Finite Group Databasebeta

← #10877190

Subgroups — #10877190

Order 768.

970 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#75964C211yesmaximal, characteristic
2D3×#384C211yesmaximal, characteristic, complemented
3C3×#2099C211yesmaximal, characteristic, complemented
4#69611C211yesmaximal, characteristic, complemented
5#69586C211yesmaximal, characteristic, complemented
6#68892C212yesmaximal, complemented
7#68880C212yesmaximal
8#68874C212yesmaximal, complemented
9#68795C212yesmaximal, complemented
10#68681C211yesmaximal, characteristic, complemented
11#68662C211yesmaximal, characteristic, complemented
12#27677—31nomaximal, Sylow
13C2×#895C2211yescharacteristic
14C2×#893—21no—
15C2×#893C2211yescharacteristic
16C2×D3×C4⋊C4—21no—
17C2×D3×C4⋊C4—21no—
18C2×D3×C4⋊C4C2212yes—
19C2×C42×D3C2211yescharacteristic
20C3×#389C2211yescharacteristic
21C3×#384C2211yescharacteristic
22C2×#821C2212yes—
23C2×(C2×D3)⋊C8C2212yes—
24C2×#818C2212yes—
25C2×Dic3⋊C8C2212yes—
26#4990—21no—
27#4987C2211yescharacteristic
28#4975—21no—
29#4975—21no—
30#4975C2212yes—
31Dic3×C4⋊C4C2211yescharacteristic
32C2×#809C2212yes—
33C2×#807C2212yes—
34#4939C2211yescharacteristic
35#4935C2211yescharacteristic
36#4666—21no—
37#4665—21no—
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#336C2211yescharacteristic
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44C4×#121—31no—
45C2×#384—31no—
46#2099—31no—
47#2099D311yescharacteristic, complemented
48#2093—31no—
49#2093—31no—
50#1947—32no—
51#1874—32no—
52#1852—32no—
53#1802—32no—
54#1797—31no—
55C22×D12—21no—
56C22×D12C2311yescharacteristic
57C22×C4×D3—22no—
58C22×C4×D3—21no—
59C22×C4×D3D411yescharacteristic
60C22×C4×D3D411yescharacteristic
61C22×C4×D3C2311yescharacteristic
62C22×C3×C8C2312yes—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—21no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—21no—
74C2×(C2×D3)⋊C4—21no—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3C2312yes—
78C2×C4⋊Dic3C2311yescharacteristic
79C2×Dic3⋊C4—22no—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—22no—
82C2×C4×Dic3—21no—
83C2×C4×Dic3D411yescharacteristic
84C2×C4×Dic3D411yescharacteristic
85C2×C4×Dic3C2311yescharacteristic
86C22×C3⋊C8C2312yes—
87#895—22no—
88#895Q814yes—
89#893—24no—
90#893—44no—
91D3×C4⋊C4—42no—
92D3×C4⋊C4—42no—
93D3×C4⋊C4—44no—
94D3×C4⋊C4—44no—
95C42×D3—24no—
96C3×#95—21no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99#821—42no—
100(C2×D3)⋊C8—24no—
101#818—42no—
102Dic3⋊C8—24no—
103#809—42no—
104#807—42no—
105C2×#121—31no—
106C2×C4×D4—31no—
107C2×C4×D4—61no—
108C22×C4⋊C4—32no—
109C22×C4⋊C4—61no—
110C22×C4⋊C4—61no—
111C22×C42—31no—
112C2×#107—32no—
113C2×#107—32no—
114C2×C4⋊C8—32no—
115C2×#102—32no—
116C2×#102—32no—
117C2×C22⋊C8—32no—
118#390—31no—
119#389C2×D311yescharacteristic
120#388—61no—
121#386—61no—
122#384—32no—
123#384—31no—
124#384—61no—
125#384—61no—
126#384C2×D311yescharacteristic
127#380—31no—
128C4×C4⋊C4—31no—
129C4×C22⋊C4—61no—
130#340—32no—
131#340—32no—
132#340C2×D312yes—
133#340C2×D312yes—
134#336—31no—
135#336C2×D311yescharacteristic
136C23×D3—21no—
137C23×D3C2×D411yescharacteristic
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2411yescharacteristic
144C22×Dic3—21no—
145C22×Dic3—22no—
146C22×Dic3C2×D411yescharacteristic
147C22×Dic3C2×D411yescharacteristic
148C22×Dic3C2×D411yescharacteristic
149C2×D12—22no—
150C2×D12—22no—
151C2×D12—24no—
152C2×D12—21no—
153C2×D12—42no—
154C2×D12—42no—
155C2×D12C2×Q812yes—
156C2×C4×D3—22no—
157C2×C4×D3—22no—
158C2×C4×D3—22no—
159C2×C4×D3—22no—
