Finite Group Databasebeta

← #10877240

Subgroups — #10877240

Order 768.

770 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74318C211yesmaximal, characteristic
2C4×#897C211yesmaximal, characteristic
3C3×#1947C211yesmaximal, characteristic, complemented
4#70645C212yesmaximal, complemented
5#70633C212yesmaximal
6#70485C211yesmaximal, characteristic, complemented
7#70475C211yesmaximal, characteristic
8#69588C211yesmaximal, characteristic, complemented
9#69467C211yesmaximal, characteristic, complemented
10#68879C212yesmaximal
11#68845C212yesmaximal, complemented
12#27682—31nomaximal, Sylow
13C2×#897C2211yescharacteristic
14C2×#892C2212yes—
15C2×C42×D3C2211yescharacteristic
16C2×C4×Dic6C2211yescharacteristic
17C2×C3×#107C2212yes—
18C2×C3×C4⋊C8C2211yescharacteristic
19C3×#384C2211yescharacteristic
20C3×C4×C4⋊C4C2211yescharacteristic
21C2×(C2×D3)⋊C8C2212yes—
22C2×#816C2212yes—
23#4987—21no—
24#4987—21no—
25#4987C2211yescharacteristic
26#4984C2211yescharacteristic
27#4977C2212yes—
28Dic3×C4⋊C4—21no—
29Dic3×C4⋊C4—21no—
30#4973C2211yescharacteristic
31C2×#810C2212yes—
32C2×#808C2212yes—
33C4×(C2×D3)⋊C4—21no—
34#4935C2211yescharacteristic
35C4×C4⋊Dic3—21no—
36C2×C4⋊(C3⋊C8)C2211yescharacteristic
37#4668—21no—
38#4653—21no—
39C3×#340C2212yes—
40#4551C2212yes—
41#4550C2212yes—
42#4529C2212yes—
43#4529C2212yes—
44C4×C22⋊Q8—31no—
45#2285—31no—
46#2101—32no—
47#2097—32no—
48#1948—31no—
49#1947D311yescharacteristic, complemented
50#1873—32no—
51#1854—32no—
52#1803—31no—
53#1803—31no—
54#1797—31no—
55C22×C4×D3D411yescharacteristic
56C22×C4×D3D411yescharacteristic
57C22×C4×D3C2311yescharacteristic
58C22×Dic6C2311yescharacteristic
59C22×C3×C8C2312yes—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4C2311yescharacteristic
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C4⋊C4C2311yescharacteristic
65C2×C3×C42—21no—
66C2×C3×C42C2311yescharacteristic
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—22no—
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3C2312yes—
75C2×C4⋊Dic3C2311yescharacteristic
76C2×Dic3⋊C4—21no—
77C2×C4×Dic3—21no—
78C2×C4×Dic3—22no—
79C2×C4×Dic3D411yescharacteristic
80C2×C4×Dic3D411yescharacteristic
81C2×C4×Dic3C2311yescharacteristic
82C22×C3⋊C8C2312yes—
83#897—22no—
84#897Q814yes—
85#892—42no—
86#892—42no—
87C42×D3—22no—
88C42×D3—22no—
89C4×Dic6—22no—
90C4×Dic6—22no—
91C3×#107—42no—
92C3×C4⋊C8—22no—
93C3×#95—21no—
94C3⋊#95—21no—
95C3⋊#95—21no—
96C3⋊#95—21no—
97C3⋊#95—21no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100(C2×D3)⋊C8—24no—
101#816—42no—
102#810—42no—
103#808—42no—
104C4⋊(C3⋊C8)—22no—
105C2×C22⋊Q8—31no—
106C2×C4×Q8—31no—
107C2×#117—32no—
108C22×C42—31no—
109C2×#107C2×D312yes—
110C2×#106—32no—
111C2×C4⋊C8—31no—
112C2×C4⋊C8C2×D311yescharacteristic
113C2×#103—32no—
114C2×#103—32no—
115C2×C22⋊C8—32no—
116#391—31no—
117#389—31no—
118#387—61no—
119#385—61no—
120#384—31no—
121#384C2×D311yescharacteristic
122#382—32no—
123#380—31no—
124#380—61no—
125#380—61no—
126C4×C4⋊C4—61no—
127C4×C4⋊C4—61no—
128C4×C4⋊C4—61no—
129C4×C4⋊C4C2×D311yescharacteristic
130C4×C22⋊C4—61no—
131#340—32no—
132#340—32no—
133#340—32no—
134#340C2×D312yes—
135#336—32no—
136C23×D3C2×D411yescharacteristic
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2411yescharacteristic
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3—22no—
146C22×Dic3C2×D411yescharacteristic
147C22×Dic3C2×D411yescharacteristic
148C22×Dic3C2×D411yescharacteristic
149C2×C4×D3—21no—
150C2×C4×D3—21no—
151C2×C4×D3—21no—
152C2×C4×D3—21no—
