Finite Group Databasebeta

← #10877188

Subgroups — #10877188

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#76291C211yesmaximal, characteristic
2D3×#384C211yesmaximal, characteristic, complemented
3C3×#2093C211yesmaximal, characteristic, complemented
4#70632C212yesmaximal, complemented
5#70615C212yesmaximal, complemented
6#70200C212yesmaximal
7#70153C212yesmaximal, complemented
8#69655C211yesmaximal, characteristic, complemented
9#69571C211yesmaximal, characteristic, complemented
10#68662C211yesmaximal, characteristic, complemented
11#68642C211yesmaximal, characteristic, complemented
12#27677—31nomaximal, Sylow
13C2×#938C2211yescharacteristic
14C2×D4×Dic3C2211yescharacteristic
15C2×D3×C4⋊C4—21no—
16C2×D3×C4⋊C4—21no—
17C2×D3×C4⋊C4C2212yes—
18C2×#67⋊C4—21no—
19C2×C42×D3C2211yescharacteristic
20C2×C3×#107C2212yes—
21C2×C3×#102C2212yes—
22C3×#390C2211yescharacteristic
23C3×#384C2211yescharacteristic
24C2×#832C2212yes—
25C2×(C2×D3)⋊C8C2212yes—
26C2×Dic3⋊C8C2212yes—
27#4987C2211yescharacteristic
28#4975—21no—
29#4975—21no—
30#4975C2212yes—
31Dic3×C4⋊C4C2211yescharacteristic
32C2×#807C2212yes—
33#4956—21no—
34#4945—21no—
35Dic3×C22⋊C4—21no—
36#4935C2211yescharacteristic
37#4928C2211yescharacteristic
38C3×#336C2211yescharacteristic
39#4551C2212yes—
40#4551C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44C4×#121—31no—
45C2×#384—31no—
46#2099—31no—
47#2099—31no—
48#2093—31no—
49#2093D311yescharacteristic, complemented
50#1947—32no—
51#1874—32no—
52#1852—32no—
53#1802—32no—
54#1797—31no—
55C22×C3×D4C2311yescharacteristic
56C22×#67—21no—
57C23×Dic3—21no—
58C22×C4×D3—22no—
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⋊C4C2311yescharacteristic
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C42C2311yescharacteristic
70C2×#217—21no—
71C2×#217—21no—
72#938—22no—
73#938Q814yes—
74D4×Dic3—24no—
75C2×(C2×D3)⋊C4—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4⋊Dic3C2312yes—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×Dic3⋊C4—21no—
82C2×Dic3⋊C4—22no—
83C2×Dic3⋊C4—22no—
84C2×Dic3⋊C4—22no—
85C2×C4×Dic3—21no—
86C2×C4×Dic3D411yescharacteristic
87C2×C4×Dic3D411yescharacteristic
88C2×C4×Dic3C2311yescharacteristic
89C22×C3⋊C8C2312yes—
90D3×C4⋊C4—42no—
91D3×C4⋊C4—42no—
92D3×C4⋊C4—44no—
93D3×C4⋊C4—44no—
94#67⋊C4—44no—
95C42×D3—24no—
96C3×#107—42no—
97C3×#102—42no—
98#832—42no—
99C3⋊#95—21no—
100C3⋊#95—21no—
101C3⋊#95—21no—
102(C2×D3)⋊C8—24no—
103Dic3⋊C8—24no—
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×#107C2×D312yes—
114C2×C4⋊C8—32no—
115C2×#102—32no—
116C2×#102C2×D312yes—
117C2×C22⋊C8—32no—
118#390C2×D311yescharacteristic
119#389—31no—
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#340—32no—
133#340—32no—
134#336—31no—
135#336C2×D311yescharacteristic
136C24×C3—21no—
137C23×D3C2×D411yescharacteristic
138C2×C3×D4—22no—
139C2×C3×D4—21no—
140C2×C3×D4—22no—
141C2×C3×D4C2×Q812yes—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2411yescharacteristic
147C2×#67—24no—
148C2×#67—42no—
149C2×#67—42no—
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3—22no—
153C22×Dic3—42no—
154C22×Dic3—42no—
155C22×Dic3—42no—
156C22×Dic3C2×D411yescharacteristic
157C22×Dic3C2×D411yescharacteristic
158C22×Dic3C2×D411yescharacteristic