160C2×C4×D3—22no—
161C2×C4×D3—22no—
162C2×C4×D3—22no—
163C2×C4×D3—44no—
164C2×C4×D3—44no—
165C2×C4×D3—44no—
166C2×C4×D3—42no—
167C2×C4×D3—42no—
168C2×C4×D3—42no—
169C2×C4×D3D814yes—
170C2×C4×D3C2×Q812yes—
171C2×C4×D3#4112yes—
172C2×C3×C8—22no—
173C2×C3×C8—22no—
174C2×C3×C8—22no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—42no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—42no—
182C3×C4⋊C4C2×Q812yes—
183C3×C42—22no—
184(C2×D3)⋊C4—24no—
185(C2×D3)⋊C4—42no—
186(C2×D3)⋊C4—42no—
187(C2×D3)⋊C4—42no—
188(C2×D3)⋊C4—42no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3—42no—
193C4⋊Dic3—42no—
194Dic3⋊C4—44no—
195Dic3⋊C4—44no—
196Dic3⋊C4—44no—
197C4×Dic3—22no—
198C4×Dic3—22no—
199C4×Dic3—42no—
200C4×Dic3#4114yes—
201C2×C3⋊C8—22no—
202C2×C3⋊C8—22no—
203C2×C3⋊C8—22no—
204C22×D4—31no—
205C22×D4—61no—
206C23×C4—31no—
207C23×C4—31no—
208C23×C4—31no—
209C23×C4—61no—
210C23×C4—62no—
211C22×C8—32no—
212C22×C8C22×D312yes—
213#121—34no—
214#121—62no—
215C4×D4—64no—
216C4×D4—124no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—32no—
221C2×C4⋊C4—31no—
222C2×C4⋊C4—62no—
223C2×C4⋊C4—62no—
224C2×C4⋊C4—62no—
225C2×C4⋊C4—61no—
226C2×C4⋊C4—61no—
227C2×C4⋊C4—122no—
228C2×C4⋊C4—122no—
229C2×C4⋊C4—124no—
230C2×C4⋊C4—124no—
231C2×C4⋊C4C22×D312yes—
232C2×C4⋊C4C22×D311yescharacteristic
233C2×C4⋊C4C22×D311yescharacteristic
234C2×C22⋊C4—61no—
235C2×C22⋊C4—61no—
236C2×C22⋊C4—61no—
237C2×C22⋊C4—61no—
238C2×C22⋊C4—61no—
239C2×C42—31no—
240C2×C42—31no—
241C2×C42—31no—
242C2×C42—64no—
243C2×C42—61no—
244C2×C42C22×D311yescharacteristic
245#107—122no—
246#107—122no—
247C4⋊C8—64no—
248#102—122no—
249#102—122no—
250C22⋊C8—64no—
251#95—21no—
252#95—61no—
253#95—61no—
254C23×C3C22×D411yescharacteristic
255C22×D3—21no—
256C22×D3—21no—
257C22×D3—21no—
258C22×D3—21no—
259C22×D3—22no—
260C22×D3—41no—
261C22×D3—41no—
262C22×D3—41no—
263C22×D3—41no—
264C22×D3—41no—
265C22×D3—41no—
266C22×D3C22⋊Q812yes—
267C22×D3C2×D812yes—
268C22×D3#13712yes—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—22no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—41no—
280C2×C3×C4—42no—
281C2×C3×C4—41no—
282C2×C3×C4C2×D812yes—
283C2×C3×C4AGL(1,ℤ/8)12yes—
284C2×C3×C4C22×Q811yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4C2×#4112yes—
287C2×C3×C4H(5,𝔽2)12yes—
288C2×Dic3—21no—
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—42no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—41no—
298C2×Dic3C22⋊Q812yes—
299C2×Dic3#12312yes—
300C2×Dic3#12312yes—
301C2×Dic3C2×D812yes—
302C2×Dic3#13712yes—
303D12—24no—
304D12—48no—
305D12—42no—
306C4×D3—28no—
307C4×D3—24no—
308C4×D3—24no—
309C4×D3—28no—
310C4×D3—88no—
311C4×D3—84no—
312C3×C8—42no—
313C3⋊C8—42no—
314C24—31no—
315C24—61no—
316C2×D4—32no—
317C2×D4—64no—
318C2×D4—62no—
319C2×D4—61no—
320C2×D4—62no—
321C2×D4—122no—
322C2×D4—122no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—22no—
326C22×C4—31no—
327C22×C4—31no—
328C22×C4—31no—
329C22×C4—34no—
330C22×C4—32no—
331C22×C4—32no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—62no—
339C22×C4—61no—
340C22×C4—62no—
341C22×C4—122no—
342C22×C4—122no—
343C22×C4—122no—
344C22×C4—124no—
345C22×C4—124no—
346C22×C4—124no—
347C22×C4D3×D411yescharacteristic
348C22×C4D3×D411yescharacteristic
349C22×C4C23×D311yescharacteristic
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—62no—
354C2×C8—62no—
355C2×C8—62no—
356C4⋊C4—21no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—62no—
364C4⋊C4—62no—
365C4⋊C4—62no—
366C4⋊C4—124no—
367C4⋊C4—124no—
368C4⋊C4—124no—
369C4⋊C4—122no—
370C4⋊C4—122no—