153C2×C4×D3—22no—
154C2×C4×D3—22no—
155C2×C4×D3—22no—
156C2×C4×D3—22no—
157C2×C4×D3—41no—
158C2×C4×D3—41no—
159C2×C4×D3C2×Q812yes—
160C2×C4×D3#4112yes—
161C2×Dic6—21no—
162C2×Dic6—21no—
163C2×Dic6—21no—
164C2×Dic6—21no—
165C2×Dic6—21no—
166C2×Dic6C2×Q812yes—
167C2×C3×C8—22no—
168C2×C3×C8—22no—
169C2×C3×C8—22no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—42no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4C2×Q812yes—
178C3×C42—42no—
179C3×C42#4112yes—
180C3×C42#4112yes—
181(C2×D3)⋊C4—24no—
182(C2×D3)⋊C4—44no—
183(C2×D3)⋊C4—42no—
184C4⋊Dic3—24no—
185C4⋊Dic3—21no—
186C4⋊Dic3—21no—
187C4⋊Dic3—22no—
188C4⋊Dic3—22no—
189C4⋊Dic3—42no—
190C4⋊Dic3—42no—
191C4⋊Dic3—42no—
192Dic3⋊C4—42no—
193C4×Dic3—22no—
194C4×Dic3—22no—
195C4×Dic3—21no—
196C4×Dic3—21no—
197C4×Dic3—44no—
198C4×Dic3—42no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C22×Q8—31no—
203C23×C4—31no—
204C23×C4—31no—
205C23×C4—31no—
206C22×C8—32no—
207C22×C8C22×D312yes—
208C22⋊Q8—34no—
209C22⋊Q8—62no—
210C4×Q8—62no—
211C4×Q8—62no—
212#117—122no—
213#117—122no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—32no—
217C2×C4⋊C4—31no—
218C2×C4⋊C4—61no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4C22×D312yes—
224C2×C4⋊C4C22×D311yescharacteristic
225C2×C4⋊C4C22×D311yescharacteristic
226C2×C22⋊C4—61no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—61no—
229C2×C42—21no—
230C2×C42—31no—
231C2×C42—31no—
232C2×C42—31no—
233C2×C42—62no—
234C2×C42—62no—
235C2×C42—62no—
236C2×C42—61no—
237C2×C42C22×D311yescharacteristic
238#107—42no—
239#106—122no—
240C4⋊C8—22no—
241C4⋊C8—62no—
242#103—122no—
243#103—122no—
244C22⋊C8—64no—
245#95—21no—
246#95—61no—
247#95—61no—
248#95—61no—
249#95—62no—
250#95—62no—
251#95—61no—
252C23×C3C22×D411yescharacteristic
253C22×D3—21no—
254C22×D3—21no—
255C22×D3—21no—
256C22×D3—21no—
257C22×D3—21no—
258C22×D3—21no—
259C22×D3C22⋊Q812yes—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—22no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—42no—
269C2×C3×C4—41no—
270C2×C3×C4—42no—
271C2×C3×C4#12312yes—
272C2×C3×C4#12312yes—
273C2×C3×C4#13512yes—
274C2×C3×C4#13712yes—
275C2×C3×C4C22×Q811yescharacteristic
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C4C2×#4111yescharacteristic
279C2×C3×C4H(5,𝔽2)11yescharacteristic
280C2×C3×C4H(5,𝔽2)11yescharacteristic
281C2×Dic3—21no—
282C2×Dic3—21no—
283C2×Dic3—22no—
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—21no—
287C2×Dic3—21no—
288C2×Dic3—21no—
289C2×Dic3—21no—
290C2×Dic3—41no—
291C2×Dic3—41no—
292C2×Dic3—41no—
293C2×Dic3—42no—
294C2×Dic3—42no—
295C2×Dic3—41no—
296C2×Dic3—42no—
297C2×Dic3—41no—
298C2×Dic3C22⋊Q812yes—
299C4×D3—22no—
300C4×D3—22no—
301C4×D3—22no—
302C4×D3—22no—
303C4×D3—44no—
304C4×D3—44no—
305Dic6—22no—
306Dic6—22no—
307Dic6—41no—
308Dic6—41no—
309C3×C8—42no—
310C3⋊C8—42no—
311C24—31no—
312C2×Q8—32no—
313C2×Q8—61no—
314C2×Q8—61no—
315C2×Q8—61no—
316C2×Q8—61no—
317C2×Q8—61no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—31no—
322C22×C4—32no—
323C22×C4—32no—
324C22×C4—31no—
325C22×C4—31no—
326C22×C4—61no—
327C22×C4—61no—
328C22×C4—61no—
329C22×C4—61no—
330C22×C4—61no—
331C22×C4—62no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—61no—
336C22×C4—62no—
337C22×C4—121no—
338C22×C4—121no—
339C22×C4D3×D411yescharacteristic
340C22×C4D3×D411yescharacteristic
341C22×C4C23×D311yescharacteristic
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—62no—
346C2×C8—62no—
347C2×C8—62no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—21no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—64no—
