22no—
74C2×C3×Dic6—24no—
75C2×C3×Dic6—24no—
76C2×C3×Dic6C2×C414yes—
77C2×C3×Dic6C2×C418yes—
78C4⋊(C3⋊Dic3)—22no—
79C4×C3⋊Dic3—22no—
80C3×C4⋊Dic3—22no—
81C3×C4×Dic3—22no—
82Dic3⋊Dic3—24no—
83(C2×D3)⋊Dic3—24no—
84(C2×D3)⋊Dic3—24no—
85Dic32—24no—
86#2310—91noSylow
87C22×C3×Q8—21no—
88C22×C3×Q8—38no—
89C22×C3×Q8—32no—
90C22×C3×Q8—31no—
91C22×C3×Q8—64no—
92C22×C3×Q8—64no—
93C22×C3×Q8C2×D311yescharacteristic
94C22×C3×Q8C2×D311yescharacteristic
95C23×C3×C4—31no—
96C23×C3×C4—32no—
97C2×D3×Q8—24no—
98C2×D3×Q8—24no—
99C2×D3×Q8Dic318yescomplemented
100C22×C4×D3C2×D312yes—
101C22×C4×D3C2×D311yescharacteristic
102C22×Dic6C2×D312yes—
103C22×Dic6C2×D311yescharacteristic
104C2×C3×C4⋊C4—31no—
105C2×C3×C42—31no—
106C2×#217—32no—
107C2×#217—32no—
108C2×(C2×D3)⋊C4—32no—
109C2×(C2×D3)⋊C4—32no—
110#926—64no—
111#926—64no—
112C2×C4⋊Dic3—31no—
113C2×C4⋊Dic3—31no—
114C2×C4⋊Dic3—32no—
115C2×C4⋊Dic3—61no—
116C2×C4⋊Dic3C2×D311yescharacteristic
117C2×Dic3⋊C4—32no—
118C2×C4×Dic3—31no—
119C2×C4×Dic3—31no—
120C2×C4×Dic3—32no—
121C2×C4×Dic3—32no—
122C2×C4×Dic3—61no—
123C2×C4×Dic3C2×D311yescharacteristic
124#892—64no—
125#872—64no—
126C3×#95—34no—
127C3⋊#95—34no—
128C3⋊#95—34no—
129C3⋊#95—34no—
130C3⋊#95—34no—
131C3⋊#95—34no—
132C3⋊#95—34no—
133C3⋊#95—64no—
134C3⋊#95C2×D314yes—
135C23×C32C2411yescharacteristic
136C22×C3×D3—22no—
137C22×C3×D3—22no—
138C22×C3×D3C22×C412yes—
139C22×C3×D3C2×D412yes—
140C22×C3×D3C2×D412yes—
141C32×Q8—28no—
142C2×C32×C4—22no—
143C2×C32×C4—22no—
144C2×C32×C4C22×C414yes—
145C2×C32×C4C22×C412yes—
146C2×C32×C4C2×D412yes—
147C2×C32×C4C2×D412yes—
148C2×C3⋊Dic3—22no—
149C2×C3⋊Dic3—22no—
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3—22no—
152C2×C3⋊Dic3—24no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3—22no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3—22no—
157C2×C3×Dic3—22no—
158C2×C3×Dic3—22no—
159C2×C3×Dic3—24no—
160C2×C3×Dic3C22×C414yes—
161C2×C3×Dic3C22×C412yes—
162C2×C3×Dic3C2×D412yes—
163C2×C3×Dic3C2×D412yes—
164C3×C4×D3—216no—
165C3×C4×D3C22⋊C4116yes—
166C3×Dic6—28no—
167C3×Dic6—216no—
168C23×Q8—31no—
169C2×#117—91no—
170C2×#117—91no—
171#391—31no—
172#391—94no—
173#391—91no—
174#391—91no—
175#391—91no—
176#381—94no—
177C24×C3—31no—
178C23×D3C22×D311yescharacteristic
179C2×C3×Q8—22no—
180C2×C3×Q8—22no—
181C2×C3×Q8—22no—
182C2×C3×Q8—22no—
183C2×C3×Q8—24no—
184C2×C3×Q8—38no—
185C2×C3×Q8—34no—
186C2×C3×Q8—42no—
187C2×C3×Q8—42no—
188C2×C3×Q8—64no—
189C2×C3×Q8—64no—
