Finite Group Databasebeta

← #10877265

Subgroups — #10877265

Order 768.

882 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#930C211yesmaximal, characteristic
2C2×#4936C211yesmaximal, characteristic, complemented
3C3×#1797C211yesmaximal, characteristic, complemented
4#70480C212yesmaximal, complemented
5#70455C212yesmaximal, complemented
6#69932C211yesmaximal, characteristic
7#69932C211yesmaximal, characteristic
8#69928C211yesmaximal, characteristic, complemented
9#69927C211yesmaximal, characteristic, complemented
10#68680C212yesmaximal, complemented
11#68644C212yesmaximal, complemented
12#27684—31nomaximal, Sylow
13C22×C3×C42C2211yescharacteristic
14C2×#930C2211yescharacteristic
15C2×C4×#67—21no—
16C22×C4⋊Dic3C2211yescharacteristic
17C22×C4⋊Dic3C2211yescharacteristic
18C2×C4×D12C2211yescharacteristic
19C2×C3×C4⋊C8C2212yes—
20C2×C3×C22⋊C8C2211yescharacteristic
21C2×C3×C22⋊C8C2211yescharacteristic
22#5213—21no—
23#5212C2211yescharacteristic
24C2×C3⋊#95—21no—
25C2×C3⋊#95—21no—
26C2×#821C2212yes—
27C2×#821C2212yes—
28C2×#818C2212yes—
29C2×#817C2212yes—
30#4940—21no—
31#4939C2211yescharacteristic
32C4×(C2×D3)⋊C4—21no—
33#4936—21no—
34#4936—21no—
35#4936C2212yes—
36#4936C2212yes—
37C4×C4⋊Dic3C2211yescharacteristic
38#4928C2211yescharacteristic
39C3×#336C2212yes—
40#4551C2212yes—
41#4551C2212yes—
42#4551C2212yes—
43#4551C2212yes—
44C4×#121—31no—
45C2×#382—31no—
46#2101—32no—
47#2092—32no—
48#1949—32no—
49#1874—31no—
50#1874—31no—
51#1853—31no—
52#1852—31no—
53#1804—32no—
54#1797D311yescharacteristic, complemented
55C23×C3×C4D411yescharacteristic
56C23×C3×C4D411yescharacteristic
57C23×C3×C4C2311yescharacteristic
58C22×#67—21no—
59C23×Dic3—21no—
60C23×Dic3—21no—
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C22×C3×C8C2312yes—
64C22×C3×C8C2312yes—
65C2×C3×C42—24no—
66C2×C3×C42D411yescharacteristic
67C2×C3×C42D411yescharacteristic
68C2×C3×C42C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×#217—21no—
71#930—22no—
72#930Q814yes—
73C4×#67—44no—
74C2×(C2×D3)⋊C4—21no—
75C2×(C2×D3)⋊C4—21no—
76C2×(C2×D3)⋊C4—21no—
77C2×(C2×D3)⋊C4—21no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3—42no—
81C2×C4⋊Dic3—42no—
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4⋊Dic3C2311yescharacteristic
84C2×C4⋊Dic3C2312yes—
85C2×C4⋊Dic3C2312yes—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×Dic3⋊C4—21no—
88C2×C4×Dic3—21no—
89C4×D12—24no—
90C3×C4⋊C8—24no—
91C3×C22⋊C8—22no—
92C3×C22⋊C8—22no—
93C3⋊#95—21no—
94C3⋊#95—21no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98C3⋊#95—22no—
99C3⋊#95—21no—
100#821—42no—
101#821—42no—
102#818—42no—
103#817—42no—
104C2×#121—31no—
105C2×C4×D4—31no—
106C2×C4×D4—61no—
107C22×C4⋊C4—31no—
108C22×C4⋊C4—31no—
109C22×C42C2×D311yescharacteristic
110C2×#107—32no—
111C2×#106—32no—
112C2×C4⋊C8C2×D312yes—
113C2×#102—32no—
114C2×#102—32no—
115C2×C22⋊C8C2×D311yescharacteristic
116C2×C22⋊C8C2×D311yescharacteristic
117#390—31no—
118#389—31no—
119#388—61no—
120#386—61no—
121#382—32no—
122#382—32no—
123#382—61no—
124#382—61no—
125#380—31no—
126C4×C4⋊C4—31no—
127C4×C22⋊C4—61no—
128C2×#95—61no—
129C2×#95—61no—
130#340—32no—
131#340—32no—
132#340—32no—
133#340—32no—
134#336C2×D312yes—
135C24×C3C2×D411yescharacteristic
136C23×D3—21no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—41no—
144C22×C3×C4—41no—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×Q812yes—
151C22×C3×C4#4112yes—
152C22×C3×C4C2411yescharacteristic
153C2×#67—24no—
154C2×#67—42no—
