Finite Group Databasebeta

← #10876160

Subgroups — #10876160

Order 768.

916 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76290C211yesmaximal, characteristic
2#74318C211yesmaximal, characteristic, complemented
3C3×#2071C211yesmaximal, characteristic, complemented
4#70199C212yesmaximal
5#70156C212yesmaximal, complemented
6#69655C211yesmaximal, characteristic, complemented
7#69589C211yesmaximal, characteristic, complemented
8#68876C212yesmaximal, complemented
9#68794C212yesmaximal, complemented
10#68692C211yesmaximal, characteristic, complemented
11#68664C211yesmaximal, characteristic, complemented
12#27587—31nomaximal, Sylow
13C22×#237C2211yescharacteristic
14C2×#892—21no—
15C2×#892C2211yescharacteristic
16C2×C42×D3C2211yescharacteristic
17C2×C3×#102C2212yes—
18C3×#390C2211yescharacteristic
19C3×#384C2211yescharacteristic
20#5221C2211yescharacteristic
21C2×#832C2212yes—
22C2×(C2×D3)⋊C8C2212yes—
23C2×#817C2212yes—
24C2×#816C2212yes—
25#4987—21no—
26#4987C2212yes—
27#4977—21no—
28Dic3×C4⋊C4—21no—
29Dic3×C4⋊C4C2212yes—
30#4973C2211yescharacteristic
31C2×#810C2212yes—
32C2×#807C2212yes—
33#4956—21no—
34#4956—21no—
35Dic3×C22⋊C4—21no—
36Dic3×C22⋊C4—21no—
37#4935C2211yescharacteristic
38#4929C2211yescharacteristic
39C3×#340C2212yes—
40C3×#336C2211yescharacteristic
41#4550C2212yes—
42#4529C2212yes—
43#4524C2211yescharacteristic
44#2297—31no—
45#2285—31no—
46#2096—31no—
47#2093—31no—
48#2072—31no—
49#2071D311yescharacteristic, complemented
50#1928—32no—
51#1926—32no—
52#1854—32no—
53#1817—31no—
54#1788—32no—
55C22×C3×D4C2311yescharacteristic
56C22×#67—21no—
57C23×Dic3—21no—
58C2×#237—22no—
59C2×#237—24no—
60C2×#237—22no—
61C2×#237—42no—
62C22×C4×D3D411yescharacteristic
63C22×C4×D3D411yescharacteristic
64C22×C4×D3C2311yescharacteristic
65C22×Dic6C2311yescharacteristic
66C22×C3×C8C2312yes—
67C2×C3×C4⋊C4—21no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C22⋊C4—21no—
72C2×C3×C22⋊C4—21no—
73C2×C3×C42C2311yescharacteristic
74C2×#217—21no—
75C2×#217—21no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—22no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2312yes—
82C2×C4×Dic3—21no—
83C2×C4×Dic3—22no—
84C2×C4×Dic3D411yescharacteristic
85C2×C4×Dic3D411yescharacteristic
86C2×C4×Dic3C2311yescharacteristic
87C22×C3⋊C8C2312yes—
88#892—42no—
89#892—44no—
90C42×D3—24no—
91C3×#102—24no—
92#832—42no—
93C3⋊#95—21no—
94C3⋊#95—21no—
95C3⋊#95—21no—
96C3⋊#95—21no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99(C2×D3)⋊C8—24no—
100#817—42no—
101#816—42no—
102#810—24no—
103#807—42no—
104C22×#41—31no—
105C2×#117—31no—
106C2×#117—61no—
107C22×C42—31no—
108C2×#107—32no—
109C2×#106—32no—
110C2×#103—32no—
111C2×#103—32no—
112C2×#102—32no—
113C2×#102C2×D312yes—
114C2×C22⋊C8—32no—
115#391—31no—
116#391—31no—
117#390—31no—
118#390C2×D311yescharacteristic
119#386—61no—
120#386—61no—
121#384—31no—
122#384C2×D311yescharacteristic
123#382—61no—
124#380—32no—
125#380—61no—
126C4×C4⋊C4—32no—
127C4×C4⋊C4—61no—
128C4×C22⋊C4—61no—
129C4×C22⋊C4—61no—
130#340—32no—
131#340C2×D312yes—
132#336—32no—
133#336—31no—
134#336C2×D311yescharacteristic
135C24×C3—21no—
136C23×D3C2×D411yescharacteristic
137C2×C3×D4—22no—
138C2×C3×D4—21no—
139C2×C3×D4—22no—
140C2×C3×D4#4112yes—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2411yescharacteristic
146C2×#67—24no—
147C2×#67—42no—
148C2×#67—42no—
149C22×Dic3—24no—
150C22×Dic3—21no—
151C22×Dic3—22no—
152C22×Dic3—21no—
153C22×Dic3—42no—
154C22×Dic3—42no—
155C22×Dic3C2×D411yescharacteristic
156C22×Dic3C2×D411yescharacteristic
157C22×Dic3C2×D411yescharacteristic