190C2×C3×Q8—632no—
191C2×C3×Q8—62no—
192C2×C3×Q8—62no—
193C2×C3×Q8C4×D314yes—
194C2×C3×Q8C2×Dic314yes—
195C22×C3×C4—21no—
196C22×C3×C4—22no—
197C22×C3×C4—31no—
198C22×C3×C4—34no—
199C22×C3×C4—34no—
200C22×C3×C4—34no—
201C22×C3×C4—32no—
202C22×C3×C4—38no—
203C22×C3×C4—32no—
204C22×C3×C4—32no—
205C22×C3×C4—64no—
206C22×C3×C4—64no—
207C22×C3×C4C22×D312yes—
208C22×C3×C4C22×D312yes—
209C22×C3×C4C22×D311yescharacteristic
210C22×C3×C4C22×D311yescharacteristic
211C22×Dic3—32no—
212C22×Dic3—32no—
213C22×Dic3—32no—
214C22×Dic3—32no—
215C22×Dic3—62no—
216C22×Dic3—62no—
217C22×Dic3C22×D312yes—
218C22×Dic3C22×D312yes—
219C22×Dic3C22×D312yes—
220C22×Dic3C22×D311yescharacteristic
221D3×Q8—232no—
222C2×C4×D3—24no—
223C2×C4×D3—24no—
224C2×C4×D3C2×Dic318yes—
225C2×C4×D3C2×Dic314yes—
226C2×C4×D3#6714yes—
227C2×C4×D3#6714yes—
228C2×Dic6—22no—
229C2×Dic6—22no—
230C2×Dic6—24no—
231C2×Dic6—24no—
232C2×Dic6C2×Dic314yes—
233C2×Dic6C2×Dic318yes—
234C3×C4⋊C4—62no—
235C3×C42—62no—
236#217—64no—
237#217—64no—
238(C2×D3)⋊C4—64no—
239(C2×D3)⋊C4—64no—
240C4⋊Dic3—22no—
241C4⋊Dic3—64no—
242C4⋊Dic3—62no—
243C4⋊Dic3—62no—
244C4⋊Dic3—122no—
245Dic3⋊C4—64no—
246C4×Dic3—22no—
247C4×Dic3—64no—
248C4×Dic3—62no—
249C4×Dic3—62no—
250C4×Dic3—64no—
251C4×Dic3—122no—
252C22×C32C23×C411yescharacteristic
253C22×C32C22×D411yescharacteristic
254C22×C32C22×D411yescharacteristic
255C22×C322−1+412yes—
256C22×C322−1+412yes—
257C2×C3×D3—24no—
258C2×C3×D3—22no—
259C2×C3×D3—22no—
260C2×C3×D3—24no—
261C2×C3×D3C2×C22⋊C414yes—
262C32×C4—24no—
263C32×C4C2×C22⋊C414yes—
264C3⋊Dic3—42no—
265C3⋊Dic3—42no—
266C3×Dic3—24no—
267C3×Dic3—42no—
268C3×Dic3—42no—
269C3×Dic3C2×C22⋊C414yes—
270C22×Q8—38no—
271C22×Q8—32no—
272C22×Q8—31no—
273C22×Q8—64no—
274C22×Q8—64no—
275C22×Q8D3211yescharacteristic
276C23×C4—31no—
277C23×C4—32no—
278#117—184no—
279#117—184no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—92no—
282C2×C4⋊C4—91no—
283C2×C22⋊C4—92no—
284C2×C22⋊C4—92no—
285C2×C42—31no—
286C2×C42—92no—
287C2×C42—91no—
288#95—34no—
289#95—94no—
290#95—94no—
291#95—94no—
292C23×C3—21no—
293C23×C3—32no—
294C23×C3—32no—
295C23×C3—32no—
296C23×C3—62no—
297C23×C3—62no—
298C23×C3C23×D311yescharacteristic
299C23×C3C23×D311yescharacteristic
300C22×D3—22no—
301C22×D3—22no—
302C22×D3C22×Dic312yes—
303C22×D3C2×#6712yes—
304C22×D3C2×#6712yes—
305C3×Q8—28no—
306C3×Q8—28no—
307C3×Q8—48no—
308C3×Q8—616no—
309C3×Q8—68no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—24no—
