Finite Group Databasebeta

← #10844795

Subgroups — #10844795

Order 768.

934 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2548C211yesmaximal, characteristic, complemented
2C2×#5072C211yesmaximal, characteristic
3#82500C211yesmaximal, characteristic, complemented
4C2×#5024C211yesmaximal, characteristic, complemented
5#71764C211yesmaximal, characteristic, complemented
6#69652C212yesmaximal, complemented
7#69650C212yesmaximal, complemented
8#69648C212yesmaximal, complemented
9#69645C212yesmaximal, complemented
10C4×#809C211yesmaximal, characteristic, complemented
11#69467C211yesmaximal, characteristic, complemented
12#25300—31nomaximal, Sylow
13C2×C3×#127C2211yescharacteristic
14C2×C3×#124—21no—
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#946—21no—
17C22×#213C2212yes—
18C2×C4×D12C2211yescharacteristic
19C3×#394—22no—
20C3×#390—21no—
21C3×#390C2212yes—
22C3×#390C2212yes—
23C3×C4×C4⋊C4C2211yescharacteristic
24C2×#832C2212yes—
25C2×#832C2212yes—
26C2×#832C2212yes—
27#5072—24no—
28#5024—24no—
29#4994—22no—
30#4987—21no—
31C2×#809C2211yescharacteristic
32C2×#809C2211yescharacteristic
33C2×#808C2212yes—
34#4939C2212yes—
35#4935C2211yescharacteristic
36#4928C2212yes—
37C2×C4⋊(C3⋊C8)C2212yes—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39C2×#803C2212yes—
40C2×C4×C3⋊C8C2211yescharacteristic
41#4529C2211yescharacteristic
42#4529C2212yes—
43#4529C2211yescharacteristic
44C2×#459—31no—
45C2×#446—31no—
46#2548D311yescharacteristic, complemented
47#2538—31no—
48#2065—32no—
49#2054—32no—
50#1972—32no—
51#1957—31no—
52#1948—31no—
53#1936—32no—
54C4×#102—31no—
55C22×C3×Q8D411yescharacteristic
56C22×C3×D4—21no—
57C22×C3×D4—21no—
58C22×C3×D4C2312yes—
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C3×#127—22no—
62C3×#127—22no—
63C3×#124—44no—
64C3×C4×Q8—22no—
65C3×C4×Q8—22no—
66C2×C3×C4⋊C4D411yescharacteristic
67C2×C3×C4⋊C4D412yes—
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C4⋊C4C2311yescharacteristic
70C2×C3×C22⋊C4—22no—
71C2×C3×C22⋊C4—22no—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C42—21no—
75C2×C3×C42C2312yes—
76C2×C3×C42C2311yescharacteristic
77#946—44no—
78C2×#213—24no—
79C2×#213—24no—
80C2×#213—42no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—21no—
83C2×(C2×D3)⋊C4—22no—
84C2×C4⋊Dic3—21no—
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2312yes—
88C2×Dic3⋊C4—22no—
89C22×C3⋊C8C2311yescharacteristic
90C22×C3⋊C8C2312yes—
91C22×C3⋊C8C2311yescharacteristic
92C4×D12—22no—
93C4×D12—22no—
94C3×#95—21no—
95#832—24no—
96#832—24no—
97#832—42no—
98C3⋊#95—22no—
99#809—22no—
100#809—22no—
101#808—42no—
102C4⋊(C3⋊C8)—24no—
103C4⋊(C3⋊C8)D414yes—
104#803—24no—
105C4×C3⋊C8—22no—
106C22×D8—32no—
107C2×#127C2×D311yescharacteristic
108C2×#124—21no—
109C2×C22⋊Q8—61no—
110C2×C4×Q8C2×D311yescharacteristic
111C2×C4×D4—31no—
112#459—64no—
113#446—64no—
114C2×#106—32no—
115C2×C4⋊C8—31no—
116C2×C4⋊C8—32no—
117C2×#102—32no—
118C2×#102—32no—
119C2×#102—31no—
120C2×#102—31no—
121C2×#102—32no—
122C2×#97—32no—
123C2×C4×C8—31no—
124#396—62no—
125#394—22no—
126#390—21no—
127#390—32no—
128#390C2×D312yes—
129#390C2×D312yes—
130#389—32no—
131#384—31no—
132#380—61no—
133C4×C4⋊C4C2×D311yescharacteristic
134#340—32no—
135#340—31no—
136#340—31no—
137C24×C3—22no—
138C23×D3—21no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—22no—
148C2×C3×D4—42no—
149C2×C3×D4—42no—
150C2×C3×D4—42no—
151C2×C3×D4—42no—
152C2×C3×D4—81no—
153C2×C3×D4—81no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2×D412yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2411yescharacteristic