371C4⋊C4D3×Q812yes—
372C22⋊C4—64no—
373C22⋊C4—122no—
374C22⋊C4—122no—
375C22⋊C4—122no—
376C22⋊C4—122no—
377C42—22no—
378C42—34no—
379C42—62no—
380C42—62no—
381C42—122no—
382C22×C3C2×C22⋊Q811yescharacteristic
383C22×C3C2×#12311yescharacteristic
384C22×C3#53511yescharacteristic
385C22×C3C22×D811yescharacteristic
386C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
387C22×C3C2×#13711yescharacteristic
388C22×C3C2×#13711yescharacteristic
389C2×D3—22no—
390C2×D3—22no—
391C2×D3—22no—
392C2×D3—22no—
393C2×D3—22no—
394C2×D3—22no—
395C2×D3—42no—
396C2×D3—42no—
397C2×D3—42no—
398C2×D3—81no—
399C2×D3—81no—
400C2×D3—81no—
401C2×D3—81no—
402C2×D3#47414yes—
403C3×C4—22no—
404C3×C4—22no—
405C3×C4—42no—
406C3×C4—81no—
407C3×C4—82no—
408C3×C4#55312yes—
409C3×C4#55412yes—
410Dic3—22no—
411Dic3—22no—
412Dic3—82no—
413Dic3—81no—
414Dic3#48014yes—
415C23—32no—
416C23—32no—
417C23—32no—
418C23—61no—
419C23—61no—
420C23—61no—
421C23—61no—
422C23—62no—
423C23—121no—
424C23—121no—
425C23—121no—
426C23—121no—
427C23—121no—
428C23—121no—
429C23C2×D3×D411yescharacteristic
430D4—64no—
431D4—128no—
432D4—122no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—32no—
439C2×C4—32no—
440C2×C4—32no—
441C2×C4—32no—
442C2×C4—32no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—68no—
452C2×C4—68no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—64no—
458C2×C4—64no—
459C2×C4—122no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—122no—
463C2×C4—121no—
464C2×C4—122no—
465C2×C4—244no—
466C2×C4—248no—
467C2×C4D3×D812yes—
468C2×C4#91412yes—
469C2×C4C2×D3×Q811yescharacteristic
470C2×C4D3×#4112yes—
471C2×C4D3×#4111yescharacteristic
472C2×C4#100912yes—
473C8—42no—
474C8—122no—
475C2×C3#248711yescharacteristic
476C2×C3C2×#47411yescharacteristic
477C2×C3#307911yescharacteristic
478C2×C3C2×#48011yescharacteristic
479C2×C3#309211yescharacteristic
480C2×C3#320511yescharacteristic
481C2×C3#321811yescharacteristic
482D3—24no—
483D3—82no—
484C22—34no—
485C22—62no—
486C22—62no—
487C22—62no—
488C22—62no—
489C22—62no—
490C22—62no—
491C22—122no—
492C22—122no—
493C22—122no—
494C22—241no—
495C22—241no—
496C22—241no—
497C22—241no—
498C22D3×C22⋊Q811yescharacteristic
499C22D3×#12311yescharacteristic
500C22#567411yescharacteristic
501C22C2×D3×D811yescharacteristic
502C22C2×#91411yescharacteristic
503C22D3×#13711yescharacteristic
504C22D3×#13711yescharacteristic
505C4—22no—
506C4—22no—
507C4—34no—
508C4—42no—
509C4—62no—
510C4—62no—
511C4—81no—
512C4—82no—
513C4—242no—
514C4—241no—
515C4#569012yes—
516C4#570912yes—
517C3#2767711yescharacteristic, complemented, Sylow
518C2—64no—
519C2—242no—
520C2#7814611yescharacteristic
521C2D3×#47411yescharacteristic
522C2#8101211yescharacteristic
523C2D3×#48011yescharacteristic
524C2#8142711yescharacteristic
525C2#8156611yescharacteristic
526C2#8174111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2099C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#2099D3
hypocenterC3#27677
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2099

Complements

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

NH ≅ G/Ntype
2. D3×#384C2semidirect
3. C3×#2099C2semidirect
4. #69611C2semidirect
5. #69586C2semidirect
6. #68892C2semidirect
8. #68874C2semidirect
9. #68795C2semidirect
10. #68681C2semidirect
11. #68662C2semidirect
47. #2099D3semidirect
517. C3#27677semidirect

Maximal subgroups

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

subgroupindex
#759642
D3×#3842
C3×#20992
#696112
#695862
#688922
#688802
#688742
#687952
#686812
#686622
#276773