159C2×C4×D3—22no—
160C2×C4×D3—22no—
161C2×C4×D3—22no—
162C2×C4×D3—22no—
163C2×C4×D3—22no—
164C2×C4×D3—22no—
165C2×C4×D3—22no—
166C2×C4×D3—44no—
167C2×C4×D3—44no—
168C2×C4×D3—44no—
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—22no—
176C3×C4⋊C4—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—42no—
179C3×C4⋊C4—42no—
180C3×C22⋊C4—42no—
181C3×C22⋊C4—42no—
182C3×C42—22no—
183#217—24no—
184#217—42no—
185(C2×D3)⋊C4—42no—
186C4⋊Dic3—21no—
187C4⋊Dic3—22no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3—42no—
191C4⋊Dic3—42no—
192C4⋊Dic3C2×Q812yes—
193Dic3⋊C4—44no—
194Dic3⋊C4—44no—
195Dic3⋊C4—42no—
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—61no—
205C22×D4C22×D311yescharacteristic
206C23×C4—31no—
207C23×C4—31no—
208C23×C4—31no—
209C23×C4—62no—
210C23×C4—61no—
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—31no—
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—61no—
228C2×C4⋊C4—122no—
229C2×C4⋊C4—122no—
230C2×C4⋊C4—124no—
231C2×C4⋊C4—124no—
232C2×C4⋊C4C22×D312yes—
233C2×C4⋊C4C22×D311yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
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—42no—
246#107—122no—
247C4⋊C8—64no—
248#102—42no—
249#102—122no—
250C22⋊C8—64no—
251#95—61no—
252#95—61no—
253#95—61no—
254C23×C3—22no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3—41no—
259C23×C3—41no—
260C23×C3—41no—
261C23×C3C22×D411yescharacteristic
262C22×D3—21no—
263C22×D3—21no—
264C22×D3—21no—
265C22×D3—21no—
266C22×D3C22⋊Q812yes—
267C22×D3C2×D812yes—
268C22×D3#13712yes—
269C3×D4—24no—
270C3×D4—42no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—41no—
281C2×C3×C4C2×D812yes—
282C2×C3×C4AGL(1,ℤ/8)12yes—
283C2×C3×C4C22×Q811yescharacteristic
284C2×C3×C4C2×#4111yescharacteristic
285C2×C3×C4C2×#4112yes—
286C2×C3×C4H(5,𝔽2)12yes—
287#67—48no—
288C2×Dic3—21no—
289C2×Dic3—21no—
290C2×Dic3—21no—
291C2×Dic3—21no—
292C2×Dic3—22no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—42no—
297C2×Dic3—42no—
298C2×Dic3—41no—
299C2×Dic3—42no—
300C2×Dic3—41no—
301C2×Dic3—84no—
302C2×Dic3C22⋊Q812yes—
303C2×Dic3#12312yes—
304C2×Dic3#12312yes—
305C2×Dic3C2×D812yes—
306C2×Dic3#13712yes—
307C4×D3—28no—
308C4×D3—24no—
309C4×D3—24no—
310C4×D3—28no—
311C4×D3—88no—
312C3×C8—42no—
313C3⋊C8—42no—
314C24—21no—
315C24—31no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—21no—
319C2×D4—64no—
320C2×D4—122no—
321C2×D4—122no—
322C2×D4D3×Q812yes—
323C22×C4—21no—
324C22×C4—22no—
325C22×C4—31no—
326C22×C4—31no—
327C22×C4—31no—
328C22×C4—34no—
329C22×C4—32no—
330C22×C4—32no—
331C22×C4—62no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—62no—
338C22×C4—61no—
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—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—32no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—61no—
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⋊C4—122no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—64no—
375C22⋊C4—122no—
376C22⋊C4—122no—
377C42—22no—
378C42—34no—
379C42—62no—
380C42—62no—
381C42—122no—
382C22×C3—42no—
383C22×C3—42no—
384C22×C3—42no—
385C22×C3—81no—
386C22×C3—81no—
387C22×C3—81no—
388C22×C3—81no—
389C22×C3C2×C22⋊Q811yescharacteristic
390C22×C3C2×#12311yescharacteristic
