Finite Group Databasebeta

← #10877256

Subgroups — #10877256

Order 768.

854 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#75964C211yesmaximal, characteristic
2#74318C211yesmaximal, characteristic, complemented
3C3×#2100C211yesmaximal, characteristic, complemented
4#69611C211yesmaximal, characteristic, complemented
5#69587C211yesmaximal, characteristic, complemented
6#68893C212yesmaximal, complemented
7#68881C212yesmaximal
8#68877C212yesmaximal, complemented
9#68798C212yesmaximal, complemented
10#68681C211yesmaximal, characteristic, complemented
11#68662C211yesmaximal, characteristic, complemented
12#27683—31nomaximal, Sylow
13C2×#895C2211yescharacteristic
14C2×#893—21no—
15C2×#893C2211yescharacteristic
16C2×#892C2212yes—
17C2×C42×D3C2211yescharacteristic
18C3×#389C2211yescharacteristic
19C3×#384C2211yescharacteristic
20C2×#821C2212yes—
21C2×(C2×D3)⋊C8C2212yes—
22C2×#818C2212yes—
23C2×Dic3⋊C8C2212yes—
24#4990—21no—
25#4987—21no—
26#4987—21no—
27#4987C2211yescharacteristic
28#4977C2212yes—
29Dic3×C4⋊C4—21no—
30Dic3×C4⋊C4—21no—
31Dic3×C4⋊C4C2211yescharacteristic
32C2×#809C2212yes—
33C2×#808C2212yes—
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—
45#2285—31no—
46#2100D311yescharacteristic, complemented
47#2099—31no—
48#2094—31no—
49#2093—31no—
50#1949—32no—
51#1875—32no—
52#1854—32no—
53#1804—32no—
54#1797—31no—
55C22×D12—21no—
56C22×D12C2311yescharacteristic
57C22×C4×D3—21no—
58C22×C4×D3D411yescharacteristic
59C22×C4×D3D411yescharacteristic
60C22×C4×D3C2311yescharacteristic
61C22×C3×C8C2312yes—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C42C2311yescharacteristic
69C2×(C2×D3)⋊C4—21no—
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—22no—
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×C4×Dic3—21no—
80C2×C4×Dic3—22no—
81C2×C4×Dic3D411yescharacteristic
82C2×C4×Dic3D411yescharacteristic
83C2×C4×Dic3C2311yescharacteristic
84C22×C3⋊C8C2312yes—
85#895—22no—
86#895Q814yes—
87#893—24no—
88#893—44no—
89#892—42no—
90#892—42no—
91C42×D3—24no—
92C3×#95—21no—
93C3⋊#95—21no—
94C3⋊#95—21no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97#821—42no—
98(C2×D3)⋊C8—24no—
99#818—42no—
100Dic3⋊C8—24no—
101#809—42no—
102#808—42no—
103C2×#121—31no—
104C2×C4×D4—31no—
105C2×C4×D4—61no—
106C2×#117—32no—
107C22×C42—31no—
108C2×#107—32no—
109C2×#106—32no—
110C2×C4⋊C8—32no—
111C2×#102—32no—
112C2×#102—32no—
113C2×C22⋊C8—32no—
114#390—31no—
115#389C2×D311yescharacteristic
116#388—61no—
117#386—61no—
118#384—31no—
119#384C2×D311yescharacteristic
120#382—32no—
121#380—31no—
122#380—61no—
123#380—61no—
124C4×C4⋊C4—31no—
125C4×C4⋊C4—61no—
126C4×C4⋊C4—61no—
127C4×C22⋊C4—61no—
128#340—32no—
129#340—32no—
130#340C2×D312yes—
131#340C2×D312yes—
132#336—31no—
133#336C2×D311yescharacteristic
134C23×D3—21no—
135C23×D3C2×D411yescharacteristic
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4—21no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2411yescharacteristic
142C22×Dic3—22no—
143C22×Dic3—21no—
144C22×Dic3C2×D411yescharacteristic
145C22×Dic3C2×D411yescharacteristic
146C22×Dic3C2×D411yescharacteristic
147C2×D12—22no—
148C2×D12—22no—
149C2×D12—24no—
150C2×D12—21no—
151C2×D12—42no—
152C2×D12—42no—
153C2×D12C2×Q812yes—
154C2×C4×D3—22no—
155C2×C4×D3—22no—
156C2×C4×D3—22no—
157C2×C4×D3—22no—
158C2×C4×D3—22no—
159C2×C4×D3—22no—
160C2×C4×D3—41no—