158#237—48no—
159#237—48no—
160C2×C4×D3—22no—
161C2×C4×D3—22no—
162C2×C4×D3—21no—
163C2×C4×D3—21no—
164C2×C4×D3—22no—
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3—22no—
168C2×C4×D3—41no—
169C2×C4×D3#4114yes—
170C2×Dic6—22no—
171C2×Dic6—22no—
172C2×Dic6—21no—
173C2×Dic6#4112yes—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C3×C4⋊C4—22no—
178C3×C4⋊C4—22no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—42no—
181C3×C4⋊C4—42no—
182C3×C22⋊C4—42no—
183C3×C22⋊C4—42no—
184C3×C42—22no—
185#217—42no—
186#217—42no—
187(C2×D3)⋊C4—42no—
188(C2×D3)⋊C4—44no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3—42no—
193C4⋊Dic3—42no—
194C4×Dic3—22no—
195C4×Dic3—22no—
196C4×Dic3—22no—
197C4×Dic3—42no—
198C4×Dic3—44no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—22no—
202C2×#41—64no—
203C2×#41—62no—
204C2×#41—62no—
205C2×#41—122no—
206C22×Q8—31no—
207C22×D4—61no—
208C22×D4C22×D311yescharacteristic
209C23×C4—31no—
210C23×C4—31no—
211C23×C4—31no—
212C23×C4—61no—
213C22×C8—32no—
214C22×C8C22×D312yes—
215#117—122no—
216#117—124no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—32no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4C22×D311yescharacteristic
224C2×C4⋊C4C22×D312yes—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—62no—
229C2×C22⋊C4—61no—
230C2×C22⋊C4—61no—
231C2×C42—31no—
232C2×C42—31no—
233C2×C42—31no—
234C2×C42—64no—
235C2×C42—61no—
236C2×C42—62no—
237C2×C42C22×D311yescharacteristic
238#107—122no—
239#106—122no—
240#103—64no—
241#103—122no—
242#102—24no—
243#102—122no—
244C22⋊C8—64no—
245#95—61no—
246#95—61no—
247#95—62no—
248#95—62no—
249#95—61no—
250#95—61no—
251C23×C3—22no—
252C23×C3—41no—
253C23×C3—41no—
254C23×C3—41no—
255C23×C3—41no—
256C23×C3—41no—
257C23×C3—41no—
258C23×C3C22×D411yescharacteristic
259C22×D3—21no—
260C22×D3—21no—
261C22×D3—21no—
262C22×D3—21no—
263C22×D3—21no—
264C22×D3—21no—
265C22×D3#12312yes—
266C3×D4—24no—
267C3×D4—42no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—42no—
276C2×C3×C4—41no—
277C2×C3×C4—42no—
278C2×C3×C4#13512yes—
279C2×C3×C4AGL(1,ℤ/8)12yes—
280C2×C3×C4C2×#4111yescharacteristic
281C2×C3×C4C2×#4112yes—
282C2×C3×C4C2×#4111yescharacteristic
283C2×C3×C4H(5,𝔽2)12yes—
284#67—48no—
285C2×Dic3—21no—
286C2×Dic3—21no—
287C2×Dic3—21no—
288C2×Dic3—21no—
289C2×Dic3—21no—
290C2×Dic3—22no—
291C2×Dic3—21no—
292C2×Dic3—21no—
293C2×Dic3—48no—
294C2×Dic3—41no—
295C2×Dic3—41no—
296C2×Dic3—41no—
297C2×Dic3—42no—
298C2×Dic3—42no—
299C2×Dic3—42no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3#12312yes—
303C2×Dic3#13712yes—
304C4×D3—28no—
305C4×D3—24no—
306C4×D3—24no—
307C4×D3—44no—
308Dic6—24no—
309Dic6—42no—
310C3×C8—42no—
311C3⋊C8—42no—
312C24—21no—
313C24—31no—
314#41—128no—
315#41—128no—
316C2×Q8—32no—
317C2×Q8—61no—
318C2×Q8—62no—
319C2×Q8—62no—
320C2×D4—22no—
321C2×D4—21no—
322C2×D4—22no—
323C2×D4—64no—
324C2×D4—122no—
325C2×D4—122no—
326C2×D4#23712yes—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—31no—
330C22×C4—31no—
331C22×C4—34no—
332C22×C4—31no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—62no—
337C22×C4—61no—
338C22×C4—61no—
339C22×C4—62no—
340C22×C4—64no—
341C22×C4—61no—
342C22×C4—62no—
343C22×C4—61no—
344C22×C4—62no—
345C22×C4—121no—
346C22×C4—122no—
347C22×C4—122no—
348C22×C4D3×D411yescharacteristic
349C22×C4D3×D411yescharacteristic
350C22×C4C23×D311yescharacteristic
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—62no—
355C2×C8—62no—
356C2×C8—62no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—62no—
363C4⋊C4—62no—
364C4⋊C4—62no—