316C2×C3×C4—22no—
317C2×C3×C4—22no—
318C2×C3×C4—32no—
319C2×C3×C4—32no—
320C2×C3×C4—32no—
321C2×C3×C4—316no—
322C2×C3×C4—34no—
323C2×C3×C4—42no—
324C2×C3×C4—42no—
325C2×C3×C4—616no—
326C2×C3×C4—62no—
327C2×C3×C4—62no—
328C2×C3×C4—64no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4C2×C4×D314yes—
334C2×C3×C4C2×C4×D312yes—
335C2×C3×C4D3×D412yes—
336C2×C3×C4D3×D412yes—
337C2×C3×C4C22×Dic314yes—
338C2×C3×C4C22×Dic312yes—
339C2×C3×C4C2×#6712yes—
340C2×C3×C4C2×#6712yes—
341C2×Dic3—22no—
342C2×Dic3—22no—
343C2×Dic3—22no—
344C2×Dic3—22no—
345C2×Dic3—22no—
346C2×Dic3—24no—
347C2×Dic3—22no—
348C2×Dic3—64no—
349C2×Dic3—62no—
350C2×Dic3—62no—
351C2×Dic3—62no—
352C2×Dic3—64no—
353C2×Dic3—62no—
354C2×Dic3—64no—
355C2×Dic3—62no—
356C2×Dic3—62no—
357C2×Dic3—62no—
358C2×Dic3—62no—
359C2×Dic3—122no—
360C2×Dic3—122no—
361C2×Dic3—122no—
362C2×Dic3—124no—
363C2×Dic3C22×Dic314yes—
364C2×Dic3C22×Dic312yes—
365C2×Dic3C2×#6712yes—
366C2×Dic3C2×#6712yes—
367C4×D3—216no—
368C4×D3#217116yes—
369Dic6—28no—
370Dic6—216no—
371C2×C32C22×C22⋊C411yescharacteristic
372C2×C32#51912yes—
373C2×C32#53611yescharacteristic
374C2×C32#53611yescharacteristic
375C2×C32#53612yes—
376C3×D3—24no—
377C24—31no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—38no—
381C2×Q8—34no—
382C2×Q8—64no—
383C2×Q8—64no—
384C2×Q8—632no—
385C2×Q8—62no—
386C2×Q8—62no—
387C2×Q8D3×Dic314yes—
388C22×C4—31no—
389C22×C4—34no—
390C22×C4—34no—
391C22×C4—34no—
392C22×C4—32no—
393C22×C4—38no—
394C22×C4—32no—
395C22×C4—32no—
396C22×C4—64no—
397C22×C4—64no—
398C22×C4—92no—
399C22×C4—92no—
400C22×C4C2×D3212yes—
401C22×C4C2×D3211yescharacteristic
402C4⋊C4—62no—
403C4⋊C4—184no—
404C4⋊C4—182no—
405C22⋊C4—184no—
406C22⋊C4—184no—
407C42—62no—
408C42—184no—
409C42—182no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—34no—
416C22×C3—62no—
417C22×C3—62no—
418C22×C3—64no—
419C22×C3—64no—
420C22×C3C22×C4×D311yescharacteristic
421C22×C3C2×D3×D411yescharacteristic
422C22×C3C2×D3×D411yescharacteristic
423C22×C3#100712yes—
424C22×C3#100712yes—
425C22×C3#100712yes—
426C22×C3#101012yes—
427C22×C3C23×Dic311yescharacteristic
428C22×C3C22×#6711yescharacteristic
429C22×C3C22×#6711yescharacteristic
430C2×D3—24no—
431C2×D3—22no—
432C2×D3—24no—
433C2×D3—22no—
434C2×D3C2×#21714yes—
435C3×C4—24no—
436C3×C4—24no—
437C3×C4—24no—
438C3×C4—34no—
439C3×C4—44no—
440C3×C4—64no—
441C3×C4—122no—
442C3×C4—122no—
443C3×C4D3×C22⋊C414yes—