391C22×C3#53511yescharacteristic
392C22×C3C22×D811yescharacteristic
393C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
394C22×C3C2×#13711yescharacteristic
395C22×C3C2×#13711yescharacteristic
396C2×D3—22no—
397C2×D3—22no—
398C2×D3—22no—
399C2×D3—22no—
400C2×D3—22no—
401C2×D3—22no—
402C2×D3#47414yes—
403C3×C4—22no—
404C3×C4—22no—
405C3×C4—82no—
406C3×C4—81no—
407C3×C4#55312yes—
408C3×C4#55412yes—
409Dic3—22no—
410Dic3—22no—
411Dic3—42no—
412Dic3—81no—
413Dic3—82no—
414Dic3#48014yes—
415C23—22no—
416C23—32no—
417C23—32no—
418C23—32no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—61no—
426C23—61no—
427C23—61no—
428C23—61no—
429C23C2×D3×D411yescharacteristic
430D4—24no—
431D4—42no—
432D4—128no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—32no—
438C2×C4—32no—
439C2×C4—32no—
440C2×C4—32no—
441C2×C4—32no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—68no—
449C2×C4—68no—
450C2×C4—61no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—64no—
455C2×C4—64no—
456C2×C4—62no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—122no—
461C2×C4—121no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—121no—
465C2×C4—244no—
466C2×C4—248no—
467C2×C4D3×D812yes—
468C2×C4#91112yes—
469C2×C4#100412yes—
470C2×C4C2×D3×Q811yescharacteristic
471C2×C4D3×#4112yes—
472C2×C4D3×#4111yescharacteristic
473C8—42no—
474C8—122no—
475C2×C3—82no—
476C2×C3#248711yescharacteristic
477C2×C3C2×#47411yescharacteristic
478C2×C3#307911yescharacteristic
479C2×C3C2×#48011yescharacteristic
480C2×C3#309211yescharacteristic
481C2×C3#320511yescharacteristic
482C2×C3#321811yescharacteristic
483D3—24no—
484C22—34no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—62no—
489C22—62no—
490C22—62no—
491C22—62no—
492C22—62no—
493C22—62no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22D3×C22⋊Q811yescharacteristic
499C22D3×#12311yescharacteristic
500C22#571511yescharacteristic
501C22C2×D3×D811yescharacteristic
502C22C2×#91111yescharacteristic
503C22D3×#13711yescharacteristic
504C22D3×#13711yescharacteristic
505C4—22no—
506C4—22no—
507C4—34no—
508C4—62no—
509C4—62no—
510C4—82no—
511C4—81no—
512C4—122no—
513C4—242no—
514C4—241no—
515C4#563612yes—
516C4#566112yes—
517C3#2767711yescharacteristic, complemented, Sylow
518C2—64no—
519C2—82no—
520C2#7802911yescharacteristic
521C2D3×#47411yescharacteristic
522C2#8096611yescharacteristic
523C2D3×#48011yescharacteristic
524C2#8128211yescharacteristic
525C2#8178511yescharacteristic
526C2#8187611yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2093C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#2093D3
hypocenterC3#27677
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2093

Complements

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

NH ≅ G/Ntype
2. D3×#384C2semidirect
3. C3×#2093C2semidirect
4. #70632C2semidirect
5. #70615C2semidirect
7. #70153C2semidirect
8. #69655C2semidirect
9. #69571C2semidirect
10. #68662C2semidirect
11. #68642C2semidirect
49. #2093D3semidirect
517. C3#27677semidirect

Maximal subgroups

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

subgroupindex
#762912
D3×#3842
C3×#20932
#706322
#706152
#702002
#701532
#696552
#695712
#686622
#686422
#276773