161C2×C4×D3—41no—
162C2×C4×D3—42no—
163C2×C4×D3—42no—
164C2×C4×D3—42no—
165C2×C4×D3C2×Q812yes—
166C2×C4×D3#4112yes—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—42no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4C2×Q812yes—
178C3×C42—22no—
179(C2×D3)⋊C4—24no—
180(C2×D3)⋊C4—44no—
181(C2×D3)⋊C4—42no—
182(C2×D3)⋊C4—42no—
183(C2×D3)⋊C4—42no—
184(C2×D3)⋊C4—42no—
185C4⋊Dic3—22no—
186C4⋊Dic3—22no—
187C4⋊Dic3—22no—
188C4⋊Dic3—42no—
189C4⋊Dic3—42no—
190C4×Dic3—22no—
191C4×Dic3—22no—
192C4×Dic3—44no—
193C4×Dic3—42no—
194C4×Dic3#4114yes—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—22no—
198C22×D4—31no—
199C22×D4—61no—
200C23×C4—31no—
201C23×C4—31no—
202C23×C4—31no—
203C23×C4—61no—
204C22×C8—32no—
205C22×C8C22×D312yes—
206#121—34no—
207#121—62no—
208C4×D4—64no—
209C4×D4—124no—
210#117—122no—
211#117—122no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—32no—
216C2×C4⋊C4—31no—
217C2×C4⋊C4—61no—
218C2×C4⋊C4—61no—
219C2×C4⋊C4C22×D312yes—
220C2×C4⋊C4C22×D311yescharacteristic
221C2×C4⋊C4C22×D311yescharacteristic
222C2×C22⋊C4—61no—
223C2×C22⋊C4—61no—
224C2×C22⋊C4—62no—
225C2×C22⋊C4—61no—
226C2×C22⋊C4—61no—
227C2×C22⋊C4—61no—
228C2×C42—31no—
229C2×C42—31no—
230C2×C42—31no—
231C2×C42—64no—
232C2×C42—62no—
233C2×C42—61no—
234C2×C42C22×D311yescharacteristic
235#107—122no—
236#106—122no—
237C4⋊C8—64no—
238#102—122no—
239#102—122no—
240C22⋊C8—64no—
241#95—21no—
242#95—61no—
243#95—61no—
244#95—62no—
245#95—62no—
246C23×C3C22×D411yescharacteristic
247C22×D3—21no—
248C22×D3—21no—
249C22×D3—21no—
250C22×D3—21no—
251C22×D3—21no—
252C22×D3—21no—
253C22×D3—22no—
254C22×D3—41no—
255C22×D3—41no—
256C22×D3—41no—
257C22×D3—41no—
258C22×D3—41no—
259C22×D3—41no—
260C22×D3C22⋊Q812yes—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—22no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4#13512yes—
275C2×C3×C4AGL(1,ℤ/8)12yes—
276C2×C3×C4C22×Q811yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C4C2×#4112yes—
279C2×C3×C4H(5,𝔽2)12yes—
280C2×Dic3—21no—
281C2×Dic3—21no—
282C2×Dic3—21no—
283C2×Dic3—21no—
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—42no—
287C2×Dic3—41no—
288C2×Dic3—42no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3—42no—
292C2×Dic3C22⋊Q812yes—
293C2×Dic3#12312yes—
294C2×Dic3#12312yes—
295D12—24no—
296D12—48no—
297D12—42no—
298C4×D3—24no—
299C4×D3—24no—
300C4×D3—44no—
301C4×D3—44no—
302C4×D3—84no—
303C3×C8—42no—
304C3⋊C8—42no—
305C24—31no—
306C24—61no—
307C2×D4—32no—
308C2×D4—64no—
309C2×D4—62no—
310C2×D4—61no—
311C2×D4—62no—
312C2×D4—122no—
313C2×D4—122no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—31no—
318C22×C4—31no—
319C22×C4—31no—
320C22×C4—32no—
321C22×C4—32no—
322C22×C4—62no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—62no—
326C22×C4—62no—
327C22×C4—62no—
328C22×C4—61no—
329C22×C4—62no—
330C22×C4—121no—
331C22×C4—121no—
332C22×C4—122no—
333C22×C4—122no—
334C22×C4—122no—
335C22×C4D3×D411yescharacteristic
336C22×C4D3×D411yescharacteristic
337C22×C4C23×D311yescharacteristic
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—62no—
342C2×C8—62no—
343C2×C8—62no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—62no—
352C4⋊C4—62no—
353C4⋊C4—62no—
354C4⋊C4—122no—
355C4⋊C4—122no—
356C4⋊C4D3×Q812yes—
357C22⋊C4—64no—
358C22⋊C4—122no—
359C22⋊C4—124no—
360C22⋊C4—122no—
361C22⋊C4—122no—
362C22⋊C4—122no—