161C22×Dic3—21no—
162C22×Dic3—22no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12#4112yes—
169C2×C4×D3—42no—
170C2×C4×D3—42no—
171C2×C4×D3—42no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C22⋊C4—44no—
179C3×C22⋊C4—44no—
180C3×C22⋊C4—82no—
181C3×C22⋊C4—82no—
182C3×C42—22no—
183C3×C42—22no—
184C3×C42—42no—
185C3×C42C2×D412yes—
186C3×C42#4112yes—
187#213—48no—
188(C2×D3)⋊C4—44no—
189(C2×D3)⋊C4—42no—
190(C2×D3)⋊C4—82no—
191C4⋊Dic3—21no—
192C4⋊Dic3—22no—
193C4⋊Dic3—21no—
194C4⋊Dic3—22no—
195C4⋊Dic3—42no—
196C4⋊Dic3—81no—
197Dic3⋊C4—44no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—22no—
200C2×C3⋊C8—22no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8C2×D414yes—
206C2×C3⋊C8#4112yes—
207C22×Q8#6711yescharacteristic
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—31no—
211C22×D4C22×D312yes—
212C23×C4—61no—
213C2×D8—64no—
214C2×D8—64no—
215C2×D8—122no—
216C22×C8—32no—
217C22×C8—31no—
218C22×C8—31no—
219#127—22no—
220#127—22no—
221#124—44no—
222C22⋊Q8—124no—
223C4×Q8—22no—
224C4×Q8—22no—
225C4×D4—62no—
226C4×D4—62no—
227C2×C4⋊C4—31no—
228C2×C4⋊C4—32no—
229C2×C4⋊C4—61no—
230C2×C4⋊C4—62no—
231C2×C4⋊C4—61no—
232C2×C4⋊C4#6711yescharacteristic
233C2×C4⋊C4#6712yes—
234C2×C4⋊C4C22×D311yescharacteristic
235C2×C4⋊C4C22×D311yescharacteristic
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—61no—
241C2×C22⋊C4—62no—
242C2×C22⋊C4—62no—
243C2×C42—21no—
244C2×C42C22×D312yes—
245C2×C42C22×D311yescharacteristic
246#106—122no—
247C4⋊C8—34no—
248C4⋊C8—64no—
249#102—62no—
250#102—62no—
251#102—64no—
252#102—64no—
253#102—122no—
254#97—64no—
255C4×C8—62no—
256#95—21no—
257#95—62no—
258C23×C3—42no—
259C23×C3—42no—
260C23×C3—42no—
261C23×C3—42no—
262C23×C3—42no—
263C23×C3—42no—
264C23×C3—42no—
265C23×C3C22×D411yescharacteristic
266C22×D3—22no—
267C22×D3—41no—
268C22×D3—41no—
269C22×D3—41no—
270C22×D3—41no—
271C22×D3—41no—
272C22×D3—41no—
273C3×Q8—44no—
274C3×D4—42no—
275C3×D4—42no—
276C3×D4—42no—
277C3×D4—42no—
278C3×D4—84no—
279C3×D4—84no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—22no—
288C2×C3×C4—41no—
289C2×C3×C4—41no—
290C2×C3×C4—41no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—41no—
294C2×C3×C4#12112yes—
295C2×C3×C4#12112yes—
296C2×C3×C4AGL(1,ℤ/8)12yes—
297C2×C3×C4AGL(1,ℤ/8)12yes—
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4C2×#4111yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C4H(5,𝔽2)11yescharacteristic
303C2×C3×C42−1+411yescharacteristic
304C2×Dic3—41no—
305C2×Dic3—42no—
306C2×Dic3—41no—
307C2×Dic3—42no—
308C2×Dic3—41no—
309C2×Dic3—42no—
310D12—22no—
311D12—22no—
312D12—41no—
313D12—41no—
314C4×D3—84no—
315C3⋊C8—24no—
316C3⋊C8—22no—
317C3⋊C8—41no—
318C24—22no—
319C24—61no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—32no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—61no—
335C2×D4—61no—
336C2×D4—61no—
337C2×D4—61no—
338C2×D4—61no—
339C2×D4—81no—
340C2×D4—81no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—61no—
344C22×C4—62no—
345C22×C4—122no—
346C22×C4—122no—
347C22×C4—122no—
348C22×C4C2×#6712yes—
349C22×C4C2×#6712yes—
350C22×C4C2×#6711yescharacteristic
351C22×C4C2×#6711yescharacteristic
352C22×C4C23×D311yescharacteristic
353D8—128no—
354C2×C8—34no—
355C2×C8—32no—
356C2×C8—61no—
357C2×C8—62no—
358C2×C8—62no—
359C2×C8—61no—
360C2×C8—61no—
361C2×C8—61no—
362C2×C8—61no—
363C4⋊C4—22no—
364C4⋊C4—24no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—61no—
370C4⋊C4—61no—
371C4⋊C4—62no—
372C4⋊C4—62no—
373C4⋊C4—124no—
374C4⋊C4—122no—
375C4⋊C4—241no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C22⋊C4—124no—