356C4⋊C4—61no—
357C4⋊C4—62no—
358C4⋊C4—61no—
359C4⋊C4—62no—
360C4⋊C4—122no—
361C4⋊C4—122no—
362C4⋊C4—122no—
363C4⋊C4—122no—
364C4⋊C4D3×Q812yes—
365C22⋊C4—64no—
366C22⋊C4—124no—
367C22⋊C4—122no—
368C42—42no—
369C42—62no—
370C42—62no—
371C42—61no—
372C42—61no—
373C42—124no—
374C42—122no—
375C42#23512yes—
376C42#23912yes—
377C22×C3C2×C22⋊Q811yescharacteristic
378C22×C3C2×#12311yescharacteristic
379C22×C3#53511yescharacteristic
380C22×C3C2×#13511yescharacteristic
381C22×C3C2×#13711yescharacteristic
382C22×C3#57511yescharacteristic
383C22×C3#57511yescharacteristic
384C2×D3—22no—
385C2×D3—22no—
386C2×D3—22no—
387C2×D3—41no—
388C2×D3—41no—
389C2×D3—41no—
390C2×D3—41no—
391C3×C4—22no—
392C3×C4—42no—
393C3×C4—82no—
394C3×C4—81no—
395C3×C4#48514yes—
396C3×C4#55311yescharacteristic
397C3×C4#55311yescharacteristic
398C3×C4#55411yescharacteristic
399C3×C4#55411yescharacteristic
400Dic3—22no—
401Dic3—41no—
402Dic3—41no—
403Dic3—42no—
404Dic3—81no—
405Dic3—82no—
406C23—32no—
407C23—61no—
408C23—61no—
409C23—61no—
410C23—61no—
411C23—61no—
412C23—61no—
413C23C2×D3×D411yescharacteristic
414Q8—62no—
415Q8—62no—
416Q8—121no—
417Q8—121no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—32no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—62no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—62no—
443C2×C4—124no—
444C2×C4—121no—
445C2×C4—121no—
446C2×C4—124no—
447C2×C4—121no—
448C2×C4—122no—
449C2×C4—121no—
450C2×C4—122no—
451C2×C4—121no—
452C2×C4—122no—
453C2×C4#89412yes—
454C2×C4#89412yes—
455C2×C4#91212yes—
456C2×C4#91812yes—
457C2×C4C2×#23511yescharacteristic
458C2×C4#100411yescharacteristic
459C2×C4C2×D3×Q811yescharacteristic
460C2×C4C2×#23911yescharacteristic
461C2×C4D3×#4111yescharacteristic
462C2×C4#100911yescharacteristic
463C8—42no—
464C8—122no—
465C2×C3#248711yescharacteristic
466C2×C3#307611yescharacteristic
467C2×C3#307811yescharacteristic
468C2×C3C2×#48511yescharacteristic
469C2×C3#309311yescharacteristic
470C2×C3#321211yescharacteristic
471C2×C3#321611yescharacteristic
472D3—42no—
473C22—62no—
474C22—62no—
475C22—62no—
476C22—121no—
477C22—121no—
478C22—121no—
479C22—121no—
480C22C2×#89411yescharacteristic
481C22#560711yescharacteristic
482C22D3×C22⋊Q811yescharacteristic
483C22C2×#91211yescharacteristic
484C22C2×#91811yescharacteristic
485C22#577311yescharacteristic
486C22#577711yescharacteristic
487C4—22no—
488C4—42no—
489C4—62no—
490C4—81no—
491C4—82no—
492C4—121no—
493C4—121no—
494C4—122no—
495C4—242no—
496C4—241no—
497C4#488814yes—
498C4#555811yescharacteristic
499C4#557911yescharacteristic
500C4#569511yescharacteristic
501C4#573011yescharacteristic
502C3#2768211yescharacteristic, complemented, Sylow
503C2—122no—
504C2#7623211yescharacteristic
505C2C2×#488811yescharacteristic
506C2#7921611yescharacteristic
507C2#8033711yescharacteristic
508C2#8048211yescharacteristic
509C2#8096411yescharacteristic
510C2#8110811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1947C2
socleC23×C3C22×D4
prime radicalC23×D3C2×D4
hypercenter#1947D3
hypocenterC3#27682
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
3. C3×#1947C2semidirect
4. #70645C2semidirect
6. #70485C2semidirect
8. #69588C2semidirect
9. #69467C2semidirect
11. #68845C2semidirect
49. #1947D3semidirect
502. C3#27682semidirect

Maximal subgroups

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

subgroupindex
#743182
C4×#8972
C3×#19472
#706452
#706332
#704852
#704752
#695882
#694672
#688792
#688452
#276823