155C2×#67—42no—
156C22×Dic3—21no—
157C22×Dic3—22no—
158C22×Dic3—22no—
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—42no—
162C22×Dic3—42no—
163C22×Dic3—42no—
164C22×Dic3—42no—
165C22×Dic3—42no—
166C22×Dic3—42no—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12—21no—
170C2×D12C2×Q812yes—
171C2×C4×D3—42no—
172C2×C4×D3—42no—
173C2×C4×D3—42no—
174C2×C3×C8—22no—
175C2×C3×C8—22no—
176C2×C3×C8—22no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C3×C42—22no—
181C3×C42—22no—
182C3×C42—22no—
183C3×C42#4114yes—
184#217—42no—
185(C2×D3)⋊C4—24no—
186(C2×D3)⋊C4—42no—
187(C2×D3)⋊C4—42no—
188(C2×D3)⋊C4—42no—
189C4⋊Dic3—21no—
190C4⋊Dic3—22no—
191C4⋊Dic3—22no—
192C4⋊Dic3—22no—
193C4⋊Dic3—22no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3—42no—
197C4⋊Dic3—42no—
198C4⋊Dic3C2×Q812yes—
199Dic3⋊C4—42no—
200C4×Dic3—42no—
201C22×D4—31no—
202C22×D4—61no—
203C23×C4—61no—
204C23×C4—61no—
205C23×C4—61no—
206C23×C4D1211yescharacteristic
207C23×C4D1211yescharacteristic
208C23×C4C22×D311yescharacteristic
209C22×C8C22×D312yes—
210C22×C8C22×D312yes—
211#121—34no—
212#121—62no—
213C4×D4—64no—
214C4×D4—124no—
215C2×C4⋊C4—31no—
216C2×C4⋊C4—32no—
217C2×C4⋊C4—32no—
218C2×C4⋊C4—31no—
219C2×C4⋊C4—31no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4—61no—
222C2×C4⋊C4—61no—
223C2×C4⋊C4—122no—
224C2×C4⋊C4—122no—
225C2×C22⋊C4—61no—
226C2×C22⋊C4—61no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—61no—
230C2×C42—24no—
231C2×C42—61no—
232C2×C42D1211yescharacteristic
233C2×C42D1211yescharacteristic
234C2×C42C22×D311yescharacteristic
235C2×C42C22×D311yescharacteristic
236#107—122no—
237#106—122no—
238C4⋊C8—24no—
239#102—122no—
240#102—122no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243#95—62no—
244#95—62no—
245#95—62no—
246#95—62no—
247#95—61no—
248#95—61no—
249#95—61no—
250C23×C3—21no—
251C23×C3—21no—
252C23×C3—21no—
253C23×C3—21no—
254C23×C3C22⋊Q812yes—
255C23×C3#13512yes—
256C23×C3AGL(1,ℤ/8)12yes—
257C23×C3C22×D411yescharacteristic
258C22×D3—22no—
259C22×D3—41no—
260C22×D3—41no—
261C22×D3—41no—
262C22×D3—41no—
263C22×D3—41no—
264C22×D3—41no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—24no—
272C2×C3×C4—24no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—44no—
280C2×C3×C4—44no—
281C2×C3×C4C22⋊Q812yes—
282C2×C3×C4#12312yes—
283C2×C3×C4#12312yes—
284C2×C3×C4C22×Q811yescharacteristic
285C2×C3×C4C2×#4111yescharacteristic
286C2×C3×C4C2×#4112yes—
287C2×C3×C4H(5,𝔽2)12yes—
288#67—48no—
289C2×Dic3—22no—
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×Dic3—41no—
299C2×Dic3—42no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—84no—
305C2×Dic3—84no—
306D12—24no—
307D12—42no—
308C4×D3—84no—
309C3×C8—42no—
310C3×C8—42no—
311C24—61no—
312C24C2×D1211yescharacteristic
313C2×D4—32no—
314C2×D4—64no—
315C2×D4—62no—
316C2×D4—61no—
317C2×D4—62no—
318C2×D4—122no—
319C2×D4—122no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—61no—
329C22×C4—62no—
330C22×C4—62no—
331C22×C4—61no—
332C22×C4—61no—
333C22×C4—122no—
334C22×C4—122no—
335C22×C4—122no—
336C22×C4—122no—
337C22×C4—122no—
338C22×C4—122no—
339C22×C4—122no—
340C22×C4—122no—
341C22×C4—122no—
342C22×C4C2×D1211yescharacteristic
343C22×C4C2×D1211yescharacteristic
344C22×C4C2×D1211yescharacteristic
345C22×C4C2×D1211yescharacteristic
346C22×C4C2×D1211yescharacteristic
347C22×C4#23712yes—
348C22×C4D3×Q812yes—
349C22×C4C23×D311yescharacteristic
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C4⋊C4—32no—