381C22⋊C4—122no—
382C22⋊C4—242no—
383C42—22no—
384C42—22no—
385C42—42no—
386C42D3×D412yes—
387C42#23712yes—
388C22×C3—82no—
389C22×C3—82no—
390C22×C3—82no—
391C22×C3—82no—
392C22×C3—82no—
393C22×C3—82no—
394C22×C3—82no—
395C22×C3C2×#12111yescharacteristic
396C22×C3#54811yescharacteristic
397C22×C3#55011yescharacteristic
398C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
399C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
400C22×C3#57611yescharacteristic
401C22×C3#57711yescharacteristic
402C2×D3—42no—
403C2×D3—42no—
404C2×D3—42no—
405C2×D3—81no—
406C2×D3—81no—
407C2×D3—81no—
408C2×D3—81no—
409C3×C4—22no—
410C3×C4—42no—
411C3×C4—42no—
412C3×C4—81no—
413C3×C4—81no—
414C3×C4#46914yes—
415C3×C4#52911yescharacteristic
416C3×C4#54211yescharacteristic
417C3×C4#54611yescharacteristic
418C3×C4#54811yescharacteristic
419Dic3—82no—
420Dic3—81no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—62no—
429C23—121no—
430C23—121no—
431C23—121no—
432C23—121no—
433C23—121no—
434C23—121no—
435C23C22×#6711yescharacteristic
436Q8—44no—
437D4—42no—
438D4—42no—
439D4—42no—
440D4—42no—
441D4—62no—
442D4—62no—
443D4—84no—
444D4—84no—
445D4—121no—
446D4—121no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—122no—
464C2×C4—121no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—244no—
468C2×C4#93212yes—
469C2×C4#93912yes—
470C2×C4#93912yes—
471C2×C4#94912yes—
472C2×C4C2×D3×D411yescharacteristic
473C2×C4C2×#23711yescharacteristic
474C2×C4D3×#4111yescharacteristic
475C2×C4D3×#4111yescharacteristic
476C2×C4#100911yescharacteristic
477C2×C4#101011yescharacteristic
478C8—64no—
479C8—62no—
480C8—121no—
481C2×C3—162no—
482C2×C3#242611yescharacteristic
483C2×C3C2×#46911yescharacteristic
484C2×C3#306211yescharacteristic
485C2×C3#336611yescharacteristic
486C2×C3#337311yescharacteristic
487C2×C3#339711yescharacteristic
488C2×C3#339911yescharacteristic
489D3—82no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22—82no—
496C22—82no—
497C22—122no—
498C22—122no—
499C22—122no—
500C22—241no—
501C22—241no—
502C22—241no—
503C22—241no—
504C22C2×#93211yescharacteristic
505C22C2×#93911yescharacteristic
506C22C2×#94911yescharacteristic
507C22#583111yescharacteristic
508C22#583411yescharacteristic
509C22#583511yescharacteristic
510C22#583611yescharacteristic
511C4—22no—
512C4—42no—
513C4—42no—
514C4—81no—
515C4—81no—
516C4—241no—
517C4—242no—
518C4#503914yes—
519C4#567611yescharacteristic
520C4#567711yescharacteristic
521C4#568011yescharacteristic
522C4#568211yescharacteristic
523C3#2530011yescharacteristic, complemented, Sylow
524C2—162no—
525C2—242no—
526C2C2×#503911yescharacteristic
527C2#8299511yescharacteristic
528C2#8337911yescharacteristic
529C2#8357611yescharacteristic
530C2#8358111yescharacteristic
531C2#8371111yescharacteristic
532C2#8371511yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2548C2
socleC23×C3C22×D4
prime radicalC2×#5072C2
hypercenter#2548D3
hypocenterC3#25300
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2548

Complements

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

NH ≅ G/Ntype
1. C3×#2548C2semidirect
3. #82500C2semidirect
4. C2×#5024C2semidirect
5. #71764C2semidirect
6. #69652C2semidirect
7. #69650C2semidirect
8. #69648C2semidirect
9. #69645C2semidirect
10. C4×#809C2semidirect
11. #69467C2semidirect
46. #2548D3semidirect
523. C3#25300semidirect

Maximal subgroups

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

subgroupindex
C3×#25482
C2×#50722
#825002
C2×#50242
#717642
#696522
#696502
#696482
#696452
C4×#8092
#694672
#253003