444C3×C4C2×#21714yes—
445Dic3—24no—
446Dic3—42no—
447Dic3—42no—
448Dic3—124no—
449Dic3—122no—
450Dic3—122no—
451Dic3—122no—
452Dic3—122no—
453Dic3—242no—
454Dic3C2×#21714yes—
455C32#231011yescharacteristic, complemented, Sylow
456C23—32no—
457C23—32no—
458C23—32no—
459C23—62no—
460C23—62no—
461C23C22×D3211yescharacteristic
462Q8—28no—
463Q8—616no—
464Q8—68no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—32no—
468C2×C4—32no—
469C2×C4—32no—
470C2×C4—316no—
471C2×C4—34no—
472C2×C4—616no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—64no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—182no—
481C2×C4—184no—
482C2×C4—182no—
483C2×C4—182no—
484C2×C4—182no—
485C2×C4C2×D3×Dic314yes—
486C2×C4C2×D3×Dic312yes—
487C2×C4D3×#6712yes—
488C2×C4D3×#6712yes—
489C2×C3—21no—
490C2×C3—22no—
491C2×C3—22no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—64no—
495C2×C3C2×D3×C22⋊C411yescharacteristic
496C2×C3#557212yes—
497C2×C3#562812yes—
498C2×C3#567511yescharacteristic
499C2×C3#572411yescharacteristic
500C2×C3#581212yes—
501C2×C3#581611yescharacteristic
502C2×C3#581612yes—
503C2×C3#581611yescharacteristic
504C2×C3C22×#21711yescharacteristic
505D3—24no—
506C22—34no—
507C22—62no—
508C22—64no—
509C22—64no—
510C22—62no—
511C22#6431512yes—
512C22#6431612yes—
513C22C22×D3×Dic311yescharacteristic
514C22C2×D3×#6711yescharacteristic
515C22C2×D3×#6711yescharacteristic
516C4—24no—
517C4—34no—
518C4—64no—
519C4—122no—
520C4—122no—
521C4—362no—
522C4—362no—
523C4D3×#21714yes—
524C3—21no—
525C3#7661011yescharacteristic, complemented
526C3#8406211yescharacteristic, complemented
527C2—64no—
528C2#1059534011yescharacteristic
529C2#1059534712yes—
530C2#1059539712yes—
531C2#1059539911yescharacteristic
532C2C2×D3×#21711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC22×C32×Q8C22
socle, cosocleC23×C32C24
prime radicalC23×C3×D3C23
hypercenterC22×Q8D32
hypocenterC32#2310
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
4. #10592928C2semidirect
5. C3×#5229C2semidirect
7. #10591261C2semidirect
8. #10591221C2semidirect
9. #10591220C2semidirect
15. C2×C3×D3×Q8C4semidirect
47. #5229D3semidirect
99. C2×D3×Q8Dic3semidirect
455. C32#2310semidirect
525. C3#76610semidirect
526. C3#84062semidirect

Maximal subgroups

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

subgroupindex
C22×C3×D3×Q82
C2×#638552
C2×#638412
#105929282
C3×#52292
#105912942
#105912612
#105912212
#105912202
#840623
#766103