365C4⋊C4—122no—
366C4⋊C4—122no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—122no—
370C22⋊C4—122no—
371C22⋊C4—124no—
372C22⋊C4—122no—
373C42—22no—
374C42—62no—
375C42—62no—
376C42—62no—
377C42—122no—
378C42—124no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—42no—
382C22×C3—81no—
383C22×C3—81no—
384C22×C3—81no—
385C22×C3—81no—
386C22×C3C2×#12311yescharacteristic
387C22×C3#52511yescharacteristic
388C22×C3#53511yescharacteristic
389C22×C3C2×#13511yescharacteristic
390C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
391C22×C3C2×#13711yescharacteristic
392C22×C3#57511yescharacteristic
393C2×D3—22no—
394C2×D3—22no—
395C2×D3—22no—
396C2×D3—41no—
397C2×D3—41no—
398C2×D3—41no—
399C2×D3—41no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—82no—
403C3×C4—81no—
404C3×C4#55112yes—
405C3×C4#55112yes—
406Dic3—22no—
407Dic3—22no—
408Dic3—41no—
409Dic3—42no—
410Dic3—82no—
411Dic3—81no—
412C23—22no—
413C23—32no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—61no—
421C23—61no—
422C23—61no—
423C23—61no—
424C23—61no—
425C23—61no—
426C23C2×D3×D411yescharacteristic
427Q8—64no—
428Q8—122no—
429D4—24no—
430D4—42no—
431D4—128no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—32no—
437C2×C4—32no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—68no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—64no—
450C2×C4—64no—
451C2×C4—61no—
452C2×C4—61no—
453C2×C4—61no—
454C2×C4—62no—
455C2×C4—124no—
456C2×C4—128no—
457C2×C4—121no—
458C2×C4—121no—
459C2×C4—121no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4#91112yes—
466C2×C4#91612yes—
467C2×C4C2×#23711yescharacteristic
468C2×C4D3×#4112yes—
469C2×C4D3×#4111yescharacteristic
470C2×C4#100912yes—
471C8—42no—
472C8—122no—
473C2×C3—82no—
474C2×C3#248211yescharacteristic
475C2×C3#309111yescharacteristic
476C2×C3#309211yescharacteristic
477C2×C3#310711yescharacteristic
478C2×C3#311511yescharacteristic
479C2×C3#320711yescharacteristic
480C2×C3#321611yescharacteristic
481D3—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—62no—
486C22—62no—
487C22—62no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22—121no—
493C22—121no—
494C22—121no—
495C22—121no—
496C22#560611yescharacteristic
497C22D3×#12311yescharacteristic
498C22#567411yescharacteristic
499C22C2×#91111yescharacteristic
500C22C2×#91611yescharacteristic
501C22#577311yescharacteristic
502C22D3×#13711yescharacteristic
503C4—22no—
504C4—22no—
505C4—62no—
506C4—62no—
507C4—81no—
508C4—82no—
509C4—122no—
510C4—121no—
511C4—241no—
512C4—242no—
513C4#565812yes—
514C4#570512yes—
515C3#2758711yescharacteristic, complemented, Sylow
516C2—82no—
517C2—122no—
518C2#7767611yescharacteristic
519C2#8031511yescharacteristic
520C2#8040911yescharacteristic
521C2#8128211yescharacteristic
522C2#8132411yescharacteristic
523C2#8156711yescharacteristic
524C2#8168711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2071C2
socleC23×C3C22×D4
prime radicalC22×#237C22
hypercenter#2071D3
hypocenterC3#27587
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2071

Complements

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

NH ≅ G/Ntype
2. #74318C2semidirect
3. C3×#2071C2semidirect
5. #70156C2semidirect
6. #69655C2semidirect
7. #69589C2semidirect
8. #68876C2semidirect
9. #68794C2semidirect
10. #68692C2semidirect
11. #68664C2semidirect
49. #2071D3semidirect
515. C3#27587semidirect

Maximal subgroups

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

subgroupindex
#762902
#743182
C3×#20712
#701992
#701562
#696552
#695892
#688762
#687942
#686922
#686642
#275873