363C42—22no—
364C42—34no—
365C42—62no—
366C42—62no—
367C42—124no—
368C42—122no—
369C22×C3C2×C22⋊Q811yescharacteristic
370C22×C3C2×#12311yescharacteristic
371C22×C3#53511yescharacteristic
372C22×C3C2×#13511yescharacteristic
373C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
374C22×C3#57511yescharacteristic
375C22×C3#57511yescharacteristic
376C2×D3—22no—
377C2×D3—22no—
378C2×D3—22no—
379C2×D3—41no—
380C2×D3—41no—
381C2×D3—41no—
382C2×D3—41no—
383C2×D3—42no—
384C2×D3—42no—
385C2×D3—42no—
386C2×D3—81no—
387C2×D3—81no—
388C2×D3—81no—
389C2×D3—81no—
390C3×C4—22no—
391C3×C4—22no—
392C3×C4—42no—
393C3×C4—81no—
394C3×C4—82no—
395C3×C4#55312yes—
396C3×C4#55412yes—
397Dic3—22no—
398Dic3—22no—
399Dic3—22no—
400Dic3—82no—
401Dic3—81no—
402C23—32no—
403C23—61no—
404C23—61no—
405C23—61no—
406C23—61no—
407C23—61no—
408C23—61no—
409C23—62no—
410C23—121no—
411C23—121no—
412C23—121no—
413C23—121no—
414C23—121no—
415C23—121no—
416C23C2×D3×D411yescharacteristic
417D4—64no—
418D4—128no—
419D4—122no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—32no—
426C2×C4—32no—
427C2×C4—32no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—64no—
443C2×C4—64no—
444C2×C4—124no—
445C2×C4—124no—
446C2×C4—122no—
447C2×C4—121no—
448C2×C4—121no—
449C2×C4—122no—
450C2×C4—121no—
451C2×C4—122no—
452C2×C4—244no—
453C2×C4#91412yes—
454C2×C4#91912yes—
455C2×C4C2×D3×Q811yescharacteristic
456C2×C4D3×#4112yes—
457C2×C4D3×#4111yescharacteristic
458C2×C4#100912yes—
459C8—42no—
460C8—122no—
461C2×C3#248711yescharacteristic
462C2×C3#307611yescharacteristic
463C2×C3#307711yescharacteristic
464C2×C3#309111yescharacteristic
465C2×C3#309411yescharacteristic
466C2×C3#320711yescharacteristic
467C2×C3#321911yescharacteristic
468D3—42no—
469D3—82no—
470C22—62no—
471C22—62no—
472C22—62no—
473C22—121no—
474C22—121no—
475C22—121no—
476C22—121no—
477C22—122no—
478C22—122no—
479C22—122no—
480C22—241no—
481C22—241no—
482C22—241no—
483C22—241no—
484C22D3×C22⋊Q811yescharacteristic
485C22D3×#12311yescharacteristic
486C22#567411yescharacteristic
487C22C2×#91411yescharacteristic
488C22C2×#91911yescharacteristic
489C22#577311yescharacteristic
490C22#577311yescharacteristic
491C4—22no—
492C4—22no—
493C4—42no—
494C4—62no—
495C4—62no—
496C4—62no—
497C4—82no—
498C4—81no—
499C4—242no—
500C4—241no—
501C4#569012yes—
502C4#570912yes—
503C3#2768311yescharacteristic, complemented, Sylow
504C2—122no—
505C2—242no—
506C2#7814611yescharacteristic
507C2#8106011yescharacteristic
508C2#8110711yescharacteristic
509C2#8132611yescharacteristic
510C2#8146711yescharacteristic
511C2#8160311yescharacteristic
512C2#8171811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2100C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#2100D3
hypocenterC3#27683
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2100

Complements

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

NH ≅ G/Ntype
2. #74318C2semidirect
3. C3×#2100C2semidirect
4. #69611C2semidirect
5. #69587C2semidirect
6. #68893C2semidirect
8. #68877C2semidirect
9. #68798C2semidirect
10. #68681C2semidirect
11. #68662C2semidirect
46. #2100D3semidirect
503. C3#27683semidirect

Maximal subgroups

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

subgroupindex
#759642
#743182
C3×#21002
#696112
#695872
#688932
#688812
#688772
#687982
#686812
#686622
#276833