357C4⋊C4—61no—
358C4⋊C4—62no—
359C4⋊C4—62no—
360C4⋊C4—62no—
361C4⋊C4—62no—
362C4⋊C4—62no—
363C4⋊C4—62no—
364C4⋊C4—122no—
365C4⋊C4—122no—
366C4⋊C4—122no—
367C22⋊C4—64no—
368C22⋊C4—122no—
369C22⋊C4—122no—
370C22⋊C4—122no—
371C22⋊C4—122no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—122no—
376C42#23714yes—
377C22×C3—22no—
378C22×C3—22no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—22no—
383C22×C3#47814yes—
384C22×C3C2×C22⋊Q811yescharacteristic
385C22×C3C2×#12311yescharacteristic
386C22×C3#53511yescharacteristic
387C22×C3C2×#13511yescharacteristic
388C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
389C22×C3#57511yescharacteristic
390C22×C3#57511yescharacteristic
391C2×D3—42no—
392C2×D3—42no—
393C2×D3—42no—
394C2×D3—81no—
395C2×D3—81no—
396C2×D3—81no—
397C2×D3—81no—
398C3×C4—22no—
399C3×C4—22no—
400C3×C4—22no—
401C3×C4—41no—
402C3×C4—41no—
403C3×C4#55312yes—
404C3×C4#55412yes—
405Dic3—42no—
406Dic3—81no—
407Dic3—82no—
408Dic3—81no—
409Dic3—82no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—62no—
415C23—121no—
416C23—121no—
417C23—121no—
418C23—121no—
419C23—121no—
420C23—121no—
421C23#89712yes—
422C23#90412yes—
423C23#90812yes—
424C23C22×D1211yescharacteristic
425D4—64no—
426D4—128no—
427D4—122no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—62no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4—122no—
448C2×C4—121no—
449C2×C4—121no—
450C2×C4—122no—
451C2×C4—121no—
452C2×C4—121no—
453C2×C4—121no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—121no—
459C2×C4—244no—
460C2×C4—244no—
461C2×C4—244no—
462C2×C4#88612yes—
463C2×C4#88612yes—
464C2×C4#89712yes—
465C2×C4C2×#23712yes—
466C2×C4C2×#23711yescharacteristic
467C2×C4C2×D3×Q811yescharacteristic
468C2×C4#100912yes—
469C8—42no—
470C8—42no—
471C2×C3—24no—
472C2×C3#248711yescharacteristic
473C2×C3C2×#47811yescharacteristic
474C2×C3#308011yescharacteristic
475C2×C3#309111yescharacteristic
476C2×C3#309411yescharacteristic
477C2×C3#321111yescharacteristic
478C2×C3#321311yescharacteristic
479D3—82no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—122no—
487C22—122no—
488C22—122no—
489C22—241no—
490C22—241no—
491C22—241no—
492C22—241no—
493C22#484714yes—
494C22C2×#88611yescharacteristic
495C22C2×#89711yescharacteristic
496C22#552511yescharacteristic
497C22C2×#90411yescharacteristic
498C22C2×#90811yescharacteristic
499C22#574811yescharacteristic
500C22#574811yescharacteristic
501C4—22no—
502C4—22no—
503C4—22no—
504C4—41no—
505C4—41no—
506C4—122no—
507C4—241no—
508C4—242no—
509C4—242no—
510C4—241no—
511C4#561812yes—
512C4#563512yes—
513C3#2768411yescharacteristic, complemented, Sylow
514C2—24no—
515C2—242no—
516C2#7402811yescharacteristic
517C2#7859111yescharacteristic
518C2#7862411yescharacteristic
519C2C2×#484711yescharacteristic
520C2#7906111yescharacteristic
521C2#7930711yescharacteristic
522C2#7934911yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1797C2
socleC23×C3C22×D4
prime radicalC2×#930C22
hypercenter#1797D3
hypocenterC3#27684
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. C2×#4936C2semidirect
3. C3×#1797C2semidirect
4. #70480C2semidirect
5. #70455C2semidirect
8. #69928C2semidirect
9. #69927C2semidirect
10. #68680C2semidirect
11. #68644C2semidirect
54. #1797D3semidirect
513. C3#27684semidirect

Maximal subgroups

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

subgroupindex
C4×#9302
C2×#49362
C3×#17972
#704802
#704552
#699322
#699282
#699272
#686802
#686442
#276843